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Ultra-Wideband, Short-Pulse Electromagnetics 5

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Ultra-Wideband, Short-Pulse Electromagnetics 5 Paul D. Smith University of Dundee Dundee, Scotland, U.K.

and

Shane R. Cloude Applied Electromagnetics Ltd. St. Andrews, Fife, Scotland, U.K.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-47948-6 0-306-47338-0

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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Preface The fifth Conference on Ultra-Wideband Short-Pulse Electromagnetics was held in Scotland from 30 May to 2 June 2000 at the Edinburgh International Conference Centre. It formed part of the EUROEM 2000 International Conference under the chairmanship of David Parkes (DERA, Malvern) and Paul Smith (University of Dundee). It continued the series of international conferences that were held first at the Polytechnic University, Brooklyn, New York in 1992 and 1994, then in Albuquerque, New Mexico in 1996 (as part of AMEREM ’96) and more recently in Tel-Aviv, Israel in 1998 (as part of EUROEM ’98). The purpose of these meetings is to focus on advanced technologies for the generation, radiation and detection of ultra-wideband short pulse signals, taking into account their propagation, scattering from and coupling to targets of interest; to report on developments in supporting mathematical and numerical methods; and to describe current and potential future applications of the technology. Since 1996 these meetings have been incorporated into the AMEREM/EUROEM biennial Conference that also includes the High Power Electromagnetics Conference and the Unexploded Ordnance Detection and Range Remediation Conference. This decision taken by the Permanent HPEM Committee in 1996 is a recognition of the interests in technology and methods of these Conferences that are common with those of the UltraWideband Short-Pulse Electromagnetics Conference. It also recognises the benefit in providing an international forum for scientists and engineers in such closely related disciplines. The next meeting will be held as part of AMEREM ’02 in June 2002 at the US Naval Academy in Annapolis, Maryland under the chairmanship of Terence Wieting. The papers in this volume report on newly emerging ideas and develop recurrent themes of earlier meetings. The topics include electromagnetic theory and scattering theory (including papers presented at a special session on fundamental solutions of Maxwell’s equations); ultra-wideband radar systems; ultra-wideband and transient antennas; pulsed power generation and propagation; ultra-wideband polarimetry; ultrawideband and transient metrology; detection and identification studies; RF interactions and chaotic effects; and biological effects. The Chairmen and Editors wish to thank all of those involved in EUROEM 2000 for their assistance and participation, especially members of the National and International Committees and their supporting institutions. We also acknowledge with gratitude the sponsorship of the Summa Foundation and the Permanent HPEM Committee, the European Office of Aerospace Research and Development (Air Force Office of Scientific Research, United States Air Force Research Laboratory), the Defense and Evaluation Research Agency (Malvern), Los Alamos National Laboratory, Lothian and Edinburgh Enterprise Limited and the Edinburgh Convention Bureau, the European Commission, and Dundee University. We acknowledge the technical co-sponsorship of the Institution of Electrical and Electronic Engineers (IEEE), the International Union of Radio Science (URSI), the Institution of Electrical Engineers (IEE), and the Applied Computational Electromagnetics Society (ACES). Paul D. Smith Shane R. Cloude

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Contents Fundamental Solutions of Maxwell's Equations and Electromagnetic Theory From Maxwell to Einstein Van Bladel, J 1 Analytical Methods for Antenna Analysis and Synthesis in the Time Domain Shlivinski, A and Heyman, E 11 Complex-Source-Point Narrow-Waisted Ray-Like Gaussian Beams for Frequency and Time Domain Radiation and Scattering Felsen, L and Galdi, V 21 Diffraction by Arrays of Complex Source Point Beams Jull, E, and Cheong, H 31 Application of Concepts of Advanced Mathematics and Physics to the Maxwell Equations Baum, C 39 Surface Discharge Cellular Automaton Model Hayakawa, H, Korovkin, N, ludin, D, Selina, E and Trakhtengerts, V 53 Green's Functions for Sheet Currents Placed Over Cylindrical Metal Surface Svezhentsev, A and Vandenbosch, G 59 UWB Analysis of EM Fields in Complex Laminates: A Multiresolution Homogenization Approach Lomakin, V Steinberg, B and Heyman, E 67 Time Domain Exact Solution of Problem of UWB Pulse Diffraction on a Conducting Half-Plane Galstjan, E 75 Spherical Wave Expansion of the Time Domain Free-Space Dyadic Green's Function Alp Azizoglu, S, Sencer Koç, S and Merih Büyükdura, O 83 On the Localization of Electromagnetic Energy Schantz, H 89 On Superliminal Photonic Tunnelling Nimtz, G, Haibel, A and Stahlhofen, A 97 Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth Seida, O, Bishay, S and Sami, G 105 Time-Domain Study of Transient Fields for a Thin Circular Loop Antenna Bishay, S and Sami, G 115 Generalized TEM, E and H Modes 127 Stone, A and Baum, C Electromagnetic Wave Scattering by Smooth Imperfectly Conductive Cylindrical Obstacle Tuchkin, Y 137 A Set of Exact Explicit Solutions in Time Domain For UWB Electromagnetic Signals in Waveguide Tretyakov, O 143 Analytical Regularization Method for Wave Diffraction by Bowl-shaped Screen of Revolution 153 Tuchkin, Y Transient Excitation of a Layered Dielectric Medium by a Pulsed Electric Dipole: Spectral Constituents 159 Tijhuis, A and Rubio Bretones, A Transient Excitation of a Layered Dielectric Medium by a Pulsed Electric Dipole: Spectral Representation 167 Tijhuis, A and Rubio Bretones, A Radar Systems A New Ultra Wideband, Short Pulse, Radar System for Mine Detection Gallais, F, Mallepeyre, V, Imbs, Y, Beillard, B, Andrieu, J, Jecko, B, and Le Goff, M

175

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CONTENTS

Ultra-Wideband Ground Penetrating Impulse Radar 183 Yarovoy, A, Van Genderen, P, and Ligthart, L Object Shape Reconstruction at Small Base Ultrawideband Radar 191 Koshelev, V, Shipilov, S, and Yakubov, V UWB Subsurface Radar with Antenna Array for Imaging of Internal Structure of Concrete Structural Elements 199 Boryssenko, A, Boryssenko, O, Ivashchuk, V, Lishchenko, A, and Prokhorenko, V Optimal Short Pulse UWB Radar Signal Detection 207 Immoreev, I and Taylor, J Experimental Results from an Ultra Wideband Precision Geolocation System 215 Fontana, R Recent Applications of Ultra Wideband Radar and Communications Systems 225 Fontana, R A Low Power, Ultra-Wideband Radar Testbed 235 Payment, T Ultra-Wideband Principles for Surface Penetrating Radar 247 Sachs, J, Peyerl, P, Rossberg, M, Rauschenbach, P and Friedrich, J Ray Tracing Assessment of Antenna Arrays and Subsurface Propagation for GPR Systems 259 Pennock, S and Redfern, M Ground Penetrating Radar System for Locating Buried Utilities 267 Pennock, S and Redfern, M Cost Efficient Surface Penetrating Radar Device for Humanitarian Demining 275 Ratcliffe, J, Sachs, J, Cloude, S, Crisp, G, Sahli, H, Peyerl, P and De Pasquale, G Some Problems of GPR soft- and hardware improving in mine detection and classification task Astanin, I, Chernyshov, E, Geppener, V, Jatzyn, A, Kostyleva, V, Nicolaev, V, Sokolov, M 285 and Smirnov, A Antennas Time-Domain Simulation Technique for Antenna Transient Radiation, Reception and Scattering Boryssenko, A, Boryssenko, E and Prokhorenko, V A Collapsible Impulse Radiating Antenna Bowen, L, Farr, E and Prather, W High-Power Ultrawideband Radiation for Radar Application Koshelev, V Broadband Operation of Tapered Inset Dielectric Guide and Bowtie Slot Antennas Hannigan, A, Pennock, S and Shepherd, P A Unified Kinematic Theory of Transient Arrays Shlivinski, A and Heyman, E Powerful Sources of UWB Pulsed Coherent Signals Kardo-Sysoev, A, Brylevsky, V, Lelikov, Y, Smirnova, I, Zazulin, S, Tchashnicov, I, Scherbak, V and Sukhovetsky, B Ultrawideband Solid State Pulsed Antenna Array Kardo-Sysoev, A, Zazulin, S, Smirnova, I, Frantsuzov, A and Flerov, A Admittance of Bent TEM Waveguides in a CID Medium Baum, C Optimization of the Feed Impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna Tyo, S Transient Fields of Offset Reflector Skulkin, S and Turchin, V A New Broad Band 2D Antenna for Ultra-Wide-Band Applications Mallepeyre, V, Gallais, F, Imbs, Y, Andrieu, J, Beillard, B, Jecko, B and Le Goff, M Time Domain Array Design Schantz, H

291 299 311 319 327 335 343 351 363 371 377 385

CONTENTS

Recent Developments in Ultra-Wideband Sources and Antennas Prather, W, Baum, C, Lehr, J, Torres, R, Tran, T, Burger, J and Gaudet, J Ultra-Wideband Sparse Array Imaging Radar Crisp, G, Thornhill, C, Rowley, R and Ratcliffe, J Cross-Field Characterization of Dipole Radiation in Fresnel Zone Badic, M and Marinescu, M Parallel Charging of Marx Generators for High Pulse Repetition Rates Lehr, J and Baum, C Pulsed Power Special Lecture: Live Fire Test and Evaluation and the RF Vulnerability Testing Mission O'Bryon, J and Carter, R Compact HPM and UWB Sources using Explosives - The Potential of Future Non-lethal Warhead Systems Ehlen, T, Bohl, J, Kuhnke, R and Sonnemann, F Sub-nanosecond Gas Breakdown Phenomena in the Voltage Regime Below 15 kV Krompholz, H, Hatfield, L, Short, B and Kristiansen, M High-Power, High-PRF Subnanosecond Modulator Based on a Nanosecond All-Solid-State Driver and a Gas Gap Pulse Sharper Yalandin, M, Lyubutin, S, Oulmascoulov, M, Rukin, S, Shpak, V, Shunailov, S, and Slovikovsky, B Upgrading of the Efficiency of Small-Sized Subnanosecond Modulators Yalandin, M, Oulmascoulov, M, Shpak, V and Shunailov, S Characteristics of Trap-Filled GaAs Photoconductive Switches used in High Gain Pulsed Power Applications Islam, N, Schamiloglu, E, Mar, A, Zutavern, F, Loubriel, G and Joshi, R Project of Semiconductor High-Power High-Repitition Rate Compact Current/UWB Pulse Generator Galstjan, E and Kazanskiy, L High Power Subnanosecond Generator for UWB Radar Prokhorenko, V and Boryssenko, A Electromagnetic Noise Emission of Industrial Pulse Power Equipment for Material Treatment Luhn, F, Zange, R, Wollenberg, G, Scheibe, H and Schätzing, W Compact Solid State Pulse Modulators for High Power Microwave Applications Gaudreau, M, Casey, J, Mulvaney, J and Kempkes, M UWB Polarimetry An Introduction to Polarisation Effects in Wave Scattering and their Application in Target Scattering Boerner, W and Cloude, S Unipolarized Currents for Antenna Polarization Control Baum, C Polarimetric Radar Interferometry: A New Sensor for Vehicle Based Mine Detection Cloude, S and Thornhill, C Modelling of the Air-ground Interface for UWB Radar Applications Lostanlen, Y, Uguen, B, Chassay, G and Griffiths, H Ultra-Wideband Polarimetric Borehole Radar Sato, M and Liu, S Buried Mine Detection by Polarimetric Radar Interferometry Sagues, L, Lopez-Sanchez, J, Fortuny, J, Fabregas, X, Broquetas, A and Sieber, A UWB & Transient Metrology An Optical Approach to Determine the Statistical Features of the Field Distribution in Mode Strirred Reverberation Chamber Baranowki, S, Kone, L and Demoulin, B

ix

393 399 407 415

423 431 437

445 453 461 467 473 479 485

493 501 519 527 537 545

555

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Influence of Variations in the Spectral Transfer Function to Time Domain Measurements Garbe, H Influence of the Precursor Fields on Ultrashort Pulse Measurements Oughstun, K and Xiao, H Dihedral Reflector Calibration for UWB Radar Systems Lambert, A and Smith, P Detection, Target Identification and Signal Processing Image Reconstruction of the Subsurface Object Cross-Section from the Angle Spectrum of Scattered Field Vertiy, A, Gavrilov, S, Salman, A and Voynovskiy, I Optimal Acoustic Measurements Cheney, M, Isaacson, D and Lassas, M Parsimony in Signature-Based Target Identification Baum, C Buried Object Identification with an Optimisation of the TLS Prony Algorithm Lostanlen, Y, Corre, Y and Uguen, B Model Problems of Pulse Sensing Velychko, L, Perov, A, Sirenko, Y and Yaldiz, E Simulation of the Transient Response of Objects Buried in Dispersive Media Hernándex–López, M, Gonzalez Garcia, S, Rubio Bretones, A and Gomez-Martin, R Electromagnetic Transient Modelling using Dynamic Adaptive Frequency Sampling Tham, C, McGowen, A, Towers, M and Poljak, D The Time Domain Numerical Calculation of an Integro-Differential Equation for Ultrashort Electromagnetic Pulse Propagation in Layered Media Sherbatko, I, lezekiel, S and Nerukh, A Marching on in Anything: Solving Electromagnetic Field Equations with a Varying Physical Parameter Tijhuis, A and Zwamborn, P Correlation of Antenna Measurements Using the Oversampled Gabor Transform Fourestié, B and Altman, Z On a Rational Model Interpolation Technique of Ultra-Wideband Signals Younan, N, Taylor, C and Gu, J

561 569 577

585 599 605 615 623 631 639 647 655 663 671

Propagation Full-Wave Solution of the Propagation of Generally Shaped Impulses and Wide Band Application in Anisotropic Plasmas 679 Ferencz, O and Ferencz, C Asymptotic Description of Ultrawideband, Ultrashort Pulsed Electromagnetic Beam Field Propagation in Dispersive, Attenuative Medium 687 Oughstun, K Dispersion Reduction in a Coaxial Transmission Line Bend by a Layered Approximation of a Graded Dielectric Lens 697 Bigelow, W, Farr, E, Prather, W and Baum, C RF Interactions and Chaos Optimal Input Signals for Driving Nonlinear Electronic Systems into Chaos Booker, S, Smith, P, Brennan, P and Bullock, R In-band Chaos in Commercial Electronic Systems Booker, S, Smith, P, Brennan, P and Bullock, R An Application of Chaos Theory to the High Frequency RCS Prediction of Engine Ducts MacKay, A Ray Splitting and Chaos in Electromagnetic Resonators Blumel, R

707 715 723

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CONTENTS

xi

Biological Effects Ultra-Wideband (UWB) Radio-Frequency (RF) Bioeffects Research at DERA Porton Down Holden, S, Inns, R, Lindsay, C, Tattersall, J, Rice, P and Hambrook, J

739

Index of Authors

749


FROM MAXWELL TO EINSTEIN

J. Van Bladel Department of Information Technology, Ghent University, and Interuniversity Microelectronics Center (IMEC) Gent, B-9000, Belgium

INTRODUCTION In 1931 Cambridge University Press published a volume commemorating the centennial of Maxwell's birth1. Among the contributors we note J.J. Thomson, Max Planck, Ambrose Fleming, Joseph Larmor, James Jeans, Oliver Lodge, Horace Lamb and Albert Einstein. Some of the authors had known Maxwell personally, and their character sketches of our patron-saint are often delightful. Fleming writes about Maxwell's lectures: "I well remember my surprise at finding a teacher who was everywhere regarded as the greatest living authority on his subject lecturing to a class of two or three students in place of the 100 or more attentive listeners he would have had in any Scottish or German University." As a lecturer, Maxwell was apparently mentally too powerful for his students. Lamb, another former student, writes: "Maxwell’s lectures had a great interest of charm for some of us, not so much for the sake of the subject matter, which was elementary, as in the illuminating glimpses we got of the lecturer’s own way of looking at things, his constant recourse to fundamentals, the humorous and unpremeditated digressions, the occasional satirical remarks, and often a literary or even poetical allusion." Lamb paid occasional visits to Maxwell's house, and remembers that "Maxwell had two toys which he would sometimes bring out to entertain fresh visitors. One was the "dynamical top", intended to illustrate various points in the classical theory of rotation. The other toy was a form of ophthalmoscope which he had independently invented. He was wont to demonstrate the use of this on himself and his friends, including his dog, which he had trained to become a patient and accommodating "subject"." Another author, Oliver Lodge, describes Maxwell’s fight with mechanical models aimed at explaining the propagation of waves in the lightcarrying ether. He writes: "In Maxwell’s model the lines of magnetic force were represented by cylinders rotating round these lines as axes. In a uniform magnetic field the cylinders would all have to be rotating in the same direction. The question was how they were to be geared together to do this. If two adjacent wheels were in contact they would rotate in opposite directions; to make them rotate in the same direction Maxwell introduced between them small spheres like ball-bearings to act as idle wheels". We know that, in an

Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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extraordinary feat of conceptual abstraction, Maxwell dropped these attempts, and decided that reality was best described by his equations.

EINSTEIN'S COMMEMORATIVE ARTICLE Einstein emphasizes the undebtedness of the scientific community to his illustrious predecessor. He writes, in particular, that "The greatest alteration in the axiomatic basis of physics since Newton originated in the researches of Faraday and Maxwell. According to Newton's system the material particle is the sole representative of reality. But the partial differential equation, which came first to theoretical physics as a servant, by degrees became its master. This process began in the nineteenth century, with the wave theory of light. Light in empty space was conceived to be a vibration of the ether, and it seemed uncalled for to regard this ether as itself a conglomeration of material particles. Here for the first time the partial differential equation appeared as the natural expression of the elementary in physics. It is true that Maxwell tried to find a basis of justification for these equations in ideal mechanical constructions, but he took none of these efforts too seriously; it was clear that the equations themselves were all that was essential, and that the field intensities that appeared in them were elementary, not derivable from other simples entities." And Einstein concludes: "Since Maxwell's time, Physical Reality has been thought of as represented by continuous fields, governed by partial differential equations, and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and the most fruitful that physics has experienced sinde the time of Newton."

TRANSFORMATION EQUATIONS BETWEEN INERTIAL FRAMES At the time of Maxwell's death the problem of including moving media in the theory was left unsolved. It was commonly believed that the ether served as a substratum for the propagation of light, and that it penetrated into bodies like water in a sponge. Some physicists assumed that moving bodies dragged the ether locally. Other believed that the ether was at absolute rest, and that the earth, for example, was swept by an ether "wind" in its motion. In this global picture an "absolute" set of axes existed, in which all true motions should be measured. In a particle left to itself would move in a straight line with constant velocity. Further, this uniform motion would also hold in all other rigid systems K which move with uniform velocity with respect to (the inertial axes). Ether was fundamentally at rest in and electromagnetic waves should therefore have a velocity c with respect to but a value different from c on earth. Experiments such as Michelson and Morley's showed the fallacy of that assumption. Extensive discussions led Lorentz to propose, in 1904, new laws for the transformation of coordinates from an inertial frame (a train for example) to another inertial frame K (a station). Taking v to be the relative velocity of with respect to K, the relevant relationships are


3

Space and time are clearly interwoven, and events which are simultaneous in at different are not simultaneous in K. In (1) hence at everyday's low velocities may be neglected with respect to 1. First-order terms in however, must be carefully kept since they play a major role in applications. From (1) it is seen that a velocity c in goes over into c in K, a revolutionary result which does away with the traditional law of addition of velocities. At about the same time, in 1905, Einstein went further2, and erased absolute motion and absolute space by postulating that the laws of electrodynamics and optics have the same form in all inertial frames that light propagates in empty space with a speed c which is independent of the state of motion of the emitting body. As a consequence, Maxwell's equations must have the same form in K and Combined with (1) this leads automatically to the transformation laws for sources, viz.

Here || and fields:

indicate components parallel and perpendicular to v, respectively. For the

At low velocities a relationship used extensively in the determination of motionally-induced currents. The transformation equations (2) (3) are of considerable interest for the electrical engineer, since they allow him to solve the field problem in a frame in which the solution is particularly simple, and to transform the results back to the frame K in which they are actually needed. This process is aptly called frame-hopping.

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Maxwell's equations must be supplemented by constitutive equations. In 1908 Minkowski3, building on Einstein's ideas, concluded that these equations should have the same form in all inertial frames in which the medium is at rest. Thus, if holds in the rest axes it must also hold in every other inertial frame in which the medium is at rest. Transforming from back to K (the "laboratory") by means of (3) gives

At low velocities:

An observer in the laboratory sees an (anisotropic) magneto-electric medium! By a similar argument, the usual boundary conditions must hold in every rest frame. Transformed back to the laboratory these conditions become (Figure 2)

For v = 0 we recuperate the "rest" form, as expected. The same form holds when i.e. at points where the motion takes place in the tangent plane.

A FEW PROBLEMS INVOLVING BODIES IN UNIFORM MOTION When the motion takes place in the tangent plane, as suggested in Figure 3, a solution directly in the laboratory axes K becomes practical4. The boundary conditions are those of a body at rest, and the motion is only felt through the constitutive equations (4).


5

In most cases, however, the appropriate strategy is to first determine the fields in the rest axes of the body, and transform them back to the laboratory. This method was already used by Einstein in his fundamental 1905 paper, in his endeavour to determine the fields reflected from a moving mirror (Figure 4). The incident field is:

Frame-hopping gives the reflected field in two easy steps. Thus,

The Doppler shift is apparent. At low velocities, in particular,

MAXWELL'S EQUATIONS IN ACCELERATED AXES Frame-hopping is also applicable to accelerated bodies. The first step is to solve the problem in the (accelerated) rest axes of the body, followed by a transformation back to the laboratory K. This step requires the form of Maxwell's equations in the accelerated axes Fundamental here is the metric tensor obtained by writing the squared elementary distance between neighbouring events - a relativistic invariant - in terms of the new coordinates. Thus, starting from (CT, X, Y, Z) in K:

J. VAN BLADEL

6

For the rotating coordinates of Figure 5

where The general form of Maxwell's equations, given is

where and are tensors4 grouping covariant and contravariant components of resp. (e, b) and (h, d). In rotating coordinates, for example, the curl e equation becomes

It is interesting to note that a tensor also appears when gravitational forces are present. Outside a spherically-symmetric mass distribution, for example,

Here

is a critical radius and m is the total mass. When lies outside the sphere, becomes singular there (the well-known "black hole" phenomenon). The theory further shows that free space acquires locally an equivalent which causes a grazing ray to be deviated (Figure 6).


7

This phenomenon, well-observed in astronomy, can be pertinent for radio-communications with or detection of space vehicles. An important problem in accelerated coordinates is the form of the constitutive equations. Firstly, the parameters may be modified by mechanical stresses and deformations. These effects can be accounted for, and corrected. More fundamental is the need for an assumption, namely that at sufficiently low accelerations the rest values of µ, hold at a moving point P when the fields are expressed in the instantaneous rest frame of P. From this working hypothesis it follows that the constitutive equations (4) still hold in the laboratory, but with a v which is now a function of r and t. By the same token the boundary conditions at a boundary point will be the same as in the instantaneous rest frame. In particular, the conditions for a body at rest hold at points where the motion is in the tangent plane. This remark allows solution of quite a few practical problems, as shown in the sequel. The validity of the rest-frame hypothesis has been checked5 on the particular case of a rotating dielectric cylinder. The dielectric is assumed to consist of electrons elastically bound to the nucleus. The analysis shows that the hypothesis holds as long as the rotation frequency is much less than the eigen-oscillating frequency of the electron-nucleus spring, a value which typically lies in the ultraviolet.

A FEW PROBLEMS INVOLVING ACCELERATED BODIES When the motion takes place in the tangent plane a solution in the laboratory axes is the best choice. The motion is now felt through the constitutive equations only. The method is applicable to a variety of configurations, for example to the rotating axisymmetric conductors shown in Figures 7a and 7b. To first order in a volume charge density and an electric field

appear in the conductor, while a voltage

is induced across the Faraday disk.

Another example is afforded by a rotating circular dielectric cylinder of radius a immersed in an E-wave. Figure 7c shows the scattering pattern obtained for and The shift in the pattern is the main effect but, given the smooth contour, it takes nonrealistic peripheral velocities to make it significant.

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When the motion modifies the spatial configuration of the body (e.g. in the case of the rotating blades of a helicopter) it becomes necessary to apply frame-hopping to and from the accelerated axes. For a rotating body this means first solving the problem "on the merry-go-round". Only a few of these problems have been solved, e.g. the rotating circular cylinder carrying a distributed surface reactance, and immersed in an H-wave6. The strong variation of the reactance gives rise to a rich Doppler spectrum (Figure 8).

THE QUASI-STATIONARY METHOD Although the rest axes method is conceptually straightforward, the actual application (for example to a vibrating mirror7) quickly becomes cumbersome. An extensively used substitute is the quasi-stationary approximation, which consists in evaluating the scattered fields in P, at time t, as if the scatterer were frozen in the position it occupies at t. The motion produces amplitude and phase modulation, and therefore an instantaneous frequency shift. The method has its obvious limitations. It would not detect a rotation effect of the type shown in Figure 7c, since all positions of the rotating circular cylinder are equivalent. In addition, the method is theoretically flawed. In the case of a uniformly moving mirror, for example, it would predict a reflected signal (Figure 4)

A comparison with the exact expression (7) shows that (a) the first-order Doppler shift is correctly predicted (b) the amplitude modulation is ignored (c) the reflected field does not satisfy the wave equation in vacuo. In particular, the propagation velocity is instead of c. Although theoretically inconsistent, the approximation yields errors of the order of which are quite acceptable when the scatterer moves little during the characteristic times of the problem. Such times are e.g. the period of a time-harmonic incident wave, of the duration of a short pulse of the type encountered in UWB applications. For for example, an airplane flying at a speed of would move only 0.3 µm during the pulse, a very short distance indeed compared with the dimensions of the target, and one that would hardly change the radar cross-section of the latter!


9

CONCLUSION It is clear that Relativity forms an integral, indispensible part of Electromagnetic Theory. The present very simplified survey does not do justice to the beauty of the fourdimensional, tensorial structure of the theory. Its sole ambition was to unfold the mechanisms which lead to a correct formulation of practical problems. From a utilitarian point of view it may be asserted, in all fairness, that most "radio" engineers can live quite happily without Relativity. Approximate methods such as quasi-stationarity suffice for their practical needs. There are exceptions, of course, particularly in the areas of high-velocity electron beams, klystrons, free electron lasers, gyrotrons, relativistic magnetrons or highenergy accelerators. Electromechanical engineers, on the other hand, are much more involved. Let us remember that Einstein worked for the Swiss Patent Office from 1902 to 1909, and that he was routinely confronted with new designs of motors and generators, some of which probably claimed perpetual motion!

He certainly meditated about the relative motion of circuits and magnets when he wrote his fundamental 1905 paper, and we may assume that the circuit law (actually based on the instantaneous rest frame hypothesis) played a role in his travails. Einstein may well have considered the teaser of Figure 9 (current or no current?), where a circuit is connected to a moving magnetized (and conducting) bar through sliding contacts4,8. It is in such an engineering atmosphere that Relativity was born! REFERENCES 1. James Clerk Maxwell, A Commemoration Volume, 1831-1931, Cambridge University 2. 3.

4. 5. 6.

7. 8.

Press (1931). A. Einstein, Zur Elektrodynamik Bewegter Körper, Ann. Phys. (Leipzig), 17: 821 (1905). H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vergängen in bewegten Körpern, Göttingen Nachrichten Math.-Phys. Klasse, 53 (1908), reprinted in Math. Annalen, 68: 472 (1910). J. Van Bladel, Relativity and Engineering, Springer-Verlag, Berlin (1984). T. Shiozawa, Phenomenological and Electron-theoretical Study of the Electrodynamics of Rotating Systems, Proc. IEEE, 61: 1694 (1973). B.M. Petrov, Spectral Characteristics of the Scatterer Field from a Rotating Impedance Cylinder in Uniform Motion, Radio Eng. Electron. Phys. (USSR) (English translation), 17: 1431 (1972). D. De Zutter, Reflection from Linearly Vibrating Objects: Plane Mirror at Oblique Incidence, IEEE Trans. AP, 30: 898 (1982). L.V. Bewley, Flux Linkages and Electromagnetic Induction, Dover Publications Inc., New York (1964).


ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS IN THE TIME DOMAIN

Amir Shlivinski and Ehud Heyman Faculty of Engineering Tel-Aviv University Tel-Aviv 69978, Israel

INTRODUCTION The ever-increasing interest in the radiation and detection of ultra-wideband (UWB) short pulses has made an impact on time domain (TD) analysis and design of shortpulse antennas. One way of analyzing the radiation and detection of these pulses would involve the traditional frequency domain (FD) antenna parameters on a frequency-byfrequency basis. FD parameterization lends itself conveniently to producing a comprehensive transmit - receive system description, yet because of the broad frequency band of the short-pulsed fields, direct treatment in the TD may lead to more efficient and physically transparent representations. The present paper is a brief overview of two formulations which have been introduced recently for TD antenna characterization: The first one is based on plane wave analysis in the TD via the slant stack transform (SST) of the current distribution [1], while the other is based on a TD multipole expansion [2]. The plane wave formulation provides a complete far-zone characterization of transmitreceive antenna systems [1]. The SST proves to be a fundamental tool in TD analysis, and is equivalent to the spatial Fourier transform of the current distribution used to evaluate the radiation field in the FD. The far field is expressed in terms of the effective height operator, which is a characteristic of the antenna, used with the source waveform to produce the far field via a convolution integral. Thus taking the receiving antenna effective height and circuit into account, we arrive at the TD transmit - receive antenna system description, consisting of a succession of convolution integrals. The TD multipole expansion, on the other hand, provides a convenient representation for the near zone properties of the field. Using this formulation we define and explore such concepts as the TD radiative and reactive fields and energies [2]. In the near zone, the TD reactive energy is a relatively strong pulse which, unlike the radiaUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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A. SHLIVINSKI AND E. HEYMAN

tive energy, discharges back into the source once the excitation pulse has ended. The radiation effectiveness is thus quantified by the ratio between the total TD reactive energy outside the antenna and the radiative energy. Borrowing from FD terminology, this ration is termed the TD Large implies large reactive energies and thereby a less effective realization. The near and far zone characteristics are, of course, related: As in the FD analysis, highly directive fields consists of high order spherical modes which requires large antennas. Realizing such modes with smaller antennas would generate very large reactive energies and These considerations pose a tradeoff between directivity, size and radiation effectiveness. In [3] we have used the measures above in order to quantify and contrast the effectiveness of several different realizations of highly collimated, short-pulse fields, and in particular, the class of space-time synthesized apertures considered in [4] THE ANTENNA EFFECTIVE HEIGHT IN THE TIME DOMAIN We first introduce the SST which is the fundamental tool in TD plane wave analysis. Given the impressed current distribution on the antenna surface, the radiating vector potential is found via the retarded potential integral where and is the speed of light. In the “TD Fraunhofer zone,” defined by where L is the source dimension and T is the pulselength, we obtain the following expression for the pulsed radiation field

where the unit vector

define the observation direction and (see Fig. 1)

The operation in (2) is the slant stack transform (SST) of be expressed as two cascaded operations [5]

in the

direction. It can

The first operation is a spatial Radon projection of at planes normal to the spectral propagation direction Fig. 1. In the second operation in (3) these planar projections are stacked with a progressive time delay associated with the propagation along the axis at the spectral speed The STT therefore extracts from the source distribution the transient plane-wave information that propagates in the direction. Further details on the SST could be found in [1, 5]. Thus form (2) the radiation pattern depends on the SST in the direction. Next we consider the transmitting antenna circuit in Fig. 2. The effective height of the transmitting antenna defines the relationship between the forward propagating input current-wave at the antenna terminals and the electric field in the far zone via the convolution integral

and we use the notation The effective height is thus the far field impulse response of the transmitting antenna. Note that in contrast to the FD definition, we define the TD effective height with respect to rather than with

ANALYTICAL METHODS FOR ANTENNA ANALYSIS AND SYNTHESIS

13

respect to the total current since the latter also includes the reflected waveform that may include multiple reflections off the antenna structure, etc. In analogy to the far field in the FD, the effective height is independent of the distance and of the radiation delay. Using the relation between E and A we obtain

where is the current distribution due to an impulsive input current and the subscript || denotes the transverse component relative to the observation direction Evaluation of (5) thus requires an electromagnetic solution of which in general needs to be calculated numerically, e.g. via the Finite Difference Time Domain (FDTD) technique, although for electrically small or large antennas some approximate model may apply. The theory above can be applied to a system consisting of a transmitting and a receiving antennas denoted, respectively, by subscripts T and R (Fig. 3). The distance between the antennas is large so that the far zone models apply. The unit vector directions from antenna T to R and from R to T are denoted, respectively, as and Using reciprocity in order to express the effective reception height of the receiving antenna in terms of its effective transmission height we obtain for the receiver load current

where denotes a vector dot product and a temporal convolution. Note that and the convolution operations may commute. Equation (6) describes a complete transmitreceive system. In [1], the theory has been further developed to provide expressions for the energy using TD gain operators. Example: Radiation from a circular disk antenna The formulation above has been applied so far in many configurations. Here we shall briefly demonstrate the use of the SST to derive a closed form expression for the TD radiation pattern from a disk of radius in the plane, carrying a uniform pulsed current distribution

where is a unit vector in the direction. Referring to (1)–(2), the TD radiation pattern in the direction can be expressed as

14


To calculate we note that the component of transverse to is where is the angle of from and is a unit vector along that coordinate. They are given by and Eq. (8) thus becomes

The integral in (9), henceforth termed reduces to the length of the line of intersection between the source disk in the plane and the slanted plane (Fig. 4). For a given the distance of this line from the origin is hence its length is giving

The expression in (10) is the TD analog of the well known FD expression for the a radiation from a circular source disk. However, unlike the FD analysis which involves spatial Fourier transforms and Bessel functions, followed by frequency transform into the TD, the TD analysis above involves only a geometrical projection of the source disk. This expression is used in Fig. 6 to verify the results of the multipole expansion. Full agreement is obtained if a sufficient number of multipoles is taken. TD MULTIPOLE EXPANSION Next we discuss the TD multipole expansion and the near-zone analysis. The analysis may be applied to any spherically stratified configuration bounded by a general conical surface whereon the boundary conditions are imposed (Fig. 5), but here we shall only be interested in radiation in free space. The field in the configuration of Fig. 5 may be decomposed into E- and H- type modes denoted, respectively, by superscript and and expressed as a sum of all these modes, with denoting the mode index and or

with a similar expression for the H field. Here spherical coordinate system, and

denotes the conventional The functions


15

and are the transversal scalar and vector mode functions which depend only on the conical cross section and are independent of The amplitude functions depend only on and account for the source excitation and the medium inhomogeneity via a solution of the radial transmission problem. For The present problem of radiation in free space, and are the well known scalar and vector spherical harmonics (i.e., with being a triple index One finds that outside the source region the field is given by

where and are calculated by projecting

with In (12), are the “TD multipole moments” which onto the spherical basis functions via

An important property of the series in (12) is that only the terms propagate without decay and with pure delay. Expressing the far zone field in the form one finds for the “TD radiation pattern”

Recall that F can also be calculated via the SST as discussed in (8). The large terms, in (12), behave like and thus dominate in the near zone. where they contribute to the “TD reactive field.” These constituents have been used in [2] to define the TD reactive power and energy. It has been demonstrated that the TD reactive power is a pulse with zero mean: At early time it propagates outward and charges the reactive energy around the antenna, while at later time, it propagates inward as the reactive energy discharges back to the source (cf. Fig. 6). The only constituents that carry a net energy are the radiative constituents Using large asymptotics of Eqs. (12)–(13) we have determined in [2] the convergence rate of the multipole expansion and the field structure in the far and in the near (reactive) zone. Example: Radiation from a circular disk antenna The concepts above are demonstrate for the example considered in (7). The excitation pulse is taken to be a twice differentiated Gaussian with pulselength and a unit energy Figs. 6(a,b) shows the radiation-pattern pulse at vs. the normalized time for source disks of radii and The figure compares the SST result (10) (solid line), which is used as an independent closed-form check, with the multipole expansion result obtained by summing up to the and term (dotted, dash-dotted and dashed lines, respectively). Note that the number of modes needed is essentially as has been determined analytically via asymptotic

16


analysis of (12)–(13) in [2]. Finally in Fig. 7 we show typical results for the radiative and reactive power-flows. Note that the radiative power pulse propagates without distortion or decay, while the reactive power pulse decays and distorted as a function of As discussed above the reactive power has a zero mean, implying that the reactive energy discharges back to the source when the excitation pulse turns off. THE TD

As discussed in the Introduction, the radiation effectiveness can be quantified by comparing the total TD radiative and reactive energies. Using the multipole expansion as discussed above, one may identify the following concepts [2]: the radiative power-flow and energy-density and respectively) and the reactive power-flow and energy-density and These quantities may be expressed explicitly in terms of the time-dependent multipole moment functions of (13). The TD factor of the field with respect to the sphere a enclosing the antenna may be defined now as

where the total radiative energy is defined by

The definition of


17

the measure of the total reactive energy is somewhat more complicated. We defined the time-averaged reactive energy outside the source support as where has been discussed above and is the RMS pulse-length of the radiative field. The general definition in (15) applies to any pulse shape with a finite energy. It also reduces to the conventional definitions of for time harmonic sources, given for example in [6]. In order to calculate the general definition in (15) for the total it may be rephrased in terms of modal quantities via [2]

where the summation involves the two mode-types and all mode indexes The modal quantities in (16), namely and are the counterpart of the total quantities defined above. These quantities may readily be calculated from the multipole moment functions which, as mentioned above, are calculated directly from Eq. (16) provides a direct mean for calculating and is the TD analog of the well known FD expression in [7]. Eq. (16) readily clarifies the tradeoff between directivity, antenna size, pulse length and effective realization. For example, large directivity requires large at the higher order modes. This, on the other hand, may cause an increase in the modal which typically increase rapidly for where and T are the antenna size and pulselength (this increase follows from the fact noted after (14) that the modal reactive energies are rapidly growing for The tradeoff noted above will be demonstrated in the simulations below. Example 1: Radiation from a circular disk antenna The concept introduced above are demonstrated here for the example of Eq. (7). We consider the radiation from both “small” and “large” source disks as well as the possible realization of the field due the “large” source by a smaller source.

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Fig. 8 shows the energy constituents for three different disks with radii and (indicated by and respectively). Fig. 8(a) depicts the radiative modal energies of (16) as a function of the mode index while Fig. 8(b) depicts the total as a function of the highest mode included in the summation (16). Note from the modal energy distribution and from the results for the total quality factor that the relevant modes are those with Beyond this value, the modal energies are small and their contributions to the total are negligible. We explore the possible realization of the field corresponding to the disk, by using a smaller source bounded by (the results are indicated by a +). The moments are taken to be those of the case, leading to the same radiation pattern and mode energies (see the * and the + marks in Fig. 8(a)). The smaller source realization, however, requires larger reactive energies and modal and thereby larger and slowly converging series for (Fig. 8(b)). As a conclusion, super resolution using short pulse fields is not feasible since it involves large TD Q.

Example 2: Collimated, space-time synthesized, source distributions Next consider a class of space-time synthesized pulsed current distributions on a circular disk of radius in the plane, given by (cf. (7))

Where the parameter will be discussed below and by the positive frequency spectrum here a twice differentiated analytic

is any analytic signal, as defined Specifically we shall use

pulse

Where T is a parameter controlling the pulse length. Also the normalization constant A is taken so that the norm of in the space-time domain is normalized to a constant ( in this case). One observes that the current decay away from the center of the aperture is affected by the imaginary part in the argument of in (17): Specifically


19

using (18), one finds that the peak of the pulse at decays like The effective width of the aperture for this distribution, defined as the diameter where the pulse energy decays to one half of its value on the axis, is found to be For large a such that the truncation in (17) may be neglected, giving rise to an iso-diffracting pulsed beam field (PB) of the type considered for example in [4], with being the collimation distance. If however, than the truncation effect is significant and the radiating field is not an iso-diffracting PB. Below we consider three different distributions, all having the same T and but with different values for and In the first and second cases, the effective aperture widths as determined above are and respectively. Since they are smaller than the disk diameter, they approximately generate iso-diffracting PBs. In the third case the aperture distribution is practically uniform over the entire disk as in (7). Fig. 9 shows the far zone electric field pulse on the as computed via the TD multipole expansion (14). Only the first 20 multipole modes have been summed up (dotted lines). In order to validate the TD multipole expansion results and the number of terms needed, the results are compared with the SST formulation (8) for the radiation pattern (full lines), which is exact in the far zone. In the large case the effective aperture is wider and thus more multipoles are needed in order to recover the correct SST result (the number of relevant modes for uniform distributions is approximately

20


The H-plane energy radiation pattern as a function of the angle from the axis (the bore sight) is depicted in Fig. 9(d). One may readily observe that the directivity increases and beam width decreases as and, thereby the effective aperture width, increase. As in Fig. 8, we explore in Fig. 10 the near field effectiveness of the three source realizations by plotting the modal energies as a function of the mode number (Fig. 10(a)) and the total TD as a function of highest mode number in the summation (16) (Fig. 10(b)). From Fig. 10(a), the dominant modes are those with while from Fig. 10(b) the number of relevant modes is (beyond this value, the modal energies are small and their contributions to the total are negligible). One also observe that the PB-type distributions in (a) and (b) are characterized by lower TD as follows from the fact that their effective source diameter is smaller.

Acknowledgements: This work is supported in part by the Israel Science Foundation under Grant No. 404/98, and in part by AFOSR Grant No. F49620-96-1-0039.

References [1] A. Shlivinski, B. Heyman and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propagat., AP-45, 1140–1149, 1997. [2] A. Shlivinski and E. Heyman, “Time domain near field analysis of short pulse antennas. Part I: Spherical wave (multipole) expansion,” and “— Part II: Reactive energy and the antenna ” IEEE Trans. Antennas Propagat, AP-47, 271–286, 1997. [3] A. Shlivinski and E. Heyman, “Energy considerations in space-time synthesis of collimated pulsed apertures,” Proc. of the URSI Trianum International Symposium on Electromagnetic Theory, Thessloniki, Greece, May 1998, pp. 602–604. [4] E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultrawideband/short-pulsed fields,” IEEE Trans. Antennas Propagat., AP-42, 518–525, 1994. [5] E. Heyman, “Transient plane wave spectrum representation for radiation from volume source distribution,” J. Math. Phys., 37, 658–681, 1996. [6] R.E. Collin and S. Rothschild, “Evaluation of antenna ,” IRE Trans. Antennas Propagat, AP-12, 23–27, 1964. [7] R.F. Harrington, “Effect of antenna size on gain, bandwidth and efficiency,” J. Res. NBS, 64D, 1–12, 1960.

COMPLEX-SOURCE-POINT NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS FOR FREQUENCY AND TIME DOMAIN RADIATION AND SCATTERING

Leopold B. Felsen1,2 and Vincenzo Galdi2,3 1

2

3

Department of Aerospace and Mechanical Engineering Boston University, Boston, MA 02215, USA Also, University Professor Emeritus, Polytechnic University, Brooklyn, NY 11201, USA Department of Electrical and Computer Engineering Boston University, Boston, MA 02215, USA Waves Group, University of Sannio, Benevento, Italy

I. INTRODUCTION Through replacement of the real spatial or spatial-temporal source locations in the frequency domain (FD) or time domain (TD) Maxwell field equations by locations in complex space or space-time, respectively, one may generate a new class of exact field solutions which convert point-source-excited fields in any environment into fields excited by Gaussian-beam-like wave objects in that environment (Deschamps, 1971; Felsen, 1976). While most applications of this elegant and physically appealing complex-source-point (CSP) technique have been concerned with ”high frequency” tracking of wellcollimated ”wide-waisted” beams, we shall be concerned here with utilizing a tight superposition of narrow-waisted ray-like CSP beams (centered on a Gabor lattice) for synthesis of FD and TD distributed aperture radiation, and the interaction of these radiated fields with complex propagation and scattering environments. Previous studies have employed this algorithm for FD distributed phased apertures (Maciel and Felsen, 1989) and for transmission of these fields through focal regions and through plane or cylindrically stratified dielectrics (Maciel and Felsen, 1990a,b). Here, we extend this algorithm to FD scattering by a moderately rough perfectly conducting boundary. We also extend the FD aperture radiation algorithm to the short-pulse TD, utilizing CSP pulsed beam wavepackets. The FD problem is considered first.

II. RADIATION AND SCATTERING IN THE FREQUENCY DOMAIN The problem geometry is shown in Fig. 1a. A two-dimensional electric field with suppressed dependence and spatial profile region in the aperture plane

time-harmonic is assumed to occupy the

This field irradiates a perfectly conducting boundary with sinusoidal height profile


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L. B. FELSEN AND V. GALDI

measured from the

plane. We first consider the aperture problem.

II.a Radiation from aperture distributions We summarize here essential results from previous publications (Bastiaans, 1980; Einziger and Shapira, 1986; Maciel and Felsen, 1989; Steinberg et al, 1991). The field radiated into the half-space from the aperture in (1) can be expressed as a line-source superposition (Kirchhoff integration)

where is the free space wavenumber; is the free space wavelength; zeroth order Hankel function of the first kind; is a scalar Debye potential; and

is the

II.b Beam discretization The aperture field is to be parameterized in terms of Gaussian beam basis functions via the rigorous self-consistent Gabor series representation

where

represents the normalized Gaussian window

With

representing the wavenumber, this representation places the beams on a discretized phase-space lattice (Fig. 1b), on which each lattice point gives rise to a Gaussian beam whose spatial and spectral (tilting) shifts are tagged by the indexes and respectively. Spatial and spectral periods are related by the self-consistency relation (configuration-spectrum tradeoff) (Bastiaans, 1980). The expansion coefficients can be computed by introducing an auxiliary function defined through the biorthogonality condition (Bastiaans, 1980),

where * denotes the complex conjugate, while (Bastiaans, 1980),

for

and

for

Accordingly

For Gaussian windows, the biorthogonal function is given in (Bastiaans, 1980). For numerical computation of the Gabor coefficients, see (Einziger and Shapira, 1986). The radiated potential field in the half-space (see (3)) can be represented as (Maciel and Felsen, 1989)

CSP NARROW-WAISTED RAY-LIKE GAUSSIAN BEAMS

where the beam functions

23

are synthesized by Gabor-weighted line-source superposition

R being defined in (4). The integral in (10) (or its spectral counterpart) can be evaluated asymptotically in the beam paraxial far zone, yielding the following complex source point (CSP) approximation (Maciel and Felsen, 1990a)1,

with representing the complex distance between the observer at point (here and henceforth, the tilde denotes a complex quantity),

and the complex source

In accord with the radiation condition, the square root is defined by The displacement parameter (Fresnel length) is related to the beam lattice period and the beam axis angle (Maciel and Felsen, 1990a), whence (11) is valid in the paraxial far-zone of each beam, For large tilt angles with the beam tilt angle is complex and the corresponding beams become evanescent.

II.c Narrow-waisted beams For narrow-waisted beams the Gabor coefficients can be effectively estimated by sampling the aperture field distribution, without performing the integration in (8) (Maciel and Felsen, 1989),

so that from (9) and (11)

where is obtained from (11) with The tilted beams in the Gabor expansion, which generate evanescent ”far fields”, are ignored in this approximation.

II.d Linearly-phased aperture Narrow-waisted beams work very well for nonphased apertures, but usually require finer aperture sampling in the presence of phasing (Maciel and Felsen, 1990a). Here we consider a linearly-phased cosine aperture field,

where is the tilt angle of the main radiation lobe with respect to the axis. In this case, a more effective discretization can be obtained by Gabor-expanding the real function only, and including the linear phasing in the beam integral (10) for the beam propagator. Accordingly, the narrow-waisted beam expansion can be recast as

1 Note that there are some sign changes with respect to (Maciel and Felsen, 1990a), since here we assume propagation into the half-space

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The beam propagator (18) differs from in (15) by the phase shift which produces the propagation-matched tilt in the beam direction. In Fig. 2, the near-zone potential field synthesized using the narrow-waisted tilted beam decomposition in (17) is compared with a calibrated computation-intensive Kirchhoff integration reference solution based on (3), and with the nontilted formulation in (15) when applied to the entire aperture field in (16). The tilted beam synthesis is hardly distinguishible from the reference solution, whereas the nontilted synthesis is less accurate in the magnitude.

II.e Reflection from a periodic perfectly conducting boundary The field radiated by the aperture distribution in (16) is now assumed to impinge on a perfectly conducting moderately rough periodic boundary described by the continuous function which is assumed to vary slowly over a wavelength scale (Fig. 1a). Moderately rough irregular dielectric interfaces separating two dielectric half-spaces are treated elsewhere (Galdi et al, 2000a). The reflected field can be constructed rigorously by complex ray tracing applied to each beam in the aperture decomposition; this requires the analytic continuation, into a complex configuration space, of all geometrical parameters involved (with the exception of the observation point). However, narrowwaisted beams can be tracked accurately and much more efficiently via a beam-tracing paraxial almost real ray-tracing scheme (Ruan and Felsen, 1986; Maciel and Felsen, 1990b)), valid in appropriately calibrated observation ranges. For the new application to a periodic boundary, we first treat the canonical problem of CSP Gaussian beam reflection from a curved segment on a conducting boundary. The problem geometry is illustrated in Fig. 3. An incident Gaussian beam is generated by a CSP at

being the beam axis real departure angle with respect to the axis. For electrically large and smooth scatterers, and when the observation point lies in the paraxial region of a reflected beam in Fig. 3), the field can be approximated in terms of the on-axis field of that beam (at ) and a complex phase correction. Denoting the on-axis parameters by the subscript zero, one finds for the potential field (Ruan and Felsen, 1986)


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Here (see Fig. 3): is the complex phase correction; is the complex (virtual) focus obtained by analytic continuation of the standard ray-optical formulas (Felsen and Marcuvitz, 1973, p. 168); the complex incidence point is approximated by the real beam-axis incidence point is the reflected-beam-axis real departure angle with respect to the axis; is the curvature radius at and is the plane-wave potential field reflection coefficient. As shown in (Ruan and Felsen, 1986), this scheme corresponds to tracing a ray along a complex trajectory from the CSP at to the intersection of the real beam axis with the real surface; from there, the path to the observer proceeds entirely in real configuration space, along the beam axis. Multiple reflections, which may occur in the configuration in Fig. la, can be incorporated by iterating (21), whereby the complex focus (either virtual or real) associated with each iteration becomes the phase reference for the next iteration (Galdi et al, 2000a). Apart from the complex ray connecting the CSP to the first real incidence point, the multi-hop path to the observer proceeds entirely in real configuration space along the beam axes, and the phase correction is applied only on the last path segment leading to the observer.

II.f Application: infinite sinusoidal boundary For a first check on the applicability of the narrow-beam algorithm to surface scattering problems, we have considered the sinusoidal boundary in Fig. 1a illuminated by a nonphased cosine aperture distribution with wavelength Extensive numerical experiments have been performed for various observation heights and aperture heights profiles with various minumum curvature radii and various beam lattice spacings All of these numerical implementations for the scattered potential field have been compared with a numerically integrated, computation intensive Physical Optics-Kirchhoff reference solution based on (3); by previous calibration, Physical Optics has been confirmed to apply to the profile parameters under consideration here. Typical results for the potential are displayed in Fig. 4. In general, we have found that the accuracy of the narrow-waisted beam algorithm improves for greater observation distance (because of the far-zone paraxial approximation), but even at moderate distances the agreement is satisfactory. We also found that the beam algorithm is quite robust with respect to scramblings of the Gabor lattice. As discussed in (Maciel and Felsen, 1990b), this feature can be exploited to obtain a priori accuracy assessments when reference solutions are not available. For the present nonphased aperture example, we found the best tradeoff between accuracy and computational cost to occur for A finer sampling may, however, be required for phased aperture field distributions. On the other hand, the accuracy gets worse as the distance of the aperture from the surface increases. We found that in order to get robust and accurate predictions, we should have where is the maximum of the boundary profile. However, this is not a very restrictive limitation since it is always possible (and computationally cheap because of the simple determination of the Gabor coefficients via (14)) to perform a multi-step Gabor decomposition for greater aperture-to-boundary distances, i.e., project the beam-computed radiated potential field onto a virtual aperture suitably close to the surface and then again apply the narrow-waisted beam algorithm. Concerning degradation of accuracy with increase in wavelength, we found that even for relatively ”low-frequency” geometries, i.e. moderate as in Fig. 4b, the beam algorithm, though no longer highly accurate, still provides reasonably good predictions (for details, see (Galdi et al, 2000a)).

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III. TIME DOMAIN RADIATION FROM DISTRIBUTED APERTURES We shall now explore the extension of the FD results for aperture radiation in Sec. II.a to timedependent excitation, in particular to short pulses. To this end, we consider a space-time aperture field distribution at with separable space-time dependence and linear time delay

where is the speed of light and is a pulse with characteristic width This distribution represents the TD counterpart of the linearly-phased time-harmonic aperture in (16). The present TD formulation is restricted to the radiated field only, and we analyze the propagation into the halfspace, wherein (23) gives rise to a space-time pulse propagating in the direction (Fig. 5). It is assumed that the normalized width of the pulse is much shorter than the aperture dimension i.e.,

III.a Reference solution Using the two-dimensional TD Green’s function

where H(·) is the Heaviside step function, the field radiated into the half-space can be represented as a space-time Kirchhoff integration (Morse and Feshbach, 1973, Sec. 7.3; Kragalott et al, 1997),

Direct numerical integration of (25) is complicated by the square-root (integrable) singularity at the upper limit and requires special care. We have used the Newton-Cotes scheme (Kragalott et al, 1997) for the numerical integration of (25), which represents our reference solution.

III.b Beam discretization The formal extension of the Gabor-based time-harmonic aperture radiation to time-dependent excitation involves a four-index Gabor series set on a discretized lattice in an eight-dimensional phase space (space-wavenumber, time-frequency). For a rigorous treatment and computational issues, see (Steinberg and Heyman, 1991). We shall explore to what extent the narrow-waisted beam approach,


27

effective for time-harmonic excitation, can be generalized to TD (short-pulse) excitation. The lineardelay aperture field distribution (23) admits the equivalent spectral representation

A TD beam discretization can be obtained by Fourier-inverting the narrow-waisted tilted beam expansion (17) for the FD linearly-phased aperture presented in Sec. II.d. In order to accommodate the evanescent spectra in the FD beam propagators (18), we use the analytic signal formulation instead of the standard Fourier transform (Heyman and Melamed, 1998). Concerning the beam lattice discretization, one can choose a frequency-independent beam lattice period (resulting in a frequencydependent beam parameter ), or a frequency-independent beam parameter (resulting in a frequencydependent ). We choose frequency-independent because it yields frequency-independent Gabor coefficients (see (17)). The TD counterpart of the narrow-waisted FD beam expansion (17) for the aperture field distribution (26), with reference to the electric field, can be thus written as

The pulsed beam propagator is the TD counterpart of the FD paraxial, far-zone beam propagator in (18), with (19)2 (for simplicity, the subscript is henceforth replaced by )

Via the analytic signal formulation, one has (Re=real part)

While the integral in (30) cannot be evaluated explicitly in general, we have found useful closed-form approximations for the important class of Gaussian pulses. In particular, we consider a Rayleigh (four-times-differentiated Gaussian) pulse

2 Note that (28), (29) are slightly different from (18), (19), since here the electric field is considered instead of the potential, the aperture plane is located at and propagation is into the halfspace.

28


but the procedure presented below can be applied to any kind of modulated or differentiated Gaussian. Since the beam lattice period has been chosen frequency independent, the beam parameter and hence the complex distance in (29) are frequency dependent. For we can approximate in the amplitude factor of (28)

rendering the distance

real. In the phase, we retain the first order paraxial correction

valid for

In the TD, the beam parameter

here

are the beam coordinates (see Fig. 5)

must be small over the entire bandwidth

of

in (31),

With these assumptions, the integral in (30) can be reduced to the generic form (the spectrum of in (31) is evaluated readily),

which can be expressed in terms of confluent hypergeometric functions (Abramowitz and Stegun, 1964, Sec. 13). Accordingly, the TD beam propagator can be written explicitly as

where and are defined in (32) and (34); is the gamma function (Abramowitz and Stegun, 1964, Sec. 6); and is the confluent hypergeometric function (Abramowitz and Stegun, 1964, Sec. 13). The above procedure can be applied to any Gaussian pulse; modulation or differentiation only affects the arguments of M. We found simple rapidly converging approximations for the functions and in the form (Galdi et al, 2000b). These functions resemble the functional form of the time pulse excitation in (31). Using these approximations the TD beam propagator in (37) can be computed efficiently. III.c Assessment of accuracy The restriction in (35) is the most serious because, for specified and observation point, it determines the maximum allowable lattice period (i.e., the minimum number of beams). The overall constraint can be expressed as (Galdi et al, 2000b)

where

is the distance of the observation plane scaled by the Fresnel distance of the aperture, is the normalized bandwidth of the pulse and determines the number of beams in the expansion (27). The nondimensional estimator expresses the range of validity of the algorithm in terms of all relevant parameters of the problem. For example, increasing the lattice period (i.e., decreasing the number of beams) can be compensated by a corresponding


29

increase of In order to assess the accuracy of the proposed TD beam expansion, we have performed computations for the linear-delay space-time aperture distribution (23) with a sine spatial tapering, excited by the Rayleigh time pulse (31). Figure 6a shows the time evolution of the electric field at a fixed observation point in the near zone of a large aperture without phase delay computed via the space-time Kirchhoff integration (25), and via the TD beam synthesis (27) with various beam lattice periods. As expected, the agreement improves as the beam lattice period decreases, and satisfactory accuracy is achieved for (for this example ). It is observed from the transverse cut in Fig. 6b that, despite the use of the paraxial paraxial far-zone approximation, the TD beam synthesis is quite accurate even in the near zone of the aperture and not only around the main radiation lobes. We found that, as the observation distance increases, a coarser discretization can be used according to (40); even at moderate distances, quite accurate syntheses can be achieved with a relatively small number of beams (~ 30) (Galdi et al, 2000b). The corresponding results for linear-delay are shown in Fig. 7 and the same considerations apply.

IV. CONCLUSIONS In many current forward and inverse scattering scenarios, there is a need for numerically efficient robust

30


forward solvers for fields excited by distributed sources in the presence of complex environments. This motivation has led us to re-visit the previously developed FD narrow-waisted Gaussian beam algorithms (Maciel and Felsen, 1989; 1990a,b) and to extend them to new FD applications as well as to the short-pulse TD. The outcomes from the rough sinusoidal scattering example here, as well as the first results in the TD, are encouraging. Further FD extensions to irregularly rough interfaces between dielectrics are already in progress, as are TD interactions with canonical scatterers to learn the new rules.

ACKNOWLEDGEMENTS We acknowledge fruitful discussions with Professor D.A. Castañon (Boston University) on forward and inverse scattering and imaging which motivated this investigation. We also acknowledge partial support by ODDR&E under MURI grants ARO DAAG55-97-1-0013 and AFOSR F49620-96-1-0028. In addition, VG acknowledges a European Union postdoctoral fellowship through the University of Sannio, Benevento, Italy.

REFERENCES Abramowitz, M., and Stegun, I.A., 1964, Handbook of Mathematical Functions, Dover, New York. Bastiaans, M.J., 1980, Gabor’s expansion of a signal into Gaussian elementary signals, Proc. IEEE, 68:538. Deschamps, G.A., 1971, Gaussian beams as a bundle of complex rays, Electron. Lett., 7:684. Einziger, P.D., and Shapira, M., 1986, Gabor representation and aperture theory, J. Opt. Soc. Am. A, 3:508. Felsen, L.B., Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams, Symp. Matemat., Istituto Nazionale, di Alta Matematica,“ Academic, London, XXVIII:40. Felsen, L.B., and Marcuvitz, N., 1973, Radiation and Scattering of Waves, Prentice Hall, Englewood Cliffs, NJ. Classic reissue, IEEE Press, Piscataway, NJ, 1994. Galdi, V., Felsen, L.B., and Castañon, D.A., 2000a, Quasi-ray Gaussian beam algorithm for scattering by, and reconstruction of, moderately rough interfaces - Part I: forward scattering (internal memorandum, in preparation for publication). Galdi, V., Felsen, L.B., and Castañon, D.A., 2000b, Narrow-waisted Gaussian beam discretization for time-dependent radiation from large apertures (internal memorandum, in preparation for publication). Heyman, E., and Melamed, T., 1998, Space-time representation of ultra wideband signals, Advances in Imaging and Electron Physics, 103:1. Kragalott, M., Kluskens, M.S., and Pala, W.P., 1997, Time-domain fields exterior to a two-dimensional FDTD space, IEEE Trans. Antennas Propagat., 45:1655. Maciel, J.J., and Felsen, L.B., 1989, Systematic study of fields due to extended apertures by Gaussian beam discretization, IEEE Trans. Antennas Propagat., 37:884. Maciel, J.J., and Felsen, L.B., 1990a, Gaussian beam analysis of propagation from an extended aperture distribution through dielectric layers, Part I - plane layer, IEEE Trans. Antennas Propagat., 38:1607. Maciel, J.J., and Felsen, L.B., 1990b, Gaussian beam analysis of propagation from an extended aperture distribution through dielectric layers, Part I - circular cyilindrical layer, IEEE Trans. Antennas Propagat., 38:1618. Morse, P.M., and Feshbach, H., 1953, Methods of Theoretical Physics, McGraw-Hill, New York. Ruan Y.Z., and Felsen, L.B., 1986, Reflection and transmission of beams at a curved interface, J. Opt. Soc. Am, A, 3:566. Steinberg, B.Z., Heyman, E., and Felsen, L.B., 1991, Phase-space beam summation for time-harmonic radiation from large apertures, J. Opt. Soc. Am. A, 8:41. Steinberg, B.Z., and Heyman, E., 1991, Phase-space beam summation for time-dependent radiation from large apertures: discretized parameterization, J. Opt. Soc. Am. A, 8:959.

DIFFRACTION BY ARRAYS OF COMPLEX SOURCE POINT BEAMS

Hong D. Cheung and Edward V. Jull Department of Electrical and Computer Engineering University of British Columbia Vancouver, BC. Canada V6T 1Z4

INTRODUCTION Scattering by an object depends not only on the shape of the object but also on the source of the incident field. Most analytical solutions (eg. Bowman et al, 1987), are for plane wave incidence; that is a source so distant that its directivity has no effect. Or, if the source is local, it is omnidirectional. Here a numerical procedure for extending local omnidirectional source solutions to those for local extended sources at any range is described. It can be applied to both low and high frequency scattering solutions with an accuracy dependant only on the number and accuracy of the basis source solutions used. A superposition of solutions for omnidirectional sources closely spaced in the aperture with amplitudes corresponding to the aperture distribution could provide a correct near field scattering solution but more efficient solutions will require larger source spacings. Then it is necessary to use beam rather than omnidirectional sources and arrange the beam sources in Gabor lattice as described by Einziger at el (1986). Both radiative and reactive aperture fields may then be represented to any accuracy at any range. Gabor (1946) proposed a series of time and frequency shifted Gaussian functions as an alternative to Fourier analysis in signal processing. Most of the impediments which delayed the implementation of Gabor analysis appear to have been overcome and its application to aperture radiation has been reviewed by Bastiaans (1998). The translated and phase shifted Gaussian functions of signal analysis become translated and directionally shifted Gaussian beams in aperture analysis. But Gaussian beams are approximate solutions to the wave equation and here it is found preferable to use complex source point (CSP) beams, which rigorously satisfy the wave equation. Then exact scattering solutions for local omnidirectional sources can be converted to exact CSP beam solutions by substituting appropriate complex coordinates for the real source coordinates. These then become the basis functions for extended source scattering solutions. Complex source point beam are paraxially Gaussian and thus fit well into the framework of Gabor analysis. When many are used with the same amplitude coefficients CSP and Gaussian beams provide virtually identical results in aperture analysis. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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32

H. D. CHEUNG AND E.V. JULL

First a brief review of the theory and the choice of beam arrangements are made. Then some numerical examples of two dimensional radiation from apertures and its scattering by local simple structures are presented to illustrate the power and simplicity of the method.

DISCRETE SOURCES IN APERTURES

Fig. 1 shows an array of linearly shifted and rotated beam sources w(x) which are to represent the two-dimensional E-polarized fields of an aperture in This aperture field can be written as a Gabor series

where L is the beam spacing along the x-axis and radians is the angular spacing of the beam directions. For a Gaussian window function w(x), the beam amplitudes can be determined by convolving the aperture distribution with a biorthogonal function defined by Einziger et al (1986) With these amplitudes, the field in

for an aperture along the

may be written as

are the elementary Gaussian or complex source point

(CSP) beam fields. For a scattering solution they are the solution for a single CSP beam incident on the scatterer. With enough beams included the above can represent the entire field of any aperture at any range, and the scattered fields for any aperture blockage. Usually the reactive fields are significant only within a fraction of a wavelength from the aperture and can be omitted. Then the bound on the summation in n is the first integer Also since the 2M + 1 beam locations spaced L must cover the aperture width the upper bound on is Finally a key parameter is the spacing in wavelengths of

DIFFRACTION BY ARRAYS OF CSP BEAMS

33

bound on m is Finally a key parameter is the spacing in wavelengths of the elemental beam. Einziger et al (1986) proposed that This places the boundary of the visible spectrum midway between the peaks of the last radiating beam and first nonradiating beam and appears to be an efficient arrangement. Einziger et al (1986) and Maciel and Felsen (1989) have investigated numerically various combinations of beam arrangements. An efficient choice of beam spacing reduces the required number beams by selecting only those that are significant. Larger beam spacings require more rotated beams (larger N) but fewer beam locations (small M) and conversely smaller require fewer rotated beams but more beam locations. We have found it most convenient to use a spacing Thus N = 0, so all beam directions are normal to the aperture, and (1) and (3) become simply

PATTERNS OF COMPLEX SOURCE POINT BEAM ARRAYS An omnidirectional electric line source is

where and and the field point at

is the distance between the source at On replacing the source coordinates with complex values

(6) becomes

which is a beam with its maximum in the direction and its half power beamwidth is

its minimum in the direction

While (9) is a far field result, in the near field of the complex source the beam behaves as the near field of an essentially Gaussian distribution in an aperture of width In the above angles are measured counterclockwise off the line of the array. Thus in Fig. 1 and if beam axes are all in the z-direction and spaced along the x-axis with a beam parameter then and each CSP beam has a half power beamwidth of 77.6° from (10). Fig. 2 shows the near and far field radiation patterns of a 5 element beam array with relative amplitudes at x = 0, respectively, representing a uniform aperture distribution. The relative amplitude of the largest coefficients omitted is All the other beam coefficients, including the tilted beams, are substantially smaller and alternate in sign. The parallel five beams produce accurate main beam and first sidelobe level patterns in the near and far field of the

34


aperture. For larger apertures more beams are required and then more sidelobes are accurately represented, errors occurring in the height of the furthest sidelobes off the main beam.

SCATTERING BY OBSTACLES NEAR APERTURES

The far field of a line source parallel to a conducting half plane can be written exactly in term of Fresnel integrals. Thus for the beam array of (5) the basis functions


35

become

where

The dashed curve of Fig. 3b is the far field pattern of Fig. 2 and the solid curve is the far field pattern when the aperture is partially blocked by a parallel conducting half plane with its edge at x = 0, that is at a range of The scattered beam is broadened and outward shifted from its original position. With a larger separation between aperture and half plane there is less broadening but slightly more shift. If the array of Fig. 2 is normally incident on a conducting circular cylinder as in Fig. 4a the array elements for the scattered field are

with replaced by in (7) and (8). Fig. 4b shows the total scattered field of a cylinder of radius with its axis at distances of and from a uniform aperture of width Pattern oscillations are due to interference between the direct and scattered fields. These do not occur in plots of the scattered field only. In this situation a single CSP beam of the same far field beamwidth as the aperture pattern provides a good approximation to the scattered field, as has been shown by Cheung and Jull (1999b). This is because only the main beam of the incident pattern in intercepted by the scatterer. The incident main beamwidth of the array broadens in the near field, as shown in Fig. 2. So also, correspondingly, does the CSP beam.

If the array of Fig. 2 is normally incident on the axis of a square conducting cylinder, with the array axis inclined at off the face of the cylinder, as in Fig. 5a,

36


If the array of Fig. 2 is normally incident on the axis of a square conducting cylinder, with the array axis inclined at off the face of the cylinder, as in Fig. 5a, the scattered field is symmetrical about The total field in can be found by the methodology of the uniform geometrical theory of diffraction (UTD) if the sides of the cylinder are not small in wavelength. Then the array elements of the total far field can be written as Here

and

are, respectively, the incident field, the field reflected from the upper

illuminated face of the square cylinder and the total diffracted field of the three edges in the angular region Shadow and reflection boundaries of the facets and edges of the square cylinder determine where these incident and reflected fields contribute and for beam sources boundaries between these regions differ slightly from geometrical optics values. These boundaries are defined in Cheung and Jull (2000) along with expressions for the uniform geometrical optics and singly diffracted fields. By comparison with corresponding numerical results obtained by the moment method, it shown that for Epolarization doubly diffracted fields from the edges are insignificant for cylinder facets of two wavelengths or more.

Fig. 5b shows the total E-polarized far field resulting from a single beam and from a beam array representing a uniform aperture distribution, both at a distance from the edge of a conducting square cylinder of side The far field pattern of the beam array is that of the lower curve of Fig. 2 and has a half power beamwidth of about 20°. At the near field range of the scatterer the incident field is similar to the upper curve of Fig. 2 with a wider beamwidth. The single beam source with has a half power beamwidth of about 20° according to (10) and in the near field this broadens corresponding to that of the array, but the pattern remains a single beam which is paraxially Gaussian. The major differences between the patterns of Fig. 5b are mainly due to the differing structures of the incident fields which, together with the scattered field, make up the far field patterns. Both patterns are almost identical in and around the forward direction where the total field is only that diffracted by the edges. The edges are illuminated by very similar incident fields. At the array representing a uniform aperture distribution has a far field pattern null whereas the single beam source does not. Consequently major differences in the total far field pattern occur. Similarly at


37

where the second null of the aperture far field pattern occurs. Short periodic oscillations in the single beam pattern are all due to interference between diffracted fields from edges off the beam axis and the incident field. This is evident from a comparison with corresponding results for a 90° wedge. The 90° wedge comparison for the aperture scattered fields exhibits also the short periodic oscillations over the reflection lobe evident in Fig. 5b and are thus due in part to inherent differences in near field wedge scattering by an extended aperture source and a single beam source.

CONCLUDING REMARKS The procedure described here seems to be the most general and efficient method for accurately dealing with practical near field scattering problems. It can use the repertoire of line or point source low and high frequency diffraction solutions for canonical structures and it can efficiently provide as much accuracy as these solutions if sufficient sources are used. The examples given here for half planes, cylinders and multiple wedges demonstrate its efficiency for these elementary scatterers. Of course a single CSP beam source is simpler and sometimes adequate for the scattered field alone. But unless this incident field closely resembles a single beam, for example if it is that of a cosine-squared aperture distribution, its total field scattering pattern will substantially differ from that of a single beam. REFERENCES Bastiaans, M., 1998, Gabor’s signal expansion in optics, Chap. 14 of Gabor Analysis and Algorithms, Feichtinger, H. G. and Strohmer T. (Eds.) Birkhauser, Boston. Bowman, J. J., Senior, T. B. A. and Uslenghi, P. L. E., (Eds.) 1987, Electromagnetic and Acoustic Scattering by Simple Shapes, revised printing, Hemisphere Publishing Corporation, New York. Cheung, H. D. and Jull, E. V., 1999a, Two-dimensional diffraction by half-planes and wide slits near radiating apertures, IEEE Trans. Antennas Propagat., 47:1669. Cheung, H. D. and Jull, E. V., 1999b, Scattering of antenna beams by local cylinders, J. Electromag. Waves Applic., 13:1315. Cheung, H. D. and Jull, E. V., 2000, Antenna pattern scattering by rectangular cylinders, IEEE Trans. Antennas Propagat., (in press) Deschamps, G. A., 1971, Gaussian beam as a bundle of complex rays, Electron. Lett., 7:684. Einziger, P. D., Raz, S. and Shapira, M., 1986, Gabor representation and aperture theory, J. Opt. Soc. Am. A, 3:508. Gabor, D., 1946, Theory of communication, J. Inst. Elect. Eng., 93III:429. Maciel, J. J. and Felsen, L. B., 1989, Systematic study of fields due to extended sources by Gaussian beam discretization, IEEE Trans. Antennas Propagat., 37:884.


APPLICATION OF CONCEPTS OF ADVANCED MATHEMATICS AND PHYSICS TO THE MAXWELL EQUATIONS

Carl E. Baum Air Force Research Laboratory AFRL/DEHE 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776

1.

INTRODUCTION

Since the pioneering work of James Clerk Maxwell [17] in establishing what we call the Maxwell equations

including both electric and equivalent-magnetic source terms, these have had a profound effect on the development of science and engineering. (Note that the divergence equations are implied by the curl equations.) In addition, some material-related parameters are needed to relate such as the constitutive parameters, for example in the form

and

to

and

Here we have introduced the common frequency-domain form so that the vector fields are dot multiplied by 3 × 3 dyadic constitutive parameters, which in time domain become convolution operators over time. More general (even nonlinear) forms are sometimes encountered. Various boundary conditions (e.g., perfectly conducting surfaces) are readily derived as limiting cases. People often think of dividing the basic and applied sides of the technological enterprise as between science and engineering, but this can lead to some confusion. I think that there is a better three-part division, which can shed some light on where electromagnetic (EM) theory fits into the structure. First, there is the basic scientific side which has electromagnetics as part of physics, and the fundamental question concerns the replacement of the Maxwell equations by something more accurate, applying to extreme conditions not normally encountered. This is not what we think of as electromagnetic theory in the usual sense. Second, we have what may be called applied science or basic engineering in which we explore the established physical laws (the Maxwell equations in this case) to see what they imply in the sense of discovering what is possible to analyze, synthesize, optimize, etc. This is distinct from the third category which might be termed applied engineering which concerns itself with the routine implementation of what is known from me second category in terms of technological products (“practicing” engineering), for example, by selection of antenna designs from a product catalog. Of course, the reader might prefer some other “diagonalization” but this should suffice for the present. So, concentrating on the second category, the role of the electromagnetic theorist (including sometimes basic experiments, particularly as demonstrations and confirmations) concerns understanding what the Maxwell equations allow one to do in the way of analysis and synthesis of the performance characteristics Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

39

40

C. E. BAUM

of various electromagnetic devices as well as understanding the behavior of electromagnetic fields in natural environments. At this point, I would like to emphasize the concept of EM synthesis. One can analyze the interaction of EM waves with arbitrary geometries of various materials. While this is a challenging task, it is not synthesis. Synthesis starts with some desired performance characteristics and asks: “Is this possible within certain general constants (e.g., passivity)?” If it is possible, then one moves on to other questions such as : “What are the best possible values of the appropriate performance parameters?”, and “What are the algorithms for designing (realizing) the device (antenna, scatterer, etc.) with the desired performance parameters?” By analogy one can recall that circuit analysis with passive lumped elements (LRC) was developed into a matrix form based on the Kirchoff laws for voltage and current as written on a network (graph). Circuit synthesis later asked (and answered) questions like [18]: “What kind of input impedances and transfer functions are possible in such networks”, and “How are such things systematically realized?” An important part of EM theory then needs to be concerned with EM synthesis. One might even think of this as a generalization of circuit synthesis In 1976 I published a review paper concerning transient EM theory [3]. In this I outlined some analytic concepts used in mathematics and physics that are not commonly being used, or just beginning to be used, in EM theory for both analysis and synthesis. Since then considerable progress has been made in exploring these concepts and obtaining useful results. In the present paper these analytical concepts and major results are summarized under the following section headings.

2.

INTEGRAL-OPERATOR DIAGONALIZATION Electromagnetic scattering is often formulated as an integral equation of the form

The notation is related to bra/ket notation in quantum mechanics, with here integration over the common coordinates type of multiplication (dot above the comma here), but with no conjugation implied since our operators are not in general Hermitian. For convenience (2.1) uses the symmetric impedance (or E-field) kernel, related to the dyadic Green function (of free space or other linear reciprocal media), but other kernels (e.g., H-field) are also used. The domain of integration can be over a volume or surface (using tangential components) as desired. As with matrices for which one finds eigenvalues and eigenvectors we can form [7,21]

and we can refer to this as the eigenmode expansion method (EEM). For cases of degeneragy (two or more equal eigenvalues) one use the Gram-Schmidt orthogonalization procedure to complete the construction of the orthonormal set. (More on this appears under symmetry.) With (2.2) we can write the kernel in the form

where represents an arbitrary power, including for inverse kernel which in (2.1) solves for the current on the scatterer. This is not the only kind of eigenmodes one can form from the integral equation, but is a natural choice for our purposes. Other kinds with other names (such as characteristic modes) are introduced by others for special purposes. At this point we can recall [3, 7] that having solved for the eigenimpedances and eigenmodes of a perfectly conducting body (for which (2.1) becomes a surface integral equation), one can also solve directly for the body loaded by some uniform, isotropic sheet impedance by the transformation

APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS while retaining the same eigenmodes. Then

can be synthesized to give desirable characteristics to the

scatterer or antenna described in the form (2.1). Given limitations of circuit synthesis one can make (poles of the response in which

41

for the unloaded body, then within the have desirable characteristics such as roots

appears) at desirable places in the s plane. These roots

can even be made second order in some cases to give critical damping to the response. Here is a clear example of EM synthesis. Here we also note that the can be split into interior and exterior parts (in electrical parallel combination) which separate the internal and exterior resonances (poles) [7]. However, the details are too elaborate to repeat here. Recently [9] a transformation like (2.4) has been found to apply to more general volumetric dielectric bodies, even those consisting of homogeneous isotropic dielectric bodies residing in an inhomogeneous dielectric space. 3.

COMPLEX VARIABLES APPLIED TO FREQUENCY

As discussed in [3] the analytic properties of the solution of the Maxwell equations as a function of the complex frequency, s, lead to several important ways to solve the Maxwell equations. For antennas and scatterers of finite size in three dimensions this leads to three methods based on expansions used in complexvariable theory. 3.1 Low-Frequency Method (LFM) In complex variables functions are often expanded in terms of a power (Taylor) series about some point where the function is analytic. In EM, this is done for scattering by expanding about s = 0. As one expects, the leading terms are related to the induced electric and magnetic dipoles, related to the incident fields by polarizability dyadics. This is extended to antenna input impedance/admittance by inclusion of a pole at s = 0 when appropriate giving leading terms which can be interpreted as inductance, capacitance, and/or resistance [21]. Here, I would like to emphasize an application of importance to antenna design, particularly the lowfrequency characteristics, concerns the matching of the electric- and magnetic-dipole moments ( and ) in transmission [1]. Defining appropriate unit vectors we have

A remarkable property of such combined dipoles is that on the axis from the antenna “center” in the direction (beam center) the electric and magnetic fields are at right angles and related by

even in the near field including and terms, i.e., all the dipole terms. This has important consequences for low-frequency illumination of large areas for EM interaction measurements, such as for simulation of the nuclear electromagnetic pulse (EMP). The pattern of such an antenna (far field) is a cardiod (radiated power proportional to where

is the angle to the observer relative to

remaining

). It has a null in me back direction

but there is a

term there with the same field ratio as in (3.2) (at right angles with Poynting vector still in the

direction, i.e., back to the antenna).

In reception such an antenna also has similarly interesting

directional properties. An important class of low-dispersion antennas (for transient/broad-band radiation/reception) are referenced as impulse radiating antenna (IRAs) [27]. These can be (and are) designed to exhibit this

42

C. E. BAUM

combined dipole behavior at low frequencies with pointing in the same direction as the high-frequency beam. This improves the directionality and modestly decreases the low-frequency roll-off frequency. 3.2 Singularity Expansion Method (SEM) Of more recent vintage (1971) there is SEM. There is already an enormous literature on this subject. Here we mention two review papers with lots of references [14, 15]. In this case a related basic complexvariable expansion is the Laurent expansions in which an expansion is found for the neighborhood of a pole. From (2.1) natural frequencies and modes are found via

Immediately we observe that natural frequencies and modes have nothing to do with the incident-wave parameters (direction of incidence, polarization) in a scattering problem. To better appreciate the above, imagine that one is performing a moment method (numerical) computation. The kernel (operator) is replaced by an N × N matrix from which we find the natural frequencies via

with the natural modes subsequently numerically determined. At this point, we can compare (3.3) to (2.2) and observe that the are roots (zeros) of the linking the index as ( root of the eigenvalue). Assuming an incident plane wave as

the current on the body is expanded as

This is the simplest form of coupling coefficient termed class 1, and it contains the information concerning the incident field. The entire-function term is applicable to early times. By judicious choice of the turn-on time it has been shown that, for perfectly conducting bodies, this entire function can be made zero [6]. In time domain the current is

possible entire function (temporal form) convolution with respect to time t so that the pole terms transform to give a simple time-domain form. While the entire function can be made to be zero the sum in (3.7) is not an efficient early-time representation. The scattered far field takes the form

APPLICATIONS OF CONCEPTS TO THE MAXWELL EQUATIONS

43

Using (2.3) one can readily express the scattering dyadic in EEM form. Here we write the SEM form as

In backscattering this takes the symmetric form

In contradistinction to the current in (3.6), except in special cases, the entire-function contribution to the scattering dyadic cannot be made to go to zero by judicious choice of Noting that the entire function is an early-time contribution one can look at the late-time response for target-identification purposes. We can summarize the major areas of SEM development: 1. description of EM response (especially transient) of various structures (currents) modeling electronic systems [19] 2. equivalent circuits for antennas and scatterers [21] 3. target identification (free space) [15] 4. identification of buried targets (mines, unexploded ordnance) [26] There are also various numerical techniques to analyze data for the SEM parameters [15]. Consulting the references one can find a huge list of references. Perhaps other major areas of SEM application will emerge in the future. 3.3 High-Frequency Method (HFM) In complex-variable theory one often deals with asymptotic expansions as the complex variable tends to infinity. In EM we can collectively refer to such techniques as the HFM [3]. This includes geometric, spectral, uniform, etc., theories of diffraction. An enormous literature exists here. While I have had occasion to consider such techniques, these have developed by many others, and I will not dwell on this. 4.

SYMMETRY AND GROUP THEORY

Group theory has long been used in physics to study the quantum mechanical properties of elementary particles, atoms, molecules, and crystal lattices based on the symmetries of the quantum wave functions. One may consider [3] whether something similar would be useful for the analysis and design of antennas and scatterers. Lewis Carroll had the Hatter ask: “Why is a raven like a writing-desk?” One might ask a similar strange question: “Why is an airplane like a hydrogen molecule?” At least the second question has an answer. They are like in two ways. A first way concerns SEM (Section 3.2). The natural frequencies (in general complex) are characteristic of the body (homogeneous problem), and are analogous to the energy levels (typically real (bound states), but also complex (radioactive decay)) of the quantum system. A second way concerns symmetry. Both objects contain a symmetry plane and the EM response (eigenmodes

44

C. E. BAUM

and natural modes) and the quantum wave functions naturally divide into two sets (symmetric and antisymmetric) with respect to the symmetry plane. In physics this property is often called parity. While the quantum symmetries are properties found in nature, the EM symmetries are of two kinds: those inherent in the Maxwell equations (duality, reciprocity, relativistic invariance), and geometrical symmetries built into objects by human beings (or aliens). The close connection between the symmetries in antennas and scatterers and the symmetries in the associated EM waves can be used to design antennas and scatterers and to identify radar targets. The reader can consult [23] for a detailed treatment of this subject, concerning which much progress has been made in recent years. Here we take a group in the form of a 3 × 3 dyadic representation as

For the point symmetry groups (rotations and reflections) these are real and orthogonal with

In some cases these can be taken as 2 × 2 dyadics (or even scalars) (e.g., for N-fold rotation axis). By a symmetric body we mean one that is invariant under transformation by each element of the group. Transforming the body by

we require that the body be unchanged after this transformation (applying to every element of the group of interest). For the body constitutive parameters (permeability, permittivity, conductivity) represented as we require

More generally, we can include the symmetries in the Maxwell equations in the transformation. For example, duality (interchange of electric and magnetic fields) can be included with the body symmetry to allow the interchange of permeability and permittivity dyadics (appropriately normalized) upon transformation by the group elements (self-dual body). The EM fields are also transformed as in (4.3) except for a minus sign in the case of the magnetic field when the transformation has a reflection (improper rotation). The eigenmodes (2.2) and natural modes (3.3) are also transformed by the while keeping the eigenvalues and natural frequencies unchanged. This leads to the symmetry-induced condition of eigenvalue (and natural frequency) degeneracy since is also an eigenmode for the same eigenvalue. In general, however, the eigenmodes so generated are not linearly independent. The number of independent eigenmodes for the same eigenvalue is the degree of degeneracy. For example, symmetry for gives a two-fold degeneracy for in the expansion in cylindrical coordinates. Small deviations from such symmetry break the degeneracy by giving small differences to the eigenvalues and natural frequencies, thereby leading to perturbation formulae.


45

Some of the recent symmetry results include:

1. placement and orientation of EM sensors on an aircraft to minimize the influence of aircraft scattering on the measurement (reflection symmetry R) ) 2. high-frequency capacitors (dihedral symmetry for e.g., an 3. nondepolarizing axial backscatter (two-dimensional rotation symmetry N-bladed propeller). 4. generalized Babinet principle (for dyadic impedance sheets) and self-complementary structures ( symmetry) 5. vampire signature (zero backscatter cross polarization in h,v radar coordinates) for mine identification (continuous two-dimensional rotation/reflection symmetry ) [10] separation of magnetic-polarizability dyadic into distinct longitudinal and transverse parts, for low-frequency magnetic singularity identification (diffusion dominated natural frequencies) of metallic targets ( symmetry for ) 7. categorization of the scattering dyadic for the various point symmetries, including reciprocity and self-dual case [28].

6.

Other types of symmetry, such as translation, also have important consequences. These include common waveguiding structures and helices, as well as periodic structures (discrete translation). Dilation symmetries (continuous as in conical structures, and discrete as in log-periodic structures and fractal structures) also give special electromagnetic behaviors. 5.

DIFFERENTIAL GEOMETRY FOR TRANSIENT LENS SYNTHESIS

In gravitational theory differential geometry is used as an integral part of general relativity. In that case, one deals with a four-dimensional space/time. One can also use differential geometry in three spatial dimensions. In this case we are looking for coordinate transformations which allow us to take a known solution of the Maxwell equations with desirable properties in a relatively simple medium, and by curving the coordinates have the same solution in a nonuniform and perhaps anisotropic medium. Bending the wave propagation in this manner gives a lens. We think of this as a transient lens because this works equally well for all frequencies (within the limits of the practical realization of such a medium). For the case of a TEM mode (dispersionless) propagating along two or more guiding conductors the conductors are also curved in the coordinate transformation and are thereby positioned as boundaries on or inside the lens medium. The theory with many examples is discussed in [22]. We imagine some as yet unspecified orthogonal curvilinear coordinate system with

The Maxwell equations as in (1.1) are taken in time domain as homogeneous, i.e., without sources. For time domain we then require zero conductivity with frequency-independent and These fields and constitutive parameters are referred to as real (indicating they can be measured), as contrasted to the formal fields and constitutive parameters. These are designated by superscript primes such that

46

C. E. BAUM

The formal parameters define a problem in the coordinates. In tensor language the the

coordinates taken as though these were Cartesian

are the covariant components of

are the contravariant components of (applying to The formal and real fields are related by

and

(applying to

and

), while

).

The formal and real constitutive parameters are related by

which for diagonal constitutive-parameter matrices reduce to

For cases considered to date then we have

where the components are referred to the coordinates. The problem is then to take some known formal fields with formal constitutive parameters, and find what coordinates exist in which we have real fields and constitutive parameters subject to constraints (realizability conditions) on the real constitutive parameters. For example, one might be considering TEM waves propagating in the direction, making and irrelevant. But then one might also like the real medium to be isotropic so that and It has been shown in such a case that constant surfaces are planes or spheres, limiting the class of acceptable coordinates. Within this class various acceptable coordinate systems, and hence transient lenses, have been found. As summarized in [22] there are several classes of solutions of these equations:

1. all six components of and nonzero for inhomogeneous but isotropic and (only two possible coordinate systems) 2. TEM waves propagating in the direction for inhomogeneous but isotropic and (coordinate systems constrained by constant surfaces being planes or spheres, examples including converging, diverging, and bending lenses)

3. two-dimensional lenses for TEM waves (only one component each of and nonzero) based on conformal transformations (resulting in only one of and µ being inhomogeneous, but both isotropic) 4. lenses with but anisotropic and inhomogeneous.


47

Since the book several new examples have been developed. An important class of these involve but inhomogeneous and isotropic, making them relatively practical for construction. Of these, an important type ofmedium is a cylindrically inhomogeneous dielectric (CID) with the permittivity distributed as

in a cylindrical

coordinate system. This admits as solutions:

5. TEM waves propagating in the direction (bending lens) with very general transmission-line cross sections (e.g., circular coax) [2]. 6.

ELECTROMAGNETIC TOPOLOGY FOR ANALYSIS AND CONTROL OF ELECTROMAGNETIC INTERACTION WITH COMPLEX SYSTEMS

A certain kind of topology, graph theory, is commonly used in electrical engineering to describe electrical networks. For circuit analysis such networks are described by nodes and branches, on which are written the Kirchoff equations which say that the sum of the currents leaving a node are zero and the sum of the voltage drops around a loop are zero. Electromagnetic topology (Fig. 6.1) begins by recognizing mat space can be divided into a set of volumes separated by boundary surfaces. For signals to propagate from one volume to another they must pass through one or more surfaces. Some of these surfaces (closed ones) can take on the role of an EM shield in the usual sense. These can be nested inside one another to form a hierarchical topology. There is a dual topology, the interaction sequence diagram, which is a graph (or network) in which the volumes are replaced by nodes (vertices) and the surfaces separating adjacent volumes by branches (edges), this also being indicated in Fig. 6.1. This is the subject of qualitative (or descriptive) EM topology, which can be used to organize the EM design of complex systems. This is contained (along with quantitative aspects) in [5, 20] which also contain numerous references. Quantitative EM topology is based on the BLT1 equation [4] which was originally stated in a form appropriate to multiconductor-transmission-line (MTL) networks as

48

C. E. BAUM

This is written on an MTL network (Fig. 6.2) consisting of junctions characterized by scattering matrices, and tubes consisting of MTLs with conductors (plus reference) connecting appropriate junctions. Each tube contains two vector waves, one propagating in each direction, indexed by for where is twice the number of tubes. The tubes here are taken as uniform (not varying along the tube) and characterized by

From these we have


In turn, the combined voltage waves are defined for each

49

wave by

with positive convention for current in the direction of increasing (For the two waves on a tube (two values of ) the current conventions are opposite.) The distributed combined sources for the uth wave are similarly

giving the source term in (6.1) as

Relating the MTL network to the EM topology, note that by shrinking the tubes to zero length the junctions can represent the volumes, the tubes the connecting surfaces, and the sources lumped equivalent sources at each surface. In this form (BLT2) then

and disappears from (6.1). An alternate way to approach this is to recognize that a tube may be represented by a junction, fitting an MTL network into BLT2 form. A more elaborate form, the NBLT (nonuniform BLT) equation [8], allows for NMTL (nonuniform MTL) tubes for which the per-unit-length parameter matrices are allowed to vary as a function of In this last case the two waves on a tube do not neatly separate, but scatter into each other as they propagate along the tube coordinate. Again this case can also be cast into BLT2 form by defining such a tube as a junction with scattering matrices and equivalent source vectors. A yet further form (BLT3) utilizes the delay property of the tubes to expand the interactionsupermatrix inverse in a geometric series which can be used for early-times in time domain [13]. These BLT networks can become rather elaborate for large electronic systems such as aircraft. Computer codes such as CRIPTE [24] have successfully modeled such systems, and further improvements are anticipated. The computation time has been recently significantly reduced by graph-theoretic techniques in which appropriate portions of the network are reduced to equivalent junctions before inverting the interaction

50

C. E. BAUM

supermatrix [11]. The successful implementation of such calculations has been from DC to several hundred MHz, pushing to a GHz. Further improvements may push this higher by modeling the cavities and cavities with transmission lines in appropriate ways that fit into the topologically-decomposed scattering-matrix formalism. Another potential improvement involves inclusion of the good-shielding approximation to break the full-system problem into smaller problems at shield/subshield boundaries, with simple matrix multiplication to reconnect the subproblems. One can also use SEM concepts to more simply evaluate the late-time behavior of the system in terms of natural frequencies and modes. Closely tied to EM topology (although one could consider this a separate subject) is the subject of the response of NMTLs [25]. For this purpose it is convenient to formulate a single NMTL via a supermatrix differential equation of the form

where currents are referenced to the +z direction. Solving this equation gives relations between voltages and currents at both ends of the tube together with equivalent sources there. This is a chain-matrix-like formulation of the problem which is later (after solution) converted int a scattering-matrix form for insertion into the BLT equation. This equation is related to the supermatrizant differential equation

Provided that we have found the supermatrizant we have the solution of (6.8) as

By choosing as one end of the tube and as the other end the terminal parameters are related and the scattering supermatrix and equivalent sources are obtained. The supermatrizant is expressed as a product integral [12]

This can be thought of as a repeated dot product (increasing

terms multiplying on left of form

by comparison to the usual sum integral. In special cases, this reduces to a sum integral as


provided

evaluated at

and

commute with each other for every pair

51

and

in the

interval One example concerns a constant matrix, which gives the result for in (6.1). Another example concerns circulant matrices for the per-unit-length parameter matrices [16]. Special results also apply to the case of uniform modal speeds as occur for nonuniform wires (size, spacing) in a uniform medium [12, 25]. The product integral is suggestive of a numerical way for evaluating the supermarizant, i.e., by dividing the interval into some number of subintervals, approximating the result for each subinterval by assuming a constant matrix there, and multiplying the results for all the subintervals. This is a staircase approximation. One can do better in some cases by allowing a smooth variation (e.g., linear or exponential) of eigenvalues with constant eigenvectors over each subinterval [12, 25]. This allows one to preserve continuity of the line parameters from one subinterval to the nect, thereby reducing reflections at such boundaries. The product integral has various special formulas analogous to those for sum integrals (e.g., integration by parts). What is called the sum rule allows one to separate

into the sum of

two terms. If one term has a readily evaluated product integral (closed form), the problem is changed to a new product integral. If the second term is suitably small, the new product integral can be readily approximated by the first two terms in a series representation of the matrizant (the first term being the identity) [25]. This gives a perturbation formula for approximating the solution of an almost uniform MTL. 7.

