Advances in Imaging and Electron Physics

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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 103

EDITOR-IN-CHIEF

PETER W. HAWKES CEMESIhboratoire d 'Optique Electmnique du Centre National de la Recherche Scientifique Toulouse, France

ASSOCIATE EDITORS

BENJAMIN M A N Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITED BY PETER W. HAWKES CEMES/Luboratoired’Optique Electmnique du Centre National de la Recherche Scient@que Toulouse, France

VOLUME 103

ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright @ 1998 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use, or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923). for copying beyond that permitted by Sections 107 or 108 of the US. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-1997 chapters are as shown on the chapter title pages; if no fee code appears on the chapter title page, the copy fee is the same as for current chapters. 1076