CONCLUDING REMARKS

So we now have a collection of modern mathematical techniques to apply to the Maxwell equations (analysis and synthesis). Much has been learned using these and I would expect that much more can be learned. This should lead to new classes of electromagnetic devices. We should continue searching for other mathematical structures which may be of use to electromagnetic theory. Noting the importance of the mathematics used in quantum mechanics, one might consider more esoteric physics such as quantum electrodynamics, string theory, etc. Not included in our discussion here, and still in its infancy is statistical electromagnetics, for which one may expect more important future results. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U. S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES 1. 2.

3. 4.

5. 6.

7.

C. E. Baum, Some Characteristics of Electric and Magnetic Dipole Antennas for radiating Transient Pulses, Sensor and Simulation Note 125, January 1971. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 405, December 1996; Proc. URSI Int’l Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. C. E. Baum, Emerging Technology for Transient and Broad-Band Analysis and Synthesis of Antennas and Scatterers, Interaction Note 300, November 1976; Proc. IEEE, 1976, pp. 1598-1616. C. E. Baum, T. K. Liu, and F. M. Tesche, On the Analysis of General Multiconductor TransmissionLine Networks, Interaction Note 350, November 1978; also contained in C. E. Baum, Electromagnetic Topology for Analysis and Design of Complex Electromagnetic Systems, pp. 467-547, in J. E. Thompson and L. E. Luessen (eds.), Fast Electrical and Optical Measurements, Martinus Nijhoff, Dordrecht, 1986. C. E. Baum, The Theory of Electromagnetic Interference Control, Interaction Note 478, December 1989; pp. 87-101, in J. Bach Anderson (ed.), Modern Radio Science 1990, Oxford U. Press. C. E. baum, Representation of Surface Current Density and Far Scattering in EEM and SEM with Entire Functions, Interaction Note 486, February 1992; Ch. 13, pp. 273-316, in P. P. Delsanto and A. W. Saenz (eds.), New perspectives on Problems in Classical and Quantum Physics, Part II, Acoustic Propagation and Scattering, Electromagnetic Scattering, Gordon and Breach, 1998. C. E. Baum, Properties of Eigenterms of the Impedance Integral Equation, Interaction Note 487, April 1992; Ch. 2, pp. 39-91, in A. Guran, R. Mittra, and P. J. Moser (eds.), Electromagnetic Wave Interactions, World Scientific, 1996.

52 8. 9.

10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

C. E. BAUM C. E. Baum, Generalization of the BLT Equation, Interaction Note 511, April 1995; Proc. 13th Int’l Zurich Symposium and Technical Exhibition on Electromagnetic Compatibility, Feb. 1999, pp. 131-136. G. W. Hanson and C. E. Baum, A Volumetric Eigenmode Expansion Method for Dielectric Bodies, Interaction Note 517, August 1996. C. E. Baum, Symmetry in Electromagnetic Scattering as a Target Discriminant, Interaction Note 523, October 1996; pp. 295-307, in H. Mott and W. Boerner (eds.), Wideband Interferometric Sensing and Imaging Polarimetry, Proc. SPIE, Vol. 3120, San Diego, Calif., July 1997. J.-P. Parmantier, X. Ferrieres, S. Bertuol, and C. E. Baum, Various Ways to Think of the Resolution of the BLT Equation with an LU Technique, Interaction Note 535, January 1998; Optimization of the BLT Equation Based on a Sparse Gaussian Elimination, Proc. 13th Int’l Zurich Symposium and Technical Exhibition on Electromagnetic Compatibility, February 1999, pp. 137-142. C. E. Baum, Symmetric Renormalization of the Nonuniform Multiconductor-Transmission-Line Equations with a Single Modal Speed for Analytically Solvable Sections, Interaction Note 537, January 1998. C. E. Baum, Extension of the BLT Equation into Time Domain, Interaction Note 553, March 1999. C. E. Baum, The Singularity Expansion Method: Background and Developments, IEEE Antennas and Propagation Mag., Vol. 28, No. 4, August 1986, pp. 15-23. C. E. Baum, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, The Singularity Expansion Method and Its Application to Target Identification, Proc. IEEE, 1991, pp. 1481-1492. J. Nitsch and C. E. Baum, Analytical Treatment of Circulant Nonuniform Multiconductor Transmission Lines, IEEE Trans. EMC, 1992, pp. 28-38. J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed., Dover, 1954 (from 3rd ed. 1891). E. A. Guillemin, Synthesis of Passive Networks, Wiley, 1957. C. E. Baum, The Singularity Expansion Method, Ch. 3, pp. 130-179, in L. B. Felsen (ed.), Transient Electromagnetic Fields, Springer-Verlag, 1976. C. E. Baum, The Role of Scattering Theory in Electromagnetic Interference problems, Ch. 13, pp. 471502, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic Press, 1978. C. E. Baum, Toward an Engineering Theory of Electromagnetic Scattering: The Singularity and Eigenmode Expansion Methods, Ch. 15, pp. 571-651, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic Press, 1978. C. E. Baum and A. P. Stone, Transient lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor & Francis, 1991. C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995. J.-P. Parmantier and P. Degauque, Topology Based Modeling of Very Large Systems, pp. 151-177, in J. Hamelin (ed.), Modern Radio Science 1996, Oxford U. press, 1996. C. E. Baum, J. B. Nitsch, and R. J. Sturm, Analytical Solution for Uniform and Nonuniform Multiconductor Transmission Lines with Sources, Ch. 18, pp. 433-464, in W. R. Stone (ed.), Review of Radio Science 1993-1996, Oxford U. Press, 1996. C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, E. G. Farr, and D. V. Giri, Review of Impulse-Radiating Antennas, Ch. 16, pp. 403-439, in W. R. Stone (ed.), Review of Radio Science 1996-1999, Oxford U. press, 1999. C. E. Baum, Target Symmetry and the Scattering Dyadic, Ch. 4, pp. 204-236, in D. H. Werner and R. Mittra (eds.), Frontiers in Electromagnetics, IEEE Press, 1999.

SURFACE DISCHARGE CELLULAR AUTOMATON MODEL

Masashi Hayakawa1, Nikolay V. Korovkin1,4, Dmitry I. Iudin1,2, Ekaterina E. Selina4, Viktor Yu. Trakhtengerts3 1

The University of Electro-Communications 1-5-1 Chofugaoka, Chofu Tokyo 182-8585, Japan 2 Radiophysical Research Institute Bolshaya Pecherskaya st. 25/14, Nizhny Novgorod 603600, Russia 3 Institute of Applied Physics, Russian Academy of Science Ulyanov st. 46, Nizhny Novgorod 603600, Russia 4 Saint Petersburg State Technical University Polytechnicheskaya st., 29, St. Petersburg 125251, Russia

INTRODUCTION The electric charge induced on a dielectric surface of an aircraft or rocket passing through a layer of cloud results in the development of the surface discharge. The distribution of the local non-uniform charge is shown in figure 1. This charge generates short wave electromagnetic radiation (down to X-ray). This radiation penetrates inside the body of an aircraft through the apertures and slits; it causes undesirable action on technical systems and biological objects. When atmospheric pressure decreases then conditions for the discharge initiation are improved, its intensity increases and the apertures are overlapped at lower voltages. The disturbances from the surface discharge in high voltage equipment are also well known. Moreover, the surface discharge impairs the characteristics of insulation and it deteriorates faster. The surface discharge on a dielectric surface is widely used in the various technical applications, for example, in electrophysical devices, pulse light sources, pumping up systems of gas lasers. At the same time the surface discharge is the basic danger for the normal functioning of the numerous dielectric structures.


53

54

HAYAKAWA ET AL.

Though the surface discharge is a frequently registered phenomenon, the methods of its mathematical description and modelling are poorly developed. The reason is that the traditional models in the form of the algebraic or differential equations do not sufficiently represent the charge distribution process. In fact, with the help of the existing models we can make only a crude estimation of mean values of the discharge current, initial voltage and radiation intensity for very simple systems.

So the existing mathematical models of the surface discharge are poorly available for the solution of the practical EMC problems. The purpose of this work is to construct a surface discharge mathematical model that reproduces in details essential features of this phenomenon. Created model should take into account the geometry of the surface, shape of the apertures and slits, characteristics of the dielectric.

PROCEDURE OF CELLULAR AUTOMATON ACTION The sliding discharge in gas along the surface of the solid dielectric appears when the normal component of the electric field intensity exceeds considerably its tangent component. Configuration of the discharge gap where this condition is fulfilled is shown in figure 2. Development of the discharge begins from the appearance of the corona discharge near the upper electrode 1. The magnitude of the initial voltage of the corona discharge initiation depends on the electrical characteristics of dielectric and cleanness of its surface. Then, corona discharge turns into the streamer form when voltage increases. The filaments of the current appear on the surface of the dielectric (see area 1 in figure 2). The surface of the dielectric is charged near the upper electrode through these filaments. Current in the filaments is closed through the dielectric as the displacement current. Transmission of the streamers into the external non-charged area (see area 2 in figure 2) goes with the generation of the branched structures. The results of the experiment show that the distribution of the charge is a treelike structure. This specific structure has the properties of scaling and self-similarity. In the other words the structure of the charge on the surface has the properties of a fractal. Simultaneously with the distribution of charge the process of charge dispersal takes place. The process of discharge of the pre-charged element of the surface occurs significantly slower. In the steady-state condition the processes of charge and discharge are in the


55

dynamic equilibrium. After charging the surface of dielectric by the treelike structure it dies. New treelike structure is growing during the distribution of charge on the uncharged area. The surface discharge modeling is done on a basis of the two-dimensional cellular automaton network where separate automaton cell represents some element of the dielectric surface with square S=d·d (see figure 3). Each element of the network (cell) has four nearest neighbors and may exist in three different states: "A", "B" or "C". Each state can be described by its action procedure. A. Waiting state. Capacitor of the dielectric surface of the cell is not charged. This is the surface of the discharge development. B. Relaxation or tolerance state. Capacitor is pre-charged up to the voltage In the state "B" it is discharged through the linear resistor with the admittance C. Current state. The transposition (distribution) of the charge along the surface takes place. Automaton cell simulates the leader channel of the surface discharge current. Procedure of the automaton action in the state “A” is the simplest among these states. An automaton exists in the state of readiness (waiting) to transition into the state “B” and informs about it by the appropriate announcement. State “A” is the initial state for all automatons except the automaton indicated in figure 3 by the circle. An automaton may turn from the state “A” only into the state “B”.

In the state “B” capacitor of the examined cell S discharges through the admittance of the element that corresponds to the leakage admittance of the dielectric cell. The process of discharge can be described by one of the following difference equations - with the order O(h) of approximation of the initial differential equation

with the order

of approximation

Here - the step of reproduction of the dynamics of an automaton action in the time domain, - initial voltage of the cell, - voltage and current in the surface element S at the point in time, n=0,1, 2, .. .

HAYAKAWA ET AL.

56

Cellular automaton passes from the state "A" to "B" if at least one of its nearest neighbors is in the state "C" and has a superfluous charge The charge satisfies the following equation: Cellular automaton passes from the state "B" to "A" at time when its voltage

becomes less than an arbitrary voltage

If the voltage

in the state "B" satisfies the inequality

is less than

and charge

, then cellular automaton generates the request on receiving a charge

portion This portion of charge can be transmitted only in the case when the nearest automaton in the state "C" has a sufficient superfluous charge. If the request of an automaton in the state “B” at the fixed time step is not satisfied then the request is canceled and is not restored later on. If several automatons in the state “B” are situated near the automatons in the state “C” with the surplus charge then it is distributed randomly between them. If there is enough charge then all neighboring automatons in the state “B” will be charged. When there are both types of automatons in states “B” and “A” among the neighbors of the cellular automaton in the state ”C” then the requests of cellular automaton in the state “B” are fulfilled first and after it the requests of cellular automaton in the state “A” are realized. If there are several cellular automaton of the type “A” among the neighbors of cellular automaton ”C” then the charge distribution passes randomly. However, even though there is an excess of the charge, not surely all requests of cellular automaton in the state “A” will be fulfilled. The number of cellular automaton that will be transferred from the state “A” into the state “B” is determined randomly. The procedure of action of the automaton in the state “C” simulates the dynamics of the surface discharge. Let us consider the continuity of one time step performance. At the beginning of the process one of the automatons (let’s denote it zero, it’s is marked in figure 3) is transferred into the state ”C” and it keeps this state unconverted during all the time of the process simulating. At each time step the null automaton receives the charge equal to m· from the external source. Further, this charge is distributed among the neighboring automatons in the state “A” and they pass into the state “B” in this way. Then, within the framework of the current step, it is determined whether the total charge received from the source is distributed. If it is so, then the time step is finished. Otherwise, the remained charge is distributed repeatedly. Herewith the automatons in the state “B” are transferred in the state “C”. The posterior charge distribution is carried through the newly created automatons in the state “C”.

CHARGE DISTRIBUTION MODELING The process of the charge distribution is made so long as the quantity of the undistributed charge stops changing. In this case the charge flows through the automatons in the state “C” simulating the streamer of the surface discharge. At the end of the process of the charge distribution at each time step the quantity of charge flowed through each automaton in the state ”C” is calculated. If automaton has transferred the more charge than some fixed value then it remains in the state “C”, otherwise it passes into the state “B”. Since the flow of charge is moved from one element S so the automatons of the “C” type generate one associated cluster. The extreme automatons of the “C” type in the cluster that have received charge but haven’t distributed it or distributed the charge less than would transfer into the state “B”. Accordingly the streamers of the surface discharge will change their shape at each


57

time step. These changes will be frequent in the periphery and more occasional in the elements that are near the source. The cellular automaton model parameters: can be simply determined from physical experiments. The model allows us to reproduce not only integrated characteristics of the discharge but also its local properties that are extremely important in applications. For example, the offered model enables us to investigate the influence of the surface irregularities (edges, apertures). The charge transferred into the system during one time step not surely will be distributed completely. In this case the null automaton accumulates it. Let the charge be distributed by the end of the time step then the discharge current and voltage at the n step can be calculated from the equations: , where M is the total number of steps in the time domain,

– the

charge which was not distributed at the step number k.

We obtain cellular automaton in "C" state if it transfers some electric charge. Having transferred a charge greater than fixed, cellular automaton remains in "C" state, otherwise it passes in "B" state. During this process the central cellular automaton is always in state "C".

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HAYAKAWA ET AL.

Temporal evolution of the tolerance cell number is shown in figure 4. Time evolution of the total current and its spectral density are show in figures 5 and 6 correspondingly. The last is power law with a slope of about -1. The cellular automaton cells of the "C" type form a connected cluster. This cluster represents a transport network that distributes electrical charge all over the grid. It is obvious that total discharge current On the other hand where correlates with the loses in the discharge channel and is the network fractal dimension. So, one may obtain that configuration is shown in figure 7.

The surface discharge typical

The leader channel is depicted in figure 7 by red color, dark-blue field correlates with the automatons in the state "A", more light colors up to yellow correspond to the automatons in the state "B". The lighter the field is, the larger the charge of the automaton is. The leader with more branches is shown in red color in figure 8, charged cells in the state "B" are pictured in blue and the cells in the state "A" – in cyan.

CONCLUSIONS 1. Surface discharge can be examined as a distribution of the charge on the surface of dielectric. The mathematical model based on finite automatons describes the behavior of this system. The dynamics of this system is close to the self-organizing chaos. 2. Properties of the surface discharge mathematical model depend greatly from the parameters of the model. These parameters can be determined by simple experiments. 3. Suggested model simulates the real process in the real technical system. This allows changing the technical system in such a way that minimizes the influence of the electromagnetic radiation of the surface discharge on the operation mode of the system.

GREEN'S FUNCTIONS FOR SHEET CURRENTS PLACED OVER CYLINDRICAL METAL SURFACE

Alexander Svezhentsev* and Guy Vandenbosch Katholieke Universiteit Leuven (ESAT-TELEMIC), Kardinaal Mercierlaan 94, B-3001 Heverlee, Belgium

INTRODUCTION In some antenna applications it is preferable to use conformal antennas in which patches are placed on non-planar surfaces, for example, cylindrical ones. In this case a necessary first step is to calculate the Green's Functions (GFs) for sheet currents in such structures. Some aspects of this problem were discussed both in approximate1,2,3 and rigorous4,5 formulations. Mainly, the resonant frequencies and radiation pattern in the far zone were investigated. In this paper the spatial GFs for sheet currents in a cylindrical structure are rigorously calculated on the cylindrical surface where those sheet currents are placed. A special technique is used to obtain the singular part of the spatial GFs in an analytical form. This approach ensures the calculation of the GFs within a reasonable time.

PROBLEM FORMULATION The structure under investigation consists of a metal cylindrical circular conductor (which is infinite in the z-direction) with radius and the current interface The cross section is shown in Fig.l. The current interface at between free space (layer i=0) and layer contains the distribution of the electric and the magnetic sheet currents where 's' stands or The problem is to find the spatial Green's functions, which connect fields and currents at the interface. To solve this problem we will pass two stages. *He has a permanent research position at the Institute of Radio Physics and Electronics of National Academy of Science of the Ukraine and is currently working with the KU Leuven as a postdoctoral researcher.


59

60

A. SVEZHENTSEV AND G. VANDENBOSCH

The first one is to find a Green's function in the spectral domain and the second one is to perform the Inverse Fourier Transform (IFT) procedure. The IFT procedure for fields in cylindrical co-ordinates for each layer i (i=0,1) is:

where is the pure imaginary unit. In each layer the z-components of the field are derived as a solution of the 2D Spectral Helmholtz equations.

The solution looks like:

where

GREEN'S FUNCTIONS FOR SHEET CURRENTS

At the current interface

61

the field components obey the boundary conditions:

where is the Fourier Transform (FT) of sheet current In (3) the transverse components of the electric and magnetic field can be expressed in terms of the z-components as:

Note that we can split fields in each layer in two independent systems (TE and TM) with only five components of the field different from zero. Due to the currents the TE and TM waves occur simultaneously. SPATIAL GREEN'S FUNCTIONS Mixed-Potential Integral Expressions for the Electric Field The electric field integral equation (EFIE) in the mixed-potential formulation is one of the forms to which a moment method procedure can be applied to investigate antenna problems. This approach was successfully applied to planar microstrip antennas6,7. Returning to the results of the previous paragraph and applying the IFT procedure to the relations between electric field and current components one can express the spatial electric field components in the form:

where

62


IFT for the Asymptotes. Spatial Green's Functions. Using handbook9 to solve some IFT integrals we can get analytical formulas for the asymptotic part of the spatial Green's functions. As a result any spatial GF can be represented in the form:

We should use the starting point n=1 for The asymptotic part of the spatial Green's functions

looks like:

where

The asymptotes

can be represented in a form:

where is the modified Bessel function. Signs (+/-) correspond to the respectively. And finally

looks like


63

the indexes 'J' and stand for current and charge, respectively. Also note that the expressions for the spectral GFs can be easily determined after satisfying the boundary conditions (3). It is seen from (4) that in the common case (dielectric layers) eight spectral GFs need to be Inverse Fourier Transformed instead of four GFs in the planar case6. In our case (without dielectric) it will be shown later that only four GFs need to be calculated. Note also that in the planar case each IFT integral is reduced to a one-dimensional integral of the Sommerfeld type. In the cylindrical case we will have to calculate not only the integral over the h parameter but also we will have to calculate the sum over the angular dependencies. Therefore the spatial GFs in the cylindrical case are always functions of two variables: and This will require more computer recourses than in the planar case. Now the problem is how to calculate the integral over h in the optimal way. Analytical investigation of the spectral GFs showed that there are regions, namely, for large values of h, where the integrand has a so-called 'bad' behaviour. A special technique will be applied. It is described in the next section. Dominant Contribution in the Neighbourhood of the Source. Asymptotes of the Spectral Green's Function. It was shown7 that the asymptotic behaviour of spectral GFs for large h determines the contribution close to the source. In order to speed up the IFT drastically it is very advantageous to subtract the asymptote (in the spectral domain) and to add (in the spatial domain) its spatial equivalent. The spatial equivalent is derived in analytic form. It is very important to stress that the situation with asymptotes (large h) in the cylindrical case is much more complicated than in the planar case because we should take into account also the spectral Green's function dependence over n. Using the approximate expressions for cylindrical functions 8 we get next approximate formulas for the spectral Green's functions which can be divided in three groups. Inside each group all asymptotes are the same. The first group of spectral Green's functions shows the asymptotes:

The asymptotes:

are valid for Signs (-/+) correspond to the The third type of asymptote is:

respectively.

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NUMERICAL RESULTS The dependence of the spatial Green's function on is shown in Fig.2 for fixed value The fact that the function has to be periodical is met. The period is

as it should be. In the plotted case the observation point lies close to the source point and also close enough to the cylinder surface. That is why we can clearly see the region of shadow, which lies opposite to the observation point vicinity with respect to the cylinder. In this region the field is very small. The calculation time mostly depends on the ratio The less this ratio the higher the calculation time. For parameters which correspond to Fig.2, the calculation time for one point is 8 second when For the calculation time of one point is 18 second for the chosen parameters. The calculations were performed on a HP workstation J-5000 (440 MHz); the processor type is PA 8500. The accuracy to calculate the integrals connected with the IFT was Note that we cannot directly sum up the expression (5) in real time without improving its convergence because the common term of this series behaves as when n goes to infinity. This function yields a convergence which is extremely slow. CONCLUSIONS An effective approach was realised for the calculation of the spatial GFs for sheet electric and magnetic currents, which are located at cylindrical interfaces over a cylindrical


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structure. This approach is based on finding the spectral GFs and calculating the Inverse Fourier Transform equivalent. It is very important to note that the singular part of the Spatial GFs is given in an analytical form. Important is that the spatial GFs depend on two variables in the cylindrical case. This means that the cylindrical case needs considerably more computer resources for this aspect than compared to planar structures. This approach can be generalised to multilayered dielectric structures. REFERENCES 1. K.-M. Luk K.-M., K.-F. Lee and J.S. Dahele. Analysis of the cylindrical - rectangular patch antenna, IEEE Trans. on Antennas and Propagation, vol. 37, N 2, February (1989). 2. M. Hamadallah. Radiation pattern of patch mounted diagonally on cylinder, Electronics letters, vol. 24, N 21, October (1988). 3. C.M. Krowne. Cylindrical-rectangular microstrip antenna, IEEE Trans. on Antennas and Propagation, vol. 31, N 1, January (1983). 4. J. Ashkenazy, S. Shtrikman and D. Treves. Electric surface current model for the analysis of microstrip antennas on cylindrical bodies, IEEE Trans. on Antennas and Propagation, vol. 33, N 3, March (1983). 5. G. Gottwald and W. Wiesbeck. Radiation efficiency of conformal microstrip antennas on cylindrical surfaces, Proceedings of the 1995 IEEE AP-Symposium, pp. 1780-1783 (1995). 6. G.A.E. Vandenbosch and A.R. Van de Capelle. Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium - application to the case of a probe current source, IEEE Trans. on Antennas and Propagation, vol. 40, N7, July (1992). 7. F.J. Demuynck, G.A.E. Vandenbosch and A.R. Van de Capelle. The expansion wave concept - part I: Efficient calculation of Spatial Green's functions in a stratified dielectric medium, IEEE Trans. on Antennas and Propagation, vol. 46, N 3, March (1998). 8. Handbook of Mathematical Functions. Edited by M. Abramowitz and I.A. Stegun. Dover publication, inc., New York, 1965. 1046 p. 9. I. Gradshteyn and I. Ryzhik. Table of Integrals, Series and Products, Academic Press, New York, 1980, (1981 printing).


UWB ANALYSIS OF EM FIELDS IN COMPLEX LAMINATES: A MULTIRESOLUTION HOMOGENIZATION APPROACH

Vitaliy Lomakin, Ben Zion Steinberg and Ehud Heyman Faculty of Engineering Tel-Aviv Univetsity Tel-Aviv, 69978, Israel

INTRODUCTION Finely layered multi-scale heterogeneity laminates can be found in many man-made structures and natural environments. The study of propagation and/or scattering of EM waves in such laminates is of fundamental importance in diverse areas of applications, ranging from material synthesis, to circuits design, to fault interrogation, and to geophysical exploration, to name a few. As the scales of the laminate heterogeneity can range from the micro- (a fraction of a wavelength) to the macro-scale (wavelength and above), the analysis of the entire set of associated wave phenomena and their detailed structure may constitute a challenge of overwhelming complexity. However, in many applications the detailed (micro-scale) structure of the EM field is of no practical importance. This can be due to the properties of the field itself (e.g. weak microscale field components), or the measurement setup (e.g. far-field or detector size). Thus, a solution strategy that constitutes only the macro-scale behaviour of the field—the field observables—is of great practical importance. The role of homogenization theory is to derive a simpler-to-solve effective formulation for the macro-scale field, which smoothes out the micro-scale heterogeneities while retaining their effect on the macro scale observables. Recently, a new homogenization theory that utilizes multiresolution analysis (MRA) has been developed and applied to time-harmonic propagation in complex laminates [1, 2]. Unlike traditional homogenization schemes that hold only for periodic micro-structures and require a large difference between the micro- and macro-scale (large “scales gap”), the new theory can handle non-periodic structures with scales continuum and enables one to choose the scale on which the fields are homogenized. In the present work we extend the multiresolution homogenization theory by allowing different homogenization scales for the medium and the field observables. We then present a full EM formulation for sources in 3D laminate configurations and explore the properties of the homogenization procedure under UWB conditions.


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FORMULATION OF THE PROBLEM We consider source-excited electromagnetic fields in complex laminates, characterized by multi-scale heterogeneities. The propagation domain (see Fig. (1)) may have any bounded or unbounded cross-section perpendicular to the stratification axis The laminate anisotropic constitutive relations are diagonal tensors whose components and and denote longitudinal and transverse components, respectively) are complex functions of z, i.e., multi-scale functions comprising of both macro and micro scales. The laminate is bounded in and may have penetrable boundaries to the surrounding homogeneous domain. The field is excited by a pulsed current J(r, t). The configuration in Fig. 1 can be analyzed via modal decomposition and transmission line theory [3, Chapter 2]. The field is expressed as a modal synthesis (either discrete or continuous) wherein the transverse and longitudinal field components, denoted by the subscript and respectively, are expressed as [3]

Here the index or denotes the E and H modes, respectively, while and are the corresponding vector and scalar mode functions for the spectral parameter . The spectral (modal) amplitudes and satisfy the spectral transmission line equations

These equations are in general difficult to solve due to the complex (multi-scale) structure of the heterogeneity functions and that are related to the constitutive parameters via

The source terms in the spectral transmission line problem are obtained by projecting the current sources J onto the spectral basis functions via

Note that the component of J, excites only E-type mode via transmission-line source, while the transverse component of J can excite both E and H modes via the source term

MULTIRESOLUTION ANALYSIS (MRA) We would like to express all the field observables on a given resolution (scale) that is typically determined by the physics of the problem (see discussion in connection with (8)). Therefore, referring to the formal theory of multi resolution analysis [4], we decompose in the following form

UWB ANALYSIS OF EM FIELDS IN COMPLEX LAMINATES

69

where the resolution space is the linear space of all functions possessing length scales between and infinity, and the wavelet space is the orthogonal complement of in We will be interested in the field components in while smoothing out all small scale field components which are contained in with The scale is a fundamental parameter in our theory, denoted as the homogenization scale (HS). Typically it is chosen such that where is the wave velocity in the medium, but it may also depend on the detector used. The resolution space is spanned by the basis where is the so called scaling function, is the dilation index representing the smallest length scale in and is a running index spanning locations. Similarly, the wavelet space is spanned by the basis with being the wavelet function. Further properties of these functions (regularity, vanishing moments, etc.) can be found in [4]. A function can be decomposed into its macro-scale (smooth) and micro-scale (detailed) components in the form

These components are found by projecting V onto and its orthogonal complement in The corresponding projection operators and are given by where where

EFFECTIVE FORMULATION As denoted above, the electromagnetic field is fully described by the functions that solve the complex-coefficient equation (2). The field observables, however, are described on a scale that is typically determined by the physics of the wave-problem (e.g., frequency, near/far field) and by the measurement setup (e.g., detector size, integration window, dynamic range and polarization). In what follows we choose this scale to be and denote it as the field homogenization scale (F-HS). In many applications we choose

although the theory below is not limited to this choice. Thus, we are only interested in the field observables It is therefore sufficient to consider a simplified (effective) equation whose heterogeneity functions are homogenized (smoother functions),

It is required though that the solutions of (9) will be equal to the true field observables in

It can be shown that this condition is satisfied if the effective (homogenized) heterogeneity functions in (9) consist only of the components of the true heterogeneities in (the index will be discussed after (12) and (13)), i.e.,

Using this and (3) one finds that the effective constitutive parameters are given by

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Following the definition of as the field homogenization scale (F-HS; see (8)) we denote as the medium homogenization scale (M-HS). For a given F-HS (i.e., ) the M-HS (i.e., ) is chosen so that the solution of the effective problem satisfies the requirement (10) with a specified error. The condition for choosing is determined by the error bound in (13). The M-HS is chosen such that i.e., what is left out of the medium after taking away its effective part is finer than the F-HS. Under this condition, the M-HS should be chosen as large as possible in order to obtain an effective medium that is as simple as possible (i.e., expressed by the smallest number of basis functions). To demonstrate these considerations we refer to Fig. 2 describing two typical media. Medium 1 possesses a continuum of scales from the macro to the micro, while Medium 2 has a gap between the macro and micro scales around the F-HS. In Medium 1 as determined by the bound in (13). For Medium 2, the M-HS (i.e. ) can be taken to be the smallest scale above the gap (i.e. M-HS > F-HS). This completely determines the effective problem. The numerical examples below and in [7] demonstrate the numerical efficacy and accuracy of this method.

Outline of the Derivation of the Effective Formulation Step 1: In order to derive the effective formulation we apply the projection operators and to the wave operator in (2). To this end we recast (2) in an integral equation form which is more convenient to handle than the otherwise unbounded differential wave operator. Furthermore, the integral equation formulation is more convenient in handling the transformation of the boundary conditions from those of the complete formulation to those of the effective one [1]. We decompose the heterogeneity functions into background and foreground components defined, respectively, as and (see (11)). One arrives at a Lippmann-Schwinger type integral equation wherein the foreground (detailed) components act as induced sources, in conjunction with the background dyadic Green’s function. The procedure is applied in two steps wherein each step treats only one of the heterogeneity functions (i.e., first and then or vice versa; for the principal idea of this procedure see [2] and references therein). Step 2: We apply and to the integral equations via a Galerkin type procedure and obtain an algebraic matrix equations for the vector of unknown coefficient and The matrix entries are obtained by projecting the Green’s function kernel onto and Via this construction, the matrix is partitioned into submatrices representing field and medium interactions across length scales. Step 3: Since our interest here is the smooth component of the field, we apply the Schur’s complement procedure and substitute the equation for into the equation for obtaining an effective equation for that contains contributions of the micro-scale heterogeneities. Step 4: In order to assess the contributions of the micro-scale heterogeneities we derive bound to the submatrices constructed in Step 2. This is done using the bounds of the inner products of the type in (7) and their dependence on the properties of the expanded kernel and of and One finds an error estimate for the effective formulation, i.e., an estimate in for the difference between the true field observables and those predicted via the effective formulation (9) (see (10)). This result can be expressed as

where (13b) is obtained from (13a) if the the F-HS is chosen according to (8).


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In these equations is the largest scale of the medium left out in the medium homogenization procedure. Referring to the discussion on the M-HS after (12) we have in general (see Medium 1 in Fig. 2) but, if the medium has a scales gap as in Medium 2 in Fig. 2, then is the index of the largest scale of the medium below the gap. Thus the term indicates that the error decays faster than linear with respect to the ratio between the F-HS and the M-HS. Choosing a smaller M-HS (larger ) reduces the error bound at the expanse of having to work with more complicated effective medium.

MRA HOMOGENIZATION UNDER UWB CONDITIONS The effective field solutions constructed via the MRA homogenization procedure discussed in the previous section can be used to construct full 3D solutions for various source configurations. The construction of such solutions involves summation over the plane-wave spectral variable Several alternative spectral formulations for this 3D field constructions have been developed and will be described elsewhere. In this paper we concentrate on the homogenization procedure under ultra wide band (UWB) conditions for a given spectral variable i.e., for a fixed plane wave direction. As discussed above in connection with (8) the F-HS index is typically chosen in connection to the frequency. These leads to two homogenization strategies: (i) A dispersive homogenization approach: Here the F-HS index is chosen for each frequency band according to condition (8). The M-HS index is chosen so that the error bound in (13) is sufficiently small for specific frequency range. The resulting effective medium is frequency dependent. This approach provides the simplest most economical effective medium for each frequency within the source operation band. As a result the field equations have to be solved on a frequency-by-frequency basis and then transformed into the time domain (if time-domain solutions are required). (ii) A non-dispersive homogenization approach: Here the F-HS index and the M-HS index are chosen to comply with condition (8) and (13) at the highest excitation frequency and then used for the entire operation band. Clearly this approach does not provide the most economical description of the medium at each frequency but on the other hand it allows for direct time domain solutions as the effective medium is frequency independent. These two approaches are demonstrated for a complex slab with and

shown in Fig. 3. It comprises a continuum of scales from 1 to The excitation signal is

spanning frequencies from to Assuming that the field observables are determined by (8), the relevant scale range is Recalling the considerations for choosing the M-HS, we show in Figs. 4a and 4b the effective medium for and respectively. Note that the effective medium is anisotropic even though the true medium is isotropic.

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Finally the transient transmitted fields corresponding to the frequency results in Pig. 5(a,b,c) are shown in Fig. 6(a,b,c), respectively. The insets zoom-in on the weak response near ct = 4 corresponding to the first round trip contribution, showing an error in the effective result (dashed line) for case (b), but accurate results for cases (a) and (c).


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EFFECTIVE RESONANCES Another significant observation concerning the behaviour of the solution under UWB excitation has to do with the complex resonances of the laminate. In view of the Singularity Expansion Method (SEM) [5] these resonances may serve as fundamental waveforms in describing the transient response as a series of decaying oscillations and thus may serve for medium classification. For the layered medium case these resonances are obtained as roots of the Wronskian the solutions and of the transmission line equation (2), satisfying, respectively, the boundary (or radiation) conditions on the left and right hand sides of the line. This condition can be stated as

where it can readily be shown that is Equation (16) may be regarded as an equation of the form It roots in the for a fixed define the guided modes supported by the slab laminate. The SBM resonances, however, are the roots in the complex for a given ( i . e . , for a given direction of incidence). Using the effective formulation one can prove the Wronskian equivalence [2]:

where the error is bounded by an expression similar to (13). It thus follows that the physical SEM roots are well described by those obtained via the effective formulation, i.e.,

at least for the frequency range at which the error bound in (13) is small. To demonstrate the equivalence in the complex plane we have used a random number generator to synthesize a laminate with µ = 1 and with a random multi-layer structure for with inner scale of approximately 1/50 (Fig. 7a). Note that due to the term in for the E mode (see (3)), it contains a scales continuum for that case. The laminate total width is 1. Using M-HS=1, we obtain constant effective properties using M-HS=0.1, we obtain the effective medium properties shown in Fig. 7b

Next, the effective poles associated with the two effective media (with M-HS=1 and 0.1) were computed for E-mode and spectral parameter (corresponding to a plane wave incidence of 48.5°). The results are compared to the corresponding true poles of the

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original medium. Fig. 8 shows the relative error for the two M-HS’s. As expected from the bounds, the error increases with frequency and is better for the smaller MHS. The work in [7] explores the dependence of the frequency-plane spectral equivalence on the heterogeneity length scales and its effect on the transient field response.

CONCLUSION In this paper we have presented an MRA homogenization approach for electromagnetic radiation and propagation in the presence of complex laminates. The formulation addresses a general 3D source configuration via plane-wave spectral expansion, yet here we have limited our study to a single plane wave (single spectral parameter ), emphasizing the UWB aspects of the theory. We have demonstrated how the medium homogenization scale (M-HS) should be chosen as to accommodate the wide frequency spectrum of the excitation. In particular we examined two alternative schemes for choosing the M-HS, a dispersive and a non dispersive scheme, and demonstrated how they are used in order to calculate the effective transient response. Finally we have used the effective Wronskian equivalence to calculate the SEM resonances of a randomly laminated complex slab. Acknowledgements The research is supported in part by a grant from the Israeli Science Foundation. References [1] B. Z. Steinberg, J. J. McCoy and M. Mirotznik, “A Multiresolution Approach to Homogenization and Effective Modal Analysis of Complex Boundary Value Problems,” SIAM J. Appl. Math, 60(3) pp. 939-966, March 2000. [2] B. Z. Steinberg, “Homogenization and Effective Properties Formulations for Propagation in Finely Structured Laminates - A Multi-Resolution Approach,” to appear in Wave Motion, special issue on Electrodynamics of Complex Environments.

[3] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Prentice Hall, 1973. [4] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM Publ., Philadelphia, 1992. [5] C.E. Baum, “The Singularity Expansion Method,” in Transient Electromagnetic Fields, L.B. Felsen, Ed. New York: Springer Berlag, 1976. [6] B. Z. Steinberg and E. Heyman, “Effective vertical modes and horizontal rays for wave propagation in complex inhomogeneous ducts, ” Proc. of URSI Trianum Int. Symp. on Electromagnetic Theory, Thessloniki, Greece, 1998.

[7] V. Lomakin and B. Z. Steinberg, “Effective Resonance Representation of Propagators in Complex Ducts - A Multiresolution Homogenization Approach,” IEEE Trans. Antennas. Propagat., submitted for publication.

TIME DOMAIN EXACT SOLUTION OF PROBLEM OF UWB PULSE DIFFRACTION ON A CONDUCTING HALF-PLANE

Eugene A. Galstjan Moscow Radiotechnical Institute of RAS Warshawskoe Shosse 132, Moscow 113519, Russia

INTRODUCTION The recent advances in the development of short-pulse (ultra wide band) communication and radar systems have given rise to an increasing interest in the electromagnetic community to formulate time-domain versions of existing frequencydomain techniques for the theory of diffraction. For this reason, exact time-domain solutions of some classical diffraction problems are of grate interest. The purpose of this paper is to obtain an exact time-domain solution of the well-known diffraction task: diffraction on a perfectly conducting half-plane. A plane electromagnetic UWB pulse is under consideration as an incident field. The pulse is arbitrary polarized with respect to the edge of the half-plane and propagates at an arbitrary angle to it. This means that the considered task is a 3D problem. Corresponding 2D problem has been considered (Galstjan, 1999) for the same pulse form and its solution coincided with the known one for the impulsive plane wave (Felsen and Marcuvitz, 1973) in a limit case.

FORMULATION OF THE PROBLEM Consider the perfectly conducting half-plane in Fig. 1. The half-plane is positioned in a rectangular xyz coordinate system with x-axis coinciding with the edge of the half-plane. The half-plane is located in the xz plane (z>0) and is illuminated by a plane pulse described by


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where are real constants, r the position vector, the time, _ the speed of light in vacuum, the polar angle, the azimuthal angle (with respect to the x axis), and the parameter of pulse duration. If the function (1) transforms into the Dirac delta function and the pulse becomes the impulsive plane wave.

First the problem is solved in the frequency-domain representation and then the solution will be transformed into the time-domain one by means of an analytical integration procedure. The frequency-domain representation of the incident pulse (1) is

with ( is the frequency) and the subscript denotes the frequency-domain quantities. Unknown diffraction fields and have to satisfy the wave equation and the boundary conditions for the total fields

Frequency-Domain Solution A frequency-domain solution of the problem under consideration is deriving by using the Wiener-Hopf technique (Noble, 1958) and will be described briefly. In this case, it is convenient to express diffraction fields and in terms of surface current densities induced on the half-plane

as follows

UWB PULSE DIFFRACTION ON A CONDUCTING HALF-PLANE

77

where are the unknown functions, the complex variable, and the components of the Hertz vector. Applying the boundary condition (3) to the total fields and using the physical condition that the current densities (4) are zero on the half-plane extension (y = 0, z < 0) we may deduce a system of dual integral equations for and . To obtain a single-valued solution of this system we have to apply to the unknown functions, and an additional condition, so called the condition on the edge,

Pass over intermediate calculations and write down the obtained frequency-domain solution for the diffraction fields as follows

TIME-DOMAIN SOLUTION OF THE PROBLEM The time-domain solution can now be derived by applying of the inverse Fourier transform to the frequency-domain solution (6). In other words, we have to take a dual integral with respect to and between the infinite limits. Let us demonstrate this process by the example of the diffraction magnetic field component, First input the substitute, and change the variable of integration from to the following result is obtained

78

where is the complex variable. Now the expression (7) is integrated over limits from 0 to , The result is

E. A. GALSTJAN

between the

It is obvious that the integrand function in this expression possess two branch points, and we have to draw two cuts beginning in the branch points to make this function single-valued. Draw the left cut line from the branch point, to so that the line is parallel to the imaginary axis of the complex variable The right cut line is directed from another branch point, to the upper half-plane of the variable in the same manner. Next step is to modify the integration path into the path along the left cut line. The resulting integral can be taken by the residue method. The final expressions for the timedomain solution are

with the functions defined by


79

Pattern of Diffraction Pulse The diffraction fields (8) are the components of a conic pulse with the axis coinciding with the edge of the half-plane and with the apex moving along the edge with the velocity For this reason the expressions (8) and (9) can be simplified and made more illustrative by using a nonorthogonal conic coordinate system. This coordinate system is a combination of the cylindrical coordinate system and a new variable In this "conic" coordinate system the expressions for the diffraction fields are

with The expressions (10) include four terms and these terms have simple physical meaning. First pair of terms describes the incident and reflected pulses (plane pulses) and second one describes properly the diffraction conic pulse. At a constant value an angle distribution of the diffraction fields is defined by a competition of these terms. The value corresponds to a conic surface synchronized with the maximum value of the incident pulse. It is convenient to obtain an approximation of the expressions (10) in a vicinity of this conic surface by expanding (10) in terms of the small parameter This approximation for the diffraction magnetic field, is defined by

with

and for the angle

First term in this

expression is much grater than the second one between the limits from to in other words the diffraction field in this transitional zone has the same nature as the incident pulse. The maximum value of the field in this zone corresponds to and is defined by

and is equal (in absolute value) approximately a half of the maximum magnitude of the incident field.

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In the shadow zone, diffraction magnetic field

we can derive a good approximation for the

The expression (12) demonstrates illustratively the diffraction conic pulse view in the shadow zone. At a given angle, the forward part of the pulse is steep, in comparison with the sloping back part Furthermore, the maximum value of the conic pulse corresponds to

It

means that the conic pulse has a time delay (proportional to the parameter with respect to the incident pulse. Figure 2 illustrates accuracy of the approximation (12) (line 1) in comparison with the exact expression (10) (line 2). The radial magnetic field distributions correspond to the case of (TE - polarization) and the values of the parameters

We have described briefly a qualitative pattern only for reasons of space. It is obvious that all fields can be calculated and illustrated graphically by using the expression (8) or (10). In closing let us describe briefly current distributions on the half-plane. Applying the dual inverse Fourier transforms to the surface current component (4) we can write the timedomain representation of the surface currents. In general case, final expressions are simple but very bulky. For this reason, we centered on expressions for the currents in special case of TE polarization of the incident pulse (1). It is convenient to express the currents


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on the half-plane in terms of currents induced by the incident pulse on a whole plane. The correspondence expressions are

where is an amplitude of the surface current on the whole plane and the superscript denots TE - polarisation.

For the sake of simplicity, the time dependence of the surface current distributions shown in figure 3 have been drown for the values of the angles and In this case, only the z-component of the current (13) is not equal zero. The lines correspond to the time values For the angle values under consideration the current pulse is moving initially to the edge of the half-plane (dashed lines, and later in the reverse direction.

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CONCLUSIONS The exact time-domain solution of the diffraction problem under consideration has been obtained by means of the reverse Fourier transform performed analytically for the special pulse form. In considered case the solution is defined by elementary functions of coordinates and time. Its simplicity makes possible to carry out a complete investigation of all diffraction fields/currents characteristics in the time-domain representation. Besides, this analytical solution can be used as a test for existing numerical or approximate time-domain methods. REFERENCES Galstjan, E.A., 1999, Diffraction of Electromagnetic Impulse on Ideal Conducting HalfPlane, Radiotechnika i Electronika, N10: 1184 (In Russian). Felsen, L.B. and Marcuvitz, N., 1973, Radiation and Scattering of' Waves, Prentice-Hall Inc., New Jersey, USA. Noble, B., 1958, Methods Based on the Wiener-Hopf Technique, Pergamon Press, London.

SPHERICAL WAVE EXPANSION OF THE TIME DOMAIN FREE-SPACE DYADIC GREEN’S FUNCTION

S. Alp

1

, S. Sencer Koç 2 and O. Merih Büyükdura 2

1

Electronic Design Group ASELSAN Inc. Ankara, 06172, Turkey 2 Department of Electrical and Electronics Engineering Middle East Technical University Ankara, 06531, Turkey

INTRODUCTION The importance of expanding Green’s functions, particularly free-space Green's functions, in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time domain scattering problems, similar expansions of Green's functions are expected to be useful in the time domain. In this paper, an expression, expanded in terms of orthogonal spherical vector wave functions, for the time domain free-space dyadic Green's function is presented and scattering by a perfectly conducting sphere is studied as an application to check numerically the validity and to demonstrate the utility of this expression. In the expression derived, in addition to the dependence on and the dependence on and is also 'separated' in the sense that each term in the expansion appears as a function of convolved with a function of Such a dependence in the Green's function is useful in a scattering formulation as it lets one set up an equation (for instance an integral equation) for some unknown quantities which in turn yield the equivalent sources. Throughout the paper the velocity of waves in free-space is taken to be so that in any frequency domain expression which appears, the wavenumber is equal to the angular frequency. are the familiar spherical coordinates of the point of observation and their primed counterparts are those of the source point. An time dependence is assumed and suppressed in the frequency domain expressions. The spherical wave expansion of the time domain scalar free-space Green's function is presented in (Buyukdura et al, 1997); therein a similar expression to our dyadic Green's function is derived for the scalar Green's function. In some previous work on the spherical wave expansion for the radiation from time dependent source distributions using multipole expansion of the sources, (Davidon, 1973), (Heyman et al, 1996) and (Shlivinski et al, Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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1999), the space-time dependence of the source enters via an nth order linear operator (n denoting the order of the associated Legendre functions) while in the present work, the same dependence enters via two (superposition (in space) and convolution (in time)) integrals.

STATEMENT OF THE PROBLEM The problem of interest is to find the dyadic Green's function which satisfies

as well as the radiation condition and causality. In Eq.(l), is the identity dyad which can be represented by a unit diagonal matrix and is the Dirac delta function. This time domain Green's function to a source equation

is useful to find the field

in free-space due

In other words the solution to the inhomogeneous partial differential

is given by

where V is the region occupied by the source.

FORMULATION AND SOLUTION OF THE PROBLEM The solution to Eq.(1) is the inverse Fourier transform of the frequency domain freespace dyadic Green's function which can be written in terms of vector wave functions that are orthogonal over a spherical surface as

In Eq.(4), q is a compact index standing for the indices p , n and

as,

WAVE EXPANSION OF TD FREE-SPACE DYADIC GREEN'S FUNCTION

where the parity index p takes on the "values" either e or

85

(standing for "even" and

"odd", respectively) and when and are the spherical vector wave functions first introduced by (Stratton, 1941). In order to find the inverse Fourier transform of Eq.(4), we write and in terms of the auxiliary spherical vector wave functions and which are independent of . Concerning the case in Eq.(4), we use the following transformation (Buyukdura et al, 1997),

where and denotes the spherical Bessel functions of order n and the spherical Hankel functions of the second kind of order n, respectively (Abramowitz et al, 1972), are the Legendre polynomials, stands for the convolution operation, represents the outgoing wave functions and is a pulse equal to unity when its argument is between -1 and 1, and vanishes elsewhere. However the transformation which is also necessary in finding the time domain expression does not exist. For this reason a new dyadic Green's function is defined in the frequency domain as,

With this definition, the solution to Eq.(2) for the electric field in free-space due to a volume current density is

where is the permeability of the vacuum, Green's function and

In order to find

first Eq.(4) is multiplied by

is the new time domain dyadic

yielding the expression for

and then is inverse Fourier transformed with the help of Eq.(6). After these trivial steps, the spherical wave expansion of the time domain free-space dyadic Green's function for can be expressed as in Eq.(10). One can go through similar steps to get the expression for

as in Eq.(11). As seen in these equations, in

addition to the dependence on and the dependence on and is also 'separated' in the sense that each term in the expansion appears as a function of convolved with a function of and that was the goal in deriving these expressions.

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RESULTS AND DISCUSSION We consider an impulsive plane wave incident from the positive-z axis (from ) which is linearly polarized in the x-direction, i.e., This incident field can be written in terms of the spherical vector wave functions with the aid of the Green's function in (11) and the field scattered by a perfectly conducting sphere of radius a centered at the origin can be expressed inspired by the form of the Green's function in (10). The unknown coefficients in the scattered field expression can be found by imposing the boundary condition on the surface of the sphere and solving the resultant equations either by direct deconvolution or by using system identification techniques. As a numerical example, the plane wave with the waveform of the derivative of a gaussian pulse, i.e.,

WAVE EXPANSION OF TD FREE-SPACE DYADIC GREEN'S FUNCTION

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incident to a perfectly conducting sphere of radius a = 1 is considered. The wave function expansion of this incident field can be obtained by convolving the waveform, with the expansion of termby-term and the scattered field can be found by convolving the scattered field due to with the incident waveform. The duration of the incident waveform is 20 ns provided that the wave velocity is that of light in vacuum. In figure 1, two solutions to the scattered field in the back-scatter direction at a distance of R = 3 m are given where the x component of the field is plotted versus time. The solid curve is obtained using the present time domain formulation (direct deconvolution is used in finding the scattered field coefficients), while the cross marks are obtained by inverse Fourier transforming the well known frequency domain solution. Both solutions are obtained by including only 4 terms in the series expansion. An observation made on the time domain solution is that the solution converges rapidly if the dimension of the scatterer is small compared to the wavelength at the highest frequency component of the incident field which is similar to the case in the solution of problems in the frequency domain.

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REFERENCES Abramowitz, M., and Stegun, I.A., 1972, Handbook of Mathematical Functions, Dover, New York. Buyukdura, O.M.,and Koc, S.S., 1997, Two alternative expressions for the spherical wave expansion of the time domain scalar free-space Green's function and an application: Scattering by a soft sphere, J. Acoust. Soc. Am., vol. 101, pp. 87-91. Davidon, W.C., 1973, Time-dependent multipole analysis, J. Phys. A: Math., Nucl. Gen., vol. 6, pp. 1635-1646. Heyman, E., and Devaney, A.J., 1996, Time-dependent multipoles and their application for radiation from volume source distributions, J. Math. Phys., vol. 37, pp. 682-692. Shlivinski, A., and Heyman, E., 1999, Time-Domain Near-Field Analysis of Short-Pulse Antennas – Part I: Spherical Wave (Multipole) Expansion, IEEE Trans. Antennas Propagat., vol. 47, pp. 271-279. Stratton, J.A., 1941, Electromagnetic Theory. McGraw-Hill, New York.

ON THE LOCALIZATION OF ELECTROMAGNETIC ENERGY

Hans Gregory Schantz Time Domain Corporation 6700 Odyssey Drive Huntsville, AL 35806 USA

INTRODUCTION This paper explores the interesting question of whether electromagnetic energy can be localized. Three specific areas will be addressed. First, the historical development of the Poynting – Heaviside theory will be traced. Then, the problems and alleged paradoxes of this theory will be examined. Finally, a method of tracking electromagnetic energy will be presented and applied to some simple examples. The challenge of short pulse electromagnetics is to understand the time evolution of a transient electromagnetic system. The aim of this paper is to demonstrate that understanding electromagnetic energy transfer can be a valuable means to that end. A BRIEF HISTORY OF ELECTROMAGNETIC ENERGY LOCALIZATION In his 1847 treatise on energy conservation, Hermann von Helmholtz introduced the concept that electric energy density depends upon the charge density and the electromagnetic potential (V):1

This approach implies that electromagnetic energy is localized with charges, in accord with the then current “action-at-a-distance” philosophy that underlay physical thought. This action-at-a-distance approach to electromagnetics had already begun to crumble at the hands of Michael Faraday who introduced the concept of a field. The implication of Faraday’s field approach is that electric and magnetic processes are distributed throughout space, not localized with sources. In 1853, William Thompson (later Lord Kelvin) introduced the idea that energy itself might be localized with the fields.2 In the context of the electric field, Thompson’s idea meant that the field carries an energy:


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This idea flowered at the hands of James Clerk Maxwell:3,4 ''The energy in electro-magnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in electrified bodies, conducting circuits, and magnets, in the form of an unknown quantity called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves....'' Not long thereafter, John Henry Poynting and Oliver Heaviside independently arrived at what came to be known as the “Poynting vector” to describe the flow of electromagnetic energy:

The mathematical relationship discovered by Poynting, “Poynting’s Theorem:”

follows unequivocally from Maxwell’s equations. Theorem as a statement of local energy conservation:

The interpretation of Poynting’s

is not so clear cut. With Hertz’s discovery of radio waves (1888-1889) however, the triumph of Faraday’s field point of view was considered complete.5 Having demonstrated that radio waves moved at the speed of light and have optical properties, Hertz proved that they were decoupled from their source. “In the sense of our theory we more correctly represent the phenomena by saying that fundamentally the waves which are being developed do not owe their formation solely to processes at the origin, but arise out of the conditions of the whole surrounding space, which latter, according to our theory, is the true seat of the energy.”6 PUZZLES AND PARADOXES OF ELECTROMAGNETIC ENERGY Later investigators noticed difficulties with the energy flow interpretation advocated by such pioneers as Poynting, Heaviside, and Hertz. Under certain circumstances, the Poynting-Heaviside theory yields seemingly nonsensical results such as closed loops of energy in otherwise static systems. For instance, consider a static point charge “q” with field:

superimposed with a static magnetic dipole with field:

ON THE LOCALIZATION OF ELECTROMAGNETIC THEORY

This “static” system has a non-zero Poynting vector:

corresponding to azimuthal loops of flux.

Further, the Poynting vector is ambiguous to a solenoidal (i.e., divergenceless) term. In other words, if S' = S + G where • G = 0, the new resulting “Poynting vector,” S', ” will still satisfy Poynting’s theorem. Thus, it is often argued that the Poynting vector has no physical significance unless integrated over a completely closed surface.7 Some observers reject the idea that the Poynting vector represents a localized flow of energy while at least accepting that the integral of the Poynting vector over a closed surface has a physical significance – the rate of change of energy in the bounded volume.8 Others, like R.W.P. King, reject the idea that energy has any physical significance whatsoever, beyond being a mathematical quantity that may be useful in calculations.9 More recent investigation has tended to uphold the original vision of Poynting and Heaviside. The seemingly implausible “loops” of Poynting flux are now generally recognized as required by the demands of angular momentum conservation. A simple example (adapted from one offered by Feynman10) serves to illustrate the point. Consider a charged dielectric hoop at rest with a bar magnet along its axis, a physical system whose fields will be similar to the point charge, point magnetic dipole described above. By Lenz’s Law, when the magnet is removed, an e.m.f. will be induced in the hoop, setting it spinning. Since the bar magnet is removed along the axis of the hoop, no angular momentum will be imparted to the system, and yet, it spins. A detailed analysis of an analogous system shows that the electromagnetic angular mometum of the original “static” fields, as predicted by the closed loops of Poynting flux, is the same as the imparted mechanical angular momentum.11,12 In fact, the torque associated with circularly polarized light has been experimentally measured. The results are also in accord with the Poynting – Heaviside interpretation.13

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The seeming ambiguity of the Poynting vector is thus subject to the constraint that any modification must not only satisfy conservation of angular momentum, but also yield the same correct value for the angular momentum. Similarly, there is a linear momentum (source of the so-called “light” pressure) known to be correctly described by the Poynting vector.14 Naturally, any modification must further satisfy conservation of energy. All these constraints severely limit alternatives to the Poynting vector for describing the local electromagnetic energy flow.15 In fact, it has long been realized that any additional solenoidal term added to the Poynting vector cannot be a function of the electromagnetic fields (or their time derivatives) and still satisfy Poynting’s Theorem.16 Since it is reasonable to assume that any measure of local electromagnetic energy flow must be a function of the electromagnetic fields, it is very difficult to imagine how there could be a physically meaningful alternative to the Poynting vector. A final consideration has been largely neglected from the debate on the physical interpretation of the Poynting vector. Virtually every communication link ever designed relies on the Friis transmission formula to predict the received power. The Friis transmission formula in turn relies on our ability to predict the power flux using the Poynting vector, not integrated over a closed surface surrounding the transmitter, but rather, over a very tiny piece of that surface. The undeniable success of the Friis formula suggests that the Poynting vector is the correct local measure of electromagnetic energy flow, or at least places very stringent limitations on alternatives. Open questions may remain, but there is no good reason to unilaterally reject the insights available from localizing and tracking electromagnetic energy.


CAUSAL SURFACES Some time ago, the author noted the possibility of identifying “causal surfaces,” surfaces on which so that there is no net flow of electromagnetic energy.17 These surfaces partition electromagnetic energy making it easier to track the source of the energy and see how it changes from one form to another. In the exponential decay of a Hertzian dipole, for instance, there is a spherical causal surface of radius (where is the time constant of the decay) on which the Poynting vector is everywhere zero. Thus, this surface bounds the field energy, partitioning it to a region inside the sphere in which energy is absorbed, and a region outside the sphere from which energy radiates away. It has also been shown that the total radiated energy which escapes to the far field is exactly equal to the total energy originally stored outside the causal surface.18

Elsewhere, this technique has been used to describe the flux of energy around a harmonic dipole.19,20 By plotting a space-time diagram of the radial energy density, the time varying causal surfaces, and the local energy velocity, the radiation of energy from a harmonic Hertzian dipole can be understood in detail. Some general observations are possible. First, no energy escapes directly from the dipole to the far field without first becoming temporarily stored or reactive. Second, energy is only emitted by the dipole half the time. The other half of the time the dipole is absorbing energy and the apparent radiation comes from energy already stored in the near fields. Finally, the average distance

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at which reactive energy is converted to radiated energy is around a result consistent with Harold Wheeler’s “radiansphere” concept21 – the observation that the near and far fields around a harmonic dipole are equal in magnitude at around This same space-time portrayal of energy flow was also applied to dipoles subject to three transient excitations: charging, discharging and a Gaussian impulse. One may also apply the concept of causal surfaces to understand how accelerating charges radiate. The process is somewhat different when a charge decelerates, so first consider acceleration. A causal surface appears where the net tangential field goes to zero. In the case of the accelerating charge, this means we need to examine the interplay of the applied external field (that causes the acceleration), and the radiation fields. If we are sufficiently far away from the charge and external field (that together are the radiating system), the dominant field is the radiation field, and the net flow of energy is outward. The radiation fields are opposite in direction to the external fields. A quick comparison of their magnitudes shows that the radiation fields are far weaker than the external fields. Somewhere around the fringe of the region where the external fields exist, radiation and external fields just cancel out yielding a causal surface. The result is that in the region where the external fields exist, the energy flows in toward the accelerating charge, adding to the magnetic field energy of the ever more quickly moving charge (see Figure 5).

Since on a sufficiently local scale the external field must always be uniform and constant, it cannot dominate arbitrarily close to the accelerating charge. A quick calculation in the low velocity limit suffices to learn where that boundary might be. In this limit, the electric field due to an accelerating charge “q” with acceleration (and with mass “m”) is:

while the external field will look something like this:

Summing the tangential is:

fields and noting that the magnitude of the force on the charge

there will be a spherical causal surface about the charge of radius:


Interestingly, the result is entirely independent of the magnitude of the external field. In fact, this result is just the so-called classical electron radius. Since the net tangential electric field is zero, energy cannot be extracted from within this surface. The surface serves as an electromagnetic analog to the event horizon of a black hole. Whatever energy is absorbed by or radiated from an accelerating charge must reside outside the sphere defined by the classical electron radius. Of course this is strictly a nonrelativistic result, but since the fields in an instantaneously co-moving reference frame should reduce to the low velocity result here, a similar result should hold for the more general case. Further, it should be noted that this is strictly a classical calculation and neglects any quantum mechanical effects. The case of a decelerating charge is slightly different. Although there will still be a causal surface at the classical electron radius, the typical source of the radiated energy is the magnetic field energy around the moving charge. The near and far magnetic fields of a decelerating charge have opposite orientations (see Figure 6). There will be a spherical surface on which the magnetic field goes to zero. The radius of this surface is:

Inside this surface, the magnetic field energy of the decelerating charge is absorbed by the external field that causes the deceleration. Typically, this means the magnetic energy of the decelerating charge is converted to the static field of the applied electric field.

To summarize, electromagnetics is not exempt from the law of energy conservation. Radiant energy does not come into being out of nothing, rather it is converted from other forms of electromagnetic energy. In the case of accelerating charges, the external field loses energy – it is the source of both the radiated energy as well as the magnetic field energy gained by the now more quickly moving charge. In the case of decelerating charges, the more slowly moving charge loses magnetic field energy – it is the source of both the radiated energy as well as the energy gained by the external field.

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CONCLUSION As Oliver Heaviside observed, “However mysterious energy (and its flux) may be in some of its theoretical aspects, there must be something in it, because it is convertible into dollars, the ultimate official measure of value.”22 The ability to track electromagnetic energy is helpful in designing antennas with low reactive field energy. Such antennas would be highly efficient with a low Q and broad bandwidth – ideal for UWB-SP applications. Electromagnetic energy localization is thus a valuable technique for UWB-SP antenna design. REFERENCES 1

Hermann von Helmholtz, The Conservation of Force: A Physical Memoir (1847) collected in Selected Writings of Hermann von Helmholtz, Russel Kahl, ed., Wesleyan University Press, Middletown, Connecticutt, (1971) pp. 3-55. 2 Sir Edmund Whittaker, A History of the Theories of Aether & Electricity,” Vol. 1, Harper and Brothers, New York, (1951) p. 222. 3 James Clerk Maxwell, A Treatise on Electricity and Magnetism Vol. II., Academic Reprints, Stanford, CA, (1953) pp. 270-271. See §631 in particular. 4 James Clerk Maxwell, On Action-At-A-Distance, Proceedings of the Royal Institution of Great Britain, Vol. 7, (1873-5) pp. 48-49. 5 Lord Kelvin, Preface to the English Translation of Hertz’s Electric Waves. 6 Heinrich Hertz, Electric Waves, Macmillan and Co., London, (1893). 7 Ronold W.P. King, Fundamental Electromagnetic Theory (2nd ed.), Dover, New York, 1963, pp. 191-192. 8 Sir James Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge: University Press, Cambridge, (1933), p. 518. 9 Ronold W. P. King, Op. Cit., p. 180. 10 Richard Feynman, The Feynman Lectures in Physics, Vol. 2, Addison Wesley Pub. Co., Reading, MA, (1964) pp. II-27-8. 11 E.M. Pugh and G.E. Pugh, “Physical Significance of the Poynting Vector in Static Fields,” Am J Phys 35, 153-156 (February 1967). 12 Gabriel G. Lombardi, "Feynman's disk paradox," Am. J. Phys., 51, 213-214 (1983). 13 Richard A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936). 14 E.E. Nichols and G.F. Hull, “The Pressure Due to Radiation,” Phys Rev17 26-50, 91-104 (1903). 15 Udo Backhaus and Klaus Schäfer, "On the uniqueness of the vector for energy flow density in electromagnetic fields," Am. J. Phys. 54, 279-280 (1986) 16 Kr. Birkeland, “Ueber die Strahlung electromagnetischer Energie im Raume.” Ann Phys 52, 357-380, (1894) 17 Hans Gregory Schantz, “The flow of electromagnetic energy around an electric dipole,” Am J Phys 63 513-520 (1995). 18 Ibid. 19 H. Schantz, “Electromagnetic Energy Around Hertzian Dipoles,” IEEE Antennas and Propagation Society International Symposium (1999), pp. 1320-4 20 Hans Gregory Schantz, “Electromagnetic Energy Around Hertzian Dipoles,” submitted to IEEE Antenna and Propagation Magazine, July 20, 1999. 21 Harold A. Wheeler, “The Radiansphere Around a Small Antenna,” Proc IRE 47 13251331 (1959). 22 Oliver Heaviside, Electromagnetic Theory, Vol. 1, Chelsea Publishing Company, New York, (1971), p. 381.

ON SUPERLUMINAL PHOTONIC TUNNELLING

Günter Nimtz1, Astrid Haibel1 and Alfons A. Stahlhofen2 1

2

II. Physikalisches Institut Universität zu Köln 50937 Köln, Germany Institut für Physik Universität Koblenz 56075 Koblenz, Germany

INTRODUCTION

The analogy of wave mechanical tunnelling are the classical evanescent modes, nowadays often called photonic tunnelling. The Helmholtz equation describing the electromagnetic fields corresponds formally to the Schrödinger equation for the wave function of particles.

Interest in the tunnelling time aroses after the first man-made electronic tunnelling structures were introduced in solid state physics: The tunnelling diode in semiconductor physics discovered by Esaki (1958) and the tunnelling in superconductor-insulatorconductor structures designed by Giaever, Josephson and others (1960). Surprisingly experimental electronic tunnelling time data are not available so far. This serious Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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dilemma has stimulated much the various studies on photonic tunnelling. The mathematical analogy between the Helmholtz and the Schrödinger equations allows to conclude that the photonic results are representative for general tunnelling. Essentially three important kinds of photonic barriers have been studied (Nimtz and Heitmann, 1997): a) Double prism with frustrated total internal reflection (FTIR), b) periodic dielectric heterostructures, i.e. photonic lattices, and c) undersized wave guides. The barriers are sketched in Fig. 1. The three barriers have different transmission dispersion relations, however, all the tunnelling modes are characterized by an imaginary wave number which means that the evanescent field does exponentially decay with distance. According to the phase time approach the field is spread out in no time. This behaviour is illustrated for the case of FTIR at the double prism. Due to the Goos-Hänchen shift D the light beam travels in two paths crossing the forbidden gap in Fig. 2. The first path of the beam is along the surface of the entrance prism, which results in the Goos-Hänchen shift and this wave solution has a real wave number After moving along the surface the beam crosses the gap by an evanescent mode with an imaginary wave number

In the case of FTIR, as sketched in Fig. 2, the electric field E is given by the relation

describing a wave propagating along the prism surface in y-direction and an evanescent mode tunnelling the forbidden gap between the two prisms in x-direction.

ON SUPERLUMINAL PHOTONIC TUNNELING

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We like to point out that the exponential field decay inside the gap is not caused by absorption since tunnelling is an elastic process. In the following chapters we are going to show that tunnelling of signals and energy can proceed much faster than light in vacuum, i.e. at superluminal velocity. This effect is observed in the .case of opaque barriers and signals containing evanescent frequency components only. (A barrier is called opaque if the transmission is dropped much more than 1/e.) In the next chapter we shall show, that it is a fundamental physical property of signals to be frequency band limited and this implies superluminal propagation. Another important property of the tunnelling process is that the tunnelling time equals approximately the reciprocal frequency of the evanescent modes in question. This property seems to be valid for all opaque barriers as shown below. SUPERLUMINAL SIGNALS The important question is: what is a physical signal? In Fig. 3 an ideal sinusoidal signal is shown. The information, i.e. the signal is presented by both the carrier frequency and the 6 oscillations. This ideal signal is not frequency band limited but it is described by a Fourier spectrum with frequencies between Quite often it has been assumed that a signal has an unlimited frequency band (e.g. Büttiker and Thomas, 1998). Only the poor technical design of transmitter and receiver systems are not able to generate and detect all frequency components.

If signals are considered as being composed of photons with energies h v then the signals finite energy implies that its frequency band has to be limited. Recently Nimtz (1999) has pointed out, that frequency band limitation is a fundamental property of any physical signal in consequence of quantum mechanics. A signal with finite energy has to have a frequency band limitation. According to the Fourier transform such a signal becomes non-causal from the mathematical point of view but not necessarily from the physical one (Nimtz, 1999). A modern signal used in optoelectronics is presented in Fig. 4. The information is given by the carrier frequency which determines the receiver’s address and the pulses halfwidth gives the number of 0 or 1 digits. The presented signal is frequency band limited to only of the carrier frequency. Such a signal can be sent through a photonic barrier without any significant signal reshaping at a superluminal speed of 4.7 times the velocity of light or even much faster (Nimtz et al., 1994; Aichmann et al., 2000). The condition sine qua non for achieving superluminal signal velocities is that all components of the signal have to be evanescent and a signal reshaping is avoided

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by a narrow frequency band width (Nimtz, 1999). The strong attenuation reduces the signals amplitude, however, not the content of information. Of course, as seen in Fig. 4 with decreasing magnitude the noise to signal ratio increases and the signal gets lost eventually. There is an instructive example of a signal in astronomy: A cosmic explosion with an outburst is analyzed by the ”carrier frequency” and by the ”width” of the pulse. For instance the radiation gives the information about the temperature and the halfwidth of the outburst gives the information on the energy involved in the cosmic event. The information (i.e. ”carrier frequency” and ”halfwidth”) is independent of its magnitude and thus independent of the distance between the cosmic event and the observer on earth.

IS THE TUNNELLING TIME UNIVERSAL ? Recently it has been observed that the tunnelling time in the case of opaque barriers has a universal property. The tunnelling time equals approximately the inverse frequency of the evanescent mode or the tunnelling photons (Haibel and Nimtz, 2000). This behaviour has been found experimentally in studies with microwaves as well as with light. Irrespective of the complete different interpretation the behaviour is also in agreement with the theoretical data of evanescent modes or tunnelling solutions of the Helmholtz and the Schrödinger equations (Enders and Nimtz, 1994). In Table 1 data of various studies are summarized. The experiments were carried out at different frequencies as well as with different barriers. Most of the data presented in Table 1 show a good agreement between the measured tunnelling time and the inverse frequency of


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the wave. The deviations are within one order of magnitude. This is not surprising since the experiments are rather sophisticated and from the theoretical point of view (e.g. Hartman, 1962) the tunnelling time depends to some extent on the wave number of the tunnelling wave.

SPECIAL FEATURES OF EVANESCENT MODES The Schrödinger equation yields a negative kinetic energy in the tunnelling case, since the potential U is larger than the particle’s total energy W:

The same happens to evanescent modes. Within the mathematical analogy, their kinetic electromagnetic energy is negative, too. The Helmholtz equation for the electric field E, for instance in a waveguide is given by the relationship

where is the cut-off wave number of the evanescent regime. The quantity plays the role of the energy eigenvalues and is negative in the case of evanescent modes. The dielectric function of evanescent modes is negative and thus the refractive index is imaginary. For the basic mode a rectangular waveguide has the following dispersion of its dielectric function, where holds and is the waveguide width, is the free space wavelength of the electromagnetic wave. According to the last equation the evanescent mode’s electric energy density given by the relationship

is

where is the electric permeability of the vacuum. In the case of tunnelling it is claimed that a particle can only be measured in the barrier with an additional particle, e.g. a photon having an energy (e.g. Gasiorowicz, 1996). This means that the total energy of the system is positive and the

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tunnelling process is stopped. The analogy between the Schrödinger equation and the Helmholtz equation holds again: it is not possible to measure an evanescent mode in analogy to a particle in a tunnel. We may also say that an evanescent mode can not be measured due to an impedance mismatch between the evanescent mode and a probe in the barrier. The impedance Z of the basic mode in a rectangular waveguide is given by the relationship where is the free space impedance. In the evanescent frequency regime the impedance is imaginary. Evanescent modes do not experience a phase shift inside the barrier. Having in mind the phase time relationship

in the evanescent regime there is an absence of time, where and are the phase shift and the angular frequency, respectively. The measured tunnelling time is spent on the entrance boundary due to the interference of the incoming and reflected wave. In the case of frustrated total internal reflection the measured total tunnelling time is spent during the travel along the surface in consequence of the Goos-Hänchen shift, while crossing the gap takes place instantaneously (Stahlhofen, 2000; Haibel and Nimtz, 2000). CONCLUSION Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000) . Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles. Acknowledgements The authors gratefully acknowledge discussions with P. Mittelstaedt and R. Vetter. REFERENCES Aichmann, H., Haibel, A., Lennartz, W., Nimtz, G., and Spanoudaki, A., 2000, Demonstrating superluminal signal velocity, in: Proc. Int. Symposium Quantum Theory and Symmetries, Goslar, 18.-23. July 1999


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Balcou, Ph., and Dutriaux, L., 1997, Dual optical tunnelling times in frustrated total internal reflection, Phys. Rev. Letters, 78:851 Büttiker, M., and Thomas, H., 1998, Front propagation on evanescent media, in: Superlattices and Microstructures, 23:781 Carey, J., Zawadzka, J., Jaroszynski, D., and Wynne, K., 2000, Noncausal time response in frustrated total reflection?, Phys. Rev. Letters, 84:1431 Desurvire, E., 1992, Lightwave Communications: The Fifth Generation, Scientific American, 266:96 Enders, A., and Nimtz, G., 1992, On superluminal barrier traversal, J. Phys. I France, 2:1693 Enders, A., and Nimtz, G., 1994, Evanescent mode propagation and quantum tunnelling Phys. Rev. E, 48:632 Gasiorowicz, S., 1996, Quantum Physics, Second Edition John Wiley & Sons, New York Haibel, A., and Nimtz, G., 2000, On the universality of tunnelling time, to be published. Hartman, Th., 1962, Tunnelling of a Wave Packet, J. Appl. Physics, 33:3427 Mugnai, D., Ranfagni, A., and Ronchi, L., 1998, The question of tunnelling time duration: A new experimental test at microwave scale, Phys. Letters A, 247:281 Nimtz, G., Enders, A., Spieker, H., 1994, Photonic tunnelling times, J. Phys. I France, 4:565 Nimtz, G. and Heitmann, W., 19997, Superluminal photonic tunnelling and quantum electronics Prog. Quantum Electronics, 21:81 Nimtz, G., 1999, Evanescent modes are not necessarily Einstein causal, Eur. Phys. J. B, 7:523. Stahlhofen, A. A., 2000, Photonic tunnelling time in frustrated total internal reflection, Phys. Rev. A in press Steinberg, A., Kwiat, P., and Chiao, R., 1993, Measurement of the single-photon tunnelling time, Phys. Rev. Letters, 71:708 Spielmann, Ch., Szipöcs, R., Stingle, A., and Krausz, F., 1994, Tunnelling of optical pulses through photonic band gaps, Phys. Rev. Letters, 73:2308


Transient Electromagnetic Field of a Vertical Magnetic Dipole on a Two-Layer Conducting Earth

Samira T. Bishay1, Osama M. Abo Seida2 , and Ghada M. Sami1 1

Mathematics Department, Faculty of Science, Ain Shams University, Abbassia,Cairo, Egypt. 2 Mathematics Department, Faculty of Education, Kafr El-Sheikh Branch, Tanta University, Kafr El-Sheikh, Egypt.

INTRODUCTION The effect of the transient responses of the dipoles on the ground subsurface has a considerable value as means of probing the earth. Sometimes ago, Wait [1] derived the transient fields of a horizontal electric loop on a homogeneous earth in a closed form expressions. An extensive analyses [2], [3] and [4], were carried out for the continuous wave and transient responses of mutual impedance between current-carrying loops or wires. Nevertheless, the transient response of dipoles over horizontally stratified ground and the interpretation of such fields in remote sensing have not been given enough attention. Therefore, a model for the ground which consists of two horizontally stratified layers with contrasting permittivities and conductivities is considered here for the case of a source on the ground with a transient current waveform. The transient electromagnetic fields are found in the frequency-domain. On the other hand, a derivation of the electromagnetic fields of a travelling current line source located over a layered conducting half-space is given [5]. Nabulsi et al. [6], using an exact numerical solution, obtained the transient response of a two-layer earth model for an obliquely incident transverse magnetic plane wave. Using a quasi-static approach, Wait [3] has derived closed-form solutions for the fields of loops above the surface of a two-layered earth which are valid at sufficiently late times. Botros et al [7], using the same method, have obtained simple expressions for the transient responses between loops laying on the surface of a two-layer earth. The late time response is obtained by an approach valid for a sufficiently large time such that all the distances encountered are compared with the significant free space wavelengths. Thus, the displacement currents in the air region may be neglected. Afterwards, Mahmoud et al [8] derived the complete timedomain fields due to a vertical magnetic dipole on the surface of two-layered ground in an analytical form. In their expressions, the displacement currents in the ground have been neglected. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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It is our purpose to obtain the frequency domain full wave solution for the fields of a small horizontal loop, i.e. a vertical magnetic dipole, on a two-layered earth's model, including the waveguide modes propagating within the earth. Concentrating on the solutions of the eignvalue equation of the problem, the displacement currents in the two-layer earth and in the air regions are accounted for.

GENERAL DESCRIPTION Referring to Fig.1, the source is a vertical magnetic dipole situated directly on the earth's surface, where the earth's adopted model consists of a homogeneous overburden slab of thickness d above a half-space. The permittivities and conductivities of the overburden and the lower half-space are ( i = 1, 2 ), respectively. The magnetic permeability is taken equal to that of the free space everywhere. The source represents a small loop of area dA carrying a circulating current I(t) which, in general, is a function of time t. Without loss of generality, we can restrict attention to situations where I(t) = 0 for t < 0. Using cylinderical coordinates for a time dependence exp(st), where in terms of the angular frequency and s is the Laplace transform variable, the frequency domain expression of the vertical magnetic field, on the first boundary where z = 0, i.e. in the air region due to the source, takes the form [9]:

where

TRANSIENT FIELD OF A VERTICAL MAGNETIC DIPOLE

and

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are the Fresnel reflection coefficients at the interfaces z = 0 and z = d,

respectively, and

is the Hankel function of the second kind and order zero.

The contour integration in (1) can be deformed in the lower half of the complex but it must pass around the two branch points at and and a discrete set of poles, as shown in Fig.2 . Here we may note that there is no branch point at and there is no contribution from the contour at infinity [8], [10] and [11].

FREQUENCY- DOMAIN SOLUTION The frequency domain expression of the vertical magnetic field, for example, can be written as

where

where

is the branch-cuts contribution modes and is given by

and respectively, and

are the branch-cut integrals at the branch points is the poles contribution, written as

and

S. T. BISHAY ET AL.

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where is the contribution of the pole at Physically, the two branch-line integrals account for a continuous spectrum of radiation in the homogeneous regions above the interface and deep within the earth, respectively. The pole contributions, of course, can be identified with waveguide modes in the stratified region of the earth.

A-Evaluation of the Integral along the Branch -Cut The branch cut integral can be evaluated at the branch point substitution

[8].

Subject

to

the

condition

using the or

the branch cut integral can be expressed as that of a half-space problem multiplied by a sratification factor. Thus, the integral in (1) can be evaluated without approximation and leads to the closed form expression

where the stratification factor

with

defined as

It is worth noting that for a homogeneous ground the stratification factors and expression (9) yields the exact result for this field. Using (10) we can put it in a more convenient form, as

where

Using a similar procedure, the contribution of the branch cut integral at obtained as

is

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where

For the other components of the electric and magnetic fields, the contribution of the branch-cut integral at is obtained as

As in the case of a lossy substratum and at a sufficiently large separation between the source and the observation point, i.e., for the overall contribution of the branch-cut integrals can be expressed as that of the integral at

only.

B- Evaluation of the Integral around the Poles The poles of (1) in the complex are the roots of the equation

Looking for the late time response, let us consider the waveguide modes under lowfrequency conditions. Hence, the ratio

and for the first few low-order

modes i.e., those having the least radial attenuation, one may assume that where

n

is

approximations are valid:

an

integer denoting Under

the these

mode order. Hence, conditions the following

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Substituting in (18) gives

where

and

B = 0or

according to whether

or

respectively. A solution of (21) is

where the plus-minus signs apply for the lower and upper halves of the complex s plane, respectively. The corresponding roots in the complex plane are given by

where

and

The residue of the pole

where

for the vertical magnetic field

is obtained as


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Taking the summation of the pole residues, the waveguide mode contribution is then given by

where and is the modified Bessel function of the second kind and the n-th order. Using the same procedure to get the other components of the electric and magnetic fields, i.e.,

NUMERICAL RESULTS The magnitude of the frequency rate of change of the vertical magnetic field due to a rectangular pulse is computed for different cases. Plots of the branch-cut contribution, the waveguide mode contribution as well as the overall frequency response are shown in Figs. 3 (a-h) for different frequencies. In these figures, the vertical scale is normalized by the factor while the normalized frequency is different, multiplied by power ten, as shown in the figures. Having determined the location of the poles in the complex the guided modes may be calculated as residues and comapred with the total field. To get the residues of the poles, we need the eigenvalues as defined by equation (18). The poles are located between the branch points and . In Figs 3(a-h) the normalized vertical magnetic field of the dipole is drawn versus the normalized frequencies. From these plots, it is seen that increasing the frequency increases the waveguide mode contribution, Fig. 3 (c,d), up to a certain limit, after which this contribution starts to diminish again Fig. 3(g).

CONCLUSION The complete frequency-domain fields due to a vertical magnetic dipole on the surface of a two-layered ground have been derived in an analytical form. In these expressions, the displacement currents in the air and in the ground are taken into

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consideration. The previously obtained approximate solution [7] is found to be the late time part of the present solution which is of overriding practical significance. It would be interesting to evaluate these fields numerically with different frequency-dependent and which is a diagnostic feature of the subsurface contaminants and hazardous fluid materials.

REFERENCES 1. J.R. Wait, "The magnetic dipole over the horizontally stratified earth", Can. J. Phys., Vol.

29, pp. 577-592, Nov. 1951. 2. J.R. Wait, and K.P. Spies, "Note on electrical ground constants from the mutual impedaance of small coplanar loops", J. Appl. Phys., Vol. 43, no. 3, pp. 810-891, 1972. 3. J.R. Wait, "On the theory of transient electromagnetic sounding over a stratified earth", Can. J. Phys., Vol. 50, no. 11, pp. 1055-1061, 1972. 4. J.R. Wait, L. Thrane, and R. J. King, "The transient electric field response of an array of parallel wires on the earth's surface", IEEE Trans. Antennas Propagat., AP-23, no. 2, pp. 261-264, 1975. 5. J.R. Wait, "EM fields of a phased line current over a conducting half-space", IEEE Trans. on Electromagnetic Comptability, Vol. EM- 38, no. 4, pp. 608-611, 1996. 6. K.A. Nabulsi, and J.R. Wait, " Ray decomposition of the pulse responses of a two-layer half-space", IEEE Trans. on Geosci. Remote Sensing, Vol.GE-35, no. 2, pp. 287-292, 1997. 7. A. Z. Botros, and S. F. Mahmoud, "The transient fields of simple radiators from the point of view of remote sensing of the ground subsurface", Radio Sci., Vol. 13, no. 2, 379-389, 1978. 8. S. F. Mahmoud , A. Z. Botros , and J.R. Wait, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer earth", Proceedings of the IEEE, Vol. 67, no. 7, pp. 1022-1029,1979. 9. J. R. Wait, "Electromagnetic fields of sources in lossy media", in Antenna Theory, R. E. Collin and F. J. Zucker, part 2, ch. 24, Mc Graw-Hill, New York, 1969. 10. J. A. Fuller and J.R. Wait, "High-Frequency electromagnetic coupling between small coplanar loops over an inhomogeneous ground", Geophysics, Vol 37, no. 6, pp. 997-1004, 1972. 11. A. Ezzeddine, J. A. Kong and L. Tsang," Time response of a vertical electric dipole over a two- layer medium by the double deformation technique", J. Appl. Phys., Vol 53, no. 2, pp. 813-822, 1982.


Time-Domain Study of Transient Fields For a Thin Circular Loop Antenna

Samira T. Bishay and Ghada M. Sami Mathematics Department, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt.

INTRODUCTION In order to probe the earth, the mechanism of the transient responses between dipoles near the ground surface must be understood. Many homogeneous earth models represented by a homogeneous half-space with a specified conductivity and permittivity are usually studied [1], [2]. For transient excitation and with the advent of operational ground penetrating radar [3],it is useful to have benchmark analytical and numerical models of layered half-space. Wait [4] had previously derived the transient fields of a horizontal electric loop on a homogeneous earth in closed form expressions. Also, the study of an elevated loop above the earth’s surface and its use in airborne electromagnetic surveying of the earth has been reported [5]. Due to the complexity of the time-harmonic solutions, closed-form expressions of the fields in the time domain, for a layered earth, are usually lacking. Hence, most frequently numerical or modeling studies have been adopted [6], [7]. Using closed-form solutions, Botros and Mahmoud [8] have obtained simple expressions for the transient responses between loops lying on the surface of a two-layered earth. Afterwards, Mahmoud et al. [9] derived the complete time-domain fields due to a vertical magnetic dipole on the surface of two-layered ground in an analytical form. Recently, Nabulsi et al. [10] have obtained the transient response of a two-layer earth model for an obliquely incident transverse magnetic plane wave using an exact numerical solution. Thus, the decomposition of the general case into ray contributions is an alternative approach that yields consistent results. In the present study, it is our purpose to obtain specific results, without approximation, for the time-domain full wave solution of the field for a small horizontal loop, i.e. a vertical magnetic dipole, on a two-layer earth’s model including the waveguide modes propagating within the earth. Cocentrating on the late time part of the solution, the displacement currents in the two-layer of the earth are accounted for.


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ANALYTICAL PART OF THE PROBLEM The geometry of the problem is illustrated in Fig.l where the air-earth interface is at z = 0. The earth’s adopted model consists of a homogeneous overburden slab of thickness d above a half-space. The upper layer for has conductivity permittivity and magnetic permeability The corresponding constants of the lower half-space are and The source is a vertical magnetic dipole situated directly on the earth’s surface. Physically, this represents a small loop of area dA carrying a circulating current I(t) which, in general, is a function of time t. Without loss of generality, we can restrict attention to situations where I(t) = 0 for t < 0.

Assuming that the time harmonic variation is the electric and magnetic field components in the time domain may then be dervied from the scalar magnetic Hertz potential at an observation point in the air region, due to the source, and are given by [11]:

and

In the region

where

the scalar magnetic Hertz potential is given as [11]:

TD STUDY OF TRANSIENT FIELDS FOR A CIRCULAR LOOP ANTENNA

where

and

117

are Fresnel reflection coefficients at the interfaces z = 0 and z = d,

respectively, and is the Hankel function of the second kind and order zero. The integration contour in (4) now can be deformed to infinity in the lower half of the plane, but it must pass around the two branch points at and and discrete set of poles, as shown in Fig.2. It is worth noting that there is no branch point at and there is no contribution from the contour at infinity. Hence, the time-domain expression of the vertical magnetic field, for example, can be written as

where

where

is the branch-cuts contribution and is given by

and respectively, and

are the branch-cut integrals at the branch points

and

is the poles contribution, written as

where is the contribution of the pole at Physically, the two branch-line integrals account for a continuous spectrum of radiation in the homogeneous regions above the interface and deep within earth, respectively. The pole contributions, of course, can be identified with waveguide modes in the stratified region of the earth.

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In the case where both the source and receiver are located above or on the surface of the earth, and when they are separated horizontally by a distance corresponding to several wavelengths in the earth, the contributions to which depend on propagation through the lossy ground, should be negligible [12]. In this case it is sufficient to evaluate

TIME-DOMAIN SOLUTION The time-domain response of the vertical magnetic field can be obtained as the inverse Laplace transform

where in terms of the angular freqency Hence, the overall response is the sum of the time responses of the branch-cut integrals contribution and the waveguide modes contribution.

A-EVALUATION OF THE INTEGRAL ALONG THE BRANCh-CUT Taking the inverse Laplace transform of [13]

where

The impulsive time response of the vertical magnetic field is obtained as

where


The normalized time

the normalized time delay

119

(c is the velocity of light),

and the prime in (15) denotes differentiation with respect to argument. The resulting inverse Laplace transforms of (16) are standard type [14] and neglecting with respect to hence

where is the unit step function which suffers a delay caused by the time delay parameter At sufficiently late time the dominant part of the solution (15) is represented by the time function Using the same previous treatment to the corresponding impulsive time response of the horizontal magnetic field and the circum- ferential electric field leads to

where

B-Evaluation of the Integral around the Poles To evaluate the integral around the poles, we take the inverse Laplace transform of the frequency domain expressions of the waveguide modes which are given by [13]

where and the n-th order.

and

and is the modified Bessel function of the second kind are given as

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where

In taking the inverse Laplace transform of the frequency domain expressions of the waveguide modes ( [13] eqs. (29) - (31) ), a difficulty arises from the factor To overcome this difficulty, it is noted that the inverse Laplace transform of these expressions take the general following form:

Expanding this summation by Taylor power separately series about the point collecting the terms of even and odd powers of then:

and

The inverse transform in (28) can be evaluated by resorting to the complex plane, taking into consideration that where is the complex conjugate of F,which take the form

where Re and Im denote the real and Imaginary parts of the integral.


121

Finally, to obtain the time-domain expressions of the electric and magnetic field, for an impulsive current excitation, we use the identity 6.6433 in [15] and the series expansion as given by the identity 9.2371 in [15],as

where

where

and

is the Eulerpsi function [15].

NUMERICAL RESULTS A computed example of pulse response of the time rate of change of the vertical magnetic field due to a rectangular pulse is computed for different cases. The branch cut and residues contributions as well as the total response are shown in Fig.3(a-d) which are denoted by and respectively. These plots for different values of the normalized radial distance

with fixed normalized overburden height

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123

124


The conductivities ratio is taken as normalized by the factor

The vertical scale is

while the normalized time

is delayed by time

delay factor Fig.3a shows that whenever X/D has small values, the waveguide mode contribution will be only effective at relatively early times. Fig.3b shows that the increase of the separating distance X would lead to the increase of the waveguide mode contribution up till a certain limit after which the reverse would occur (Figs. 3c and 3d). In these latter figures, the oscillations manifested by the branch-cut contribution when the values of X are equal to 10 and 31.6,respectively, deserve an explanation. The observed deviations are due to the second, third and fourth terms in the square bracket of (15). These terms affect the total pulse response considerably, especially at early times, leading to the observed oscillations. It is noted that these oscillations do not accurately represent the response at early times since late time approximations were used in deriving the function However, they could be regarded, on a qualitative basis, as due to the differentiation of the imput pulse. Fig.4 shows the plot of the late time part of the solution and the effect of the earth layer’s conductivities. As the waveguide mode contribution diminshs at the late time, the branch-cut contributins are the only ones plotted for a wide range of these conductivities. The late time decay of the response is also shown to be strongly affected by these conductivities. Fig.5 shows the response on a homogeneous earth

which is also computed

and plotted. It is noteworthy that this half-space response shows two peaks with time locations which largely depends on the seperation parameter X. On the other hand, the time occurrence of the last peak in the layered earth response is dependent largely on the overburden height D. This latter observation is quite important in remote sening applications.

CONCLUSION The time-domain fields originating from a vertical magnetic dipole placed on the surface of two-layered ground have been derived in an analytical form. The displacement currents in both the two ground layers have been taken into consideration. A previously obtained approximate solution [8] thus represents the late time part of the present solution. It is noteworthy to mention that the term "late time" used in the present work is not considered independently. It depends on both the separation and depth parameters X and D. The condition defining the late time term in [eq.(44), 8] thus seems to be overrestrictive in the case of the layered ground. The pulse and step responses of the induced voltage in a horizontal receiving loop would lead to definite conclusions about the properties of the ground. These conclusions could be deduced from Figs. 3-5. Thus, both the overburden height D parameter and the conductivities ratio could be determined. Figs. 3a and 3d, for example, could be used to measure the parameter D as the time occurence of the peak in the pulse response is greatly dependent on it. Also, Fig. 4 could be used to determine the ratio from the late time decay of the pulse response.


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REFERENCES 1. J.R. Wait and C. Froese, "Reflection of a transient electromagnetic wave at a conducting surface (a half-space)," J. Geophys.Res., Vol. 60, no.l, pp. 97-103,1955. 2. S.H. Dvorak, H.Y. Pao, D.G. Gudley, and M.Sheikh, "Use of the short-space time Fourier transform in the extraction of the electrical properties for a conducting half- space," Porc. Progress in Electromagnetic, Res. Symp, Seattle, WA, July 1995, p. 99. 3. J.A. Pilon,Ed. "Ground penetrating radar," Paper 90 - 4, Geolog. Survey Canada, 1992. 4. J.R Wait, "The magnetic dipole over the horizontally stratified earth," Cand. J.Phys., Vol. 29, pp. 577-592, Nov. 1951. 5. J. R.Wait and R.H. Ott "On calculating transient electromagnetic fields of a small currentcarrying loop over homogenous earth," Rev. Pure Appl. Geophys., Vol. 95, pp. 157162, 1972. 6. J. Ryu, H.F. Morrison, and S. H. Ward, " Electromagnetic field above a loop source of current," Geophysics, Vol. 35, pp. 862-896, 1970. 7. V. K. Gaur, "Electromagnetic model experiments simulating an airborne method of prospecting," But.Nat. Geophys. Res. Inst. (India) Vol. 1, pp. 167-174, 1963. 8. A. Z. Botros, S. F. Mahmoud, "The transient fields of simple radiators from the point of view of remote sensing of the ground subsurface", Radio Sci., Vol. 13, no. 2, pp. 379-389, 1978. 9. S. F. Mahmoud, A. Z. Botros and James R. Wait, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer earth", Proceedings of the IEEE, Vol. 67, no.7, pp. 1022 - 1029, 1979. 10. K. A.Nabulsi and J. R. Wait," Ray decomposition of the pulse responses of a two-layer half-space," IEEE Trans, on Geosci. Remote Sensing, Vol.GE-35, no. 2, pp. 287292, 1997. 11. J. R. Wait, "Electromagnetic fields of sources in lossy media", in Antenna Theory, R. E. Collin and F. J. Zucker, Eds part 2, ch. 24, McGrawHill, New York, 1969. 12. J. A. Fuller and J. R. Wait," High-frequency electromagnetic coupling between small coplanar loops over an inhomogeneous ground", Geophys., vol. 37, no. 6, pp. 9971004, Dec. 1972. 13. S. T. Bishay, O. M. Abo Seida and G. M. Sami, "Transient electromagnetic fields of a vertical magnetic dipole on a two-layer conducting earth " (Sent for publication). 14. S. R. Murray," Laplace transforms", Schaum’s Outline Series, McGraw - Hill Book Company, New York, 1965. 15. I. S. Gradshteyn and I.M. Ryzhik, "Tables of Integrals, Series and Products", New York: Academic Press, 1965.


GENERALIZED TEM, E, AND H MODES Carl E. Baum

Alexander P. Stone

Air Force Research Laboratory Directed Energy Directorate

Department of Mathematics and Statistics University of New Mexico Abstract

Previous papers have considered transient lenses for propagating TEM modes without dispersion. This paper considers the properties of E and H modes in such lenses. The presence of longitudinal field components brings in additional constraints on the allowable coordinate systems, limiting the cases of transient lenses supporting E and H modes to a subset of those supporting TEM modes.

1. INTRODUCTION A technique developed by C. E. Baum1 for the design of EM lenses utilizes the expression of the constitutive parameters and and Maxwell’s equations in a general orthogonal curvilinear coordinate system, yielding what we will call the formal quantities. These are customarily denoted by affixing primes as superscripts. The line element is

and the coordinates are The scale factors, relate the formal parameters and to the real world parameters and in the diagonal case via the equations

The scale factors also relate the formal fields via the equations

and

to the real fields


and

127

128

for

C. E. BAUM AND A. P. STONE

Maxwell’s equations for the formal fields are

Here we use the two-sided Laplace-transform variable or complex frequency. This suppresses the time-derivatives for our convenience in notation, and furthermore allows the constitutive parameters to be frequency dependent if desired. This last point is significant only in the case of dispersive media, which need not concern us here. Since we are not going back and forth between the time and frequency domains, we do not need to indicate the fact that all fields are Laplace transforms (functions of complex frequency). Thus if we assume diagonal forms for the tensors and Maxwell’s equations in expanded form become

and

These equations will be the starting point in our search for conditions on the parameters and in the case of E (or TM) and H (or TE) modes. A general discussion of differential geometric lens synthesis appears in Baum and Stone2 .

2. TEM CASE (formal fields) We recapitulate the results obtained,in the formal case, for a TEM wave propagating in the coordinate direction. These results, which are consistent with examples described in earlier work 3,4 , will suggest the approach to be taken in the case of an E-wave or an H-wave. We begin with the assumption that the parameters are in the form

and

GENERALIZED TEM, E, AND H MODES

We will think of our

129

coordinates as though they are cartesian and allow

and to be inhomogeneous and anisotropic. Our TEM plane wave is to propagate in the direction and all fields will be assumed to have propagation factors which account for this. Thus if is the propagation direction, then and are irrelevant. At this point we assume a form for the fields and constitutive parameters that factors the dependence into products of the form a function of and (transverse coordinates) times a function (longitudinal or propagation coordinate). For the constitutive parameters this is (all terms real and positive)

Note that there are not separate functions of for each of the 1 and 2 components, this being an assumption of invariance to transformation (e.g., rotation) of the coordinates. We then seek TEM solutions of the form

with

Maxwell’s equations then take the form

These equations may then be rewritten as, using (9) and (10),

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We note that in the above equations we have “separability” in the sense that these equations may be reexpressed in a form where we have a function of and equal to a function of only. The immediate result is then the fact that both functions are equal to a constant (i.e., independent of the spatial coordinates). This same reasoning, it will be recalled, is used in the “separation of variables” technique in partial differential equations. Thus we define constants and from (13) by

Thus

We also note, from (14), that wave since

and

need not be mutually perpendicular, for a TEM

Instead, orthogonality holds between and and also between and a property associated with allowing the medium to be anisotropic. Whether one considers the uniform formal (prime) medium as anisotropic or isotropic, there is still the question of scaling to the real coordinates and fields. One can then again ask the question of whether the formal permittivity and permeability can be isotropic or anisotropic. In previous work2 , the assumption of isotropic media parameters led to the result that surfaces of constant could only be spheres or planes. Since we now have the possibility of anisotropic and the formal medium need not be isotropic and surfaces of constant may take more general shapes. In particular if then the permeability is both uniform and isotropic, and we have a case of practical significance 2 , that of a bending lens.

3. E-WAVE CASE (TM) We now take parameters and

in Maxwell’s equations and seek conditions on the formal which lead to solutions of the formal Maxwell equations. As usual


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in the case of waveguides we seek solutions for the formal-field components in terms of some operator on which we will later take as some mode function of and (transverse coordinates) times some propagation function of Thus if we obtain

The above equations will lead to restrictions on the formal parameters and as well as solutions for the formal fields and We assume that the formal constitutive parameters have the forms specified in (9) as

for

with

irrelevant since

is the assumed propagation direction. We take

and try solutions of the form

Now for physical realizability and are real, nonzero and frequency independent. The propagation functions and are in general complex functions of the complex frequency (exponential like) and all have derivatives nonzero except possibly at special frequencies like or degenerate cases like propagation perpendicular to (waveguide cutoff). The functions of are taken as independent of the complex frequency Thus we obtain

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and

Moreover, since

we have

and also, from

we have

Constants may be defined since we can separate variables in (21) and (22). Thus we set

Here and are included to give a dimension of inverse length (in the coordinates) to balance the derivatives. Later we will find that these are related to the propagation constant or wave number. As such and may be functions of complex frequency while is not a function of A constant velocity is also included to make the units work out. The equations in (21) can be written in a more compact two-dimensional form (transverse coordinates only for vectors/dyadics) as

where


and since there is a function

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such that

we may find an equation for

which involves only the electric field. It is

where

One is then able to obtain an equation

by a change of variable from

to

given by

with The product is an eigenvalue giving a transverse wave number which can be computed. This applies to waveguide solutions with on some closed contour in the plane. The main result is that we can have E modes in the same media as the TEM modes in Section 2, provided we have an additional constraint on the part of the permittivity. Specifically varies reciprocally with respect to for the special case of a constant. However, the medium can now be both inhomogeneous and anisotropic. The factor is specified by We note that a limiting case of an E mode is a TEM mode. As in a typical waveguide with perfectly conducting walls, the ratio of the longitudinal electric field to the transverse electric field tends to zero for a given mode. This result is derivable under the assumption that two or more independent E modes exist.

4. SCALING TO REAL MEDIUM FOR E-MODES As in the TEM case we may now consider the scaling of the coordinates to something other than cartesian coordinates (for which the results in Section 3 are directly applicable). The scaling relations are given, as in Section 1, by

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where the constitutive-parameter dyadics (matrices) are assumed diagonal in the coordinate system. The scale factors and line element are

We also have

If the real medium is constrained to be isotropic we have

so that we have five relevant constitutive-parameter components to consider, one more than in the TEM case. In particular we have

with surfaces of constant limited to spheres and planes. In the case that (uniform real permeability) we find that and so the surfaces of constant can only be planes. Such a case of a bending lens with constant surfaces is considered in Baum3 . If we have isotropic real and formal media then which gives the case which admits only 2 types of solutions (cartesian coordinates and their inversion). This is a very restrictive case and so we can allow only the formal permittivity to be anisotropic.

5. H-WAVE CASE (TE) In this section we summarize the results obtained in the H wave case. We impose the condition that the formal field component, vanishes and then look for conditions on the formal parameters, and which lead to solutions of the formal Maxwell equations. The analysis is dual to that of section 3, in which the E-wave case was studied, and the results will be dual. Duality is the symmetry on interchange of electric and magnetic parameters. Thus solutions will be sought for the formal field components in terms of an operator on which will eventually be taken as some mode function of the transverse coordinates, and multiplied by a propagation function of The assumptions made in Section 2 on the form of the formal parameters remain in effect. The parameter is irrelevant since we take as the propagation direction, while Just as in the E wave case, and are real, nonzero,


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and frequency independent. Similarly the propagation functions and are in general complex functions of the complex frequency (of exponential type). Thus by separating variables in Maxwell’s equation, a dual collection of constants arise with the interchange of media parameters and electric and magnetic field components. Analogous to (33) there is a differential equation for given by

with appropriate boundary conditions for the magnetic field. Thus we would require that the normal derivative of be zero on some closed contour in the plane. Consequently we have a waveguide problem for with assuming the role of an eigenvalue (a transverse wave number or a propagation constant). Hence we will obtain one result that H modes can propagate in the same media as TEM modes under a constraint, similar to that for E modes, on a parameter The introduction of a scaling in the coordinates then leads to results similar to those obtained in the case of E modes. For example, if the real medium is constrained to be isotropic we have and In this event and surfaces of constant are sphere or planes. In the practical case of uniform real permeability with isotropic the results obtained are

Thus in this case surfaces of constant

can only be planes.

6. CONCLUSIONS We now have a significant set of results for TEM, E, and H modes. The basic form for those is found by separating out the (propagation) coordinate from the (transverse) coordinates, and requiring propagation in one direction without reflection. Various assumptions (constraints) on the constitutive parameters lead to constraints on the allowable coordinate systems. We can note that these results apply only to such modes, and not to all possible solutions of the Maxwell equations which may include additional contributions (e.g., hybrid HE modes). A related problem is treated by Friedman6 . Those results have some similarities to and differences from the present results. There only was considered. His results are based on those of Bromwich7 . The results had the decomposition of the fields into unique E- and H-mode parts. Only cartesian and spherical coordinates were considered. There was found to be a function of only, but our present results allow for more general specifically as a function of and as well. Nevertheless, there are some remarkable similarities in the results. In particular is independent of Constraining the practical case of we all have

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with some freedom for In our case, however can be a function of and This leads to a nontrivial example of a bending lens in which surfaces of constant are nonparallel planes. The present results also allow for more general anisotropic real and/or formal media to be considered, including for the case of TEM modes. This may lead to other interesting cases for transient lens design. Note the fundamental assumption of and each having both and components. This could be relaxed by allowing the fields to have only one transverse component (e.g., and ) as in Baum and Stone5 . So there are various possible other cases that can be considered. An expanded version of this paper will appear in a forthcoming Sensor and Simulation Note. The mathematical details omitted in this paper will appear in the SSN report. This work was sponsored in part by the Air Force Office of Scientific Research, Arlington, VA.

REFERENCES 1. C. E. Baum, A Scaling Technique for the Design of Idealized Electromagnetic Lenses, Sensor and Simulation Note 64, August 1968. 2. C. E. Baum and A. P. Stone, Transient Lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor and Francis, 1991. 3. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 405, December 1996; URSI International Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. 4. C. E. Baum and A. P. Stone, Synthesis of Inhomogeneous Dielectric, Dispersionless TEM Lenses for High-Power Application, Electromagnetics, 2000, pp. 17-28. 5. C. E. Baum and A. P. Stone, Unipolarized Generalized TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Similation Note, 433, January 1999. 6. B. Friedman, Propagation in a Non-homogeneous Medium, pp. 301-309, in R. E. Langer (ed.), Electromagnetic Waves, U. Wisconsin Press, 1962. 7. T.J.I’a Bromwich, Electromagnetic Waves, Philosophical Magazine, 1919, pp. 143164.

ELECTROMAGNETIC WAVE SCATTERING BY SMOOTH IMPERFECTLY CONDUCTIVE CYLINDRICAL OBSTACLE

Yu. A. Tuchkin 1,2 1

Institute of Radiophysics and Electronics, NAS of Ukraine, Kharkov, Ukraine 2 Gebze Institute of Technology, Gebze, Turkey

INTRODUCTION New strong in mathematical sense and numerically efficient method for investigation of two-dimensional boundary value problem of electromagnetic wave diffraction by infinite and homogeneous in longitudinal direction imperfectly conductive cylinder of arbitrary smooth cross section is suggested. The imperfect conductivity is modelled by the boundary condition of the third kind, for example, by Leontovich condition with impedance, which is supposed to be an arbitrary smooth function of points of the obstacle cross section contour. The method of investigation is based on generalisation of Analytical Regularisation Method, which is developed in our previous papers (Tuchkin, 1985,1987,1997; Tuchkin, and Shestopalov, 1990; Shestopalov at al). A few qualitatively different cases are considered, which can be qualified as regular and a few types of singular perturbation of Neumann boundary value problem (BVP). As a result, the BVPs considered are equivalently reduced to a few different equation of the second kind in of type with the operator H compact in

BOUNDARY VALUE PROBLEM OF THE THIRD KIND Let us consider an infinite cylinder of arbitrary cross-section homogenous in longitudinal direction (see figure 1). This obstacle is illuminated by wave which does not depend on z – coordinate and it is necessary to find the scattering field (evident that it does not depend on z–coordinate either) as a solution of the following two–dimensional BVP of the third kind.


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where D is an open domain bounded by contour and and and are supposed to be (infinitely) smooth functions of point are normalized as

is total field, and they

Boundary condition of type (2) is used widely in diffraction theory for simulation of obstacle’s finite conductive material, small regular or irregular inhomogenities of its surface, its coating by dielectric layers of variable thickness, etc. From mathematical point of view, it is necessary to consider functions and for as uniform on S limits:

where

is unit outward normal to contour S in point

GENERALIZED POTENTIALS OF SINGLE AND DOUBLE LAYER Utilization of the theory of generalized potentials and relevant Green’s formulae technique give us the possibility to obtain following representation of scattering field:

where

and for arbitrary smooth function gle and double layer respectively:

operators

and

are generalized potential of sin-

ELECTROMAGNETIC WAVE SCATTERING BY CYLINDRICAL OBSTACLE

where

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is Green’s function of two dimensional free space:

and

(z) is zeroth-order Hankel function of the first kind. Formulae (4)-(6) enable us to obtain integral representation of functions

and

where

and the dashed operators and are direct value of and on S, and means the limiting value of the same sense as in formula (3). Using notation (5), formulae (2) can be rewritten as follows with evidently defined and known function From other side, substitution of the formulae (8) and (9) into (2) gives the following relation, with known function which incident field defines: Thus, BVP (1) and (2) is reduced to the system (11) and (12) with unknown functions and It has to be pointed out, that all integral transforms in equation (12) have singular kernels of different singularity. Operator has the most singular kernel, and the next, less singular one, has operator Kernels of operators and have the minor singularity. We suppose that some smooth parameterization of contour S is given and where and mean the derivatives of order Such a parameterization can always be constructed numerically with any necessary accuracy. DIRICHLET BOUNDARY VALUE PROBLEM We start with the simplest BVP: with Dirichlet boundary condition, when and for convenience Hence, from (11) and from (12) the unknown function satisfies the equation: where is some known function. By means of parameterization reduced to the integral equation of the following kind

with the unknown function

where

equation can be

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We denote the Fourier transform which maps arbitrary function to infinite vector-column of its Fourier coefficients by F; the inverse transform - by

We will use special infinite diagonal matrix

It can be proved that kernel in (14) is rather smooth function (after its periodical continuation on infinite plane). Thus, operator in (13) describes the most singular part of integral operator in (14). Due to this, equation (14) could be rewritten as follows:

where I is the identical matrix-operator of space and M is matrix-operator formed by Fourier coefficients of function It can be proved, that is a compact (and even Hilbert-Schmidt) operator in space That is why, one may define operators and and infinite vector-column (consequently ). Acting by operator L on left and right-hand sides of (18), following equation of the second kind in can be obtained: with compact operator

in

NEUMANN BOUNDARY VALUE PROBLEM The case of Neumann BVP is (11) immediately follows and function.

and we took for convenience From satisfies the following equation with known

It can be proved that by means of parameterization the integral-differential equation of the type

Kernel has singularity proportional to (21) can be rewritten as follows.

equation (20) can be reduced to

only. That is why, equation

It can be proved that matrix-operator has the same qualitative properties in the operator compact and even Hilbert-Schmidt operator in As above, defining operators reduces (22) to equation of the second kind in

infinite vector-column

BOUNDARY CONDITION OF THE THIRD KIND: REGULAR CASE

like

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ELECTROMAGNETIC WAVE SCATTERING BY CYLINDRICAL OBSTACLE

At first, we consider the simplest, so called regular, case:

where is some (not “very small” - see below) constant. Elimination of function from (3) by means of relation (11) gives after some transformation an equation of the same kind as equation (22), with another kernel which has the same qualitative properties as kernel That is why the considered problem can be reduced to equation just of the same qualitative kind as equation (23).

BOUNDARY CONDITION OF THE THIRD KIND: SINGULAR CASE The alternative condition to (24) is the case

when function

can be very small and even equal to zero on some part of contour S. This situation is arising, for example, in analysis of E-polarized wave diffraction by well conductive cylinder. In this case of small values of boundary condition (2) can be considered as singular perturbed Dirichlet condition: At first, we consider the special case of small values of but when condition (24) is valid for certainly small after that we will consider the general case. Elimination of function from (12) by means of (11) gives after some transformation the following integral-differential equation, with the unknown function

where is a function with the same qualitative properties as kernel in (14). Here is the differential operator of the kind below, with directly connected with function

We introduce small parameter and new normalized function scribes the “shape” of function and parameter describes its values:

where

de-

According to suppositions made above, is infinitely smooth function where is a set of infinitely differentiable functions on with coinciding derivatives of all orders in points Thus, the construction of regularizator of equation (25) can be separated, as it was proved, on two problems. The first one is constructing the resolvent operator of ordinary boundary value problem: The second problem (when the first one is solved) is analytical regularization of the equation of the kind (14) with another but qualitatively the same kernel Consequently, this problem can be solved in the same way as one for equation (14). The resolvent of equation (28) can be constructed numerically (if is not extremely small) by means of standard numerical technique. If is small enough, the resolvent can be constructed analytically on the basis of different well-known asymptotic methods. As a result, equation (25) has been equivalently reduced to the equation of the second kind in of the type with compact in operator which is uni-

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formly bounded and regular one when if The last is natural condition from point of view of possible physical applications. In addition, the corresponding perturbation theory of power series about has been constructed. More general than (24) for small is condition outside some arc (the generalization to a few such arcs is evident):

In this case the only difference is to change the boundary condition in (28) by the following ones: where and corresponds to end points of contour L, parameterized by function and it is evident that equation degenerates into for

CONCLUSION The problem of wave diffraction by impedance cylindrical smooth surface is solved. The initial boundary value problem is reduced to a few different algebraic systems in of the kind This gives relevant basis for efficient numerical algorithm construction for most part of possible physical and engineering applications. The constructed method includes the most complicated case of imperfectly but well conductive cylinder. REFERENCES Yu. A. Tuchkin, 1985, Wave scattering by unclosed cylindrical screen of arbitrary profile with Dirichlet boundary condition, Soviet Physics Doclady, 30. Yu. A. Tuchkin, 1987, Wave scattering by unclosed cylindrical screen of arbitrary profile with Neumann boundary condition, Soviet Physics Doclady, 32. Yu. A. Tuchkin, and V. P. Shestopalov, 1990, A wave diffraction by screens of finite thickness, Soviet Physics Doklady, 35. Yu. A. Tuchkin, 1997, Regularization of one class of system of integral-differential equations of mathematical physics, Doclady of The Ukrainian National Academy of Sciences, ser. A, No. 10, pp.47-51 (in Russian). V. P. Shestopalov, Yu. A. Tuchkin, A. Ye. Poyedinchuk and Yu. K. Sirenko, 1997, Novel methods for solving direct and inverse problems of diffraction theory, vol. 1: Analytical regularization of electromagnetic boundary value problems, Kharkov: Osnova, (in Russian)

A SET OF EXACT EXPLICIT SOLUTIONS IN TIME DOMAIN FOR UWB ELECTROMAGNETIC SIGNALS IN WAVEGUIDE

Oleg A. Tretyakov1,2 1

2

Department of Theoretical Radio Physics Kharkov National University Kharkov-77, 310077, The Ukraine Department of Electronics Engineering Gebze Institute of Technology 41400, P.K. 141, Gebze, Kocaeli, Turkey

INTRODUCTION: TWO APPROACHES TO TIME DOMAIN System of Maxwell’s equations may be shortly written as an abstract operator equation where M is Maxwell’s operator, – electromagnetic field sought for, – given function of impressed forces. Classical approach to study of transient fields corresponds to separation of Maxwell’s operator on two parts as where A is a remainder of M. Unbounded linear differential operator has a set of eigenfunctions that is complete in the sense of Fourier theorem. Just this set serves as a basis for development of Electromagnetics in the Frequency Domain. The solution sought may be presented as a direct Fourier transform

which means projecting of the solution onto the basis elements. Remainder A of M can supply then with a problem for the vector Fourier coefficients (with using of the inverse Fourier transform) provided that A is a linear operator as well, however. The disadvantages of the classical approach were scrutinized by Hillion recently (1993). Evolutionary approach has been developed starting from another separation of Maxwell’s operator (see our cited book). For the waveguide problem under consideration, separation of M has been made as M = W + B where W is liberated as a linear self-adjoint operator acting on waveguide transverse coordinates of the position vector at the argument of the solution sought. Operator B is a remainder of M; it may be linear or nonlinear in the general case as the Constitutive Relations involved in M dictate it. Since operator W is self-adjoint (due to involved Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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in its definition the boundary conditions over the perfectly conducting waveguide surface), it has the eigenvectors specified as the eigensolutions of the operator eigenvalue equation where S – a waveguide cross-section. Spectrum is discrete since operator W is bounded. Eigenvector set is complete and it originates a basis in a Hilbert space chosen as the space of solutions. Projection of the solution sought onto the basis elements looks as follows

where the scalar coefficients are unknowns. A problem for them can be obtained via projecting of Maxwell’s equations themselves onto the same basis elements This procedure supplies with a system of evolutionary (i.e., with time derivative) partial differential equations for Implementation of this scheme with the mathematical details needed the reader can find in our cited book. In this paper some new results will be presented which are obtained within the scope of the evolutionary approach. N.B. Equations (1) and (2) obtained within the scope of different approaches give distinct presentations for the same field sought for.

PROJECTION OF THE FIELD SOUGHT ONTO A MODAL BASIS The vector eigenvalue equation for basis elements in can be scalarized what yields well studied Dirichlet and Neumann boundary eigenvalue problems for Laplacian, supplemented with the proper normalization conditions, namely:

where L is a contour bounding the waveguide cross section S, and subscripts each put the eigenvalues in order of increasing of their values. Then formal equation (2) can be rewritten in terms of potentials and for components of electromagnetic field sought as follows

where are the free space constants. Herein the coefficients with argument are equivalent to from the formal equation (2).

EXACT EXPLICIT SOLUTIONS FOR UWB SIGNALS IN WAVEGUIDE

Terms with a fixed subscript taken from all the series over electromagnetic field of TM waveguide modes as

145

originate jointly

where subscript the same as in equation (3), identifies the mode. The fields of TE– modes reveal themselves in a similar way as

Term from equation (5) generates a specific mode with The basis resulting in the modal decompositions (4), (5) can be named as a modal basis, and the self-adjoint operator W generating as a modal operator, respectively. WAVEGUIDE EVOLUTIONARY EQUATIONS Equations for amplitudes of the modal field components have a simple form when a waveguide is filled with a lossless medium where the following constitutive relations hold

provided that and are arbitrary differentiable functions other than zero at any point in plane Projecting of Maxwell’s equations onto the modal basis yields ultimately a differential equation for the amplitudes of longitudinal mode components and direct formulae for the amplitudes of transversal fields as

where the speed of light in the free space, and the functions of impressed forces we put as zero for simplicity sake. Amplitude from the first equation (5) satisfies a pair of equations resulting in In fact, this system of evolutionary equations is exact form of Maxwell’s equations which they acquire after projecting onto the modal basis. When a waveguide is hollow, i.e., then the differential equations from (9), (10) turn into well studied Klein-Gordon equation (KGE)

where while and while In such a sense, differential equations with variable coefficients exhibited in (9), (10) is a generalization of KGE. Van Bladel (1985) was the first apparently who obtained

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the evolutionary equations for a hollow waveguide but he has made it starting from another position. Evolutionary approach admits to obtain waveguide evolutionary equations as well when the constitutive relations for lossy nonlinear medium hold. However, analysis of this situation needs much more place, therefore it will be considered elsewhere.

SEPARATION OF VARIABLES IN KGE AND ANALYTICAL RESULTS Separation of variables in KGE is evident: it results in monochromatic waveguide waves. A question arise: is it possible to introduce some new variables in such a way which allows to obtain a solution of equation (11) in the form of product as

Direct substitution of

in equation (11) yields

It is clear that substitutions must be very special to give a solution in the form (12). Fortunately, this problem can be solved successfully using study of KGE by means of group theory methods made by Miller (1977). These results will be used below successively. Variables must be real. That is why a pair of substitutions taken separately is unable to cover all the plane completely Therefore an additional dual pair of substitutions is necessary (see Cases 1 and 2 and the rest below). Case 1. These substitutions turn partial differential equation (13) into a pair of standard equations

here and henceforth is a constant (possibly complex-valued) of separation of variables Solution of equation (13) obtained in the form (12) looks as

where are linearly independent Bessel functions, constants; the constant plays role of a free parameter. Example: excitation of a waveguide when of time. Let us first put for and for We may choose then as a set of integers: then with these constants is Bessel function

– arbitrary is a given function

When is complete


and it may be used for Neumann’s expansion of follows

147

in a series of Bessel functions as

Coefficients can be calculated easily. For example, when is Heavyside step function then For arbitrary function solution of starting KGE (11) can be written as

where coefficients

are the same as in equation (16). In fact, set of functions

plays role of an evolutionary basis here. Some comments may be relevant. Solution (17) is available for amplitudes of longitudinal modal components and both. However, one should take for and for Condition means that either or is given as an arbitrary function of time at the waveguide cross section Since formula for the longitudinal modal amplitudes is obtained as the highly convergent series (17), the amplitudes of transverse modal components can be calculated analytically by differentiations given in (6), (7). Substitutions considered above admit to cover two quadrants only in plane where inequality holds. To cover two other quadrants where inverse inequality holds the following dual substitutions are necessary. Case 2. In terms of these variables, equation (13) has a solution as

where – arbitrary constants and plays role of a free parameter as before; – modified Bessel functions. This solution is convenient for solving of Cauchy problem when function is given at initial moment of time as a function of coordinate This problem has been studied by Kristensson (1995) in another manner. Case 3: In terms of these variables, equation (13) supplies a pair of Bessel equations in the following forms:

which have as their solutions

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with arguments

Case 4: In this case

where

Possibly, a pair of similar substitutions be found useful in some situations. Case 5: when

where

Case 6: when

where


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Case 7: This substitutions convert equation (13) to a pair of equations for parabolic cylinder functions as

where

There is an interesting particular case when the parameter where Then

where

take on the values

– Hermite orthogonal polynomials.

Case 8: In this case we have the other forms of equations for parabolic cylinder functions as

where

Case 9: These substitutions give equations for Airy functions

with arguments

Case 10: We have also equations for Airy functions

but their arguments are distinct from previous

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Case 11: These give equation for Mathieu periodic functions

where

and

Next and the last four substitutions yield different versions of modified Mathieu equations.

Case 12:

where

Case 13:

Case 14:


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Case 15:

Waveguide Modes in a Time-Variant Medium Let the same waveguide be filled with a time variant but spatially homogeneous medium having where are constants and is an arbitrary integrable function. Equations (9) and (10) can be rewritten as

where while and while Its solution is evident: it is a sum of products of the amplitude factors proportional to and ”phase” factors where a constant of separation of variables and When is real, the waveguide modes have instant frequencies depending on time as

In regard to KGE obtained with variables, the same procedure of introducing and successive separation of and variables can be repeated. It supplies a new set of the Cases 1 - 1 5 regarding to the waveguide modes in the time-variant medium.

CONCLUSION Main results may be listed as follows. 1. Evolutionary Approach to Electromagnetics has been proposed for waveguide theory in Time Domain as an alternative to the classical Method of Complex Amplitudes. 2. The Approach opens a prospect for partial evolutionary (i.e. with time derivative) differential equations as a powerful tool for development of Electromagnetics in Time Domain. 3. A set of analytical results valid for hollow waveguides and for waveguides filled with a special time-variant medium as well are obtained via the Approach proposed.

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Acknowledgments The author gratefully acknowledge the support of The Royal Swedish Academy of Sciences by dint of the Contract between Sweden and the former Soviet Union, and the personal support of Swedish Professors S. Ström and G. Kristensson.

REFERENCES Hillion, P., 1993, Some comments on electromagnetic signals; in Essays on the Formal Aspects of Electromagnetic Theory, ed. A. Lakhtakia World Scientific Publ. Co., Singapore. Tretyakov O. A., 1993, Essentials of nonstationary and nonlinear electromagnetic field theory; in Analytical and Numerical Methods in Electromagnetic Wave Theory, ed. by M. Hashimoto, M. Idemen, and O. A. Tretyakov, Science House Co., Tokyo. Van Bladel, J., 1985, Electromagnetic Fields, Hemisphere, Washington. Miller, W., Jr., 1977, Symmetry and Separation of Variables Addison-Wesley Publ. Co., Massachusetts. Kristensson G., 1995, Transient electromagnetic wave propagation in waveguides J. Electromag. Waves Applic., 5/6:645.

ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION BY BOWL-SHAPED SCREEN OF REVOLUTION

Yu. A. Tuchkin1,2 1

Institute of Radiophysics and Electronics, NAS of Ukraine, Kharkov, Ukraine 2 Gebze Institute of Technology, Gebze, Turkey

INTRODUCTION New mathematically strong method for solving of the boundary value problem (BVP) of wave diffraction by axially symmetrical bowl-shaped screen of arbitrary profile is suggested. The approach used is based on generalization of Analytical Regularization Method developed in our previous publications (Tuchkin, 1985,1987,1997; Tuchkin, and Shestopalov, 1990; Shestopalov at al, 1997) and on ideas, presented in (Vinogradov at al, 1978,1980,1981), which are devoted to solving the problem of wave diffraction by arbitrary shaped unclosed or closed cylindrical screens. The approach used is based on Green’s formulae technique and reduction of corresponding integral or integral differential equations to the equivalent dual series equations. The regularization procedure for such dual series equations is described in our previous papers. As a result initial Dirichlet and Neumann BVP ‘s are reduced to equivalent infinite algebraic systems of the second kind: with the operator H compact in the space of square summable sequences. DIRICHLET PROBLEM Let us consider bowl-shaped screen of revolution, which is formed by rotation of plane unclosed (or closed) contour L (see Fig. 1). Without loss of generality we suppose that L is a part of smooth non-self-crossing contour S which has ends placed on z-axis. Additionally we suppose, that some smooth parameterization is given, where and are corresponding cylindrical coordinates of point which corresponds to the parameter value and


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Let us consider Dirichlet diffraction problem: it is necessary to find scattering field which satisfies homogenous Helmholtz equation and Dirichlet boundary condition as follows:

where is given incident field which is supposed to be smooth in some vicinity of surface. As well known, the problem considered can be reduced to the integral equation of the first kind

where space:

is unknown function (two side current density) and G is Green function of free

Due to axial symmetry of the screen, the integral equation (5) can be separated into an infinite set of integral equations over contour L, where each equation corresponds to value m=0,+l,±2,±3,... of azimuth index. By means of parameterization each equation can be reduced to the following equation:

with unknown function

where point

is the end-point of contour L and,

ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION

In particular, kernel

where,

155

for spherical cap is the following:

are orthonormal associative Legendre functions (chosen according to the

rule Let us define new known and unknown functions as follows:

Substituting expressions (11) and (12) into equation (7), one arrives to the following integral equation (m=0,±l,±2,±3,...),

with new unknown function Equation (13) is the subject of Analytical Regularization Method application. The first step of it is reducing equation (13) to dual series equations of special kind. With such a purpose, we define, at first, new function,

that allows us to receive from (13) the relation of the kind,

Let us represent

and

by means of their Fourier-Legendre series:

Substituting the formulae (16)-(18) into equation (13) and using the orthonormal property of functions one obtains the first series equation:

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where

Y. A. TUCHKIN

is matrix-operator formed by matrix elements

are infinite

vector-columns, is n-th component of vector-column The second series equation comes from formula (16) and the second equality in formula (14):

According to standard terminology, equations (19), (20) are dual series equations involving function where are orthonormal Jacoby polynomials. Dual series equations of this type were investigated in our previous papers (Vinogradov at al, 1978,1980,1981; Shestopalov at al, 1997). The relevant regularization procedure reduces equations (19), (20) to an infinite algebraic system of the second kind: where is compact operator in space The compactness of the operator

follows from sufficiently fast decreasing of coefficients

when p,

The last inequality is the result of proper change (11) of unknown function and proper separation of function from the kernel of integral operator in formula (13). NEUMANN BOUNDARY VALUE PROBLEM Let us consider the same diffraction BVP with the exception that condition (4) is changed by Neumann boundary condition:

It can be proved that in the case considered, one obtains integral-differential equation of the following new type:

instead of equation (13). Here is unknown function, function is defined by formula (10) and is some function, which is less singular in comparison with where is the following differential operator (“main” part of Legendre differential operator):

As well known,

ANALYTICAL REGULARIZATION METHOD FOR WAVE DIFFRACTION

157

Using the same technique as for equation (13) and after taking relation (25) into account, one obtains dual series equation approximately of the same kind as equations (19), (20). Consequently, these new dual series equations can be regularized by means of the same technique (Vinogradov at al, 1978,1980,1981; Shestopalov at al, 1997). As a result of this regularization, Neumann BVP is reduced to an infinite linear algebraic system of the second kind: where is compact operator in space

CONCLUSION Analytical Regularization Method is generalized on the case of three dimensional diffraction BVP for bowl-shaped screen of revolution. Both Dirichlet and Neumann BVP are solved, i.e. they are reduced to corresponding infinite linear algebraic systems of the second kind in space with compact operator in space H. These systems can be used for construction of numerically efficient algorithms that gives solutions with arbitrary necessary accuracy REFERENCES S. S. Vinogradov, Yu. A. Tuchkin and V. P. Shestopalov, 1978, The effective solving of dual series equation involving associative Legandre functions, Doklady AN SSSR, , v.242, n.1, pp. 80-83. S. S. Vinogradov, Yu. A. Tuchkin, V. P. Shestopalov, 1980, Investigation of dual series equations involving Jacoby polynomials, Doklady AN SSSR, v.253, n.1. S. S. Vinogradov, Yu. A. Tuchkin, and V. P. Shestopalov. 1981, On the theory of wave scattering by unclosed spherical screens, Doklady AN SSSR, v.256, n.6, pp. 13461350. Yu. A. Tuchkin, 1985, Wave scattering by unclosed cylindrical screen of arbitrary profile with Dirichlet boundary condition, Soviet Physics Doklady, 30. Yu. A. Tuchkin, 1987, Wave scattering by unclosed cylindrical screen of arbitrary profile with Neumann boundary condition, Soviet Physics Doklady, 32. Yu. A. Tuchkin, 1997, Regularization of one class of system of integral-differential equations of mathematical physics, Doclady of The Ukrainian National Academy of Sciences, ser. A, No. 10, pp.47-51 (in Russian). V. P. Shestopalov, Yu. A. Tuchkin, A. Ye. Poyedinchuk and Yu. K. Sirenko, 1997, Novel methods for solving direct and inverse problems of diffraction theory, vol. 1: Analytical regularization of electromagnetic boundary value problems, Kharkov: Osnova, (in Russian)


TRANSIENT EXCITATION OF A LAYERED DIELECTRIC MEDIUM BY A PULSED ELECTRIC DIPOLE: SPECTRAL CONSTITUENTS

Anton G. Tijhuis1 and Amelia Rubio Bretones2 1

Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands 2 Departamento de Electromagnetismo, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

INTRODUCTION Spectral methods are the obvious choice for modeling the transient excitation of a continuously layered, plane-stratified dielectric half space. To arrive at an efficient numerical implementation, we consider the evaluation of the spectral constituents, which are governed by two sets of coupled transmission-line equations for the E and H modes. These equations are solved for a fixed space discretization, where the number of subintervals across a slab embedded in between two homogeneous half spaces does not depend on the value of the spectral parameters. The relative error in the constituents increases with increasing frequency and spectral wave number, but this does not affect the accuracy of the obtained space-time results. This general idea was first proposed by Tijhuis et al. (1989), and has since then been applied successfully to a large number of scattering configurations. In this paper, three different schemes are presented to compute the spectral fields. Applying central differences to the transmission-line equations on a staggered grid and eliminating one of the discretized unknowns results in a three-point rule in the interior of the dielectric slab. Boundary conditions at the interfaces are approximated by single-sided three-point rules. Second, we use reciprocity to derive two coupled integral equations and a repeated trapezoidal rule to discretize them. Third, we derive a second-order differential equation for a single unknown. From this equation, we derive a contrast-source integral equation, which is discretized by a combination of central differences and a repeated trapezoidal rule. In all three cases, the discretized equation(s) can be solved by a fully recursive procedure with a computational effort that is proportional to the number of subintervals across the slab. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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FORMULATION OF THE PROBLEM We consider a horizontally stratified medium of thickness embedded in between two homogeneous half spaces. The location of the half spaces and the value of the corresponding constitutive parameters are specified in Figure 1. In all cases, the constitutive parameters may be complex and frequency-dependent.

Transmission Line Equations For this configuration, the spectral electric fields can be decomposed in their transverse and longitudinal parts according to

where the dependence on the spectral parameters and is assumed implicitly. Next, the transverse field components are expressed in components parallel and orthogonal to the direction

and are the spectral amplitudes for E or TM and H or TE modes, respectively. Substituting the definitions (2a) and (2b) into the longitudinal part of Maxwell’s equations directly results in the identification

where we have restricted ourselves to a source-free region. The spectral amplitudes satisfy two coupled transmission-line equations of the form

where when

and where The spectral impedance are defined as and

with with and the corresponding admittance

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Exterior Half Spaces and Boundary Conditions In view of the linearity of the problem, we restrict the analysis to the excitation by a properly normalized plane wave incident from In we only have a wave that travels and/or is attenuated in the positive Dropping the superscript and taking as the fundamental unknown, we can express the behavior in the exterior half-spaces as

where depends on the current density in shown in Figure 1. For we have from (4a)

and where

as

From (5) and (6) we can derive the following boundary conditions for which are consistent with the interpretation of Y as a spectral admittance. Since is continuous in (7) can also be used to derive boundary conditions for

The aim of the computation is to solve the system of differential equations (4a,b) subject to the boundary conditions (7) or (8a,b).

METHODS OF SOLUTION For a general stratification, the system of first-order differential equations (4a,b) must be solved numerically. Perhaps the most accurate way to achieve this is to use a Runge-Kutta type scheme with an adaptive step (see e.g. Tijhuis (1987), Section 2.4.2). This leads to a fixed relative error in the obtained result at the cost of an increasing computational effort for increasing values of the spectral parameters and Alternatively, we can use the decrease in the spectrum of the generating current or incident field to compensate an increasing relative error. This means that we can choose a fixed space discretization, independently of and and still arrive at a reliable space-time result. In this category, perhaps the most straightforward choice is to approximate the continuously layered medium by a stack of M homogeneous layers of thickness and use a transfer or scattering matrix formalism (Lekner and Dorf (1987), Ko and Sambles (1988)). This procedure, however, has the disadvantage that the accuracy of the solution is at best of i.e., the results are at best first-order accurate. In this section, we review three methods that are second-order accurate, while the computation time remains of which is the best we can hope to achieve for computing field values.

Finite Difference Spectral Domain In the first approach, we use finite differences to discretize the differential equations

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(4a,b). Considering again as the fundamental unknown, we sample V at with and I at with and we approximate derivatives by central differences. Eliminating the sampled current values then results in the three-term recurrence relation

which is valid for and where a subscript refers to sampling at The difference equations for M follow from the boundary conditions (8a,b). For the evaluation of the derivative, we use a three-term forward difference rule at and a three-term backward difference rule at This results in:

To solve this system of equations, we first remove the element with from the last equation, with the aid of (9) with We then obtain the matrix equation

where for and where A is the right-hand side of (10a). In (11), is an element of the diagonal, and and are elements of the first sub- and superdiagonals. denotes element of row 0 after row operations. These operations use rows 1, 2 , . . . , M to successively remove the leftmost nonvanishing elements of row 0. Since the right-hand sides of (9) and (10b) are zero, the right-hand side of (11) is not affected. The necessary computations can be summarized in terms of a recursive procedure for these elements:

This procedure leads to a matrix equation, where the only nonvanishing elements of the system matrix are located on the diagonal and the first two superdiagionals. From this equation, the sampled values are readily determined by back substitution. That procedure can be written as

The corresponding values for may be obtained from the difference equation corresponding to (4a). At a first glance, it would appear that such an operation enhances the numerical error in the computed values However, (9) was obtained


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by eliminating the and is therefore the exact solution of the discretized firstorder differential equations. As a consequence, the error in the numerical solution of this system of equations will be comparable for both quantities. Finally, it should be observed that the procedures given by (12) and (13) both require M + 1 steps, which confirms the estimate of the computational effort given above. Two Coupled Integral Equations The FDSD approach described in the previous subsection has two disadvantages. First, the differential equations (4a,b) are formally not valid when or has a discontinuity. Second, the boundary conditions (8a,b) are only taken into account in an approximate form. Both problems can be circumvented by deriving two equivalent integral equations. To achieve this, we include source terms and in the righthand sides of (4a) and (4b). For any two solutions of the resulting inhomogeneous equations, referred to as states A and B, we have the following reciprocity relation:

To arrive at the desired integral equations, we make the following choices. State A is the actual state with sources in and material parameters as specified in Figure 1. State B is an auxiliary state with and or and Configuration B is an infinite medium with and for and satisfy the radiation condition as Finally, we integrate (14) over and interchange and For the case with and we then arrive at the integral equation

where the boundary conditions (7) have been used to eliminate I(0) and For and we find a similar equation for In the space discretization, we introduce a spatial grid with with and and we enforce both equations at the grid points. A repeated trapezoidal rule then results in a discretized equation of the form

In (16),

and

are 2 × 2 matrices, is the solution vector, accounts for the excitation in and for the propagation in the reference medium. To solve this system of equations, we perform three row operations in closed form.

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We end up with a matrix equation that closely resembles the one in (11). The difference is that the elements of row 0 and and of rows are now as such 2 × 2 matrices. The tridiagonal elements and can be expressed in terms of the matrices in the discretized integral equation (16):

The method of solution follows the same lines as the one specified in (12) and (13) and can be formulated as

and

In (18) and (19), the “multiplications” and “divisions” are matrix operations that must be carried out exactly in the order as indicated. Each step involves the inversion of a 2 × 2 matrix as well as a number of matrix-vector multiplications. Second-Order Differential Equation and Associated Integral Equation Finally, we present an integral-equation scheme that combines the efficiency of the FDSD with the capability of handling the boundary conditions (8a,b) in closed form. For both types of modes, we can choose either or as the fundamental unknown. However, the computation becomes more efficient when we choose the amplitude that gives rise to a longitudinal field component. We thus obtain a second-order differential equation of the generic form

where either and or and For this choice of the fundamental unknown, the boundary conditions (8a,b) for and the corresponding conditions for can be written as

As in the previous section, Green’s function is introduced as the solution of the secondorder differential equation

that satisfies the radiation condition as In (22), are the constitutive coefficients of an arbitrary homogeneous reference medium. Using Green’s second


theorem over and substituting the boundary conditions (21a,b) then leads to the integral equation:

Like (15), (23) expresses the unknown amplitude in in terms of plane waves that propagate in the infinite reference medium. The space discretization proceeds along the same lines as in the case of the two coupled integral equations. We introduce a spacial grid with approximate the integrals by a repeated trapezoidal rule, express at in terms of and evaluate at interior points by central differences. This leads to a discretized equation of the form:

where Carrying out the same row operations for (24) as were carried out for (16) results again in a tridiagional system matrix for The equation thus obtained can be solved as described above, but with the difference that all multiplications and divisions are now scalar operations. Special care is needed when the constitutive parameter is discontinuous in and the evaluation of the remaining spectral amplitudes and is no longer straightforward.

A SIMPLE EXAMPLE The algorithms outlined in the previous section have been validated for various choices of and Because of space limitations, we restrict ourselves here to an incident plane wave with To provide a true test for the algorithm based on (23), we must have a configuration where Therefore, we cannot consider the customary example of a homogeneous slab. Instead, we consider a reflectionless configuration with and For the exterior medium we choose free space. The algorithm based on the two coupled integral equations (18) and (19) gives an almost vanishing for any value of M. This can be explained from the property that I, V and are treated in a completely symmetrical manner. Therefore, we consider for which corresponds to nine complete oscillations in and hence to In Table 1, we present for an increasing value of M for each of the three algorithms. It is observed that the error is indeed of for increasing M. For this particular configuration, all three methods deliver a comparable accuracy. However, this is not true in all situations. For the configuration described above, the accuracy test is more severe for the single integral equation, where and are treated in a different manner, while their effects must cancel out. Further, the advantage of having exact boundary conditions becomes more important for increasing Another

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criterion is the flexibility of the algorithms. The coupled integral equations are better in this respect, but their solution requires more computational effort.

CONCLUSIONS In this paper, we have presented three algorithms for the efficient evaluation of spectral fields in continuously inhomogeneous, plane-stratified media. These algorithms have in common that field values are obtained in steps with an accuracy of i.e., doubling the computational effort leads to a four times better result. The effectivity of the approach was demonstrated for a simple test geometry. Possible applications include the analysis of buried objects, antenna arrays, frequency-selective surfaces and feed networks. The algorithms presented in this paper have already been used to analyze the excitation of a plane-stratified medium by a pulsed dipole, one or two straight thin-wire antennas and a circular loop antenna, and the design of an airgap in a rectangular waveguide. In all cases, the spectral representation as such remains to be developed. One example is the excitation of a plane-stratified medium by a pulsed dipole, which is discussed in a companion paper.

REFERENCES Lekner, J. and Dorf, M., 1987, Matrix methods for the calculation of reflection amplitudes, J. Optical Society of America A, 4: 2092. Ko, D. and Sambles, J., 1988, Scattering matrix method for propagation in stratified media: attenuated total reflection studies of liquid crystals, J. Optical Society of America A, 5: 1863. Tijhuis, A.G., 1987, Electromagnetic Inverse Profiling: Theory and Numerical Implementation, VNU Science Press, Utrecht, the Netherlands. Tijhuis, A.G, Wiemans, R., and Kuester, E.F., 1989, A hybrid method for solving time-domain integral equations in transient scattering, J. Electromag. Waves Applic., 3:485.

TRANSIENT EXCITATION OF A LAYERED DIELECTRIC MEDIUM BY A PULSED ELECTRIC DIPOLE: SPECTRAL REPRESENTATION

Anton G. Tijhuis1 and Amelia Rubio Bretones2 1

2

Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands Departamento de Electromagnetismo, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

INTRODUCTION Spectral methods are the obvious choice for modeling the transient excitation of a continuously layered, plane-stratified dielectric halfspace. Such methods typically involve an inverse spatial Fourier transformation and the evaluation of the constituents. In this paper, we consider the spectral representation. The idea is to normalize the spatial wavenumber with respect to frequency. Compared with the Cagniard-De Hoop method, our approach is different in the sense that we keep the frequency real, and allow the time variable to become complex. In this respect, our work also resembles the spectral theory of transients. We restrict the temporal Fourier inversion to nonnegative frequencies by expressing the time-domain signal as the real part of a dual analytic signal. Reversing the order of the temporal and spatial Fourier inversions then leads to the so-called time-domain Weyl representation for the reflected field. In this representation, accumulated guidedwave poles give rise to an additional branch cut. The representation thus obtained is used to derive a suitable combination of Gaussian quadrature rules for the evaluation of the spectral integral. FORMULATION OF THE PROBLEM We consider a configuration that consists of an isotropic, linearly and instantaneously reacting, horizontally stratified, lossy dielectric medium embedded between two homogeneous, lossless dielectric half-spaces and (Figure 1). The slab is loUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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cated in the domain The configuration is driven by a pulsed electric dipole whose current density is given by

with a unit vector, and a causal time signal. We are interested in the electromagnetic field in the upper half space since that is the field that can be detected by an antenna. Once this field is known, we can apply the superposition principle to determine the effect of a more general current distribution. For this configuration, Maxwell’s equations can be simplified to

Spectral Fields To exploit the fact that the constitutive parameters in (2a,b) depend only on the we solve these equations in the spectral domain. To this end, we introduce the following temporal and spatial Fourier transformations:

where and are vectors in the transverse plane. We decompose the electromagnetic fields into their transverse and longitudinal parts, according to

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and express both parts in terms of spectral amplitudes We arrive at

and

where The spectral amplitudes satisfy the transmission-line equations (Felsen and Marcuvitz (1994))

and a similar system of equations for and where with and when The solutions determined by and and and are indicated as E or TM and H or TE modes, respectively. The systems of first-order differential equations can be reduced to a single secondorder differential equation of the form

where only the dependence on has been indicated explicitly. In (7), is a constitutive parameter, is the unknown function and is the forcing function. In both cases, we choose the quantity that corresponds to a longitudinal flux density as the fundamental unknown. For a vertical dipole, we have For a horizontal dipole pointing in the

we find

where is the derivative of the delta function. For this excitation, the differential equation (7) must be solved twice for each combination of and

LONGITUDINAL BEHAVIOR The first step towards solving the problem formulated above is solving the differential equation (7) for the forcing functions given in (8) and (9a,b) for a set of parameters that allows the evaluation of the integrals in the inverse transformations of the ones given in (3a) and (3b). In this section, we address some aspects of this solution. We restrict ourselves to the longitudinal electric field caused by a vertical dipole, but the procedure runs along the same lines for the remaining field components.

Exterior Half Spaces In the dielectric medium is homogeneous and lossless. For the forcing functions specified above, we can therefore find a closed-form solution of the differential

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equation(7). For the vertical dipole, we have

where is an unknown reflection coefficient. Since (7) is linear, we may restrict the evaluation of the longitudinal behavior to determining the response to a unit-amplitude plane wave. The multiplicative factors in (10) can then be included in the evaluation of the spectral integrals. We therefore consider the normalized solution which, in the homogeneous half-spaces, .behaves as

where and where Obviously, the direct field for must also be adapted. However, for the homogeneous upper half-space considered in this paper, this field is available in closed form. Asymptotic Solution To obtain for we need to solve the system of differential equations (6a,b) or the second-order differential equation (7) numerically. Three possible methods are discussed in a companion paper. For large values of the parameters and/or a first-order asymptotic solution can be derived. By scaling the spatial wave vector according to

where is the speed of light in free space, we can handle both limits simultaneously. We specify the results for i.e. for E-modes. Equation (7) can be written as

where the prime denotes differentiation, where and where is the scaled counterpart of The logarithmic derivative of in (13) is expanded in a geometrical series in powers of and we derive a first-order WKB approximation. Following Erdélyi (1956), we introduce two linearly independent solutions that correspond with

where is the free-space wave impedance. In (14) the superscript ± indicates the direction of propagation, and the wave originates from the point With this result, we can formally write:

The coefficients at

and and

are found by enforcing the continuity of and For the reflection coefficient for example, we


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then obtain the following first-order approximation:

where

and

are asymptotic reflection and transmission coefficients.

Singularities in the Complex As a function of complex the spectral constituent has two types of singularities. In the first place, there are two branch cuts in the upper half of the complex with branch points at and These branch cuts are associated with the choice of the “physical” root in the attenuation coefficients and which occur in the radiation conditions in and In the transform domain, the problem is completely defined by these conditions and the second-order differential equation (7) in the interval This differential equation only contains a term therefore no extra branch cuts are introduced for the interior of In the second place, we need to consider the occurrence of so-called guided-wave poles. For such poles only occur in the interval

where is the local refractive index in provided that this interval exists. Each pole corresponds to a homogeneous solution of Maxwell’s equations that propagates in the transverse direction. The location of the poles, and their number, depends on For the poles occur in the first quadrant of the complex plane, and approach the same interval as Now, the integration contour may be considered as running just below the real Therefore, numerical problems will occur for in that interval, since neither of the equations (6) or (7) has a unique solution for values of that correspond to guided-wave poles. TRANSFORMATION TO THE SPACE-TIME DOMAIN Once has been computed, the longitudinal electric field can be calculated by carrying out the inverse Fourier transformations corresponding to (3a) and (3b). We restrict the discussion to the reflected field in but the ideas presented in this section are also directly applicable to the total fields in and In the transform domain, the longitudinal reflected-field component is given by.

where is the reflection coefficient introduced in (10). To speed up the calculations, we cast the Fourier inversion into a special form that was also used in the half-space configurations considered in Rubio Bretones and Tijhuis (1995, 1997). Weyl Representations First, we restrict the temporal Fourier inversion to nonnegative frequencies by expressing the time-domain signal as

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with Here, represents the dual analytic signal corresponding to This signal is an analytic function in the lower half of the complex and its real part approaches when The restriction to allows us to use the normalized spatial transform vector introduced in (12). Further, since only depends on we also change over to the cylindrical coordinates The spatial Fourier inversion corresponding to (3b) then assumes the form:

where are cylindrical coordinates in actual space. Combining (18), (20) and (21) and reversing the order of the temporal and spatial Fourier inversions then leads to the so-called frequency-domain Weyl representation for the reflected field:

where the complex time delay is given by

and where Because the cosine in (21) is periodical, the variable in the angular integration has been changed to and does not depend on Apart from a constant amplitude and a factor of the term in braces in (22) is of the same form as the right-hand side in (20), and the complex time argument cannot have a positive imaginary part. Therefore, this term can be identified as the time derivative of a dual analytic signal. Contour Deformation and Quadrature This observation is used to derive a suitable combination of Gaussian quadrature rules for the evaluation of the integral over To achieve this, we analyze the situation in the complex We first consider the dual analytic signal

for a fixed with the reflection coefficient

The singularities in the integrand in (24) originate from As remarked above, we have two types.

First, there are two branch cuts in the complex-wavenumber plane. After the normalization of carried out above, these singularities show up as the branch cuts in the normalized attenuation coefficients and These branch cuts are present for all frequencies and, hence, will also be observed in


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Second, there are guided-wave poles. For these poles are located on the interval specified in (17), provided that this interval exists. The integration over in (24) reduces the influence of the poles to an extra jump discontinuity along this interval. For each point in the subinterval is a condensation point for guided-wave poles at different values of Therefore, the behavior along the real will be similar to that in the lossless case. The definition of was chosen such that the integration contour may be regarded as running along the real axis in the fourth quadrant of the complex Therefore, the situation in the complex may be envisaged as indicated in the left half of Figure 2, where it is assumed that To ensure that the transmission-line equations have a unique solution, we deform the contour into the lower half of the complex as shown in the right half of Figure 2. We choose large enough to avoid possible problems in the numerical solution of the transmissionline equations and in the numerical integration along the interval between and To investigate to what extent this is allowed, we write the frequency-domain Weyl representation (22) in terms of the dual analytic signal introduced in (24). This leads to the so-called time-domain Weyl representation for the of the electric field in the case of the vertical dipole:

Since all singularities in the integrand in the right half of the complex are located on or above the real no extra contributions are encountered in this deformation. However, the definition (24) of may only be applied for To satisfy this condition, we take into account the complex time delay in (25). With the aid of the asymptotic expansions it follows that, for a suitable definition of in (1), the imaginary part of the time argument in (25) must be nonpositive for all and This implies that the limiting contour is given by where is a real-valued length parameter. Solution of this equation leads to the Cagniard contour for a point on the interface at with respect to the source point at The representation (25) allows the derivation of a composite Gaussian quadrature rule that is valid for all and and for smaller than a given maximum offset. For ex-

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ample, in the semi-infinite subinterval we substitute and use the asymptotic approximation and the large-argument behavior of analytic signals to derive that the integrand of (25) decays as independently of the values of and This allows the application of a fixed Gauss-Laguerre quadrature rule. Once the quadrature rule is derived, the integral over in (22) is replaced by the discretized form derived for (25). Next, the integral over is expressed as a linear combination of the Bessel functions with Finally, the integral over is truncated, discretized with the aid of a repeated trapezoidal rule, and cast into the form of an FFT operation.

AN EXAMPLE Numerical results were obtained for a variety of configurations. We restrict ourselves to an example for a Gaussian pulsed horizontal dipole located above an inhomogenous slab in free space. In Figure 3, we show the time signature of the of the reflected electric field for two inhomogeneous, nonmagnetic slabs with the same relative permittivity profile with and The dipole is located at a distance above either of the slabs and the observation point is (1, 0, –0.5)m. As expected, the two signals are identical until the arrival of the fields reflected at and beyond.

REFERENCES Erdélyi, A., 1956, Asymptotic Expansions, Dover Publications, New York, Chapter 4. Felsen, L.B. and Marcuvitz, N., 1994, Radiation and Scattering of Waves, Second Edition, IEEE Press, New York, Section 2.2. Rubio Bretones, A. and Tijhuis, A.G., 1995, Transient excitation of a straight thin wire segment over an interface between two dielectric half spaces, Radio Science, 30:1723. Rubio Bretones, A. and Tijhuis, A.G., 1997, Transient excitation of two coupled wires over an interface between two dielectric half spaces, Radio Science, 32:25.

A NEW ULTRA WIDEBAND, SHORT PULSE, RADAR SYSTEM FOR MINE DETECTION

F. Gallais1, V. Mallepeyre1,Y. Imbs1, B. Beillard1, J. Andrieu1, B. Jecko1 M. Le Goff2 1

Institut de Recherche en Communications Optiques et Microondes IRCOM - 7 rue Jules Vallès -19100 Brive la gaillarde (France) E-mail: [email protected] 2 CELAR (DGA) GEOS/SDM - BP 7419 - 35174 Bruz Cedex (France) E-mail: [email protected]

INTRODUCTION An experimentation is described for measurement of UWB transient scattered responses from different targets. The measurements are performed using a new UWB Synthetic Aperture Radar (SAR) for the detection of placed atop soil targets at first, and then, buried in soil targets. The aim is to use lower frequencies for penetrating foliage, vegetation, soil and ultra wide band for high resolution SAR image. This study comes after a precedent work which was the realization of a low frequency Radar Cross Section (RCS) measurement facility in the time domain1. This facility is funded by the DGA to meet CELAR requirement2. First results will be presented about lied on soil targets.

MEASUREMENT FACILITY The measurement configuration is an UWB (100 MHz - 1 GHz) transmission and reception system implanted on a mobile boom. This boom which can reach about ten meters high, is installed on a truck and moves sideways along the test area. The system is presented on figure 1 and figure 2. The general synchronisation is carried out by a sequencer. This device is remote by a coder wheel behind the truck which defines measurement step in azimuth along the moving direction. A receiving system between receiving antenna and the digital sampling oscilloscope, is needed to protect the oscilloscope input from high level tension and to optimise the measurement dynamic. The effective signal is insulated by time windows. The receiving Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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system gain is programmable from -20 dB to 40 dB in 1 dB step. This receiving system has got a limited frequency bandwidth. As a matter of fact, its bandwidth is 700 MHz at - 3 dB but it will be upgraded with better components. The oscilloscope must acquire measurements during the moving with the best bandwidth in monopulse mode. It's the Lecroy LC 584 which was chosen for his adequate data transfer rate (GBIP): until 1 Mpoints at 280 Ko/sec. But its bandwidth is DC - 1 GHz (risetime ~ 350 ps) with a 8 Gsample/s rate and a 8 bit dynamic. A tachometer is installed on the area test to provide the position of a near antenna fixed theodolite. This device is a Total Positioning System from Leica Geosystem. It is used in motion compensation to estimate the antenna phase centre position at each acquisition. Then, it is needed to calculate radar distance between pixel and antennas. All the data are sent to a PC recording Unit for storage and data processing is subsequently done with Unix workstation.

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Pulse Generator The generator used is a KENTECH generator based on a PBG3 which has a pulse output voltage of 8 kV, a 10 -90 % risetime better than 120 ps and a 50 % pulse duration (full-width at half-maximum) better than 460 ps. This pulse has a frequency range from DC to about 2 GHz (at -20 dB). See below on figure 3.

Antennas Two identical antennas are used. The dimensions (less than a 60 cm ridge cube) are limited by the mechanical structure of the boom. The choice of the study was to design a 2D antenna to reduce weight and volume. That is why, antennas will be more directive in the antenna arm plan. The dimensions are also 1 m by 0.6 m. Two types of antennas have been selected : "Vivaldi" and "Scissors"3 (see figure 4). Antennas are resistively loaded on the upper half-length of each arm to match impedance in low frequencies, and to limit back scattered field. Only a pair of "Scissors" antennas are currently used in measurements.

The "scissors" antennas have been designed with the space time integral equation method. The computed radiated pulse is shown on figure 5. A differential coaxial balun4 has been designed to feed correctly the antenna. The purpose is to make a transition between a coaxial cable and the two wire feed line and, to feed symmetrically the antenna.

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MEASUREMENT CONFIGURATION A theoretical study enables to estimate the influence of the site angle (from horizontal) on the maximum level scattered field by a buried target for normal and parallel polarisation. The scattered electric field has been calculated with a 2D FDTD method and a plane wave excitation. Dry, fairly wet and wet soil are considered with Debye model5. The normal incidence is the most favourable in free space but with a soil, the specular soil echo is very high and the target response is mixed with this soil echo. The maximum field level globally decreases if the site angle also decreases at a constant radar distance configuration (15 m). This phenomena is less important in case of wet soil for the parallel polarisation. Then, it is better to work with an site angle greater than 30°. Thanks to facilities of this Radar, the first measurements had been done at 8 meter high with targets lied on soil at 6 m site angle ~50°), 10m and 17 m from the truck. Antennas are oriented in VV polarisation. Another study is needed to know the radar displacement L length to collect the most of scattered information from the target for SAR data processing. This distance L depends on the position of the target, the nature of target and the antenna radiation. But, in worse case, the radar and the target may have omnidirectionnal characteristics. The punctual target M is supposed to diffract a pulse as a Dirac pick in order to estimate the received scattered field level v(y) at each position y (see figure 6).

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The radar distance between target and radar is

with d0 the

shortest distance between radar and target If the attenuation distance is only taken into consideration, the electric field level can be expressed by: the maximum level is measured when the radar is in front of the target and it is normalised, so v(ym) = 1 . The SAR data processing used is based on coherently summing pulse responses Sr measured along the area test4. For a pixel I(xi,yi), the algorithm identifies the sample Sr(d) corresponding to the radar distance (radar to pixel) for each measurement Sr(p) and sum them.

For a punctual target M(xm,ym,zm), an analytic approach is presented for calculating the pixel level I(xm,ym) corresponding to the target M : with y1 < y2 y1 start position and y2 end position of radar With this variable change,

the M pixel image of the M point is :

If the radar moves symmetrically along the target on the length L (see figure 6.), then the pixel level is:

In case of the radar displacement is very longer than the shortest radar distance d0, the level pixel should reach : The L distance to get K percent of the maximum pixel level is :

This result shows that if the radar covers twice the shortest distance (d0) between radar and target, the calculated pixel level is half of the maximum level In configuration measurements described before, furthest targets are at the distance (d0) of 19 m. The radar moves on the length L of 90 m, so 75 % of information is collected for the furthest target (see figure 7).

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The radar resolution can be estimated with classical SAR expressions. The distance resolution depends on width impulse and incidence angle.

In the measurement configuration, the distance resolution is about 20 cm ( ns et ). Once proceeded, theoretical signals have been computed to estimate the azimuth resolution The coherently summing treatment allows a good azimuth resolution which is about 20 cm and depends on width impulse.

EXPERIMENTAL RESULTS First measurements have been realised outdoor on January 2000. The test area was grass and the humidity soil rate was high about 30 %. More, roughness and homogeneity were not controlled because of some hole, mole-hill and little water puddle on soil surface. So, measurement conditions were realistic but rather unfavourable for target detection. Several target types have been measured but two types are presented. Nine trihedrons were lied on soil at three distances (x = 6m, 10 m and 17m) with three different length ridges (22 cm, 30 cm, 49 cm). And, three metallic mines which are 11 cm high and a 27 cm diameter large were lied on mown grass. There are also false targets like mole-hill around mines. The scene is presented on figure 7. Measurements show a high low frequency signal (less than 250 MHz). This is repeatable on each position measurements. It means that this signal may be the coupling signal between antennas and soil clutter. This undesirable information will be attenuated by filtering signal with a high pass numerical filter and by subtracting background. After, the coherently summing is proceeded with the correction of the distance attenuation. The trihedron images are accurate and shows the correct position of targets due to motion compensation system (see figure 8). A zone between 5 m and 12 m (along distance x) seems to be more perturbed by soil clutter. The effective signal has been insulated by time window. No signal has been acquired before the distance on soil of x = 6 m. So, noise does not


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appear highly in nearby zone image (x < 5 m). For further zone (x > 12 m), clutter is less important if the angle site is lower than 40°.

The metallic mine image is encouraging. Lied on mines (y = -44m) are detected as well as mole-hills which can be considered like half conductive sphere lied on the soil because of the very high humidity rate (see figure 9). With theses smaller targets, the azimuth and distance can be estimated about 20 cm from mine image on figure 9. Signature on image presents side lobes along the distance axe. It can be explained by the temporal wave form target response and by the high pass filtering operation. But theses side lobes can be a criterion to distinguish small target from noise on image. Theses first results show that contrast environment and permittivity discontinuities are detected.

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CONCLUSION This paper presents the first results obtained in rather unfavourable measurement condition with some metallic targets as trihedron and mine. Lied on soil targets are detected and good localised although roughness soil clearly appears on image. Theses results are encouraging and the next step of works can be decomposed on three points. Data processing is based on coherently summing of measurements and it will be improved with radio frequency interference rejection, specific treatments for rejecting soil clutter and coupling, calibration with a canonical target. Sub banding image will be tested to reduce soil clutter and to discriminate target. Then, some special measurement configurations will allow to analyse and reduce high coupling signal between antennas, truck and soil. The resolution will be estimated. The bandwidth of receiving system will be upgraded. And more exhaustive measurements are planed with more favourable conditions, different type of soil and real mines. Targets will be buried. Results will be analysed and compared to FDTD simulations.

REFERENCES 1. Chevalier Y., Imbs Y., Beillard B., Andrieu J., Jouvet M., Jecko B., Le Goff ML., Legros E., "UWB measurements of canonical targets and RCS determination", Ultra Wide Band Short Pulse Electromagnetics, vol 4. 2. Le Goff M., Pouligen P., Chevalier Y., Imbs Y., Beillard B., Andrieu J., Jecko B., Bouillon G., Juhel B., "UWB short pulse sensor for target electromagnetic backscattering characterization", Ultra Wide Band Short Pulse Electromagnetics, vol 4. 3. V. Mallepeyre, F. Gallais, Y. Imbs, B. Beillard, J. Andrieu, B. Jecko, M. Le Goff, "A new broadband 2D antenna for UWB applications", Ultra Wide Band Short Pulse Electromagnetics, vol 5. In press. 4. The baluns were made by the EUROPULSE company (Cressensac Lot, France). 5. P. leveque, A. Reinex, B. Jecko, "Modeling of dielectric losses in microstrip patch antennas : application of FDTD method", Electronics Letters, vol. 28, n°6, mars 1992, pp 539-541

ULTRA-WIDEBAND GROUND PENETRATING IMPULSE RADAR

Alexander G. Yarovoy, Piet van Genderen and Leo P. Ligthart International Research Centre for Telecommunications-Transmission and Radar, Faculty of Information Technology and Systems, Delft University of Technology, 2628 CD Delft, The Netherlands

INTRODUCTION It is widely believed that ground penetrating radar (GPR) should be a key component of any system designed for humanitarian demining. GPR-part of such sytem should satisfy two crucial demands: high probability of object detection and low false alarm rate. While detectability of the GPR can be improved by means of improving the resolution and the dynamic range, decrease of the false alarm rate can be achieved only via localization, classification and identification of detected targets. Solution of the latter problem requires accurate measurements of the electromagnetic field scattered from the subsurface. This qualitatively new demand makes the principal difference between usual GPR and GPR for landmine detection: the first one should just detect the field scattered from a buried target (i.e. distinguish this field from all other electromagnetic fields) while the second one should measure accurately the scattered field (i.e. determine magnitude of the field as a function of time). Different inverse scattering methods can be used later to determine localization, size, shape and even spatial distribution of dielectric permittivity within the buried target from the measured values of the scattered field. Taking into account this principal demand of high accuracy and using its long experience of near-field antenna measurements the International Research Centre for Telecommunications-transmission and Radar (IRCTR) in the Delft University of Technology has developed two GPR systems dedicated to landmine detection: a video impulse system and a stepped-frequency continuous wave system. In this paper the main guidelines of the video impulse system design are presented. The stepped-frequency system is described elsewhere. The impulse GPR system developed in IRCTR for landmine detection comprises a pulse generator, an antenna system, a receiver (which consists of a signal conditioner and a sampling converter) and a processing software. All these items will be briefly described below. At the beginning of each section we first formulate demands to each item from the Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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system point of view. After that we describe what has been developed in order to satisfy these demands.

GENERATOR For the impulse radar choice of waveform and duration of the pulse is critical. We have chosen a monocycle waveform. The advantage of a monocycle in comparison with a monopulse is that the frequency spectrum of the first one decreases to zero at low frequencies, which cannot be efficiently transmitted via the antenna system, while the frequency spectrum of the second one has a global maximum there. As a result, the magnitude of the field radiated by an antenna system fed by a monocycle is considerably larger than the magnitude of the field radiated by the antenna system fed by a monopulse with the same magnitude. Besides, the energy, which is not radiated from the transmit antenna, reflects back into the feeding line frequently causing additional ringing of the transmit antenna. As far as the pulse duration is concerned we have chosen 0.8ns (a period of approximating ideal monocycle). The frequency spectrum of such pulse covers interval from 210MHz till 2100MHz on 10dB level. At frequencies below 1GHz, attenuation losses in the ground are small (Daniels, 1996) and considerable penetration depth can be achieved. However, landmine detection requires down-range resolution (in the ground) of the order of several centimeters, which can be achieved using frequencies above 1GHz. It was found experimentally that the 0.8ns monocycle satisfies penetration and resolution requirements. The spectrum of this pulse has a maximum at frequencies where the attenuation losses in the ground start to increase. So the spectral content of the monocycle below this maximum penetrates deep into the ground and the spectral content above this maximum provides sufficient down-range resolution.

The pulse generator for the radar has been delivered by SATIS Co. The unique feature of this generator is its small trailing oscillations, which are below 2.4% of the maximum amplitude during the first 2ns and below 0.5% afterwards (Fig. 1a). The generator spectrum covers a wide frequency band from 420MHz till 1.67GHz on 3dB level (Fig. 1b).

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ANTENNA SYSTEM The antenna system is one of the most critical parts of every GPR, because the performance of the whole radar depends strongly on it. The antenna system should satisfy a number of (sometimes contradictory) demands. We list them separately for transmit and receive antennas. In order to achieve sufficient signal-to-clutter ratio and to use same signal processing for detection of surface laid and buried objects the transmit antenna should: 1. radiate short ultra-wideband (UWB) pulse with small ringing; 2. produce an optimal footprint on the ground surface and below it (size of the footprint should be large enough for SAR processing but at the same time it should be small enough to reduce surface clutter and in order to filter out undesirable backscattering from surrounding objects); 3. keep constant the waveform of the radiated field on the surface and in different types of the ground (i.e. with different dielectric permittivity). In order to allow successful SAR processing for the given frequency band the scattered field should be measured with a cross-range step 3cm maximum. The measurement plane should be sufficiently elevated above the ground surface in order to avoid influence of evanescent fields. Together with operational demands for landmine detection it means elevation of the receive antenna at least 10cm above the ground. Thus the receive antenna should: 1. receive the field in a local point (effective aperture should not be sufficiently smaller than 9cm2); 2. provide sufficient sensitivity in order to receive very weak scattered fields; 3. be elevated at least 10cm above the ground surface. Additionally a possibility of isolation of the direct air wave from the ground reflection by the time windowing and a possibility to measure simultaneously backscattered field in two orthogonal polarizations are desirable. To satisfy demands NN.2-3 for the transmit antenna it was decided to implement the far-field approach, meaning that the transmit antenna is elevated sufficiently high above the ground. The demands NN.1-2 can be satisfied if a good transient antenna with reasonably high directivity is used. Such antennas are not commercially available and design of such an antenna is extremely difficult. In close collaboration with SATIS Co. (Russia) a dielectric filled TEM horn (DTEM) has been designed (Yarovoy, Schukin and Ligthart, 2000), which is ultrawideband, has linear phase characteristics over the whole operating frequency band, has constant polarization and possesses short ringing. This antenna is based on a dielectric wedge. Such design reduces the sensibility of the antenna for external EMI and reduces the antenna’s physical dimensions. The shape of the metal flare has been optimized so that the characteristic impedance in each cross-section of the antenna gradually changes from 50Ohm (impedance of the feeding line) near the feed point to (impedance of free space) near the aperture. More specifically we have tried to minimize reflection from all antenna cross-sections, so that only reflection from the aperture can take place. The latter will not cause late time ringing if the antenna is perfectly match to the feeding line and there are no other centers of reflection within the antenna. The waveform of the electric field radiated from this antenna fed by the 0.8ns monocycle generator is presented in Fig. 2a. The footprint of the antenna measured in the plain 54cm from the aperture has an elliptic shape with halfaxes 21cm and 27cm at 3dB level. The waveform of the radiated field remains the same within the whole footprint (on 20dB level). For the receive antenna a small loop antenna has been chosen. This antenna has an aperture of the same order as a linear dipole (so the demand N.1 for receive antenna is satisfied), but unlike the dipole the loop possesses a very small ringing. As the loop is

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transparent for the incident wave, the loop has been placed just below the transmit antenna on its main axis (to satisfy demand N.2). By choosing proper elevation of the loop antenna above the ground we easily satisfy the demand N.3. As a result we have arrived at a new antenna system, which has a number of advantages over usual GPR antenna systems with two (or several) identical antennas elevated to the same height above the ground.

The developed antenna system has an ultrawide frequency band. The spectrum of the signal passed through the antenna system (perfectly conducting flat ground calibration) is presented in Fig. 2b. The developed antenna system has been patented (de Jongh et. al (1999)). The disadvantage of the developed antenna system is a high magnitude signal due to the direct wave from Tx to Rx antenna. The magnitude of this signal is considerably larger than the reflection from the ground, so it determines the upper level of the system dynamic range. RECEIVER According to our simulations and practical experience GPR receiver should satisfy following demands: 1. its bandwidth should be larger than the bandwidth of the received pulse measured at -40dB level; 2. within this bandwidth the receiver should have linear phase characteristic; 3. the linear dynamic range of the receiver should be larger than 60dB to detect both AT and AP mines in typical ground conditions; 4. the sampling time should be considerably smaller than that defined by Nyquist criterion with respect to the highest frequency in the received pulse spectrum (our experience shows that for 0.8ns pulse the optimal sampling time lies between 15ps and 20ps); 5. finally, to perform SAR-like data processing the receiver parameters should be extremely stable (for 0.8ns pulse the time drift should not exceed 5ps within the whole measurement of the C-scan).


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In order to satisfy these demands we developed the receiver, which consists of the signal conditioner and stroboscopic sampling recorder. The signal conditioner should improve signal to noise ratio and should allow to use the whole dynamic range of the ADC. Besides, the received signal (except of its part caused by the direct wave) should be processed linearly otherwise the surface laid target response cannot be distinguished from the surface reflection. These demands can be satisfied if the signal conditioner will clip the high peak due to the direct wave from Tx to Rx antenna and will amplify the ground reflection signal up to the maximal level linearly acquired by the ADC. Thus the developed signal conditioner combines LNAs and a limiter (with very short recovery time) for voltage clipping. Our approach differs from the conventional one in which a variable gain amplifier is used. The main drawback of the conventional approach is that a variable gain amplifier changes the waveform and the spectrum of the received signal. Such changes can be acceptable if the final aim of the radar is target detection, but the task of target identification is not compatible with any nonlinear signal processing. In our approach the signal conditioner behaves linearly from the moment of arrival of the ground reflection. The four channel sampling converter from GeoZondas Ltd. (Lithuania) with a sampling rate 100kHz (by one channel operation) allows to measure transient signals with an accuracy of about 1% in the bandwidth from 100MHz up to 6GHz. Maximal error in time scale linearity is around 1%. The precision of the sampling converter is sufficiently high to do accurate measurements of the scattered transient field. Using averaging the linear dynamic range as high as 80dB can be achieved.

DATA PREPROCESSING Both the direct air wave from the transmit to the receive antenna and the surface clutter create a background which often masks the response of the target. In order to remove this background pre-processing of data has been used. This pre-processing includes subtraction of the system response due to the direct air wave, averaging within the footprint of the transmit antenna and subtraction of the averaged ground reflection from each A-scan. In order to limit the magnitude of artifacts due to subtraction of time domain signals, before the subtraction the time drift is numerically compensated within each A-scan. As a reference signal for the time drift compensation the direct air wave has been used. Despite of fluctuations in the arrival time of the direct air wave due to mechanical vibration of the system and other factors, the quality of compensation was found to be good.

EXPERIMENTAL RESULTS The GPR system has been tested in different environments, e.g. sand, clay, forest ground, etc. (de Jong, Lensen and Janssen, 1999). Examples of the B-scans over flash buried antipersonnel mines and a deeply buried antitank mine are presented in Fig.3 - 4. In Fig. 4 deformation of the ground surface above the mine is clearly visible.

CONCLUSION On the basis of numerical simulations of different GPR scenarios and field experience the technical requirements for the impulse GPR system specialized on detection and identification of small and shallow buried objects have been formulated. These

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requirements have been partly realized in the video impulse ground penetrating radar system developed in IRCTR. Experimental trials confirmed that the system can detect small dielectric and metal targets at a depth up to 50cm. Images build from the acquired data allow to determine position, size, shape and (sometimes) internal structure of the buried objects (Groenenboom, Yarovoy(2000)). In the next step of the program we shall develop polarimetric antenna array system. Using a multi-channel sampling scope we shall simultaneously acquire data in several different positions and for several polarizations. New software for image processing, localization and identification of targets will be also developed and implemented into the system.

Acknowledgements This research is supported by the Technology Foundation STW, applied science division of NWO and the technology program of the Ministry of Economic Affairs of the Netherlands. The authors wish to acknowledge contributions to the system development by G.Hermans, J. van Heijenoort, S.v.d.Laan, I.L.Morrow and B.Sai (all IRCTR), N.Budko


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and J.Groenenboom (TU Delft), I.Kaploun and A.Schukin (SATIS Co., Russia) and B.Levitas and A.Minin (GeoZondas Ltd., Lithuania).

REFERENCES Daniels, D.J., 1996, Surface-Penetrating Radar, IEE, London. Groenenboom J. and Yarovoy A.G., 2000, Data processing for a landmine detection dedicated GPR, in: Proceedings on Ground Penetrating Radar Conference, Gold Coast, Australia, 23-26 May 2000. Jong, W. de, Lensen H.A. and Janssen Y.H., 1999, Sophisticated test facility to detect land mines, Detection and Remediation Technologies for Mines and Minelike Targets IV, SPIE Proc., 3710:1409. Jongh, R.V. de, Yarovoy A.G., Schukin A.D., Ligthart L.P., Morow I.L., 1999, Penetrating air/medium interface microwave radar, Filing number 1013661-NL. Yarovoy, A.G., Schukin A.D. and Ligthart L.P., 2000, Development of dielectric filled TEM-horn, in: Proceedings on Millennium Conference on Antennas & Propagation, Davos, Switzerland, 9-14 April 2000.


OBJECT SHAPE RECONSTRUCTION AT SMALL BASE ULTRAWIDEBAND RADAR

Vladimir I. Koshelev, Sergey E. Shipilov, and Vladimir P. Yakubov Institute of High Current Electronics RAS, 4, Akademichesky Ave., 634055 Tomsk, Russia INTRODUCTION Recently, owing to the considerable progress in the field of generation and radiation of short ultrawideband (UWB) electromagnetic pulses1,2 the interest to the analysis of the ultrawideband radar potential possibilities including the radar object (RO) recognition has increased. The solution of the object recognition problem is frequently related directly to the possibility to obtain the data concerning its shape. In case of a small-base radar system the application of the well-known methods of RO shape reconstruction becomes problematic. The investigations in this direction previously carried out by the authors3 with the use of a Lewis-Boyarsky transformation allowed to obtain a 10° angular surveillance base being the utmost minimum value for the given approach at a four-fold frequency overlapping of a sounding UWB pulse. The noise level should not exceed 1%. In the paper presented here, the way of the problem solution of the RO shape reconstruction in the time-domain approach at the essential limitation to the angular surveillance base and the utmost minimum number of observation aspect angles is considered taking into account measurement noise. A small angular base doesn’t allow to obtain a spatial RO resolution directly. Application of the short UWB pulses for sounding gives a high temporal resolution of reflected signals. The solution of a shape reconstruction problem is related to a possibility to re-calculate the temporal resolution in the object spatial resolution and makes possible to minimize the angular base. The realization of the given approach for the RO shape reconstruction is based on the use of the so-called genetic functions (GF) as the temporal images of its different fragments. An idea to use the GF for the recognition of the 2D images was suggested previously4. In the given paper, the potential possibilities of a RO shape reconstruction algorithm using the GF are presented. The change of the electromagnetic radiation pulse waveform reflected by the sounded object contains the information concerning the shape of the object and its other parameters. A number of functions approximating the signals reflected by the RO in the given observation aspect angle will be understood as an object set of the GF.


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where S(t) is the signal in a receiver, is the set of the GF for the given aspect angle having the amplitude coefficients and the position at the temporal axis Each GF presents a signal scattered by a definite geometric object. On the basis of the available a priori information concerning the RO a supposition about the composition of GF being a part of the set (1) and about their form is made. An important stage of the suggested reconstruction method is the GF composition determination for description of the signal reflected by the RO. For this operation the distance between the receivers necessary both for the determination of the coordinates of the object fragments with the corresponding GF and for the subsequent RO shape reconstruction is not required. Thus, even application of a monostatic radar scheme allows to evaluate the GF composition and to use this information in the problem of the RO recognition. SHAPE RECONSTRUCTION BY THE SET OF GENETIC FUNCTIONS The problem of the RO shape reconstruction is divided into two parts: 1) the GF composition determination for the approximation of the signals scattered by the RO in the given observation aspect angle; 2) carrying out calculations of coordinates of the object fragments corresponding to the obtained GF set. The first part of the problem comes to the solution of the matrix equation of the type

relative to the unknown values and Here, is the signal reflected by the RO and received by the main receiver of a receiving-transmitting system (Figure 1). A linear part of is found by the solution of the system of linear

algebraic equations

Here, is the pseudoreciprocal matrix from A with the elements obtained with the use of a singular decomposition method. The unknown comes into the equation (2) nonlinearly. In order to find them, the iteration method of successive

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approximation is used where the values of following step of

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obtained a step earlier are used for the

The iteration process is ended when the value becomes less than the given calculation error. At such approach an essential influence on the solution convergence is made by the choice of the initial approximation. The RO parts in which the surface curvature radius essentially differs from the neighboring parts will correspond to the local S(t) maxima. When the RO aspect angle is known and a priori information about the object type is available, the identification of the signal maxima position with the initial approximation of the GF is possible. A set of the GF presents a databank of the previously calculated signals scattered by the RO fragments as well as their variations by the dimensions and shape at a given waveform of the sounding pulse. The GF corresponding to the RO fragments are calculated in the far-field zone. Such presentation makes them independent on the distance to the object. A complete databank is composed of the GF obtained at all the aspect angles of the surveillance. The delays of the signals corresponding to the same GF measured by means of the receivers that are apart from each other can be used to determine the coordinates R of the object corresponding to it. As the initial point of counting it is convenient to use the GF delays in the main receiver which is situated at the beginning of coordinates in Figure 1. The calculation of a relative delay where j is the receiver number is made according to the following equation: Here, corresponds to the main receiver delay. When the receivers have the coordinates (b,0,0), (0,b,0), (0,0,b) relative to the main receiver, the equation for the components of the vector R of the n-th fragment with a corresponding GF is the following

Here, is the distance to the n-th fragment of the sounded object determined by the delay between the pulse radiated by the transmitter and the n-th local maximum of a received signal in the main receiver; c is the velocity of light. The object uniting all the fragments corresponding to the found GF with the coordinates calculated by means of the equation (5) is a RO with the reconstructed shape. COMPUTER SIMULATION To confirm a capacity for work of the suggested algorithm, a computer simulation of the shape reconstruction of the RO in the shape of a stylized 3D model of an airplane (Figure 2) with a linear dimension L at sounding by the UWB pulses of different length has been carried out using a receiving-transmitting system consisting of an ultrawideband pulse transmitter and four space-distributed receivers (Figure 1). To make the calculation more convenient, the transmitter and the main receiver were combined and disposed at the beginning of the coordinates. In the course of the computer simulation the sounding bipolar UWB pulses at the radar output had the length and 4 ns. A situation was considered when a single pulse reflected by the object which had the coordinates X=500 m,

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Y = 1000 m, Z = 1000 m. The distance to the object center was R = 1500 m. The distance between the receivers was given as b = 50 m that corresponded to 2° of the angular surveillance base. A direct problem of calculation of the field reflected by the object with an arbitrary shape was solved by a Kirchhoff method for the nonstationary diffraction problems in the approximation of a single scattering. In the process of calculation the influence of the object self-shadowing was taken into account and the influence of the intershadowing of its separate parts was neglected. Owing to the complex shape of the radar object, the integration by the surface was substituted by the summation by the elementary areas. The dimensions of the areas were chosen much less than the sounding pulse spatial length. Using this approach, all genetic functions for the given object aspect angle composing the databank for each pulse were calculated and used later for the RO shape reconstruction. All the set of the GF was divided into five classes corresponding to the RO main fragments. Each class contained three GF corresponding to the shape variations of the given fragment. One of variations completely corresponded to the fragment by the shape composing the sounded object. The problem of the object shape reconstruction was solved by the reflected signals (Figure 3) calculated for four receivers. To simulate the real

measurements, the uniformly distributed noise with a zero mean value and the given dispersion value was added. The dispersion value was determined relative to the signal maximum in the receiving system. By the signal in the main receiver the positions of its first five (by number of the used classes) local maxima used as the initial approximations of (n = 1, 2, ...5) for solution of the equation (2) were determined. Using the obtained approximations, the amplitude coefficients for the GF were determined from the equation (3). A filtration operation was carried out to minimize the GF set when one GF with a maximum amplitude was chosen from each class. An iteration problem (4) for a more precise definition of the obtained approximation was solved for the resulting set. Then the obtained solution was tested for stability. The delays of calculated in that way were used as the initial approximations to calculate the delays in the rest of the receivers. The same set of the GF obtained as a result of the filtration operation and stability test but with the aspect angle corresponding to the given receiver was used at their calculation. The next stage was the transformation of the obtained delays of the genetic functions into the object fragment coordinates corresponding to the found GF according to the equation (5). The following a priori information was used when reconstructing the shape with the use of the given algorithm: the object type and its aspect angle relative to the receivingtransmitting system were supposed to be known. This allowed to form the GF set for the approximation of the signal reflected by the object and to determine the belonging of the maxima in the reflected signal to one or another GF class. If the information concerning the


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RO aspect angle is absent, its determination is possible through the analysis of the object motion during a definite period of time. In the course of the computer simulation the dependence of the measurement noise influence on the accuracy of the object shape reconstruction was investigated at the different lengths of the sounding pulses. The reconstruction accuracy was determined as

where is the shape projection of the given object to the OXY plane (Figure 2), is the shape projection of the reconstructed object to the same plane. The shape reconstruction accuracy of a model object having the length L = 4.5 m decreased with the increase of the noise level and the pulse length. For the pulse length of 1, 2 and 4 ns and noise level the object reconstruction accuracy was 65%, 55% and 45%, respectively. The accuracy was calculated under condition that the solution of the equation (2) is convergent. The solution convergence denotes that at the given initial approximations the GF composition will be determined at which the summary contribution of the fragments corresponding to it into the reconstructed object shape gives no less than 70% of the whole projection of the given object shape to the plane OXY. In the course of the numerical experiment it was determined by 1000 realizations that the solution comes to the real if the evaluation comes into the interval where when bipolar pulses are used. For such condition is fulfilled approximately in 90% of cases at the noise level not exceeding a 20%. The databank of the GF for the different object fragments should be formed for a whole set of angular directions. The choice of the angular step between the nearest aspect angles with which the databank is formed is directly related to the object dimension. Besides, discretization by the angle results in the error at the determination of a real object aspect angle. Maximum value of such error equals to the half of the chosen angular step. Maximum change in the two neighboring aspect angles of a relative delay between two object points having a dimension L in the far-field zone is determined from a simple geometric consideration by is the angle between the two nearest aspect angles (in radians). At the same time, the error in determination of the coordinates of the object fragments is related to the error in the relative delay determination as The error in the object aspect angle determination doesn’t depend on the distance to the object. The summary error in the determination of the coordinates of the n-th object fragment will be formed from the error in the object aspect angle determination as well as the error of related to the availability of the measurement noise not allowing to determine a relative delay exactly:

Here, is the error of determination of the n-th GF relative delay. The value of is directly proportional to the sounding pulse length and rises with the increase of the noise level and discretization step of the received signal. The equation (6) can be used for the evaluation of the error upper boundary of determination of the object fragment coordinates and hence the accuracy lower boundary of the object shape reconstruction. The numerical experiments at the model presented in Figure 2 have shown that for a complex object the equation (6) is modified as

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Here, is the error averaged by all the fragments; is the coefficient depending on the object shape and aspect angle; is the coefficient dependent on the level of the measurement noise level and discretization step d. Figure 4 presents the dependence of the reconstruction accuracy on the ratio of the object dimension to the mean error of determination of the fragment coordinates for the object used in the calculations. Hence, if

a limitation to the error of the fragment coordinate reconstruction is set, then a maximum step by the angle when forming the GF databank can be determined under condition that the characteristic dimension (as a rule, a priori) of the object is known. E.g., at b=50 m, L=4.5 m, R=1500 m, d=1/20 , the coefficients and were calculated for the given aspect angle by the results of a numerical simulation. In the calculations a step by the angle was At such error of evaluation of the fragment coordinates the accuracy of the object reconstruction was This evaluation of the accuracy is true at where is the dimension of the least RO fragment described by means of the GF. In the course of a numerical experiment, the dependence of the object shape reconstruction accuracy on the ratio of the object dimension to the sounding pulse spatial length at the absence of the aspect angle determination error was obtained (Figure 5). Curves 1 and 2 are calculated at for the noise levels and respectively; curves 3 and 4 are calculated at for the same noise levels. The obtained results show that the reconstruction accuracy has a strongly expressed dependence on that is confirmed by a good match of the curves 1 and 3, 2 and 4. The reconstruction accuracy weakly changes with the rise of ratio of the object dimension to the sounding pulse spatial length at This approach for the object shape reconstruction requires knowing the object aspect angle. By means of the equation (7) and results presented in Figures 4, 5 the dependence of the angular step value on the ratio at a constant accuracy of the object reconstruction can be evaluated qualitatively. This dependence has a form similar to the dependence presented in Figure 4. The investigation of the dependence of on for the fixed values and have shown that at the angular step was


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The decrease of by a factor of two results in the decrease of by an order of magnitude and increase by an order of magnitude gives the increase of by a factor of two. The numerical experiment results confirmed the efficiency of the RO shape reconstruction method based on the GF use. Their application gave a possibility to study the 3D object shape reconstruction. In comparison with the approach previously used by the authors3 and based on the Lewis-Boyarsky transformation, the suggested algorithm allows to minimize the angular surveillance base no less than by a factor of 5 and to decrease it to For the higher noise levels the suggested approach gives rise in the reconstruction accuracy: the accuracy of the given approach for was at and and for the previous method at and At the object dimension of 50 m at using a 1-ns sounding pulse and a 10% noise level, a 1° step by the angle between the two nearest aspect angles in the databank, and a 50-m dimension of the receiving system, the distance to the object at which the reconstruction accuracy exceeds 60% is 30 km. At the distance rise up to 100 km, the reconstruction accuracy decreases to 17% that is insufficient for the object recognition on the basis of its reconstructed shape. In this case, the suggested approach is applicable for the determination of the composition of the GF describing the signal reflected by the object. The information about the GF composition can be used in the tasks of the RO recognition. The decrease of the sounding pulse length, increase of the distance between the receivers in the receiving system, increase of the signal/noise ratio, decrease of the angular step and increase of the object aspect angle determination accuracy allow to increase the accuracy of the RO shape reconstruction and the distance to the object at the given accuracy. CONCLUSION The investigation that had been carried out has shown a possibility to use the genetic functions in a small aspect angle radar for the RO shape reconstruction. The reconstruction accuracy essentially depends on the measurement noise, the ratio of the object dimension to the electromagnetic pulse spatial length, the angular step with which the databank of the genetic functions is formed, the accuracy of the aspect angle determination of the object and the angular base of the receivers in a measuring system. The accuracy of a 4.5-m long object reconstruction at a 1-ns pulse length, a 2° angular base of the receivers and a 10% noise level was 80% for the suggested method. The suggested method is applicable for a monopulse radar when a RO displacement during a pulse can be neglected and it can be generalized for the case of sounding by the series of the UWB pulses with synthesizing the angular base owing to the data summation at the object motion. REFERENCES F.J.Agee, C.E.Baum, W.D.Prather, J.M.Lehr, J.P.O’Loughlin, J.W.Burger, J.S.H.Schoenberg, D.W.Schoflied, R.J.Torres, J.P.Hull, and J.A.Gaudet, Ultra-wideband transmitter research, IEEE Trans. Plasma Sci. 26:860 (1998). 2. Yu.A.Andreev, Yu.I.Buyanov, A.M.Efremov, V.I.Koshelev, B.M.Kovalchuk, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B. Zorin, Gigawatt-power-level ultrawideband radiation generator, in: Proc. of 12 IEEE Inter. Pulsed Power Conf., C. Stallings and H. Kirbie, eds., Monterey, CA, USA. 2:1337 (1999). 3. V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Reconstruction of the object shape at small aspect angle ultrawideband radiolocation, Radiotechn. Electr. 44:301 (1999). 4. V.M.Ginzburg, Presenting of images by means of genetic geometrical functions, Doklady Akademii Nauk. 244:580 (1979). 1.


UWB SUBSURFACE RADAR WITH ANTENNA ARRAY FOR IMAGING OF INTERNAL STRUCTURE OF CONCRETE STRUCTURAL ELEMENTS

Anatoliy O. Boryssenko, Elena S. Boryssenko, Vladimir A. Ivashchuk, Alexander N. Lishchenko, Vitaliy P. Prokhorenko Research Company “Diascarb” Kyiv, P.O. Box No. 222, 02222, Ukraine

INTRODUCTION Basic design principals implemented in the advanced time-domain UWB radar including real/synthetic array and full-polarimetric schema with 100 – 900 MHz effective band are considered here. The radar is installing on a moveable platform for remote operation on the territories with radioactive pollution. This probing system should be employed near the Chernobyl nuclear power plant, block No. 4 damaged in 1986, that is situated not far from Kyiv, Ukraine,. The key task is searching and localisation of the nuclear-fuel contained materials (NFCM) inside the Cascade Wall around nuclear reactor. UWB radar described below is under testing now.

PROBLEM BACKGROUND Basically problem of localization if the NFCM inside the Cascade Wall with concrete can be solved by nonstandard ground penetrating radar (GPR) technique rather than by commercially available radars. The challenger features of the treated problem involve: i) sounding medium is layered concrete one up 10 meters and more thickness; ii) this medium has strong signal attenuation determined by internal water/ionic content; iii) it is inhomogeneous medium due to chaotic internal filling with a large variety of different objects including dispersed by explosion elements of the destroyed reactor; iv) the NFCM should be detected can be presented as solidified masses of molted concrete with patches of radioactive substances; v) it is expected that in some cases those patches may have a cylindrical shape as original fuel-assemblies; vi) searching areas on the horizontal plane of the Cascade Wall have rectangular shape where edge regions near vertical walls will cause strong diffraction interference (figure 1); vii)due to radioactive pollution of searching areas only completely automatic radar operation can be applied. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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Actually any GPR maps changes within material under survey due to contrasts in the electromagnetic properties. So the NFCM inside concrete can be recognized by set of its inferential features. For the most of known GPR applications measured data are presented as distorted images or pseudo-cross-sections of the subsurface regions for visual analysis, without any further necessity of processing (Olhoeft, 2000). However, the considered problem requires more detailed and precise information about type of each hidden object, its depth, orientation, composition etc. Such data can be obtained by properly acquired by advanced subsurface radar. One can find some useful approaches to be applied here. There are such fields of GPR applications as land-mine and UXO detection (Daniels, 1999), nondestructive testing of concrete in civil engineering (Boryssenko et al, 2000) and for noninvasive profiling of asphalt and concrete pavement and bridge desks (Wahrus et al, 1994). Key problems that need to solve, when GPR is applied for land mine detection, are connected with small electrical contrast of mines like plastic ones and influence of environment on radar performances. The last is due to irregular stratified structure of sounding media, rough interfaces, the presence of clutters like tree roots, rocks etc. Microwave inspection of concrete structures with is powerful tool with a wide range of applications. This is a rapid technique for nondestructive detection of defects in reinforcement concrete, for multi-layered road structure and so on. Note that all listed above application are connected with concrete probing up to several meters while the considered problem requires more deep down-range operation under specific conditions. We conclude that successful detection and discrimination of the FCM objects deeply inside the concrete construction of the Cascade Wall can be reached by application of the advanced GPR rather than ordinary hand-held GPR. Such advanced GPR should possess a high dynamical range and optimal operating frequency band. Also physical aperture technique should be employed beyond the synthetic aperture method. Polarimetric and other signal processing approaches have to be incorporated in this radar project. High procession of antenna space positioning is also required for exact subsurface mapping. The last can be fulfilled by robotic platform where radar is housed and that is urgently needed for safety operation on the area with radioactive pollution. Some of the formulated above features of the presented GPR project are under consideration in this paper.

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GENERAL DESIGN APPROACHES The key principal employed in this design is that such system must be realised in the short term with available technologies and examined before solutions. The presented radar possesses features originated from previous research and design projects in UWB radar especially for subsurface probing based on the time-domain technology. Optimal Frequency Band The key factor should be care is electrical properties of concrete as medium for electromagnetic wave propagation. Those properties in 1 MHz - 10 GHz band have been studied by many researchers (Robert, 1998). Figure 2 summarizes those data and illustrates the high frequency limit for GPR probing in concrete. Generally there exists a common tradeoff between resolution and depth range for choice of GPR operation frequency band. Increasing of resolution by using of high frequencies, can be achieved by the price of dramatic rise of radar potential and appearance on radar images redundant details and speckle structure (Daniels, 1999). There is also low frequency limit for optimal detection of target with specific shape and size like expected NFCM to minimize signal-to-clutter ratio besides requirement of high resolution (Brock and Patitz, 1994). Due to above reasons 100-900 MHz bandwidth is optimal one for the designed GPR.

Radar Energy Potential Here we express the radar performance factor as usual ratio of the peak radiated power to the smallest detectable receiver signal (Wahrus, et al, 1994; Daniels, 1999). Whereas the first factor is a constant value, additional coherent processing including averaging in time and space can increase the second one. Impulse generator of transmitter forms bell-like pulse on the 10-Ohm impedance antenna as a load with peak power up to 400 kW (56dB). The minimum discernible signal (MDS) is expressed as (Wahrus, et al, 1994): MDS = k·T·BW·NF·SNR, where k – Boltzmann constant, T maximum operation temperature, BW = 800 MHz – bandwidth, NF – Noise Factor of the receiver, SNR – Signal-to-Noise Ratio. Assume that NF = 1.3 dB and SNR = 15 dB. At this case we have MDS = -96 dB. We are considering peak power rather energy because in high-resolution radar system shape of transmitting signal is precisely registered and processed.

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Additional gain of system performance factor is following as stated above from radar response coherent processing like averaging in time with factor up to 2048 (33dB), array processing for physical aperture (9dB) and synthetic aperture (12-20 dB). Therefore the total energetic potential reaches of 180 dB magnitude. Sufficient rise of the last figure by processing is possible due to radar operation with discrete profiling when system has high level of stationary determined by platform positioning at each measurement point. The high system energy potential enables high depth range of radar. In this case for typical 5% internal water content the two-way loss magnitude is about 12 dB/m in accordance to figure 1. The last corresponds to depth range of about 15 m that give opportunity to search through all thickens of the Cascade Wall. Radar signals We paid particular attention to optimising transmitter/receiver array subsystems including optimal choice of signal waveforms and equivalent effective spectra shown in figure 3. Transmitting antenna driving pulse has a rise time of 1 ns and 5-ns fall time. Radiated electromagnetic signal is mainly determined by the current excitation waveform, the antennas and ground-coupling factor. The last has stochastic disturbance effect on both transmitter and receiver radar antennas resulted in registered waveform distortion. Also effect of multiple cooperative /non-cooperative scattering (Astanin et al, 1994) by specific shape targets will further modify received signal making its different from idealised one in figure 3c. Other factors change the registered signal waveform including: roughness of surface interface, antenna shielding from side of upper half-space, non-uniformity of antenna patterns at the angles off vertical directions etc. Finally the common measured two-pass waveform is depicted in figure 4.


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SYSTEM ARCHITECTURE OVERVIEW Generally presented UWB radar system includes two principal components, i.e. electronic and mechanical ones, the multi-channel radar and moveable robotic platform. The multi-channel UWB radar consists of three main subsystems: 1) sensor receiving array; 2) transmitter/receiver electronics; 3) data and command transfer via optical link, 3) operation control and data processing/storing. Transmitter electronics is based on drift step recovery diode diodes and special driving circuits. Lower pulse repetition rate (PRR) are employed in this radar for the reason of increasing of life time of compact high-voltage transmitter units, maintaining its optimal temperature regime for stable operation without external cooler and for power supply minimization. Receivers are built with using low-noise input circuits and stroboscopic time-domain sampling as for conventional pulse GPR. This approach and 16-bit ADC circuit enable high-accuracy registration of time-scaled signal waveforms. Such operation require some time with low PRR and long signal averaging but it is not drawback for the presented system because total data collection time at each point of scanning line is lower that time required for movement and positioning of mechanical part of radar. Radar control and data storing by remote host PC are fulfilled by the data/command transmission subsystem. The last forms a local network between computer on robotic platform and the host computer installed on the safe distance up to 0.5 km from the radaroperating zone. Physical interface of the data exchange between GPR and control PC unit is based on fiber optic communication link with full-duplex data transfer possibility. The radar sensor array is installed on the robotic platform for remote operation. The robotic platform is in fact a distantly controlled vehicle for programmed radar replacement over searching territory. In process of radar survey the sensor array is moving in discrete mode over the searching area. We employed as basic for robotic platform design electrical and pneumatic module components of the FESTO Company (http://www.festo-usa.com/). Table 1 lists the principal characteristics of advanced UWB subsurface radar.

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ANTENNA ARRAY AND PROCESSING TECHNIQUES Inherently the beam pattern of subsurface radar antennas is widely spread (Daniels, 1999) and to improve it the synthetic aperture technique (SAR) technique (Soumekh, 1999) is widely employed. The SAR is based on moving of radar antennas along scanning line laid over searching area (Finkelstein, 1994). In this way processing of radar data forms image of subsurface region. Actually migration processing (Morah, 2000) is basic imaging technique, which rearranges reflected data so that reflection and diffraction events are plotted at the origin locations. Effectiveness of the SAR depends on radar performances and specific features of sites under radar survey. For the considered problem the most unfavorable factor is limited size of the horizontal plane of searching areas due to the edge effects on SAR as shown conditionally in figure 1. In order to overcome limitations of existence GPR technique, such as for hand-held radar, the physical aperture combined with accurate SAR should be employed. In this way a near-field beam-forming with real physical aperture array is implemented by adjusting time delay magnitudes in receiving channels for array focusing on definite space point (spot) inside volume covered by antenna array (Rappaport and Reidy, 1994). Array structure, in figure 5a, implements a high-resolution post-processing array method for radar imaging with improving of signal-to-noise ratio, down-look radar range and suppers interference signal with out of interest arriving. Main functional goal of radar on the survey site is detection and discrimination of the NFCM as stated before. In the case of target with definite geometrical shape the value of scattered response will depend on target position with respect to polarization. Reliable radar system should be insensitive to this factor and designed to separate transmitter and receiver polarization states for arbitrary located buried target. In another words, the polarimetric radar measurement schema is urgently required. Figure 5b shows the dual polarized radar array antenna with separated two transmitting and eight receiving antennas. Besides target characterization radar polarimetric technique improves quality of radar image of subsurface region by reducing its speckle components (Stiles et al, 1999). There are two basic configurations i.e. Vivaldi tapered slot antenna and TEM ridge horn antenna. All antennas are resistively loaded for prevent ringing antenna effect and optimise signal waveform. The transmitter and receiver modules are directly terminated to elements of antenna array. The radar array has upper shielding for improving system interference immunity.


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DATA COLLECTION AND IMAGING Presented radar system implements the following in situ radar measurement that based on described above principal moments. Robotic transport platform after installation, mounting radar on it and connecting to external power, control and data transfer subsystems begins operation. In process of operation platform with radar is moving in step mode for discrete probing. Total trajectory of movement over searching area will cover it by the set of scanning lines. At each step the radar array unit is precisely positioned on surface. Plane coordinates of antenna reference point is measured by platform positioning subsystems and fixed in data file of the measurement records. Then radar measurement is consequently accomplished for both polarizations of transmitters when all eight receivers are working to register the scattered signal. Measurement at each point can be accomplished with different scenario of radar operation including time-varying gain control, data averaging, nonuniform sampling and so on. The host remote PC unit stores results of measurement at each point with coordinate and radar-setting information. Radar team can visualize data received at each point that based high-resolution antenna array technique with time-domain beam forming. In this way searching area covered by antenna physical aperture can be preliminary visualized in the 2-D or 3-D form. Next the transport platform should move along given scanning line and measurement procedure described above would be repeated. Much of the effectiveness of GPR technology is a function of the skill of radar team including operator and data interpreter. GPR equipment must be designed to provide in situ maximum effectiveness of radar data collections. Radar data are analyzed from a graphical presentation in 2-D forms like of B-scan (vertical section) and C-scan (horizontal section) or in 3-D from. Those graphical representations contains focused images of subsurface medium showing its various features originated from the contrast of electrical properties, i.e. dielectric constant, conductivity and magnetic constant rarely. Other factors governing features of scattered signals are attenuation, RCS, polarization etc. Each subsurface scattering phenomenon is characterized by a set of those parameters. The last is not quit unique and actually used for probabilistic discrimination of the scattering events in subsurface medium inside the concrete Cascade Wall. Simulated (left) and real field result (left) for focused radar 2-D image of subsurface scatterer like reinforcement bar in concrete are illustrated in figure 6.

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CONCLUSION Some of the principal components of the presented radar system have been manufactured and are under testing now. Firstly the radar with two-receiver array for copolar and cross-polar operation has been examined. The most resent experiments involve four-receiver array for the signal processing algorithms with TD beam forming. The next will include complete radar installation on platform for remote operation. At the moment of this paper preparation the final decision about using of such GPR system was unknown in the frame of the International Shelter Project (SIP). If this decision will be positive the radar system will be assembled completely on the robotic platform for its final testing, updating and field changing. REFERENCES Astanin, L. Yu., Kostylev A. A., Zinoviev Yu. S., Pasmurov A. Ya., 1994, Radar Target Characteristics: Measurement and Applications, CRC Press, Boca Raton. Boryssenko, A. A., Boryssenko, E. S., Lishchenko, A. N., Prokhorenko, V. P., 2000, Inspection of Internal Structure of Walls by Subsurface Radar, Submitted to 2000 GPR International Conferernce, Queensland, Australia. Brock, B. C., Patitz, W.E., 1994, Factors Governing of Operation Frequency for Subsurface-Imaging Synthetic-Aperture Radar, in: Proceedings of SPIE Conference, 2217: 176. Daniels, D. J., 1999, System Design of Radar for Mine Detection, in: Proceedings of SPIE Conference on Subsurface Sensors and Applications, 3752:390. Finkelstein, M.I, Ed., 1994, Subsurface Radar, Radio i Svjaz, Moscow (In Russian). Morah, M. L., Greenfield, R. J. Arcone, S. A., Delaney, A. J., 2000, Multidimensional GPR array processing using Kirchgoff migration”, J. Applied Geophysics, 43:281. Nathanason, F. E., Ed., 1991, Radar Design Principles, McGraw-Hill, Inc. Olhoeft, G. R., 2000, Maximizing the Information Return from Ground Penetrating Radar, J. of Applied Geophysics, 43:175. Plumb, R.G., Noon D.,A., Longstaff, I. D., Stickley G. F., 1998, A Waveform-Range Performance Diagram for Ground-Penetrating Radar, J. of Applied Geophysics, 40:117. Rappaport, C. M., Reidy, D. M., 199?, Focused Array Radar for Real Time Imaging and Detection, in: Proceedings of SPIE Conference, 2747:202. Robert A., 1998, Dielectric Permittivity of Concrete Between 50 MHz and 1 GHz and GPR Measurements for Building Materials Evaluation, J. of Applied Geophysics, 40: 89. Skolnik, M. I., Ed., 1990, Radar handbook, McGraw Hill, 2nd edition. Soumekh, M., 1999, Synthetic Aperture Radar Signal Processing with Matlab Algorithms, A Willey-Interscience Publications. Stiles, J. M., Parra-Bocaranda, P., Apte, A., 1999, Detection of Object Symmetry Using Bistatic and Polarimetric GPR Observations, in: Proceedings of SPIE Conference on Detection and Remediation Technologies for Mines and Minelike Targets IV, 3710: 992. Wahrus, J. P., Mast J. E., Johnson E. M., Nelson S. D., 1994, Advanced Ground Penetrating Radar, in Proceedings of SPIE Conference, 2275:177.

OPTIMAL SHORT PULSE ULTRA-WIDEBAND RADAR SIGNAL DETECTION

Igor I. Immoreev1 and James D. Taylor2 1

Moscow Aviation Institute Gospitalny val, home 5, block 18, apt 314 Russia 105094, Moscow [email protected] 2 J.D. Taylor Associates 2620 SW 14th Drive Gainesville FL 32608-2045, USA [email protected]

INTRODUCTION High-resolution ultra-wideband (UWB) radar systems will produce a multiple time scattered return from large targets. Any individual return will be difficult to detect and meaningless for surveillance radar purposes. Large target detection requires building a filter for integrating multiple time-scattered returns into a single detector output. This filter can provide optimal detection of signals with unknown parameters. We will discuss short duration (1 ns) video pulse type signals, however the concept may apply to other highresolution signals such as pseudorandom noise and FM chirp. References 1,2,3, and 4 describe the advantages of video pulse UWB radar. ULTRA-WIDEBAND RADAR SIGNALS AND LARGE TARGETS For high-resolution video pulse radar, the target geometry and viewing angle determine how each impulse will be reflected toward the receiver. Each target will have a set of N bright points. The target electromagnetic characteristics determine the returned video pulse shape and the effective scattering point radar cross section (RCS) determines the reflected impulse amplitude. If the radar instantaneous bandwidth approaches 100% of the center frequency, and the radiated pulse length is about 1 ns, then the space signal duration is 30 cm, and much


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smaller than targets such as vehicles and aircraft. Some target points may be bright, and others may change the signal waveform according their electrical properties.1 As a result a returned signal is transformed into the sequence of pulses with random parameters such as shown in Figure 1. QUASI-OPTIMAL DETECTORS FOR UWB SIGNALS As shown in Figure 1, a video pulse signal scattered from an extended target has an intricate pattern. Reflected signal parameters, such as the duration and the number, location and amplitude of signal maximums will be unknown. Unknown signal parameter information makes it impossible use correlation processing. G. A. van der Spek proposed a processing algorithm for optimal detection of unknown targets. To understand this approach, suppose the target has a length L and occupies N resolution cells, in space. The signals scattered by bright points are present in K cells, and the other cells are empty. Processing all combinations from N elements on K bright points can provide optimal detection of the unknown signal. This algorithm works because one of these combinations must coincide with the extended target return.. Figure 2 shows a schematic diagram for such an optimal detector. The problem is that a practical realization of 6 this scheme requires many processing channels.

For example, if there are N = 40 resolution cells within the observation interval for a 1 GHz bandwidth signal, and the number of expected bright points is K = 8, then the number of processing channels required will be The structure of such detector is complex and cannot be realized using present day electronic components. G. van der Spek proposed two simpler algorithms for quasi-optimal processing of unknown signals.6 The first algorithm uses the changes of the energy at the detector output when a target scattered signal is received. If N = K, where a scattered signal is present in all resolution cells within the observation interval, the optimal detector of Figure 2 is modified into a quadratic detector with a linear integrator in Figure 3.8 In this case the integration is performed over all N resolution cells, so there is no need to have prior information about the presence and location of K bright points. This detection scheme is called the “energy detector.” If we use this detector when additional losses result from the summation of noise in “empty” resolution cells within an observation interval. By increasing the number of bright points K within this interval, the energy detector approaches the optimal detector performance for fully known signal.

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For case of K2, untethered UWB transceivers (Beacons) and an untethered UWB Rover to resolve time-of-flight measurement ambiguities to determine position. The UWB geolocation system is self-synchronizing, and does not require the existence of a clock distribution system and associated cabling. The system operates independently of GPS, thereby providing operation under conditions in which satellite coverage is unavailable or is blocked by obstructions or shielding (e.g., wartime operation, in buildings, urban environments, under heavy canopy, next to large obstructions such as vertical mine walls, etc.). Measurement standard deviations of less than one foot have been demonstrated for inbuilding operation, and less than a few inches for line-of-sight operation, using a 400 MHz bandwidth (2.5 ns duration) ultra wideband waveform. ACKNOWLEDGMENTS The author wishes to thank Dr. Edward Richley for his assistance with the implementation of the optimization algorithms, and Mr. Robert Mulloy for his help with the experimental phase of this effort. REFERENCES Fontana, R.J., 1997, A novel ultra wideband (UWB) communications system, in Proceedings MILCOM 97. Gikhman, I.I. and Skorokhod, A.V., 1969, Introduction to the Theory of Random Processes, Dover Publications, Mineola, NY. Ross, G.F., 1984, Comments on baseband or carrier-free communications, Technical Memorandum, ANRO Engineering, Inc., Lexington, MA.


RECENT APPLICATIONS OF ULTRA WIDEBAND RADAR AND COMMUNICATIONS SYSTEMS

Robert J. Fontana, Ph.D. President Multispectral Solutions, Inc. Gaithersburg, MD 20855 USA http://www.multispectral.com

INTRODUCTION Ultra wideband (UWB) technology, well-known for its use in ground penetrating radar, has also been of considerable interest in communications and radar applications demanding low probability of intercept and detection (LPI/D), multipath immunity, high data throughput, precision ranging and localization. After a very short introduction to the history and theory of ultra wideband technology, we describe the current state-of-the-art (within the United States) in this emerging field by way of examples of recently fielded UWB hardware and equipment. Multispectral Solutions, Inc. (MSSI) is a pioneer and an established industry leader in the development of ultra wideband systems and has been actively involved in UWB hardware and system development since 1984. AN (ULTRA) SHORT HISTORY OF UWB TECHNOLOGY The origin of ultra wideband technology stems from work in time-domain electromagnetics begun in 1962 to fully describe the transient behavior of a certain class of microwave networks through their characteristic impulse response (Ross (1963, 1966)). The concept was quite simple. Instead of characterizing a linear, time-invariant (LTI) system by the conventional means of a swept frequency response (i.e., amplitude and phase measurements versus frequency), an LTI system could alternatively be fully characterized by its impulse response h(t). In particular, the output y(t) of such a system to any arbitrary input x(t) could be uniquely determined by the well-known convolution integral

However, it was not until the advent of the sampling oscilloscope (Hewlett-Packard c. 1962) and the development of techniques for subnanosecond (baseband) pulse generation Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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(to provide suitable approximations to an impulse excitation) that the impulse response of microwave networks could be directly observed and measured. Once impulse measurement techniques were applied to the design of wideband, radiating antenna elements (Ross (1968)), it quickly became obvious that short pulse radar and communications systems could be developed with the same set of tools. While at the Sperry Research Center, then part of the Sperry Rand Corporation, Ross applied these techniques to various applications in radar and communications (Bennett & Ross (1978)). The invention of a sensitive, short pulse receiver (Robbins (1972)) further accelerated system development. In April 1973, Sperry was awarded the first UWB communications patent (Ross (1973)). Through the late 1980's, this technology was alternately referred to as baseband, carrier-free or impulse – the term "ultra wideband" not being applied until approximately 1989 by the U.S. Department of Defense. By that time, UWB theory, techniques and many hardware approaches had experienced nearly 30 years of extensive development. By 1989, for example, Sperry had been awarded over 50 patents in the field covering UWB pulse generation and reception methods, and applications such as communications, radar, automobile collision avoidance, positioning systems, liquid level sensing and altimetry. Today, literally hundreds of patents, papers, books and bibliographical references exist on all aspects of UWB technology. An excellent, recent compendium of this material was put together by Dr. Robert Fleming of Æther Wire & Location, Inc. and can be found on the Web at http://www.aetherwire.com/CDROM/Welcome.html. As shown by the extensive list of references provided in this compendium, there have been numerous researchers involved with UWB technology over the past 38 years, and it is virtually impossible to even begin to list the other important contributors within the scope of this short paper. Within the United States, much of the early work in the UWB field (prior to 1994), particularly in the area of impulse communications, was performed under classified U.S. Government programs. Since 1994, much of the work has been carried out without classification restrictions, and the development of UWB technology has greatly accelerated. The purpose of this paper is to provide the reader with a short overview of where UWB technology is today, albeit from the perspective of one U.S. company that has been active in this field for the past 15 years. The applications and hardware discussed below are relatively new, all having been developed and demonstrated within the past 5 years. Many of these programs came to exist only because of recent breakthroughs in UWB source, receiver and antenna technologies. UWB APPLICATIONS Figure 1 illustrates an ultra wideband, handheld transceiver that was designed for full duplex voice and data transmission at rates of up to 128 kb/s (CVSD) and 115.2 kb/s (RS232). The radio has an operational center frequency in L-band (1.5 GHz) with an instantaneous bandwidth of 400 MHz (27% fractional BW). Peak power output from the UWB transceiver was measured at 2.0 Watts, with a resultant average power (worst case) of 640 µW. This results in a worst case power density of 1.6 pW/Hz. These units have a range of approximately 1 to 2 km (with small antennas shown and line-of-sight), and an extended range of 10 to 20 miles with small gain antennas.

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Figure 2 illustrates a rather unique UWB radio designed for non line-of-sight communications utilizing surface or ground wave propagation. To excite such propagation modes, the frequency of operation needs to be well below 100 MHz (e.g., Skolnik, 1990). Thus, this system was designed to operate in the frequency band from 30 to 50 MHz (50% fractional BW) and utilized a peak power output of approximately 35 Watts.

As in the above example, this radio was capable of both digital voice and data transmission to 128 kb/s and had an operational range over water of approximately 10 miles using a standard SINCGARS (30-88 MHz) VHF military antenna. Operational range over land depended upon terrain, but was observed to be from 1 to 5 miles with intervening foliage, buildings and hills. [Note that multipath cancellation is a serious problem over water, even with UWB impulse technology, because of the typically low grazing angles (resulting in small differential delays between direct and reflected paths) and strong, outof-phase reflection from the water surface. Thus, both higher frequency UWB systems as well as more conventional narrowband VHF/UHF radios were unable to provide the requisite performance.] Also unique to the receiver design was the addition of anti-jam circuitry to prevent loss of sensitivity due to strong, in-band interferers commonly observed in this frequency range. The units could also be operated in a "digipeater" mode in which packet store-and-forward techniques were used to transmit data from one radio to another via an intermediate repeater link. Figure 3 illustrates a high-speed (up to 25 Mb/s) UWB radio designed for transmission of compressed video and command & control information across an asymmetric, bi-directional link. This system was also designed to operate in the 1.3 to 1.7 GHz region (27% fractional BW) with a 4W peak power output. An earlier design, developed under funding from the U.S. Defense Advanced research Projects Agency (DARPA), operated with a 500 MHz instantaneous bandwidth in the C-band region (5.4 to 5.9 GHz).

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The ultimate goal of this design was to provide for up to 60 nautical mile, line-of-sight transmission to/from an unmanned aerial vehicle (UAV). A small parabolic dish antenna was used at the ground platform. A variant of the above system is illustrated in operation in Figure 4 below. In this figure, a 2 Mb/s asymmetrical UWB link is used to transmit compressed video from a small unmanned ground vehicle (robot) through a UAV (unmanned helicopter) relay to a soldier ground station. The command & control signal (115.2 kb/s) to the robot is relayed through the UAV; while compressed robot video transmissions (1-2 Mb/s) are relayed through the UAV to the soldier. Ranges to UAV and robot were a few kilometers.

Another unique application for UWB communications is illustrated by the tagging device shown below in Figure 5.

This system, dubbed Vehicular Electronic Tagging and Alert System (VETAS), was designed for the U.S. Department of Transportation to provide a means for keeping problem drivers (i.e., drivers who have repeated been convicted of traffic accidents or violations due to driving while under the influence of alcohol) off the road. The concept was to tag the vehicle with a device which relays a picture of the driver, together with


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information on the driver and the vehicle, to a roadside sensor in a police vehicle. The tag would be installed in lieu of impounding the vehicle or placing the convicted driver in jail. Ultra wideband technology was considered for this application because of its ability to transmit large amounts of data at high speed in a mobile, multipath-prone environment. The UWB tag operated in the L-band region (1.4 to 1.65 GHz) and had a peak output power of approximately 250 mW for a demonstrated range of over 300 meters. The image of the driver was stored as a compressed JPEG file, together with additional ASCII data, in EEPROM and periodically transmitted at a 400 kb/s burst rate to a UWB receiver with display. The tag operated off of two AAA batteries (3.0V) and, in an operational scenario, was mounted behind the front grill of the automobile. Figure 6 illustrates a set of prototype UWB transceivers designed for the U.S. Navy to provide a wireless intercom capability on-board Navy aircraft. The prototype UWB transceivers provide multichannel, full duplex, 32 kb/s digital voice over a range of approximately 100 meters. An ultra wideband waveform was selected because of its ability to operate in severe multipath (created by multiple RF reflections inside and around aircraft), and because of its non-interfering, low probability of intercept signature.

Current intercommunications systems (ICS) designs for aircraft utilize lengthy, and often unwieldy, cords to physically attach the crewman’s headset to a distributed audio (intercom) system. Such physical attachment presents a safety hazard to personnel, impedes movement throughout the platform and reduces mission effectiveness. Replacement of these mechanical tethers with wireless RF links is a desirable alternative. Frequency of operation for the WICS transceivers was again in the L-band region (1.2 to 1.8 GHz). One of the unique features of the WICS design was the use of a frequency division multiplex, time division multiple access (FDM/TDMA) strategy for full duplex, multi-user operation. Because of the extremely short duration pulsewidths and resulting low energy densities, UWB systems are much less vulnerable to intercept and ECM attack than conventional RF communications systems. As a consequence, they also minimize interference to other on-board electronics, such as sensitive flight control systems, GPS, etc. With an extremely low duty cycle, a very low power drain can be achieved, thereby providing communications capability for mission life exceeding 12 hours. The WICS program has recently received additional funding to further improve and miniaturize the design. One of the most recent applications of UWB communications technology is to the development of highly mobile, multi-node, ad hoc wireless communications networks. Figure 7 illustrates such a system currently under development for the U.S. Department of Defense. The system is designed to provide a secure, low probability of intercept and

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detection, UWB ad hoc wireless network capability to support encrypted voice/data (to 128 kb/s) and high-speed video (1.544 Mb/s T l ) transmissions.

A parallel effort, currently funded by the Office of Naval Research under a Dual Use Science and Technology (DUS&T) effort is developing a state-of-the-art, mobile ad hoc network (MANET) based upon an Internet Protocol (IP) suite to provide a connectionless, multihop, packet switching solution for survivable communications in a high link failure environment. The thrust of the DUS&T effort is toward commercialization of UWB technology for applications to high-speed (20+Mb/s) wireless applications for the home and business. A UWB application which bridges the gap between communications and radar is that of precision geolocation. Also see accompanying paper (Fontana, 2000). Figure 8, for example, illustrates a system designed to provide 3-dimensional location information utilizing a set of untethered UWB beacons and an untethered, mobile UWB rover. Precision location is derived from round trip, time-of-flight measurements using packet burst transmissions from the UWB rover and beacon transponders.

The system in Figure 8 utilizes a 2.5 ns, 4 Watt peak, UWB pulse, again operating in the 1.3 to 1.7 GHz region. Line-of-sight range for the system is better than 2 kilometers utilizing small, omnidirectional vertically polarized (smaller) or circularly polarized (larger) antennas. Within a building, the range becomes limited by wall and obstacle attenuation; however, ranges exceeding 100 meters inside have been attained. An unique feature of the system is the ability to detect the pulse leading edge through the use of a charge sensitive, tunnel diode detector. Leading edge detection is critical to the resolution of the direct path from the plethora of multipath returns produced from internal reflections. The UWB geolocation system was originally developed to permit a soldier to determine his or her position to within 1 foot resolution in an urban environment. It is currently being used to augment a video capture system for 3-D modeling, and for materiel location onboard a Navy ship.


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Figure 9 illustrates an ultra wideband system designed as a precision altimeter and obstacle/collision avoidance sensor. Originally developed for the U.S. Marine Corps' Hummingbird unmanned aerial vehicle, the sensor has proved capable of detecting small diameter (0.25" or 6.35 mm) suspended wires to ranges beyond 250 feet. With a peak output power of only 0.2 Watts, the system operates in the C-band region from 5.4 to 5.9 GHz (8.9% fractional BW) and has an average output power at 10 kpps of less than Range resolution of the radar was better than one foot utilizing the leading edge detection capability.

For the Hummingbird application, the system incorporated a linear forward-looking phased array (cf. Figure 9 right), and broad beamwidth side-looking antennas, for use in autonomous control. Interestingly, a predecessor of Hummingbird was developed for the U.S. Naval Air Systems Command as a multifunction precision altimeter, collision avoidance sensor and low data rate communications system. A 1 Watt version of the radar operated as a precision (1 foot resolution) radar altimeter to an altitude of better than 5000 feet. Several variants of the Hummingbird radar have also been developed. For example, Figure 10 illustrates an ultra wideband backup sensor for the detection of personnel, vehicles and other objects behind large construction and mining vehicles.

Operating with approximately 250 mW peak in the C-band region from 5.4 to 5.9 GHz, the backup sensor utilizes a dual antenna configuration for the detection of objects as close as 1 foot to beyond 350 feet from the vehicle. Ultra wideband provides a significant advantage for this application because of the ability to provide precision range gating to eliminate clutter which, with conventional Doppler-based sensors, often results in large false alarm rates. This sensor was developed for the National Institute of Occupational Safety and Health. Another variant of the Hummingbird collision avoidance sensor was developed as part of an electronic license plate for the U.S. National Academy of Sciences' Transportation Research Board (Figure 11 below). The UWB Electronic License Plate provides a dual function capability for both automobile collision avoidance and RF tagging for vehicle to

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roadside communications. Collision avoidance functions are achieved with a miniature, 500 MHz bandwidth C-band UWB radar; and RF tagging functions are accomplished with a low power, 250 MHz bandwidth L-band system.

The UWB C-band radar utilized a 0.2W peak power (4 µW average) waveform to achieve a range of better than 100 feet against other vehicles, with an accuracy of better than 1 foot. The L-band tag operated with a 0.3W peak power (500 µW average) packet burst transmission to achieve a data throughput of 128 kb/s over a range exceeding 800 feet. An ultra wideband solution was chosen for the Electronic License Plate because of its precision ranging capability (radar mode) and high multipath immunity (tag mode). Another short range radar, this time operating in the X-band region of the spectrum, is shown below in Figure 12. This prototype sensor was developed for the U.S. Army Missile Command as a low probability of intercept and detection (LPI/D), anti-jam, radar proximity sensor for medium caliber, small caliber and submunition applications. The system exhibited an operational bandwidth of 2.5 GHz with a 10 GHz center frequency. Specifically designed for very short range applications (less than 6 feet), the UWB sensor has a 6 inch range resolution. With an average output power output of less than 85 nanowatts, a -4 dBsm target could be detected at a range of approximately 15 feet using small, microstrip patch antennas.

A variant of the X-band UWB radar fuze is currently being developed for DARPA's Micro Air Vehicle (MAV) program under a Phase II Small Business Innovation Research (SBIR) contract. Figure 13 illustrates a mockup of a 4 inch micro helicopter with an array of four X-band UWB antennas. Weight and size are obviously driving factors for this design, and a UWB chipset is being developed for an onboard collision and obstacle avoidance sensor.


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Figure 14 illustrates a UWB intrusion sensor radar which was designed for throughthe-wall penetration. With an L-band operational frequency and 33% fractional bandwidth, this system utilizes a 1 Watt peak UWB waveform to detect personnel through several intervening walls.

Broad area surveillance coverage was provided for both in-building and outdoor field environments. An extended range system was also developed to detect and track human targets at distances exceeding 1000 feet. Figure 15 illustrates the switched antenna array used with this broad area surveillance system. Target azimuth and distance are determined and used to point a camera in the direction of the target.

CONCLUSIONS Ultra wideband technology has its origins in the development of time-domain (impulse response) techniques for the characterization of linear, time-invariant microwave structures. The advent of the time-domain sampling oscilloscope (Hewlett-Packard c. 1962) and the development of techniques for subnanosecond (baseband) pulse generation provided the requisite tools for further basic research. In the early 1970's, impulse or baseband techniques were applied to a large number of potential applications ranging from low cost, high resolution radar to specialized communications systems having low probability of detection and low interference potential.

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Within the United States, much of the early work in the UWB field (prior to 1994), particularly in the area of impulse communications, was performed under classified U.S. Government programs. Since 1994, much of the work has been carried out without classification restrictions, and the development of UWB technology has greatly accelerated. This paper has illustrated a number of recent UWB developments in the fields of communications, radar and localization. A graphical summary of some of these applications, for both the military and commercial markets, can be seen below in Figure 16.

ACKNOWLEDGMENTS The author wishes to thank Dr. Gerry Ross for his introduction in 1984 to this fascinating technology. He also wishes to thank Mr. Robert Mulloy, MSSI, for his continued support in the development of new applications for this technology over the past 15 years. REFERENCES Bennett, C.L. and Ross, G.F., 1978, Time-domain electromagnetics and its applications, Proceedings of the IEEE, Vol. 66, No. 3, pp. 299-318. Fontana, R.J., 1997, A novel ultra wideband (UWB) communications system, in Proceedings MILCOM 97. Fontana, R.J., 2000, Experimental results from an ultra wideband precision geolocation system, in EuroEM 2000. Robbins, K., 1972, Short Base-band Pulse Receiver, U.S. Patent No. 3,662,316. Ross, G.F., 1963, The transient analysis of multiple beam feed networks for array systems, Ph.D. dissertation, Polytechnic Institute of Brooklyn, Brooklyn, NY. Ross, G.F., 1966, The transient analysis of certain TEM mode four-port networks, IEEE Trans. Microwave Theory and Tech., Vol. MTT-14, No. 11, pp. 528-547. Ross, G.F., 1968, A time domain criterion for the design of wideband radiating elements, IEEE Trans. Antennas Propagat., Vol. 16, No. 3, p. 355. Ross, G.F., 1973, Transmission and reception system for generating and receiving base-band duration pulse signals for short base-band pulse communication system, U.S. Patent 3,728,632.

A LOW POWER, ULTRA-WIDEBAND RADAR TESTBED

Tim Payment Time Domain Corporation Huntsville, Alabama USA

ABSTRACT The availability of custom timer and correlator chips has enabled the design and development of a general purpose UWB research instrument by Time Domain Corporation. One configuration of this tool serves as a basic UWB Radar Testbed suitable for research on antennas and algorithms aimed at specific applications of UWB radar. The system includes the UWB radar unit, an antenna assembly, and a personal computer for user interface and data storage. The radar unit houses the UWB transmitter and receiver under the control of an embedded processor that accepts commands from and provides data to the controlling PC over an Ethernet link. The antenna assembly consists of separate transmit and receive antennas mounted on a back reflector and cabled to the radar unit via connectors to allow experiments with alternate antennas. The system can be viewed as a radar response or received waveform capture device. The waveform is displayed and the user can save the received waveform data to a file for post processing and/or application-specific algorithm development. This radar is particularly useful as a design and development system where a commercial product is envisioned as the ultimate end goal. While the form factor of the Radar Testbed (with its PC, Ethernet, radar unit, and external antenna unit) does not resemble an end product, the fundamental UWB building blocks in this system are applicable to a variety of products. Similar versions of this basic UWB radar research tool are in development to support research into communications and geo-location applications. INTRODUCTION A Radar Testbed system for investigating low power ultra-wideband (UWB) radar applications has been configured as shown in Figure 1. The Radar Testbed consists of a personal computer running a custom application under Microsoft Windows NT, a PulsON™ Application Demonstrator (PAD) in a radar configuration, and a radar antenna assembly. The Radar Testbed operates at a low RF power level. Low power in this context is defined as RF transmissions on the order of 10 µW. Radar Testbeds are operated by TDC Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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under Special Temporary Authority (STA) to support the development of products in compliance with the FCC Waiver of June 1999 [ref. 1]. Waiver compliant devices are limited to 30 nW / MHz (effective radiated power) along with other operational constraints. At these power levels the energy from many transmitted signals must be accumulated to form useful radar returns. This paper provides an overview of the Radar Testbed System, describes its key modules and subsystems, shows sample waveforms, outlines some research directions, then concludes showing how similar subsystems might fit in a candidate product. First, some terms are defined. Definition of Terms and Acronyms

UWB

Ultra-WideBand - RF systems whose signal bandwidth is 25-100% of the center frequency TDC Time Domain Corporation, Huntsville, Alabama, USA PulsON™ Trade name given to TDC’s time modulated, UWB pulse technology PAD PulsON™ Application Demonstrator - Three board set of hardware, software, and firmware in an enclosure designed to interface to a Personal Computer. Provides the basic UWB Radio Frequency (RF) functions needed to investigate communication, geopositioning, and radar applications. SMA Sub-Miniature A connector for coaxial cable Pseudo-random Noise PN STA Special Temporary Authority issued by the US Federal Communications Commission (FCC) to allow otherwise restricted emissions at a specific location for a specific period of time. monocycle short pulse approximated by the first derivative of a gaussian function FPGA Field Programmable Gate Array Low Noise Amplifier LNA Application Program Interface API Application-Specific Integrated Circuit ASIC User Interface UI DESCRIPTION OF RADAR TESTBED MODULES AND SUBSYSTEMS The personal computer used in the Radar Testbed system is a conventional, Pentium™-based computer running Microsoft Windows NT™ 4.0 operating system. A high capacity, removable storage device facilitates data transfer when the testbed is used in a stand-alone mode of operation. A second Ethernet card supports a network connection

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when available. A custom Radar Testbed application, developed in Visual Basic, serves the returned waveform acquisition, review, and archive needs of a radar engineer. The antenna assembly for this system is adapted from a previous, prototype UWB radar unit developed by TDC in 1998 and designated as the RV1000 [ref. 2]. Both the Tx and Rx antenna elements are a magnetic slot design with horizontal polarization. These magnetic antenna elements are mounted 1.3 inches from an aluminum plate, which serves as a back reflector. SMA connectors allow alternate antenna/back reflector assemblies to be substituted for the basic assembly as appropriate for investigation of specific applications. The key module of this testbed is the PulsON Application Demonstrator (PAD). The PAD was first described by Petroff [ref. 3] in Sep 1999. A simplified block diagram of a PAD in a radar configuration is shown in Figure 2. A connection is established between

the embedded processor and the host PC via the Ethernet link. The PAD accepts commands that configure the fixed and programmable logic, executes in accordance with those commands, and returns waveform data to the PC. The gate array provides a register interface between the embedded processor and the hardware. The gate array implements pseudo-random noise (PN) coding and provides 24 digital timing control signals to each of the two Timer subsystems. The PN coding serves to smooth the emitted RF spectrum, mitigate range ambiguity, and enable co-site operation of multiple units with different codes. One Timer subsystem provides the trigger signal to the Pulser which generates a wideband monocycle pulse (add figure??) to the Tx antenna element. The other Timer subsystem provides the trigger signal to the Correlator subsystem for sampling the received waveform at a precise time with respect to the transmitted pulse. Both timers work in conjunction with the 20 MHz master clock. A low noise amplifier (LNA) and a filter provide signal conditioning of the received RF prior to the Sampler subsystem. The Timer and Sampler subsystems under control from the FPGA form the heart of the PAD. Each of these subsystems use first generation, TDC-developed ASICs. The Timer subsystem is depicted in Figure 3. Digital signals from the gate array provide inputs to the Timer ASIC both directly and via high-speed D/A converters such that timing trigger pulses can be placed at a resolution of 3 picoseconds. System jitter limits the timing accuracy to approximately 20 picoseconds. The details of this Timer ASIC and its

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operation are described by Rowe et al. [ref. 4]. For the case of a Tx timer, the trigger is routed to the Pulser circuit which drives the Tx antenna element. A duplicate Timer subsystem provides the trigger to the correlator. The Sampler subsystem is diagrammed in

Figure 4. The trigger from the Rx Timer subsystem is coordinated with the Tx Timer subsystem such that a small (~180 ps) sample of the RF signal is taken by the Correlator ASIC at a precise time difference relatative to the transmitted pulse. More details of the Correlator ASIC are described by Dickson and Jett [ref. 5]. This analog sample is converted to a digital word and read by the gate array as a raw sample. Many (typically 100’s to1000’s) raw samples are summed by the gate array to form an integrated sample for each point in time relative to the transmitted pulse.

The Field Programmable Gate Array (FPGA) provides the real time control signals for the Timer subsystems and reads the raw samples from the Sampler subsystem as shown in Figure 5. Various other control signals from the gate array to the subsystems are omitted for clarity. The gate array is closely coupled to the embedded StrongARM® processor by a number of registers. The embedded StrongARM® processor provides the next higher level of control and gathers a sequence of integrated samples to form the return waveform. These waveform segments are passed to the controlling PC. The embedded software on the StrongARM®


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is linked with an Application Program Interface (API) module on the PC via the two-node Ethernet. From an application viewpoint, the PAD and its API module on the PC can be viewed to a great extent as a black box. Figure 1 is redrawn as Figure 6 to emphasize such a black box view. Other applications are under development at TDC that take a similar view of the PAD and its API, but take advantage of comparable PAD functions for communications and geo-positioning instead of radar waveform functions.

RADAR TESTBED SYSTEM OPERATION The users of this system are Radar Engineers. The Radar Testbed system allows the Radar Engineer to specify how a UWB waveform is to be measured, view the returned waveform (as measured or differenced from a reference), and file the results for use by other analysis tools. The parameters listed in Table 1 are user specified. Pulse Repetition Frequency (PRF) controls the average time between RF transmission of each pulse. A PRF of 5 MHz sets an average of 200 ns between pulse transmissions. In theory, return waveforms could be 200 ns in duration (corresponding to 100 ft or 30 m in range), however the PN coding reduces the useful response time to 175 ns as the instantaneous time between pulses is varied from 175 to 225 ns. The code file values control these instantaneous variations. Code values and the length of the code file are selected to smooth

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the RF spectrum and allow co-site operation of multiple testbeds without interference. The Rx Start and Stop times allow the user to focus on a particular portion of the returned waveform. The units are fundamentally time, however the Radar Testbed application allows the user to alternately specify these values as distances (feet or meters). Similarly, Step Size allows the radar engineer to specify the granularity of the return

waveform sampling in time (or distance). Regardless of how the user specifies Start, Stop, and Step, the values are converted to the nearest integral number of internal clock counts prior to scanning the waveform. Integration specifies the number of pulses and received raw samples to be used in the measurement of each point in the returned waveform. Integration can be viewed as a dwell duration or pulse count for each point. The final user parameter is the Rx Gain. The correlator ASIC includes an RF gain stage that can be varies from 0-30 dB. This gain is adjusted to keep the received signal level at a measurable level. A portion of the measured waveform from a loop-back test with an SMA cable and attnuators from the pulser output to the sampler input is shown in Figure 7. The positive pulse has a width of approximately 700 picoseconds.


With transmit and receive antennas connected in place of the loop-back cable, Figure 8 shows a typical response with and without a designated target. Annotations on the figure identify the coupling from the transmit to receive antennas, and the response of the

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environment, including the target of interest. With the addition of the Tx and Rx antenna elements and a back reflector, the waveform of Figure 7 gets somewhat more complex as seen in the antenna coupling region of Figure 8. Two waveforms are plotted, a reference waveform taken without the target, and a response waveform with the target. The target in this case is a 14” square aluminum plate perpendicular to boresight at a range of 6 feet. Only minor differences in the two waveforms can be seen in the vicinity of 6 feet. By saving a reference waveform taken without the target present, the Radar Testbed can be set to show only the difference between the current scan and the stored reference. Such a difference waveform is shown in Figure 9.

The Radar Testbed Systems are serve as TDC internal tools for UWB radar research. Data from these systems are being used to complement previously measured values from earlier systems. A number of experiments are planned, including signal propagation through walls, antenna element comparisons, and target polarity response. This information will be compared with alternate UWB propagation results measured using a scanning receiver as reported by Withington, et al. [ref.6] and used to support system design trade studies for specific applications. To support these experiments, Radar Testbed systems are cart-mounted to facilitate movements between labs and positioning of antennas. Figure 10 identifies the major components of a radar testbed system.


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The form factor of the Radar Testbed is appropriate for a research tool, but differs significantly from the form of a commercial product. However, the key subsystems used in the Radar Testbed can be directly applied to candidate products. Algorithms and user interfaces developed on the PC of the Testbed would be embedded in a final product. Figure 10 shows a candidate mapping from the Radar Testbed to a Radar Product. The performance of the key RF subsystems in the product can be directly predicted from their performance in the testbed system. The Testbed provides the means to validate the antenna configuration, algorithms, and user interface prior to the product decision. The areas not proven by the Testbed are the actual performance of algorithms on the embedded processor, the quality of the user interface on the embedded display, the product power system, and the product enclosure. While these items are by no means trivial, their specification and development are not unique to UWB systems. Considerable expertise exists in the implementation of these non-UWB areas.

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CONCLUSION The pieces for UWB products are falling in place. First generation ASICs have given rise to Timer and Correlator Subsystems. These subsystems have been implemented in a general-purpose box under control of software and firmware. This box, under control of a PC and equipped with appropriate antennas, serves as a UWB testbed for (in this case) radar. The path from testbed to product is straightforward. Just as yesterday’s circuits became today’s ASICs, today’s subsystems and applications are earmarked for tomorrow’s ASICs to reduce the size, power, and cost of future products. ACKNOWLEDGEMENTS The author wishes to thank the entire engineering team at Time Domain Corporation. Without the first discrete radios, the timer and correlator circuits would not have existed. Without those circuits, the ASICs could not have been defined. Without those ASICs, the subsystems, firmware, and software being implemented in the PADs and other systems would not be possible. These subsystems and modules allow meaningful research into system applications and the development of products. REFERENCES

1.

FCC Waiver to TDC, Letter to David Hilliard from Dale Hatfield dated 29 June 1999.


2.

3. 4.

5.

6.

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M. Barnes, “Covert Range Gated Wall Penetrating Motions Sensor Provides Benefits for Surveillance and Forced Entries,” Presentation at 1999 ONDCP International Technology Symposium, Washington, D.C., March 1999. A. Petroff, “Time Modulated UltraWideband Performance on a Chip,” Presentation at UWB Conference, Washington, DC, September 1999. D. Rowe, B. Pollack, J. Pulver, W. Chon, P. Jett, L. Fullerton, and L. Larson, 1999, “A Si/SiGe HBT Timing Generator IC for High-Bandwidth Impulse Radio Applications,” Custom Integrated Circuits Conference, San Diego, CA, May, 1999. D. Dickson and P. Jett, “An Application Specific Integrated Circuit Implementation of a Multiple Correlator for UWB Radio Applications,” S38P6, IEEE MILCOM 1999, Atlantic City, NJ, November 1999. P. Withington, R. Reinhardt, and R. Stanley, “Preliminary Results of an UltraWideband (Impulse) Scanning Receiver,” S38P3, IEEE MILCOM 1999, Atlantic City, NJ, November 1999.


ULTRA-WIDEBAND PRINCIPLES FOR SURFACE PENETRATING RADAR

J. Sachs*, P. Peyerl**, M. Roßberg*, P. Rauschenbach*, J. Friedrich** *Ilmenau Technical University, Germany [email protected] **MEODAT GmbH, Germany [email protected]

INTRODUCTION Surface Penetrating Radar (SPR) uses the properties of RF- and microwave pulses to penetrate into soil and most non-metallic building materials. Obstacles in the way of propagation cause reflections which may be received outside the body of investigation. Thus SPR can be used to detect hidden objects and to investigate the internal composition of manifold structures. The SPR antennas are moved over the surface of the body of interest touching it or retaining a certain distance depending upon the type of employment. The difficulty of the method is in the interpretation of the gathered data as the waves are sensible to all variations of the permittivity and conductivity within the body and not only to the objects searched for. Furthermore the relative long wavelengths (cm and dm range) of the sounding waves provide radar images which are not immediately accessible to the common optical interpretations of the human being. It is to be expected however that these drawbacks will be overcome in the future by using sophisticated software tools for data interpretation and an improved method of data gathering. Thus, SPR will be a powerful tool in many applications such as non-destructive testing in civil engineering, testing and surveillance of transport routes (roads, railways, bridges, tunnels etc.), environmental protection, detection of anti-personal landmines etc. The assumption of successful data processing is based on high quality data which could also include multistatic and polarisation information according to the specific application. In what follows, aspects of wideband measurement technique for SPR data gathering will be emphasised. It should however be noted that high quality data in terms of SPR not only means precision measurements of scattering amplitude as a function of time (or frequency) but also of space. This - the antenna positioning - will not be regarded here. A SPR device will be considered as an LTI-system (linear time invariant). The most important technical parameters will be derived attempting to find a common base in order to compare


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the different working principles. The basic wideband principles will be evaluated and a new principle will be introduced.

LTI-MODEL OF A SPR ARRANGEMENT The aim of a SPR-device is to gather information from an object under test by the use of the scattering properties of electromagnetic waves. In order to do this, the object is touched by a wave and its reaction to this wave is measured. In the simplest case, two antennas are used (see Figure 1) - one for sending and one for receiving. The use of one antenna for both - sending and receiving - is rare because of antenna mismatch. The application of arrays with more than two antennas will however be increasingly seen as an area may be scanned in a faster way and multistatic and polarimetric data may provide more information about the body under test. It will be usual to introduce two interfaces in a SPR arrangement. From the standpoint of an SPR-user and a simple interpretation of the images, the radiators are referred to virtual point sources (see Figure 1) which are considered as sources of spherical waves. In contrast to that, the measurement plane, to which we will restrict further consideration, is more common from the standpoint of the radar electronics. It is defined by the input/output channels and for antenna arrays respectively). The antennas are attributed to the measurement object by embedding the real object of interest. It is referred to [1] for relating between both interface concepts. Regarding the measurement plane, the radar electronics represent a two-port measurement device (N-port-device in case of an array) and the body under test plus its embedding (further called system under test) may be looked upon as a linear two-port (N-port). Assuming the antenna displacement during the measurement time (observation time) is negligible

time independent behaviour can be supposed and classical network theory can be applied. In (1), c means the propagation velocity of the wave, the maximum displacement speed of the antennas and B is the bandwidth of the sounding wave. At fixed antenna positions, the system under test is completely determined by its N by N scattering matrix S: for the frequency domain

or for the time domain Herein a is a column vector of the normalised guided waves incident to the antenna feeds, b is a column vector of the normalised guided waves leaving the antenna feeds and S is the scattering matrix of the system under test. S(f ) represents a set of Frequency Response Function (FRF) and S(t) a set of Impulse Response Functions (IRF). They are mutually referred by the Fourier Transform. Underlined symbols mean complex valued functions and refers to the convolution. The individual functions of the S-matrix will permanently change through the moving of the antennas over the ground. They represent the reflection behaviour (monostatic mode) or of the antenna i at position and the behaviour of the transmission path or between the antennas i and j (bistatic mode) at positions and In practice, these functions form the so called radargrams (B-

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scan) and radar volumes which serve to interpret the inner structure of the body under test. Generally the time representation of the measurement results are preferred because it is more accessible to human imagination. The Fourier transform however permits the change of the domains at will so that software procedures may also act upon frequency domain data if advantageous. Regarding the current state of development, it should be stated that for simplicity only the transfer characteristic or, between one antenna pair is measured. The use of several antenna pairs at the same time is rare in practice. But a further improvement of the SPR-technology supposes antenna arrays and will only arise if sophisticated correction of systematic device errors is applied. Especially the last point demands the knowledge of the full S-matrix of the array. Only the network analyser which is not useful to employ in the field currently meets the stability requirements for error corrections with respect to random fluctuations and drift. Stable and integrated ultra wide band electronics with excellent noise suppression are required for further development of highly sophisticated SPR devices.

PERFORMANCE PARAMETERS The key features describing the performance of the radar electronics in an SPR device refer to its spatial resolution in range and cross range to the observation range R (unambiguity range), its sensibility for detecting weak reflecting objects and to the measurement rate These parameters have to transform to corresponding properties of the IRF or FRF measured by the radar electronics. For details on the performance of the whole radar device, the reader is referenced to [2]. To illustrate the facts, Figure 2 indicates an idealised curve of the IRF of the transmission path resulting from the simple situation in Figure 1. Pulse like sections appear which provide clues as to overall length and attenuation of the individual

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propagation paths. These results are finally used to reconstruct the inner structure of the body. The first two impulses in Figure 2 will merge if the antennas are in contact with the surface.

Spatial resolution. The range resolution depends upon the capability to distinguish between two adjacent pulses of equal amplitude. It is evident from Figure 2 that this may be expressed as:

whereas τ is the half-value width of the pulse envelope and B is the corresponding bandwidth. The effective usable bandwidth B of a SPR device is not only determined by the bandwidth of the stimulus signal and the antennas but also from the time jitter arising from instabilities in the transmitter and receiver circuit. The cross range resolution also strongly depends upon the pulse width if synthetic aperture processing is applied as well as the beamwidth of the antennas.

It should be noted that the carrier frequency does not apparently play any role in equations (4) and (5). It is however very critical since the stimulation band should be at frequencies as low as possible to avoid unnecessary damping of the sounding waves in the body under test. This leads finally to the requirement of a large fractional bandwidth for the electronics and the antennas which is not always easy to achieve in practice. Observation range: The observation range R (unambiguous range) depends upon the length of the time window for which the impulse response is measured. In the case of periodical stimulation signals, it is limited to its period T to avoid time aliasing.


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Measurement rate: The time which is needed to gather all the data which is included in a complete IRF or FRF will be called observation time As such, the repetition rate for the measurement is

In case of an antenna array of L elements the measurements rate factor L [3][4].

may be reduced by the

Detection limit: The detection limit describes the capability to find still small scattering amplitudes in the IRF that are caused either from small scatterers with poor dielectric contrast or by propagation loss. The sensitivity of an SPR device is limited by all deviations from a straight flat base line within the IRF if not any scatterers are present. This means, referring to Figure 2, that only the first impulse may appear. All deviations from that represent errors which limit the sensitivity. Depending upon the error phenomenon, several parameters are usual to characterise these deviations by the notation signal-to-perturbation-ratio. Let us suppose that the maximum detectable amplitude of the IRF is then the following three values can be defined: signal to noise ratio

peak to side lobe ratio

peak to spurious ratio is the effective value of random noise , the maximum side lobe amplitude and the maximum spurious lobe amplitude. Note that the values above are in respect to an IRF which may also result from signal processing, thus can be much higher than the real measured signal amplitudes. Corresponding holds for the perturbation values and which may be reduced in the digital domain by averaging, deconvolution and certain kinds of error correction. It is distinguished between and for the following reason: is caused by linear effects (ripples in the stimulation band, abrupt cutting of frequency band, device internal reflections etc.) and is the result of non-linear effects in the receiver and crosstalk by clock lines or similar. Very hard constraints with respect to may arise for shallow target detection since the scattering peak is located very close to the main lobe where the strongest side lobes can also be found. Note however that the antenna is often the most critical component in this respect. Finally, the maximum depth / minimum size of a scatterer may be estimated from the effective system performance

1 Note, that system performance is seen as a ratio between max. transmitter amplitude and effective noise amplitude, whereas refers to a more usable value respecting receiver overload, non-linearity etc.

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corresponds to the attenuation of the strongest transmission path, e.g. antenna breakthrough or surface reflection. The key to a high side lobe and spurious suppression is sophisticated hardware design. These phenomena are however systematic errors, thus they may be corrected by an appropriate calibration routine. With regards to linear effects, the corresponding methods are well known from network analyser theory (response-, 3-term-, 12-term-calibration etc.). Comparable procedures seems also to be applicable to reduce the influence of spuriousness. It must however be noted that software corrections are only successful if all the required data2 is available and the system is working stable over time – that is equidistant sampling and poor in random noise, jitter and drift.

MEASUREMENT PRINCIPLES Three basic measurement principle are known which are mainly distinguished by the kind of stimulus signal that is applied. In what follows, they will be shortly introduced for the example of a transfer measurement and their advantages and disadvantages will be mentioned. Impulse technique: The impulse technique uses the fact that the convolution (3) may be simplified if the stimulus represents a Dirac-like pulse Referring to the notations in Figure 1 this provides

Since can be measured by an oscilloscope, the image of its screen represents the IRF if the bandwidth of the stimulus pulse is larger than the bandwidth of the antennas. Nearly all commercial available SPR-devices work on this principle because of its simple technical implementation. In order to concentrate the energy of the stimulus in the pass band of the antennas, monocycles are rather used than pulses. The impulse generation is largely based on avalanche transistors or step recovery diodes and sequential sampling circuit serves as receiver front-end. In the following, some problems of pulse systems are summarised. Often, the measurement rate is relatively low because the avalanche transistors need time to recover from the pulse shocks and moreover only one data sample is captured per pulse. This limits the use of the method in large arrays and high-speed applications. It is reported in [6] on a module with improved measurement rates to overcome this drawback. Peak power is however lost. The mean energy of a pulse is very low even for relative high amplitudes and the noise bandwidth of the sampling gates is very large. The method is therefore sensitive to random noise. The noise influence may be suppressed either by averaging (further reducing the measurement rate) or by generating extreme high voltage impulses having a peak power up to 100 MW and more [7]. This however represents no practical solution for an industrial application of the SPR method. The sampling gate control is undertaken by voltage ramps. Thus, any inadequateness of these ramps such as non-linearity, temperature drift or noise translates to errors of the

2

For correction of linear errors this means for example, that the full S-matrix must be known which drastically increases the system complexity. 3

The vectors and of antenna location and the matrix representation for multi-channel systems will be omitted for simplicity in the following


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time axis (drift, jitter and non-equidistant sampling). However, the ability of the sampling control to blank-out the troubling antenna breakthrough and surface reflex is often of advantage in pulse methods for deep sounding. On the other hand error corrections as mentioned above are less promising in cause of the drift susceptibility of sequential sampling. Possibly new system designs may overcome these drawbacks [5]. Sine Wave Technique: The sine wave technique determines the IRF roundabout way over the complex

by the

Herein are IFT{} the inverse Fourier transform, the complex valued cross spectrum between stimulus and receiving signal and is the real valued auto spectrum of the stimulus which should be constant over the bandwidth of the antennas. In the technical implementation, a sine wave is stepped or continuously sweeped over the band of interest and the cross spectra is measured via quadrature modulators in the IFband of an heterodyne receiver (network analyser, vector receiver). The step width of the frequency steps determines the unambiguous range. Sometimes, for deep sounding purposes, gated network analysers are used in order to blank out leakage signals such as ground reflection or antenna coupling [8]. Attention should be paid to the inverse Fourier transformation if the stimulation band is smaller than the antenna bandwidth. The side lobes in the impulse response are no longer determined by the antenna response but rather from the abrupt breakdown of the stimulation spectrum These side lobes may be suppressed by windowing the data before transformation but this results in slightly reducing the range resolution. The potential of the method is its excellent drift stability and random noise suppression because of the narrow band receivers as well as its flexibility within the choice of the stimulation band. It is however also the most expensive and slowest method. The FMCW-radar represents an attractive alternative to the stepped frequency radar because of its simplicity, measurement speed and dynamic range. It is based on a homodyne receiver using a stimulus continuously sweeped over an appropriate band. Particular problems arise by a non-linear VCO characteristic thus further expense is necessary (PLL, reference delay or similar). A FMCW-radar is only able to determine the real part of the FRF due to the lack of a quadrature modulator. Consequently, sophisticated calibration routines such as for network analysers are not available and the IRF can only incompletely calculated. A network analyser is more robust against spuriousness caused by a nonlinearity in the mixers than the FMCW-principle since its narrow IF-filters may partly reject intermodulation products.

Correlation Technique: From the theoretical standpoint, the correlation technique is the most flexible method of system identification since it is not fixed to a certain kind of test signal. Comparable to equation (12) the IRF of a device may also be determined by:

in which the correlation functions

and

are defined according to

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The side condition in equation (14) is much more weaker than that in equation (12), because it does not demand a specific shape of the test signal. It refers only to a flat spectrum: constant. In practice this means that the auto correlation should be short compared to the impulse response and with few side lobes. The opportunity to choose different types of test signals opens the possibility to optimise the measurement method with respect to several aspects. The favourable noise suppression of the correlation is based for example on this feature because high energy signals with low amplitudes (small crest factor) can be applied. The handling of such signals is also often easier than that with high peak power. The problem is however to find an appropriate procedure for solving the correlation integral (15). Several solutions are known from which the matched filtering is the most common but these filters cannot be built for ultra wideband purposes with low centre frequencies. Thus matched filtering by analogue filters is not applicable for SPR purposes. White random noise is a good choice for a stimulus signal if interactions between different radar devices should be prevented or if the device is working in a non-cooperative environment (military application). In that case, the conversion of (15) is based on a mixer, a stepped or sweeped delay line and a low-pass filter [9], [10]. The bottleneck is however the delay line which is expensive to manufacture and its properties limit the overall device behaviour to a large extent. This makes it unattractive for industrial SPR use. The answer for ultra wideband principles can only be to carry out the correlation/matched filtering in the digital domain. The simplest way to do this is the so-called polarity correlator. It captures only the zero crossings of the signals. Multiplication, delay and summation is undertaken by digital circuits (XOR, shift register, counter for bit-wise summing). The method could not be realised up till now despite its simplicity. One reason may certainly be found in its time consuming data capturing. A new ultra wideband principle working on a digital correlation/matched filtering will be shortly introduced in the following (see also [3], [4] for more information). MBC-RADAR As noted above, the correlation technique opens the possibility to freely choose the test signal. Thus a great deal of other constraints may be taken into account in finding an optimum solution: Wideband signal (general requirement), Low crest factor signal in order to generate and handle high (mean) power signals (high signal to noise ratio ) by simple electronics, Periodic signal in order to apply undersampling for signal acquisition and averaging for noise suppression, Simple and stable generation in the RF- and microwave range, Simple and stable generation of acquisition clock, Simple and fast correlation algorithm (digital matched filtering), Ability to simply synchronise and to control multi-channel arrangements, High measurement rate, Integration friendly electronics, and High flexibility with respect to the technical implementation. The Maximum length Binary sequence Correlation Radar (MBC-Radar) meets these requirements. The maximum length binary sequence (MLBS) is a special kind of binary random code. Its time shape, auto correlation function and spectrum is indicated in Figure 3.


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An MLBS of order n may be generated by an n-stage shift register using an appropriate feedback. The MLBS period is where is the period of the system clock. Regarding the spectrum in Figure 3, it is useful to fix the equivalent sampling frequency of the receiver circuit to the clock frequency that means one sample per elementary pulse of the sequence. Thus the usable bandwidth B is limited to the range of the MLBS-spectrum. As long as the maximum clock rate of the shift register is respected, there are no limits in the choice of this rate. As such, the measurement system may be simply adapted to a specific measurement situation by varying the clock rate. Table 1 compares different circuit technologies with respect to the maximum bandwidth within reach.

One of the most important features of the new method is, that the sampling frequency in the receiver may be derived in stable manner from the clock rate by an m-stage binary clock divider so that Since the period length T of an MLBS always differs by one clock period from a power of two periods of the MLBS are needed to acquire the complete data set with the equivalent sampling frequency The practical implementation of the principle is demonstrated in the block diagram of Figure 4. Except for the emphasised part, all components are low cost commercial ICs. The

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whole system is triggered by a stable clock generator that pushes an n-stage shift register and an m-stage binary divider. The shift register generates the MLBS stimulus and the divider delivers the sampling clock which drives the S&H circuit, the ADC and an averager. Finally, a digital signal processor (DSP) calculates the cross correlation which is approximately equal to the IRF of the system under test:

Due to the periodicity of the MLBS, the function represents a cyclic cross correlation function which can be calculated very fast by the Hadamard-Transform.

The p-fold averager matches the signal acquisition rate to the processing speed of the DSP by reducing the data rate to Simultaneously, it increases the dynamic range of the captured signal. The dynamic range with respect to random noise and the overall observation (correlation) time results in:

Here b is the number of effective bits (ENOB) of the acquisition circuitry (S&H and ADC). The last two terms in (17) represent the signal processing gain by which the dynamic of the real signals is improved. is infinity for an ideal MLBS, independent from its order and is mainly determined by the linearity of the receiver circuit (ADC, S&H). The maxi-


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mum measurement rate of the method is mostly fixed by the DSP-hardware. The determination of 16 000 IRFs (consisting of 511 points) per second and more seems to be possible with modern signal processors. The structure of the MBC-radar is divided into three domains each having its own processing speed (see Figure 4). The interconnection between the different sections is formed by programmable digital blocks (divider, averager). This provides a great flexibility to adapt the system parameters to the requirements of the actual measurement and it permits a high flexibility within system design and manufacture. It should be further noted that the simple clock scheme and the ability to enable/disable the MLBS shift register simplifies the creation of multi-channel arrangements like antenna arrays since multiplexing of RF-lines is no longer needed.

CONCLUSION SPRs will achieve wide industrial employment if they succeed in offering adapted solutions for a specific class of tasks. These solutions will be based on sophisticated methods of digital data processing which require stable, high quality data, increasingly gathered by antenna arrays. The future challenge on RF-electronics is to meet these requirements. The main demands on the RF-electronics of a SPR device results from a large fractional bandwidth and a high dynamic range. Several wideband methods were presented. Impulse principles are the most frequently used followed by the FMCW-radar. Sine wave techniques are very flexible. They are particularly suited to laboratory experiments but their low measurement speed may limit field use. A new ultra wide band principle was introduced which is based on a maximum length binary sequence. It is a promising method to capture high quality data and it is suited for application in antenna arrays.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10]

R. Zetik, J. Sachs, B. Schneegast: Evaluation of antenna pattern for radiation in solid media. Proc. of IRS 98, vol. II, p. 629-38 T. Scullion, C. L. Lau, T. Saarenketo: Performance Specification of Ground Penetrating Radar. Proc. of GPR'96, p. 341-6 J. Sachs, P. Peyerl, M. Rossberg: A New UWB-Principle for Sensor-Array Application. Proc. of IMTC/99, vol. 3, p. 1390-5 J. Sachs, P. Peyerl: Ein neues Breitbandmeßverfahren für das Basisband. Workshop of German IEEE/AP Chapter on Short Range Radars, Technical University Ilmenau, July 1999( http://www.meodat.de/veroeff.htm) A. Schukin, I Koploun, A., Yarovoy, L. Ligthart: Evolution of GPR Antennas, Pulse Generators and Sample Recorders. Proc. of AP2000, Davos, Switzerland J. Warhus, J. Mast, S. Nelson: Imaging Radar for Bridge Deck Inspection. http://wwwlasers.llnl.gov/lasers/idp/mir/files/warhus_ spie/spiepaper.html and other publications around the MIR-module. P.R. Bellamy: Ultra Wideband Radar: Current and Future Techniques. Proc. of EUROEM 1995, p. 1620-6 G. F. Stickley, D. A. Noon, M. Cherniakov, I. D. Longstaff: Current Development Status of a Gated Stepped-Frequency GPR. Proc. of GPR'96, p. 311-5 R. M. Narayanan, Y. Xu, P. D. Hoffmeyer, J. O. Curtis: Design and performance of a polarimetric random noise radar for detection of shallow buried targets. Proc. of SPIE, vol. 2496, p. 20-30, Orlando 1995 R. Stephan, H. Loele: Ansätze zur technischen Realisierung einer Geschwindigkeitsmessung mit einem Breitband-Rausch-Radar. Workshop of German IEEE/AP Chapter on Short Range Radars, Technical University Ilmenau, July 1999


RAY TRACING ASSESSMENT OF ANTENNA ARRAYS AND SUBSURFACE PROPAGATION FOR GPR SYSTEMS

Steve Pennock and Miles Redfern Department of Electronic and Electrical Engineering, University of Bath, Claverton Down, BATH, BA27AY, UK.

ABSTRACT The design of antenna arrays for Ground Penetrating Radar systems has proven to be as much an art as a science. The assessment of the design of the antenna and its housing, together with the mapping of the subsurface propagation has been greatly assisted using the well established technique of ray tracing. In this work its use is extended to include a complex arrangement of electromagnetically different objects, such as is typical of the cluttered subsurface environment that may be encountered by a GPR system.

RAY TRACING ANALYSIS Ray tracing analysis methods are frequently applied in examining radio propagation. The size of the features within the environment, typically buildings or urban areas, are rather smaller than the signal wavelength and the ray tracing analyses are based on geometrical optics assumptions. Dielectric slab waveguides have been studied in the past, with a propagating ray interpretation of the analysis. The Green’s function for the slab can be written in terms of a summation over the finite number of discrete bound modes (N) and an integral over the continuous radiation spectrum [1, 2]:

where

is the transverse wavenumber. As shown in Figure 1 the discrete bound modes


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are guided along the slab by repeated total internal reflection, and the radiation modes can propagate at all possible angles in the region surrounding the slab. Hence by launching rays out from a source at numerous angles, we can then approximate the radiation integral, and explicitly launch the discrete modes.

The environment being studied in the 2-D model used here is divided into a number of equal sized rectangular cells. Each cell is assigned its own permittivity and conductivity. The simulation is based on geometric optics and a large number of rays are launched from the transmitter at equal angle increments. Each ray is traced while its field strength is more than a pre-defined power cut off level. The field of the ray in a cell is:

where K is an amplitude constant, is the cumulative pathlength to that point and is the cumulative attenuation due to reflection and transmission coefficients during transmission of the ray. This is:

where and indicate the reflection coefficient and transmission coefficient respectively whenever the ray is reflected or refracted during its path. and refer to the cumulative attenuation and phase constants as the ray has propagated through the cells. These propagation constants are defined by:

where is the pathlength of the ray in medium is the attenuation constant of medium and is the phase constant of medium i. Both reflection and refraction effects are taken into account. Continuity of propagation constant is used between adjacent cells as in the studies of dielectric slab waveguides and some derivations of the Fresnel equations. This is used to evaluate the wave characteristic impedances in each cell and from this the amplitude and phase of the reflection and refraction coefficients are determined for all combinations of permittivity and conductivity between adjacent cells. The usual spreading loss is applied as the ray propagates and in lossy media the additional conduction loss is also included. The total group delay of each path is also recorded, allowing for simple reconstruction of the delay spread profile at any point in the environment.

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In some instances, particularly close to the transmitter point, many rays will enter a cell that represent the same wavefront. All but one of these are eliminated by examining the effective source positions and cumulative delay of the rays. In addition, refracted rays that originated at adjacent angles from the source, and have the same history of reflection and refraction through the environment, are also reduced to a single ray. This can prevent duplication in later calculations and reduce the filestore space that is used. The information needed to reconstruct the field strength pattern at any point within each cell, such as the angle of entry, distance from source and polarization of each ray are recorded at the boundary of every cell. The effects of all rectangular obstacles, be they lossless dielectric, lossy dielectric or finite conductivity conductors, are included in the model. This model is also able to consider particular transmitter and receiver antenna radiation patterns if needed. The total field at a position is simply the complex sum of the fields of the M individual incident rays:

The model has been seen to compare well with measured radio propagation data in the past [3] and in the study of Inset Dielectric Guide antennas [4]. In addition, as it is not restricted to considering only a finite number of re-reflections, it is very well suited to modeling complex environments.

ANTENNA ANALYSIS The ray tracing and field reconstruction analysis has been applied to the study of an antenna array for a GPR ground mapping system. The system, illustrated in Figure 2, uses one transmitter and 2 receiver antennas mounted within dielectric blocks.

Data available in the literature shown that typical subsurface media, sand, soil, rocks etc have permittivities in the range [5]. A system radiating from air into this will naturally encounter a reflection at the air/ground interface. To reduce this we examine the situation where the antennas are realised in a dielectric block. In particular the standard PCB substrate FR4 has a permittivity of which gives a reasonable match to typical ground media.

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Considering a half wavelength dipole antenna say, its length is reduced by a factor of Hence, a 500MHz half wavelength dipole is about 14 cm long in FR4, rather than 30cm in free space. In addition, if a back reflector is used behind the dipole, the spacing is reduced by the same factor. With this implementation in mind the antenna blocks considered here are typically 24cm × 24cm × 5cm for dipole antennas radiating between 200 M H z and 800 M H z.

A 3 head antenna is considered that consists of 3 blocks with 1.5 cm air gaps between adjacent blocks. These are positioned on the surface of the ground. The ground is modeled using data for relatively wet sand or loam. The calculated field is shown in Figure 3. While a considerable amount of the radiation enters the ground, there is also a significant radiation into the air region over the unit. This gives cause for concern due to its EMC implications and also for the fact that the GPR system would be rather sensitive to objects in the immediate vicinity of the GPR system. In an urban area these objects might well be moving during the measurement cycle and complicate the signal processing. One basic method of alleviating this problem is to place a ground plane on the back of the blocks. The field pattern for this configuration is shown in Figure 4. Clearly the radiation into the air is considerably reduced. The field pattern in the ground is quite complicated and can vary rapidly, indicating a multipath interference situation. The ray path diagram shown in Figure 5 clearly shows that the edges of the blocks are acting as secondary sources. In these diagrams the multipath nature of the signal is apparent, most particularly when there is a target under the unit. The case where there is no target present shows that the multipath interference changes with position under the unit, and so the response to a target must vary with the position of the target. This will complicate the signal processing, as a simple target signature or template cannot


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be used for all target positions.

To alleviate the multipath problem small conductor blocks are placed on either side of the antenna housing dielectric blocks. The field patterns for this configuration are shown in Figure 6. The ray tracing analysis used here does not include diffraction effects at the edges of the conductor blocks and is limited in the prediction that the receiver is perfectly isolated from the transmitter in the absence of a target. The field pattern under the unit is however more uniform as some of the additional effective sources have been reduced or eliminated. Multipath pattern are apparent towards the edges of the illumination area. Introducing the metal target results in a finite signal in the receiver block to the left of the transmitter as desired. Multipath interference is

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more apparent on the left. This is caused by re-reflection of signal from the antenna head back into the ground, and is comparable to the reflection that would occur at the ground/air interface if no antenna head were present. Introducing these conductor blocks into the antenna array has improved the illumination into the ground under the unit, but multipath patterns are apparent towards the edge of the illuminated area. In the system considered there are two receivers, which provides the opportunity to use convolution techniques to enhance the system response to targets directly below the unit center line while reducing the response to off-center targets that lie in the multipath affected areas.

EXAMPLE TARGET SCANS The analysis has been used to simulate the response of the two receiver GPR system over an example environment. In this 3 targets were positioned at horizontal locations of -60cm, 0cm and 40cm, with respective depths of 40cm, 15cm and 40cm. The response of the system was modeled as it was moved in 2.5cm increments over a range of 2 metres. The unit operates as a FMCW radar sweeping from 250MHz to 750MHz. The basic raw data produced by the system is shown in Figure 7. The direct path signal between the transmitter antenna and receiver antenna is apparent at all measurement positions, occurring at a constant frequency bin number of 12. Higher frequency bin numbers correspond to greater depths within the ground. The classic hyperbolic signatures of the targets are apparent with peak signal levels about 10 dB below the direct path. The situation is somewhat cluttered due to the interference between the various signal paths in this environment. Using the data at a horizontal location of 100cm as a reference, which is subtracted from all the other measurements, the influence of the direct path can be virtually removed and concentrates the system on the buried targets. In addition a gaussian symmetry filter is applied where the signal is multiplied by the gaussian function of the difference between the two channels. When the two channels receive the same signal, as is the case when the target lies directly under the center line of the system, the signals are not affected as they are multiplied by one. When there is imbalance, as in the case of off-center targets, the signals are reduced in amplitude by the gaussian function.


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The net response for one of the channels is shown in the right hand plot in Figure 7. Clearly the direct path has been greatly reduced and the hyperbolic signatures of the targets is now clearly visible. The reference was taken at 100cm and the targets were still visible to the system at that point. This produces the constant depth signatures in the plot near frequency bins 100 and 140. Convolving the signals from both of the two measurement channels produces the data plot in Figure 8. The constant depth false targets appear in the two individual channel responses at different depths, and are greatly reduced by the convolution process. The major responses in these plots are then clearly at the target positions, as desired .

CONCLUSIONS The modelling of the GPR radar antenna array, together with the propagation of the microwave signals through the subsurface medium has produced a clear insight into the operation of the system and its response to buried targets. As a result improvements to the antenna housing and signal processing system have been identified and evaluated.

References [1] T. Rozzi & M. Mongiardo, “Open Electromangetic Waveguides”, IEE Electromag-

netic Waves Series 43, London, 1997 [2] S.R. Pennock & P.R. Shepherd, “Microwave Engineering with Wireless Applica-

tions”, MacMillan Press, London, 1998 [3] Ch. Ghobadi, P.R. Shepherd & S.R. Pennock, “A 2D Ray Tracing Model for Indoor

Radio Propagation at MM Frequencies and the Study of Diversity Techniques” IEE Proc.-Microw.Antennas Propag. Vol. 145, No. 4, p349-353, August 1998. [4] S.R. Pennock, Y. Weizhong & T. Rozzi, “Circuit and Antenna Properties of Diodes

Mounted in Inset Dielectric Guide”, Proc. 23rd European Microwave Conference, Madrid, Sept 1993, Paper A10.5 p553-555.

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[5] Daniels DJ et al. Introduction to subsurface radar. IEE Proc. Pt. F, 135(4):278– 320, August 1988.

GROUND PENETRATING RADAR SYSTEM FOR LOCATING BURIED UTILITIES

Steve Pennock and Miles Redfern Department of Electronic and Electrical Engineering, University of Bath, Claverton Down, BATH, BA2 7AY, UK.

ABSTRACT In this paper the design and operation of a ground penetrating radar system based on the Frequency Modulated Continuous Wave (FMCW) technique is described. Simulation and example measurements over a test site containing three targets at separations typical of those occurring for real buried utilities are presented. It is seen that the position of the targets is accurately found, and that quite closely spaced targets can be distinguished. The operational range of the system is approximately up to depths of 2 metres, depending on the ground conditions.

INTRODUCTION With the growth in buried energy and communications utilities in our pavements and roads, the demand for mapping these facilities has increased to ensure both human safety and to minimise disruption to services during trenching operations. More recently, the burying of optical fibre ducting for cable communications has added a further complication that is difficult to detect using established mapping techniques. Unfortunately, although breaking these links presents few safety problems, it can have significant financial implications. This has produced a need for mapping tools that can locate optical fibre ducting as well as pipes and electricity cables in the range of just below the surface to one or two meters in depth. Ground penetrating radar is one of the few technologies capable of locating this variety of buried objects [1, 2]. The use of pulsed radar is well established using the sub-nanosecond pulses that are Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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required for this short range probing. These systems unfortunately tend to be expensive and their operation generally requires skilled surveyors to interpret the results. With the dramatic growth in the availability of microwave electronics for the mobile telecommunications market, inexpensive devices are readily available for low energy radar techniques. The Frequency Modulated Continuous Wave (FMCW) technique offers the promise of inexpensive ground penetrating radar implementations. This coupled with the availability of inexpensive and very powerful portable computing systems makes user-friendly systems a real prospect.

THE SYSTEM A basic FMCW ground penetrating radar system broadcasts a simple linear frequency sweep of an RF signal, and measures the difference between the frequency of the return from the target and the currently transmitted signal. This frequency difference between the transmitted and received signals provides an indication of the presence of buried objects and the distance of the object from the transmitter/receiver antennas. As such it is very amenable to detecting short range targets by suitable choice of the frequency sweep rate.

The addition of a second receiver antenna, as shown in Figure 1, provides a differential facility that is able to highlight an object located directly beneath the transmitter/receiver antenna array, or on an axis at an angle from this centre line. The transmitted frequency range is dictated by the propagation characteristics of the media in which the objects are buried. Although high frequencies are preferred to provide higher resolution of the location of an object, lower frequencies are preferred for their lower signal attenuation through the sub-surface media. A compromise is therefore required in the choice of operational bandwidth. The attenuation characteristics of an infill medium are very variable, however typical data is available [1]. From this, practical trials, and the desire to have a system dynamic range of 70dB between transmitted and received signal, an operating frequency range of from 250 MHz to 750 MHz was chosen. In order to minimise concerns over potential interference with electronic equipment, the transmitter output power was limited to 23dBm.

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The transmitter and receiver antenna use a basic tapered dipole design, with the three antenna embedded in a dielectric of similar characteristics to the infill in which the objects were buried. This ensures a reasonable match between the antenna array and the infill reducing the complications caused by surface reflections.

The electronic system is shown in Figure 2. This and the antenna array were configured to be symmetrical around the transmitter channel. The only compromise to this symmetry was the use of a ’simple’ single end feed to the antenna rather than a balun connection. Experience demonstrated that the consequences of this were minimal. The non-ideal characteristics of the relatively inexpensive VCO required compensation using a non-linear control drive signal. The exact waveform of this control signal was determined using a pre-programmed calibration procedure. This takes measures over a known and matched RF path that is switched in to replace the transmit and receive antennas, and then deduces the required variation in control voltage from a Hilbert transform analysis. The frequency modulation period was chosen to provide a de-modulated signal in the audio frequency range of dc to 2.5kHz for targets buried up to 2 metres in typical infill. The sweep duration was about 20 msec, and this enabled fairly simply signal conditioning, sampling and processing in a commercial personal computer. The use of a PC also provided an easily used man-machine interface. During the sweep the gain of the transmitter and receiver amplifiers are altered to compensate for the fact that the signal propagation through the soil to the target and back suffers from an attenuation that increases with RF frequency. This is a relatively simple process that applies a simple gain control voltage profile with time during the sweep. This can easily be adjusted to suit local soil conditions. In a pulsed system such compensation would require an RF pulse compression filter to be applied to the typically used sub-nanosecond width pulses. Adjustment for soil types would require

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control of the pulse compression filter characteristics.

SIGNAL PROCESSING The central processor unit controls the VCO output frequency, and by using the calibration data derived for the VCO, produces an output signal to the transmit antenna whose frequency varies linearly with time from 250MHz to 750MHz. Amplitude control is realised by controlling the gain of the receiver and/or transmitter amplifiers. The basic aim of this is to produce a linear ramp sweep with constant amplitude. The received and transmitted signals are multiplied in the mixer, and the resulting product is low pass filtered to produce the output If the reflected response from the target is constant with frequency, the output from a single target is a constant frequency sinewave burst whose duration is that of the original ramp. The frequency spectrum of this output signal is the combination of two sinc functions [3, 4]:

where is the delay for the signal to propagate to the target and back over the path length d in the soil medium whose refractive index is The rate of change of output frequency with time is the basic parameter that is set to ensure that the return from the buried targets is within a prescribed bandwidth. Each buried target will produce such a response, with the peak frequency revealing the time delay and hence depth to that target. Combining the outputs of our left and right channel measurements as:

the resulting response concentrates on targets below the center of the unit, while reducing the response from those off the center line. The line of concentration can be made to rotate under the unit, and thereby ‘scan’ the subsurface targets, when the combination of the left and right channels is:

Accurate reproduction of the angular positions of the targets requires a knowledge of the refractive index of the subsurface medium, which is generally unknown. Information is however available as to which side of the unit the target is, allowing the unit to be re-positioned over the target and thereby accurately finding the lateral position of the target. The response from the targets can be further enhanced by suitable matched filtering to suppress random noise, and by clutter filtering to remove particular received signals. Such a signal that can cause difficulty is the direct path signal between the transmitter and receiver antenna. In the experimental system, this direct path signal


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was further suppressed by means of the antenna design [5].

TARGET SCANS The first target scan, shown in Figure 3, shows the results from one receiver channel to three objects buried in a test site at known positions. As such this shows the unit operating as per Equation 2. The three targets were located at horizontal position of 290 cm, 330 cm and 390 cm and at depths of 30 cm, 10 cm and 30cm respectively. To the practiced eye, two of the three objects are identified while the image of the third, although identifiable, is masked by a shallow disturbance in the infill. Such an output can hardly be considered user friendly.

The second scan, shown in Figure 4, shows the result of subtracting a reference signal measured at the site, but displaced to the side. This reduces the contribution of the direct path between the antennas, making the 3 targets more apparent. Figure 5 shows the effect of combining the responses of the two receiver systems and thereby concentrating on the targets directly beneath the antenna array. This shows the unit operating as per Equation 3. In this, the two clear objects are easily highlighted and the third object is more clearly seen than before. The positions of these objects are also more clearly defined. We have also evaluated the improvements offered by matched filter and clutter filters. Figure 6 shows the result of applying clutter filtering to one of the channels. The clutter response used was that of the reference signal taken on one side of the main

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measurements. Clearly the targets can be accentuated, but the process can also throw up extra responses that the user or interpreting software need to take into account. The time involved in the extra signal processing is quite minimal, being less than a second per channel on a DX4-100MHz 486 processor. While the processing time for producing the scans shown here is relatively long, about 1 minute, this is seen as a short term problem. Computer processing speeds are continually improving, and the software can be further optimised. The entire process of data capture and processing is governed more by the time taken to move the system over the targets and capture the data.

CONCLUSIONS An FMCW Ground Penetrating Radar system has been described that is suitable for mapping utilities buried in typical street environments, together with examples of the responses obtained using a real test site. Through appropriate choice of the operating parameters, targets spaced at typical separations for buried utilities are clearly discernible in the display. The images may be improved on by the use of filtering, with some extra processing delay, but this need not be onerous. The system has demonstrated the ability to identify targets, and coupled with the availability of inexpensive RF and microwave frequency electronic components, the prospects for providing a user friendly mapping tool are encouraging.

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References [1] Daniels DJ et al. Introduction to subsurface radar. IEE Proc. Pt. F, 135(4):278– 320, August 1988. [2] Michiguchi Y et al. Advanced subsurface radar system for imaging buried pipes. IEEE Transactions on Geoscience and Remote Sensing, GE-26(6):733–740, November 1988.

[3] Olver AD et al. Portable FMCW radar for locating buried pipes. In Radar 82, volume 1, pages 413–418. IEE Publ. No. 216, 1992. [4] Yamaguchi Y et al. Synthetic aperture fm-cw radar applied to the detection of objects buried in snowpack. In IEEE Transactions on Geoscience and Remote Sensing, volume GE-32(1), pages 11–18, January 1994. [5] D. Park and J. Ra, Decoupled Planar Dipoles for Subsurface CW Interface Radar Microwave and Optical Technology Letters Vol 7, No. 8, pp367-370, 1994

COST EFFECTIVE SURFACE PENETRATING RADAR DEVICE FOR HUMANITARIAN DEMINING

J A Ratcliffe (DERA), J Sachs (TUI), S Cloude (AEL), G N Crisp (DERA), H Sahli (VUB), P Peyerl (Meodat), G De Pasquale (IDS)

INTRODUCTION A consortium of European companies and universities1 are collaborating on an EC supported project to develop a novel hand-held Surface Penetrating Radar (SPR) device which is able to detect Anti-Personnel Landmines (APLs). The sensor utilises a new Ultra-Wideband (UWB) radar principle based on Maximal Length Binary Sequence (MLBS) techniques, already successfully tested in acoustic wide band devices. A 6 element multi-static linear antenna array has been designed to fully exploit the radar technology. The array consists of unique planar bow-tie elements with distributed resistive loading. The data processing techniques investigated include the study of inverse problems, adapted to extract features of the reflecting targets, and to eliminate non-target related influences such as antenna and soil characteristics. Pseudotomographic methods extract measurements of the size and shape of the reflectors. This information is combined in a robust target classification algorithm. Acoustic impulse time-of-flight techniques are being used to register the movement of the antenna array in the X-Y plane. In this paper we present an overview of the key principles and techniques which are being exploited in the development of the device. THE DEMINE PROJECT The DEMINE project is one of many R&D projects partially backed by ESPRIT, an integrated programme of industrial R&D projects managed by the European Commission. Several of these projects are collectively aimed at researching, developing and testing new systems for detecting anti-personnel landmines. This paper specifically relates to one of these projects, entitled DEMINE. The DEMINE consortium consists of 4 companies, 2 universities and 1 demining organisation1. The consortium aims to develop a hand held SPR mine locator, which is


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capable of distinguishing between mine and non-mine targets. The handling of the device is to be as close as possible to that of the commonly used metal detector.

SYSTEM CONFIGURATION The planned operation of the device is shown in figure 1. The common use of a metal detector, with a safe demining lane delineated by a marker stick is adhered to. When the exposed area has been cleared the marker is moved forward. Ultrasonic beacons are located on the marker frame to allow the monitoring of the sensor position. This is useful both in the data processing and for marking the position of a potential mine.

DEMINE SPR PRINCIPLE Wideband Surface Penetrating Radar (SPR) is an attractive method for locating APLs as it is sensitive to metallic and non-metallic materials. Also, with careful selection of the operating frequency, adequate soil penetration can be achieved whilst not compromising the wave scattering from an APL-like object with respect to its size, shape and composition. However, the operating frequencies which may be used are still too low to generate an image which corresponds directly to the known optical appearance of the APL. As a consequence, more sophisticated characteristics of the scattering behaviour must be exploited for classifying buried APLs. The idea underpinning the DEMINE project is to gather a large amount of diverse information about a buried object by use of a sensor array. This then allows for the extraction of features with which to classify the object as ‘No mine’ or ‘Possible mine’. The classical SPR approach is shown in figure 2, using a single bistatic pair of antennas. One radar scan is performed at each position as the sensor is passed over the target. Time records are stacked together to form a Radargram or B-scan. A static signal

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from the ground is usually seen and the target returns form a hyperbola. Only one record is available from each sensor position and thus limited information is available.

The DEMINE approach is illustrated in figure 3. An array of antenna pairs is used which are arranged in a line. At each antenna position, 21 different scans may be made. That means, for each transmitting antenna, radar records may be recorded for all the receiving antennas.

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This yields much more information about the target’s scattering characteristics over different angles, thus allowing a more robust classification to be made. Secondly, the scanning requirements placed upon the operator are greatly eased due to the increased ground coverage of the array. Multi-static array processing may be employed in the lateral direction, while the longitudinal scanning motion allows a Synthetic Aperture process to be employed to build a 3-D image.

DEVELOPMENT TASKS Preliminary Examinations The signature of an R2M2 AP mine target buried in dry sand was simulated. The polarisation response, shape and spectral response have all been considered as potential classifiers of the target. In polarisation analyses it was shown that the signature of the target is dominated by an oblate spheroidal dipole response and hence polarisation is not a strong discriminant between mines and symmetric clutter objects such as stones and boulders. With respect to the shape of the target, it was shown that high resolution 2 or 3-D imaging may be useful for target recognition. The spectral transfer function was found to be similar to a mixed first and second derivative function with no significant resonant behaviour across the bandwidth considered. The full bistatic scattering characteristics of the mine were numerically modelled in detail. The scattering data was analysed in both the time and frequency domains. The conclusion was that an HH antenna polarisation would provide the highest signature variation with respect to bistatic angle. This is a desirable feature, as it will give diverse information about the target, thus aiding classification. The modelling showed that the most useful specular scattering information is only obtained above ~2GHz, and that below this the target behaves like a volume dipole scatterer. The frequency range for the sensor was therefore set at 1-4GHz. System Conception The DEMINE device is designed to be operated in a similar fashion to a metal detector. As such the proposed external hardware is illustrated in figure 4. The headphone and head display are included as a result of discussion with the demining partner who suggested that an audible alarm and a clear visual indication of ‘mine’ or ‘no-mine’ would be the best way to pass information to the user. The final device will utilise a backpack for the power supply and miniaturised computing hardware, but the prototype will instead make use of a separate base station with an umbilical cord to the backpack thus easing the weight considerations.


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UWB Antennas The simplest structure which satisfies the requirements of the DEMINE device is the planar Bow–Tie antenna. These antennas are non-dispersive and the input impedance depends only on the angle of the metal plates in a well defined way. Thus it is possible to match the impedance to the DEMINE radar chip by correct choice of bow-tie angle. However, optimisation of bow-tie antennas for a UWB application requires distributed resistive loading along the antenna plates as shown in figure 5. The resistance has the effect of absorbing the pulse propagating out along the antenna, so that when it arrives at the outer edge there is very little energy left. Thus there is no reflected pulse and the impedance will be flat over a wide frequency range.

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Also, the transmitted pulse, coming from the feed point alone, will be a much more faithful version of the input signal. The optimum antenna blade dimensions and resistive loading profile were found by simulation. As the DEMINE device will utilise an array of antennas, issues of antenna cross-talk and rear lobe screening were also considered. These results were then used as a basis for a practical antenna design, in which the blades were split into discrete radial segments to closely approximate the simulated design. The antennas were manufactured using PCB derived technology and are light, compact, and inexpensive. The antenna elements were then incorporated into a pre-prototype array, for the purposes of testing and gathering training data for algorithm development. The preprototype array was made up from two sub-arrays, each with only one active element. This allowed the simulation of the full linear array by adjusting the relative position of the two active antennas. Parasitic elements were added on either side of the active elements and terminated as though connected to the radar head, to accurately represent the interactions between adjacent antennas in the final prototype array. The sub-arrays were mounted on a computer controlled x-y displacement rig and connected to a network analyser via wide bandwidth baluns. Algorithm training data was gathered for targets in free space, targets on a soil surface and targets buried up to 15cm. Radar Head The DEMINE radar head hardware is fabricated in an integrated circuit using Silicon-Germanium technology. There will be one transmit-receive chip for each antenna pair in the array. Fabricating the radar hardware in this way allows easy integration with the antenna array without numerous lengths of cable and without a complicated R.F. switching network. The radar measurement utilises a Maximal Length Binary Sequence (MLBS) technique, which requires simple transmit and receive electronics. The operation of the radar head is described in detail in another paper2. Position Measurement Measurement of the position of the sensor head during data acquisition is essential for the construction of the 3-D radar image. Accurate instantaneous measurement of the position in the X-Y plane is the most critical aspect, while knowledge of the height change of the sensor head over each scan is also desirable for good quality imaging. The attitude of the sensor head during data acquisition is also an issue. Technologies assessed for feasibility in this application included accelerometer measurements, acoustic methods, RF methods, GPS, mechanical methods and optical methods. An acoustic method was considered the most appropriate for the X-Y measurement in terms of technical suitability, practicality, and cost. Figure 6 shows generically the concept that is employed. Three sets of ultrasonic transceivers measure the distance between 3 points on the ground and the 2 ends of the sensor head. This is achieved by measuring the time of flight of short bursts of ultrasonic energy. By utilising 3 points on the ground separated in height as well as laterally it is possible to remove errors caused by variations in the sensor height.


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However, the height baseline utilised on the ultrasonic system is not large enough to measure height with the accuracy required for the 3-D image construction. An accelerometer/magnetometer system is being investigated for this purpose. The system uses magnetometer data to correct tilt induced accelerometer errors. In order to alert the operator of excessive tilt during the scanning procedure, a simple fixed accelerometer will be used with filtering to remove accelerations caused by height change. Data Processing & Classification The DEMINE system will use a well defined and recognised data processing route as shown in figure 7. Raw data includes all the radar data plus all position data, both with time stamps. Data correction is operated on the radar data and is designed to remove unwanted artefacts before image reconstruction. This includes antenna cross-talk removal, equalisation of the different transfer functions of the different measurement channels and correction of certain known non-linear effects introduced by the radar chips. At the image construction stage the position data is combined with the radar data during multi-static array processing and synthetic aperture processing. Also the ground parameters (permittivity, attenuation, wave speed etc.) are estimated and included in these processes in order to enhance image quality. Certain features in the responses of targets and clutter objects can be exploited in order to tell them apart. These include spatial features which are dependant upon the size and shape of the objects, which are extracted using pseudo-tomographic methods, and time and frequency domain features which can yield other information about the scattering properties.

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A series of feature vectors is then available which may be used to classify the target. These are represented by a set of points in a multidimensional feature space. Each such point represents a given observation that must be classified according to its location in the feature space. The classification process thus reduces to the association of points in feature space with corresponding classes. Several classification processes have been considered, including: tree based classifiers, automatic adaptive classifiers, parametric classifiers, non-parametric classifiers and distribution based classifiers. A combination of parametric and distribution based processes has been considered the most appropriate, although this will be dependant upon well clustered features being found, so that simple classification boundaries can be constructed. However once such features have been found, the use of a parametric form allows relatively sparse training data to be used to train the classifier. Exploitation While the successful operation of an APL detector of this type would provide a great humanitarian benefit, the market for such a specialised device is small, especially given the limited resources and funding available to humanitarian demining organisations. This makes the unit cost of such a system comparatively high. So in order to make a system which is financially viable and affordable to the demining community, other commercial applications for the device or its subsystems must be found. So along with the main product of an SPR mine detector, the following additional products have been identified as having potential for future exploitation:


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Vehicle Based Mine Detector General SPR detector (pipe inspection, NDT, monitoring historic buildings...) UWB antennas and array New software for commercial applications (CAST) Integrated UWB radar chips Positioning system SPR detection / classification algorithms These and other possibilities will be assessed in a future phase of the project. CONCLUSION The current technical status of the project is summarised by the following main points: The concept of the SPR device is developed Preliminary analyses including simulations are complete UWB antennas are designed, manufactured and evaluated RF chips are designed, manufactured and evaluated Positioning system is under development The control system and man-machine interface is designed and under realisation The data processing route has been fully defined and is under development. In the next phases of the project the main focus will be on software development for data processing. Other aspects of work will be the assembly of the full array and connection with the radar chips, completion of the position measurement system and full integration of the sub-systems to complete the SPR device. Laboratory based trials are planned to test the device under known soil conditions and against known targets. Subsequent field tests against real targets in real environments are also planned depending upon the outcome of the laboratory-based trials. REFERENCES

1. The DEMINE Consortium: Applied Electromagnetics (UK) Defence Evaluation & Research Agency (UK) Ingegneria dei Sistemi S.p.A (Italy) Menschen gegen Minen e.V. (Germany) Messtechnik, Ortung und Datenverarbeitung GmbH (Germany) Technische Universitat Ilmenau (Germany) Vrije Universiteit Brussel (Belgium) THATI GmbH Erfurt (Germany) 2. J Sachs et al., Ultra-wideband principles for surface penetrating radar, EUROEM, (2000)


SOME PROBLEMS OF GPR SOFT- AND HARDWARE IMPROVING IN MINE DETECTION AND CLASSIFICATION TASK

Astanin L.Yu, Chernyshov E.E., Geppener V.V., Jatzyn A.S., Kostyleva V.V., Nicolaev V.A., Sokolov M.A., Smirnov A.B. Radioavionika Corp. P.O.B. 111, St-Petersburg, 198103, Russia (812) 251 3875, (812) 251 2743 E-mail: [email protected]

Introduction New ultrawideband (UWB) impulse ground penetrating radar (GPR, georadar) has been developed in Radioavionica Corporation. Alongside with broad possibilities of UWB georadars for civil application (pavement evaluation, building control, pipelines and cable detection, soil and water table depth mapping etc.) there are essential premises to use them in mine detection and classification [4]. Georadar is an effective tool for detection of subsurface object which dielectric permeability differs from a permeability of an environment. The main advantage is the possibility of detection not only metal but also small size (from 50 mm) low contrast dielectric objects due to high resolving power of UWB signals. Some results of theoretical and experimental researches carried out in Radioavionica Corporation will be described.

Modeling A processing package "GEO" was developed for synthesis and analysis of sequences processing procedures. It consists of interacted units realise different signal and image processing algorithms [3]. Brief description of program units is indicated below: «Spectral processing» - system of the objects spectral descriptions on basis of FFT spectra, autoregressive methods (Yule-Walker and Berg algorithms) and Prony method; also perform objects selection. «2D filtration» - system of the objects selection based on two-dimensional filtration. «Static subtraction» - system of the objects selection based on static subtraction of a base signal with matching. «Precision subtraction» - system of the objects selection based on precision subtraction of a base signal with matching. «Dynamic subtraction» - system of the objects selection based on dynamic subtraction of a base signal. «Signals modelling» - system of modelling signals reflected from target. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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«Transformation» - system of a grey scale images two-dimensional filtration and histogram transformation. «Segmentation» - system of a grey scale images segmentation and features extraction. «Classification» - system of a classification including discriminant analysis algorithms and a decision making based on fuzzy logic. The carried out researches have shown a preliminary possibility of a mine detection and classification task solution. The worked out optimal technological schemes were practically realise in original UWB radar.

Mine detection problem The problem of mine detection is divided into two stages with rather different requirements. At the first stage the problem of real-time detection is under consideration. The second stage deals with further processing for recognition and classification of detected object. Different methods are used for the solution of this problem. In GPR systems every receiving waveform is a superposition of a number of components: a direct coupling, a ground surface reflection, reflections from target or/and false target and multiple reflections, which are signals that make multiple round trips within layers of soil and antenna aperture. Ground surface reflection amplitude exceeds plastic object reflection amplitude for ten times. Commonly used methods are based on the whole waveform processing and provide the stable successful detection in the experimental setup comprises a sandbox and positioning system but they are not effective in practical demining.

Real-time Signal Processing The new approach has been developed for the solution of mine detection problem – step-by-step decomposition method. The essence of it is the reduction of uninformative components by subtracting model signal of every component from each of receiving waveform. The deducted signal is interpolated in advance for increase processing exactitude. So the term “precise subtracting” is used. The algorithm works with use of correlation technique for minimization of remains. The followed filtration effectively reduce the errors of subtracting. The method under consideration provides level of remains better than - 20 dB relative peak amplitude of raw signal. Step-by-step decomposition procedures for antitank metal and antipersonnel plastic mines are shown on Figure 1,a-c and Figure 2,a-c. Then not a single signal but an ensemble of signals is taken under examination and effected with spatial-time processing procedures. Algorithm of dynamic subtraction based on gradient filtering along the scan line provides the determination of object edges (Fig.1,d; Fig.2,d). Some statistical estimates were made for optimal choice of filter parameters. The segmentation procedure is applied to the energetic envelope of obtained signals. The results of this operation are used for automatic estimating of object size and depth of location. The obtained estimations serve for automatic selection of objects as ‘minelike’ or ‘not minelike’. If ‘minelike’ object is arranged then the system provides an audio alarm signal.

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As target reflection from antipersonnel plastic mines is comparable with noise value and antitank metal mine reflections exceed it to a marked degree the adapted threshold is used. The main advantage of suggested technique is the possibility of real-time processing in developed portable georadar. Hardware improving The signal/clutter ratio increases by hardware improving also. The separation of transmit/receive channels in georadar may be achieved both in space (biposition antenna system) and in feeder line (single antenna). For the second mode wideband microwave bridge is used where one arm is resistive loaded (Fig. 3,4).

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In the single antenna mode there is no direct coupling effect and the level of clutter depends on the matching of antenna aperture to air. Therefore it is rather simple to obtain time window free of clutter (Fig.5).

In spite of waste of power in monoposition antenna mode the levels of received signals prove to be comparable if the square of single antenna aperture is equal to the sum of squares of transmit antenna and receive antenna apertures in bipositional mode. Monoposition antenna mode is notable for more symmetrical antenna pattern and gives the possibility of operating more clearly to ground subsurface as the ground reflection signal prove always to be later in time window than reflection from antenna aperture.

Recognition and classification After ‘minelike’ object is arranged an operator interrupts the real-time processing. The last 100 waveforms stored in RAM are visualized on EL display. The ensemble of waveforms is presented as a gray scale coded intensity plot with depth on the y axis calibrated in time units and linear position on the orthogonal x axis. We implement image enhancement and several filtering techniques to improve the presentation of object. If there is no stable detection of object an operator may repeat scanning in order to get good performance. Multilevel segmentation algorithm is applied to radio image of object to obtain a number of slices corresponding to the different energy levels. The searching objects are man-made ones and they have a sophisticated inner structure (elements of construction). Therefore the obtained slices include the peculiarities which are usual for definite object (Fig..6, 7). It is necessary to note that the low level slices have numerous of peculiarities but such slices are noisy and thus their description is more difficult. The high level slices are similar for all types of objects and look like a spot.

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The problem of classification is rather complicated and includes a number of subtasks [1-2]: spatial-time signal processing, selected object features extraction, generation of standard tables of features, construction of deciding rules. The worked out subsurface object classification methods are based on the analysis of separate segments of image descriptions. The geometric type features are used: square, perimeter, lengths of the circumscribed rectangle sizes; parametrical features of an object contour; curvature features, camber and concavity features of the selected object contour. For the feature description the determined and fuzzy procedures are offered. Figure 8 demonstrates the segmented image samples for four mine types involved in tests. Table 1 gives the brief description of every type.

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The classification procedures are based on discriminant analysis methods, fuzzy logic and neural nets. In Table 2 some results of testing classification are presented. The solutions obtained on the various feature descriptions and various levels of segmentation are integrated.

Conclusion The described methods of step-by-step decomposition and dynamic subtracting provide real-time detection and further classification in portable georadar developed in Radioavionica Corporation. It makes a progress in the solution of demining problem.

References 1. Astanin L.Yu., Kostylev A.A.: ‘Fundamentals of ultrawideband radar measurements’. -Moscow: Radio i Svyaz, 1989; 2. Astanin L.Y. Kostylev A.A. Ultrawideband Radar Measurements: Analysys and Processing. By: The Institution of Electrical Engineering, London, UK, 1997. 3. L.Yu. Astanin, V.V. Geppener, V.A.Nikolaev, V.N. Kaftas'ev and M.A.Sokolov, «Methods for Visualization and Classification of Undersurface Objects and Structures Based on Broad-Band Radar Probing», Pattern Recognition and Image Analysis, Vol.8, No.3, pp.384-386, 1998. 4. Brushini K., Gros B., Guerne F., Piece P-Y., Carmona O. Ground penetrating radar and imaging metal detector for antipersonal mine detection. Journal of Applied Geophysics, v. 40, 1998, p.59-71.

TIME-DOMAIN SIMULATION TECHNIQUE FOR ANTENNA TRANSIENT RADIATION, RECEPTION AND SCATTERING

Anatoliy O. Boryssenko, Elena S. Boryssenko, Vitaliy P. Prokhorenko Research Company “Diascarb” Kyiv, P.O. Box No. 222, 02222, Ukraine

INTRODUCTION This paper gives an insight on some features of transient electromagnetic events related to antenna and scattering problems involving the classical aspects of transient electrodynamics and engineering issues. Such properties like near-field range effects, peculiarities of transient antenna in radiation, reception and scattering modes and others, which are not considered enough in literature, will be treated here. Reaching this goal rigorous and asymptotic analytical bounds for linear and wire-grid modeled antennas will be introduced. There are a variety of intuitively evident definitions here like pulse, ultra-wide band (UWB), transient, non-sinusoidal, non-stationary electrodynamics. Generally those phenomena can be treated from the point of view of energy beams (Zialkowski, 1992) as well as with its time history (Smith, 1997) or time-harmonic presentation. However inherent distortion of signal waveform is principal moment for electromagnetic pulse (BMP) simulators, high-resolution radars, spread-spectrum communications, electromagnetic compatibility (EMC) issue, VLSI and printed board design and so on. Generally each element of such system effects on signal waveform passing through it (Harmuth, 1990). Resulted signal is not rather simple replica of input waveform like in case of narrow-band or sinusoidal signal. Due to these reasons time-domain (TD) modeling of transient electromagnetic events is more preferable than frequency-domain (FD) techniques despite their mathematical equivalence due to the Fourier transform. Traditionally numerical approaches to the transient electromagnetic problems are applied like FD method of moments with the next Fourier transformation or FDTD (Taflove, 1995). Also Baum (1965, 1971) developed analytical approaches with the Laplace transform for some asymptotic cases. Generally numeric studies, mostly applied, have principal drawback followed from sufficient programming and computing efforts. Finally the physical meaning of the most numerical solutions is not initially evident. Therefore we developed simple mathematical models, which enable numerical simulations with universal mathematical software like Maple, Mathcad, Matlab etc. The result of such simulations will illustrate the major points of our study. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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BASIC RESEARCH APPROACHES All known analytical approaches and techniques (Smith, 1996; Martin et al, 1999; Shivinski et al, 1997) are based mostly on: i) far-filed asymptotic solutions; ii) using of first time derivative of exciting signal for radiated field characterization; iii) analysis of electrically short and geometrically simple antennas; iv) consideration of transient antennas in transmitting mode. Sometimes far-field asymptotic solutions may be enough for such applications like wireless and radar systems, while other fields like EMC aspects of VLSI and high-speed printed circuit design, subsurface probing radar etc. demand more deep glance on near-field transient phenomena. Also it valuable involve in exploration, beside canonical structure like monopole and center-fed dipole, other antennas including biconical, V-shape, bow-tie, TEM-horn antennas. Those antennas are schematically shown with their wire-grid models in figure 1a. Analysis of such antenna structures in transmitting, receiving and scattering modes is practically important too. Modes of antenna excitation resulted from its edge loading (Boryssenko and Tarasuk, 1999) should be carefully treated, figure 1b. Please note that number of pulse passing along antenna can be connected with number of time derivatives affected on initial exciting signal discussed by Ziolkowski (1992). This can be done explicitly for electrically short antennas and far-filed range operation while finite length antennas and not far-range operation is characterized by quite different, rather not simple, signal transformations. Our primary goal in the presented study is straightforward expressions for characterization effects in mentioned above antennas and their operations modes. A relatively simple mathematical technique is developed here. Finally minimal programming efforts with Matlab, Maple, Mathcad are required for simulation to receive numerical results with productive physical meaning.

GENERAL VECTOR ANALYTICAL SOLUTIONS We start our exploration from a simple case of radiated monopole by introducing, as usually (Baum, 1968; Martin et al, 1999), the vector magnetic potential. That vector has for linear radiator, figure 2a, only tangential, z-axis, nonzero component (1) with respect to the observation point (2) in the given coordinates. Corresponding magnetic (3) and electric (4) field vectors are followed from Maxwell’s equations (Franceschetti, 1997).

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Let note that we do not consider here the scalar potential function and the result of the Lorentz gauge application that is included in (4). Also we do not study antenna excitation in detail that is specific boundary problem but assume that waveform of current in antenna is same as initial one (Martin et al, 1999). At this point the expressions (1)-(4) can directly transformed in numerically effective Maple code (Boryssenko, 2000) by using rampfunction approximation for antenna current proposed by Thomas et al, 1987. This approach is based on the Maple enhanced opportunities in symbolic computations. Results received in such mode are demonstrated later in this paper later. Also for the problems in figure 2 we developed here other analysis technique based on analytical transformations of (1)-(4). Many researchers (Baum, 1971; Smith, 1997; Martin et al, 1999) have done same before. But we will study numerically the complete range of solutions, not only far-field asymptotic. Following this way one can receive after mathematical manipulations formulas for the magnetic (5) and electric fields (6) in the Cartesian coordinates. Related expressions like (7) and (8) give integro-differential operators applied to the original antenna exciting waveform. Such operators are more general than the slant transform (Shivinski et al, 1997) and define the waveform transformation more exactly than time differentiation (Ziolkowski, 1992).

We introduced also additional definitions (9) including slow-wave factor to modify signal velocity in antenna and a retarded time with respect to observation point given by R quantity and antenna point, q, which is integration variable in (6)-(7):

Introducing now in (5)-(9) a slow-wave factor, and latter in (10) an attenuation factors, are important for studying special class of resistively loaded antennas to control antenna waveform by maintaining its single passing excitation, figure 1b. Next the current (10) in antenna, figure 2a, should be determined for its arbitrary excitation. One can do this by considering equivalent transmission line model with standing wave in straightforward mode or by incorporating the standard analysis technique with the Laplace transform.

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The reflection coefficients characterize antenna edge loading and [...] is a common floor operator, which gives the number of pulse reelections from both antenna ends. Receiving antenna, figure 2b, can be considered with simple boundary condition for conductive monopole illuminated by incident arbitrary, not only plane, electromagnetic wave where tangential component of electric field is taken into consideration. In contrast to dual radiating problem, figure 2a, in this case we have distributed antenna excitation in this case and final integrating along the antenna gives induced current (11) with respect to the point, p on the antenna where this current is evaluated.

The complete mathematical structure of (11a) is similar by main features to that (10) and not shown here due to limitations in paper space. One can use the expressions (5)-(l 1) for numerical simulation with Mathcad, Maple etc. It can be done with numerical integration and differentiation. Some obtained in that way results are shown and discussed later. Approximate, enough frequently for design practice, models for other antenna structures like those in figure 1a can be developed by using linear superposition of vector fields produced by each wire-grid monopole element, which composes entire antenna.

FAR-FIELD RANGE ASYMPTOTIC One can obtain a far-field region asymptotic by limiting transition (Baum, 1968) in (7)-(8). It has simple mathematical presentation in the spherical coordinates, figure 2. In this way elevation electric component of the radiated monopole field with single passing excitation (figure 1b) is expressed with (6).

Similarly, a current (voltage) at a load of monopole receiving antenna, figure 2b, exited in single passing mode by incident plane electromagnetic wave with waveform E(t), is determined with (13). Please note that expressions (12) and (13) are similar by some general features but quite different concerning the transformation of primary waveform of exciting signal due to integration -(13a). Some differences between transmitting and receiving antennas are summarized in Table 1 where two auxiliary functions (14) and (15) are used:

for a pattern factor (14) and a complex pattern function (15) of traveling-wave linear antenna with sinusoidal excitation. and are complex spectra yielded from the Fourier transform of the exciting signals s(t) and E(t). Table 1 illustrates also connection of antenna representation in TD and FD, as well as behavior of electrically short antennas used ordinary as a simplest field probe.


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SOME SIMULATION RESULTS AND THEIR DISCUSSION Near-Field Range Effect Without any loss of generality we consider simple case of monopole antenna with single passing excitation by Gauss-shape pulse of 1-nanosecond (ns) duration. These enables receiving clear physical picture with principal features common for all antennas and not complicated due to more complex geometry or excitation. Results of numerical simulations with Maple and Mathcad are shown in figure 3 and 4. Figure 3 with data computed with Mathcad illustrates near-field range effect when space observation point is chosen at different distances from antenna. One can observe asymmetric waveform of radiated field near antenna due to dc and low-frequency spectral component. Especially the presence or not of dc component can be used to characterize antenna operation range. Traditional Rayleigh criterion valid for sinusoidal signal can not be applied here due to broadband radiation (Zialkowski, 1992). We can introduce some criteria from the physical point of view that far-range field should demonstrate properties of an outward spherical waves. Such properties involve the amplitude change inversely proportional to radial distance R. Also ratio of principal electrical component amplitude to that of magnetic in free space must equal to (Boryssenko, 2000).

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Other illustration of the near-range effect in transient antenna is given in figure 4 for the observation point with elevation angle equals figure 2a. These data are computed with Maple (Boryssenko, 2000) for the case with same distance from antenna to the observation point but antenna length is different. Generally discrimination of antenna near/far field properties depends from ratio of antenna physical length and spatial length of exciting pulse. Notice in case of more complex antenna excitation, different from simple single passing or travelling wave of current (figure 1b) that exciting signal has a long time history. The last can results in more expansion in space of near field radiation as pointed out before by Ziolkowski (1992). Filter Network Presentation For far-field range system formed by pair of center-fed, pulse-driven, linear dipole elements (one terminated to transmitter and other to receiver) Zialkowski (1992) introduced the equivalent network presentation where main feature is a specific number of time derivatives applied to input waveform. So far we concentrated on the near-field range effects in antenna we present transient radio channel model with three same antennas operating in transmitting, scattering and receiving modes without any limitations concerning near or far range, antenna type and its excitation. Such generalized system is shown in figure 5 and can be simulated with presented above models. Each antenna in figure 5 is characterized by its own transformation operator A1,2,3. For example, figure 6 demonstrates results of Mathcad simulation with respect to the notations in figure 5. We have in this case three center-fed dipole antennas with double passing excitation and the effect of near-field range is clear visible in this figure.


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Figure 7 shows cross-link effect for a pair of closely spaced transmitting and receiving bow-tie antennas. Simulated waveform data, figure 7a, has been computed with Matlab wire-grid model while experimentally measured one is given in figure 7b. Both, computed and measured, curves have good agreement in early time behavior but different late time history due to effects in real system do not included in the presented models like non-ideal broadband antenna matching etc.

CONCLUSION Time-domain interpretation of non-stationary electromagnetic events, including nearrange effects, which is not enough shown in literature, is discussed here. Generally the results obtained with time domain simulations demonstrate more physical meaning and are more clearly dependent on the influence of problem parameters than those in frequencydomain. All presented above regularities are important for UWB or transient antenna design. Inherent transformation of signal waveform passing through components of UWB system especially its antennas should be carefully treated. The last is ordinary achieved with complex numerical computing. In this sense the benefits of proposed physically meaningful straightforward technique with easy Matlab, Maple etc. simulation seems valuable for

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research and engineering practice as well for academic goals. Generally the presented above approach allows the next steps in research when time-domain antenna arrays can be considered that are in progress now.

REFERENCES Baum, C., 1968, Some Limiting Low-Frequency Characteristics of a Pulse-Radiating Antenna, Sensor and Simulation Notes, 65. Baum, C., 1971, Some Characteristics of Electric and Magnetic Dipole Antennas for Radiating Transient Pulses, Sensor and Simulation Notes, 125. Boryssenko, A. A., Tarasuk, V. M., 1999, Ultra-Wide Band Antennas for Subsurface Radar Applications, in: Proceedings of Antenna Application Symposium, Monticello, IL, 478. Boryssenko, A. A., 2000, Time-Domain Vector Representation of Monopole Transient Electromagnetic Radiation by Using Maple Software, Submitted to IEEE Antenna and Propagation Magazine. Franceschetti, G., 1997, Electromagnetics: Theory, Techniques, and Engineering Paradigms, Plenum Press, New York. Harmuth, H. F., 1990, Radiation of Nonsinusoidal Electromagnetic Waves, Academic Press, Boston. Martin, G. R., Rubio, A. B., Gonzalez, S. G., 1999, Some Thought about Transient Radiation by Stright Thin Wires, IEEE Antennas and Propagation Magazine, 41: 24. Shivinski, A., Heyman, E., Kastner R., 1997, Antenna Characterization in the Time Domain, IEEE Trans, on Antenna and Propagation, 45:1140. Smith, G., 1997, An Introduction to Classical Electromagnetic Radiation, Cambridge University Press, Cambridge. Taflove, A., 1995, Computation Electrodynamics – The Finite Difference Time-Domain Method, Artech House, Boston. Thomas, D. E., Hutchins, R. L., Wiggins III, Nickei, F., S., 1987, Time-Domain Calculation of Radiated Fields, in: AP-S International Symposium Digest: Antennas and Propagation, Blacksburg, VA, 954. Ziolkowski, R.W., 1992, Properties of Electromagnetic Beams Generated by Ultra-Wide Bandwidth Pulse-Driven Arrays, IEEE Trans. on Antenna and Propagation, 40:888.

A COLLAPSIBLE IMPULSE RADIATING ANTENNA

Leland H. Bowen1, Everett G. Farr1, and William D. Prather2 1

Farr Research, Inc. 614 Paseo Del Mar, NE Albuquerque, NM, USA 87123 2 Air Force Research Laboratory, Directed Energy Directorate 3550 Aberdeen Ave. SE Kirtland AFB, NM 87117-5776

INTRODUCTION A reflector Impulse Radiating Antenna (IRA) consists of a parabolic reflector with a TEM feed. The IRA provides broadband coverage with a narrow beamwidth. This class of antenna has a considerable body of literature associated with both its analysis and measurementsl,2,3. Farr Research, Inc. has developed a Collapsible Impulse Radiating Antenna (CIRA) with outstanding RF characteristics. The approach selected by Fan Research for the FRI-CIRA-1 utilizes an umbrella-like design, with a reflector sewn from a very tough, electrically conductive mesh fabric. The CIRA is lightweight, compact, and easily portable with low wind loading and high mechanical ruggedness. The reflector for the FRI-CIRA-1 is 1.22m (4 feet) in diameter with a focal length of 0.488 m (F/D = 0.4). The antenna, when collapsed, measures 102 mm (4 in) in diameter by 810 mm (32 in) long. The antenna weighs 2 kg (4.5 lb.). We measured the characteristics of the antenna using the time domain outdoor antenna range of Farr Research. The time domain data were processed to obtain the normalized time domain impulse response (TDIR) as described by Farr and Baum4 and summarized in the next section of this paper. We made pattern measurements at 2.5° intervals in both the H and E planes and converted them to effective gain. The conversion from impulse response to effective gain is based on the derivation given by Bowen et al,5 and summarized later in this paper. We present the impulse response characteristics in both the time and frequency domains. We also present the effective gain on boresight as a function of frequency. Finally, we present the effective gain as a function of angle in the principal planes, at multiple constant frequencies.


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Normalized Impulse Response Before we get into the details of the antennas, we first review the parameters used to describe them. We can describe antennas in the time domain with an impulse response, of the form Here we provide a quick review of the derivation by Farr and Baum.4 We use somewhat simplified antenna equations that show only the response to dominant polarization on boresight without the propagation terms. It is straightforward to add these other effects later. In transmission mode, the antenna radiates a field on boresight, which is described by

where is the impedance of free space, is the impedance of the feed cable, r is the distance to the observation point on boresight, is the source voltage measured into a load, c is the speed of light in free space, and the “ ° ” symbol indicates convolution. In reception mode, the antenna is described by

where is the incident electric field on boresight. Note that the normalized impulse response, completely describes the behavior of any antenna in both transmission and reception. If we have both a transmitting and receiving antenna, we can relate the received voltage to the source voltage by combining the above two equations as

where is the normalized impulse response of the receive antenna and is the corresponding response of the transmit antenna. To calibrate our measurement system, we use two identical TEM sensors. In this case, the combined antenna equation becomes

The normalized frequency domain impulse response of the sensors can be extracted from (4) as

Once a calibration has been performed with two identical antennas, then we can measure the response of an antenna under test (AUT) by replacing one of the sensors with the antenna under test. The impulse response of the antenna then becomes

and the time domain normalized impulse response is found with an inverse Fourier transform. When making measurements on a focused aperture antenna, we normally extract an aperture height, which can be related to the physical parameters of the AUT. However, is difficult to measure directly, without making assumptions about the antenna’s feed impedance, so we will first find the effective height, The effective


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height is useful since (at midband) it relates the incident electric field strength (in V/m) to the voltage into a scope ( in volts) by a simple proportionality This expression is valid only when the full-width half-max (FWHM) of is much less than the measured signal. The midband effective height can be determined from (2) and the integral of the impulsive portion of the normalized impulse response, and it is calculated as

where the integral is over the impulsive part of it to using

Once we have

we can convert

where

In these equations, is the antenna feed impedance on the TEM portion of the feed arms, and is the transmission coefficient between the TEM feed arms and the feed cable. Note that the quantity is meaningful only in the context of a wideband antenna with a TEM feed. Note also that the quantity is impossible to measure directly – one can only approximate it after assuming that the feed impedance is a constant across the relevant bandwidth, and is purely resistive. On the other hand, we can measure without making approximations, as long as the time domain impulse response has an impulse-like wave shape. For this reason, provides a more useful description of antenna performance than Note also that when there is a balun in the circuit, a different form of the transmission coefficient must be used. For example, in the case of the IRA, there is a balun that matches the impedance of the antenna to a cable, using two sections of transmission line connected in parallel at one end and in series at the other end. In this case, the voltage is halved, so the transmission coefficient, is 0.50. The use of avoids this difficulty, by avoiding the need to define the transmission coefficient. EFFECTIVE GAIN It is frequently desirable to convert the impulse response developed in the previous section to frequency domain gain as defined by in IEEE Std 145.7 As we will see, for wideband applications the effective gain is a more useful quantity, because it accounts for impedance mismatch between the antenna port and feed cable. We provide here the derivation of effective gain from the normalized impulse response, We begin with the standard expressions in the frequency domain. Thus, the power received into a 50-ohm feed cable is where is the incident power density in is the effective area, and is a power transmission coefficient that accounts for the impedance mismatch between the antenna port and 50-ohm feed cable. Absolute gain is related to effective aperture by

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where is the effective gain, the gain after accounting for the impedance mismatch between the antenna port and the 50-ohm feed cable. The term “effective gain” has not yet been recognized by the IEEE Std. 1457, but it is in common use8. Combining the above two equations, we have

By taking the square root, and recasting into voltages, we find

where space,

is the cable impedance (generally

) and

is the impedance of free

Let us now compare the above equation to the standard equation for reception. Thus, we convert (2) of this paper into the frequency domain, obtaining

where is the normalized antenna impulse response expressed in the frequency domain. The normalized impulse response, is already known. To convert it to effective gain, we combine equations (14) and (15),

This formula allows us to convert our time domain normalized impulse response to effective gain, and this is the formula that is used in the remainder of this paper. Effective gain is simply absolute gain, as defined by IEEE Std. 145, multiplied by a transmission coefficient that accounts for mismatch between the antenna and feed line. For ultra-wideband (UWB) antennas, this is a far more useful version of gain than simple antenna gain (or absolute gain) as defined by IEEE Std. 145. This is due to the fact that impedance mismatch between the antenna and feed line is a large part of the challenge inherent in UWB antenna design.

CIRA DESCRIPTION Let us now provide details of the design of the CIRA. The cross section of the CIRA is shown in Figure 1, and a photograph of the open CIRA is shown in Figure 2. The reflector for the FRI-CIRA-1 is 1.22 m (48 in) in diameter with a focal length of 0.488 m (F/D = 0.4) The parabolic reflector is constructed of a very tough electrically conductive mesh fabric. This fabric is silver and nickel plated and has a resistivity of less than The wind loading on the antenna is low due to the high air permeability of the fabric. The reflector has 12 sections or panels that are supported on an umbrella-like frame with fiberglass stays. The stays are connected to the support at the rear of the antenna by aluminum pivots. The antenna is opened by sliding the yoke on the center rod toward the rear of the antenna. Turning the knob on the yoke locks the antenna open.


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An aluminum enclosure located at the rear center of the antenna houses the RF splitter and acts as the support for the antenna. The splitter was provided by Prodyn Technologies. It consists of a input impedance connector, which then splits into two cables. The cables attach to the feed arms at the feed point in a series/parallel configuration as is standard for IRAs with 4 feed arms. The feed arms are made from a combination of conductive rip-stop nylon and resistive polyester fabric. The feed arms are attached to copper tips that facilitate the necessary solder connections at the feed point. The copper tips are attached to a Teflon support on the end of the center rod. A cover made from ultra-high molecular weight polyethylene (UHMW) protects the electrical connections at the feed point. An SMA connector on the side of the splitter enclosure provides a connection to the antenna. A bracket attached to the splitter enclosure provides a standard 3/8"-16 thread tripod connection. The antenna can be rotated easily to either horizontal or vertical polarization by repositioning the tripod support bracket. The backside of the antenna with the splitter enclosure and tripod mount is shown in Figure 3. The large black knob shown to the right of the picture can be loosened to reposition the tripod bracket. Also shown in this figure is one of the coaxial cables between the splitter and the feed point. The collapsed antenna is shown in Figure 4. When collapsed, the antenna measures 102 mm (4 in.) diameter by 810 mm (32 in.) long. The antenna weighs 2 kg (4.5 lb.) and can be easily transported and set up by one person.

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CIRA DATA We measured the characteristics of the CIRA-1 using the time domain outdoor antenna range of Farr Research. The FRI-TEM-2-100 horn sensor used for these measurements is a standard sensor manufactured by Farr Research. These sensors are ultra-wideband electric field sensors designed for low dispersion and high sensitivity. They are calibrated using two identical sensors and the normalized impulse response procedure described earlier. The impulse response of this sensor has a FWHM of 47 ps. The clear time is 4 ns. The midband effective height of the sensor is 42 mm. These sensors overcome the problem of making fast impulse field measurements with derivative sensors, which have very low sensitivity and small effective areas at high frequencies. The TEM sensors are a half TEM horn mounted on a truncated ground plane. Four versions of TEM horn sensors are available from Farr Research9. We used a Picosecond Pulse Labs 4015C step generator to drive the TEM horn antenna. This step generator has a 4 V output with a 25 ps risetime. The response of the CIRA-1 was recorded using a Tektronix 11801B Digital Oscilloscope with a SD-24 TDR/Sampling Head. The distance between the antennas was 20 m and the height was 3 m. We measured the antenna pattern in the H and E planes at 2.5° increments from 0° to 45° off boresight. Also, the effective gain is computed and plotted on boresight as a function of frequency and at various frequencies as a function of angle in the principal planes. The test data for the CIRA-1 are as follows. The TDR of the CIRA is shown in Figure 5. The TDR at the feed point and along the feed arms is the best (flattest) we have achieved on this type antenna. In Figures 6–10 we show the on-boresight characteristics of the CIRA-1. The data were clipped just before the arrival of the ground bounce signal and then zero-padded out to 20 ns to provide frequency domain information down to 50 MHz. The FWHM of the normalized impulse response (Figure 7) is 70 ns. The CIRA proved to be usable from below 50 MHz to above 8 GHz, as shown in Figures 8 and 9. When deciding the distance at which to place the sensor, one has to realize that the far-field begins at a distance that is dependent upon the smallest FWHM one expects to measure. We expected a FWHM of around 100 ps, so we expected that a distance of 20 meters would be adequate. However, we were pleasantly surprised by the 70ns FWHM measurements of the improved CIRA. This narrower impulse width extends the far field to around 25 m, using the formula antenna radius, c is the speed of light in free space, and

where a is the is the FWHM of the


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radiated impulse response. While there was no opportunity to make new measurements at a greater distance, we believe the measurement error due to antenna spacing is small. Next, we provide the effective gain vs. frequency in Figure 10. These data show that at lower frequencies the response of this antenna is quite flat and that the highfrequency response is approximately smooth to 8 GHz. The peak effective gain of the CIRA is 23 dB at 4 GHz. The midband effective height of the antenna is found from the integral of the normalized impulse response shown in Figure 9 to be approximately 0.28 m This is 71 % of the theoretical value of 0.396 m.

In Figure 11 we show the cross-polarization (crosspol) response of the CIRA. The effective gain on boresight for the crosspol case is shown in Figure 12. The crosspol response is 10–20 dB below the copol response from Figure 10. This data may be of interest due to recent work suggesting improvements in the IRA that would result in improved gain and reduced crosspol.6 This is accomplished by placing the feed arms

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at ±30° from vertical, instead of ±45°, which we currently have in the CIRA. We have recently demonstrated this principle on an 18-inch diameter IRA with an aluminum dish with great success. Since each panel of the CIRA is 30° wide, it will be straightforward to incorporate the new feed arm positions into future versions of the CIRA.

Next, in Figures 13–14 we show the antenna pattern in the H and E planes, based on the peaks of the raw voltage measurements. The half-voltage beamwidth is 5.1° in the H plane and 6° in the E plane. If we choose to use the half-power beamwidth, we have ~3° in both the H and E planes. Samples of the raw data from the H and E plane scans are shown in Figures 15–16.

In Figures 17–18 we show the principal plane pattern cuts of the antenna at various frequencies. At low frequencies, the pattern is quite smooth and flat as expected. At high frequencies the high gain and narrow beam width become evident.


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CONCLUSION The FRI-CIRA-1 provides broadband antenna coverage in a single compact package that is easily portable. The antenna has outstanding RF characteristics in both the frequency and time domains. In the time domain, it has an impulse response with FWHM of 70 ps and mid band effective height of 30 cm. In the frequency domain, the peak gain at 4 GHz is 23 dB, and the antenna is usable from 50 MHz to 8 GHz. The reflector for the FRI-CIRA-1 is 1.22m (4 feet) in diameter with a focal length of 0.488 m (F/D = 0.4). The umbrella-like frame covered with conductive mesh fabric provides a very practical, lightweight, and easy-to-use antenna. The collapsed antenna measures 102 mm (4 in) diameter x 810 mm (32 in) long. The antenna weighs 2 kg (4.5 lb.). The antenna can be easily transported and set up by one person and can be attached to a variety of military and COTS transmitters and receivers. The FRICIRA-1 is now commercially available from Farr Research, Inc.

Acknowledgements We wish to thank Drs. Carl E. Baum and George H. Hagn for helpful comments on this work. This work was sponsored in part by the Air Force Office of Scientific Research, Arlington, VA, and in part by Air Force Research Laboratory, Directed Energy Directorate, under contract F29601-98-C-0004.

Patent Notice A patent is pending on the antenna described in this note.

REFERENCES 1. C. E. Baum and E. G. Farr, Impulse Radiating Antennas, pp. 139-148 in H. L. Bertoni et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics, New York, Plenum Press, 1993. 2. E. G. Farr, C. E. Baum, and C. J. Buchenauer, Impulse Radiating Antennas, Part II, pp. 159-170 in L. Carin and L. B. Felsen (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 2, New York, Plenum Press, 1995. 3. E. G. Farr, C. E. Baum, and C. J. Buchenauer, Impulse Radiating Antennas, Part III, pp. 43-56 in C. E. Baum et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 3, New York, Plenum Press, 1997. 4. E. G. Farr and C. E. Baum, Time Domain Characterization of Antennas with TEM Feeds, Sensor and Simulation Note 426, October 1998. 5. L. H. Bowen, E. G. Farr, and W. D. Prather, Fabrication and Testing of Two Collapsible Impulse Radiating Antennas, Sensor and Simulation Note 440, November 1999. 6. J. S. Tyo, Optimization of the Feed Impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna, Sensor and Simulation Note 438, November 1999. 7. IEEE Standard Definition of Terms for Antennas, IEEE Std 145-1993. 8. George Hagn, Personal communication. 9. L. H. Bowen and E. G. Farr, Recent Enhancements to the Multifunction IRA and TEM Sensors, Sensor and Simulation Note 434, February 1999.


HIGH-POWER ULTRAWIDEBAND RADIATION FOR RADAR APPLICATION Vladimir I. Koshelev Institute of High Current Electronics RAS 4, Akademichesky Ave., 634055 Tomsk, Russia INTRODUCTION In the recent years, investigations directed to the creation of high-power ultrawideband (UWB) radiation sources are made intensively in many countries. In a considerable degree, this is due to the interest to the UWB radar development intended to solve various tasks. Each of the areas of radar application makes its requirements to the parameters of the UWB sources. However, there is a common base of the UWB radars and namely obtaining the information concerning the objects from a comparison of the sounding and reflected pulse waveforms. This information can be used for reconstruction of the object shape. This is a principal feature of the UWB radars which causes the high interest to the development of such radars alongside with a high spatial resolution due to a short pulse length. Our research team used to develop methods and technology of the UWB radars since 1993. The development of methods for reconstruction of the object shape allows to specify the requirements to the parameters of the UWB radiation sources. Understanding the physical and technological limitations of the UWB source parameters allows to specify the direction for searching the methods acceptable for solution of the practical tasks. The main efforts were concentrated on the research and development of new UWB radiators with a constant phase center, high-power UWB sources on the basis of single radiators and antenna arrays, methods of shape reconstruction of single complicated objects with a perfectly conducting surface. This paper presents a short review of the results of these investigations. UWB COMBINED ANTENNA High-power UWB radars require radiation sources on the basis of multielement antenna arrays. An antenna in such arrays should be compact, it should have a constant phase center and radiate high peak power The antenna pattern should be either cardioid for electron steering by a wave beam in a wide angle range or high-directed for the arrays with mechanical steering. Theoretical investigations1 allowed to suggest a new approach to the creation of a combined antenna satisfying the requirements made to the element of the UWB steering antenna arrays. The approach consists in the combination of the near-field zones of two small-dimensional radiators with a constant phase center having a common input but different reactive energies. If the electrical energy prevails in a near-field zone of one Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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radiator, then magnetic energy should prevail in a near-field zone of another radiator. At the fulfilment of definite energetic, frequency and phase conditions, the realization of a combined antenna having a wide band matching with a feeder, constant phase center, cardioid pattern of linearly polarized radiation is possible. Theoretical investigations of a combined antenna consisting of one electrical and two magnetic dipoles have shown a possibility to match the antenna with the feeder at the voltage standing-wave ratio (VSWR) in the frequency band ( is the lower frequency boundary determined by VSWR, is the upper frequency boundary determined by the pattern distortion). It is wellknown that the utmost matching band of the electrical dipole with a feeder by the level at the reactive energy store in the near-field zone equal to zero is A theoretical analysis of the idealized model of a combined antenna has shown that the interval of the parameters at which the matching band of such antenna with a feeder is close to the utmost one is sufficiently narrow. Therefore, the experimental realization of combined antennas with the utmost matching band is a sufficiently difficult and interesting task.

Figure 1 presents the variants of a combined antenna design1. The antenna consists of a plane electrical monopole 1 with a length and magnetic dipole 5. In contrast to the idealized symmetrical combined antenna consisting of an electrical dipole and two magnetic dipoles, the realized antenna is nonsymmetrical. This is due, first of all, to the necessity to use a high-voltage nonsymmetrical feeder and antenna design simplicity. Note that the matching band of the realized antennas with a feeder by the level was that was far from the utmost one and presented a subject of our further investigations. Special experimental investigations2 confirmed the conclusions of the theory concerning the increase of the matching frequency band of a combined antenna. Before starting our investigations in 1993, we have chosen a bipolar waveform pulse (one cycle of a sinusoid) for the antenna excitation. This is due to the possibility to realize a more high energetic efficiency of the bipolar pulse radiation in comparison with antenna excitation by a monopolar pulse. The latter is explained by the difference in the spectra of a monopolar and bipolar pulses. This was confirmed by the experimental investigations1 carried out for different antenna designs. In all the designs of the antennas the length of the electrical monopole was where is the bipolar pulse length at the antenna input, c is the velocity of light. High-voltage antennas were realized for the bipolar pulses with The experimental investigations were carried out with the use of the monopolar and bipolar low-voltage pulses. The monopolar pulse length was two times less than the bipolar pulse one in one series of experiments. The investigations have shown that the

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energetic efficiency of the bipolar pulse radiation was that was more than two times higher than the energetic efficiency of the monopolar pulse radiation An important parameter is the antenna efficiency by the peak power which is determined as the ratio of the peak power of a linearly polarized radiation to the peak power of the electrical pulse exciting the antenna. The value of depends on power of a cross-polarized radiation, radiated pulse waveform and dependence of the radiated pulse waveform on the radiation direction. The evaluations, taking into account the abovementioned, have shown that is less than by a factor of 1.5-2 and for bipolar pulses is approximately equal to 0.4-0.6. Detailed experimental investigations1 have shown that in a wide angle range relative to the pattern main direction (±60° in an azimuthal plane and ±30° in a meridional plane) the waveform of the radiated pulse is preserved and the antenna phase center is constant in a frequency band of the radiated pulse.

ANTENNA ARRAYS Theoretical and experimental investigations2-4 of radiation of the arrays consisting of the equal combined antennas were directed to the influence evaluation of the array geometry, distribution of the amplitudes of the exciting pulses by the array elements, interaction of radiators in the array on the characteristics of the radiation and arrays. Theoretical investigations3 were carried out in the approach of the absence of the interaction between the antennas in the array with the number of elements 8x8. Linear2 (up to 4 elements) and rectangular4 (2x2) arrays at the antenna excitations by bipolar pulses with the length were carried out experimentally. The following main results have been obtained: i) the pattern width decreases and the background radiation increases with the increase of the distance between the radiators; ii) the radiated pulse waveform depends on the angle relative to the pattern main direction, zeroes and diffraction lobes at the wave beam steering are absent; iii) the background radiation level decreases with the increase of the number of elements in the array and it is minimum at the equiamplitude distribution of the exciting pulses in the array; iv) the interaction of the radiators in the array results in the array efficiency decrease by the energy and peak power in comparison with the efficiency of a single antenna; between v) the increase of the number of radiators in the array at the distance them increases the electromagnetic field strength in the pattern main direction proportionally to the number of radiators. Let’s discuss in short the two last results. The array energetic efficiency drops with the increase of the number of radiators not so essentially as the efficiency by the peak power. The latter is due to the more essential dependence of the pulse waveform radiated by the array on the angle in comparison with a single antenna. For a 2x2 array and the energetic efficiency decreased by 13% in comparison with a single antenna and the efficiency by the peak power decreased by 1.5-2 times. The electrical field strength increase in the pattern main direction is proportional to the number of the radiators in the array due to the time delay of the interaction of radiators and allows to increase the field strength of the sounding pulses at the array efficiency decrease by the energy and peak power. New possibilities for solution of the radar tasks will be opened at the development of the methods and devices for synthesizing the electromagnetic pulses of an arbitrary waveform in free space, in particular, for essential increase of the radiation spectrum width in comparison with the radiation spectrum width of single antennas or arrays consisting of equal antennas. One2 of possible ways to expand the spectrum of UWB pulses is application of the arrays consisting of the unequal combined antennas excited by bipolar

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pulses of different length. The calculations have shown that the least changes of the radiation parameters due to the jitter of the switches are realized at the bipolar pulse synchronization by the moment of time corresponding to the change of polarity of all pulses. HIGH-POWER SOURCES OF UWB RADIATION The first source of high-power UWB radiation on the basis of a single combined antenna excited by a bipolar pulse with a 100-kV amplitude, 4-ns length and 50Hz pulse repetition frequency was created in 1994. The subsequent high-power (0.1-1 GW) UWB sources of this series with the length and a 100-Hz pulse repetition frequency are described in detail in the papers4,5. In all the UWB sources the formers of the bipolar pulses developed on the basis of a Vvedenskii scheme6 with one and two gas switches were used. In the UWB sources with a single antenna and two-element array, the formers with one switch5 and the wave impedance and respectively, were used. In the UWB source with a four-element array, the former with two switches4 and was used. In a primary charging circuit the Tesla transformers were used. To decrease the amplitude instability of the output pulses the formers were charged by the short pulses A multichannel commutation regime was realized due to the high velocities of the voltage rise The radiation power increase from 0.1 to 1 GW resulted in the decrease of the time of continuous operation from several hours to 20 minutes with the subsequent cooling during 1.5 hours. The limitation is due to the heating of the insulators installed in the coaxial lines. To provide a stable operation of the UWB sources, the antennas were placed into the dielectric containers at a low gauge pressure (0.1-0.6 atm) of The thickness of the dielectric containers was much less than the spatial length of the exciting pulse and they practically had no influence on the radiation parameters.

METHODS OF OBJECT SHAPE RECONSTRUCTION AT SMALL ANGLE BASE OF ULTRAWIDEBAND RADAR Recognition of radar objects (RO) is related first of all to the possibility to obtain the information concerning their shape. The efforts were concentrated on the developments of methods of the object shape reconstruction at a small angle base of the UWB radars. This is due to our interest to the application of the UWB radars for recognition of the objects disposed at a large distance. In all our works the objects in the shape of a stylized 2D and 3D airplane models with different linear dimensions L were used. The objects were sounded by the bipolar pulses of electromagnetic radiation having a different length It was suggested that at the propagation of the electromagnetic pulses in the air their waveform was not distorted. The problem of calculation of the electromagnetic field reflected by the object was solved by a Kirchhoff method in the single scattering approach. To simulate the real measurements, the uniformly distributed noise with a zero mean and given value of dispersion was added to the signal scattered by the object. The value was determined relative to the signal maximum in a receiving system. Tomographic Method It is well-known that the relation between the joint scattering coefficient object shape is determined by the Lewis-Boyarsky equation

and


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Here, V(p) is the object transmission function, vector corresponds to the direction to the object, is the unit vector, is the circular frequency. Function is equal to 1 and 0 at and respectively. At the large object sounded by the pulses from the ground we fail to record the signals scattered by the upper invisible object part and that’s why the relation (1) can’t be used directly for the object shape reconstruction. In order to eliminate the stated limitation it was suggested to substitute the unknown shadow part of the object for a fragment of the body for which V*(-p) is known. The tests of the suggested approach were carried out at a 2D airplane model with a linear dimension L = 36 m. The object shadow part was substituted for an ellipsoid (dashed line, Figure 2).At the simulation, two radars with the angle base were used. The UWB pulse spectrum was limited by the frequencies and

Figure 2a, b presents the results of the object shape reconstruction at the noise level and angle base and 10°, respectively. The accuracy of the reconstruction was determined by the equation

where is the reconstructed object shape. For the data presented in Figure is equal to 94% and 70%, respectively. The calculations have shown that the reconstruction accuracies of the sounded (unknown) and shadow (known) object parts are close to each other. This allows to evaluate the reconstruction accuracy of the sounded object part beforehand. The accuracy decreases with decrease and increase and equals to for and This is the boundary for the object recognition by its reconstructed shape. The essential increase of the object shape reconstruction accuracy in a tomographic method is related to the spectrum width increase It can be realized only at the pulse waveform synthesizing in free space with the use of multielement arrays2.

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Method of Brilliant Points It is well-known that at the pulse scattering by a complicated object containing the parts with a different curvature, a sharp increase of the reflected signal into the direction of the receiver from the local object parts getting the name of brilliant points is possible. It is obvious that the measurement of coordinates of the brilliant points allows to evaluate the RO shape. An aperture synthesis of the angle base in the process of the moving object sounding by the pulses with the pulse repetition frequency F allows to use only one radiator, to measure the object motion velocity, its position in space and, respectively, its track, and to increase the accuracy of the object shape evaluation by a large number of measurements. To measure the coordinates of the brilliant points, it was suggested8 to use a radar system installed into a Cartesian coordinate system. At the beginning of a coordinate system there are placed the radiator and the main receiver. Three receivers had the coordinates (b,0,0), (0,b,0), (0,0,b). The equation for the vector R components of the n-th brilliant point is the following:

Here, is the distance to the n-th brilliant point of the sounded object obtained by the delay between the radiated pulse and the n-th local maximum in the reflected signal. The relative delay where j is the receiver number, is the signal delay in the main receiver. When this approach was tested, a 3D airplane model with a linear dimension L= 10 m was used. The aperture was synthesized during 10 seconds with the pulse repetition frequency F = 2 Hz. The object was moving at a 1-km constant height with the velocity V=200 m/s. During calculations the values of b = 10-50 m, varied. The best results presented in Figure 3 were obtained at b = 50 m. Note, that at the moving object synthesizing, the coordinates of the brilliant points were re-calculated relative to the beginning of the coordinates corresponding to the first brilliant point. It is obvious from Figure 3 that the object shape approximation accuracy by means of the brilliant points deteriorates as increases.


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Method of Genetic Functions Any complicated object consists of simple fragments. A genetic function (GF) presents a signal scattered by a fragment. A signal S (t) reflected by the complicated object in the given direction can be approximated by the GF set

where is the GF set for the given aspect angle having the amplitude coefficients and position at a time axis The problem of the RO shape reconstruction consists9 of determination of the GF composition and measurement of the fragment coordinates corresponding to these GF. The GF set presents a databank of the previously calculated signals scattered by the RO fragments at the given waveform of the sounding pulses and all the surveillance aspect angles. A radar system described in the previous section and relation (3) is used to measure the coordinates of the n-th GF. This approach was tested at the shape reconstruction of a 3D airplane model having a linear dimension L = 4.5 m and placed at the distance R = 1.5 km. The object was sounded by the single bipolar pulses and was supposed to be motionless during the pulse length. The distance between the receivers b = 50 m and the angle base Figure 4 presents the dependence of the object shape reconstruction accuracy (shape projection to the plane OXY) calculated in accordance with the relation (2), on the noise level averaged by 10 realizations for It is obvious from the Figure 4 that decreases with the increase of and The accuracy for and that is essentially higher than for a tomographic approach. Calculations have shown that essentially depends on the ratio (Figure 5). The shape reconstruction accuracy weakly changes with the rise of the ratio of the linear object dimension to the spatial pulse length at

CONCLUSION The obtained results present a realization of the first stage of the research program on the development of the methods and technology of the ultrawideband radars. Subsequent investigations are related to the development of combined antennas excited by the bipolar

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long pulses, high-power UWB sources on the basis of multielement arrays with the increased pulse repetition frequency, receiving antennas for investigation of a polarization structure of the UWB pulses. An important aspect of our activity is the development of codes for simulation of the UWB pulses and arrays, scattering of pulses by complicated objects. The development of new approaches for object recognition and more detailed study of the previously suggested approaches for the object shape reconstruction is carried out. Acknowledgments The author is thankful to all colleagues for a successful research team work resulting in presenting the given review paper.

REFERENCES 1.

2. 3.

4.

5.

6. 7. 8. 9.

V.I.Koshelev, Yu.I.Buyanov, B.M.Kovalchuk, Yu.A.Andreev, V.P.Belichenko, A.M.Efremov, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, High-power ultrawideband electromagnetic pulse radiation, in: Proc. of SPIE Inter. Conf. on Intense Microwave Pulses V,H.E.Brandt, ed., San Diego, CA, USA. 3158:209 (1997). Yu.A.Andreev, Yu.I.Buyanov, V.I.Koshelev, V.V.Plisko, and K.N.Sukhushin, Multichannel antenna systems for radiation of high-power ultrawideband pulses, in:Ultra-Wideband Short-Pulse Electromagnetics 4, J.Shiloh and E.Heyman, eds., Plenum Press, New York, in publication. V.P.Belichenko, Yu.I.Buyanov, V.I.Koshelev, V.V.Plisko, Short electromagnetic pulse formation by plane antenna array, in: Proc. of Conf. on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, M.I.Andiychuk, ed., Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the Ukrain National Academy of Sciences, Lviv. 43 (1997). Yu.A.Andreev, Yu.I.Buyanov, A.M.Efremov, V.I.Koshelev, B.M.Kovalchuk, V.V.Plisko, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, Gigawatt-power-level ultrawideband radiation generator, in: Proc. of 12 IEEE Inter. Pulsed Power Conf. C.Stallings and H.Kirbie, eds., Monterey, CA, USA. 2:1337 (1999). Yu.A.Andreev, Yu.I.Buyanov, A,M,Efremov, V.I.Koshelev, B.M.Kovalchuk, K.N.Sukhushin, V.A.Vizir, V.B.Zorin, High-power ultrawideband electromagnetic radiation generator, in: Proc. of 11 IEEE Inter. Pulsed Power Conf., G.Cooperstein and I.Vitkovitsky, eds., Baltimore, MD, USA. 1:730 (1997). Yu.V.Vvedenskii, Tiratron generator of nanosecond pulses with universal output, Izv. Vuzov. Radiotekhnika. 2:249 (1959). V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Reconstruction of object shape at small aspect angle ultrawideband radiolocation, Radiotekhn. Electr. 44:301 (1999). V.I.Koshelev, S.E.Shipilov, and V.P.Yakubov, The problems of small base ultrawideband radar, in: Ultra-Wideband Short-Pulse Electromagnetics 4, J.Shiloh and E.Heyman, eds., Plenum Press, New York, in publication. V.I.Koshelev, S.E.Shipilov, V.P.Yakubov, Use of genetic function method for object shape reconstruction at small aspect angle ultrawideband radiolocation, in:Radiotekhn. Electr., in publication.

BROADBAND OPERATION OF TAPERED INSET DIELECTRIC GUIDE AND BOWTIE SLOT ANTENNAS

Andrew B. Hannigan, Steve R. Pennock and Peter R. Shepherd Department of Electronic and Electrical Engineering University of Bath Bath, BA2 7AY, United Kingdom

INTRODUCTION In this paper two promising broadband radiating structures, the inset dielectric guide tapered slot antenna (IDG-TSA) and the bowtie slot antenna, are evaluated. For each type of antenna, the general radiation behaviour is established and the input match characteristics over a broad frequency band are discussed. A recently developed analytical method for prediction of IDG-TSA far field radiation is described, and is shown to provide results which compare well with measured values across a wide range of observation angles. New types of feed for the IDG-TSA are proposed which extend the useable bandwidth of the structure as well as allowing much easier integration with planar circuitry. The operation of both antennas with wideband pulsed excitation is investigated experimentally and, for the IDG-TSA, using a finite-difference time-domain (FDTD) simulation. The IDG-TSA performs well with such excitation, showing little pulse distortion in the far field and low reflection back into the feed. The bowtie slot antenna shows more pulse distortion than the IDG-TSA, but performance is reasonable.

THE INSET DIELECTRIC GUIDE TAPERED SLOT ANTENNA The IDG-TSA is a slow wave travelling wave structure consisting of a tapered dielectric-filled slot cut into a ground plane, as depicted in figure l(a). As such, the antenna can be described as a tapered length of inset dielectric guide (IDG). The simplicity of this structure provides a number of advantages; as well as being inexpensive to manufacture, the IDG-TSA is lightweight, rugged, and presents a flat profile which allows it to be flush mounted in a surface. The width and orientation of the main radiated beam is determined by the slot and ground plane geometry; main beams at


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elevation angles from near-endfire to near-broadside may be achieved. In this paper, only vertically polarised shallow slot antennas are considered.

Analysis An accurate analytical method for predicting the far field radiation characteristics of the IDG-TSA has recently been developed (Hannigan et al, 1999). This method uses the fields of the H-guide, which can be written as simple closed form expressions, to model the shape of the IDG fields in the slot. The longitudinal propagation constant is obtained using a rigorous transverse resonance diffraction technique. A full model of the fields on the aperture and ground plane surfaces is thus built up, and Fourier transform relationships are used to obtain the far field radiation patterns of the antenna. The H-guide fields are in general an acceptable approximation to those of IDG. The approximation breaks down, however, at the slot edges where the H-guide analysis assumes the existence of vertical metal plates. The new method uses static field theory to describe the singular behaviour of the fields near to the IDG edge and thus to provide an analytical link between the ground plane fields and the fields on the slot surface. The static field description is linked to the H-guide fields within the dielectric close to the slot edge thereby fixing the magnitude of the ground plane fields. Figure 2 serves to demonstrate the accuracy of the new analytical method over a wide range of observation angles, and also to establish the near-endfire radiation behaviour of a long IDG-TSA. In the figure, the elevation angle is measured from the antenna surface and the azimuth angle is measured from the longitudinal axis of the structure. This moderately directive pattern is in contrast to that of the much shorter structure depicted in figure 3, which gives a single broad beam in the forward direction. Figure 2 also includes a comparison to an FDTD result, generated by a model similar to that presented by et al (1996). It can be seen that the new analytical method provides results that are at least as accurate as FDTD, but the computer model runs in a fraction of the time required by FDTD. For instance, the FDTD model for a short IDG-TSA takes 20 minutes to run on a Silicon Graphics Origin 2000 machine, whilst the new analytical method takes approximately one minute on the same machine. Feed Structures Consider now the methods by which the IDG-TSA can be fed. In the past, rect-

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angular waveguide (RWG) has been used for this purpose; this arrangement is shown in figure l(a). This has disadvantages in that it can introduce a sharp discontinuity at the feed, the available bandwidth of the structure can be reduced, and the scope for integration with other circuitry is limited. As a result of these shortcomings, other feeding options have now been considered. These are empirical designs based on a knowledge of the fields of IDG and on the work carried out by Ma et al (1990). In order to couple to the strong vertically directed electric field component in shallow slot IDG, a microstrip line can be placed vertically on the open end of the antenna. Backto-back measurements reveal that this transition has a lower than the RWG feed but that transmission is also lower, being -3 to -4dB between 9 and 12GHz compared to -1 to -2dB for the RWG feed. These results indicate that the microstrip feed is unsatisfactorily radiative. Using the microstrip feed as a basis, a less radiative structure has been developed based on the well known co-axial to RWG transition. This consists of a stripline feed placed over the open end of the IDG-TSA with a 5.5mm length of shorted IDG fixed behind it. Figure 3 shows the elevation plane patterns of a short IDG-TSA fitted with each of the three feed types for the complete 360° angular range. In this figure 0°, coinciding with the antenna surface, is defined from the right hand horizontal axis so that the top right quadrant shows the forward radiation above the plane in which the aperture sits. The measurements show that the RWG feed produces the least back radiation, whilst that from the microstrip feed is high. The stripline feed, however, has significantly cut the back radiation, giving performance close to that of RWG. Measurements of stripline fed antenna impedance show reasonable broadband per-

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formance. For a short IDG-TSA, is better than - l0dB between 7.5 and 13GHz, between -5 and -8dB from 13 to 16GHz, and better than -8dB between 16 and 20GHz. Thus, the use of the stripline feed allows the IDG-TSA to be operated over a wider range of frequencies without serious losses to unwanted radiation. Indeed, a short IDG-TSA has been operated in a predictable manner up to 21 GHz using this feed, thus displaying an operating bandwidth of approximately 3:1. Wideband Pulsed Excitation of the IDG-TSA The performance of the IDG-TSA with wideband pulsed excitation has been investigated using PDTD simulation and by experiment. The FDTD model uses a standard algorithm (Yee, 1966) modified using a contour path technique to model the tapered floor of the IDG (Jurgens, 1992). Twofold Mur’s first order absorbing boundary conditions are used (Mao, 1998) and the radiated far field in the time domain is found using the method of Luebbers et al (1991). Only the top surface of the structure is modelled and the antenna is fed using a section of uniform inset dielectric guide. An electric field wall is used for wave launching; an incident transverse electric field is allowed to appear in the dielectric region at the start of the structure which has the same field pattern as the desired mode at the centre of the frequency band. The absorbing boundary conditions at this end of the model are only enabled after the pulse has left the excitation plane. The time dependence of the excitation is shown in figure 4(a). It is a bandpass Gaussian pulse with a temporal width of 80ps (full width at half maximum, FWHM), centred on 20GHz and with a width of approximately l0GHz (FWHM) in the frequency domain. Figure 4(b) depicts the reflected component (directed across the slot) calculated by FDTD monitored 6.35mm prior to the start of a short IDG-TSA with an of 2.08. The antenna aperture is 30mm long with 14.6mm of ground plane extending beyond the slot end. The uniform IDG feed is 69mm long, so that the FDTD test point is 62.65mm away from the excitation plane. For the first 360 time steps negligible field can be seen at the test point, as the pulse is travelling along the uniform feed section. The excitation pulse can then be observed, the trailing edge of which is distorted due to reflection from the discontinuity


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at the taper start. This reflection is approximately 14dB down on the incident pulse. After 1000 time steps, further reflections can be seen, approximately 20dB down on the incident pulse. These reflections come around 0.36ns after the incident pulse, which corresponds to approximately twice the time required for a wave travelling at the speed of light to reach the end of the structure (0.17ns). Thus, these reflections can be seen to be due to a combination of the discontinuities presented by the slot and ground plane ends. Figure 5 shows the calculated elevation plane far field radiation in the time domain at two angles for the antenna considered above. Comparison to the incident pulse, figure 4(a), shows that it is being radiated without serious distortion. At both elevation angles, radiation from the feed point can be seen to be combining with the main radiated pulse, though at the higher angle there is a greater time separation between these two components.

Finally, figure 6 shows the measured transmission between a short IDG-TSA and an X-band horn of a more narrowband pulse. These results support the calculated data given above in that very little distortion of the transmitted pulse can be seen.

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THE BOWTIE SLOT ANTENNA The bowtie slot antenna is a planar version of the wideband biconical antenna. The structure under consideration here is shown in figure 1(b), and consists of a bowtie etched onto a metalised substrate fed by a microstrip line printed on the reverse side of the board, terminated in a radial stub. In this work, antennas were constructed on substrate with an of 2.2 with two structures fabricated on a 50.8 by 25.4mm board so that isolation between two closely spaced antennas could be tested. It has been found that isolation between two antennas with 90° bow angles whose centres are 40mm apart is better than 25dB between 5 and 20GHz. However, it has been observed that radiation from the microstrip lines on the rear of the board is high. The situation has been improved by enclosing the rear of the board in a metal box, although this is at the expense of reduced isolation between the antennas, which is now closer to 15dB across the 5 to I5GHz band. Measurements have shown that the of a boxed bowtie with 30° bow angle is poor compared to the stripline fed IDG-TSA; better than -5dB between 5.5 and 7GHz, between 9 and 12GHz, and from 15.5 to 19GHz. However, power loss measurements (i.e. of ) show significant loss from the structure, greater than 60%, over the aforementioned frequency ranges. With the rear of the structure boxed, this power is being radiated predominantly in the forward direction. Figure 7 shows the measured radiation pattern of two bowtie slot antennas at different frequencies. The structure tends to produce a broad beam, with 3dB bandwidths of approximately 100° and 40° in the elevation and azimuth planes respectively, although there is significant ripple along the beam in the elevation plane. Figure 8 depicts the measured transmission of a wideband pulse between two bowtie slot antennas with 90° bow angles. There is considerably more pulse distortion than was seen in the IDG-TSA measurement in figure 6, albeit for a pulse with a more limited frequency content, but the shape of the incident pulse has been preserved to a large extent. The ‘echoes’ of the main pulse, following 0.4 and 1.2ns behind it, correspond to the pulse propagating for further distances of between 120 and 360mm. These dimensions are consistent with reflections from within the enclosure on the rear of the antenna, which is 50.8mm long, or from the open circuit end of the microstrip radial stub feed line, which is 32mm long.


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CONCLUSIONS The radiation characteristics of the IDG-TSA and the bowtie slot antenna have been established. The efficacy of a new analytical method for the IDG-TSA has been demonstrated. A novel IDG-TSA feed structure has been proposed which gives performance close to that of rectangular waveguide but which allows greater exploitation of the broadband properties of the antenna, and easier integration with planar circuitry. The operation of both antennas with wideband pulsed excitation has been investigated for the first time. Both structures perform well, with particular promise being shown by the IDG-TSA. REFERENCES Hannigan, A.B., Pennock, S.R., and Shepherd, P.R., 1999, Improved modelling of tapered IDG antennas, in: Proc. 29th European Microwave Conf. Jurgens, T.G., Taflove, A., Umashankar, K., and Moore, T.G., 1992, Finite-difference time-domain modeling of curved surfaces, IEEE Trans. Antennas Propagat., 40:357. Luebbers, R.J., Kunz, K.S., Schneider, M., and Hunsberger, F., 1991, A finite-difference time-domain near zone to far zone transformation, IEEE Trans. Antennas Propagat., 39:429.

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Ma, L., Rozzi, T., and Pennock, S., 1990, Linear arrays realised in IDG, in: Proc. of IEE Colloq. on Components for Novel Transmission Lines, 6:1. Mao, J., 1998, Twofold Mur’s first-order ABC in the FDTD method, IEEE Trans. Microwave Theory Tech., 46:299. V., Pennock, S.R., and Shepherd, P.R., 1996, The use of the FDTD method in the design of IDG antennas, in: Proc. 3rd Conf. on Computation in Electromag. Yee, K.S., 1966, Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., 14:302.

A UNIFIED KINEMATIC THEORY OF TRANSIENT ARRAYS

Amir Shlivinski and Ehud Heyman Faculty of Engineering Tel-Aviv University Tel-Aviv 69978, Israel

INTRODUCTION The main objective of this paper is to explore the kinematic properties of transient arrays, in order to define the parameter range for an antenna-array realization of collimated pulsed beam fields. Such arrays are driven by pulsed waveforms and are controlled by the inter-element time shifts. The properties of the radiated field depend on the problem parameters: the center frequency and the fractional bandwidth, the pulse repetition rate, the inter-element spacing and the total number of elements. Following [1] we present a unified parameterization for the kinematic properties that covers the entire parameter range, from the conventional monochromatic dense array to the ultra wideband sparse array. The lobe structure of the radiation pattern is a parametric interplay of several mechanisms: In addition to the side-lobes (SL) and the grating-lobes (GL), which are mainly quasi-monochromatic phase-interference phenomena extended to the wideband regime, we also identify cross-pulse-lobes (CPL) which are an intrinsic wideband phenomenon. It is also shown that under certain conditions one may design a sparse array (with inter-element spacing much larger than the wavelength for all frequencies in the band) which basically does not suffer from sever side lobe problem. THE PHYSICAL MODEL We consider a linear N-element array of identical antennas along the axis with interelement spacing The elements is driven by the signal where the inter-element delay controls the main beam direction. is a pulse-train consisting of M pulses with pulse repetition rate and possible modulation It can


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have one of the following forms:

or

In (1a) all the pulses share a single (coherent) carrier as in digital communication systems. Note that in general, namely unless where is an integer and the modulation within consecutive pulses in (1a) have different phases. The form in (1b), on the other hand, describes systems where the pulses are formed separately, say by triggering an emitting device. Note that in this model all the pulses have the same shape, including the modulation. It will be convenient to parameterize the pulse in (1) as follows:

i.e., the pulses have unit energy, pulse length T and are centered at the origin. Clearly We shall distinguish between three bandwidth regimes: Narrow band (NB) or quasi-monochromatic regime: Intermediate regime: (Note that different phenomena are obtained here if or if which are both possible here). Ultra-Wideband (UWB) regime: As will be shown below, the intermediate regime posses both NB and UWB kinematic phenomena. THE TIME-DEPENDENT RADIATION PATTERN The time dependent radiation pattern in the

direction is given by [1, eq. (8)]

where denotes the element effective radiation height [1, eq. (10)], measures the angle from the axis, is the interelement delay as defined in connection with (1), and the symbol denotes a temporal convolution. To simplify the presentation we shall consider only the “array pattern” i.e. the summation in (3). A contracted characterization of the array is obtained if we eliminate the temporal dependence by projecting (3) on a normed detector that measures the field. The radiation pattern is then expressed as

where denotes the within a consider only energy and peak detectors,

“observation window.” Henceforth we shall or respectively, with


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It will be demonstrated that the type of the detector may affect the radiation lobes magnitudes but not the grid structure (Fig. 5). KINEMATIC PROPERTIES: LOBE STRUCTURE The far field radiation pattern of the antennas array, whether observed directly from the field in (3) or using the normed detectors in (4), is dominated by an interference between signals emanated by the individual radiating elements. The resulting radiation lobe structure is controlled by the bandwidth, the pulse repetition rate and the interelement distance. There are basically two mechanisms for this interference: The first is a “phase interference” causing the side-lobes (SL), nulls and grating-lobes (GL), while the second is “pulse-interference” that causes what will be termed here “cross pulse lobes” (CPL) and is relevant only if M > 1 in (1). The CPL is dominant only in the UWB regime, where the exact conditions will be discussed below. A general formulation A radiation lobe is formed whenever a constructive interference between (two or more) signals emanating from different sources occurs. The condition for can be states as

and it is tagged by the triple index Here represents the distance between the sources that emit the interfering signals, hence represents the index of the interfering pulses hence it is denoted as the CPL index and is bounded by Finally describes the phase difference between the pulses, hence where is a measure for the number of oscillations within and thus is related to the fractional bandwidth. Relation (5) completely characterize the lobe structure in the far field as a function of and the signal parameters. Without loss of generality we shall only consider henceforth the case Examining eq. (5) we note that the parameters and have the same functional role, hence we may start by considering two extremes. In the NB regime where we consider only the case Eq. (5) then defines the conventional phase interference condition for the GL and SL. In the UWB regime, on the other hand, hence there are no oscillation and no phase interference phenomena. We shall therefore set in (5), obtaining the condition for a CPL. In the intermediate regime all indexes in (5) have to be considered, implying the coexistence of both GL and SL with CPL. below we shall consider these cases in more details. Quasi-monochromatic regime: Side lobes (SL), nulls and grating lobes (GL) First we consider the lobe phenomena associated with phase interference. It is simpler to consider first the limit where the interference occurs within a single long pulse (i.e., in (5)). In this case the interfering oscillations have essentially the same magnitude, hence a pure null occurs when the relative phases of all the element contributions sum up to an integer multiple of as described by the condition

which is the same as (5) with The SL are obtained between two nulls. A GL is obtained when the contributions from all elements are in phase, i.e,

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giving in (5) These monochromatic phenomena can also be discerned in the transition regime as long as there are several oscillations within the finite duration pulse In the main-beam direction all the element-contributions arrives simultaneously hence the radiated signal is At other directions these contributions arrive at slightly different times, yet the oscillations may interfere constructively to form a GL via (7) as long as the pulses are partially overlapping. However, due of the relative delay between the pulses, the interfering oscillations have different magnitude and thus the GLs become weaker and wider. This is demonstrated in Fig. l(a) where the full and the dotted lines describe the contributions arriving at the second GL direction from the first and second elements. From the same reason the sharp nulls become wider local minima and eventually the SL-nulls structure disappears completely. Fig. 1(b) depicts the radiation patterns obtained from a sparse 11-element array driven by a single pulse with either NB or UWB spectral content. UWB regime: Cross pulse lobes (CPL) CPL are formed by an interference of two or more pulses emitted at different times from different elements (i.e. pulses with different indexes in (1)). In these directions the pulses in the radiating signal are enhanced while in all other directions the radiating signals consists of a weak pulse-train. The CPL phenomenon is possible only for sparse arrays in the sense that

A CPL is obtained if the temporal delay between the and pulses emerging from the and sources located at equals to the relative propagation delay The resulting condition is (recall that is assumed


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throughout)

This condition is similar to (5) where we identify as the CPL order, and The main beam is tagged as a CPL order giving The temporal signal and the number of the interfering pulses at a given CPL direction depends on the indexes and on the time. In order to demonstrate this process we shall consider, for simplicity, the case (see Fig. 2). One finds that during the first interval, there are equi-distance pulses arriving from the first sources. During the second there are equi-distance doublesize pulses corresponding to the interference of the second pulse of first group with the first pulse of the second group. At the next the magnitude of the pulses is tripled. The number of time that this process continues depends of the length M of the time series and on the array size N (Fig. 2). After intervals the pulse series arrives to a steady state which lasts after which the decaying signal is symmetrical to the initial transient. Referring to (5), the first CPL near the main beam is the one with index This CPL is the weakest since the interfering signals comes only from the first and last elements ( and N). The strongest CPLs are those with the smallest possible and that still yield in (5) (see . The strongest possible CPL is since then the pulses emitted by all the sources merge on the time axis. Such CPLs are obtained (if at all) far from the main beam (this CPL is outside the visible spectrum in Fig. 3(a)). Increasing the inter-pulse index may yield CPL solutions to (5) which, however, are located further away from the main beam. The CPL can also be viewed as the UWB extension of the GL, if one regards the latter as a constructive interference of sinusoidal pulses emerging from adjacent elements with relative time shift, thus replacing by in (9).

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The lobe structure in the transition zone So far we discussed the CPL independently of the pulse-shape. If the pulse contains oscillations (i.e., if ) then there might also be lobes due to the interference of these oscillations as can be noted in Fig. 3. If the array is sparse in the sense of (8) and supports CPLs, then GL and SL can be generated by phase interference of the modulation across pulses. Specifically, if then there are several GL around each CPL as can readily be observed in Fig.3(b). In the same parameter regime it may also be observed that for the signal model in (1b), all the CPL have the same spatio-temporal shape, whereas for the signal model in (1a) the shapes of each CPL is slightly different since the pulses in (1a) have in general different modulation phases. However, the CPL-grid is mainly unaffected by the choice of the signal models in (1). CPL CONSIDERATION FOR DENSITY AND SPARSITY From the discussion in connection with (9) it follows that a CPL of order (with ) is possible only if Thus in order to avoid CPL of that order it is required that Practically, however, we shall try only to avoid the strongest CPL with and demand that

This condition guarantees that there are no strong CPL. Yet there might be phase interference phenomena (i.e., GL and SL) if around each CPL and in particular around the main beam (see Fig. 3(b)). The well known condition for avoiding these


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GLs is

(this condition is in fact for broad-side radiation with For end-fire radiation with the condition is For UWB signals, however, where the phase-interference phenomena disappear whence there are no lobes as long as (10) holds. Noting that can be much larger than T, this implies that in the UWB regime the array may be ultra sparse in the conventional frequency domain sense. Specifically one finds that for most frequencies within the pulse spectrum such that where from (10) Note that the CPL are formed only if there is a sufficient number of repetitive pulses. Fig. 5 (a) explores the CPL of the normed radiation pattern of (3) for the same array used in Fig. (3(b)), but with several values of M: M = 1,... ,6. One readily note that the CPL’s and their associate GL’s occupy the range while in the range we find the main beam and its GL. The number of CPL’s and their magnitude grow as the number of pulses increases. The figure also compares the radiation pattern for two types of normed detectors: square law detector and peak detector A notable similarity between these detectors is observed, the only difference is in the “shape” of the lobes. CONCLUSION We have present a unified framework for the kinematic analysis of transient arrays. In addition to the side-lobes (SL) and the grating-lobes (GL), which are due to phaseinterference, we also identified cross-pulse-lobes (CPL) which are an intrinsic wideband phenomenon. It has been shown that the lobe structure is an interplay of several mechanisms and controlled by the problem parameters: the center frequency, the fractional bandwidth, the pulse repetition rate, the inter-element spacing and the total number of elements. It has been shown that the conventional frequency domain dense-array condition where is the frequency, which is used to avoid the formation of GL, is irrelevant in the UWB case and in fact the UWB signal spectrum does not have

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to satisfy this condition. Thus, under these conditions one may use sparse array (with inter-element spacing much larger than the wavelength for all frequencies in the band) without suffering from sever GL problems.

Acknowledgements This work is supported by in part by the Israel Science Foundation under Grant No. 404/98, and in part by AFOSR Grant No. F49620-96-1-0039.

References [1] A. Shlivinski, E. Heyman and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propagat., AP-45, 1140–1149, 1997.

POWERFUL SOURCES OF ULTRAWIDE BAND PULSED COHERENT SIGNALS A.F.Kardo-Sysoev1, V.I.Brylevsky1, Y.S.Lelikov1, I.A.Smirnova1, S.V.Zazulin,1 I.G.Tchashnicov2, V.I.Scherbak2, B.I.Sukhovetsky2 1 Ioffe Physico-Technical Institute RAS St. Petersburg, Russia 2 Pulse Systems Group, St. Petersburg, Russia INTRODUCTION Coherent ultrawide band signals are used in UWB radard and communication systems. A short pulse at low PRF is the such simplest signal to be processed at receiving end, but to get enough average power a very high peak power is needed, which strongly complicates pulser, antenna designs and interference problem. New generation of semiconductor fast power switches, developed by us, opens the way to trade peak power for high repetition rate, which eliminates the problems, mentioned above. New semiconductor switches show many advantages compared to other switches used in UWB such, as high peak and average powers, very high time stability, lifetime and so on. The switches can be separated into two classes: 1. Closing switches with faster turn-on process of transition from opened to closed state. 2. Opening switches with faster turn-off process of transition from closed to opened state. Closing switches should be used with capacitive energy storage systems. Opening switches utilise inductive storage systems, which provides far higher stored energy density than capacitors. Generally speaking, any semiconductor (and even not semiconductor) switch may be thought off as a some medium between two electrodes, which may be filled by highly conducting electron – hole plasma or be depleted. There are only two ways to fill medium that is initially depleted of current carriers, (this medium is usually the Space Charge Region (SCR) of a p-n junction under blocking bias): a. To inject carriers into the SCR from its borders via additional pn and np junctions. In the case of field drift, the minimal turn-on time is proportional to SCR thickness and to the maximum blocking voltage: i.e. the higher voltage, the longer the turn-on time. b. To generate carriers inside the SCR by external irradiation (light) or by internal impact ionisation. The turn-on time may be independent of the thickness of the SCR if the rate of carrier generation is enough high. There are also two ways to remove plasma from medium: a) To pull off plasma into the same borders from which carriers were injected; or b) To let electrons and holes recombine within the material In practice it is possible to make injection and generation processes fast enough and controllable by external means (e.g. the well-known case of generation by intense light). It must be noted that the requirement of good external control of the pulse's time position (coherence) is very important for UWB communication and radar systems. But it is very hard to control the withdrawal of high-density plasma due to the "trapping” effect and, practically, it is impossible to control (to "trigger") the recombination process. Therefore, at short pulse repetition periods, it is plasma decay processes, which cannot be well controlled externally, that determine the position of the each successive pulse and, accordingly, the stability of the pulse position in the coding pulse train. Here we discuss new classes of devices that overcome these drawbacks. I. Drift step recovery devices, which include Drift Step Recovery Diodes (DSRD) and Drift Step Recovery Transistors (DSRT). II. Devices based on Delayed Ionisation - Silicon Avalanche Shapers (SAS). Also the new approach to radiation of short pulses by the Folded Horn type antennae will be considered. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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1. DRIFT STEP RECOVERY DEVICES - OPENING SWITCHES. In these devices the plasma injection and plasma removal processes are organised in such a manner that restoration of the SCR and voltage on the device (turn-off process) occurs due to fast withdrawal of plasma into the SCR borders. Two such devices are discussed in this paper, DSRD and DSRT: 1.1. DRIFT STEP RECOVERY DIODES (DSRD)

Practical use of a semiconductor diode as an opening switch includes two steps (Fig. 1): Initially plasma is pumped to the diode by current in forward direction; Then the current's direction is reversed and plasma is removed. Due to the plasma injected at the first "pumping" step, the diode remains at high conductivity during some time and the voltage drop is small. Then at the moment the space charge region appears the voltage rises, and the current decreases. For good switching properties the transition between high (turn-on) and low (turn-off) conducting states should be as sharp as possible, the turnon state time should be long, and the voltage restoration time should be short. The sharpest transition and the shortest (i.e. the highest dV/dt rate) are possible if two conditions are fulfilled: Cond.1. At the moment all injected non-equilibrium carriers are removed from essentially all of the diode, and the equilibrium carriers occupy all p and n layers. Cond.2. The velocity of the SCR’s widening has maximum possible value (Fig.2) which may be reached at current density where - doping level, q - electron charge. Under these conditions the voltage rise rate dV/dt reaches its theoretical maximum

where – maximum breakdown field. For silicon, Equation (2) yields It should be noted that is independent of the maximum voltage of a diode time of restoration may be defined as For example: for

Equation (3) yields

for

The minimal

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The main question is how to fulfil both of these conditions in practice? One approach has been used since the in step recovery diodes (SRD): the llifetime of the carriers is made short, and so is the diffusion length where D - diffusivity, and plasma accumulates only in a thin layer. So the initial space charge region, appearing during plasma removal may only be thinner than and the additional voltage drop on it is small, as may be seen in Fig.2. This moment of switching cannot be controlled just by an external pumping current, both because of the strong temperature dependence of life time and because of high life time dispersion in different diodes due to poor manufacturing control of life time. Due to poor control of it is very hard to increase switching voltage by connecting many diodes in series. We developed two new approaches that overcome these drawbacks: i. To make plasma pumping time short, while carriers lifetime is very long. In this case the enriched diffusion region is determined by the same relation as in Equation (4), where pumping time replaces life [1], Under this condition the moment of switching is strictly controlled by an external circuit which provides current until the moment With such good control it is possible to connect many pn junctions in series, by assembling them in a stack, to increase voltage hundreds times while keeping the turn-off time of the stack equal to that of single p-n junction. Thus the end user sees such a diode stack as a single two-electrode device with larger size. But still, determined by Equation (4), can be very short and for a high voltage p-n-junction is less than 50 ns for a 1 kV device. ii. To fabricate a special distribution of pumped carriers so that the plasma removal stage converges to the distribution, which satisfies both cond.1 and cond.2. In this distribution the injected plasma should be stored in both the p and n layers in appropriate proportions. Such distribution can be realised in a p-n junction with special impurity profiles. For this case the duration of the pumping pulse and the high conductivity state may be increased several times, which simplifies the circuits and increases efficiency. We named these devices Drift Step Recovery Diodes, after SRD. In Fig. 3 we show the most simple and very efficient circuit to realise the current sequences shown in Fig.1. Initially all energy is stored in C1, C2 capacitors. At the moment the primary switch S1 closes. At the second primary switch S2 closes. Any type of closing switch (thyristor, bipolar or field effect transistor and so on) with suitable parameters may be used for S1, S2. At time the energy is stored in inductors L1, L2. When the diode opens (breaks current) this inductively stored energy is switched into the load R1. Voltage at the load may be many times higher than the initial voltage at capacitors C1, C2, i.e. in this circuit the high voltage exists only for a short time, which strongly decreases the danger of breakdown. Load current's rise time is determined by the diode voltage restoration the current decay time is It may be shown that the minimal decay time is which corresponds to a pulse width (FWHM) close to decay time. In this circuit, until time current's shape is controlled by the LC circuits, which easily may be made very stable. As long as the moment is controlled only by external current, the stability of generated pulses is very good. This circuit (Fig.3) may be considered as a pulse compressing circuit, which compresses energy in time from down to with a corresponding increase of peak power. Other circuits for utilising the advantages of the DSRD have been developed. In some cases to get better efficiency, two or more stages of pulse compression in LC circuits may be used. The minimum pulse repetition period for a DSRD is determined by

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and may be made close to But in practice is controlled by the external circuit as well, which generates the desired current shapes. In the case of the circuit in Fig.3, and must be many times more than the turn-on times of the primary switches Additional limiting factors are: Maximum pulse repetition rate of the primary switches; Time needed to charge capacitors C1, C2. So this circuit, while being very efficient, has severe limitations on maximum pulse repetition rate, i.e. far below the DSRD’s capabilities. Using the DSRT as the primary switch, this limitation may be solved by: 1.2. DRIFT STEP RECOVERY TRANSISTORS (DSRT) The Drift Step Recovery Transistor can be fabricated with an n+pnn+ structure. For step restoration of the voltage on the p+n junction (collector) of a DSRT, the same cond.1 and cond.2 as in a DSRD must be satisfied during the process of plasma removal from the n-layer. But there is large difference in the plasma pumping process. The electron current across the collector p-n junction is controlled by balancing two currents: the emitter gating current Ig and the maximum collector current which is limited by the external circuit. If an electron-hole plasma region appears in the n-layer near the p-n junction. But in the DSRT, unlike the DSRD, the electrical field force, moving holes, works against the diffusion force. So the size of plasma enriched region may be controlled only by balancing the emitter and collector currents. (Recall that in the DSRD this size is controlled by or ). Additionally carriers are stored in the pbase. It turns out that for special geometry of the n+pnn+ structure and certain doping impurities profiles, and by adjustment of both currents it is possible to get a pumped plasma distribution, which is quite similar to that for the DSRD. At the point of plasma removal, thereby satisfying cond. and cond.2 In the collector of the DSRT the base current flows only across part of the p-n junction area, due to the presence of the additional emitter electrode. Therefore the maximum dV/dt for the DSRT is less than in DSRD, per Equation (2), usually by a factor of 2 to 3. The simplest circuit for a high repetition rate pulse generation is shown in Fig.4. A triggering pulse pumps both DSRT (Q1) and DSRD (D2) during the period (Fig.5), then plasma dissi-


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pation begins in the DSRT During time the inductors L1, L2 store energy. Due to the separating diode D1, the DSRD (D2) is pumped only via the inductor L3. When the DSRT opens (breaks current), the recharging of the collector p-n junction begins. The current in L2 starts to flow via the separating diode, D1, and the DSRD, D2. The current in L1 changes its direction and combines with that in L2 (Fig. 5). The combined reverse current removes plasma from the DSRD. At the moment the DSRD opens (breaks current) and all the energy from both inductors L1, L2 is switched into the load Rl. As in Fig. 3, the load pulse rise time is determined by the DSRD turn-off time and for decay one has for the case L1 = L2 and L3>>L1. In the DSRT both the externals triggering circuit and the collector circuit rigidly control plasma pumping and removal, and correspondingly control the voltage restoration moment and the pulse position. To improve the efficiency of the pulse forming circuit, other modifications of the circuit (Fig.4) have been ddeveloped as well. The minimum pulse repetition period for a DSRT/DSRD circuit is determined by In practice some period should be added to damp any stray oscillation of energy between circuit's parts. At high power output levels, the need for cooling may limit maximum repetition rate. A typical output pulse is shown in Fig.5. Output is 2.7 kV into 50 Ohms, front 0.7 ns, FWHM ~ 1.5 ns, total efficiency Maximum PRF is 2 MHz. Many other pulsers with different combinations of DSRT and DSRD have been built; some examples are shown in Table 1.

2 DEVICES BASED ON DELAYED IONIZATION (SILICON AVALANCHE SHAPERS SAS) As mentioned in the Introduction, generation of plasma inside the space charge region (SCR) of a device can result in the fastest switching. We generate high density plasma by the process of delayed ionization, which was made feasible in practice by use of high dU/dt sources based on drift step recovery devices [2]. We can utilize this phenomenon to design a very effective closing switch, the Silicon Avalanche Shaper (SAS). When a fast rising voltage is applied to a diode biased in the blocking direction (Fig.6), the high-density displacement current j appears in the SCR. The field in the SCR increases and the border between the SCR and the neutral region (NR) moves with the velocity which is equal to the velocity of electrons in the NR. It may be shown that, when the condition (where is the initial maximum field at p-n junction) is fulfilled, the field in the NR exceeds the value at which the velocity o f electrons reaches the saturated value When U’ is high enough, the maximum field at the p-n junction can reach and exceed the static breakdown threshold If the diode leakage current is small, the number of initial carriers in the SCR that could start ionization is small as well, and no ionization happens in the SCR, even when is much higher than E (overvoltaged region (OR) appears). In the neutral region due to very high electrons concentration the rate of impact ionization generation may be high in spite of relatively low electric field (compared to the SCR) where coefficient). Holes, created by the ionization, drift into the

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SCR (Fig.6), reach the overvoltaged region of the SCR, where and cause a very high ionization rate due to the very high field. Hole-electron plasma, also created by the ionization, sharply decreases the field in the OR which is thereby displaced toward the neutral region. Due to hole current from the NR, the extensive ionization starts again, the field decreases, the OR is displaced again, and so on. A fast running wave appears, the velocity of which may exceed the carriers saturated velocity by orders of magnitude. So the ionization wave quickly fills the volume of the SCR by highly conducting plasma and turns on the diode, i.e. it acts as a closing switch. The total measured turn- on times for high voltage diodes are 50200 ps for diodes in the 3-15 kV voltage range, and turn-on time strongly depends on dV/dt rate (for higher dV/dt one obtains faster turn-on). It may be seen from Equations (2) and (6) that the value of dV/dt needed to start super fast switching of diodes is close to the value of dV/dt for super fast voltage restoration. So these two effects ideally match each other. Thus the fast voltage restoration provided by a DSRD is ideally suited to switch a diode based on delayed ionization. Such diodes we have named Silicon Avalanche Shapers (SAS), and are very suitable for connecting to the pulse output of DSRT+ DSRD circuits for a final stage of pulse compression. Due to plasma " trapping", the pulse repetition period of the SAS is relatively long, but repetition frequencies up to 100 kHz are possible. Some examples of pulsers utilizing this DSRT+ DSRD+ SAS concept are shown in Table 2.


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3. RADIATION OF SHORT PULSES BY THE FOLDED HORN ANTENNA Antenna theory shows that a uniformly excited aperture radiates the shortest pulse. The beam width is inversely proportional to the size of the aperture. The most efficient embodiment of such a transmitter is a horn antenna. It is evident that the beam width angle cannot be less than the apex angle of the horn, even for large aperture size. So ideally the apex angle would be small and total length of the horn would be large (Fig.7), which limits the use of such an antenna. We have proposed to fold the horn, as shown in Fig.7, by a series of steps; each step folds the initial horn to half the length into smaller subhorns. Each level of subhorns has different length, to help the field distribution recover after distortion at "sharp" points A. All subhorns are fed synchronously via a pulse splitter or by separate synchronised pulsers. We have designed and tested two types of folded horns: 1. Flat (or planar horn), which may be made even on a two- sided printed circuit board. 2. Two-dimensional folded horn in which the periodicity of different sub-horn levels shown in Fig.7 exists in both dimensions and the size in H plane is comparable or larger than pulse length. Such a flat folded horn (FH) was tested while radiating the 2 ns pulse which shape is close to shown in Fig.5. Aperture span is 1.5m, beam width for the E plane ~55°, H plane ~ 150°, and the forward/back ratio is 20 dB. The radiated pulse is shown in Fig.8. With a 20 kV (8 MW) feeding pulse amplitude, the field intensity reached 6.345 kV/m at a distance of 12.4 m. With a 2.7 kV (150 kW) HFPG pulser feeding the FH, the field intensity was 1100 V/m at a distance of 4.5 m from the FH. It is well known that for aperture antennas in far field zone maximum field intensity is where pulse output power, r distance, A - quality factor of an antenna (included efficiency and beam width). For our FH, from the experimental data noted above, we have This factor may be compared with the well-known Impulse Radiating Antenna (IRA) [3], which has a

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slightly lower reported value, It should be noted that the disklike IRA is 3.6m in diameter and dozens of times heavier than the FH. Fig. 9 shows radiated field for two dimensional FH. One sees that radiated pulse is very close to derivative of feeding pulse. The field was measured by E-dot sensor. The pulser generates 50 Volts in to 50 Ohm antenna input at PRF up to 10 MHz average.

Conclusions We have developed new and efficient semiconductor devices and circuits for generation of powerful short pulses in nano- and subnanosecond ranges with practically unlimited lifetime (number of shots) at high repetition rate. We have also developed new portable and efficient pulse radiating antennas. References 1. I.V.Grekhov, V.M.Efanov, A.F.Kardo-Sysoev, S.V.Shenderey, "Formation of high nanosecond voltage drop across semiconductor diode". Sov. Tech.Phys.Lett., vol.9 (1983), n4. 2. I.V.Grekhov, A.F.Kardo-Sysoev, "Sub-nanosecond current drops in delayed breakdown of silicon p-n junction". Sov.Tech.Phys.Lett.,vol.5(1979), n8, pp.395-396. 3.Sensor and Simulation Notes, Note 382, 4 July 1995. A Reflector Antenna for Radiating Impulse-Like Waveforms. D.V.Giri and H.Lackner. Pro-Tech, 1308 Mt.Diablo Blvd, Suite 215 Lafayette, CA 94549. I.D.Smith and D.W.Morton, Pulse Sciences, Inc., 600 McCornic Street, San Leandro, CA 94577 and C.E.Baum, .R.Marek, D.Scholfield, and W.D.Prather Philips Laboratory, Kirtland AFB, NM 87117.

ULTRA WIDE BAND SOLID STATE PULSED ANTENNA ARRAY

A.F.Kardo-Sysyoev1, S.V.Zazulin1, I.A.Smirnova1, A.D.Frantsuzov2, A.N.Flerov2, 1 Ioffe Physico-Technical Institute RAS St. Petersburg, Russia 2 Pulse Systems Group, St. Petersburg, Russia INTRODUCTION Phased Antenna Arrays (PAA) are widely used in all fields of modern electronics. Their main advantages are: 1. High total microwave power levels may be achieved by combining of powers of many, small inexpensive sources, 2. Fast electronic deflection of a radiated beam. It is only PAA, which gives means to build powerful all solid state (semiconductor) microwave radiators due to strong limitations on power of semiconductor microwave devices. At the same time for the case of semiconductor devices PAA approach is natural because: 1. Price for a unit sharply drops with production rate increase, 2. Jitter is very low, especially for bipolar devices. But semiconductor devices have inherent slow temperature drift, which, practically, may be orders of magnitude larger than jitter and can disrupt synchronization some time after initial adjustment. The well known practical approach to suppress drift is Closed Loop Techniques CLT [1] which tracks time difference between time positions of a power output and a reference signal and minimizes it. This approach may be used with both narrow band continuous CW and ultrawide band (gaussian monopulse) signals. Whereas CLT is widely used for CW signals, there is little work on UWB. For example in [1], synchronization of four sources had been considered, each of which radiated 3 rf cycles of 2.6 GHz central frequency at 1 kW peak power. CLT used mixing of radiated signal with 2.3 GHz CW reference oscillation. So this system could track the time difference inside only one cycle = 0.4 ns and could be considered as some intermediate step between CW and real monopulse CLTs. Here we consider experimental pulsed PAA system comprising four 50-Ohm input antennas, individually fed by 1,1 kV, 200 ps FWHM pulses at variable up to 12 kHz PRF. 1. PAA SYSTEM 1.1. GENERAL APPROACH The pulsed PAA system is outlined in Fig. 1a. Externally triggered pulser PG feeds antenna module. A pulser's circuit is shown in Fig.2. Synchrogenerator generates four reference signals Time position of each signal may be fine tuned in respect to others inside 00.5 ns range. A pulser PG has main 1.1 kV into 50 Ohm output and 13 V coupled synchronizing output (Fig.3). The and are fed to time discriminator TD (Fig. 1). TD output level depends on the sign of time position difference between and . The TD output via low pass filter (LPF) and amplifier controls Voltage Controlled Delay (VCD). VCD's output triggers Pulse Generator.


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Internal delay between triggering pulser of and output is around 200-210 ns. Internal minimal delays of VCDs are around 40 ns. To compensate the delays and their variations, stable delay ns is added to TD's reference signal. So when the loop is closed all four output pulser's positions are reduced to the time position delayed by 280 ns from SG output It is clear that this approach may be applied to a PAA with any number of elements without degradation of time synchronization. To evaluate time stability, the system shown in Fig. 1, was tested on matched dummy loads. RMS of time deviations around is less than 15 ps at PRF = 12 kHz and still is less than 20 ps at PRF ~ 1 kHz. 1.2. PULSE GENERATOR. Pulse Generating circuit is shown in Fig.2a. Generally, its operation is described in [2]. New fast power switches used in the circuit are Drift Step Recovery Transistor (DSRT), Drift Step Recovery Diode (DSRD) and Silicon Avalanche Shaper (SAS) described elsewhere [3]. Initially DSRT is opened and electron hole plasma is pumped into the collector by driving pulse of ~ 100 ns length via transformer Inductor's L1, L3 current rises linearly at the rate After the end of the driving pulse, DSRT still is opened sometime due to pumped plasma. During all DSRT’s conducting time DSRD D3 is

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pumped via L4, R9 circuit. When, pumped into DSRT, plasma is pulled out, collector's conductivity current stops very abruptly. Inductor current starts to flow via separating diode D4 and DSRD, which is still in conducting state due to previous pumping. Inductor L3 current oscillates in a circuit comprising collector capacitance of DSRT, L3, D4, and DSRD D3. At the second half period of oscillation when both currents of L3 and L1 have the same direction, DSRD breaks this total current and switches it to charge peaking capacitor C4 via L2 inductor. When C4 voltage reaches threshold voltage of SAS D5, SAS turns on and discharges the capacitor C4 into the load. Output pulse is shown in Fig.3. Rise time is less than 70 ps, FWHM is ~200 ps. Maximum pulse repetition frequency 12 kHz is limited by overheating. Addition of heat sinks increased PRF up to 50 kHz. Energy efficiency is better than 15 %. 1.3. ANTENNA Antenna array consists of 8 horns, 2 columns (E plane) by 4 rows, shown in Fig. 4. Each horn's impedance is 100 Ohm. Each two horns in a column are fed by a power splitter (PS) with 50 Ohm impedance. A pulse generator feeds each splitter. All edges of adjacent horns are electrically connected. For the case, when all pulses reaches the aperture simultaneously, therefore low cut off frequency of the array is determined by overall aperture size and total efficiency of radiation is high. Energy, reflected from antenna to PG, is less than 20%. Each connected pair of horns, may be considered as two times folded long horn with the same apex angle. This folded horn (FH) approach provides smaller variations of EM wave time arrival at different points of aperture. In our case the difference is less than 50 ps (1.5 cm in distance). So taking

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into an account the tolerance of synchronization, mentioned above (RMS 2 m, which coincides


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We used the next procedure to check the efficiency of summation of field. We put a switch to lock/unlock loops. Initially a loops were unlocked and pulses took random positions inside the range 25-10 ns prior to the synchronized position not interleaving each with other. Then one loop was locked and corresponding pulse took position with field amplitude Then the next loop was locked and so on. In Fig. 7 the field intensity is shown for different numbers of locked loops. One sees that field summation is perfect. In Fig. 8 fields at different angles in H plane are shown. Pulse width increases, and at large angles (>20°), the pulse splits into two pulses. The splitting angle is determined by condition where 1 - pulse length 10 cm, Z - 26 cm distance between the centers of adjacent horns. Fig.9 shows antenna patterns in H and E planes for both positive and negative field amplitudes. H plane beam width is at level 0.7. Background noise due to stray scattering and interference at is under 20 db level. Forward /back ratio is far better than this 20-

dB level. The difference between positive and negative amplitude is less than 20% and may be explained by interference of the main pulse with after pulse oscillations (see Fig.8). E plane beam width is at 0.7 level. To scan beam, the corresponding delays to triggering pulses were added. Array patterns are shown in Fig. 10 for sets of delays corresponding given scanning angles.

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The picture clearly demonstrates the well known for the case of CW rule: array pattern may be represented as a product of a single element pattern and an array multiplier. For


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two columns array, of course, the scanning angle is not large. CONCLUSION Pulsed synchronized antenna array 2x2 has been built and tested. Array radiates monocycle of 0.5 ns length, ~2 GHz center frequency. Total radiated power ~ 80 kW, efficient potential ~11 kV, beam width at 0.7 level. The array showed perfect summation of field intensity of partial radiators. REFERENCES 1. Gerald F.Ross, “The far field synchronization of UWB sources by closed loop techniques”, Proceeding of an International Conference on Ultra-Wideband, Short-Pulse Electromagnetic, October 8-10, 1992, at WRI, Polytechnic University, Brooklyn, New York.. 2. A.F.Kardo-Sysoev, S.V.Zazulin, V.M.Efanov, Y.S.Lelikov, "High Repetition Frequency Power Nanosecond Pulse Generation ", 11th IEEE International Pulsed Power Conference, Baltimore, Maryland, p. 107, 1997. 3. V.M.Efanov, A.F.Kardo-Sysoev, I.G.Tchashnikov, "Fast Power Switches from Picosecond to Nanosecond Time Scale and Their Application to Pulsed Power ", Tenth IEEE International Pulsed Power Conference, Albuquerque, New Mexico, pp.342-347, 1995


ADMITTANCE OF BENT TEM WAVEGUIDES IN A CID MEDIUM


1.

INTRODUCTION

Various solutions have been developed for the propagation of TEM waves in an inhomogeneous dielectric medium with permittivity proportional to in a cylindrical coordinate system with propagation in the direction [1-5, 7]. Experimental work is underway to approximately synthesize such a medium with guiding conductors to form a TEM-transmission-line bend [6]. There are also cases of TEM waves propagating in other directions in such a medium [8]. The general procedures are differentialgeometric lens synthesis as discussed in [11]. This medium then has very special properties. So let us give it a name: cylindrically-inhomogeneous dielectric or CID for short. (El Cid is a Spanish title, roughly translating as lord or sir.) The particular form of inhomogeneity as

is the form to be implied by this name. Here and are convenient reference permittivity and radius respectively. In the present context, these refer to the middle ofthe lens cross section. With as the center of the waveguide with respect to the coordinate, let the guide extend a maximum distance on either side of In the present cases, the fields are all confined to Then we have

This

is an important parameter and as we shall see the guide admittance has a correction proportional to

for appropriately symmetrical guide cross sections. As discussed in [4, 5] the electric potential for the TEM mode satisfies

which is similar to a Laplace equation on a cross section (constant ). Here is a function of and z, but other coordinates such as cylindrical coordinates centered on the cross section are more appropriate for the circular coax case. Note that there is a propagation function for the coordinate not included in the above, Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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as well as a waveform function. One can also write an analogous equation for the magnetic potential, but we will not need it here. Further details are included in the references. In the limit that the waveguide becomes straight and the dielectric is uniform in the guide cross section, changing (1.3) into a true Laplace equation as

with whatever transverse (subscript t) coordinates one wishes. This reference case is denoted by the subscript 0 and we later have occasion to write

as a perturbation. For our cross-section boundary-value problem we take our electric conductors as

or 0

potential (volts) giving boundary conditions for and having zero for boundary conditions. The current I, found from a line integral of the magnetic field is used to define the characteristics admittance as

The electric field for the cross section is given by

with coordinates as appropriate. The magnetic field for the cross section is given by

where propagtion has been taken in the uniformly in the guide.

direction. Note that for

the wave admittance become

The basic problem considered in this paper is the variation of the characteristic admittance as a function of After two simple canonical cases, we concentrate on the important case of a circular coax. 2.

SYMMETRY CONSIDERATIONS

To aid in the analysis let us consider certain kinds of symmetrical guide cross sections. Specifically we consider symmetries under which reflection and or rotation of the bent guide reverses the direction of bend while conserving the characteristic admittance of the guide. Consider first reflection symmetry in the cross section about a symmetry line ( axis) at as in Fig. 2.1 A. Consider first that as looking into the page the guide is bent to the left (positive positive ). Now consider reflection of the entire bent guide through a plane containing the axis and perpendicular to the page. The guide has a rotation axis a distance to the right of the original rotation axis. The guide now bends right as one moves into the page, which we can consider as a negative normalized curvature i.e., in the transformation. Using on the plane measured from the axis the potential transforms from case 1 (left bend) to case 2 (right bend) as

This is a mirroring of the potential as discussed in [9]. The associated fields transform as


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The permittivity transforms like the potential, i.e.,

The potential and the current I (from a line integral around the center conductor) being unchanged in the transformation, the characteristic admittance is unchanged, i.e.,

which is an even function of the bend curvature. A similar thing happens if the guide cross section on the

plane has a two-fold rotation axis

the plane), as illustrated in Fig. 2.1B. Again, case 1 has the guide bending to the left. Case 2 is found by rotating the entire bent buide by (180°) about Now the guide has rotation symetry about the axis which we can interpret as a negative normalized curvature The potential fields on the cross section transform as

The permittivity transforms as in (2.3). The characteristic impedance is unchanged and the conclusion in (2.4) applies again. From (1.6) and (2.4) we have

Let us define a normalized form of the admittance as

When

the lens reaches the rotation axis where we can assume that

series in

The fact that

is an analytic function of is even in

a singularity occurs in the lens medium. For i.e., that it can be expanded in a power (Taylor)

means that only even powers are allowed in the expansion, giving

where the second index above the summation indicates the increment (2 in this case) in the summation index for successive terms. Keeping the first two terms we have

which can be used as an approximation valid for small

3.

CANONICAL H-PLANE BEND

This case with two electrically conducting boundaries of width 2b and spacing 2d is illustrated in Fig. 3.1 A. The lens region is closed at the edges of the conductors by magnetic boundaries to give the case discussed in [2(Section 4.1)]. Here we have the fields

which are integrated to give


For

we have a uniform dielectric medium giving the simple result

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C. E. BAUM

from which we find the normalized characteristic admittance

Expanding this for small

we find [10]

Consistent with the symmetry results of Section 2. Figure 3.1B shows the case of three conducting boundaries (coax-like) with the outer conductors at zero potential and the “center” conductor at potential With the dimensions as indicated (same overall dimensions as the previous case) the problem is readily solved as the parallel combination of two 2-conductor problems with spacing d and width 2b. This gives as 1/4 of that in (3.2) and the same as in (3.4) and (3.5). 4.

CANONICAL E-PLANE BEND

Figure 4.1 A shows the case of an E-plane bend where now the electric conductors are of width 2d with spacing 2b. Again the lens region is closed at the edges of the conductors by magnetic boundaries to give the case discussed in [2(Section 4.2)]. The fields are now

which are integrated to give

So the characteristic admittance is not a function of a very simple result. The normalized characteristic admittance is simply

This is the case for all

and the expansion for small

gives

with no remaining error terms. This is a special case which is still consistent with the symmetry results of Section 2. Figure 4.1B shows the more coax-like case with three conducting boundaries with outer conductors at zero potential and the “center” conductor at potential With the dimensions as indicated the problem is solved as the parallel combination of two 2-conductor problems with spacing b and width 2d. However, these two problems are not identical owing to the different permittivities in the two regions. Our previous results can be used by replacing by the value of Y (the wave admittance) at the center of each of the two regions giving and conjunction with (4.2) gives

for the left and right regions respectively. Using this in


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Again we have the convenient result of a four-fold increase in the characteristic admittance, but now with a second order correction in

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C. E. BAUM

COAXIAL BEND

Now consider a bend in a circular coax as illustrated in Fig. 5.1. The inner conductor (radius a ) has potential and the outer conductor (radius b) has potential 0. Note the coordinates on the plane centered on the coax with

Here the fields are given by

The characteristic admittance is

where this applies to any between a and b. We note for later use

which applies for all

where

We also write

applies to the case of

and

has boundary conditions 0 on both conductors.

5.1 Straignt Coax For we have the well-known case of a straight coaxial waveguide (cable) with a uniform dielectric medium. In this case we have

5.2 General Solution Through Second Order From (5.3) and (5.6) we have


Divide through by

and integrate with respect to

Now expand as a simple geometric series

which is valid for all

with

giving

In this form we can look at the terms given by each

to give

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the

C. E. BAUM Next apply symmetry for as in Section 2. The circular coax has cross-section symmetry on plane of all rotations and reflections known as This includes both the reflection with

respect to the axis in Fig. 2.1A and the rotation with respect to from left bend (case 1) to right bend (case 2) as

in Fig. 2.1B. From (2.5) we have

Both cases must give the same result for the admittance, so let us take the average giving

Now change the angular variable as

Noting that any

interval for

and

will do due to the periodicity, we then have

Now we note that

Integrating over gives zero for odd functions of only even terms. We can then write

For the

terms first integrate over

Then all terms for odd

in (5.14) give zero, leaving

and invoke (5.4) giving

The leading term 1 is now separated out and terms for (even) are left as corrections. For the term let us first integrate by parts over to give


Now write

361

as in (5.5) where

We then have [10]

Combining these results we have

Noting the form the

must take as in (2.9) then the

(goes to zero faster than

) is combined with

to give

5.3

Special case of small b- a

An interesting special case has

This limit of 1/2 can also be found from a physical problem of integrating the admittance per unit closely spaced coaxial cylinders with the CID medium between them as

of two

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C. E. BAUM

So the general result is consistent with this simple check. The case of small b-a also gives a bound for since [10]

Note for small a/b we have logarithmically. This is associated with the fact that for a small center conductor the electric field is relatively large near the center conductor where is close to The permittivity near

6.

then dominates this case.

CONCLUDING REMARKS

We now have some canonical results for the characteristic admittance of a bent TEM waveguide in a CID medium. The H-plane and coax bends show an increase of the characteristic admittance over that of the straight waveguide with taken as the permittivity in the center of the bend guide. However, the twoconductor version of the E-plane bend shows no change in the characteristic admittance with bending; the three-conductor version shows some increase with bending. More general guide cross sections may also be considered. Here we have the general result that for certain symmetries in this cross section the correction to the characteristic admittance is second order in the bend curvature. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U. S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES C. E. Baum, Two-Dimensional Inhomogeneous Dielectric Lenses for E-Plane Bends of TEM Waves Guided Between Perfectly Conducting Sheets, Sensor and Simulation Note 388, October 1995. 2. C. E. Baum, Dielectric Body-of-Revolution Lenses with Azimuthal Propagation, Sensor and Simulation Note 393, March 1996. 3. C. E. Baum, Dielectric Jackets as Lenses and Application to Generalized Coaxes and Bends in Coaxial Cables, Sensor and Simulation Note 394, March 1996. 4. C. E. Baum, Azimuthal TEM Waveguides in Dielectric Media, Sensor and Simulation Note 397, march 1996. 5. C. E. Baum, Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation note 405, December 1996; Proc. URSI Int’l. Symposium on Electromagnetic Theory, Thessaloniki, Greece, May 1998, pp. 636-638. 6. W. . Bigelow and E. G. Farr, Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Material, Sensor and Simulation Note 416, January 1998. 7. W. S. Bigelow and E. G. Farr, Impedance of an Azimuthal TEM Waveguide Bend in a Graded Dielectric Medium, Sensor and Simulation Note 428, November 1998. 8. C. E. Baum and A. P. Stone, Unipolarized Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis, Sensor and Simulation Note 433, January 1999. 9. C. E. Baum, Interaction of Electromagnetic Fields with an Object Which Has an Electromagnetic Symmetry Plane, Interaction Note 63, March 1971. 10. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th Ed., macmillan, 1961. 11. C. E. Baum and A. P. Stone, Transient Lens Synthesis: Differential Geometry in Electromagnetic Theory, Taylor & Francis, 1991.

1.

OPTIMIZATION OF THE FEED IMPEDANCE FOR AN ARBITRARY CROSSED-FEED-ARM IMPULSE RADIATING ANTENNA

J. Scott Tyo Department of Electrical and Computer Engineering US Naval Postgraduate School Monterey, CA 93943

INTRODUCTION Impulse radiating antennas (IRAs) are members of a class of antennas that are designed for the radiation of ultra-wideband (UWB) electromagnetic impulses. Through a combination of a non-dispersive transverse electromagnetic (TEM) feed structure and a focused aperture, IRAs act like differentiators for the early-time portion of the waveform. When excited by a fast-rising step, the radiated field closely resembles a narrow impulse. While the nature of the focusing optic and the feed structure do affect the features of the radiated waveform before and after the prompt impulse, the fast part of the radiated signal for a general IRA is [1]

V(t) is the applied voltage waveform, is the peak of the applied voltage waveform, and the surface integral is over the transverse components of the TEM mode in the aperture defined by The radiated field can also be described in terms of the geometric impedance factor of the TEM transmission and the aperture height [2]

A number of metrics have been proposed to compare the performance of antennas operating in the time domain [3,4,5]. The difficulty in comparing performance arises from the non-unique choice of a norm for time domain comparisons. In [4] and [5], performance metrics are defined in terms of the or the peak radiated field. Farr and Baum define a power normalized gain and a voltage normalized gain


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that are used to compare the performance of antennas under constant input power and voltage conditions, respectively. Optimization of is typically is accomplished by allowing to go to zero, an impractical scenario that requires infinite input power and results in infinite fields in the aperture [4]. For that reason, the power normalize gain is often a better metric; however, as defined in (3) has units of length, and care must be taken when applying it to an optimization problem in order to make a fair comparison between antennas of different sizes, as can be increased simply by increasing the physical size of the antenna. For example, when optimizing the feed impedance of a lens IRA constrained to fit within a circular aperture of fixed radius, Farr and Baum [6] used to conclude that low-impedance horns were undesirable. This result is true given the imposed constraint, but the result is dominated by the fact that the area of the aperture of low impedance horns that fit inside a circle of fixed radius goes to zero as Buchenauer, et al., [5] introduced the dimensionless quantity of prompt aperture efficiency defined as

where A is the area of the aperture defined by and is the principle component of the electric field in the aperture, taken without loss of generality as being parallel to the y-axis. Because of the area normalization in (4), aperture efficiency is the preferred metric for comparing the inherent performance of classes antennas regardless of physical size. In contrast to the result presented in [6], it was demonstrated in [5] that low impedance horns are actually more aperture efficient than high impedance horns, and they can be used to efficiently fill a given aperture by arraying. Aperture efficiency and power normalized gain are related by Regardless of the metric used to compute the optimum, it is clear from (4) that the optimum antenna for a fixed input impedance is the one that maximizes the aperture height. An important class of IRAs is the set of antennas that are fed by self-reciprocal feed structures. Self reciprocal antennas are discussed in [7], and have feed geometries that are unaltered by the reciprocation operation where is the position vector in the aperture plane (after stereographic projection) and a is the radius of the circle of symmetry. The coplanar feed IRAs discussed in this paper are examples of self-reciprocal apertures, as shown in fig. 1. Self reciprocal apertures have a number of interesting properties, but the most important ones for this study are 1) exactly half of the power on the transmission line propagates outside the circle of symmetry, 2) the total charge on the feed arms inside the circle is equal to the total charge on the feed arms outside the circle of symmetry, and 3) all contiguous points on the circle of symmetry that are not occupied by conductor lie on a single field line. For the important class of self reciprocal apertures, which are typically confined to focusing the circle of symmetry, the aperture area A is constant for all configurations, and and are equivalent metrics. Aperture efficiency will be the parameter used in this study to optimize the feed configuration in crossed coplanar fed IRAs, primarily because of its dimensionless property and ready interpretation [5].

OPTIMIZATION OF THE FEED IMPEDANCE

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COMPUTATION OF THE TEM MODE DISTRIBUTION It is well known that the field distribution of the TEM mode on a multi-conductor transmission line can be computed as the gradient of a scalar potential that satisfies the Laplace equation. For many classes of feeds, the potential can be calculated using a combination of the stereographic projection and conformal transformations [1]. However, for arbitrary geometries, the conformal map may not exist in closed form. In this study, it is assumed that the stereographic projection has already been carried out, and the conically symmetric feed structure has been transformed to a longitudinally symmetric structure as discussed in [8]. The important properties of the stereographic projection for this class of antennas are discussed in greater detail in [8]. The asymmetrically crossed coplanar feed structure depicted in fig. 1 can be described in terms of successive conformal mappings, but the Schwartz-Christoffel transformation integrals for the asymmetric cases have not been performed analytically. When the analytic form of the conformal transformation is not known, a numerical approximation can be obtained by employing a Laplace equation solver such as the method of moments or finite element method (FEM). The properties of self-reciprocal symmetry discussed in section I.B. above make the geometry depicted in fig. 1 ideally suited to analysis by the FEM. After numerical calculation of the fields as described in [9], the integrals in (1) – (4) can be evaluated directly or by casting the aperture integral into one of the alternate contour integral forms presented in [2]. The modeling method was

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validated by comparing the computed solutions for

solution is known, namely

and

for the case where the analytic

[9].

CALCULATION OF ANTENNA DESIGN PARAMETERS Using the FEM method described in the previous section, the feed impedance, aperture height, and aperture efficiency were calculated at values of ranging from 3° to 87° and at values of ranging from 0.02 to 0.97. Figure 2 presents curves of for several values of and fig. 3 presents the value of as a function of to achieve popular values of the feed impedance. As expected, as (limit as the spacing between the electrodes goes to zero) and as (limit as the electrodes approach infinitesimal wires). The family of curves in fig. 2 has been fitted to the functional form

where m is given in [4]. The values of the coefficients A, B, C, and D are tabulated for the values of investigated in this study in table 1. The sum of (5) was obtained by analyzing the solution for The first term is the low-impedance limiting form, the second term is the high impedance limit. The third term represents the error due to the addition of the two asymptotic solutions. Because the actual form of is expected to be given in terms of elliptic integrals and elliptic functions the coefficients in (5) do not have a convenient representation in terms of elementary function of Figure 4 presents curves of for several values of Figure 5 presents the aperture height as a function of for popular values of the feed impedance. As which is one-half of the mean charge separation for a four-wire transmission line [2]. As The family of curves in fig. 4 is fit by the functional form

The tabulated values of valid only over the range DISCUSSION

and

are given in appendix 1. The form given in (6) is and strictly does not fit the solution as

The data presented in figs. 2 - 5 provide all of the information needed to design an antenna with a particular aperture efficiency, feed impedance, or feed arm angle Curves of can be plotted for distinct values of as is done in fig. 6. Analysis of fig. 6 provides two interesting results. First, for any particular value of feed impedance, there is a unique geometry that provides the optimum aperture efficiency. The curves in fig. 6 can be used as a design tool to select a particular feed geometry to match the impedance of an individual source. Second, it is clear from the figure that as increases, the optimum aperture efficiency occurs at higher and higher impedances. The peak value of the aperture efficiency for each is plotted in fig. 7, and the feed impedance corresponding to this peak is plotted as a function of in fig. 8. The relationship between optimum feed impedance and appears to be linear so the data points were fit using a


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constrained least squares linear regression (setting the intercept to 0). The equation for the line in fig. 8 is

assuming that the medium is free space. This linear relationship between optimum feed impedance and was unexpected, and if closed form expressions for the conformal mapping can be obtained, they might provide some physical understanding of the interaction between feed arm angle, extent of the electrodes, and aperture efficiency. It is worth noting that (7) predicts an optimum impedance of for the case of when the two pairs of crossed coplanar feeds are at the same location, corresponding to the case of a single pair of coplanar feeds. This geometry was optimized by Farr and Baum in [4], and the optimum impedance was found analytically to be a difference of less than 2%. In this paper, the optimization was considered for the aperture efficiency (or equivalently, the power normalized gain of [4]). However, for many UWB systems, the quantity that should be maximized is the prompt radiated field, which scales like or the voltage normalized gain of [4]. As mentioned in the introduction and found in [4], the voltage normalized gain is optimized by allowing Not only is this impractical for current-flow reasons (since current on the antenna goes like ), but the wide feed arms needed to obtain low impedances may be expected to significantly enhance feed blockage [10]. However, the importance of for maximizing the radiated field should not be overlooked in designing a system, and hence the value of (normalized to the aperture radius a) is plotted as a function of in fig. 9 for several important values of the feed impedance. The optimization reported here was for the 4-arm IRA, but there is nothing preventing a similar analysis of N-arm IRAs. The optimum aperture efficiency for the 2arm IRA can be computed using the results from [4] and is 27%. The optimum aperture efficiency for the 4-arm case considered here is 35%. Intuitively, the addition of nonblocking feed arms will continue to optimize the aperture efficiency. This is true because additional feeds cause the field distribution in the aperture to be more uniform, hence increasing aperture efficiency. However, the late-time field will be pulled down more rapidly as more feed arms are added, affecting the nature of the radiated pulse, even in the absence of feed blockage. CONCLUSIONS The study described in this paper has provided three principle results. First, the entire design space for reflector IRAs fed by crossed coplanar feeds with reflection symmetry has been sampled. Curves are presented in figs. 2 and 4 with corresponding empirical fits in (5) and (6), that allow ready prediction of feed impedance, aperture height, and aperture efficiency as a function of the geometric parameters of the antenna. These relationships provide more flexibility in IRA design beyond what was possible using configurations with known analytic solutions [4]. Second, the data presented in this paper allows the optimization of the aperture efficiency for any value of the geomterical properties. It has been shown that a distinct optimum exists for any feed arm angle and that an absolute optimum configuration exists at and ( in free space). Finally, the results presented in figs. 6 and 8 show that for any specific value of feed impedance, there is a unique optimum configuration that will maximize aperture efficiency. The feed arm angle is linearly related to the desired


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impedance by (7). This is important in that once the input impedance of the antenna is specified, the antenna can be optimized without impacting upstream components of the system by selecting the appropriate values of and presented in this paper. The method used in this report are general in that they can be applied to any focused aperture system to calculate feed impedance and aperture height (and hence any of the performance metrics described above). While best suited to the analysis of self reciprocal apertures, iterative boundary condition methods have been developed that allow computation of open TEM modes [5]. The method can be easily modified to include the effects of aperture blockage in the evaluation of (2), allowing analysis of geometries for the feed arms that are not coplanar plates, including circular-cross-sectioned feed arms, curved plates, or other arbitrary configurations. REFERENCES

1.

C. E. Baum and E. G. Farr, “Impulse Radiating Antennas” in Ultra Wideband/Short Pulse Electromagnetics H. L. Bertoni, C. E. Baum, and L. B. Felson, Eds., pp. 139147, Plenum Press, New York, 1993 2. C. E. Baum, “Aperture efficiencies of IRAs” Sensor and Simulation Notes #328 (USAF Phillips Lab, Albuquerque, NM, 1991) 3. E. G. Farr and C. E. Baum, “Extending the Definitions of Antenna Gain and Radiation Pattern Into the Time Domain” Sensor and Simulation Notes #350, (USAF Phillips Lab, Albuquerque, NM, 1992) 4. E. G. Farr and C. E. Baum, “Optimizing the Feed Impedance of Impulse Radiating Antennas Part I: Reflector IRAs” Sensor and Simulation Notes #354 (USAF Phillips Lab, Albuquerque, NM, 1993) 5. C. J. Buchenauer, J. S. Tyo, and J. S. H. Schoenberg, “Aperture Efficiencies of Impulse Radiating Antennas” Sensor and Simulation Notes #421 (USAF Research Lab, Albuquerque, NM, 1997) 6. E. G. Farr, “Optimization of the Feed Impedance of Impulse Radiating Antennas, Part II: TEM Horns and Lens IRAs” Sensor and Simulation Notes SSN#384, (USAF Phillips Lab, Albuquerque, NM, 1995) 7. E. G. Farr and C. E. Baum, “Radiation from Self-Reciprocal Apertures” Chapt. 5 in Electromagnetic Symmetry, C. E. Baum and H. N. Kritikos, Eds., Taylor and Francis, Bristol, PA, 1995 8. E. G. Farr and C. E. Baum, “Prepulse Associated with the Feed of an Impulse Radiating Antenna” Sensor and Simulation Notes #337, (USAF Phillips Lab, Albuquerque, NM, 1992) 9. J. S. Tyo, “Optimization of the feed impedance for an Arbitrary Crossed-Feed-Arm Impulse Radiating Antenna” Sensor and Simulation Notes #438 (USAF Research Lab, Albuquerque, NM, 1999) 10. E. G. Farr, Personal Communication, October 1999

TRANSIENT FIELDS OF OFFSET REFLECTOR

Sergey P. Skulkin 1 , Victor I. Turchin2 1

Radiophysical Research Institute Nizhny Novgorod, Russia 2 Institute of Applied Physics Russian Academy of Science Nizhny Novgorod, Russia

INTRODUCTION Reflectors have figured prominently in the history of electromagnetic radiation, antennas and communications. The latter part of the 20th century has seen significant changes and improvements in the design and practice of reflector antennas. Offset configurations of parabolic reflector antennas are widely used for many applications. Transient fields radiated from such antennas1 (which is fed by transient step or deltapulse) can be useful not only for transient ultra wideband antennas. In many cases time domain calculation techniques are simple and can be used for frequency domain fields calculations 2. In this paper a closed form analytical solution is developed for predicting the transient electromagnetic fields which radiated by a perfectly conducting offset parabolic reflector antenna when it is fed by an elementary dipole. We suppose here that the projection of offset reflector surface to the plane has a circular shape. The calculation technique described below is used for any reflector surface with circular projection and the special case of such geometry is also prime-focus parabolic antenna.

ANALYTICAL DEVELOPMENT IN THE TIME DOMAIN Consider a reflector which is a cut of the symmetrical parabolic surface where F is the focus, is Cartesian coordinate system. Let us assume that the antenna aperture (projection of the cut to plane) is the circle described as where are coordinates of its center and is the radius. It is seen when the origin of coordinate system is placed outside the circle, and the cut of the parabolic surface represents an offset reflector. When the system represents a symmetrical reflector. It is assumed that the reflector is fed by an elementary dipole placed at the focus point (0,0, F) and oriented along the unit vector If the reflector antenna is fed by input signal the electric field at any observation point which is defined by the radius-vector can be written as

where denotes the convolution over the time, and represents the transient field at the observation point which is the response on the input signal at the Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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point The exact calculation and analysis of from any reflector antenna require complex solution of nonstationary diffraction problem. However, we suppose that the most part of energy of the input signal spectrum is concentrated in the interval and In this case the usually approximations may be used. In particularly, the electrical current on the reflector surface can be represented as

where is the magnetic field on the reflector surface, is the normal to the surface. Note that If the input signal represents the current on the reflector surface can be found as where the vectors and determine the coordinates of the reflector surface and focus respectively, For instance, when the dipole is directed along the Thereafter the transient field

can be written in the next form

where

space. Each polarization component of vector

and is the impedance of the free will be represented as:

The dependencies of on time for each polarization can be obtained from (7) immediately, using the integration technique for a of complex argument2. Using this technique the integral over the surface can be represented as the integral over the curve

where is a part of plane contour L, belonging to the plane area is determined by the equation

the contour L


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and

It was found that for parabolic surface (13) represents the circle of the radius depending on time: with the center at the point

where The integral over

where

in (12) is written after substituting of (15) as

where in (19) lie on the circle and if i.e. the current point of L lies inside and in the opposite case. As follows from (19), the behavior of considering as time function depends on mutual location of L and There are four cases: (i) if i.e. the observation time is less then minimal travel time from the focus to the reflector and from the reflector to the observation point; it can be shown that this case is realized if or (ii) the circle L lies inside the aperture (iii) only a part of L lies inside the aperture (iv) the circle L lies outside this case corresponds to as (i). The above simple rules determines the difference between the transient fields of the off-set reflector and symmetrical reflector as it will be demonstrated below. NUMERICAL RESULTS AND DISCUSSION As the first example, consider the pulses when the observation point lies in front of the aperture center, i.e. In the case of symmetrical reflector Taking into account that we obtain that the contour L and the aperture boundary are the concentric circles, and when Because when the time boundaries of the transient field are written as

Note that only the cases (i),(ii), and (iv) are realized for this symmetrical scenario which results in the appearance of two in the transient field of the symmetrical reflector: see Figure 1. Compare this result with the transient function of the offset

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reflector: When observation time increases the radius increases starting with zero and the center of L moves simultaneously from the point to the origin which results in the appearance of the case (iii) and corresponding decreasing of the magnitude of the second pulse: see Figure 1. The above properties are illustrated in Figure 1 showing the normalized transient field for symmetrical and offset reflectors

Figure 2 shows the time domain dependencies of for prime-focus reflector antenna. Here and projection of observation point is placed out of


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aperture.

Figure 3 shows the time domain dependencies of for prime-focus reflector antenna. Here and projection of observation point is placed out of aperture. From these figures we notice that fields of prime-focus and offset reflector antennas in near-field region are different. CONCLUSIONS The transient fields for offset reflector antenna have been obtained for all points of the half-space in front of the aperture. We illustrated that the structure of the spatial-temporal field distribution is quit complex, especially in the near-field region. We show that in this region there is a difference between transient fields of prime-focus and offset parabolic antennas. REFERENCES 1. C. E. Baum, E. G. Farr, and D.V. Giri ”Review of Impulse-Radiating Antennas”, in book Review of Radio Science 1996-1999, ed. by W. R. Stone, Oxford University Press, 1999. 2. S. P. Skulkin, V. I. Turchin, ”Radiation of nonsinusoidal waves by aperture antennas,” Proc. EUROEM ’94 Symposium, Bordeaux, France, part2, pp.1498-1504, May 1994.


A NEW BROAD BAND 2D ANTENNA FOR ULTRA-WIDE-BAND APPLICATIONS

V. Mallepeyre1, Y. Imbs1, F. Gallais 1, J. Andrieu 1 , B. Beillard1, B. Jecko1 M. LeGoff 2 1

2

Institut de Recherche en Communications Optiques et Microondes IRCOM - 7 rue Jules Vallès - 19100 Brive la Gaillarde (FRANCE) E-mail : [email protected] CELAR (DGA) GEOS/SDM - BP 7419 - 35174 Bruz Cedex (FRANCE)

INTRODUCTION In this article, the theoretical analysis and the experimental results of a new broad band 2D antenna are presented. This study was supervised by the CELAR (French Technical Centre for Armament Electronics) and the IRCOM (Research Institute of Microwave and Optical Communications) in order to develop a radar demonstrator to localise buried or surfaced targets. The purpose of this study is to design an antenna to radiate and receive ultra short pulses covering the frequency range 100 MHz-1 GHz. Even more this antenna must be contained in a minimum volume. The antenna must be well matched in the frequency range, have maximum gain and also preserve the rise time and duration of the generated pulse signal. PURPOSE AND STUDY The aim of the study is to realise an antenna whose requirements are to radiate and receive ultra short pulses (about 1 ns) with high voltage level (up to 20 kV). One important point is that the antenna does not distort the generator pulse, which will degrade the resolution of the radar if the impulse response of the antenna is significantly extended. We determine a distortion coefficient as the extension of the transient radiated pulse divided by the input pulse duration at the feed point of the antenna. The calculation of the distortion coefficient requires the knowledge of the expressions below: = time when 95% of the entire signal energy of the generator is reached


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V. MALLEPEYRE ET AL.

= time when 95% of the entire signal energy of the radiated signal is reached

Suitable distortion coefficient of the antenna should be less than 1.5 . Broad band antennas like TEM horns, ridged horns or log-periodic, don’t have that quality. For instance the distortion coefficient is greater than 15 for ridged horns, 30 for TEM horns and 120 for log-periodic antennas. Also the new concept(figure 1) is an original conducting wire aerial which covers required bandwidth and could radiate ultra short pulse with a distortion coefficient equal to 1.4 .

THEORETICAL STUDY – INTEGRAL FORMULATION TEM antennas consist in two perfectly conducting wire structures with symmetrical fed. The electromagnetic field is formed in the feeding alimentation wire plane, and propagates along conducting arms. Vertical plane contains electric field (E-plane) and horizontal plane magnetic field (H-plane). The theoretical study of perfectly conducting wire structure behaviour can be resolved in the same way as a problem of transient electromagnetic wave diffraction by an obstacle1. This generates a space-time integral equation 2 verified by the induced currents. For the wire structure, currents are obtained by solving the equation (1).

with

and

A NEW BROAD BAND 2D ANTENNA FOR UWB APPLICATIONS

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s and refer respectively to the observation points and the source points. represents the image source points, a and are the radius and the wire outline. are the unit vectors. is the applied field corresponding to the source voltage. The electric field is deduced from currents with the equation (2).

NUMERICAL PROCESSING The integral equation (1) is discretised with the method-of-moments. It uses a second order Lagrange interpolation polynomial and is numerically resolved in the time domain2. It can be written in a matrix system as :

with : the wire structure is divided into segments of length the time is divided into equal intervals The unknown is the column vector at the time The current at the structure segment at the time depends on diffracted field at the time The latter is calculated from the induce currents on the structure at the anterior times and from the incident electric field which is known at this time. Then, the matricial system is solved numerically by successive time steps.

APPLICATION TO THE NEW ANTENNA The antenna is directly fed by the bifilar line. Electric field is also guided in the line, before being radiated in space. Electric field polarisation is principally vertical rectilinear and a single 90° degrees rotation of the antenna gives the horizontal rectilinear polarisation. The entire device is contained in a single plane (figure 2) , so that there is no crosspolarisation. Electromagnetic properties of the antenna (feed impedance, gain, radiation patters, frequency range, distortion) depend essentially on geometrical dimensions such as length or aperture angle.

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SPECIFICATIONS Aperture angle calculation Adaptive formulas exist in literature for similar antenna design like V-dipole which consists in only two wires. Empirical formulations allow to determine inner optimum angle with maximum axial gain, function of s dimension and wavelength

If

then it’s possible to make an extrapolation from above formulas.

It appears useful to add to V-dipole many additional wires connected at their ends (figure 2) whose geometric forms have been optimised step by step to increase electromagnetic qualities of the device, especially to stabilise feed point impedance on the bandwidth, to improve directivity, to intensify field level in axial direction and to completely eliminate cross-polarisation. Electromagnetic fields stay between the two planar lines. Feed point impedance Feed point impedance depends on antenna geometry, on resistive loads and on wire radius. A small r radius strengthens inductance effects of the wires, whose are increasing the inductance imaginary part with the frequency. At the opposite, a more important radius allows to keep a small imaginary part on the bandwidth. So for the simplicity of the input matching of the system, it’s essential to choose a suitable radius (1 centimeter for instance). End of wires matching Classical antennas present reflections of currents from the ends of wires whose are deteriorating their performances. This resonance is responsible for an important lengthening of transient radiated pulse and a weakening of standing wave ratio at the antenna feed-point. The problem is solved with resistive loads distributed on the upper part of each arm of the antenna. The currents propagation on each conductive wire is progressively reduced until nullifying and decreasing parasite reflections. For instance, the evolution law of distributed resistance uses the Wu and King non-reflecting principle3 which is written below :

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with wire length resistively loaded, position of the resistive element on the wire, first resistance at The value is choosen between and Each resistance takes place every 5 centimetres from each other. Resistive values are not critical, so it’s possible to use a hyperbolic law approximation. Then simple and easy to use devices have been built with the association of many standard resistances in parallel along each extremity of wires. The main disadvantage of this technique is the global efficiency of the antenna which is fairly reduced. So to avoid gain degradation, only the upper part of each wire is resistively loaded. The length of this part is between s/3 and s/2.

EXPERIMENTAL RESULTS Geometric dimensions of the antenna given below, are calculated from design rules presented before: L =

s= s’ = s’’ =

1 m 1,044 m 0,3m 0,744 m

l= l’ = r=

0,65 m 0,35 m 0,01 m

To match this antenna (input impedance ) to the feed line (characteristic impedance ), a differential balun has been constructed by Europulse4. This device has been designed to feed the antenna with a symmetrical pulse too.

Reflection coefficient determination Figure 4 shows reflection coefficient of the antenna with the The delivered signal has a maximum of -13dB for in the frequency range 200 MHz - 1.6 GHz.

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Radiated pulse determination One of the antennas is transmitting, fed by a HMP/F Kentech generator [distributed in France by ARMEXEL society] (output voltage of 4 kV, risetime of 220 ps, 50% pulse duration (full-width at half-maximum) of 700 ps, output impedance). This gaussian pulse has a frequency range from DC to about 1.5 GHz (-20 dB/maximum) (figures 5 and 6).

The other is receiving, connected to a digital sampling Tektronix sequential acquisition oscilloscope TDS820 (6 GHz bandwidth). On figure 7, theoretical and measured signals are presented when two “scissors” antenna are facing each other at a 5.80-meter distance. The graph is normalised to allow comparison. The measured voltage peak at feed point is about 50 Volts. The distortion rate is less than 1.4. The measured signal spectrum (figure 8) exhibits a bandwidth from 80 MHz to 1.2 GHz of -20 dB below the maximum.


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Frequency gain and radiation patterns determination As the two antennas are perfectly identical, the axial gains will be identical too. Thus, it is possible to determine the gain for different azimuthal and elevation angles by the following expression:

with

excitation signal spectrum at the drive point of the emitting antenna, measured pulse spectrum at the drive point of the receiving antenna, R : distance between antennas.

So, the determined gain depends on the measured pulses at the drive point of the emitting and receiving antennas, on the electrical length and on the distance between antennas.

The comparison between computed and measured axial gain is given on the figure 9 into the V-V and H-H polarisation with a good agreement. “Scissors” antenna pattern results from combination of each own radiated field wire. However, the main lobe is maximum in axial direction, but there are, in site, side lobes whose levels are usually smaller. The radiation patterns in E-plane and H-plane are shown in figure 10 and 11. In Hplane, the principal lobe has a half aperture angle of 45° at 500 MHz. In E-plane, the lobe has for the same frequency a half aperture angle of 13°. Side lobes, in this plane, are about 8dB smaller (for 500 MHz) from the maximum level. The backscattered field is -15 dB below field measured in axial direction.

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Antenna performances are in good agreement with requirements for distortion coefficient, for bandwidth matching and for high frequency gain.

CONCLUSION A study was led by IRCOM to design antennas capable to radiate and receive ultra short transient pulses with a minimal distortion in the frequency range from 100 MHz to 1GHz. It has resulted an original conducting wire aerial with suitable geometric dimensions for outdoor applications. Resistive loading used in upper-part of antenna wires improves matching input of the antenna and transmitted transient pulse. This plane-antenna has been successfully and easily integrated in the experimental CELAR radar demonstrator PULSAR5.

REFERENCES 1. Miller E.K., Poggio A.J. and Burke G.J.; «An integrodifferential equation technique for time domain analysis of thin wire structures» . Journ. Of Comp. Physics, pp. 12-24, n°48(1973) 2. «Transient EM fields» Topics in Applied Physics. Ed. : L.B. Felsen-Vol. 10-Springer Verlag, New-York(1976). 3. Maloncy J.G., Smith G.S., «A study of transient radiation from the Wu-King resistive monopole-FDTD analysis and experimental measurements». IEEE Transactions on Antennas and Propagation Vol.41-n°5-May 1993. 4. The baluns were made by the EUROPULSE Company (Cressensac, Lot, France) 5. F. Gallais, V. Mallepeyre, Y. Imbs, B. Beillard, J.Andrieu, B. Jecko, M. Le Goff, «A new Ultra Wideband short pulse, radar system for mine detection» Ultra Wide Band Short Pulse Electromagnetics, vol.5, in press.

TIME DOMAIN ARRAY DESIGN

Hans Schantz Time Domain Corporation 6700 Odyssey Drive Huntsville, AL 35806 USA

INTRODUCTION A short pulse ultra-wideband system imposes different constraints on array design than the usual continuous wave narrow band system. The goals of this paper are threefold: to explore these differences, to show how to model a short pulse array, and to discuss how best to graphically describe the performance of an ultra-wideband antenna system. First, this paper will discuss the differences between narrow band and wide band array design. The implications of wide band/short pulse systems to such array properties as grating lobes will be considered. Second, this paper will discuss how to model short pulse arrays under the assumption that there is no mutual coupling between elements. As an example, the behavior of a particular end fire array will be calculated and compared to experimental measurements. Finally, traditional narrow band depictions of antenna performance are ill-adapted for ultra-wideband radiators. Several alternate methods will be presented, including portrayal of the angular dependence of peak instantaneous power, the angular dependence of average power, and the angular dependence of the time domain pulse waveform. CW LINEAR ISOTROPIC ARRAY Linear arrays are a standard topic in any antenna or electromagnetics text.1 Assume a linear arrany has n isotropic sources as shown in Figure 1. Then, the radiation intensity in the direction of the principal maximum is times the maximum intensity of a single oscillator. Since power is divided among n sources, array gain goes as The total phase difference of the fields from adjacent sources is:


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where

if the sousrces are excited simultaneously.

If the field due to an individual source is given by given by:

then the total field of the array is

In a sparse array (one whose inter element spacing is grating lobes will occur as waves are “aliased” integer and half integer periods apart, thus causing constructive and destructive interference, respectively.

UWB LINEAR ISOTROPIC ARRAY A short pulse, sparse UWB array is inherently grating lobe resistant. In the limit as a single pulse is transmitted, there are no earlier or later cycles to be aliased. Grating lobes will show up as pattern sidelobes. The number and intensity of these sidelobes depend upon the complexity of the waveform being transmitted. Consider the ultra wideband “diamond” dipole invented by Larry Fullerton for example.3 When excited by a broadband source, this antenna emits a waveform similar to a Gaussian third derivative:

where is the center frequency and where the waveform has been normalized to have unitary peak amplitude. A comparison of this model to a waveform measured from a diamond dipole is shown in Figure 2. Because of the short duration of this waveform, grating lobes are strongly suppressed away from the main beam.

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As a specific example, consider a two element broadside array with a 26” separation (about at the pulse center frequency of 1.85 Ghz). The roughly 7° halfpower beamwidth is well modeled by conventional continuous wave (“CW”) array theory, but the predicted grating lobes are absent except for a single minor sidelobe on either side. A surface plot of the pulses propagating out in time is shown in Figure 3.

An experiment was performed to verify this grating lobe suppression in simple two element UWB arrays. A Picosecond Pulse Lab Model 4050 Pulser was used to excite a variety of different two element UWB arrays. A 1-18 GHz ARL Horn antenna was used to receive the resulting waveforms and feed them to an HP 54750 Sampling Digitizing Oscilloscope. The array was placed on a rotator and spun around by a computer which gathered waveform and peak power information for each angular

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position. A custom built data acquisition (DAQ) program running in HP Vee was used. Details of the experiment are shown in Figure 4.

The peak power patterns for three arrays are compared to the CW array predictions in Figures 5a-c. Notice the suppression of all but the closest grating lobes. As in a CW array, the intensity in the direction of the principle maximum is times the maximum intensity of a single element. In the present experiment, the short range (“R”) raised the off main beam background level above the –6 dB that would otherwise be expected for a two element array with d 0). The plane y = 0 is the surface of the sand. There is a scattering center D under the surface. Generally, this center can be exited by the field of electromagnetic wave, source of which is placed in region 1. In our case the generator G (from Network analyzer) exits the center D by using coaxial line with the transmission coefficient The dipole D radiates electromagnetic wave into the party of the surface of the half-space 2. We can characterize the propagation of this wave by the transmission coefficient of dielectric layer between the dipole D and the surface of the half-space 2. The wave radiated by the dipole D creates on the surface of the half-space 2 the distribution of electromagnetic field. If we will take Fourier transform of this function we will obtain the angle spectrum of electromagnetic field in the half-space y < 0. Using Network analyzer we can measure the complex transmission coefficient T between the reference planes (generator) and

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(generator) and (detector) for each angle is integer numbers; (2m +1) is the number of angles) at the scanning by the antenna A along the circle C of the radius R. This transmittance for can be written in the form

where

is complex amplitude of electromagnetic field in reference plane

antenna A is placed at

when

is complex amplitude of electromagnetic field in

reference plane is the coefficient of coupling of the field with the plane wave propagating in the direction of the angle is the transmission coefficient of the free-space layer between the plane

and reference plane

of

antenna

is the transmission coefficient of

antenna A . Similar equations can be written for any position C

Here the transmission coefficient

of antenna A on the circle

is defined for the free-space layer between the plane

IMAGE RECONSTRUCTION OF THE SUBSURFACE CROSS-SECTION

and reference plane

of antenna A at the position

is the coefficient of coupling of

the field with the plane wave propagating in the direction of the angle transmission coefficient can be written in the form (Goodman, 1968)

The coefficients of coupling

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The

are defined by equations

where are complex amplitudes of the plane waves propagating in directions respectively. They are the angle spectrum of electromagnetic field in the space y < 0. Let us suppose that in the calibration process we can determine the complex coefficient then

and

as for reference planes tangential to the line C. With using definitions (13), (14), the equation (16) can be rewritten as

From (17) one can see that if we will measure the transmission coefficient in each n - th position of the antenna A, we will determine the angle spectrum of electromagnetic field in half-space y < 0 up to the complex constant

After that it is possible, using tomographic algorithm (Chommeloux, Pichot, Bolomey, 1986; Vertiy, Gavrilov, 1998), obtain the image of the center D.

EXPERIMENTAL SETUP The scheme of experimental setup one can see in Figure 3.

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The experimental setup has next parts: 1) 2) 3) 4) 5) 6) 7) 8) 9)

Network analyzer; Microwave amplifier; Receiver antenna; Radiating source (dipole); Semicircle scanner; Stepping motor control block; Box filled by sand; Table for the fixing of the scanner; Absorber.

The setup operates as follows. Signal from Network analyzer, operating in “transmission “ regime, is increased by the microwave amplifier and after that exited the source The transmitted signal is received by the antenna which is fixed on the scanner and moved along semicircle of the radius with constant the angle step The plane wave component from the plane wave spectrum of the radiated field propagating in direction of antenna is received of antenna. The signal from an output of the antenna acts on an input of the analyzer. The dipole was placed in different points on the surface and under surface of the sand. Scheme of the construction, the VSWRfrequency characteristic and the radiation pattern of the dipole are shown in Figures 4, 5a), b), respectively. Before measurements the antenna is placed in the center of the circle the dipole is placed under antenna on the surface of the sand and the system is calibrated such way, that on all 32 frequencies from the work band taken with constant the frequency step complex transmission coefficient On the


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following step the antenna is reseted (n = –m) and the data acquisition begins. The data are real and imaginary parts of complex transmission coefficient for each angle of the antenna at all 32 frequencies.

RESULTS OF EXPERIMENT Several schemes of the dipole position were studied in experiment. The images of the radiating centers were reconstructed using described above the measurement method and tomography algorithm. The image function is the modulus of the normalized polarization current |K| distributed in the sand around the radiating center. In Figures 6a), b), c) one can see the images of the source placed on the surface of sand. Figure 6a) shows the source shifted on distance from the center of the circle C (See scheme on the right); in Figure 6b) and for Figure 6c). The schemes on the right show the position of source and values of the shift

*

Victor Spetanyuk has done design and realization of this dipole antenna.

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Figures 7a), b), c) illustrate the images of the source in the case when the source is placed under surface of the sand on the depth Schemes of positions and values of and d are shown on the right. It is possible to see that the described method allows reconstructing the image of radiation–producing center and to find the positions of the center. The shape of the investigated source close to tomographic image of the point scattering center (Chommeloux, Pichot, Bolomey, 1986). The images have the sizes in depth large, than in transversal direction. The positions of images of the source are pursuant to actual positions of the source. There is an interest to investigate the resolution of the described method. Results of such experiment are presented in Figures 8a), b), c). This Figures show the images of two identical dipoles placed on the surface (Figure 8a)), under surface (Figure 8b), c)). In Figures 8a), b) dipoles are shifted in opposite directions along the x -axis on the distance (Fig.8a), b)) and in the depth (Figure 8c)). In last case the dipole 1 is placed at

In this figure one can see good resolution “in

cross-range” (along x -axis). The deference in the depth is not enough for the reconstruction of the images of two sources 1, 2. The real positions of the sources are shown on the right.

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Except for described experiments, the experiments with a metal scattering rod were conducted. The rod was placed inside a marble plate near to its surface or on depth Schemes of these experiments are shown in Figures 9a), b), c) on the right. Figures 9a), b), c) show the reconstructed images of scatterers 1, 2. These scatters are 1) 1-empty hole, 2 -hole with the inserted metal rod; 2) 1 - hole with the inserted metal rod, 2-empty hole; 3) 1-empty hole, 2-empty hole. The hole 1 was made close to surface and the hole 2 was made at The holes have diameter Figure 9a) illustrates the case when the metal rod is placed in hole 1; Figure 9b) - the metal rod is placed in the hole 2 and Figure 9c) – the metal rod is absent. In the last case two empty hole can be considered as scatterers and Figure 9c) illustrates this case. It was used rectangular adapter AD for the excitation of electromagnetic wave propagating inside of the marble plate. Into the adapter was inserted the end of the marble plate with thickness The adapter AD associates the coaxial line with the dielectric waveguide and excites inside of the dielectric plate the waveguide mode which propagates along the plate. This wave is scattered by the scatterers 1 and 2. The results of the reconstruction presented in Figures 9a), b), c) show that in the considered case one can see size (shape) of scatterers and they positions. In the case Figure 9c) the two scatterers are not resolved. For observing two scatterers distributed In depth it is need the distance between scatterers larger, then CONCLUSION Thus the conducted experimental investigation in the frequency range of 3÷4GHz showed that described setup and signal processing methods allow to figure the shape and to estimate the cross-section size of objects buried in sand. We can also evaluate depth on which the researched object is loaded using the object cross-section image in the plane perpendicular to the medium surface. The results obtained may be applied in practical microwave imaging systems for detection and observation of different undersurface objects.

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REFERENCES Chommeloux L. Pichot, Ch. Bolomey, J. – Ch., 1986, Electromagnetic modelling for microwave imaging of cylindrical buried inhomogeneties, IEEE Trans. Microwave Theory Tech., Vol. MTT – 34, No. 10, pp. 1064 – 1076. Vertiy A. A., Gavrilov S. P., 1998., Modelling of microwave images of buried cylindrical objects, InternationalJournal of Infrared and Millimetre Waves, Vol. 19, No. 9, p p. 12011220. Vertiy A. A., Gavrilov S. P., Tansel B, Voynovskyy I.V., 1999, Experimental investigation of buried objects by microwave tomography methods, Part of the SPIE Conference on Subsurface Sensors and Applications, Denver, Colorado, SPIE Vol. 3752, pp. 195-205. Goodman J.W., 1968. The book. Introduction to Fourier Optics.-McGraw-Hill Book Company, San Francisco-New York-St.Louis-Toronto-London-Sydney.


OPTIMAL ACOUSTIC MEASUREMENTS

Margaret Cheney1,2, David Isaacson1, and Matti Lassas3 1

Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY 12180 USA 2 Department of Electromagnetic Theory Lund Institute of Technology, Lund, S122 00 Sweden 3 Department of Mathematics P.O. Box 4, 00014 University of Helsinki, Finland

INTRODUCTION This paper is motivated by the question “How can we design the best possible system to do acoustic imaging?” If we want to make the best possible images, we must begin with data that contain the most possible information. In particular, since all practical measurements are of limited precision, some scatterers may be undetectable because their scattered fields are below the precision of the measuring instrument: our data will contain no information about them. What incident fields should we apply that will result in the biggest measurements? There are many ways to formulate this question, depending on the measuring instruments. In this paper we consider a formulation involving wave-splitting in the accessible half-space: what downgoing wave will result in an upgoing wave of greatest energy? A closely related question arises in the case when we have a guess about the configuration of the inaccessible half-space. What measurements should we make to determine whether our guess is accurate? In this case we compare the scattered field to the field computed from the guessed configuration. Again we look for the incident field that results in the greatest energy difference. This optimal measurement problem has been studied for fixed-frequency problems in electrical impedance tomography (Isaacson, 1986) and acoustic scattering (Mast, Nachman, and Waag, 1997). For time-domain problems, the issue of optimal time-dependent waveforms in a special 1 + 1 – dimensional case was studied Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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in (Cherkaeva and Tripp, 1997), where a time-harmonic waveform was found to be optimal. In this paper we study the question of optimal time-dependent waveforms in the 3 + 1 – dimensional case. In particular, we consider the half-space geometry: we imagine that a plane divides space into accessible and inaccessible regions, and we assume that we can make measurements everywhere on the plane. We show that the optimal incident field can be found by an iterative process involving time reversal “mirrors”. The connection between optimal measurements and an iterative time-reversal process was first pointed out in (Mast, Nachman, and Waag, 1997), (Prada and Fink, 1994), and (Prada, Thomas, and Fink, 1995); in all these papers, the analysis was carried out a a single frequency. In this work we consider time-dependent fields. We show that for band-limited incident fields and compactly supported scatterers, the iterative time-reversal algorithm converges to a sum of time-harmonic fields. This provides a theoretical foundation for the pulse-broadening observed in the computations of Cherkaeva and Tripp (1997) and in the time-reversal experiments of Prada and Fink (1994), and Prada, Thomas, and Fink (1995). Section 2 contains a careful formulation of the idealized problem: the wave equation model, the measurements, the notion of “biggest”. Section 3 gives the adaptive experimental algorithm that can be used to find the optimal field even if the scatterer is unknown. In general the iterates converge to a time-harmonic field. The paper concludes with a discussion. Full details can be found in (Cheney, Isaacson, and Lassas, 2000).

PRECISE FORMULATION OF THE PROBLEM We consider the constant-density acoustic wave equation

in the case in which everywhere in the upper half-space Here This model includes neither dispersion nor dissipation. We consider the half-space geometry, in which the lower half-space is inaccessible and the upper half-space is accessible. The measurements we consider are those in which we send a downgoing wave into the lower half-space and measure, on the plane the corresponding upgoing wave. “Upgoing” and “downgoing” can be defined by means of Fourier transforms (in time and in the lateral space variables see (Cheney, Isaacson, and Lassas, 2000) for details. We denote by S the scattering operator that maps the downgoing wave to the upgoing wave on the plane The scattering operator for a reference configuration is denoted

by In order to find the downgoing field that maximizes the difference between the scattered waves and we want to maximize the quotient

Here it is necessary to divide by energy we can apply is always finite.

to account for the fact that in practice the

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In (2), we need to define the measured of “size” || · ||. We do this with the help of the energy identity

where the right side corresponds to the increase in energy in the volume V and the left side is the energy flux through the surface We thus use as our measure of “size” the total (time-integrated) energy flux into the region

(In the electromagnetic case, the corresponding quantity would be the integral of the Poynting vector, dotted with the unit normal vector integrated over the plane and integrated over all time.) Thus the best downgoing field is the one that maximizes

AN ADAPTIVE METHOD FOR PRODUCING THE BEST FIELD To maximize (4) in the case when the medium is unknown, we can use the following adaptive method. 0) Begin with any downgoing wave 1) Send into the lower half-space; measure the resulting upgoing field

2) Calculate the corresponding scattering from the reference configuration Calculate the difference field

3) The next downgoing wave is the (normalized) time-reversed difference

add one to j; go to step 1). To understand why this iterative process converges to the right answer, imagine the case of a single point scatterer, and take the reference medium to be empty space. Any incident wave will scatter from the point scatterer into an outgoing spherical wave. If this outgoing spherical wave is time-reversed, it becomes a spherical wave that focuses on the point scatterer. Focusing all the energy of the incident wave on the scatterer gives rise to the largest possible scattered wave.

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For fixed frequencies, the iterative time-reversal algorithm has been analyzed by Prada, Thomas, and Fink (1995). They considered the case of a finite array of transducers and the corresponding “transfer matrix” in which the element of the matrix is the scattered field measured at the transducer resulting from the activation of only the transducer. Prada et al. (1995) showed that the iterative time-reversal algorithm converges to the spatial eigenfunction corresponding to the largest eigenvalue of the transfer matrix. Their paper contains diagrams that show what happens when the scatterering region contains a strong scatterer and a weak scatterer. The strong scatterer returns a stronger wave, which, when time-reversed, becomes a stronger wave focusing on the stronger scatterer. After a few iterations, the signals from the weak scatterer disappear because all energy becomes focused on the stronger scatterer. Here we consider the question of the time-domain waveform of the optimal field. In (Cheney, Isaacson, and Lassas, 2000) we analyze the problem by Fourier transforming to the frequency domain. At each fixed frequency, the scattering operator has eigenvalues, and these eigenvalues depend analytically on the frequency. Suppose the largest eigenvalue is biggest at the frequency Then in general, the iterative time-reversal algorithm converges to a time-harmonic (fixed-frequency) wave with the frequency The spatial shape of the waveform is given by the eigenfunction corresponding to this largest eigenvalue. This prediction is consistent with the experiments of Prada, Thomas, and Fink (1995), in which pulse broadening and frequency shifts are observed as the iterative algorithm proceeds. This prediction is also consistent with the one-dimensional work of (Cherkaeva and Tripp, 1997), in which the optimal time-domain waveform was found to be a time-harmonic wave.

CONCLUSIONS AND OPEN QUESTIONS This approach to the optimal measurement problem decouples the measurement issue from the problem of forming an image. However, this work shows only how to find the single best measurement for determining whether an unknown scatterer is identical to a particular guess. It is not clear how to extend this work to find a full set of measurements that would be necessary for forming an image. This analysis shows that the iterative time-reversal work of Prada and Fink (1994) and Prada, Thomas, and Fink (1995) provides an experimental method to obtain optimal fields. Moreover, this analysis explains the pulse-broadening seen in (Prada, Thomas, and Fink, 1995) and (Cherkaeva and Tripp, 1997): the optimal time-domain waveform is a time-harmonic one. This analysis suggests that the commonly-used pings and chirps are not optimal from the point of view of providing the biggest signal to distinguish an unknown scatterer from a particular guess. There are many open questions related to this work, some of which are probably easy and others hard. In particular, problems that seem to be relatively straightforward are extending this work to Maxwell’s equations and to materials in which the medium parameters depend on frequency. Extending the theory to measurements made in a limited aperture is probably not difficult, but will involve detailed modeling of transducers or antennas. More difficult problems include extending the work to dissipative media, and

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finding a full set of optimal measurements for forming an image. In the fixed-frequency case, it is clear that a full set of optimal measurements are the eigenfunctions of the operator or But the eigenvalues of these operators depend continuously on frequency, which means that there is no “next best” incident field. Thus it is not clear what finite set of measurements contain the most possible “information” in a broader sense.

ACKNOWLEDGMENTS This work was partially supported by the Office of Naval Research. M.C. would like to thank a number of people for helpful discussions: Gerhard Kristensson and his group in Lund, Jim Rose, Claire Prada, and Isom Herron.

REFERENCES 1 Cheney, M., and Isaacson, D., “Inverse Problems for a Perturbed Dissipative

Half-Space”, Inverse Problems 11 (1995) 865–888. 2 Cheney, M., Isaacson, D., and Lassas, M., “Optimal Acoustic Measurements”,

preprint (2000). 3 Cherkaeva, E., and Tripp, A.C., “On optimal design of transient electromagnetic

4 5 6

7 8

waveforms”, SEG97 Expanded Abstracts, 67th Annual Meeting of Soc. Exploration Geophys. (1997) 438–441. Isaacson, D., “Distinguishability of conductivities by electric current computed tomography”, IEEE Trans, on Medical Imaging MI-5(2):92-95, 1986. Lassas, M., Cheney, M., and Uhlmann, G., ”Uniqueness for a wave propagation inverse problem in a half space”, Inverse Problems 14, 679-684 (1998) . Mast, T.C., Nachman, A.I., and Waag, R.C., “Focusing and imaging using eigenfunctions of the scattering operator”, J. Acoust. Soc. Am. 102, Pt. 1 (1997) 715–725. Prada, C. and Fink, M., “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media”, Wave Motion 20 (1994), 151–163. Prada, C., Thomas, J.-L. and Fink, M., “The iterative time reversal process: Analysis of the convergence”, J. Acoust. Soc. Am. 97 (1995) 62–71.


PARSIMONY IN SIGNATURE-BASED TARGET IDENTIFICATION


1.

INTRODUCTION

In signature-based target identification [20] the scattering dyadic of each type of target is characterized by a set of functions based on a particular scattering model (e.g., complex exponentials for the late-time response, or delta, step, ramp, etc., for the early-time response). Each of the functions is characterized by a small number of parameters (e.g., complex natural frequencies ) including a scaling coefficient (scalar, dyadic) to adjust the amplitude (perhaps including vector orientation). Each of the targets is represented in a target library by an appropriate set of such functions with parameter values particular to the individual target types (e.g., a particular type of aircraft such as a 707). The approach is to associate these parameter values with the characteristics of the electromagnetic waves scattered (usually backscattered, but not necessarily so) from the target by some appropriate radar. The problem then is to distinguish one target from another by the differences in the parameters in the scattering model(s) inferred from the measured scattered fields. Given the presence of noise in any measurement there is some ambiguity in the declaration of a particular target type because of errors in the parameter estimation. Particularly as the parameter values for one target type approach those of another type the discrimination becomes increasingly difficult. Let us distinguish between two types of parameters. One type (fixed parameters) assumes particular values (scalar, vector, dyadic) for each target type. A second type, which we might call variable parameters is adjusted as part of the process of fitting the scattering model to the data. Such variable parameters are typically the coefficients of the fitting functions, which adjust the amplitudes (not necessarily scalars) of the fitting functions to best fit the data. Since we wish to discriminate between various targets we would like it to be difficult to fit the wrong target parameters to the data. So we would like to reduce the number of variable parameters (and the range of their variation) as much as we can. This is aided by constraining these scaling coefficients (e.g., pole residues) to values appropriate to the target type to the degree practical. This leads to a principle of parsimony as [25]: “Use as few feature variables as possible to provide consistent classification.” In the present context feature variables are interpreted as variable parameters, or parameter values that aapply to multiple target types (at least approximately). 2.

FITTING WITH GENERAL FUNCTION SETS To illustrate the problem of fitting data with too many parameters, consider some target response (before or after decomvolution for impulse response) along with some noise

some parametric scattering model for which we have a set of functions infinitely many) appropriate to the mth target type. Then we try to approximate


for

We have (perhaps by these functions as

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where L is available for us to choose. The

are chosen to minimize an appropriate norm of the

difference of the two sides in (2.1). Then define

where the norm || || can be defined in various ways [3, 17], including the use of various weighting functions [18] if desired. While the above functions are written as functions of time t, they can be transformed to complex frequency domain or to a wavelet/window-Fourier-transform domain [9, 31 ] and the norm can be defined in such terms.

Having chosen our norm we define the set of

difference as in (2.2) as the set

that minimizes the

There may be more than one such set in which case

we choose one of these sets at our convenience. Then we have

What now if we try again with L + 1 functions? We obtain some set This gives

Now one choice of the (before minimization) is just the with This gives as in (2.3). this choice might give the minimum in (2.4), but generally gives something larger than the minimum. Hence we can conclude

This does not necessarily imply that as but it does show that adding more functios in our target-signature set with adjustable weights makes the mth target-type signature more closely match the data from the nth target type (with or without noise). Suppose however, that the

for

form a complete set on the support of interest

(time interval, frequency interval, or even some function can be approximated by this set and

phase space). Then any reasonably well behaved

However, the are for the mth target type and we are approximating the waveform for the nth target (plus noise). In this case minimization of the norm for large L cannot distinguish between the nth and mth targets. So it is important that the

not be complete for the domain of time, freq., etc., for successful

target discrimination. We want the to apply to only the mth target (for all m in our library). Note that a complete set of functions need not be orthogonal (zero inner products on the support). Of course a nonorthogonal set can be converted to an orthogonal one (Gram-Schmidt orthogonalization [26]). Alternately, if the function set

is complete, it is important that L be limited (parsimony) so

that only terms that are dominant for target m are included. Similarly the set

needs to include only

terms that are dominant for target n. This is also a question of how to best order the i.e., which is labelled by etc. Presumably they should be placed in the order of decreasing dominance. In (2.2) and (2.3) the norms do not take account of varying signal strength as the same target is measured at various distances from the radar. Using the far-field approximation (incident and scattered fields


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varying as 1/r with the same waveforms) one can normalize these expressions as the dimensionless expressions

However, this does not account for variation in the strengths of the signals between different targets (large vs. small scatterers). So one may wish to instead normalize as (one factor of r for the incident wave, one for the scattered)

assuming one has a measure of the range r to the target. In this form the combinations

also give

information concerning the strength of the various scattering modes and can perhaps be constrained (parsimony) to give better target discrimination. Combining these ideas we can define

so that the are now range invariant. The are, however, in general still aspect (polarization, angle of incidence) dependent. While the discussion in this section has been in terms of scalar functions with scalar coefficients, vector and dyadic forms are readily considered in the same expressions, with appropriate attention to the norm used. Such forms are appropriately used with multiple radar measurements to give the scattering dyadic of the target, which is then fit by the above procedure. 3.

EXPONENTIAL FUNCTIONS

A common set of functions used for representing a time-domain signal f(t) is exponential functions (in general complex). This is readily seen through the two-sided Laplace (or Fourier) transform as two-sided Laplace transform

Bromwich contour parallel to

axis

Here the Bromwich contour is taken to the right of any singularities of f(s) in the s plane. So already we have a restriction that f (t) must be passive, i.e. with Re[s] > 0 is not allowed in representing f(t). However, as discussed in the previous section, our signal from the nth target includes noise which does not necessarily include this constraint. This implies some constraint (parsimony) on the allowable functions to represent a target and discriminate against noise. Note that the Bromwich contour extends over implying an infinite set of functions to represent any target in the library (not parsimonious). The integral over the Bromwich contour can be represented by a sum as

where now the all lie on or to the left of the axis. In this form we can see the effect of a finite sum of exponentials. As the number of suchfunctions any well-behaved passive f (t) can be represented. Instead of an infinite interval in time where the signal is eventually lost in the noise, one might consider a finite interval oftime The transform f(s) of f(t) can then be replaced by a Fourier series with The are now discrete, but generally infinite in number. Furthermore, as is well known any reasonably behaved f(t) (not necessarily passive, and including noise) can be accurately represented by such a Fourier series. So this is also not a good choice.

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Following [27] we can deform the Bromwich contour into the left half plane. In Fig. 3.1 A we see the singularities of a passive system lying in the left half plane (LHP). Since our f(t) is real valued the singularities not on the negative axis must occur in complex conjugate pairs, both for their locations and amplitudes (i.e., pole residues). As we deform our contour to the left these singularities are isolated to give separate functions which for poles (shown as first order, but not necessarily so) gives a representation as

where the entire functions corresponds to the singularity as In time domain it is an early-time contribution to the response [4]. Hence the pole series is used to represent the response for late times after the incident and scattered waves have had time to transit over the target. Already we see some parsimony at work. The poles corresponding to the mth target do not well approximate those belonging to the nth target for So discrete pole locations in the LHP are more parsimoneous than a continuous distribution (or large number) of such locations on the axis. Even more parsimoneous would be some restrictions on the residues Carrying the contour deformation yet further, suppose that our targets of interest have no singularities away from the negative axis. Then the contour collapses as in Fig. 3.1B to include only singularities there. While this can include in principle a branch cut there [4], there is an important class of targets which can be well-approximated by first order poles there. These are the diffusion poles in highly (but not perfectly) conducting metal targets of finite linear dimensions [6]. In this case the response takes the form

where the entire function is now a constant in frequency domain or a delta function in time domain. Comparing (3.4) to (3.3) we can see that (3.4) is more parsimonious in the sense that it cannot represent poles in the third and fourth quadrants of the s plane far from the negative axis. For this special class of targets appropriate to magnetic singularity identification (MSI) the form in (3.4) is more general corresponding to a decomposition of the magnetic-polarizability dyadic [6]. The dyadic residues rotate together with the target allowing constraints on them [13, 15, 16], thereby giving more aspect independent parameters (besides the to aid in the target discrimination. This is then even more parsimonious.

PARSIMONY IN SIGNATURE-BASED TARGET IDENTIFICATION 4.

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RELATION TO PRONY-LIKE FITTING OF DATA

Much work has been done extracting damped sinusoids out of data [19]. Recently, some improvement has been made with what is called matrix pencil [21 ]. Restricting the number of terms and only considering those with large coefficients (residues) helps. A more robust procedure may consist of fitting the data with only the specific exponential sets corresponding to targets in the library to see which fits best. Of course, if the real target is not in the library, consideration of how good is the fit (size of residuals) may be needed to establish this fact. Filters such as the E/K. pulse [9, 19] are one way of doing this. One has for the mth target type which annihilates the late-time response (except for noise) when it is convolved with the response of the nth target only when n = m. This is achieved by setting (for all significant natural frequencies

of the mth target type).

Prony-like fitting may still be needed to generate the natural-frequency sets for the target library from experimental data. Of course, this can be achieved from many careful measurements under more ideal conditions. Such measurements may be eventually supported by accurate numerical computations of natural frequencies from integral and/or differential scattering equations. Preliminary estimates of the can perhaps be refined by adjusting them to optimally annihilate the late time response for many target aspects with one 5.

USE OF POLARIZATION WITH TARGET SYMMETRY

With two choices of polarization one can reconstruct the backscattering dyadic. If one has the target impulse response dyadic (2 × 2) for a particular angle of incidence, one can rotate the target about this incidence direction (or equivalently rotate the radar) to match the stored data and thereby also orient the target by an angle about the incidence direction. Of course the rotation can be accomplished by a rotation of the scattering dyadic in a computer. In the process one can identify this rotation angle modulo since the scattering dyadic is invariant to a rotation by (sign reversal of incident and scattered fields). From a parsimony point of view this is a single real parameter varying over a restricted interval. The constraint of a known angle of incidence has greatly reduced the number of fitting parameters. The amplitude is also assumed to be constrained by independent knowledge ofthe distance to the target. 5.1 Target Symmetry Plane Passing Through Observer:

Symmetry

A special case of interest is that of a target with a symmetry plane (such as a typical fixed-wing aircraft) [1, 2, 29]. Referring to Fig. 5.1, let this symmetry plane lie along the direction of incidence at the target. With the usual h, v radar coordinates we have the directions (unit vectors)

with

forming a right handed system. Note that

The target symmetry plane ized by a reflection dyadic

is rotated from the vertical by an angle

giving

identity (three dimensional)

symmetrycharacter-

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If one measures the backscattering dyadic (delta-function response) by removing the antenna characteristics and range dependence, this will have three separate terms in the 2 × 2 scattering dyadic removed) since the fields of concern only have h and v components and we have

Defining a rotation matrix [30] as

coordinate


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gives a positive (counterclockwise) rotation of the coordinates as in Fig. 5.1. Suppose we have measured in the h,v coordinate system. Then we can compute

where we have rotated the scattering into the as, sy system corresponding to the antisymmetric part and symmetric part

These two parts only (parsimony) are needed to characterize this scattering,

being reduced to vectors parallel to

and

on the symmetry plane. Note that this decomposition is

frequency/time independent. Natural frequencies decompose into two sets and etc. If is not known a priori from some other measurement, then rotating the data by varying discovers to make (5.6) hold, thereby learning the roll angle. Note, however, then adding integer multiples of to also produces this diagonalization. generally has more symmetry than the target, in this case two symmetry planes giving the symmetry group

There is generally some error in target alignment such that the symmetry plane does not pass exactly through the observer (radar). This means that will generally not be perfectly diagonal but the off-diagonal components will have some minimum (norm sense over time/frequency) for particular Noting this alignment error one may wish to restrict the range of frequencies. If the maximum transverse dimension target is d one may wish to restrict radian wavelength such that

where is the angular error. If say D is 10 m and is radians then frequencies to less than about 400 MHz, with even lower uppermost frequencies as related to glint or angle noise in traditional radar systems [28]. 5.2

Body-of-Revolution Target, Including the Nearby Media:

restricting is increased. This is

Symmetry

A yet higher degree of symmetry is that of a body of revolution with axial symmetry planes giving symmetry [11]. Here the axis of revolution

is taken as perpendicular to the earth surface

which we take as the z = 0 plane, as in Fig. 5.2. The earth constitutive parameters are allowed to vary with z (layering) provided the

symmetry is preserved, including the earth.

In this case the h,v coordinates are established with the traditional convention that and

lies in the plane of incidence containing

is parallel to

and the rotation axis. In this case we have [ 11, 24]

for all making it appropriate for synthetic aperture radar (SAR) as the radar is moved with respect to the target. This is called the vampire signature due to the lack of reflection in the h, v “mirror”. Here we use the nulling of a parameter (the cross polarization) as an identifier of a class of targets. However, it does not discriminate among the various target types with this symmetry. One can go on to consider other details of the and (only two remaining now) to achieve yet further discrimination

[11].

C. E. BAUM

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6.

PARSIMONY IN MAGNETIC-SINGULARITY IDENTIFICATION

For magnetic singularity identification (MSI) we have the general form of the magnetic polarizability dyadic as [6, 16]

= real unit vector for mode , (all negative real natural frequencies)

= real scalar

This applies to highly, but not perfectly, conducting targets. The correspond to exponential decays in time domain. The frequencies of interest are rather low, corresponding to diffusion into the metal targets. The incident fields are now not in the form of a plane wave, but the near fields of loops. Similarly the scattered near fields are sensed by loops. As such three-dimensional information concerning the target is available. The and are all aspect independent while the and rotate with the target, i.e. are fixed in a target-based coordinate system. As discussed in [16] by rotating the MSI signature in the target library and moving the target location (say under the earth surface) with respect to the observer location one can attempt to match the library entries to the data. In so doing there are six real parameters to adjust: three Euler angles for target orientation, two angles for direction to target, and one for distance to the target. Thereby the aspect-dependent parameters are constrained by their mutual orientation relationships. The situation is further simplified in the case of target symmetry [7, 13, 15]. For a target with symmetry with (N-fold rotation axis with no assumption of symmetry planes) the magnetic polarizability dyadic becomes


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where the rotation axis is taken as Here the unit vectors have all conveniently lined up to give two distinct sets, each with common orientations (common aspect dependence) of the unit vectors characterizing the pole residues. In this case there are only needed two real scaling parameters (more parsimonious) to multiply the axial and transverse functions characterized by constrained poles and residues. 7.

EFFECTS OF VARIATION IN EMBEDDING MEDIA

For targets in a uniform well-characterized medium one can use the a priori knowledge of this medium to constrain the target signatures. In particular, aspect independent parameters can be considered constants, i.e., not variables depending on the medium parameters. Such is the case for targets effectively in free space, e.g., flying aircraft and missiles. For targets in a variable medium such as earth the situation is more complicated since the signature in the scattering (e.g., natural frequencies) can be significantly affected by the constitutive parameters of the nearby earth [14]. In this case we are concerned primarily with frequencies such that wavelengths in the external medium are of the order of the target dimensions (used in electromagnetic singularity identification (EMSI)). If one has independent knowledge of the earth parameters (particularly the permittivity ), say by a nearby measurement, then one can attempt to compute the effect of such parameters on the library signatures (e.g., natural frequencies) before fitting these to the radar data. Alternatively, one can use one or more of these parameters as variable fitting parameters with the library signatures to best fit the data (less parsimonious). For targets that can be approximated as perfectly conducting in a uniform isotropic earth, there are exact scaling relationships for natural frequencies and associated modes and residues [5], making the shifting of library parameters fairly simple. For dielectric targets in such a medium the situation is more complicated. However, if the target permittivity is less than that of the surrounding earth there are applicable perturbation formulas simplifying the situation somewhat [8,10]. Realistically, earth is not uniform. In particular the earth surface can be near the target of interest, significantly changing its signature [22,23]. If the target is not too close to the interface (earth surface, either above or below) perturbation formulas can also be used [12]. In this case, distance from the interface is the parameter to be adjusted. 8.

CONCLUDING REMARKS

Parsimony in target identification then seeks to constrain the representations of the target scattering (the target signatures) so as to make it difficult for the target signature of the mth target type represent that of the nth for This implies that there be as few variable fitting parameters as possible. It is generally helpful to have aspect-independent parameters to the extent feasible because these can be constrained as a priori constants (e.g., natural frequencies), not having to assume a large number of different values for the various possible (a priori unknown) directions of incidence and polarizations. To the extent that direction of incidence and polarization with respect to the target orientation are known from other measurements, one can constrain the various scaling constants for the target-signature functions, making it harder to fit the data with the wrong target type. Target symmetry also plays a useful role in parsimony. Symmetry planes allow the 2 x 2 backscattering dyadic to be diagonalized, reducing the number of elements to be considered from three to two and giving orientation information in the process. For low-frequency MSI characterized by the magneticpolarizability dyadic we have found that the number of scaling constants can be reduced from six in the general nonsymmetrical case to two for targets characterized by symmetry for [15]. Part of the problem in target identification is the corruption of the scattering data by noise. In fitting the data with target signatures one is also fitting the noise with such signature functions. Of course we would like the fit to the noise to be poor. Furthermore, if we could distinguish the functional form of the noise (random?) and model it or remove it we might reduce this signature-fitting problem. This work was supported in part by the U. S. Air Force Office of Scientific Research, and in part by the U.S. Air Force Research Laboratory, Directed Energy Directorate. REFERENCES 1. C. E. Baum, Scattering, Reciprocity, Symmetry, SEM, and EEM, Interaction Note 475, May 1989. 2. C. E. Baum, SEM Backscattering, Interaction Note 476, July 1989. 3. C. E. Baum, The Theory of Electromagnetic Interference Control, Interaction Note 478, December 1989; pp. 87-101, in J. Bach Anderson (ed.), Modern Radio Science 1990, Oxford U. press, 1990. 4. C. E. Baum, Representation of Surface Current Density and Far Scattering in EEM and SEM with Entire Functions, Interaction Note 486, February 1992; Ch. 13, pp. 273-316, in P. P. Delsanto and A. W. Saenz (eds.), New Perspectives on Problems in Classical and Quantum Physics, Part II, Acoustic Propagation and Scattering, Electromagnetic Scattering, Gordon and Breach, 1998.

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5. C. E. Baum, The SEM Representation of Scattering from Perfectly Conducting Targets in Simple Lossy 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Media, Interaction Note 492, April 1993; Ch. 3, pp. 47-79, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Low-Frequency Near-Field Magnetic Scattering from Highly, but not Perfectly, Conducting Bodies, Interaction Note 499, November 1993; Ch. 6, pp. 163-218, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, The Magnetic Polarizability Dyadic and Point Symmetry, Interaction Note 502, May 1994; Ch. 7, pp. 219-242, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Concerning the Identification of Buried Dielectric Targets, Interaction Note 504, July 1994; Ch. 4, pp. 81-102, in C. E. Baum (ed.), Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1998. C. E. Baum, Direct Construction of a from Natural Frequencies and Evaluation of the Late-Time Residuals, Interaction Note 519, May 1996; pp. 349-360, in G. Heyman et al (eds.), Ultra-Wideband, Short-Pulse Electromagnetics 4, Kluwer Academic/Plenum Publishers, 1999. G. W. Hanson and C. E. Baum, Perturbation Formula for the Internal Resonances of a Dielectric Object Embedded in a Low-Impedance Medium, Interaction Note 520, August 1996. C. E. Baum, Symmetry in Electromagnetic Scattering as a Target Discriminant, Interaction Note 523, October 1996; pp. 295-307, in H. Mott and W. Boerner (eds.), Wideband Interferometric Sensing and Imaging Polarimetry, Proc. SPIE, Vol. 3120. G. W. Hanson and C. E. Baum, Perturbation Formula for the Natural Frequencies of an Object in the Presence of a Layered Medium, Interaction Note 532, October 1997; Electromagnetics, 1998, pp. 333351. C. E. Baum, Application of Symmetry to Magnetic-Singularity Identification of Buried Targets, Interaction Note 543, June 1998. C. E. Baum, Target-Vicinity Scattering Parameters, Interaction Note 546, August 1998. C. E. Baum, Use of Residue and Constant-Dyadic Information in Magnetic-Singularity Identification, Interaction Note 547, August 1998. C. E. Baum, Magnetic Singularity Identification of Nonsymmetrical Targets, Interaction Note 549, December 1998. C. E. Baum, Norms of Time-Domain Functions and Convolution Operators, Mathematics Note 86, December 1985; Ch. 2, pp. 31-55, in H. N. Kritikos and D. L. Jaggard (eds.), Recent Advances in Electromagnetic Theory, Springer-Verlag, 1990. C. E. Baum, Energy Norms and 2-Norms, Mathematics Note 89, April 1988; Ch. 11.4, pp. 498-508, in H. Kikuchi (ed.), Environmental and Space Electromagnetics, Springer-Verlag,, 1991. C. E. Baum, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, The Singularity Expansion and Its Application to Target Identification, Proc. IEEE, 1991, pp. 1481-1492. C. E. Baum, Signature-Based Target Identification and Pattern Recognition, IEEE Antennas and Propagation Mag., Vol. 36, No. 3, June 1994, pp. 44-51. T. K. Sarkar and O. Pereira, Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials, IEEE Antennas and Propagation Mag., Vol. 37, No. 1, February 1995, pp. 48-55. E. J. Rothwell and M. J. Cloud, On the Natural Frequencies of an Annular Ring Above a Conducting Half Space, J. Electromagnetic Waves and Applications, Vol. 10, 1996, pp. 155-179. S. Vitebskiy and L. Carin, Resonances of Perfectly Conducting Wires and Bodies of Revolution Buried in a Lossy Dispersive Half-Space, IEEE Trans. Antennas and Propagation, 1996, pp. 1575-1583. L. Carin, R. Kapoor, and C. E. Baum, Polarimetric SAR Imaging of Buried Landmines, IEEE Trans. Geoscience and Remote Sensing, 1998, pp. 1985-1988. H. Rothe, Approaches to Pattern Recognition, pp. 1-1 through 1-29, RTO Lecture Series 214, Advanced pattern Recognition Techniques, RTO-EN-2, 1998. J. N. Franklin, Matrix Theory, Prentice-Hall, 1968. C. E. Baum, Toward an Engineering Theory of Electromagnetic Scattering: The Singularity and Eigenmode Expansion Methods, Ch. 15, pp. 571-651, in P. L. E. Uslenghi (ed.), Electromagnetic Scattering, Academic press, 1978. D. D. Howard, Tracking Radar, Ch. 18, pp. 18.1-18.60, in M. I. Skolnik, Radar Handbook, 2nd Ed., McGraw-Hill, 1990. C. E. Baum, SEM and EEM Scattering Matrices and Time-Domain Scatterer Polarization in the Scattering Matrix, Ch. 1-9, pp. 427-486, in W.-M. Boerner et al (eds.), Direct and Inverse Methods in Radar Polarimetry, Kluwer Academic Publishers, 1992. C. E. Baum and H. N. Kritikos, Symmetry in Electromagnetics, Ch. 1, pp. 1-90, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995. C. E. Baum, Symmetry and Transforms of Waveforms and Waveform Spectra in Target Identification, Ch. 7, pp. 309-343, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 1995.

BURIED OBJECT IDENTIFICATION WITH AN OPTIMISATION OF THE TLS PRONY ALGORITHM

Lostanlen Y., Corre Y., Uguen B. Groupe Détection Image Diffraction Laboratoire des Composants et Systèmes pour les Télécommunications Institut National des Sciences Appliquées , Rennes 35043 Cedex, France

INTRODUCTION Radar systems emitting an ultra-short pulse, corresponding to an ultra-wide frequency bandwidth, (UWB-SP) are used in many applications including detection of buried objects (landmines, unexploded ordnances UXO) or stealth targets. (Bertoni, 1993; Carin, 1995; Baum, 1997; Shiloh, 1999). Such systems are designed to operate at low frequencies allowing a deeper penetration in a dispersive ground, while combining a very large bandwidth, hence providing a better resolution. Although these systems have been found to detect buried objects quite well, the resolution is rather poor preventing a good radar imaging of the scene. Many false alarms occur when interrogating the subsoil. Therefore there is a great need for a reliable identification tool. Many targets have a simple shape and only a few complex natural resonances (CNR) dictate the late time behaviour of the scattering response. Many authors have studied this late-time backscattered response in an attempt to identify and discriminate targets by studying the CNR. The first efforts were based on the use of Prony’s method, but this failed to give any satisfying result in real environments. However, improvements of the method were conducted by Rahman (1987). Therefore the purpose of the paper is to present the TLS-Prony method we have implemented (based on Rahman’s work) and to propose some criteria to optimise the input parameters of the method. After having validated our method on a sphere, we will illustrate an application


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of this method to real GPR data. THE PRONY AND TLS-PRONY METHODS For many radar targets, the late-time impulse response may be expressed in the time-domain as the sum of a finite number of resonant modes as follows:

is the early-time duration, are the complex poles on the s-plane, and the residues (or mode amplitudes). Note that where are the damping factors and the resonant pulsations. The complex poles are invariant target characteristics (provided that the target is perfectly conducting). Only the amplitudes depend on the target orientation or the excitation source. The discrete time signal, sampled at interval is written as a function of the complex poles lying in the z-plane:

The object of the Prony method is the resolution of this non-linear system, and the extraction of poles and residues, at least in the absence of any perturbation. By means of the Vandermonde matrix Z, the system is written in its matrix form:

The Prony method leads to a unique and exact solution when the number of sampled data N is twice the number of complex modes P. The basic idea is to consider the characteristic polynomial

whose

roots are the poles of the system. We can easily demonstrate that the signal solution of the following autoregressive difference equation:

Finally, the different steps of the Prony method are: The polynomial coefficients

are determined by solving Eq. 5.

is

OBJECT IDENTIFICATION WITH THE TLS PRONY ALGORITHM

The poles

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are deduced from the roots of

System 4 is solved to obtain the residues However this method is not very well suited when dealing with data perturbed by noise. In that case the system 4 is ill-conditioned, which in turn introduces substantial errors in the pole estimations. The first way to improve the method performance is to choose N > 2P and solve the polynomial coefficients in the difference Eq. 5 in a least square error sense. That is known as the Least Square (LS) - Prony method. In many cases, especially for data with low Signal to Noise Ratio (SNR), the Total Least Square (TLS) approach is more appropriate (Dowling, 1994; Chen, 1996). This method takes into account the fact that noise introduces perturbation in both sides of Eq. 5 which tends to minimize its effect on the pole estimations. If we note L the order of the characteristic polynomial B(z), chosen greater or equal to the number of poles P, system 7 may be written as follows:

For simplicity, the matrix is noted The TLS solution is found by means of the Singular Value Decomposition (SVD) of the matrix as follows:

where U and V are unitary matrices, and the elements are the singular values of the matrix; they are supposed to be arranged in descending order. If the data under test is free of noise, is not full rank and only part of the singular values are greater than zero. In practical cases, all the values are nonzero, and the effect of noise is reduced by considering the values that are inferior to a threshold as negligible.

The choice of the threshold value is not obvious and will be the object of the following section. In Rahman (1987) it is shown that the polynomial coefficients may be deduced from the matrix where is the column of V.

where

The determination of L pole locations results from the TLS algorithm: P true poles and L – M extraneous poles. A common way to minimise the number of extraneous poles is to choose only the poles lying inside the unit circle (on the z-plane). A second method consists in the selection of the poles corresponding to the most energetic modes. The energy of the resonant modes is given by:

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The described method gives very good results in ’normal’ conditions. However the low SNR, of real signals disturb considerably the extraction process. Thus the TLS-Prony method needs to be modified.

THE MODIFIED TLS-PRONY Many difficulties arise when the TLS-Prony algorithm is applied to real radar signals with low SNR. The choice of the threshold which determines the number of singular values that must be retained after the SVD decomposition is delicate. Indeed, a non-appropriate choice may affect the integrity of the algorithm, or simply gives rise to large errors in the location of the poles. On one hand, the calculation based on a too small number of singular values does not permit the extraction of a sufficient number of poles (only the main resonant modes are recognised). On the other hand, the intervention of the smallest singular values in the calculation increases the effect of noise and degrades the algorithm accuracy. Moreover, as illustrated later in this section, a small variation of the threshold may produce results with extremely variable precision. That gives a random aspect to the conventional TLS-Prony method, and represents a serious drawback. A second difficulty is the presence of extraneous poles. To prevent false alarm, or to permit effective target detection and identification, a perfect distinction between real poles and poles introduced by the algorithm is necessary. The rejection method based on the energy of the poles removes a part of the extraneous poles, but is not sufficient. The impulse response of the perfectly conducting sphere is a good basis for the validation of the TLS-Prony method efficiency. The sphere impulse response illustrated in Fig. 1 presents a highly resonant part comprised between 1.5 ns and 2 ns, resulting from the creeping wave. The Singularity Expansion Method (SEM) shows that the response may be considered as an infinite sum of resonant modes. In the s-plane the complex poles associated to those resonant modes lie into infinite branches centred on the real axis Lavenant (1994), Chen (1981). Whereas an infinite number of poles are needed for an accurate reconstitution of the impulse part, only a few poles located in the first and second branches are necessary to obtain a good approximation of the creeping wave contribution.

To estimate the efficiency of the TLS-Prony method, we compute the sphere re-


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sponse by means of the Mie series. We then compare the poles extracted by the algorithm to the theoretical poles given by the SEM method. Lets consider the impulse response of the sphere of radius computed with the sampling period A gaussian noise is added, so that the SNR is equal to 55 dB (the SNR is given by the ratio of the magnitude of the creeping wave contribution over the noise standard deviation). A 5th order lowpass Butterworth filter is used in order to simulate the effects of anti-aliasing filters. Then the algorithm is applied to the late-time part of the signal, that is the 350 samples placed after instant As we have no priori knowledge about the number of singular values that must be retained after SVD, we calculate the poles and residues for different singular value thresholds. The poles located in the vicinity of a theoretical pole are counted; the errors about the frequency, the damping factor, and the residue are determined. As shown in Fig. 2, up to 9 poles which correspond to the theoretical resonant modes are identified. At first, when the singular value threshold increases, the number of identified poles increases as well, continuously. But, afterwards, that number is fluctuating, and some of the threshold values lead to a very poor identification. Fig. 2 represents also the error on the frequency for the three first poles of the sphere. Note that the frequency accuracy is extremely variable. Nevertheless, intervals such as [25 – 30] or [40 – 45] offer acceptable solutions. Finally, if we combine the information provided by both figures, the threshold indices that lead to the most precise and complete solutions are 27 and 28.

The plot of the singular values with a logarithmic scale (in Fig. 3) presents, in the vicinity of indices 25 and 26, a breaking point separating two linear areas. It accurately indicates the optimal singular value threshold, and the singular values placed after it may be considered as negligible. However, as shown in the next section with experimental data, that point is not always so distinct and so relevant. Nevertheless it remains a useful indicator. We decide to carry out a tracking process on the poles that have been extracted with successive threshold indices, in order to study their evolution. Some of the extracted poles appear only for one threshold value; others are very unstable: they are rejected. This permits to put aside many extraneous poles. The error relative to the frequency, and the residue magnitude of the remaining poles compared to the theoretical ones are calculated. The square of these errors is averaged for each threshold value to obtain the global frequency and residue behaviour. Fig. reffigGlobVar presents

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both frequency and residue global variations. The figures let appear small variations for the threshold indices inferior to 30, and large variations for higher values. From this observation, we have developed a modified and iterative TLS-Prony method, which leads to the extraction of a great number of poles with reasonable accuracy, and to the rejection of most extraneous poles: SVD decomposition of the matrix pole extraction for a range of singular value thresholds, pole tracking, variation measurement, and pole rejection, the optimal threshold is chosen such that the variations are small and the number of poles is maximal

The result for our noisy sphere impulse response appears in Fig. 5. The extracted poles are represented by points, the theoretical ones by circles. (Note that the singular value threshold chosen by the algorithm is 25).


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REAL DATA Detecting targets buried in the ground with a GPR device requires the ability to identify or remove the clutter contribution. Besides, the discrimination of inoffensive stones from hazardous objects such as antipersonnel landmines may be performed by the extraction of features which positively characterise the target. Therefore the modified TLS-Prony method may be found to be a useful processing tool. The method has

been applied to a set of data resulting from measurements. An exhaustive study on the processing techniques to characterise and remove the clutter has been carried out on the same kind of data set by Brooks (1999). The data set consists in a C-scan of GPR returns regularly measured from an antenna moving above a sandbox, in which different objects are buried (5 cm below the surface). Fig. 5 and 6 show the frequencies, residues and poles extracted with the TLSProny algorithm at different antenna positions. We observe that some of the frequencies are constantly present; they are probably due to the correlated clutter and the radar

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system. On the other hand the poles which occasionally appear result from soil local particularities or buried objects. CONCLUSION This paper has presented an application which uses CNR signatures to identify buried objects. In particular, we have explained the improvements we have made in the automatic determination of the number of singular values. Finally we applied our optimised method to real data. In many practical cases, the clutter generated in the vicinity of the target causes major errors in the extraction process of CNR. We are currently working on the generation of this clutter by the interface and in the close surrounding of the targets in order to provide a useful input for the target classification. Acknowledgements The authors gratefully acknowledge the support of the Région Bretagne. REFERENCES Baum, C.E., Carin, L., Stone, A.P., 1997, Ultra-Wideband, Short-Pulse Electromagnetics 3, Kluwer Academic/Plenum Press. Bertoni, H.L., Carin, L., Felsen, L.B., 1993, Ultra-Wideband, Short-Pulse Electromagnetics, Kluwer Academic/Plenum Press. Brooks J. W. , Van Kempen L., Sahli H., 1999, Ground penetrating radar data processing: clutter characterization and removal, IRIS TR 0059. Carin, L., Felsen, L.B., 1995, Ultra-Wideband, Short-Pulse Electromagnetics 2, Kluwer Academic/Plenum Press. Chen H., Van Huffel S., Dowling E. & DeGroat R. D., 1996, TLS based methods for exponantial parameter estimation, 2nd international workshop on TLS and errors-in-variables modeling in Leuven Chen K.-M. & Westmoreland D., 1981, Impulse Response of a conducting sphere based on singularity expansion method, Proc. IEEE 69 Dowling E. M., DeGroat R. D. & Linebarger D. A., 1994, Exponential parameter estimation in the presence of known components and noise, IEE Transactions on antennas and propagation 42:5 Kergall J., Extraction des pôles naturels de résonance de cibles radar, dans le domaine temporel, Mémoire pour DEA (1998) Lavenan T., 1994, Contribution à al discrimination d’obstacle radar en zone de résonance, Thèse de Doctorat Rahman MD. A.,Yu K.-B., Total Least Squares Approach for frequency estimation using linear prediction, IEEE Trans. on acoustics, speech, and signal processing., 35:10 (1987). Shiloh J. , Mandelbaum B., Heyman Ehud, 1999, Ultra-Wideband Short-Pulse Electromagnetics 4, Kluwer Academic Press.

MODEL PROBLEMS OF PULSE SENSING

Lyudmyla G. Velychko,1 Andrey O. Perov,1 Yuriy K. Sirenko,1,2 and Ercan Yaldiz2 1

Department of Mathematical Physics Institute of Radiophysics and Electronics 12 Acad. Proskura st., Kharkov, 61085, Ukraine 2 Department of Electronic Engineering Gebze Institute of Technology P.K. 141, 41400, Gebze/Kocaeli, Turkey

INTRODUCTION The radio-locating means for determination of layered media parameters and visualization of objects or voids hidden in them have gained wide-spread acceptance in various fields, namely, civil engineering, geodesy, use of hydraulic-engineering structures and of oil-pipe-lines, etc. The solution of the engineering problems arising therewith requires the development of essentially new approaches. Traditional theoretical and experimental methods have made possible only the first generation of radars with both small signal penetration depth and incomplete retrieval of useful information from remote sensing data. Quite reasonable hopes for improvement of the situation are widely associated with the construction of radars based on nonsinusoidal waves. A central theoretical problem here consists in correct interpretation of the results of measurements, demanding for its solution adequate mathematical models (and algorithms) describing the transient processes in the part of space under research. Model problems of pulse sensing are initial boundary value ones in unbounded regions with inhomogeneous compact objects. The key to the effective solution of them lies with a proper truncation of the computational domain in finite-difference methods, i.e. in a limitation of such kind that, on the one hand, reduces the essentially open problem to the closed one, and, on the other hand, does not effect on accuracy and reliability of the data obtained. Our paper is just devoted to the analysis of this problem. In the first section, the problem of scattering of nonsinusoidal waves by a perfectly conducting compact object imbedded in an inhomogeneous half-space with an irregular boundary is considered. The second section is devoted to the extension of the results to other situations peculiar for pulse sensing: an inhomogeneous compact object and a stratified dielectric structure in a field of nonsinusoidal waves and a pattern-forming structure. All model problems are two-


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dimensional and scalar ones (E-polarization), and yet the results can be generalized and modified as applied to solution of three-dimensional scalar (acoustic) or vector (electromagnetic) problems. In the third section of the paper, an example of correct incorporation of exact conditions on virtual boundaries into a standard computational scheme of the finite-difference method is given, the results of numerical experiments focused on testing and approbation of the constructed algorithms are presented. The theoretical and methodological foundations for the approach being developed here are laid in the papers by Y. K. Sirenko, A. O. Perov, and N. P. Yashina,1,2 which are devoted to origination of rigorous and efficient techniques for the analysis of electromagnetic transient processes in open periodic resonators (gratings) and waveguide resonators. We follow these papers, extending the results obtained there to another class of topical problems of electromagnetic theory that are of practical significance.

MODEL PROBLEM OF UNDERSURFACE PULSE SENSING Let us consider a problem with the geometry presented in Figure 1 ,a. We have to find a function such that

Here Q is a part of the plane bounded by contours S, and are relative permittivity and specific conductivity of the medium sensing respectively, and

It is supposed that the functions F,

which are finite in the region Q, satisfy the conditions of the theorem on

single-valued solvability of problem (1) in the Sobolev space

.3

The supports of all these functions at all instants considered are concentrated in the region Above the boundary separating the homogeneous and inhomogeneous media, the condition is valid. The tangential components of the field intensity vectors are continuous on For efficient use of the finite-difference method one has to close the area of the analysis by introduction of virtual boundaries. The most-used approach is the construction of approximate absorbing boundary conditions (ABC).4,5 However, these conditions do not “pass” completely the field incident on the boundary. The wave is partially reflected by the imaginary boundary. The free propagation of the field is distorted, so the calculation error caused by it cannot be estimated analytically. Too many factors affect its value. The alternative approach is based on the use of exact radiation conditions for secondary fields.1,2,6 It does not lean upon any heuristic assumption on the field structure near the virtual boundaries and reflects the essence of the simulated process. In our paper we construct the boundary condition using the representation of the field as expansion in elements of an evolutionary basis of nonstationary signals.1,2 In the region the solution of initial boundary value problem (1) represents a nonstationary wave “outgoing” from the area containing both the sources and the effective


scatterers. Separation of variable

and makes

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in (2) results in the following representation for

elements of the evolutionary basis of wave U subjected the conditions

Multiplying (2) by semiaxis we obtain

and performing Fourier-Bessel transform in

for the images

of functions

Here

is the Bessel function of order

Heaviside function, and

on the

is the

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Convolution of the fundamental solution and the right-hand side of (3) results in7


of the operator

Going back to the originals we obtain

Performing differentiation in (4) and taking into account properties of Legendre functions, we come to the following exact representation of in the region for

This relationship gives one of the possible forms of representation of explicit radiation conditions for the solution of problem (1).

EXACT CONDITIONS ON VIRTUAL BOUNDARIES IN OTHER MODEL PROBLEMS OF PULSE SENSING The investigation of the scattering of E-polarized nonstationary electromagnetic field by a dielectric object in a free space (see Figure 1,b) is reduced to solution of the initial boundary value problem similar to (1) but with the following condition The area of the analysis here is different from that considered above. It covers the whole plane of variables and and condition in this connection, is replaced by the condition of periodicity. Obviously the technique for deriving the exact radiation conditions is not subjected to serious changes. The following transformations are similar to those for problem (1). The resulting condition on the virtual boundary takes the form

Here

and


are

the

eigen

values

appropriate

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to

the

eigen

functions

Let us consider one more problem in Cartesian coordinates (Figure 1 ,c). The layer ( and are functions of only one variable z) is excited by the sources F, and of nonsinusoidal E-polarized waves with compact in the region supports. It is assumed that in all instants t these supports belong to We have to find the function U(y,z ,t) such that

Here

). By carrying out Fourier-transform in y for (7) we obtain a new problem

whose solution determines U uniquely in taking inverse Fourier-transform. For represents the “wave” outgoing from the area containing the sources and the scatterers and satisfies homogeneous initial boundary value problem (8) with and . With the use of cosine transform in z on the semiaxis z>0, we obtain the Cauchy problem for images and we have for its solution

(the technique is identical with that used for deriving (4)) or

On the virtual boundary z = a, after simple transformations in (17), we obtain

(the region as well as does not contain either sources or scatterers). A few words about an antenna problem whose geometry is exemplified in Figure 1,d. The virtual boundary (dashed lines) closing the area of wave propagation consists of two parts. On the first part of it, which is located in the region z > 0, one can use, without changes, condition (5) obtained for problem (1). On the second part coinciding with the cross section of the waveguide, the standard technique makes it possible obtaining the explicit conditions of the same type, as well as in the case with a periodic structure.1

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FINITE-DIFFERENCE SCHEMES AND SOME NUMERICAL RESULTS Further steps in algorithmization of model problems differ from standard ones only by the set of expedients in the technique.1 The computational experiments, within the framework of which the exact conditions following from (9), (13), (18), and (19) were tested, are practically identical to ones realized by Perov and Sirenko1 for gratings. Omitting technical details we present only one result describing the situation in general. Figure 2 allows us to compare errors caused by the use of both exact conditions (5) (dashed lines) and classical heuristic ABC of the first approximation order4 (full lines). Two problems of type (1) with simple geometry (see Figure 2,a) and are considered. For the data from the left column

is valid, for the data from the right column


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is valid. The local errors D(n,m,l) are calculated at grid points on the imaginary boundary The value D(l) determines the averaged global error on the l th time step within the limits of the whole grid. The discretization step in and (indices n and m) is equal to in time it is equal to The second approximation order is considered. The results of the experiment (Figure 2) clearly demonstrate that the truncation of computational domain by introducing virtual boundaries with exact radiation conditions does not increase calculation errors caused by the difference algorithm proper. Some more examples of implementation of the algorithms considered in this paper we present in Figures 3 and 4. The curves in Figure 3 are the amplitudes of pulses

arising in the reflection zone of the stratified structure (Figure 1,c) excited by the signal

The function

is the solution of the initial boundary value problem (8) with and

The response of the structure from Figure 4,a illuminated by the pulse

is shown in Figure 4,b. The pulse is depicted by a dashed line. The perfectly conducting boundary is described by the function

REFERENCES 1. 2. 3. 4. 5. 6.

7.

A.O. Perov and Y.K. Sirenko, Nonstationary model problems of the electrodynamic theory of gratings, in: Radiofizika i Electronika, Institute of Radiophysics and Electronics, Kharkov. 2, No.2: 66 (1998). Y.K. Sirenko and N.P. Yashina, Nonstationary model problems for waveguide open resonator theory, Electromagnetics, 19,No.5: 419(1999). O.A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York (1985). B.B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation. 31, No. 139: 629 (1977). G. Mur, Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetics field equations, IEEE Tr. on EMC. 23, No.4: 377 (1981). A.R. Maykov, A.D. Poyezd, A.G. Sveshnikov, and S.A. Yakunin, Difference schemes of initial boundary value problems for Maxwell equations in unbounded region, Comp. Math. and Math. Physics, 29, No.2: 239 (1989). V.S. Vladimirov. Equations of Mathematical Physics, Dekker, New York (1971).

SIMULATION OF THE TRANSIENT RESPONSE OF OBJECTS BURIED IN DISPERSIVE MEDIA

M. A. Hernández López 1, S.González García2, A. Rubio Bretones2 and R. Gómez Martín2 1

Departamento de Física Aplicada Universidad de Salamanca Plaza de la Merced, s/n 37008 Salamanca (Spain) 2 Departamento de Electromagnetismo y F. de la Materia Universidad de Granada Fuentenueva s/n 18071 Granada (Spain)

INTRODUCTION The study of the response of dispersive media excited by a transient electromagnetic signal is of interest in areas such as the simulation of Ground Penetrating Radar, the stimulation of biological tissues and broadband communications. This paper is focused on the simulation of a three-dimensional short-pulse ground penetrating radar (GPR) to study the transient response of objects buried in dispersive media. Since these systems often include thin wire antennas (Montoya and Smith, 1999), the hybrid technique combining the Finite Difference Time Domain method (FDTD) and the Method of Moments in Time Domain (MoMTD), described in (Rubio Bretones et al, 1998), has been extended to deal with the frequency dependence characteristics of the media, modeled with a non-uniform discretization and truncated with Berenger’s Perfectly Matched Layer (PML) conditions. An in-house tool based on the commercial program AutoCAD™ is used to specify the geometry and the general parameters of the full problem, thus providing a versatile way to define complex geometries involving several kinds of media.


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DESCRIPTION OF THE HYBRID METHOD In the hybrid MoMTD-FDTD method, the inhomogeneous region of the problem is modeled using the Finite Difference Time Domain Method (FDTD), which is based on the direct solution of Maxwell's curl equations and is capable of dealing with complex geometries with arbitrary electrical properties. However, the application of FDTD to an arbitrarily oriented thin-wire antenna presents some difficulties in modelling the thin-wire antenna features precisely, and so the MoMTD is used. The interaction between the antenna and the object is carried out by means of the surface equivalence theorem (Huygens’ principle). Since the method is implemented entirely in the time domain, it can efficiently generate information over a wide frequency band. The hybridization begins by dividing the original problem into two separate ones. The first of these, the thin-wire antenna, is solved by using the MoMTD, while the second one, the dielectric body, is handled via the FDTD scheme (Fig. 1). The basic steps in the hybridization are: 1) An imaginary closed Huygens’ surface S is located around the thin-wire antenna. Equivalent sources on S are deduced from the fields radiated by the antenna in free space, which are obtained solving the electric field integral equation (EFIE) by the MoMTD for the currents on the antenna. 2) The incident fields on the dielectric body are computed from the equivalent currents on S. The FDTD algorithm is applied to compute the fields at any cell inside the FDTD computational domain, in both the total field zone (outside S) and the scattered field zone (inside S). 3) The FDTD solution inside S when the antenna is not present is, by definition, the incident field on the antenna needed to solve the EFIE by the MoMTD and hence compute the currents induced on the wire. Spatial and temporal linear interpolation is applied to calculate the incident field at specific locations on the antenna and specific times. NON-UNIFORM FDTD-PML FOR DISPERSIVE MEDIA The classical FDTD method is based on the approximation of Maxwell’s curl equations replacing the time and space derivatives by centred differences on a uniformly discretized space using the field distribution given by Yee's cube (Yee, 1966). A drawback of this algorithm appears when high permittivity and permeability media are present, since an overall fine discretization must be used, with the consequent increase in memory and CPU time requirements. The non-uniform FDTD algorithm (Kim and Hoefer, 1998) overcomes this problem by using first-order non-centred differences to approximate the spatial derivatives in a graded spatial mesh with the same relative field distribution given by Yee's cube. This allows us to use a fine discretization where needed, maintaining a coarser one in the rest of the computational space.

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633

In order to incorporate a realistic model of the soil in FDTD, it is necessary to take into account its frequency-dependent dielectric behaviour, as well as its conductivity, which depend mainly on its water concentration. Let us assume a Debye model in frequency domain

where is the relative infinite frequency permittivity, the relative static permittivity, is the relaxation time, is the free space permittivity, and its constant conductivity. The time domain relationship between equation as

and

can be written in the form of a differential

Then, the source-free Maxwell’s curl equations in time domain for non-dispersive magnetic linear media of permeability µ, are written as

Eqs. (2) and (3) are discretized with second order accurate centred differences for the time derivatives and first order non-centred differences for the spatial ones, obtaining an explicit-in-time advancing scheme. Finally, Berenger’s PML absorbing boundaries (Berenger, 1994) have been chosen to truncate the computational space. Following the formulation described in (González, Villó et al, 1998), a material-independent form (Zhao et al, 1998) of the PML valid for dispersive and conductive media has been obtained. In time domain the governing equations are

634


where the subscript s denotes any split field in the PML related

to

the

corresponding

maxwellian

field

at

the

interface

through

It can be proved (González, Villó et al, 1998) that a plane wave with wavevector propagating in a maxwellian medium, and inciding on a PML medium, for instance at is not reflected if the transmitted wave inside the PML is enforced to propagate with the same wavenumbers in the x and y directions as the incident one Since there is no constraint on the wavenumber in the z direction this can be properly chosen to attenuate the transmitted wave at a desired rate by setting

If the PML is backed by a perfect conductor at a depth at

the normal reflection coefficient

when the incidence medium is not dispersive is

with

being the

frequency independent phase speed in the z direction. Since for dispersive media the wavevector is complex and frequency dependent, the normal reflection coefficient now depends on frequency and takes the form So must be chosen to provide the desired reflection coefficient in the less favourable case (maximum frequency). and related by

in Eq. (5) are the electric and magnetic conductivity matrices, which must be

to achieve perfect matching. The matrix elements must fulfill freely chosen positive number in order to match waves impinging at

with

being a


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MoMTD

The method of moments is based on the solution of the EFIE derived by enforcing the boundary condition on the tangential electric field over the surface of the wires. It has been widely used for analyzing thin-wire structures, both in the frequency and directly in the time domain (MoMTD). For a detailed explanation of the method see, for instance, (Rubio et al, 1989). MODELING

The geometry of the problem, as well as the general FDTD parameters, is fully specified in a drawing created with the aid of AutoCAD™, which provides a very flexible way of defining a complex geometry. A set of AutoLISP™ routines translates AutoCAD database information into suitable FDTD input parameters. The tool, which is inspired by the modeling technique used by the EMPIRE™ FDTD simulator1 (González, Baggen et al, 1998), allows among other capabilities: The specification of the materials involved in the problem through rectangular boxes, cylinders and spheres which can be interpenetrated by means of a system of increasing priorities. The definition of voltage and current sources with 3D boxes wherein the fields are enforced to the desired values. The specification of Huygens’ surfaces on which the equivalent currents are placed to separate total from scattered field zones. The definition of the wire antenna geometry inside the Huygens’ boxes. The placement of observation boxes. RESULTS

In order to test the tool, we simulated several typical GPR problems. In the first one a metallic pipe, with a 4 cm square cross section and running indefinitely in the x-direction, is buried 2.5 cm beneath the surface of a wet soil A pair of crosswise wire antennas separated vertically 2.8 cm, are located 3 cm above the soil. A full FDTD model (Fig. 2), incorporating an inverse-with-the-distance field variation at the vicinity of the wires in the FDTD equations, is used. Firstly, the antenna aligned with the pipe was excited with a pulse of 5 GHz of bandwidth, keeping the perpendicular one passive; and secondly their roles were interchanged. Fig. 3 represents the subtraction between the currents at the centre of both antennas when the pipe is buried in the soil and when the soil is empty, showing, as expected, that for the transversal polarization, the current is smaller than for the longitudinal one, thus providing a technique to detect the orientation of the pipe. In none of the cases were the cross-polarized reflections significant. In order to test the hybrid technique, the analytical expression of the fields created by a simple Hertzian dipole was used to obtain the equivalent currents on a Huygens’ surface, and used to illuminate a 24x24x12 cm plastic mine with low permittivity The mine was buried at a depth of 12 cm in the same soil of the previous example, whilst the dipole was placed 24 cm above the soil and excited with a 3 GHz bandwidth gaussian pulse. An observation point was placed inside the Huygens’ box 1

Developed by the Institut für Mobil-und Sattellitenfunktechnik (IMST) in Germany

636


(scattered field zone) above the dipole. Fig. 4 shows the reflected copolarized electric field component dispersed when the mine is or not buried. The results obtained for the crosspolarized components were also non-significant with respect to the copolarized ones and are not shown in the figures.


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Finally, the hybrid FDTD-MoMTD method was used to simulate a GPR problem involving a V-antenna. A metallic box of dimensions 20x20x6 cm was buried at a depth of 50 cm. in a purely dielectric soil The antenna was excited at its centre by a 3 ns. gaussian source voltage. An observation point was located 8 cm from the vertex of the antenna, and on a plane parallel to the one containing the antenna. The solid line in Fig. 5(left) represents the copolarized scattered field without the object, and the dashed line shows it when the object is present. Fig. 5(right) shows the same results but when the Vantenna is resistively loaded with a Wu-King profile (Sánchez García et al, 1998). It can be seen that the loading enables the reflections from the mine to be distinguished in time domain quite clearly.

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CONCLUSIONS A time-domain tool to simulate GPR problems has been developed and tested. It uses an automatic geometry preprocessor based on AutoCAD to define the full problem in a versatile way. The kernel of the tool is a non-uniform FDTD simulator which can handle inhomogeneous dispersive media terminated by PML conditions, as well as wire structures, through the incorporation of the MoM solution of the wire problem, coupled to the FDTD zone via the equivalence principle. We have used this tool to study the transient excitation of inhomogeneous bodies by thin-wire broadband antennas and specifically to simulate a short-pulse GPR. Acknowledgements This work is partially supported by the CICYT (Spain) through project TIC99-0624.

REFERENCES Montoya, T. P., Smith, G. S., Land Mine Detection Using a Ground-Penetrating Radar Based on Resistively Loaded Vee Dipoles, IEEE Trans. on Antennas and Prop., vol. 47, no. 12, 1999 Rubio Bretones, A., Mittra, R., and Gómez Martín R., A new Hybrid Method Combining the Method of Moments in the Time Domain and FDTD., IEEE Microwave and Guided Wave Letters, vol. 8, n.8, 281-283, 1998. Yee, K.S., Numerical solution of initial boundary problems in isotropic media, IEEE Trans. on Antennas and Propagation, vol. 14, pp. 302-307, 1966. Kim, I. S. and Hoefer, W. J. R., A local mesh refinement algorithm for the time domain finite difference method using Maxwell's curl equations, IEEE Trans. on Microwave Theory and Techniques, vol. 38, no. 6, pp. 812-815, 1990. González García, S., Villó Pérez, I., Gómez Martín, R., García Olmedo, B., Extension of Berenger's PML to adapt bi-isotropic media, IEEE Microwave Guided Wave Lett. , vol. 8, no. 9, 1998. Zhao, A. P., Renko, A., and Rinne, M. A., Material-independent PML absorbers for arbitrary lossy anisotropic dielectric media, Proc. of 1998 Int. Conf. on Microwave and Millimeter Wave Technology, pp. 978-981, Beijing, CHINA, August 1998. González García, S., Baggen, L., Manteuffel D., and Heberling D., Study of Coplanar Waveguide-Fed Antennas Using the FDTD Method, Microwave and Optical Technology Letters, vol. 19, no. 3, 1998 Sánchez García, I., Rubio Bretones, A., and Gómez Martín, R., Pulse Receiving Characteristics of Resistively Loaded V antennas, IEEE Electromagnetic Compatibility, vol. 40, n. 2, pp. 174176,1998

ELECTROMAGNETIC TRANSIENT MODELLING USING DYNAMIC ADAPTIVE FREQUENCY SAMPLING

Choy Yoong Tham 1, Andrew McCowen 1, Malcolm S Towers 1 and Dragan Poljak 2 1

Department of Electrical and Electronic Engineering University of Wales Swansea Singleton Park, Swansea, SA2 8PP, UK 2 Department of Electronics FESB, University of Split bb, 21000 Split, Croatia

INTRODUCTION In frequency domain transient modelling, the inverse Fourier transform (FT) process is usually implemented by the fast Fourier transform (FFT) algorithm. In the usual approach, the frequency sampling interval is established by the relation where is the time window of interest [1]. In this relation the sampling interval, depends on an arbitrary parameter which requires empirical insight in its selection to yield accurate result. For transients with a relatively smooth spectrum such as the current on a thin wire scatterer [1] or a dipole in free space [2, 3], the empirical approach does not pose serious difficulty. The sampling interval may vary over a wide margin while the result remain reasonably accurate. However with a resonant structure, the spectrum has sharp and narrowly defined peaks. Coarse sampling intervals are not sufficient to resolve the peaks and will result in error. When used to model the transient of a highly resonant structure the conventional FFT technique may fail to give the correct result. This is demonstrated [4] in the case of a long single conductor transmission line lying very close to a ground plane with one end short circuited. The short circuit current induced by an obliquely incident EMP plane wave was modelled with the conventional FFT technique which yielded totally erroneous result. These observations highlight the weakness of the frequency domain FFT method of transient analysis. When modelling a structure without a priori knowledge of its frequency response, guess work is often being relied on to select the values for and which may not yield successful result. In this paper a systematic and objective procedure is proposed. A new technique using dynamic adaptive sampling to take samples of the spectrum at nonuniformly spaced frequency intervals is formulated. A modified inverse discrete FT (DFT) formula is used to process the data into the transient waveform. This proposed technique Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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improves the computational efficiency of frequency domain analysis by reducing the number of samples substantially. ILLUSTRATING THE PROBLEM OF USING UNIFORM SAMPLING Consider the current induced in a parasitic wire by a voltage pulse in an adjacent parallel wire. Fig. 1 shows the geometry of the two parallel wires over a perfect ground plane. The wires are of length L = 1.0 m., radius a = 2.0 mm., separated by a distance s = 0.5 m. and lying at height h = 0.25 m. over the ground plane. One wire is excited at its centre with a Dirac delta function voltage pulse. The coupling effect and the presence of the ground plane induces a current with a resonant spectrum in the parasitic wire.

The frequency response spectrum for the current at the centre of the parasitic wire is shown in Fig. 2. The spectrum exhibits the high Q characteristic of a resonant structure. The spectrum for an identical single wire in free space is also shown to contrast the resonant point of the parasitic wire. To study the transient, a Gaussian voltage pulse of the form where and is used as the excitation source. For a transient waveform duration of 30 ns., the parameters for the inverse DFT, rounded to convenient figures are N = 128 and Figs. 3 and 4 show the frequency response spectra for the imaginary part of the

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current near the resonant point. For the case of the single wire in free space, virtually the same spectrum is obtained with the frequency interval 13.020833 and 6.510417 MHz. corresponding to N = 64, 128 and 256 respectively. However, for the parasitic wire, the frequency response spectrum obtained with where N = 512, shows an error against a more closely sampled spectrum. This error is significant and will lead to an erroneous transient response which will be shown in Section IV. In Section III a method is developed to accurately represent such a frequency response without incurring the penalty of needing a large number of frequency samples.

FORMULATION OF THE METHOD A convenient way to derive a transient waveform from frequency domain data is by taking the inverse FT of the frequency response spectrum. The impulse response so obtained is then used to carry out a convolution with the excitation waveform in the time domain. This procedure avoids the end effect error of using the alternative convolution theorem in the frequency domain. The inverse DFT is given by the expression below [1] –

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The inverse DFT is usually implemented with FFT routines which compute results more rapidly than the DFT. However, standard FFT routines process only uniformly spaced samples where the number of samples N in equation (1), may be large and is usually, though not always, be a power of 2. In the new procedure proposed here, the time step is determined from sampling theory as (2 × bandwidth of the excitation waveform) and the frequency sampling interval by –

The inverse DFT formula in (1) may be written in terms of the frequency samples and expanded by using Euler’s identity as,

Equation (3) implements the Fourier integral over a period from - to by the trapezoidal rule. In computing the inverse FT, the product of each frequency sample with its corresponding sin or cos function contributes to the summation of equation (3). For a resonant spectrum the error from sampling is very much accentuated by the high Q characteristic. Sampling error at the resonant point carries disproportionate weight into the computed inverse DFT. This is due to the fact that samples away from the resonant point have negligible amplitudes and contribute little to the overall sum. A test is necessary to ensure the frequency samples characterise a resonant spectrum accurately at their respective positions. This is done by the method for the adaptive integration of an unknown function using the trapezoidal rule [5]. In trapezoidal rule integration the local error is given by –

where h is the panel width and

are the integration limits. This is illustrated in Fig. 5.

The error expressed in equation (4) is a function of the second derivative which implies large errors at the sharp resonant peaks. The error diminishes rapidly when the sample spacing is reduced. Using this principle a sampling interval is subdivided


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progressively with intermediate samples to achieve the desired accuracy. As the sample spacing is reduced, successive integral values will converge towards the true value. The difference of the integration values obtained by two successive set of samples is used for convergence test. Unlike in standard adaptive integration, there is no a priori knowledge of the amplitude for an unknown electromagnetic spectrum nor even its order, to fix an absolute tolerance limit. Instead a relative criterion as shown in (5) is used for the test.

is the most recently evaluated integral, is the previous value and Tol is the tolerance limit. When the pre-selected tolerance limit is reached further subdivision is stopped. The algorithm is implemented by first computing three initial samples using the interval determined from the Nyquist rate and the time window. Taking three samples at a time, is the area of the single trapezium formed by the first and the third samples and that obtained by adding the two individual trapeziums together as shown in Fig. 5. If convergence is not reached a fresh sample midway between the first and second is computed. The test is then repeated on the two subintervals. When the procedure is used the first time, it is necessary to conduct a numerical experiment to find the optimum convergence criterion. It has been found that 10% - 20% are highly precise limits with which very accurate results can be expected in all cases tested. In this parasitic wire example, it is found that a limit of 50% will produced transient results of good accuracy. With more relaxed convergence limits, the accuracy of the result deteriorates rapidly. It is also found that the saving in the number of samples by specifying a larger tolerance limit is relatively marginal. Table 1 shows the result of the numerical experiment. In the table the accuracy indicated is that of the transient waveform comparing against a benchmarked result validated by using an independent time domain method described later in Section IV.

There are three points in the inverse DFT formula of equation (3) that need modification to process the non-uniformly spaced samples. Firstly each of the samples is assigned a weight inversely proportional to its sampling interval. Secondly the sin and cos phase terms associated with a sample are calculated using the integer index “n” which is the sample number. This is no longer correct with non-uniform intervals and is

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instead calculated by using the ratio of the sample’s frequency to twice the bandwidth; i.e. instead of n/N to obtain the phase displacement. Thirdly, to use the trapezoidal rule on differently weighted samples the integration is carried out piece-wise. In practice the samples although taken at non-uniform intervals often occur in bands in which the interval is constant. This is due to the fact that away from the resonant points most electromagnetic spectra remain relatively smooth. Taking advantage of this nature, the piece-wise integration is implemented by summing the samples in bands with constant weighting. The modified inverse DFT formula is given in equation (6).

NUMERICAL RESULTS The frequency domain electromagnetics code, MoM3D, used in this paper is based on the method of moments applied to the electric field integral equation (EFIE) [6]. A time domain solver for wires which uses the finite element integral equation method (FEIEM) [7], is used to compared the results.

For the single wire radiating in free space, the conventional FFT is able to produce the transient waveform without noticeable error with only 64 uniformly spaced samples of the spectrum. This result together with another waveform obtained with 128 samples is shown in Fig. 6 as compared to that obtained with the FEIEM. It is to be noted that the 64 samples yield only a 19.2 ns. transient, not enough to


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cover the whole specified time window of 30 ns. In the case of the parasitic wire problem, as discussed in Section II, the conventional FFT procedure produces erroneous result with the 128 samples. Increasing the number of samples to 256 or even 512 the transient waveform is still in error as shown in Fig. 7. Examining the set of samples taken with dynamic adaptive sampling it is found that the frequency interval at the resonant point is reduced to to achieve convergence. Using this value of for the FFT over the whole bandwidth would result in 2048 samples. The transient obtained by using the FFT on 2048 uniformly spaced samples is shown in Fig. 8. This result is in good agreement with that obtain by the time domain FEIEM.

Using adaptive sampling, just 176 samples or less than 10% of 2048 are sufficient to produce an accurate transient current waveform. Fig. 9 shows this latter result compared to the waveform derived from 2048 samples using the conventional FFT. With the new technique the sampling intervals ranging from to 13.0208 MHz. There is hardly noticeable difference between the two results. The effectiveness of this technique in saving computer time can be gauged by looking at the time taken to generate each frequency sample. In this problem the solution uses a total of 106 nodal current unknowns on the two wires. The effect of the perfect ground plane is implemented by image theory. The single precision MoM3D code runs on a SUN Spare Ultra 1 workstation and takes about 9 sec. to compute one sample. The total time projected for the 2048 uniformly spaced samples for the conventional FFT is over 5 hours

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as compared to only about 26 minutes using this new technique.

CONCLUSION The main difficulty in modelling transients via the frequency domain in highly resonant structures is to extract the complete information from its frequency response spectrum. The selection of suitable values for the number of samples and the frequency resolution determines the accuracy of the result. However, neither of these parameters are known prior to simulation. A high Q spectrum requires high resolution samples which if taken at uniform intervals for the FFT will result in a large number of samples. The proposed dynamic adaptive sampling technique takes the samples at intervals according to the requirement at different points of the spectrum. It will, without a priori knowledge of the spectrum, concentrate samples around resonant frequencies. The method has successfully overcome the difficulty without incurring the penalty of increasing computer time. REFERENCES 1 2 3 4 5 6 7

Tijhuis, A G, Peng, Zq and Bretones, A R, “Transient excitation of a straight thin-wire segment: a new look at an old problem”, IEEE Trans. on Antennas Propagat., vol. 40, no. 10, Oct. 1992, pp. 1132 -1146. Miller, E K, Poggio, A J and Burke, G J, “An integro-differential equation technique for the time-domain analysis of thin wire structure. II Numerical results”, J. Comput. Phys., vol. 12, no. 1, June 1973, pp. 210 - 233. Miller, E K and Van Blaricum, M L, “The short-pulse response of a straight wire”, IEEE Trans. on Antennas Propagat, May 1973, pp. 396 - 398. Tesche, F M, Ianoz, M V and Karlsson, T, EMC Analysis Methods and Computational Models, John Wiley & Sons, Inc., 1997, pp. 342 - 345. Gerald, C F and Wheatley, P O, Applied Numerical Analysis, 4th ed. Addison-Wesley Publishing Co., 1989. The MoM3D Code, Dept. of Electrical and Electronic Engineering, University of

Wales Swansea, UK, 1997. D Poljak and V Roje, “Time domain calculation of the parameters of thin wire antennas and scatterers in a half-space configuration”, IEE Proc. Pt. H, Microw. Antennas Propag., vol. 145, no. 1, Feb. 1998, pp. 57 – 63.

THE TIME DOMAIN NUMERICAL CALCULATION OF AN INTEGRO-DIFFERENTIAL EQUATION FOR ULTRASHORT ELECTROMAGNETIC PULSE PROPAGATION IN LAYERED MEDIA

Igor V. Scherbatko 1, Stavros Iezekiel 1 and Alexander G. Nerukh 2 1

Institute of Microwave and Photonics, School of E&E Engineering, University of Leeds, Leeds, UK 2 Department of Mathematics , Kharkov Technical University of Radio&Electronics, Prosp. Lenina 14 Kharkov, 310726, Ukraine

INTRODUCTION Continuous progress in computing power and numerical methods has led to considerable growth in the simulation of short and ultrashort pulse propagation in media that exhibit dispersion, loss, gain and nonlinearity 1,2,3 . Finite-difference time-domain (FDTD) methods are popular due to their clarity and elegant realization in numerical algorithms. Indeed, it is difficult to identify areas in electromagnetic theory where FDTDbased methods have not been applied. Unfortunately, an undesirable feature of FDTD schemes is the time instability during long integration times. For electrically large domains and late-time analysis, the classical Yee scheme is limited by accumulation of phase errors, as shown by Young 4 . One possible solution is to provide a specific boundary treatment, by introducing a simultaneous approximation term, or to use a low-pass filter that eliminates the destabilizing Fourier components (numerical noise) from the results 5 . The high-order and leap-frog integrator methods also have been used to overcome these shortcomings 2,4,5. However, the leap-frog scheme is inapplicable to the time-dependent Maxwell equations and filtering leads to loss of high-frequency information in the signal, which is unacceptable for some applications. One reason for insufficient stability of FDTD-based methods, in some cases, lies in the high sensitivity of numerical differentiation to computational errors, especially for higherorder derivatives. Moreover, finite digit representation in computer memory limits the improvement of such calculations. As a result, most calculations need to be supported by double-precision or even higher levels of accuracy, which demands more computer memory and increases computation time. Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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The integral equations approach seems to have a more promising future in terms of both reflection-free realization and also stability due to elimination of errors related to differentiation. An advantage of the direct time integration methods is that they can be generalized to nonlinear dispersive materials. Although the integral-equation approach avoids the stability problems, they require the solution of a sparse matrix equation at each time step and need special treatment of the singular and hypersingular integrals that arise 6 . The integro-differential Volterra equation approach takes an intermediate place between the differential and integral methods, because it consists of both differential and integral operations. The most attractive feature of this method is that the Volterra integral equation is of the second kind, which means that unlike integrals of the first kind, it neither possesses singularities nor requires special regularization treatment. In addition, the derivatives presented here have only the lowest order that can lead to higher stability. Using a resolvent method for solving Volterra equations that describe electromagnetic transients allows the formulation of the evolutionary recursion method 7 . The evolutionary algorithm developed by Scherbatko8 for the direct numerical calculation of nonstationary electromagnetic field in active media is based on a spatial-time discretization scheme for the 1-D case. The algorithm imposes restrictions neither on the signal shape and duration nor on the temporal behaviour of the medium and is therefore generic. This scheme was applied recently to simulation of wavelength conversion of infrared optical pulses in a semiconductor9.

VOLTERRA INTEGRO-DIFFERENTIAL EQUATION The propagation of an electromagnetic signal into a dispersive dielectric with conduction is described by the integro-differential Volterra equation of the second kind 7

where

and E(t,r) is the time and spatial dependent electrical intensity,

is the

velocity of light in the initial medium, are the electric and magnetic permittivity of vacuum, is the nabla operator, is the Dirac delta function, is the Heaviside unit function, and x = (t,r). Beginning from an initial time (assumed to be t = 0), the permittivity and conductivity are determined by functions and respectively. Before this initial time, the permittivity and conductivity of the medium were and zero respectively. For the one-dimensional case, where the medium’s parameters depend only on time and the spatial coordinate x, the vectors have components perpendicular to the x-axis. Then after integrating over transverse coordinates in the equation (1) we have the following scalar integro-differential equation:

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In (4) the following notation is used: We can switch to normalized variables in (4): scaling factor with the wave-number dimension. Then we get:

where

is the

where

In terms of complex refractive index n = n'+in" , these parameters have a form 8 :

where

is normalized frequency if the initial electromagnetic wave is taken as Since the electric field exhibits jumps when the medium parameter changes abruptly, it is appropriate to consider the electric flux density:

which remains continuous. Rewriting the integro-differential Volterra equation in terms of electric flux density allows the stability of the numerical algorithm to be improved. Finally we will have

where

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This equation describes the evolution of the process and determines explicitly the field magnitude at a point through its magnitudes at points preceding the given moment on the lines THE MARCHING SCHEME To overcome the weak stability of the previous method8 an enhanced numerical scheme was developed. The essence of the proposed methodology is the sequential resolution of equations (7)-(11) step by step in time. The main difference between the previous technique and the one proposed here is the introduction of an smaller internal subgrid into square cells to satisfy the Courant condition ( or for the normalized variables) and the use of a prediction-correction technique. As in previous developed scheme8, we assume a uniform external grid on the coordinate plane with equal time and spatial steps: The integration paths in equations (8-11) are then straight diagonal lines, which pass through the nodes in the grid. If we assume that electric flux density and functions and are known for the previous time step the electric flux density at node we obtain

where the expressions (8)-(11) were transformed according to the recursive scheme described in 8 and have the form:

The integrals in equations (13) - (16) can be calculated numerically using the internal time subgrid introduced here. The subgrid nodes are chosen according to the Chebyshev quadrature scheme at the time interval

where are tabulated abscissas, and N is the number of points in the subgrid (usually no more that 9). Chebyshev’s quadrature scheme has equal weights and therefore minimizes the error for the case of integrated data that undergo uniformly distributed stochastic


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deviations. The only problem now is that the electric flux density is unknown for An intuitive way to solve this problem is through extrapolation of the function D at the interval to

about the point

If we apply a second-order Taylor series expansion with respect we will have:

where the derivatives at the prediction stage can be estimated through simple two-point scheme.

VALIDATION OF THE PROPOSED SCHEME The test problem relates to plane wave propagation and reflection from a dielectric slab with thickness d. Simulation of such a problem allows simple estimation of amplitude and phase errors in the calculated reflected and transmitted signals, because the phase error is transformed into amplitude error by interference inside the dielectric layer. The analytical solution for a multilayered medium and plane harmonic wave is well known and described, for example, in 10 . In our case, the initial signal is incident on a dielectric layer with complex refractive index n = n'+in" and has the form of a semi-infinite travelling wave with unitary magnitude and wavelength

It is assumed that the medium outside the dielectric layer has refractive index Therefore, the electric flux density of the reflected and transmitted signals is the same as the electric strength that is traditionally used in electromagnetic notation. The reflection and transmission coefficients for the dielectric layer with complex (in the general case) refractive index n and width d are given by the following expressions:

Figure 1 shows the spatial distribution of the electric flux density at for two special cases of electromagnetic wave propagation through the dielectric layer with real refractive index The solid line represents the case of total internal transmission. When the interference inside the layer totally suppresses the back reflection. The value of the layer’s width in this special case corresponds perfectly to the theoretical prediction (with accuracy ). The power of the reflected signal was about 40 dB below the initial one, which also corresponds to perfect coincidence with theory. In contrast with the previous case, the open squares represent the distribution of D for the maximum reflectivity case, when For this situation the calculated absolute values of reflection and transmission coefficients (for magnitude) are and respectively. Note that the expressions (20) give slightly different values, namely 0.303 and 0.956 respectively. This means that the electromagnetic transients still continue inside the dielectric layer. Electromagnetic pulse propagation through a dielectric layer presents a more interesting situation. Comparing the reflection of the continuous wave from a dielectric slab

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and the pulse signal, we see a noticeable difference. For example, the totally suppressed back reflection, which is observed for the continuous wave, does not occur for the pulse signal. If we select a Gaussian pulse

with subcarrier frequency for the incident signal, the minimum in reflection will be the same at However, the reflected power is strong enough. This situation is shown in Figure 2. The maximum magnitude of the reflected signal is 0.15, which corresponds to –16 dB power suppression only.

CONCLUSIONS An improved direct numerical algorithm for solving an integro-differential Volterra equation in the time domain for time- and spatially-varying dispersive dielectric media was developed. The technique is based on a marching scheme and introduces an internal subgrid that results in an enhanced Courant stability condition. The algorithm performs refinement of the results by a prediction-correction scheme. Results show good correlation with existing theoretical solutions and considerable improvement in precision over previous methods. The influence of electromagnetic transients in the lossless dielectric layer on reflection and transmission coefficients is investigated.


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Acknowledgments This investigation has been carried out under the Royal Society/NATO postdoctoral fellowship awarded to Dr. Igor Scherbatko for joint research with the Institute of Microwaves and Photonics at the University of Leeds. REFERENCES 1. A. S. Nagra, and R.A. York, “FDTD Analysis of wave propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. AP, vol. 46, pp.334-340, Mar. 1998.

2. J.L Young, “A higher order FDTD method for EM propagation in a collisionless cold plasma,” IEEE Trans. AP, vol. 44, No.9, pp.1283-1289, Sept. 1996.

3. M. Morgan, “Ultrawideband impulse scattering measurements,” IEEE Trans. AP, vol. 42, No.6, pp.840-846, June, 1994.

4. J.L Young, D. Gaitoude, and J.S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach, ” IEEE Trans. AP, vol. 45, No. 11, pp. 1573-1580, Nov. 1997. 5. J.S. Shang, “High-order compact-difference schemes for time-dependent Maxwell equations,” J. of Computat. Physics., No.2, pp.312-333, Aug. 1999. 6. M.D. Pocock, M. J. Bluck and S.P. Walker, “Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations,” IEEE Trans. AP, vol. 46, No.8, pp. 1212 -1219, Aug. 1998. 7. A.G. Nerukh, I.V. Scherbatko and D.A. Nerukh, “Using the evolutionary recursion for solving electromagnetic problem with time-varying parameter”, Microwave and Optical Technology Letters, Vol. 14, No.1, pp. 31-36,1997. 8. A.G. Nerukh, I.V. Scherbatko, and O.N. Rybin, “The direct numerical calculation of an integral Volterra equation for an electromagnetic signal in a time-varying dissipative medium,” J. of Electromag. Waves and Applications, Vol.12, pp.163-176, 1998. 9. I. Scherbatko, “Double-Doppler wavelength conversion of infrared optical pulses by moving grating of refractive index in semiconductors,” Optical and Quantum Electronics, Vol.31, pp.965-979, 1999. 10. S. L. Chuang, Physics of Optoelectronic Devices, -(Wiley series in pure and applied optics), A Wiley-Interscience Publication , 1995.


MARCHING ON IN ANYTHING: SOLVING ELECTROMAGNETIC FIELD EQUATIONS WITH A VARYING PHYSICAL PARAMETER

Anton G. Tijhuis1 and A. Peter M. Zwamborn2 1

2

Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, the Netherlands TNO Physics and Electronics Laboratory, P.O. Box 96864, 2509 JG ’s-Gravenhage, The Netherlands.

INTRODUCTION In this paper, we consider the determination of electromagnetic fields for a (large) number of values of a physical parameter. We restrict ourselves to the case where the linear system originates from one or more integral equations. We apply an iterative procedure based on the minimization of an integrated squared error, and start this procedure from an initial estimate that is a linear combination of the last few “final” results. When the coefficients in this extrapolation are determined by minimizing the integrated squared error for the actual value of the parameter, the built-in orthogonality in this type of scheme ensures that only a few iteration steps are required to obtain the solution. The paper is organized as follows. We first describe the general approach. Second, we give an overview of various practical applications. Third, the iterative procedure is illustrated for scattering by a two-dimensional dielectric cylinder in free space. For that example, finally, we outline the use of the algorithm in transient scattering, in linearized and nonlinear inverse-scattering algorithms, and in scattering by an object in a more general environment. Results for all four applications are available, but cannot be included because of space limitations. METHOD OF SOLUTION In the computational modeling of electromagnetic fields for practical applications, typically a large system of linear equations must be solved. This system originates from Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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spatially discretizing Maxwell’s differential equations (in “finite” or “local” techniques) or equivalent integral equations (in “global” techniques). In formal notation, such a system can be written as where = = = =

a linear operator, the unknown field, the forcing function, a physical parameter.

The operator originates from discretizing the integral operator in the continuous equation, is a discretized field and corresponds to an impressed source or an incident field. We are interested in the situation where this problem must be solved for a large number of sampled values of the parameter e.g., with Iterative procedure In this subsection, we consider the iterative procedure that is used to solve the system of equations (1). We summarize the classical description of Van den Berg (1985). The basic idea is to construct a sequence of functions such that the norm of the residual in the operator equation (1), i.e.,

decreases with increasing

At each step of the iterative procedure, we write

where is a suitably constructed correction function. We start the procedure with an initial guess with corresponding residual and a suitably chosen variational function Let We now determine the scalar

such that

is minimized. This leads to

In subsequent steps, we let

for follows that

and

In (6),

is again a suitably chosen variational function. It now is minimized when

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With equations (3)–(8) the iterative scheme based on error minimization has been defined. The interpretation of this procedure is that the forcing function is projected on the set of functions This can also be seen from the fact that

for

and

respectively.

Variational functions

In principle, the iterative procedure outlined above works for any choice of the expansion functions Improvement is obtained as long as the coefficient differs from zero, i.e., when Our choice of expansion functions is inspired by the fact that depends in a wellbehaved manner on the parameter Therefore, it should be possible to extrapolate, by choosing for where N = 2 (linear extrapolation) or N = 3 (quadratic extrapolation). For we follow the conjugate-gradient method, and generate the expansion functions from the residual:

Because of the built-in orthogonality of the iterative procedure, we are certain that this procedure does not search for components of in the space spanned by the “previous” functions The iterative procedure is formulated such that the variational functions can be determined from the residual in each iteration step. However, with (11), the first N variational functions are available at the start. Therefore, the relevant coefficients can be determined directly from the second orthogonality relation in (9). If we substitute

we find directly that the coefficients

with initial estimate

where the

can be found from the system of linear equations

This is equivalent to a conventional conjugate-gradient scheme with

are found by minimizing the squared error

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This modification leads to a more efficient implementation. However, the original version of the procedure provides a better explanation of its effectiveness.

POSSIBLE APPLICATIONS Examples of physical parameters for which this idea can be applied are: Frequency: as in conventional time-marching solutions, a fixed space discretization is introduced for the integral equation. However, the subsequent time discretization, which introduces the well-known stability problem, is replaced by a temporal Laplace or Fourier transformation. This leads to a system of linear equations of a fixed dimension, which must be solved for increasing with integer Afterwards, the desired time signals are obtained by a straightforward FFT operation (Tijhuis and Peng (1991)). Angle of incidence: in EMC applications it may be necessary to obtain the electromagnetic plane-wave response of a – usually conducting – object for a large number of directions of incidence. When such a computation is needed at a single frequency, we can again choose a fixed space discretization. In fact, the operator products required in the minimization procedure for the initial guess are now already available from carrying out the iterative procedure at “previous” angles. Source position: in conventional iterative techniques for solving multi-dimensional inverse-scattering problems, it is often necessary in each step to compute the result of a point- or line-source excitation for a varying position of the source. This can be achieved in the same manner as the variation in angle of incidence mentioned above. Obstacle position: when an object moves with a non-relativistic velocity with respect to a fixed source, the electromagnetic-field computations can be carried out by considering the problem in the rest frame of the obstacle. We then end up with a configuration with a varying source position. Contrast: for penetrable objects, the iterative solution of contrast-source integral equations may be a quite efficient way to determine the electromagnetic field, especially when the convolution structure of the original integral equation is preserved in the space discretization (Zwamborn and Van den Berg (1994)). In that case, the operator products can be evaluated with the aid of FFT operations, which has led to the designation “CGFFT method”. However, the convergence may deteriorate when the contrast in material parameters between the scatterer and the surrounding medium increases. This can be remedied by gradually increasing this contrast, and using the field values for lower contrasts to generate the successive initial estimates until the complete object has been “beamed up”. Shape: in the design of microstrip circuits, it may be necessary to “tune” the dimensions of one or more of the elements of the structure. A similar situation arises when the shape of a scattering object of known constitution needs to be reconstructed or changes gradually in time. The approach was originally conceived as an alternative for marching-on-in-time computations, and has therefore been called “marching on in frequency”. Compared with time-marching computations, the main advantage is that using results for “previous”


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frequencies accelerates the computations, but does not influence the final result. Therefore, error accumulation is inherently avoided. Compared with conventional frequencydomain techniques, the main advantage is that the space discretization is fixed for all frequencies. Therefore, the computational effort does not increase for increasing frequencies. Since its publication, the scheme has been applied as well to march on in angle, source position, contrast and object dimension. Therefore, the name “marching on in anything” now seems more appropriate. 2D DIELECTRIC CYLINDER: FORMULATION The applications mentioned in the previous section have by now all been demonstrated for one or more configurations. In this section, we outline the implementation of the iterative procedure for an infinitely long, axially uniform lossy dielectric cylinder embedded in free space (Figure 1). We follow Peng and Tijhuis (1993). In the next section, various applications will be described. An electrically polarized two-dimensional wave with electric-field strength is normally incident on the cylinder. The aim of the computation is to determine the corresponding total field component

The cornerstone of our method of solution is the integral representation

which holds for all In (16), denotes the modified Bessel function of the second kind of order zero, and When (16) reduces to an integral equation of the second kind for inside the cylinder. The space discretization of (16) is obtained as follows. The logarithmically singular behavior of as is substracted by breaking the integral over on the right-hand side of (16) up into

The first integral in (17) contains an almost regular part of the integrand of (16). The logarithmic singularity is confined to the factor of in the integrand of the second integral.

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The cross section is enclosed within a rectangular region consisting of square subregions with mesh size The grid points of the simple cubic mesh are located a t with for and for Solving (16) now amounts to determining an approximation of at the grid points The discretization of the integrals in (17) is based on approximating suitable parts of the integrands by piecewise-bilinear approximations, and integrating analytically over polygons determined by the boundary of and the grid. For the first integral, approximating the entire integrand results in

as In (18), is a discretized kernel, a weighting coefficient that contains shape information, and a sampled, extrapolated susceptibility. For the second, integral in (17), we use the piecewise-bilinear “filtered” approximation

and we integrate over

The resulting approximation assumes the form

where the discretized kernel is independent of frequency. Combining the approximations (18) and (20) then results in the discretized integral equation

where the convolution-type structure of the continuous equation (16) has been preserved. This makes this equation suitable for the application of the conjugate-gradientFFT method.

2D DIELECTRIC CYLINDER: APPLICATIONS As mentioned above, the solution of the two-dimensional scattering problem described above has been used in various applications. In the present section, we give an overview. Transient fields: the most straightforward application of the procedure is the solution of a transient scattering problem. In the marching-on-in-frequency method, (21) is solved repeatedly for with and


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From the frequency constituents thus obtained, the time-domain field is obtained by numerically evaluating a Bromwich inversion integral of the type

with the aid of an FFT operation. In this procedure, is determined by the duration of the time interval of interest, and N by the bandwidth of the incident pulse. Linearized inverse profiling: in inverse profiling, the cylinder is excited by an electric line source on a contour outside the cylinder (Figure 2). The aim of the computation is to determine from scattered-field values at a single frequency Let be the field caused at by a line source at Linearized inversion schemes like the distorted-wave Born and NewtonKantorovitch iterative procedures essentially proceed as follows. In step we first determine by taking the permittivity equal to i.e., the estimate available from the previous step. Subsequently, we obtain the next estimate by minimizing a cost function involving the difference between the simulated and known scattered fields on In the field computation, we march in angle in the first few iteration steps and in contrast once the inversion begins to converge. Nonlinear optimization: the availability of the full scattering matrix also offers the possibility to overcome some of the limitations of linearized schemes. The idea is to replace the determination of by a line search in a nonlinear optimization scheme. We search in the so-called gradient direction, which can be determined from the scattered field. There is no need of inverting matrices or solving adjoint problems; only straightforward integrations are involved. Until now, the expression for the gradient direction has been rarely used in practice, because it was too time consuming to compute the full scattering matrix with

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conventional solution methods. With the fast forward scheme described in this paper, that is no longer a problem. We march on in angle in the first two steps of each line search, and in search direction in subsequent steps. Embedding: the most recent idea is to consider objects in a more general environment. The CGFFT method can only be applied to objects in a homogeneous environment. Practical measurements, e.g. for biological tissue, must be carried out in a more complicated set-up like a water-filled metal cylinder. Such a set-up is presently being developed at CNRS/Supélec. An impression is given in Figure 3. In that case, a direct computation of the field is relatively complicated from a computational point of view. Hence, the idea is to use the procedure outlined above to determine the complete scattering response of the estimated object in a homogeneous environment, and subsequently use a spectral formulation to “embed” this object in the actual environment. That environment then needs to be characterized only once. This will make it feasible to solve the corresponding inverse problem by one of the approaches outlined above.

REFERENCES Van den Berg, P.M., 1985, Iterative schemes based on the minimization of the error in field problems, Electromagnetics, 5:237. Tijhuis, A.G. and Peng, Z.Q., 1991, Marching-on-in-frequency method for solving integral equations in transient electromagnetic scattering, IEE Proc. H, 138:347. Peng, Z.Q. and Tijhuis, A.G., 1993, Transient scattering by a lossy dielectric cylinder: marching-on-in-frequency approach, JEWA, 7:739. Zwamborn, A.P.M. and Van den Berg, P.M., 1994, Computation of electromagnetic fields inside strongly inhomogeneous objects by the weak conjugate gradient FFT method, JOSA A, 11:1414.

CORRELATION OF ANTENNA MEASUREMENTS USING THE OVERSAMPLED GABOR TRANSFORM

Benoît Fourestié and Zwi Altman France Telecom R&D 92794 Issy les Moulineaux FRANCE

INTRODUCTION Antenna measurements in amplitude and phase inside anechoic and semi-anechoic chambers have important applications in antenna characterization and electromagnetic compatibility. The analysis of antenna radiation characteristics, the characterization of electromagnetic properties of materials such as absorbers, and the analysis of the properties of the test site itself, in terms of spurious reflections and resonances, are only few examples. Typically, a measurement setup of two antennas facing each other is used and a network analyzer performs frequency measurements in amplitude and phase. The receiving antenna measures both the direct propagating wave component as well as the reflected components. By identifying the reflections from the measured signal, one can assess the influence of the test site. In addition, if one can suppress the reflected components, it is possible to retrieve the signal one would have measured in a perfect anechoic chamber. In most cases Fourier analysis fails to provide enough resolution to separate a signal from its propagating components due to the lack of frequency bandwidth, and other powerful signal processing techniques are required, such as super-resolution or multi-resolution techniques. In this work we propose to apply the Oversampled Gabor Transform (OGT) to perform a high resolution analysis of a measured discrete and complex signal. The Gabor Transorm (GT) (Gabor, 1946) is a gaussian windowed short time Fourier transform, and in certain cases it may suffer from instabilities. Recently, the GT has been reformulated using frame analysis for continuous and discrete signals (Zibulski and Zeevi, 1993, 1994, 1997), rendering the GT stable to the condition that it is oversampled. An important gain in resolution has been obtained by allowing the time-frequency windows to be located at small time and frequency intervals from each other, in an overlapping tile-like structure in the time-frequency coefficient domain. Although less resolution is obtained using the OGT analysis compared to other super-resolution techniques such as the Matrix Pencil (Fourestié et al 1999, 2000), the OGT provides a more robust decomposition of the signal into its propagating wave components.


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FRAMES AND OVERSAMPLED GABOR TRANSFORM The OGT is based on frame analysis which is an extension of the concept of bases (Daubechies, 1990). The definition of a frame is given as follows: A sequence in a Hilbert space H constitutes a frame if there exist positive numbers A and B called frame bounds, such that for all we have

were denotes the scalar product corresponding to the Hilbert space, For an orthonormal basis we have A = B = 1. Given a frame in a Hilbert space H, we define the frame operator S as follows

is the dual frame of with frame bounds represented by the frame as follows:

and

Unless the frame is a basis, the representation coefficients

Every

can be

are not unique. The

choice of the dual frame for computing the representation coefficients yields the minimal energy solution for the coefficients. In the context of antenna measurements, discrete representation of frames is of particular interest since measurement results are obtained in a discrete form. Consider a signal f of L samples which belongs to an L-dimensional space of complex vectors. The finite set of vectors is represented as the columns of a matrix X, and the inner product of the elements of and f by a vector d, If is a frame, then is nonsingular, and we can write the frame operator as

Next, we write (3) in matrix form. The dual frame vectors and the vector made of the coefficients

are the columns of can be written as

is the Moore-Penrose or the pseudo-inverse of X (Golub, and Van Loan, 1989) which is a simple and elegant way for finding the dual frame. The continuous GT of a function f(t) is given by

where

The decomposition (5) is stable for ab < 1, and we choose ab = p/q < 1, where p and q are prime integers. This choice is referred to as the oversampled scheme, or the OGT. Assume that constitutes a frame; then the coefficients can be found using the dual frame of which are also referred to as the bi-orthogonal functions

CORRELATION OF ANTENNA MEASUREMENTS

Having found the

we derive the coefficients

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using the following scalar product:

For the discrete OGT we consider a discrete L-periodic signal f. Let M and N be two divisors of L satisfying L = N ’M = M’N, with M, N, M’ and N’ positive integers. We define the discrete Gabor representation of f(i)

with the discrete Gaussian window function

As in the continuous case, we write and consider the oversampled case with for which the Gabor representation is stable. A small value for N’ is translated into oversampling in the time domain, with big overlapping of adjacent Gaussian windows. Similarly, a large value for N corresponds to a big overlapping of the windows in the frequency domain. The expansion coefficients can be derived as follows:

where (11) can be calculated using a DFT-based algorithm. Equations (9-11) are formulated in the signal domain. The continuous and discrete OGT have first been formulated in the Zak Transform (ZT) domain (Zak, 1967) which is equivalent to the formulation in the signal domain. The ZT of a function f(t) is defined as follows:

with a fixed parameter Oversampling in the GT can considerably increase the resolution of the analyzed signal as explained below. To each coefficient corresponds a time-frequency window of size with constant area of The sides and are the root-mean-square radii of g in the time and frequency domain respectively. By oversampling one can locate the windows close to each other in an overlapping tile-like structure. It is then possible to set the window approximately at the center of any event in the time-frequency plane, such as a maximum of a function. Without oversampling, the maximum of a function may occur at the center of four adjacent windows; its energy will be split amongst the four windows, resulting in a very poor resolution. The application of the OGT to antenna measurements is described in the next section.

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ANTENNA MEASUREMENT ANALYSIS In this section we propose to perform a high resolution analysis by applying the OGT to antenna measurements in amplitude and phase inside a semi-anechoic chamber. The objective is to identify and suppress the reflected wave components in order to remove the influence of the test site from the measured results and to obtain the signal one would have measured in a folly anechoic chamber. The measurement setup consists of two log-periodic antennas with a nominal frequency bandwidth of 100-1,000 MHz in a semi-anechoic chamber. The two antennas are in horizontal polarization, 2 m above the ground and 3 m apart, and the corresponding parameter is measured using a network analyzer. The transmission coefficient is measured in the semi-anechoic chamber in the frequency range of 100-1,000 MHz and with a sampling step of Denote by the transmission coefficient in the same folly anechoic chamber which can be measured by adding absorbing materials on the ground. The propagation delay of the direct wave component decreases logarithmically with frequency. This behavior is due to the migration of the active zones of the log-periodic antennas towards their apexes with the increase of frequency. The propagation path for the direct wave component is illustrated in Figure 1, with the distances and defined in the Figure. The wave travels the distance inside the coaxial line or the waveguide portion of each antenna and the distance in free space. In a first order of approximation, and are logarithmically decreasing functions of frequency (Mittra, 1969).

The magnitude and phase of

are plotted in Figure 2. We can observe that the

phase period of increases with frequency. This behavior is due to the decrease of the propagation path and the propagation time between the two antennas with frequency.


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The complex signal is decomposed using the OGT and the results are presented in Figure 3. The decomposition is performed with M = 20 and N = 700, i.e. an oversampling factor of 20, to obtain the highest possible resolution in time. The width of the gaussian window is chosen with d = 100 samples. Two distinct strips in the timefrequency plane corresponding to the direct and reflected wave components are clearly identified above 300 MHz.

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At low frequencies the OGT coefficients are smeared on the time-frequency plane with insufficient resolution to separate out the propagating components. We now proceed and force to zero the coefficients corresponding to the reflected wave on the ground. Denote by the corresponding reconstructed signal.

The results obtained for the

reconstructed signal will be validated by comparing the reconstructed signal with measured in the fully anechoic chamber. The suppression of the reflected component is performed using the following automatic procedure: be the vector formed by the magnitudes of coefficients for a (i) Let given m, i.e., a given frequency location and of the two peaks of corresponding to the (ii) Calculate the indices direct and reflected components (iii) Find the index of the minimum of between and

(iv) Force coefficients

to zero for It is assumed in (iv) that the width of the reflected component does not exceed 20 samples over the entire frequency range. In Figure 4 the results for the decomposition of in the time-frequency plane is presented. For frequencies above 300 MHz, the contribution of the reflected wave component has been identified and removed using the automatic procedure.

The decomposition of the reference signal, this decomposition is very close to that of the signal

is shown in Figure 5. We can see that in Figure 4.


Finally, the signal

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is reconstructed in the signal domain from the time-frequency

coefficient data and compared to in the frequency range of 300-1,000 MHz (Figure 6). We can see that the strong oscillations of more than 7 dB in magnitude (Fig. 2) which correspond to the interference pattern of the direct and reflected wave have practically disappeared. The difference between the two signals is less than 1.3 dB, with an average of 0.44 dB, and a standard deviation of 0.33 dB. Since the correction process is fully automatic, it is possible to know the frequency range of validity of this approach, which makes it a very robust technique for antenna measurement analysis.

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CONCLUSION The OGT has been successfully applied to the analysis of antenna measurements in amplitude and phase in the frequency domain. The measured signal can be easily processed in the transform, time-frequency coefficient plane, and its reconstruction from the timefrequency data is straightforward. The OGT allows to perform a robust and high resolution analysis of the measured signal leading to the identification and separation of its propagating wave constituents. The component reflected on the ground has been systematically identified and removed to retrieve measurements performed in a fully anechoic chamber. It is interesting to note that the difference between the reconstructed signal and the signal measured in the anechoic chamber remains less than 1 dB above 350 MHz, i.e. in the major part of the frequency bandwidth under consideration. Since a difference of 1 dB corresponds to the standard criterion for Measurement Test Site equivalence, the proposed method can be used to correlate measurements at different test sites. The OGT seems promising in the context of antenna testing, site characterization and, more generally, signal component discrimination.

REFERENCES Daubechies, I., 1990, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36:961. Fourestié, B., Altman, Z., and Kanda, M., 2000, Efficient detection of resonances in anechoic chambers using the Matrix Pencil Method, IEEE Trans. Electromagn. Compat., 42:1. Fourestié, B., Altman, Z., Wiart,J., and Azoulay A., 1999, On the use of the Matrix Pencil method to correlate measurements at different test sites, IEEE Trans. Antennas Propagat., 47:1569. Gabor, D., 1946, Theory of communication, J. Inst. Elec. Eng., 93:429. Golub, G. H., and Van Loan, C. F., 1989, Matrix Computation, Hopkins University Press, Baltimore. Zak, J., 1967, Finite translations in solid state physics, Phys. Rev. Lett., 19:1385. Zibulski, M. and Zeevi, Y.Y., 1994, Frame analysis of the discrete Gabor-scheme, IEEE Trans. Signal Processing, 42:942. Zibulski, M. and Zeevi, Y.Y., 1997, Discrete multiwindow Gabor-Type transforms, IEEE Trans. Signal Processing, 45:1428. Zibulski, M., and Zeevi, Y.Y., 1993, Oversampling in the Gabor scheme, IEEE Trans. Signal Processing, 41:2679. Mittra, R., Log-Periodic Antennas, 1969, chap. 22, in: Collin et al, ed., Antenna Theory, Part 2, 22., McGraw-Hill.

ON A RATIONAL MODEL INTERPOLATION TECHNIQUE OF ULTRA-WIDEBAND SIGNALS

N. H. Younan, C.D. Taylor, and J. Gu Department of Electrical and Computer Engineering Mississippi State University Mississippi State, MS 39762

INTRODUCTION Applications of ultra-wideband pulsed RF energy is increasing. These applications range from the treatment of cancer and other maladies to non-destructive testing, remote sensing, and ultra-wideband weapons. Monitoring the time history and the energy spectrum of ultrawideband pulses pushes the state of the art in electronic instrumentation. This paper examines how limited data can be used to infer data outside the range of the instrumentation. Generally, discrete-time or discrete-frequency data are obtained via uniform sampling. In some cases, this may not be practical. For instance, logarithmic spacing is often used to limit the samples to a practical number in frequency domain data, collected for frequencies over several decades. Moreover, for high frequency measurements, gaps in the frequency domain data occur as a result of skipping certain frequency bands. Accordingly, interpolation is used to characterize the frequency domain response. Interpolation techniques require a data model to restore the unknown data samples. Often, the model is quite simple. Consequently, interpolation is related to function approximation. In general, interpolation schemes presume some degree of smoothness for the function to be interpolated. However, this may not be valid for noisy data. Moreover, if the interpolating function is fitted to the known data points with additive noise, significant interpolation error may occur. Various interpolation schemes have been used to restore unknown data samples. However, most of them have been performed on time-series data using models like AR, MA and ARMA1-3. Techniques for interpolating complex frequency data have received little attention. Traditional techniques, such as linear, cubic spline, and Lagrange interpolation, have been shown to be data dependent and are generally not satisfactory. They become highly unstable for data corrupted with noise. Linear interpolation is perhaps the most widely applied and the simplest technique. However, when applied to noisy data, inaccurate results are obtained when a spectral peak or null occurs between samples. The Lagrange technique uses a polynomial function to


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interpolate between data samples, it causes instability at high order degree even for data without noise; therefore, it is not suitable for noisy data4. The Cubic Spline interpolation uses a polynomial fit between each pair of known data points. The coefficients are determined nonlocally. For noise-free data, splines tend to be more stable than polynomials, with less possibility of wild oscillation between the known data points. However, this technique becomes highly unstable for data corrupted with noise5. A regressive rational function interpolation for noisy data is presented. This technique incorporates the use of the singular value decomposition method and a statistical measure of goodness-of-fit to obtain the best estimate of the model coefficients. Additional features include a self-tuning ability to obtain an optimum model order and noise reduction. Compared to traditional interpolation techniques, this method is shown to be more robust. Results are obtained for noisy data with low signal-to-noise ratios to ascertain the validity and robustness of the proposed technique.

REGRESSIVE FIT MODEL INTERPOLATION In general, rational functions interpolation methods have shown superiority over polynomials because of their ability to model functions with poles. Accordingly, rational functions can be viewed as ARMA type models for frequency domain applications. An ARMA model is described as6,

where,

equation (1), represents the actual received frequency data samples and the right-hand side represents the ARMA model used to fit the corresponding frequency data. In traditional interpolation techniques, the coefficients are usually determined by forcing the interpolating function to match the known samples7. For the technique presented here, a rational function interpolation is incorporated with a least-square method that uses a regressive fit to obtain the rational model parameters8. Once the coefficients are known, then the complete frequency response can be reconstructed at any frequency value. In general, the square error is defined as,

Where N is the number of data samples. Although this error criterion can be used to calculate the coefficients, it has some deficiencies. First, if the frequency response has to be determined for frequencies extending several decades, the lower frequencies can not be fitted well since they have very little influence. Second, if has poles in the complex S-plane such that could vary widely throughout the experimental points, large errors would be

A RATIONAL MODEL INTERPOLATION TECHNIQUE OF UWB SIGNALS

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introduced. The stated deficiencies can be overcome by an iterative procedure where, at the stage of iteration, the square error is modified to9,

where the model coefficients, are obtained by setting the derivative of with respect to the coefficients to zero. This yields a set of simultaneous equations of the form of Ax = B. Here A and B are a 2n×2n matrix and a n×1 vector formed from the known frequency data and x consists of the model coefficients. In solving the linear algebraic equations, the singular value decomposition (SVD) technique is incorporated to eliminate the noise components. At each step, the SVD is used to solve for the coefficients via the leastsquare technique and the singular values are truncated according to the Akaike Information Criterion (AIC)10, i.e.,

Here K is the rank of the matrix A and corresponds to the i th eigenvalue. The optimum order is chosen as the value of m that minimizes the AIC. SIMULATION RESULTS To determine the limitations and accuracy of the presented technique, a damped sinusoid with additive white-Gaussian noise is used to function as the received signal, i.e.,

Here, L is the number of sinusoids, T is the sampling period, and w(n) is the noise sequence. The noise level is quantified by the signal noise ratio (SNR), which is defined as

Fast Fourier Transform (FFT) is then used to obtain the frequency samples. To illustrate the validity of the technique, a signal with a spectral peak at 20 MHz is generated with and The sampling rate is chosen to be A worst case data gap, namely points centered around the resonant peak, is considered. Figures 1 and 2 ilustrates the magnitude and phase plots of the noisy simulated data with SNR = 20 dB. The resulting magnitude and phase plots obtained by applying the regressive fit interpolation to the one peak noisy signal with a 3-point gap together with the

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magnitude and phase plots of the noise free signal are shown in Figures 3 and 4. These figures indicate how well the presented technique performed, i.e., the reconstructed signal matches well with the original signal and the noise component is successfully suppressed. Similar results are obtained for the one peak noisy signal with SNR=10dB. The corresponding magnitude and phase plots comparisons are illustrated in Figures 5 and 6 respectively. Note that the relative error for the 20 dB case is approximately 9% compared to 57% and 33% relative errors obtained from Lagrange and Spline interpolation schemes. In addition, a slight degradation in performance is obtained for the 10 dB case, but it is still reasonably acceptable.


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To further demonstrate the applicability of the interpolation technique, simulated data of length of 1024 samples with two spectral peak at 10MHz and 30MHz are. In this case, and and the

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sampling rate is chosen to be 600 MHz. Similar to the single peak signal, a 3-point gap around each spectral peak is considered. The restored magnitude and phase plots for the 20 dB case are shown in Figures 7 and 8 respectively. It is clearly seen that the proposed technique performs well.


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CONCLUSIONS A regressive rational function interpolation technique for noisy frequency domain data is presented. This technique is based on using a modified ARMA model to fit the transfer function of the underlying noisy spectral data. The coefficients of the model are then calculated based on minimizing a weighted least square criterion iteratively. This procedure incorporates the singular value decomposition technique to accurately estimate the order of the model and at the same time to suppress the noise components, which corrupt the actual data samples. Simulation results are presented to demonstrate the applicability of the interpolation technique.

REFERENCES l.

R. Steele and F. Benjamin, “Sample Reduction and Subsequent Adaptive Interpolation of Speech Signals,” The BELL System Technical Journal, Vol. 62, No. 6, July-August 1983. 2. Ronald E. Crochiere and Lawrence R. Rabiner, “Interpolation and Decimation of Digital Signals: A Tutorial Review”, Proceeding of The IEEE, Vol. 69, No. 3, March 1981. 3. A.J.E.M. Janssen, R.N.J. Veldhuis, and L.B. Vries, “ Adaptive interpolation of Discrete-Time Signals That Can Be Modeled as Auto-regressive Processes,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-34, No. 2, April 1986. 4. Clayborne D. Taylor, Nicholas H. Younan, and Shinfeng D. Lin, “EMP Data Analysis and Software Development”, Final Report for TRW Inc., Space & Defense Sector, January 1990. 5. Richard L. Burden, J. Douglas Faires, Numerical Analysis, six edition, 1996. 6. S.L. Marple, Jr., Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, New jersey, 1987. 7. C. Daniel and F.S. Wood, Fitting Equations to Data, Wiley-Interscience, 1971. 8. C.K. Sanathanan and J. Koerner, “Transfer Function Synthesis as a Ratio of two Complex Polynomials,” IEEE Transcations on Automatic Controls, pp 56-58, January 1963. 9. A.W.M.v.d. Euden and G.A.L. Leenknegt,” Design of Optimal IIR Filters with Arbitrary Amplitude and Phase Requirements,” Proceedings EUSIPCO, Signal Processing III: theory and Applications, 1986. 10. Steven M. Kay, Modern Spectral Estimation: Theory & Application, Prentice Hall, 1988.

FULL-WAVE SOLUTION OF THE PROPAGATION OF GENERALLY SHAPED IMPULSES AND WIDE BAND APPLICATION IN ANISOTROPIC PLASMAS

Orsolya E. Ferencz1 and Csaba Ferencz1 1 Eötvös University, Dept. Geophys., Space Research Group H-1117 Budapest, Pázmány P.s. 1/A., Hungary E-mail: [email protected]

INTRODUCTION One of the most interesting wave propagational problems is the investigation of arbitrarily shaped signals, i.e. impulses in different media. This problem is actual, because in space research and applications the signals of natural origin detected by space vehicles or ground stations are generated by general shaped sources (e.g. lightening strokes, atmospheric currents, seismic events), or the artificial signals used in man made applications must have a general form with extra-wide frequency band or these signals are very short impulses. The most common media in space applications are the anisotropic plasmas. Therefore the paper deals with propagation of impulses in inhomogeneous anisotropic plasma-models using a plane-wave, applying of which the solution of Maxwell's equations can be obtained in accurate and closed formed time-space function of the impulse-type signal, without any monochromatic or quasi-monochromatic assumption or restriction. The example, in the case of which the model is presented, is the phenomena of magnetospheric (upper-atmospheric) signals excited by lightening stroke impulses and detected in ELF-VLF and HF-UHF bands ("whistlers" and related phenomena). The propagation of impulses and other generally shaped signals in plasmas are commonly described by approximate theories based on the monochromatic solution of Maxwell's equations in the given medium (dielectric tensor, refractive index, AppletonHartree formula, wave packet, group velocity calculations etc.) – e.g. by Budden (1961) or Walker (1993). These descriptions of the wave-propagational situation do not always result in a satisfactory interpretation of the electromagnetic phenomena caused by transient effects, e.g. the physical, theoretical explanation of some observations. The real character of the phenomena contains definite starting point according to time and/or space. The problem is similar to the well known one in the network analysis as transient effects. In these cases the introduction of functionals becomes necessary.


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The solving process is fully analytical, starts directly from the Maxwell's equations and applies the multi-dimensional Laplace-transformation managing the transient character of the signals. The main part of the solution uses the Method of Inhomogeneous Basic Modes (MIBM) deriving the propagating field strength components of the electromagnetic signal. A great advantage of this method is hidden in the fact, that the "transfer function" of the medium – the system-answer given for a Dirac-delta functional – can be available, which makes possible to develope a very flexible model for linear propagational problems, which type of models are often used in network-engineering problems. THEORY The task is to derive a general-shaped, real solution of the Maxwell's equtations, using the MIBM. The main steps of the solution process are presented below. Let the medium be considered in which the medium-parameters suddenly change at some open or closed surfaces having no intersection (Figure 1). The medium-parameters within the volumes are described by continuous functions joining each other with jumps at the surfaces where the medium is characterised by functionals (distributions). The signal in each volume will be disintegrated into n modes. During the derivation process all of the possible modes must be determined in every volume. Let the form of the solution looked for in be the following

where means the components of the electromagnetic field. Furthermore, let the 1(x) Heaviside (or unit step) and the Dirac-delta distributions (functionals) be introduced. denotes the distribution, the value of that varies from 0 to 1 at the surface i.e. where the vector represents the the parametric equation of the surface Using the functionals it is possible to create the following distributions (or "gate-functions") from them

which has 1 value in the region between surfaces everywhere out of that (Figure 2).

and

while it has 0 value

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According to MIBM the complete solution can be obtained using equations (1) and (2) as

where M is the number of the continuous regions. Using this form of solution in the Maxwell's equations the derivative of the "gate function" will appear, which is

where is definitely the normal vector of surface, points outward (to in the present case). Substituting (3) into the Maxwell's equations, they can be automatically decomposed into two groups. One group is valid within the regions and the other has to be fulfilled at the surfaces In the first step the equations valid in the volumes will be solved. Applying the common notations of electrodynamics this equation-system is

where and are the source current and charge densities in the given region, if they exist there. The solutions of eqations (5) can be obtained for each layer in different ways, e.g. using the Laplace-transformation. If the medium is inhomogeneous in these regions known methods of solutions can be used, see Ferencz (1978). If the application of the (generalised) W.K.B. philosophy is necessary, one can apply it at the end of the solving process just before the inverse-Laplace transformation. Using the derivated form of the Laplace-transformed amplitudes a W.K.B.-type relation can be defined between the electric and magnetic amplitudes. The other group of the equations is valid at the boundary surfaces delivering the connection among the regions. This is called coupling equation system

In equation system (6) it is necessary to use every modes existing in the volumes Solving (6), the complete solution can be obtained.

SOLUTION OF THE PROBLEM IN MAGNETOIONIC MEDIA As an example of application the propagation of an electromagnetic signal in an anisotropic plasma and generated by a source current existing beyond the plasma is presented here. This is the case of whistlers and related phenomena. The whistlers are remarkable bursts of electromagnetic energy in ELF~VLF bands produced by ordinary lightning. Related phenomena are energy bursts appearing in higher frequency bands with special character, e.g. the Transionospheric Pulse Pairs (TiPP's) in the 20~100 MHz bands in the Earth's atmosphere. Using the traditional magneto-ionic theory based on the examination of harmonic signals (e.g. Budden, 1961; Walker 1993), numerous hypotheses were born regarding the generation and propagation mechanism of the whistler signals, but an exact time-space waveform of the e.m. field was missed up to now. Because of the

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contradictions between these monochromatic-type model-calculations and the physical reality, many ideas and explanations of the measured effects were fundamentally false. Besides this no acceptable idea was born up to now about the generation mechanism of the TiPP's. This problem can be solved deriving a real full wave solution using the theory summerised above.

For the investigation of this situation one can use a relatively simple model structure (Figure 3). The main atmospheric payers are modelled in Cartesian coordinate system by two volumes, i.e. by two half-spaces and or medium and medium – and the plane-surface between these regions at the place of Medium is a usually vacuum-type, infinite half-space modelling the Earth's surface - ionosphere waveguide, where the arbitrarily shaped source current density generated by the lightning discharge impulse arises in the region It is important to emphasise that there are not any 'traditional' wave-packet models or other monochromatic-type approximations in the model. The shape of in time and in space is general and has a switch-in character. The starting time of the excitation is Medium is an inhomogeneous or homogeneous, anisotropic plasma in the infinite half-space, as a usual model of the magnetosphere. In this half-space no source current density exists. The plasma can be considered to be lossless or lossy (characterised by collision frequency), cool, tempered, anisotropic, magnetised, time-invariant and linear electron-plasma or a plasma-model containing arbitrary ion-distributions too. The superimposed (geomagnetic) field can be parallel to the direction of propagation (i.e. strictly longitudinal propagation; ), or not (oblique propagation; ). The different plasma-effects are included in the motion equtions of the electrons and ions. The supposed wave-pattern of the solution is a plane-wave (i.e. plane-signal) having definite temporal and spatial starting points. At the begining of the deriving process let the Laplace-transformation be applied for the equations (5) written in ( t) domain in medium and respectively. The solution in medium is not too complicated and the result contains the source current density The next step is the inverse Laplace-transformation, which produces the space-time functions of exsisting modes in medium – The solution process in the


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medium is more complicated. If the propagation is oblique, i.e. three independent variables appear (x, y, t) which turn into transformed variables (p, l, s) by the Laplacetrasformation. If the medium is inhomogeneous weakly, the conditions of the W.K.B. approximation are given and the amplitudes of the exsisting modes can be correct by a generalised W.K.B. philosophy: for example, if the pole of an e.m. mode is after an transformation, than the W.K.B. amplitude of the mode is

The inverse Laplace-transformation can be executed in the way known from the engineering applications. In the last step the equation-system (6) will be solved at the boundary plane for all existing e.m. modes. At the end of these investigations it is possible to apply the same signal analysing methods (FFT, curve fitting, matched filtering etc.) for the computed signals, for the computed field strengths, which are commonly used for the signals measured on board of satellites or at terrestrial stations. Further, as an example of the derived solutions, the strictly longitudinal modes will be presented here, propagating in an inhomogeneous, multicomponent plasma along the magnetic field. In every cases two modes propagate into one direction, in this case into the +x direction. Here only one electric component of these two modes is shown:

where

is the wave impedance for vacuum, n = 1, 2 and

and are the plasma frequency of the electron and the different ions in the plasma respectively, and are the giro-frequency of the electron and the different ions respectively; if n=1, the upper signs are valid, if n=2, the lower signs are valid in (9); and the excitation defines the quantity as

The excitation is a Dirac-delta in order to get the transfer function of the medium (of the whole propagational situation), and in this case

Another useful excitation is a simple rectangular impulse at the place x = x *, i.e.

The relation (10) determines the excitational effect independently of the shape of source signals.

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APPLICATION OF THE RESULTS It is a simple fact that using these computed signals it is possible to calibrate and verify the analytical methods, i.e. softwares and systems, used in wave analyses and research. Processing the transfer functions of various propagational models and calculated signals generated by different excitations the real answers and artifacts can be seen, produced by the wave-processing method itself. This is an important new possibility in method-calibration. Another new possibility in wave-investigations that it is possible to separate the propagational and excitational effects. Comparing the measured signals to the transfer functions and signals computed by different excitations, it can be determine the effects of the propagatinal path and the possible (probable) excitation (Figure 4).

Another example: Seeing the relation (12) it is clear that in the case of a rectangularimpulse excitation the signal contains amplitude minima which are strictly periodic on the frequency-axis and the place of the first minimum is If the excitation is a double Dirac-delta, the wave-pattern will be similar, but the places of this strictly periodic amplitude minima will be different. However, if the signal is a resultant of two transfer functions originating from the same source but propagating on slightly different paths, the resultant signal contains amplitude minima which are non-periodic on the frequency-axis. Using the oblique propagational solution one can compute a branch of signals (transfer functions) originating from the same source but propagating with different angles to the superimposed magnetic field. In some cases on board of satellites it was registrated whistler branches with nearly the same character (Figure 5). The satellite registrated these signals on a small sequence of the orbit, nearly at one given place. Therefore these signals were at a given time at a given region and could not propagate longitudinally along different earth magnetic field lines, which was supposed in some former theories. This investigations open the way for new ideas in ELF~VLF magnetospheric propagation.


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The application of these solutions in the wave-investigations is a new an effective way in the research (in space research) as well, as in practical applications.

RESULTS IN EXTREME WIDE-BAND IMPULSE PROPAGATION Using the results presented by equation (8) it was demonstrated (Ferencz, 1999) that the first mode represents the electron-whistler and the second mode belongs to the ionwhistlers in the magnetospheric propagation in the ELF~VLF bands. (This fact gives an essentially new explanation of ion-whistler generation.) However, expanding the frequency bands of investigation up to the high frenquencies (e.g. up to 100 MHz) it can be found a consistent and common explanation of the electron- and ion-whistlers, the Faraday rotation and the TiPP-events. The TiPP events have been registrated by satellites in high frequency bands (Rodger, 1999). In every cases a pair of dispersed signals appear on the FFT dynamic spectrum and both has a doubled fine structure. No acceptable interpretation of this phenomenon was presented up to now. In order to give explanation of these events, the solutions (8) in the extreme wide band from ELF to 100 MHz can be computed, using a rectangular type impulse excitation, where the length of this impulse in time is equal to the time difference of the main signal pairs in the TiPP event. The two exsisting modes will propagate with different time delay. This phenomenon produces the electron- and ionwhistlers in the lower frequencies, and this will appear in the higher frequency-bands above the plasma and giro-frequencies of the medium as a doubleness inside of the first and second main part of the TiPP event. This small time-delay between this two modes produces the Faraday-rotation if the propagating signal is monochromatic (Figure 6). However, seeing the evolution of a signal excited by a rectangular impulse it can be demonstrated that an essential doubleness appear in the signal during the propagation (Figure 7a). This inherent structure of the signal will appear in the high frequency bands in the dynamic spectrum as twin pair of signals with a time difference equal to the temporal length of the impulse (Figure 7b). The time delay of the two modes in the high frequency-band is much smaller than the length of the source impulse. Therefore the upward directed edge of the impulse causes the

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first main part of a TiPP with an inner doubleness and the downward directed edge of this impulse-origined signal forms the second part of these twins.

A final conclusion is that this new method of the solution of Maxwell's equations and the solutions itselfs form very effective and productive way in wave-propagational research and applications. REFERENCES Budden, K.G., 1961, Radio Waves in the Ionosphere, Cambridge Univ. Press, London, E. Ferencz, O., 1999, Electromagnetic wave propagation in different terrestrial atmospheric models, Ph.D. Thesis, Budapest University of Technology and Economics, Budapest, Hungary. E. Ferencz, O., 1999, Origin of ion-cyclotron whistlers; Proc. of XXVIth Gen. Assembly of URSI, pp.534, H4.P9, 13-21 Aug. 1999, Toronto, Canada. Ferencz, Cs., 1978, Electromagnetic wave propagation in inhomogeneous media: Method of inhomogeneous basic modes, Acta Techn. Ac. Sci. H., 86(1-2), 79-92. Rodger, C.J., 1999, Red sprites, upward lightning, and VLF perturbations, Reviews Geophys., 37, 3, 317. Walker, A.D.M., 1993, Plasma Waves in the Magnetosphere, Springer-Verlag, Berlin.

ASYMPTOTIC DESCRIPTION OF ULTRAWIDEBAND, ULTRASHORT PULSED ELECTROMAGNETIC BEAM FIELD PROPAGATION IN A DISPERSIVE, ATTENUATIVE MEDIUM

Kurt Edmund Oughstun College of Engineering & Mathematics University of Vermont Burlington, Vermont 05405-0156 [email protected]

INTRODUCTION

The asymptotic description of the coupled spatio-temporal dynamics of an ultrawideband, ultrashort pulsed electromagnetic beam field that is propagating in a dispersive, attenuative material occupying the halfspace is obtained from the angular spectrum of plane waves the paraxial representation1. The analysis leads to a comparison between approximation and the generalized Sherman expansion2,3 of that exact integral representation. Both approaches rely upon an expansion of some aspect of the propagation kernel that appears in the angular spectrum representation, each approach differing from the other in its particular method of expansion. The generalized Sherman expansion provides a spatial series representation of a pulsed, source-free electromagnetic beam field. This spatial series representation explicitly displays the temporal evolution of the pulsed beam wave packet throughout the halfspace through a single contour integral that is of the same form as that obtained in the Fourier-Laplace integral description of a pulsed plane wave field that is propagating in the positive z-direction in the dispersive medium. In particular, the spatio-temporal pulsed beam-field evolution is shown to depend upon the transverse spatial position in the propagated field through the spatial derivatives of the initial field boundary values at the plane As a consequence, it is found that the Sommerfeld and Brillouin precursor fields, which are a characteristic of the dispersive material, cause the ultrashort pulsed beam field to break up into several localized subpulses which travel at their own characteristic velocity through the dispersive material.


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ANGULAR SPECTRUM REPRESENTATION

Consider the evolution of a freely-propagating electromagnetic field in the half-space that is occupied by a homogeneous, isotropic, locally linear, temporally dispersive material. The term ‘freelypropagating’ is used here to indicate that there are no external sources for the field throughout this half-space, the source then residing somewhere in the half-space It is unnecessary to know what the source is provided that either the electric field vector or the magnetic field vector is a known function of time and the transverse position vector in the plane The freely-propagating electromagnetic field throughout the half-space is then given by the angular spectrum of plane waves representation1

for both the electric and magnetic field vectors. It is assumed here that the two-dimensional transverse spatial Fourier transform and temporal Fourier-Laplace transform of each field vector at the plane exist, as given by

where is the transverse wavevector. If the initial time dependence of the field vectors at the plane is such that for all for some finite value of then the time-frequency transform pair relation appearing in Eq. (2) is a Laplace transformation and the contour of integration is the straight-line path with the real constant a being greater than the abscissa of absolute convergence1,4 for the initial time behavior of the field and with varying from 0 to if not, then it is a Fourier transform. The spatio-temporal spectra of the electric and magnetic field vectors at the plane satisfy the transversality conditions1

so that Here is the complex wavevector for propagation into the positive half-space with the associated complex wavenumber given by

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where and are the plane wave propagation and attenuation factors, respectively, and where is defined as the principal branch of the expression

with

Here µ is the constant magnetic permeability and

is the complex permittivity of the dispersive medium with frequencydependent dielectric permittivity and electric conductivity Notice that both cgs and MKS units are employed here through the use of a conversion factor that appears in the double brackets in each equation affected. If this factor is included in the equation it is then in cgs units provided that while if this factor is omitted the equation is then in MKS units. If no such factor appears, then the equation is correct in both systems of units. The integrand appearing in the angular spectrum representation (1), namely

where with corresponds to a time-harmonic plane wave field that is propagating away from the plane at each angular frequency and transverse wavevector that is present in the initial spectra of the electric and magnetic field vectors at that plane with just one significant difference: the wavevector components and are always real-valued and independent of while is, in general, complex-valued. Hence, each spectral plane wave component appearing in the angular spectrum representation (1) is attenuated in the z-direction alone. The angular spectrum representation (1) explicitly displays the manner in which the temporal and spatial dynamics of the field are coupled. Even for the special case of an initial field whose temporal and spatial properties are separable as with spatiotemporal spectrum the temporal and spatial dynamics remain coupled through the complex factor as

K. E. OUGHSTUN

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For a monochromatic field of angular frequency angular spectrum representation of the propagated field becomes

the

for all

Real Direction Cosine Form of the Angular Spectrum Representation The plane wave propagation factor appearing in the angular spectrum representation (1) may be cast into a more geometric form under the change of variable defined by the set of relations3

where

with magnitude

and phase so that the direction cosines p and q are real-valued. With these substitutions, Eq. (5) yields

and the angular spectrum of plane waves representation (1) becomes3

Here

and

for all

Paraxial Approximation of the Angular Spectrum Representation In the paraxial approximation one expands in a binomial series for small with3 and retains only the first two terms, with the result

With this substitution, the angular spectrum representation given in Eq. (12) becomes

for all


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GENERALIZED SHERMAN EXPANSION

Wave fields in attenuative media that contain only inhomogeneous plane wave components with for all in their angular spectrum representation are called source-free wave fields3. In lossless media, these inhomogeneous plane wave components become homogeneous plane wave components with it is for this case that source-free fields were first defined by Sherman2. In this approach, the exponential propagation kernel

is replaced by its Taylor series expansion

where the complex variables and been introduced for notational convenience, and where

have

With this substitution in Eq. (12), one obtains the spatial 3series representation of a pulsed, source-free electromagnetic beam field as

where

Here

and

for integer values of m > 0. The generalized Sherman expansion given in Eq. (18) explicitly displays the temporal evolution of an electromagnetic beam field through a single contour integral that is of the same general form as that obtained in the description of a pulsed plane wave field that is propagating in the positive z-direction in the dispersive medium1. In particular, the temporal pulse evolution is seen to be dependent upon the transverse position in the field through the even-ordered spatial derivatives of the transverse beam profile at the plane The

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remaining contour integral may then be directly evaluated as an asymptotic approximation using well-defined saddle point methods1.

NUMERICAL RESULTS

As a specific example of a dispersive, attenuative material, consider triply-distilled water. The Rocard-Powles model5 of the frequency dispersion of the dielectric permittivity of this material results in the frequency dispersion of the complex wavenumber that is presented in Fig. 1. At the paraxial approximation of the complex quantity as given in Eq. (13), is found to be quite accurate for as seen in Fig. 2. In contrast with the lossless case where there is a discontinuous change from the lossless, homogeneous wave region to the evanescent wave region notice the smooth transition in the exact behavior of from the “low-loss” region to the “high-loss” region where at A comparison of the exact behavior of the real and imaginary parts of the propagation kernel G(p,q, ) with both the paraxial approximation (13) for and the quadratic approximation

of the Taylor series expansion (16) is presented in Fig. 3. The paraxial approximation of the propagation kernel is seen to remain reasonably accurate for all while the quadratic approximation (21) of the Taylor series expansion is accurate only for Unfortunately, this series expansion converges very slowly so that the


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inclusion of each higher-order term only provides a very slight improvement in the accuracy of the resultant approximation over a slightly expanded (p,q) domain.

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Because of the increased material absorption as the frequency is increased further into the absorption band of triply-distilled water (the peak of which occurs at approximately 10THz], improved results are obtained for the paraxial approximation at The paraxial approximation (13) of is now accurate for all as seen in Fig. 4. Finally, the paraxial approximation of the propagation kernel G(p,q, ) is now reasonably accurate for all while the quadratic approximation (21) of the Taylor series expansion (16) is still accurate only over the very limited domain as is clearly evident in Fig. 5.

SUMMARY AND CONCLUSIONS

The results presented here have shown that the Sherman expansion of the angular spectrum representation of a pulsed 3,electromagnetic beam field, while being exact for a source-free wave field 4, is unfortunately of minimal practical use for computational purposes because of its slowly convergent character that only decreases as the propagation distance increases above zero. Nevertheless, it may have some practical application for near-field calculations. The paraxial approximation, on the other hand, has been shown to improve in its accuracy as the material absorption increases. As a consequence, it provides a convenient method with which to evaluate the general features of ultrashort, ultrawideband pulsed beam field propagation in any causally dispersive material. For example, the spatial part of the paraxial approximation (14) of the angular spectrum representation may be directly evaluated for a Hermite-gaussian beam, the remaining contour integral over the angular frequency may then be evaluated using well-defined asymptotic techniques. Because of the


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precursor fields that are a characteristic of the particular dispersive medium1, the propagated ultrawideband, ultrashort pulsed beam field will break up into several spatio-temporally localized packets. For a Rocard-Powles (or Debye) model dielectric, the asymptotic description of the propagated beam field may be expressed as

as where Here describes the low-frequency Brillouin precursor portion of the propagated beam field, and describes the carrier frequency portion of the propagated wave field. Because of the lower frequency content of the Brillouin precursor, the space-time packet will have a larger diffractive spread than does the carrier signal packet For a single resonance Lorentz model dielectric an additional term due to the high-frequency Sommerfeld precursor appears in the asymptotic representation

as Because of its very high frequency content, this additional space-time packet would have near minimal diffractive spreading.

ACKNOWLEDGMENT

This research has been supported by the United States Air Force Office of Scientific Research Grant #F49620-94-1-0430.

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REFERENCES 1. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Springer-Verlag, Berlin (1994).

2. G C Sherman, Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves, J Opt Soc Am, 59:697 (1969).

3. K E Oughstun, The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuattve media, J Eur Opt Soc A, 7:1059(1998). 4. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941). 5. J. E. K. Laurens and K. E. Oughstun, Electromagnetic impulse response of triplydistilled water, in: Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum and J. Shiloh, eds., Plenum, New York (1999).

DISPERSION REDUCTION IN A COAXIAL TRANSMISSION LINE BEND BY A LAYERED APPROXIMATION OF A GRADED DIELECTRIC LENS

W. Scott Bigelow,1 Everett G. Farr,l William D. Prather,2 and Carl E. Baum2 1

Farr Research, Inc. 614 Paseo Del Mar NE Albuquerque, NM 87123

2

Air Force Research Laboratory / DEHP Directed Energy Directorate Kirtland AFB, NM 87117

INTRODUCTION A bend in an electrically large geometry poses a problem for high-voltage UWB pulse transmission. Dispersion is introduced by the dependence of propagation path length on the local radius of curvature within the bend. This dispersion increases the risetime and broadens the pulse width of the transmitted signal, thus degrading system bandwidth. This source of dispersion can be reduced by use of a gradient index lens to compensate for the electrical path length differences through the bend. This approach has been suggested in a number of papers which develop solutions for a TEM wave propagating in the in a cylindrically inhomogeneous dielectric (CID), with permittivity, proportional to in a cylindrical coordinate system [1, 2, 3, 4, 5, 6, and 7]. We take the particular form of the CID inhomogeneity as

where n is the refractive index, is the relative permittivity, and is the radius at which the index becomes unity. In such a perfect CID medium, the electrical path length and pulse transit time, through a bend, are independent of the radius. In [8 and 9], we described our approximate synthesis of a CID medium to compensate a strip transmission line bend. That experimental effort employed five coarsely graded layers of uniform dielectric materials to approximate a CID lens. The compensated bend achieved a 70 ps reduction in pulse risetime, a 30% improvement, when compared to an identical air-filled line. The performance of the strip line was limited by propagation of fringe fields ahead of and behind the main pulse. Here, we summarize our latest implementation of a layered approximation of a graded dielectric lens, this time, employing square coaxial hardware [10]. Use of a coaxial geometry eliminates the fringe fields that plagued the strip line results. Although our design approach for a layered coaxial bend employed a heuristic technique, we subsequently perUltra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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formed a global optimization, which demonstrated that our design was very nearly optimal. To guide the design effort, we obtained the line impedance from a finite element model of the cross section [6]. In contrast to the modest dispersion reduction achieved with the strip line bend, compensation by a layered approximation of a CID medium dramatically improved impulse transmission through the coaxial transmission line bend.

LAYERED DIELECTRIC LENS DESIGN We began the design of a transmission line bend, approximately compensated by graded layers of dielectric material, by assuming a cross section and maximum radius of curvature for a 90° bend. To simplify manufacture of the layered dielectric material, we chose a square cross section, which we assumed to be 5.08 cm on each side. For consistency with an earlier design approach, we chose 14.68 cm as the outer radius of curvature. With these choices, the centerline path length of the bend was about 19 cm. Next, we specified a performance goal to use in assessing the need for bend compensation. Our criterion was that the signal transit time through the bend should not vary by more than about 100 ps. The transit time at any radius is

where is the bend angle in radians and c is the speed of light. For an uncompensated bend (constant index of refraction), the difference in transit time between the inner and outer radii of curvature is completely determined by the width of the bend and by the index of refraction. For an air-filled, 5.08 cm wide, 90° bend, that time difference is 266 ps. It increases linearly with the index of refraction. Since the 100 ps limit is exceeded for the proposed bend, even in air, compensation is required. To begin the design of a compensating dielectric laminate, we needed to identify the materials to be used. We selected a series of low-loss, cross-linked polybutyldiene resins, commercially produced at half-integer increments of the dielectric constant, from 3.0 to 10.0. Although the materials can be readily machined with sharp carbide tools, they are somewhat brittle, especially at the higher dielectric constants. The manufacturer recommended extreme care if layers thinner than about 0.64 cm were to be machined. Therefore, we chose to avoid layers thinner than this limit. If all layers were of this minimal thickness, eight would be required to fill the radial extent of the bend. Since manufacturing cost increases with layer count, we sought to use the minimum number of layers capable of meeting the assumed 100 ps transit time variation limit. In a layered bend, there are two sources of transit time variation. Across the width of each layer, the transit time changes linearly because the dielectric constant is uniform, while the radius of curvature changes. Across adjacent layers, the transit time changes abruptly because the dielectric constant is discontinuous at the layer interface. In designing the bend, we applied the same 100 ps limit to both sources of transit time variation. Finally, we sought a design in which the laminate of uniform dielectric layers would closely approximate the inverse square relationship between the dielectric constant and radius of curvature that characterizes a CID medium. To this end, we attempted to place the midpoint of each layer as close as possible to the correct radius for the dielectric constant of the material comprising that layer. A heuristic algorithm used to design the dielectric laminate, subject to the constraints outlined above, has been described in detail in [10]. Application of that algorithm led to a six-layer design employing materials with dielectric constants of 3.0, 3.5, 4.0, 4.5, 5.0, and 6.0. The layer-thickness-weighted average dielectric constant for the laminate is 4.1. As the plots in Figure 1 demonstrate, the design represents a good approximation to a CID profile

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699

with (9.636”). In the plot of transit time variation, note that is defined as the radius-dependent difference between the predicted transit time and the constant transit time through an ideal, 90° CID bend. Thus, in the plot, (2) has been evaluated at and to obtain

where for the 90° bend. Note that, between inner and outer radii of curvature, an uncompensated bend, filled with a material having a dielectric constant of 4.1, would exhibit more than five times the postulated 100 ps limit on transit time variation.


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Although the heuristic design algorithm works well, it provides no indication as to whether its result is anywhere near optimal. It is also tedious to apply. In [10], the rootmean-square of (3), over all dielectric layers, was chosen as an objective function for use with a nonlinear global optimization routine. Specifically, for a bend consisting of

The optimal set of dielectric constants and layer-bounding radii is that which minimizes this objective function, subject to all applicable constraints. The minimizing set was found to be nearly identical to the set generated by the heuristic algorithm. The associated minimum value of is 27 ps.

INTEGRATION OF LENS AND COAXIAL TRANSMISSION LINE BEND We integrated the dielectric lens laminate with a square coaxial TEM transmission line bend test fixture. To minimize reflections and enhance high-voltage performance, we required the bend and its interfaces to have identical conductor sizes in addition to having the same nominal impedance. Since the layer-thickness-weighted average dielectric constant within the laminate is 4.1, material with a dielectric constant of 4.0 was selected to fill straight (tapered) sections interfacing to the bend. There is a parametric expression available for the impedance of a uniform straight square coaxial line with cylindrical center conductor [11]. For a dielectric constant of 4.0, it predicts when the length of the side of the outer conductor is five times the diameter of the inner conductor. Since an outer conductor matching the laminate is 5.08 cm on each side, 1.016 cm was selected for the diameter of the inner conductor. To predict the impedance of the layered transmission line bend, we modeled its cross section using finite elements and calculated the impedance as described in [6]. There we proved that the characteristic impedance could be found (approximately) from

Here, is the impedance of free space, T is the vector transpose operator, and S and U are, respectively, the finite element matrix and solution vector for the two-dimensional potential equation for a rotationally symmetric geometry,

The electrical potential, u, is a function only of

and

and

where and are unit vectors. For a layered bend, (5) is valid to the extent that the layers represent a good approximation of (1). The model result was The cross section of the layered bend model, showing the position of the center conductor, is shown in Figure 2. A top view of the bend layers is shown in Figure 3.


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In addition to the bend test section, we also designed and built a comparable 19 cm long straight coaxial section, as well as a pair of square pyramidal taper sections. The latter provided a constant impedance transition between the 5.08 cm square coaxial test sections and a standard N-type female connector. Details of the taper design are to be found in [10]. A photograph of the complete transmission line test fixture with 90° bend is shown in Figure 4. The diagram in the lower portion of the figure is a z-plane cross section at the fixture centerline, with layered dielectric filling the bend.

EXPERIMENTAL PROGRAM Equipment Setup and Signal Processing The equipment setup for the coaxial transmission line measurements is shown in Figure 5. The device under test (DUT) was either a 90° bend or straight section loaded with dielectric material. After propagation through the test fixture, the raw impulse response was detected by the SD-24 sampling head, and stored on the Tektronics 11801B oscilloscope. To obtain a system response, the DUT was removed and the two tapered sections were directly connected. Signal processing was used to isolate the response of the DUT by deconvolving the system response from the raw transmission data. Impedance measurements employed the 0.25 voltage step of SD-24 operating in TDR mode, while a terminator replaced the PSPL 5210.


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Measurements The TDR measurements of the test fixture are shown in Figure 6. With the straight section of transmission line loaded with material having a dielectric constant of 4.0, the impedance is nominally throughout the fixture. It is extremely flat within the straight section, which is delimited by the small bumps at 10.3 and 12.8 ns. The larger features on either side of the straight section arise primarily from SMA-to-N adapters at the ends of the tapered sections. A minor machining error is responsible for the broad hump at the input taper; the output taper does not exhibit this feature. The TDR with the layered bend installed in the test fixture has similar features, and also exhibits nominal impedance.

Impulse transmission through the straight section of transmission line provides a reference, against which transmission through the 90° bend can be compared. In Figure 7, the top pair of plots shows the raw impulse response of the straight section in both time and frequency domains. The FWHM of the time domain signal is 55 ps. The middle pair of plots shows the impulse response for the 90° bend, filled with the same uniform dielectric.

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Instead of a narrow impulse, the uncompensated bend transmits a broad bipolar signal spanning approximately 500 ps. The bottom pair of plots shows the impulse response for the same 90° bend when compensated by the layered dielectric lens. The FWHM of the time domain signal is 70 ps, only 15 ps wider than observed for the comparable straight section of transmission line filled with a uniform dielectric.

The raw impulse response data for the straight transmission line section and for the 90° layered bend were processed to correct for the system response. The FWHM of the system response, shown in Figure 8, is 50 ps. A 5 th order modified Butterworth filter with a 15 GHz cutoff was used in deconvolving the system response from the raw data. The


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corrected impulse responses are shown in Figure 9. The FWHM of the time domain response of the straight section is 40 ps. For the layered bend, the peak is only slightly wider, 44 ps. Thus, the layered dielectric lens has been extremely successful in compensating the transmission line bend.


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CONCLUSIONS In an electrically large square coaxial transmission line, we demonstrated preservation of a narrow impulse during propagation through a relatively tight 90° bend compensated by a layered approximation of a CID lens. By use of a non-linear global optimization algorithm, we established that the layered dielectric design was essentially optimal. In terms of the observed, raw impulse response, the compensated bend increased the pulse width by only 15 ps, to 70 ps, from the 55 ps observed for an equivalent length of straight transmission line. In contrast, without compensation, the transmission line bend distorted the impulse into a bipolar signal spread over a 500 ps time interval. With a compensating dielectric lens like the one described here, even electrically large transmission line bends can be implemented with minimal dispersion. Use of such compensated bends may permit more compact, efficient designs of high-voltage, UWB systems. ACKNOWLEDGMENTS This work was funded in part by the Air Force Office of Scientific Research, Alexandria, VA, and in part by the Air Force Research Laboratory, Directed Energy Directorate, under contract F29601–98–C–0047.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

C. E. Baum, “Two-Dimensional Inhomogeneous Dielectric Lenses for E-Plane Bends of TEM Waves Guided Between Perfectly Conducting Sheets,” Sensor and Simulation Note 388, 14 October 1995. C. E. Baum, “Dielectric Body-of-Revolution Lenses with Azimuthal Propagation,” Sensor and Simulation Note 393, 9 March 1996. C. E. Baum, “Dielectric Jackets as Lenses and Application to Generalized Coaxes and Bends in Coaxial Cables,” Sensor and Simulation Note 394, 23 March 1996. C. E. Baum, “Azimuthal TEM Waveguides in Dielectric Media,” Sensor and Simulation Note 397, 31 March 1996. C. E. Baum, “Use of Generalized Inhomogeneous TEM Plane Waves in Differential Geometric Lens Synthesis,” Sensor and Simulation Note 405, 5 December 1996. W. S. Bigelow and E. G. Farr, “Impedance of an Azimuthal TEM Waveguide Bend in a Graded Dielectric Medium,” Sensor and Simulation Note 428, 21 November 1998. C. E. Baum, “Admittance of Bent TEM Waveguides in a CID Medium,” Sensor and Simulation Note 436, 2 May 1999. W. S. Bigelow and E. G. Farr, “Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Material,” Sensor and Simulation Note 416, 5 January 1998. W. S. Bigelow, and E. G. Farr, “Minimizing Dispersion in a TEM Waveguide Bend by a Layered Approximation of a Graded Dielectric Lens,” p. 213-219 in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, and J. Shiloh (eds.), Kluwer Academic / Plenum Publishers, New York, 1999. W. S. Bigelow, E. G. Farr, and W. D. Prather, “Compensation of an Electrically Large Coaxial Transmission Line Bend by a Layered Dielectric Lens,” Sensor and Simulation Note 445, 17 July 2000. Reference Data for Engineers: Radio, Electronics, Computer, and Communications, 8th Ed., p. 20-22, SAMS, Prentice Hall Computer Publishing, Carmel, Indiana 46032, 1993.

OPTIMAL INPUT SIGNALS FOR DRIVING NONLINEAR ELECTRONIC SYSTEMS INTO CHAOS

Stuart M. Booker1, Paul D. Smith 1 , Paul V. Brennan2 and Richard J. Bullock2 1

2

Department of Mathematics University of Dundee Dundee DD1 4HN, UK Department of Electronic and Electrical Engineering University College London Torrington Place, London, WC1E 7JE, UK

INTRODUCTION Nonlinear feedback loops have found a wide variety of applications in modern electronic systems. Many such circuits are vulnerable to chaos if driven by an appropriate input signal. Furthermore, since nonlinear circuits cannot be characterised simply by their spectral response, the onset of chaos is extremely waveform dependent. The question arises: which input signal is the most effective at driving the circuit into chaos? It has been shown recently (Levey and Smith, 1997; Booker, 2000) that an answer can be found to this fundamental question for one particular class of electronic systems. This class comprises those nonlinear electronic systems whose dynamical equation resembles a perturbed Hamiltonian system. In fact, this is a very broad class of electronic systems and includes many practical circuits. For this class of electronic systems a pair of optimal input signals may be determined. These induce the onset of homoclinic chaos in the dynamics of the circuit with the smallest possible amplitude, or power, for which homoclinic chaos can be found. These optimal input signals allow in-band disruption of the nonlinear electronic systems concerned. In this paper we review the approach of Booker (2000) with particular reference to a realistic test circuit, a second order phase-locked loop (PLL).

OPTIMAL FORCING FUNCTIONS Many of the nonlinear electronic circuits used in modern communication systems Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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can be described by a dynamical equation of a particular form. This can be written

where the forcing term is a T–periodic function. It is important to know how small this forcing term can be and still cause disruption of the circuit. Such information would allow the best chance of disrupting the circuit in-band, and would also suggest a design strategy for minimising any unwanted disruption. The other important property of the circuit is that: if and are set identically equal to zero then the resulting dynamical system is Hamiltonian and possesses a homoclinic orbit connecting a saddlepoint to itself. The dynamics of the circuit, as defined by equation. (1), can then be viewed as a perturbed Hamiltonian system. This allows us to make a quantitative analysis of the circuit. Such a homoclinic connexion is depicted in figure 1(a) and results from a coincidence of the unstable and stable manifolds, and of the saddle-point; a useful introduction to these ideas is given by Drazin (1992). If damping and forcing are applied to this system, as in equation (1), the manifolds separate and one of two situations arises. Either, the manifolds intersect nowhere (as depicted by the dash curve of figure 1(a)), or, the manifolds intersect an infinite number of times (as depicted in figure 1(b)). The latter situation is termed a homoclinic tangle and gives rise to complex dynamics, including subharmonic, quasiperiodic and chaotic dynamics. Such dynamics will completely disrupt the behaviour of any circuit designed to exploit a simple periodic signal. If we can determine an input signal which is optimal at inducing a homoclinic tangle, then we will know the best signal for disrupting the operation of the circuit. This fact allows us to determine two important input signals: the signal of smallest amplitude which will disrupt the circuit, and that of smallest power. In order to determine these we assume that has or norm 1, respectively, and that is therefore a direct measure of the amplitude, or power, of the input signal. Our aim is thus: to determine the unit norm waveform, which will induce a homoclinic tangle in the dynamics of equation (1) with the smallest possible value of This can be achieved using an approach due to Mel’nikov (1963) who proved that the distance between the unstable and stable manifolds is proportional to a T–periodic function defined by

The functions and define the homoclinic orbit of the Hamiltonian system which results by setting and to be identically zero; these functions are usually well known for electronic circuits of this form. Obviously, if this function is zero for any value of then the unstable and stable manifolds of the saddle-point intersect and a homoclinic tangle occurs in the dynamics of the circuit. This in turn gives rise to circuit disruption. Since we know that is constrained to have unit norm, we may use some elementary functional analysis to determine the smallest possible value of for which a zero occurs in equation (2). We may also determine the waveform which is optimal at inducing such disruptive dynamics. In general, the optimal forcing waveform, of smallest power is defined by special functions relating to the function however, the optimal forcing waveform of smallest amplitude is usually a square-wave.


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THE PHASE-LOCKED LOOP (PLL) In order to demonstrate the validity and usefulness of the approach described above, we consider a simple test problem: how to best disrupt a second order phaselocked loop (PLL) FM demodulator. The PLL is a simple nonlinear feedback loop and is depicted in figure 2. The input signal is subject to a baseband modulation which it is the purpose of the loop to recover; for a more detailed account of the PLL the reader is referred to the book by Brennan (1996). Whether this can be done or not depends upon the phase error the phase difference between the input signal and that fed back by the loop. If the phase error is a simple periodic oscillation about zero then the modulation can be recovered. More complex dynamics, such as a subharmonic, quasiperiodic or chaotic responses, will effectively disrupt the operation of the PLL. Chaos has been demonstrated in the PLL by Endo and Chua (1988) and other authors (Endo et al, 1988; Chu et al, 1991) but in each instance a sinusoidal modulation was assumed to allow an analysis of the circuit. However, the sinusoid is only one of many possible modulations, many of which may be better at disrupting the PLL. Using the approach described above, the PLL may be analysed subject to the optimal modulation for inducing homoclinic chaos. We consider a typical second order PLL design, for which the phase error satisfies a non-dimensionalised dynamical equation of the form

The baseband modulation which excites the system has form where is a unit amplitude waveform and is the depth of modulation. We assume a phase detector characteristic of the form

It is evident that the PLL satisfies a dynamical equation with the form of (1). Furthermore, if we assume that and are both identically zero, then the dynamics of the undamped, unforced PLL form a Hamiltonian system. In this case we find that a saddle-point exists at the point which is connected to itself by two homoclinic orbits. The underlying dynamics of the undamped, unforced PLL are depicted in Figure 3; and denote the two homoclinic connexions. Note that is and that, hence, the points and are identical. Disruptive dynamics can result from the break-up of either homoclinic connexion. Clearly, the approach of Booker (2000) may be applied to this circuit, in order to determine a waveform of least modulation depth with which to disrupt the PLL. If this approach is taken then it is found that this optimal waveform takes the form of a squarewave. Other effective disrupting waveforms can also be found by this method, such as the sawtooth, which proves to be a particularly effective disruptor at low frequencies of modulation (Booker et al, 2000). In order to demonstrate the effectiveness of our approach for designing disruptive signals, we next contrast the effectiveness of three different modulations. These are: the sinusoid (the signal commonly assumed in the study of nonlinear circuits), the sawtooth (expected to be particularly effective at low frequencies of modulation), and the square-wave (expected to be the optimal disruptor of the PLL).


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In figure 4 we contrast the effectiveness of these three different modulation waveforms at inducing a homoclinic tangle in the dynamics of the PLL. These results are due to an analysis of the Mel’nikov function for each waveform, as given in equation (2). The figure depicts the threshold value of the modulation depth, at which a homoclinic tangle can first be observed in the PLL’s dynamics. The parameters assumed in this analysis correspond to those of the test circuit design described below. The threshold for the sinusoid (solid curve) is very similar to the linear limit for the PLL. This is the limit of operation usually assumed for the circuit; see Brennan’s (1996) account for detail. It may be clearly seen, however, that the sawtooth waveform is much more effective at inducing a homoclinic tangle at low frequencies of modulation than is the sinusoid. It is also clear that the square-wave is a significantly better disruptor than either the sinusoid or sawtooth. This is just what we expected.

Numerical Results In order to demonstrate that the onset of a homoclinic tangle in the dynamics of the PLL really does disrupt the operation of the circuit, numerical simulations were performed. The dynamical equation of the PLL was solved numerically for a wide range of initial conditions across a range modulation frequencies. The parameters assumed here were again taken from the experimental test circuit design discussed below. In figure 5 we depict the threshold for disruption observed in the PLL subject to each modulation. The threshold marks the depth of modulation, which is required to induce complex dynamics in the circuit; below the threshold only period 1 oscillatory responses are observed, but above each curve the relevant modulation induces chaos and other complex dynamics.

Experimental Results A test circuit for the second order PLL was designed, with a damping parameter and a natural frequency of 5 kHz. The response of the PLL was then measured when subject to a range of modulation frequencies from 1 to 10 kHz, with a modulation of sinusoidal, sawtooth and square-wave form. The experimental set-up for measuring the PLL is depicted in figure 6. The measured response of the PLL was found to be in remarkable agreement with the numerical simulations described above; even the finer details of a rich array of dynamics were found to be in agreement. A comparison of the numerical and experimental results obtained for sinusoidal and sawtooth modulation have been reported (Booker et al, 2000) and demonstrate the sort of agreement which was found, in general, between the numerical and experimental data. This agreement clearly demonstrates that our original model for the PLL (3) is a very accurate representation of the PLL’s dynamics. The resulting thresholds for disruption of the PLL, subject to sinusoidal, sawtooth and square-wave modulation, are depicted in figure 7. Below each threshold curve only a period 1, oscillatory response can be observed, whilst above each threshold a wide variety of complex dynamics are found. The results depicted in figures 4, 5 and 7 clearly corroborate the approach to circuit disruption which has been described in this paper.

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CONCLUSION In this paper we have described a novel approach to the problem of disrupting nonlinear electronic circuits; this approach is applicable to a wide range of nonlinear electronic systems. It allows us to calculate input signals of smallest amplitude and power which will induce homoclinic chaos and, hence, disruption in the circuit. Furthermore, as is clearly demonstrated by the results described above, this approach allows us to develop in-band signals with which to disrupt such circuits. It may also be noted that the strategy which we have exploited in this paper, for a certain class of analogue circuit, can be extended to the problem of disrupting digital nonlinear systems. Using the method described above it is possible to: develop optimal waveforms for disrupting certain nonlinear electronic systems; and, develop a design strategy to minimise the risk of unwanted disruption of such circuits.

Acknowledgements The authors gratefully acknowledge the support of the Defence Evaluation and Research Agency, DERA Fort Halstead, UK.

REFERENCES Booker, S.M., 2000, A family of optimal excitations for inducing complex dynamics in planar dynamical systems, Nonlinearity, 13:145. Booker, S.M., Smith, P.D., Brennan, P. and Bullock, R., 2000, The effect of the forcing function on disruption of a phase-locked loop, to appear in Int. J. Bifurcation and Chaos, September, 2000. Brennan, P.V., 1996, Phase-locked Loops: Principles and Practice, McMillan, London. Chu, Y.H., Chou, J.H. and Chang, S., 1991, Chaos from third order phase-locked loops with a slowly varying parameter, IEEE Trans. Circuits and Systems, 37:1104. Drazin, P.G., 1992, Nonlinear Systems, Cambridge University Press, Cambridge. Endo, T. and Chua, L.O., 1988, Chaos from phase-locked loops, IEEE Trans. Circuits and Systems, 35:987. Endo, T., Chua, L.O. and Narita, T., 1989, Chaos from phase-locked loops – Part II: high dissipation case, IEEE Trans. Circuits and Systems, 36:255. Levey, D.B. and Smith P.D., 1997, The core of chaos in the dynamics of phase-locked loops, in: Proc. 5th Int. Workshop on Nonlinear Dynamics in Electronic Systems (Moscow, June 26-27, 1997), Moscow Technical University of Communication and Informatics, Moscow. Mel’nikov, V.K., 1963, On the stability of the center for time periodic perturbations, Trans. Moscow Maths. Soc., 12:1.

IN-BAND CHAOS IN COMMERCIAL ELECTRONIC SYSTEMS

Stuart M. Booker1, Paul D. Smith1, Paul V. Brennan 2 and Richard J. Bullock2 1

2

Department of Mathematics University of Dundee Dundee DD1 4HN, UK Department of Electronic and Electrical Engineering University College London Torrington Place, London, WC1E 7JE, UK

INTRODUCTION In this paper we describe a method for inducing chaos in a standard, commercially available electronic sub-system using an in-band signal. In particular, we show how, with relatively little knowledge of the commerical circuit design, an in-band signal can be determined which will disrupt the operation of the circuit and drive it into chaos. The problem which we consider is the disruption of an INMARSAT standard-A terminal. This device, which employs a phase-locked loop (PLL) FM discriminator, is designed to accommodate a 3 kHz modulation bandwidth with 12 kHz of peak FM deviation. In order to model this commercial system we consider a PLL test circuit with an identical specification. This PLL circuit provides a test problem with which analysis and numerical modelling can be compared, the INMARSAT terminal design being, in essence, a ‘black box’. Having developed a successful strategy for disrupting the test PLL circuit we then employ our knowledge on the INMARSAT terminal. Experimental measurements are reported on an actual INMARSAT standard-A terminal which validate our approach.

A SCHEME FOR CIRCUIT DISRUPTION The problem which we shall investigate is: how best to disrupt the operation of a standard, commercially available electronic system, using only a limited knowledge of the system itself. In particular, we consider how to disrupt the operation of an INMARSAT mobile communications terminal, for use with standard-A analogue voice


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transmissions. The knowledge which we shall exploit is limited: that the terminal comprises an RF/IF module and phase-locked loop (PLL) FM discriminator, and that the device is designed to accommodate 3 kHz modulation bandwidth with 12 kHz of peak FM deviation. The INMARSAT terminal contains two sub-systems for which chaos has been reported in the literature: an automatic gain control (AGC) loop in the IF board, and the phase-locked loop (PLL). Although chaos has been reported for one particular AGC loop design (Chang et al, 1993) the design investigated was highly unusual; more common AGC designs are known to be stable, and robust (Green, 1983). Using a suitably amplitude modulated waveform to induce chaos in the INMARSAT terminal is unlikely to prove successful as a strategy for disruption. By contrast, the PLL is known to be much more prone to chaos. Standard PLL FM demodulator designs have been shown to exhibit chaotic dynamics, if driven by suitable frequency-modulated signals (Endo and Chua, 1988; Endo et al, 1989; Chu et al, 1991). For this reason we choose the following strategy to disrupt the operation of the INMARSAT terminal: (i) design a PLL test circuit with identical specifications to the INMARSAT terminal, (ii) determine a frequency modulation suitable for disrupting the PLL test circuit given our knowledge of its design, (iii) validate the effectiveness of the disrupting signal on the PLL test circuit, (iv) validate the effectiveness of the disrupting signal on a test circuit comprising RF/IF module and PLL, (v) validate the effectiveness of the disrupting signal on the INMARSAT terminal itself.

PLL TEST CIRCUIT The PLL FM discriminator test circuit can be modelled as a nonlinear feedback loop whose dynamical equation resembles a perturbed Hamiltonian system, when modulated by a periodic input signal. For such an electronic system it is possible to determine an input signal of smallest amplitude (or modulation bandwidth, in this context) which will allow the onset of homoclinic chaos in the circuit (Booker, 2000). A review of this approach has been given in an accompanying paper in this volume (Booker et al, 2000a) with particular reference to an example problem: the PLL FM discriminator circuit whose specification matches that of the INMARSAT terminal. For details of the PLL test circuit, and an outline of how the optimal modulation waveform is determined, the reader is referred to the accompanying paper. Analysis of the PLL test circuit design reveals that a square-wave modulation is the optimal signal for inducing a homoclinic tangle in the dynamics of the PLL; a useful introduction to this kind of dynamics is given by Drazin (1992). This feature of the circuit’s dynamics is responsible for inducing a chaotic response in the circuit, together with other complex dynamics such as subharmonic and quasiperiodic responses. All of these dynamics will effectively disrupt the operation of the PLL test circuit and, hence, the onset of a homoclinic tangle in the PLL test circuit corresponds to the threshold at which disruption can be expected. In figure 1 we depict the threshold at which a homoclinic tangle occurs in the test PLL circuit when modulated by: the sinusoid (solid curve), the sawtooth (dot-dash curve), and the square-wave (dash curve). The squarewave is chosen as the optimal disruptor of the PLL circuit, whilst the sawtooth and sinusoid are given for comparison. The sinusoid is the signal conventionally assumed for the investigation of a nonlinear circuit. Below each curve a simple period 1 oscillatory response can be expected, given the appropriate modulation, whilst above the curve complex and disruptive dynamics may be expected. Numerical and experimental tests


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of the PLL test circuit design revealed these predictions to be extremely accurate (Booker et al, 2000a).

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EFFECT OF RF/IF MODULE In an application such as the INMARSAT terminal the PLL FM demodulator is preceded by an IF board. It is necessary, therefore, to ensure that this additional sub-system has no significant effect on the waveforms used to disrupt the PLL. To this end a simple RF/IF test module was designed in order to mimic, as closely as possible, a physical implementation of the PLL circuit. The design of this module is depicted below, in figure 3; it consists of a front-end RF amplifier followed by a frequency convertor, IF filter and IF amplifier with automatic gain control (AGC).

The test/IF input allows a suitable IF signal to be directly introduced to the module before the IF filtering stage; this enables us to directly assess the effect of IF filtering on our disruptive waveforms. The test IF filter was designed for a bandwidth of around 27kHz, to comply with Carson’s rule, and with a centre frequency of 1 MHz. The AGC loop bandwidth must be significantly less than the IF frequency and a value of 5 kHz, equal to the PLL FM demodulator loop natural frequency, was chosen. The series of experimental measurements which had been performed on the PLL test circuit alone were repeated with the RF/IF module preceding the PLL FM demodulator. The limits of disruption found with the RF/IF module included were very close to those for the PLL test circuit alone. The dynamics observed above the limit of disruption were also similar to those observed for the PLL alone. The only significant difference due to the introduction of the RF/IF module was an increase in the amount of chaos observed above the limit of disruption (as opposed to subharmonic, quasiperiodic responses etc) when a sawtooth modulation was applied to the circuit. This may be accounted for by the fact that the sawtooth resembles the zero modulation frequency limit of an important waveform: the optimal disrupting modulation of least rms modulation depth (Booker et al, 2000b). At non-zero frequencies of modulation, however, this waveform resembles a sawtooth waveform with smoothed peaks. The effect of IF filtering is thus likely to render the sawtooth even more effective at inducing chaos then it would otherwise be.


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THE INMARSAT TERMINAL Having determined an optimally disruptive modulation waveform for the test PLL circuit, and having proven that the effectiveness of this disruptor is undiminished by the presence of an RF/IF module, we are now in a position to consider the actual circuit: the INMARSAT standard-A terminal. A series of experimental measurements were performed on the INMARSAT terminal which allowed its limit of disruption to be measured. The test signal was introduced before the IF board of the INMARSAT terminal (rather than immediately before the PLL FM discriminator) ensuring receiver behaviour identical to that encountered in normal operation. Three different modulation waveforms were used: the sinusoidal, sawtooth and square-wave signals which had been used in assessing the PLL test circuit. The range of dynamics observed in the disrupted response of the INMARSAT terminal was identical to that observed with the PLL test circuit, including chaos, subharmonic responses and cycle-slipping (or rotaional-type periodic responses). A summary of these experiments is given in figure 2 which depicts the limit of disruption for the INMARSAT terminal modulated by: a sinusoid (solid curve), a sawtooth (dot-dash curve), and a square-wave (dash curve); triangles are used to denote the data points corresponding to actual measured values for the threshold of disruption. The results of figure 2 can be compared directly with the threshold for which a homoclinic tangle becomes apparent in the dynamics of the test PLL circuit, figure 1. Bearing in mind that neither figure is scaled to fit the agreement between the two is remarkable. The sinusoid is the waveform commonly used to investigate the dynamics of nonlinear circuits; also, the usual calculation for the linear limit of a PLL (Brennan, 1996) gives a limit almost identical to the threshold for disruption of a sinusoidally modulated PLL. The limits of operation for a PLL, or PLL-based circuit, are thus effectively defined by our results for sinusoidal modulation. As may be seen, however, the sinusoid proves to be remarkably poor at inducing disruption in the INMARSAT terminal at low frequencies of modulation. The sawtooth is a much better disruptor in this regime. However; the square-wave proves to be significantly better than either of the two test waveforms across all frequencies of modulation. Indeed, at low frequencies of modulation the square-wave is several times better then the sinusoid at inducing disruption in the INMARSAT terminal. Obviously, the square-wave provides an excellent waveform for inducing in-band disruption of a circuit like the INMARSAT terminal, at relatively low powers. It achieves this highly desirable effect by exploiting to the full the potential of the PLL circuit to be driven into homoclinic chaos. In figures 4, 5 and 6 we compare the limits of disruption of the INMARSAT terminal with those measured experimentally for the test PLL circuit, and with the predictions of numerical simulations of the test circuit. Figure 4 depicts the limit of disruption measured for the relevant circuit modulated by a sinusoidal waveform, whilst figures 5 and 6 depict results for sawtooth and square-wave modulation, respectively. The slight differences between the experimentally measured results for the test circuit and the INMARSAT terminal are easily accounted for by two facts. Firstly, the IF filtering stage will cause some slight distortion of the input signal. Secondly, the INMARSAT PLL FM discriminator may have design parameters which differ slightly from those assumed for our test circuit; in particular, if the damping parameter of the INMARSAT PLL was slightly different from that of the test PLL circuit, then we would expect their respective thresholds for disruption to differ slightly in magnitude, although possessing the same shape.

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CONCLUSION In this paper we have presented an approach to the problem of disrupting effectively a commercial nonlinear electronic system. Our approach is based on a minimal understanding of the circuit concerned, the basic design and operational specifications, rather than a detailed knowledge of the design implementation itself. By designing a test circuit with identical specifications to the commercial circuit and concentrating on one method of in-band disruption, we have been able to determine an input signal which proves to be extremely effective at disrupting the system in practice. Analytical, numerical and experimental study of the test circuit confirms the effectiveness of our approach at inducing disruption in a realistic PLL design. Experimental investigation of the commercial circuit itself demonstrates: that our disruptive signal is effective against commercial equipment as well as a test circuit, that it is the PLL of the INMARSAT terminal which gives rise to disruptive dynamics, and that our approach to circuit disruption is of practical value. From the results presented in this paper it is clear that in-band disruption of commercial systems can be achieved using relatively low power levels. In consequence, the ideas outlined in this paper offer a methodology for two important tasks: the design of signals with which to disrupt commercial systems in-band, and the design of signals with which to ensure that any commercial circuit is robust against such disruption.

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Acknowledgements The authors gratefully acknowledge the support of the Defence Evaluation and Research Agency, DERA Fort Halstead, UK.

REFERENCES Booker, S.M., 2000, A family of optimal excitations for inducing complex dynamics in planar dynamical systems, Nonlinearity, 13:145. Booker, S.M., Smith, P.D., Brennan, P.V. and Bullock, R.J., 2000, Optimal input signals for driving nonlinear electronic systems into chaos, Ultra-Wide band Short-Pulse Electromagnetics 5. Smith, P.D. and Cloude, S.R., ed., Plenum Press, New York. Booker, S.M.; Smith, P.D., Brennan, P. and Bullock, R., 2000, The effect of the forcing function on disruption of a phase-locked loop, to appear in Int. J. Bifurcation and Chaos, September, 2000. Brennan, P.V., 1996, Phase-locked Loops: Principles and Practice, McMillan, London. Chang, F.J., Twu, S.H. and Chang S., 1993, Global bifurcation and chaos from automatic gain control loops, IEEE Trans. Circuits and Systems, 40:403. Chu, Y.H., Chou, J.H. and Chang, S., 1991, Chaos from third order phase-locked loops with a slowly varying parameter, IEEE Trans. Circuits and Systems, 37:1104. Drazin, P.G., 1992, Nonlinear Systems, Cambridge University Press, Cambridge. Endo, T. and Chua, L.O., 1988, Chaos from phase-locked loops, IEEE Trans. Circuits and Systems, 35:987. Endo, T., Chua, L.O. and Narita, T., 1989, Chaos from phase-locked loops – Part II: high dissipation case, IEEE Trans. Circuits and Systems, 36:255. Green, D.N., 1983, Global stability of automatic gain control circuits, IEEE Trans. Circuits and Systems, 30:78.

AN APPLICATION OF CHAOS THEORY TO THE HIGH FREQUENCY RCS PREDICTION OF ENGINE DUCTS

Andrew J. Mackay1 1

Defense Evaluation and Research Agency (DERA), Malvern. St Andrews Road, Great Malvern, WR14 3PS, UK.

INTRODUCTION The accurate prediction of the radar cross section (RCS) of electrically large ducts and cavities is a particularly difficult problem but one of significant interest for a number of applications. Most general purpose methods employ variants of shooting-andbouncing (SB) ray tracing (e.g. Burkholder et al, 1991) since more accurate modal, finite element or finite difference methods are computationally too intensive. I have previously shown (Mackay, 1998; Mackay, 1999) that SB ray tracing in straight ducts of moderate length is inherently non-convergent for most duct geometries. This involves a direct application of well established results in chaos theory used in the study of classical and quantum dynamical systems. In the previous work I established a convergence bound in terms of the characteristic duct dimensions, the angle of incidence and the Lyapunov exponent for a straight duct terminated by a flat conductor. Here, we summarise these findings and extend the investigation to describe the RCS properties of a long stadium cross section duct in a regime where SB convergence is not achievable. I investigate the use of random fields to construct an ensemble of statistically equivalent RCS predictions and compare with an accurate modal solution.

DETERMINISM IN RCS PREDICTIONS Radar engineers seldom require a precise lobe-for-lobe convergence of predicted RCS at high frequencies. Firstly, measurements seldom agree with each other for complex targets at high frequencies since tolerances are rarely maintained to a small fraction of a (short) wavelength. Secondly, high frequency predictions are almost always approximate. This is primarily because such algorithms only provide an approximate solution to Maxwell’s equations, but also because of approximations in the mathematical representation of the geometry for complex structures (commonly using standard


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computer aided design meshes). Quite often it is sufficient only to require the mean RCS over an angular window, small compared with 360° but large compared to the average lobe width. For other applications, e.g. for imaging purposes, it is also necessary to obtain the correct autocorrelation of the RCS taken as a function of incidence or scatter angles. It is our contention that, at high frequencies, deterministic “exact” RCS predictions which provide lobe for lobe correspondence are not required for nearly all applications. In any measurement system there is always a finite sensor resolution and experimental uncertainty associated with structure dimensions and orientations. For service applications, e.g. for vehicles in transit, these uncertainties can be very large and considerably limit any requirement for deterministic predictions. Thus, although we have shown that SB ray methods can not generally achieve convergent RCS predictions, there are grounds for believing that SB methods can probably be employed in a non-convergent setting to obtain statistically correct RCS predictions. In this paper I report some work to support this belief and show how the fields at the end of a long straight stadium duct can be modelled by a sum of randomly directed plane waves. For chaotic structures of this kind it then becomes possible to decompose the exit field by a “deterministic” component predicted using convergent ray tracing and a “random” component to represent the non-convergent behaviour. A more detailed account of these findings is given elsewhere (Mackay, 1998-1999).

DYNAMICAL CHAOS AND RAY TRACING Chaos theory is concerned with the properties of formally deterministic systems (i.e. without an intrinsic random element) which evolve unpredictably with the gradual increase of some parameter such as time. Here, a small perturbation in the initial conditions of the system grows exponentially with time. There is considerable literature proving that ray tracing within a closed or ‘nearly closed’ structure is in general (but not always) a chaotic process (e.g. Berry, 1989; Heller, 1989; Gutzwiller, 1990). Here, by ‘nearly closed’, we mean a system where at least some rays eventually escape after a finite time. This is the case for many scattering systems including engine ducts, open cavities and certain other classes of concave perfectly conducting structures. Not all structures resulting in multiple ray bounces exhibit chaos. In particular, geometries which are fully separable under the wave equation do not (e.g. Lichtenberg and Lieberman, 1983). For example, a straight duct with circular or elliptical cross section is fully separable and hence does not give rise to chaotic ray tracing. Because fully separable geometries allow exact solutions in terms of standard functions, these are just those geometries which are often used for software validation. It is thus rather too easy to arrive at incorrect general conclusions on the use and accuracy of ray tracing algorithms. There are two further points from chaos theory which are important for our application. Firstly, in order to exhibit chaos, the scatterer must be concave or composed of disconnected elements. An example of the latter is the three cylinder problem (Svitanovick and Eckhardt, 1989; Smilansky, 1989) which also gives rise to fractal scattering in the high frequency limit. Straight ducts, as we indicate below, can give rise to chaotic but not fractal scattering. Secondly, the scatterer must have non-zero curvature over at least some of its domain. For example, a duct with polygonal cross section with a finite number of sides may or may not be separable under the wave equation but can not give rise to chaos in the high frequency limit.

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Suppose a wave enters a straight perfectly conducting duct of arbitrary cross section and length L, terminated by a flat perfectly conducting plate normal to the duct axis. This problem can be unfolded, using mirror symmetry, and is equivalent to an open duct of length 2L with a separate exit and entry plane. We consider an incident plane wave entering the duct at the entry plane at an angle to the normal and an azimuth angle as illustrated in figure 1, below.

If we assume an SB first order ray method, then the plane wave can be represented as a set of zero divergence ray pencils each with the same direction as the incident wave contiguously spanning the entry aperture. A given ray pencil will make N bounces before escaping from the exit plane. N will generally be different for each ray pencil but for a straight duct, the total transit time for any ray between the exit and entry plane is given by, where C is the speed of light in free space. This is independent of the duct cross section and it follows that a straight duct can not trap rays; i.e. it is not possible to choose a ray trajectory where the time approaches infinity for a finite length duct away from grazing incidence. A consequence of this is that fractal scattering is impossible. However, it is important to note that for bent ducts (1) is not valid, can approach infinity and fractal scattering is possible for non-separable geometries in the high frequency limit. Within the duct let us define the “billiard ray” as the projection of a ray within the duct on to the cross section, where the cross section will sometimes be referred to as the “billiard plane”. Let be the angle made between the billiard ray and the inward pointing normal and let be the chord distance in the billiard plane between the and ray bounce for a given ray pencil. If we define as the expected (mean) value of then for large N,

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We will now consider a special class of ergodic chaotic systems, where every ray state is equally likely for large N. For an ergodic system the average geometrical path length of a ray between successive bounce points is equal to the average chord length. A result, probably going back to Euler, shows that the average chord length is given by for a general billiard shape where A is the area of the billiard and S is its perimeter. Hence, for large N, While this result is not strictly valid for non-ergodic ducts, it is likely to hold for most straight ducts to a reasonable approximation. In the tangent mapping formulation one can define a deviation matrix (Berry, 1981), which relates a small change in the ray characteristics between one bounce and the next. One deviation matrix is of special use and relates the deviations and where is a deviation in the width of a billiard ray normal to the direction of the ray in the billiard plane at the ray bounce and is the divergence angle of the billiard ray. has a specially simple form and describes the change of the first order ray characteristics between successive bounces, analogous to the Deschamps formulation (Deschamps, 1972). The deviation matrix is defined by,

where

where

is the Abbe invariant at skew incidence at the

reflection defined by

and is the surface curvature in the billiard plane at the bounce point, defined positive if the local boundary and the billiard ray lie on the same side of the tangent plane and negative if they lie on opposite sides of the tangent plane. The Lyapunov exponent (strictly, the maximum Lyapunov exponent) is an important measure of the level of chaos in a dynamical system which can be connected with ray tracing and hence with high frequency RCS prediction. This is defined as the limit of where

where

is the maximum eigenvalue magnitude of the compound deviation matrix defined with respect to the mapping by,

This result is independent of the coordinate system to an accuracy of order O(1/N), so we may employ Suppose that the initial values of a ray, specifying its origin and direction on the incidence plane, are known to a relative accuracy of for some positive This can be interpreted in several ways. In terms of the SB ray algorithm, we can define


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is the width of a ray pencil and D is the maximum distance between any two points on the duct aperture in which case defines the convergence error. Alternatively, we can define where is a small change in azimuth angle associated with the incident plane wave. In this case is a measure of angle of incidence error. We can also relate to a small relative change in the shape of the boundary. Defined in terms of an azimuth angle error or uncertainty in the geometry, and given the general uncertainties in RCS prediction and geometrical design specification, it is unlikely that can exceed 3 or 4 in simulation work and may be rather smaller. Defined as a mesh convergence error the same is true, since on a square grid the number of ray pencils is of the order of resulting in launched rays for In all cases can be regarded as an error estimate of the initial or first reflected rays. The growth of this error from bounce to bounce will allow precise knowledge of the state of the system only as long as where is our estimate of the Lyapunov coefficient for a finite number of ray bounces, N. We can thus set the limit of deterministic predictability when,

This implies that the maximum number of ray bounces permitted, before we can no longer know their position and direction, is given by

This provides an important inequality for straight ducts. In particular, deterministic predictions can only be made if,

The ratio is a measure of the aspect ratio of the duct where can be estimated from (3), although strictly valid only for ergodic ray tracing for large N. The Lyapunov exponent can be determined using the eigenvalue definition, but it may also be estimated from the intensity of a ray. In particular we can show that if is the base-10 logarithm of the ray intensity of the ray at the bounce (defined either per unit ray cross section or per unit divergence angle) then,

with increasing certainty for large N. More accurate estimates are also possible (Mackay, 1998,1999) and we have a simple means to estimate A using the standard Deschamps ray tracing formulation. RANDOM WAVE REPRESENTATIONS In this section we show how a field distribution down a duct can be represented by a sum of randomly directed plane waves, valid in a statistical sense when convergent ray tracing is not possible. We assume a fixed incidence angle, away from any planes of symmetry, and consider the nature of the bistatic RCS resulting from such a distribution. The extension of the model to allow variation with incidence angle, and hence monostatic RCS, is the subject of other work which will not be reported here. Our hypothesis, based on those in the field of quantum chaos (e.g Heller, 1989), is to suggest that in the high frequency limit the fields within a chaos-inducing duct can

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be represented as a coherent sum of randomly directed plane waves with random phase. Such a field can, we suggest, be used to replace the contribution of non-convergent rays assuming incident ray pencil widths which are small but not very small compared to a wavelength. Such fields form members of a statistical ensemble, each member of which is a valid representation of the ‘true’ field in the high frequency limit. The fact that such fields do not satisfy the boundary conditions at the surface of the duct is not important in this limit. We then assume that the random field model can be applied at finite frequency over the frequency regime for which ray tracing is traditionally employed. The random wave component of a transverse electric field in a straight duct is written by,

for independent random variables and to The polarisation is defined by,

uniformly distributed over the interval 0

for the two different states of incident polarisation. The normalisation constant defined by,

is

where is the number of ‘bad’ rays which have diverged past some preset threshold and is the total number of launched ray bundles at the duct entry plane. M must be sufficiently large to satisfy the Nyquist sampling criterion on the boundary, for cross section perimeter and wavelength (not to be confused with the Lyapunov exponent ). The total transverse field is then taken as the sum of the random field and the ray field from the convergent rays. For a sufficiently long ergodic duct, e.g. the stadium duct, the convergent ray contribution will approach zero except at special angles (i.e. when ). As an example we consider a stadium duct which is an example of a structure exhibiting ergodic behaviour and which may be representative of more realistic engine intakes. A stadium duct is defined by a duct whose cross section is composed of two semi-circles of radius a, smoothly joined by two straight sections of length I consider a stadium duct where unfolded duct length 2L = 30.0 a and incident angles for a plane incident wave polarised in the direction. In this special case all rays have diverged beyond computational limits and the field is composed only of the random wave components. Predictions are given assuming a wavenumber Each figure shows a co-polar (top) graph, labled as ‘phi-directed’ and a cross-polar (bottom) graph, labled as ‘theta-directed’. RCS is shown in dB relative to a square metre assuming metre. This is shown as a function of azimuthal scattering angle with elevation scattering angle Figures 2, 3 and 4 show three members of the ensemble of possible random wave predictions whilst figure 5 shows a modal solution (assuming a Kirchhoff approximation) which, neglecting non-Kirchhoff edge diffraction, can be regarded as accurate. These results are typical of those for a long stadium duct and demonstrate the feasibility of using random waves for RCS predictions of chaotic ducts.


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CONCLUSIONS We establish a connection between the Lyapunov exponent and SB ray methods which places a limit on the ability to make numerically deterministic or convergent predictions. When this limit is reached the ray fields are essentially random. One method, valid for ergodic straight ducts, is to represent the random ray field component by a field composed of randomly directed plane waves. We demonstrate that predictions made under this assumption have very similar bistatic RCS characteristics to those of an accurate modal solution. Estimates of the Lyapunov exponent are directly related to average ray divergence and hence average ray intensity. For ergodic ray tracing this can be determined by the tracing of a single ray in order to place limits on the ability to achieve convergence. The computational cost of estimating the Lyapunov exponent is usually negligible compared to the cost of a full RCS prediction where all the rays on the entry plane must be traced. REFERENCES G.A.Deschamps, September 1972, ‘Ray techniques in electromagnetics’, Invited paper, Proceedings of the IEEE, Vol. 60, No. 9. A.J.Mackay, July 1988, ‘Chaos theory and first-order ray tracing in ducts’, IEE Electronics Letters, Vol. 34 No. 14, pp1388-1389 A.J.Mackay, April 1998, ‘Chaos theory applied to first order ray tracing in ducts’, DERA Malvern, UK, report DERA/SN/R/TR980002/1.0, (available through DRIC) A.J.Mackay, March 1999, ‘New representations of ray tracing, chaos theory and random waves in straight ducts’,DERA Malvern, UK, report DERA/S&P/RAD/CR990108/1.0, (available through DRIC) A.J.Mackay, December 1999, ‘Application of chaos theory to ray tracing in ducts’, IEE Proceedings Radar, Sonar and Navigation, Vol 146, No 6, pp298-304 M.C.Gutzwiller, 1990, ‘Chaos in classical and quantum mechanics’,Springer Verlag, Interdisciplinary Applied Mathematics Series. M.Berry, 1989, ‘Some quantum to classical asymptotics’,section 4 pp251, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991. E.J.Heller, 1989, ‘Wavepacket dynamics and quantum chaology’,section 9, pp548, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991. A.J.Lichtenberg and M.A.Lieberman, 1983, ‘Regular and stochastic motion’,Springer Verlag, Applied Mathematical Sciences 38. P.Cvitanovic and B. Eckhardt, August 1989, ‘Periodic quantization of chaotic systems’, Physical Review Letters, Vol 63, No. 8, pp823-826 U.Smilansky, 1989, ‘Theory of chaotic scattering’, section 7 pp371, Les Houches 1989 Session LII, Chaos and quantum physics, North Holland publishing 1991 M.Berry, 1981, ‘Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard’, Eur.J.Phys.2 , pp91-102. Doc. ref. DERA/S&E/RAD/CP000180 ©Crown Copyright 2000 /DERA.

RAY SPLITTING AND CHAOS IN ELECTROMAGNETIC RESONATORS

Reinhold Blümel Department of Physics Wesleyan University Middletown, CT 06459-0155, USA

INTRODUCTION Rays are a fundamental concept in any wave theory. They describe the particlelike propagation of radiation in the small-wavelength limit of geometrical optics. There is no conservation law for rays. Quite the contrary. Whenever the properties of the propagation medium change on a scale small compared with the wavelength of radiation, rays proliferate as they split into transmitted and reflected components. This phenomenon is called ray splitting. Ray splitting occurs in many fields of science and technology. The best-known example is the splitting of a light ray as it encounters the interface between two media with different indices of refraction. The implications of ray splitting in acoustics and quantum mechanics were first investigated by Couchman et al (1992). The presence of ray splitting in a wave system has important consequences for the mean and fluctuating parts of the level density of the system. It was shown by Prange et al (1996) that the average number of states in a resonator – Weyl’s famous mode-counting function – has to be modified in the presence of ray splitting. The essence of ray splitting is best conveyed with the help of a simple onedimensional quantum system, for instance the one shown in figure 1 (see also Bauch et al, 1998). A quantum particle is confined between two impenetrable walls a distance apart. A potential step of height and width produces ray splitting of the rays representing the quantum particle bouncing between the walls. If the energy E of the particle is smaller than the particle (neglecting tunneling) bounces between the left-hand wall and the potential step periodically traversing the orbit D. This is precisely what a “Newtonian” particle would do. Therefore the periodic orbit D is called a “Newtonian” orbit (characterized by N in figure 1). At the quantum particle has many more options for its motion. Starting on the left, it might decide to reflect Ultra-Wideband, Short-Pulse Electromagnetics 5 Edited by P. D. Smith and S. R. Cloude, Kluwer Academic/Plenum Publishers, 2002

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off the potential step resulting in the orbit C. This kind of motion does not occur in Newtonian mechanics since a Newtonian particle at passes straight over the potential hump heading for the right-hand wall. Thus orbit C is a non-Newtonian orbit (characterized by NN in figure 1). The orbit B is non-Newtonian as well, while the orbit A corresponds to what we would expect according to Newtonian mechanics.

Does quantum mechanics know about the existence of the non-Newtonian orbits B and C in figure 1? It certainly does. In order to demonstrate, let us consider the scaled quantum problem defined by where is a scaling constant. We will see later that this is not an artificial, academic problem. It is tightly connected with the spectral problem of electromagnetic resonators partially filled with dielectric substances. Defining the Schrödinger equation of the scaled quantum


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problem leads to the transcendental equation

where is the wave number of the particle. The roots of (1) determine the particle’s energy spectrum Choosing and units such that where is the mass of the quantum particle and is Planck’s constant divided by the real part of the discrete Fourier transform

of the first M = 162 roots is shown in figure 2 as a function of We see a regular sequence of peaks at 0.6 and 0.9 which can be associated with multiples of the optical path length of the non-Newtonian periodic orbit B. The peak at is the (optical) path length of the non-Newtonian orbit C. The largest peak in figure 2, at corresponds to the Newtonian orbit A. The remaining two peaks are explained as composites of the Newtonian orbit A plus one or two additional (non-Newtonian) above-barrier bounces of the particle. As a result we obtain that quantum mechanics is able to identify every single one of the classical orbits of the ray-splitting system of figure 1, whether Newtonian or not. Since Schrödinger’s wave mechanics is essentially a classical field theory, and the system shown in figure 1 is essentially a quantum resonator, we expect similar phenomena to be present in other confined-wave systems, for instance in electromagnetic resonators.

ELECTROMAGNETIC RAY-SPLITTING RESONATORS Electromagnetic resonators partially filled with dielectric substances are excellent examples of ray-splitting systems. We focus here on flat cylindrical resonators operated below the cut-off frequency for the onset of the first non-trivial axial mode. The cross section of a typical ray-splitting resonator is shown in figure 3.

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The resonator has a generic shape. One part of it (perimeter L, area A) is filled with a dielectric with dielectric constant the other part (perimeter area ) is filled with a dielectric with dielectric constant The interface between the two dielectric substances is assumed to be sharp. It is called the ray-splitting boundary (“RS boundary” in figure 3). Electromagnetic ray-splitting cavities of the type shown in figure 3 were first considered by Prange et al (1996) who show that the spectral problem of a ray-splitting resonator is formally identical with a scaled quantum problem. This is demonstrated in the following way. Since the resonator is assumed to be thin in axial direction (assumed to be the direction here) and operated below the axial cutoff frequency, the electrical field in direction is homogeneous, vanishes at the boundary of the resonator (Dirichlet boundary condition) and is a function of and only. If we denote the phase velocities in sections A and of the resonator by and respectively, the wave equation for takes the form

where is the Laplacian in two dimensions and is a function that takes the value 0 if is in A, and the value 1 if is in Identifying the term with the scaling constant shows that the dielectric resonator problem is indeed formally equivalent with a scaled quantum problem. Thus electromagnetic raysplitting resonators can be used as analog computers to solve scaled quantum problems. They can also be used to predict new electromagnetic and quantum phenomena.

CHAOS INDUCED BY RAY SPLITTING Suppose the perimeter of the resonator shown in figure 3 is circular and i.e. the resonator is empty. Then there is no problem solving for its resonance frequencies. They are essentially given by the zeros of the Bessel functions of the first kind. We may also imagine a classical point particle bouncing in this resonator imposing perfectly elastic (specular) reflection at the walls of the resonator. The particle will trace out a regular pattern; its motion is simple and perfectly predictable. But now imagine the circular resonator split down the middle and one of its parts filled with dielectric. A narrow laser beam bouncing in this cavity traces out a completely erratic, chaotic path. Thus ray splitting can induce chaos in an otherwise regular resonator. The quantum analog of the split-circle resonator was studied in detail by Blümel et al (1996). Apart from the split-circle resonator, other resonator geometries were studied that are analytically solvable (regular) in the absence of ray splitting, but become chaotic if dielectrica inducing ray splitting are present. Examples are the triangular ray-splitting resonator studied by Kohler et al (1997) and the annular ray-splitting billiard studied by Kohler and Blümel (1998).

LATERAL RAYS There are special types of orbits that occur only in ray-splitting systems. One of them are lateral rays. Lateral rays lead to periodic lateral-ray orbits whose signatures are found in the Fourier transform of ray-splitting resonators. Kohler and Blümel (1998) identified the signature of a family of periodic lateral-ray orbits in the spectrum of a rectangular quantum ray-splitting resonator. Indications of a lateral periodic orbit were


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also found in the split-circle ray-splitting resonator (Blümel et al, 1996). A periodic lateral ray hits the ray-splitting boundary at the critical angle of total internal reflection, runs along the ray-splitting boundary, emerges from the ray-splitting boundary under the critical angle, and completes its path by joining up with its starting point by specular reflections off the resonator’s boundary, if necessary. Lateral periodic orbits are non-Newtonian orbits.

STADIUM-SHAPED RAY-SPLITTING RESONATOR: EXPERIMENTAL RESULTS Are the signatures of Newtonian and non-Newtonian periodic orbits actually observable in the Fourier transform of the frequencies of dielectric-loaded ray-splitting resonators? This question was answered recently by the experiments of Sirko et al (1997) and Bauch et al (1998).

The ray-splitting cavity used in these experiments is shown in figure 4. It consists of a metallic enclosure in the form of a stadium and a movable (and removable) teflon bar insert a distance d away from the tip of the stadium. In a first set of experiments the teflon bar was removed and 50 resonances were measured. The Fourier transform of the measured resonances is shown in figure 5. The signatures of the simplest Newtonian periodic orbits are clearly visible as peaks in figure 5. No peaks associated with nonNewtonian orbits appeared (control experiment). The signatures of non-Newtonian orbits appeared as soon as the teflon bar was inserted. The simplest non-Newtonian periodic orbits are the “internal bounce orbit” that bounces inside of the teflon bar, and the orbit that bounces between the upper edge of the teflon bar and the tip of the stadium. Figure 6 shows that both non-Newtonian orbits can actually be identified in the Fourier transform of the experimental data. The mobile teflon bar makes sure that the peaks in move in the correct direction as

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a function of d (see figure 6).

R. BLÜMEL


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SUMMARY AND CONCLUSIONS The above discussion shows that flat electromagnetic resonators partially filled with dielectric substances are excellent systems for studying ray-splitting phenomena. Due to a formal equivalence of the stationary Maxwell equations and the quantum Schrödinger equation, flat ray-splitting cavities are also suitable for the study of ray splitting and quantum chaos in mesoscopic devices, important components of the next generation of super computers. The experiments of Sirko et al (1997) and Bauch et al (1998) demonstrate that the Fourier transform of the frequency spectrum of a raysplitting cavity shows peaks that correspond to periodic orbits of a novel non-Newtonian mechanics that forms the semiclassical backbone of ray-splitting systems in the shortwavelength limit. In addition the work of Prange et al (1996) and Kohler and Blümel (1998) showed that ray splitting contributes novel universal terms to the Weyl formula describing the number of modes in electromagnetic and quantum resonators. Raysplitting induced chaos and the emergence of new ray-splitting phenomena, such as periodic lateral rays are the most promising directions for future theoretical and experimental research. Acknowledgement The author gratefully acknowledges financial support of the National Science Foundation, grant number 9900730. REFERENCES Bauch, Sz., A., Sirko, L., Koch, P.M., and Blümel, R., 1998, Signature of non-Newtonian orbits in ray-splitting cavities, Phys. Rev. E, 57:304. Blümel, R., Antonsen, Jr., T.M., Georgeot, B., Ott, E., and Prange, R.E., 1996, Ray splitting and quantum chaos, Phys. Rev, Lett., 76:2476; Phys. Rev. E, 53:3284. Couchman, L., Ott, E., Antonsen, Jr., T.M., 1992, Quantum chaos in systems with ray splitting, Phys. Rev. A, 46:6193. Kohler, A., Killesreiter, G.H.M., and Blümel, R., 1997, Ray splitting in a class of chaotic triangular step. billiards, Phys. Rev. E, 56:2691. Kohler, A., and Blümel, R., 1998, Annular ray-splitting billiard, Phys. Lett. A, 238:271. Kohler, A., and Blümel, R., 1998, Signature of periodic lateral-ray orbits in a rectangular ray-splitting billiard, Phys. Lett. A, 247:87. Kohler, A., and Blümel, R., 1998, Weyl formulas for quantum ray-splitting billiards, Ann. Phys. (N. Y.), 267:249. Prange, R.E., Ott, E., Antonsen, Jr., T.M., Georgeot, B., and Blümel R., 1996, Smoothed density of states for problems with ray splitting, Phys. Rev. E, 53:207. Sirko, L., Koch, P.M., and Blümel, R., 1997, Experimental identification of non-Newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity, Phys. Rev. Lett., 78:2940.


Ultra-Wideband (UWB) Radio-Frequency (RF) Bioeffects Research at DERA Porton Down S Holden, RH Inns, CD Lindsay, JH Tattersall, P Rice and JL Hambrook Chemical and Biological Defence, Defence Evaluation and Research Agency (DERA), Porton Down, Salisbury, Wiltshire SP4 0JQ, UK Abstract Current guidelines controlling exposure of personnel to radiofrequency fields are not specifically relevant to situations where subjects may be exposed to ultra-wide band (UWB) pulses. In addition, there is an increasing awareness that pulses of low mean power may induce cellular effects below the thermal threshold, with effects being found in windows of a particular frequency. The Biomedical Sciences group of the Chemical and Biological Defence (CBD) sector undertakes fundamental and applied research on the biological effects of RF radiation. An ongoing programme is addressing whether personnel may be at risk from exposure to radio frequency radiation of low mean power (both pulsed and continuous wave). The main areas of research are: Physical and computer based dosimetry modelling; Physical models using dosimetry phantoms filled with tissue simulants; Computer based models using computational electromagnetic methods such as Finite Difference Time Domain (FDTD), Transmission Line Matrix (TLM) and Quasi-optical methods.

In vitro human and animal tissue studies; Human and animal cell cultures exposed to UWB; Rat hippocampal brain slice exposed to CW (microwave). This paper will describe details of the work programme and provide examples of findings to date. Introduction Currently, RF health risks are attributed to thermal and “sub-thermal” effects. “Subthermal effects are further divided into the general categories of microthermal (no measurable heating, any transient heating is compensated by homeostatic mechanisms) and non-thermal (no measurable temperature rise). Thermal effects due to RF are well known to produce deleterious effects and are used as the basis of UK/US safety guidelines at MHz/GHz frequencies. “Sub-thermal” effects are much more contentious and are not sufficiently proven for guidelines1 to have been issued. There are some established physiological effects at sub-thermal levels (e.g. microwave hearing) but it is not known if they are hazardous. Presently, sub-thermal effects and in particular non-thermal effects are a major source of investigation worldwide. 1

See recent Independent Expert Group an Mobile Phones (IEGMP) report Mobile Phones and Health - “Stewart Inquiry”


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Ultra-wideband (UWB) sources: The MoD2 problem Novel UWB RF sources are under development in the DERA. Current national guidelines in the UK, produced by the National Radiological Protection Board (NRPB) and the IEEE Standard in the USA do not address human exposure to novel sources such as UWB. This presents a problem, as the MoD is required to develop safety guidelines for MoD/DERA staff to demonstrate “duty of care” as an employer. UWB sources are characterised by the production of very short RF pulses ( 10GHz) is problematical. In particular, the memory requirements of differential methods are high. The number of discrete cells required in differential methods is normally 10 cells per minimum wavelength modelled. If the frequency to be modelled is doubled this can mean the memory requirement can go up 8 fold! Using expertise within DERA it was decided that a hybrid computer model was the best way forward using differential methods for most of the band of interest and quasi optical methods for the high frequency end of the spectra. Quasi-optical methods can be applied in this regime as they have much lower memory requirements and the electromagnetic properties of human tissue do not change much above at the upper end of the UWB spectra.

2

MoD, the Ministry of Defence of the United Kingdom

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The DERA Porton Down hybrid model will consist of a geometric representation of the human being which will have the appropriate material properties assigned to it. Then a series of analysis codes will be applied, consisting of: Differential methods covering most of the band of interest (~ 8GHz): Transmission Line Matrix (TLM) algorithm; Finite Difference Time Domain (FDTD) algorithm. Quasi - Optical methods, based on differential ray tracing, to extend to higher frequencies (>8GHz). To develop codes and assemble the model straight away has an element of risk associated with it as the model could provide incorrect predictions. The approach taken is to undertake model development in conjunction with a validation programme. Validation targets referred to as common target models (CTM) have been designed to allow comparison during model development of: Different computer-based solution methods; Analytical solutions; Direct physical measurements. The CTM will address the two main problem areas encountered when modelling humans over a wide range of frequencies: Modelling frequency dependent lossy dielectrics accurately and; Modelling large dielectric objects at high frequencies. Code development -differential methods At present we expect to have two differential codes developed that will have frequency dependant material models implemented and will be able to perform calculations on parallel machines. The parallel approach has been taken due to the size of the problem and associated memory usage and processing time requirements. To date, the TLM code has an implementation of a Debye, frequency dependent, dielectric model of human tissue (see Figure 1) with 43 tissue types at frequencies of up to 20 GHz, the upper frequency limit being dependent on computer resources. A Computer Aided Design (CAD) interface is available for the software to input human geometry and parallelisation of the code is underway. However, more work is required after parallisation to reduce the memory requirement. This is being investigated by relaxation of the cell requirement to

3

Note: the TLM method has a degree of redundancy in the method of calculation. This approach is acceptable for preliminary investigations.

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The FDTD code also has a Debye tissue model under development and, in addition, parallelisation of the code has almost been completed. Preliminary experiments using parallel differential codes on a DERA SGI Origin 2000 are underway. The model being used is a humanoid with homogenous tissue properties (known as CTM-4). A 4th order (spatial) algorithm is being used which should reduce the cell requirement to cell whilst giving the equivalent accuracy. This should reduce the number of cells to be calculated from cells to cells. Given the memory on the Origin 2000 (32 GBytes) calculations up to ~10 GHz are expected.

In-vitro neurophysiology - brain slice tem cell model Computer-based modelling is being applied to in vitro studies. The in vitro studies consist of investigations into RF effects on organised tissue (brain tissue, heart) and individual cells in culture. An important element of the overall work programme is to relate computer modelling results with in vitro test systems. This is to ensure that field strengths derived from computer models are the same as the in vitro systems are exposed to. Studies have been targeted at understanding the processes involved in exposing in vitro systems to RF and associated dosimetry. Initial studies on in vitro neurophysiology have been made using continuous wave sources. Slices of rat hippocampus have been exposed to RF within a TEM cell (see Figure 2).

UWB RF BIOEFFECTS RESEARCH AT DERA PORTON DOWN

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Dosimetry of the brain slice has been undertaken using non-invasive thermometers (fibre optic), analytical calculations and now computational methods4 (FDTD, see figure 3).

Figure 4 shows a slice of brain tissue of the type used in the experiments at DERA Porton Down. This part of the brain, the hippocampus, is involved in learning and memory as well as epilepsy. An electric stimulus is delivered via an electrode: this elicits an electrical response in the tissue (marked PS). The top graph shows the amplitude of the PS response plotted against time. The lower graph shows the RF field intensity to which the tissue is exposed. Low exposures, at 700MHz, caused a slight increase in the response, the highest intensity caused a dramatic, reproducible and reversible reduction of the electrical response. This effect is at very low field intensities (SAR5 1.5 mW.kg-1) and no temperature rise associated with gross temperature changes was observed. However, since a metal stimulating electrode was used to evoke the response, there is the possibility of an artefact. This issue was addressed in further experiments using a chemical stimulant rather than electrical by introducing a chemical that causes an epileptic like seizure. It was found the RF at 700 MHz and the same field strength, caused an inhibition of spontaneous seizure activity. This reinforces results from the previous study on electrically evoked potentials. It is unlikely the measurement electrode (used in both experiments) consisting of an ionic solution in a glass tube is causing an electrode artefact. However, we are currently investigating optical methods such as calcium ion imaging to confirm this. Other investigations have taken place on isolated heart and ion channels. No effects were observed on isolated heart by UWB or CW fields and no effects were observed on nicotinic ion channels6 when exposed to CW fields at 10GHz.

4

This study is also being used to look at TEM cells with differing apertures (for measurement) and to relate dose to that observed in published head models. Confusingly for radar engineers the unit of RF energy absorption is the Specific Absorption Rate or SAR. 6 Nicotinic ion channels facilitate the propagation of electrical impulses. 5

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Rf-lnduced stress response in the nematode Caenorhabditis elegans

C. Elegans is a nematode worm widely used in biological research. It has recently had its entire genome sequenced. Genetically modified variants of C.elegans have been used for environmental monitoring. One form of modification causes the nematodes to change colour when their cells produce Heat Shock Proteins (HSP). HSP are produced by cells as a protective response to one (or more) of a range of stressors, e.g. heavy metal ions and not just heat as the name implies.

Figure 5 shows the sensitisation of HSP16 expression in C.EIegans with increasing temperature. The upright triangles show sham exposures and indicate the response in the absence of exposure to radio frequency radiation. The downward pointing triangles indicate results after exposure to CW RF at 750MHz. There was a marked difference between exposed compared to unexposed nematodes over the temperature range of 24.5 – 27°C. The results indicated a reproducible biological effect (in this case a stress response) in a well defined model system. Note that there was no measurable increase in temperature during or after exposure to RFR i.e. the effect was not due to gross heating caused by the radiation. Work is now proceeding to expose the worms to UWB.

In Vitro Cell Culture Studies In addition to studies involving organised tissue experiments have also been performed on cell cultures. This is an important element to understanding the biological effects of RF as it permits use of cells of human origin including growing cell cultures from organs “at risk”. Four cell culture models of relevance to current epidemiological concern were used: 2 human bone marrow leukaemia stem cell lines (KG1 and MEG01; non-adherent); Human foetal fibroblast line (HFFF2; adherent); Rabbit corneal cell line (SIRC; adherent - no human equivalent available). The cultures were placed 70cm from impulse radiating antenna (IRA) and exposed to UWB pulses with peak field 5 kV/m, rise times

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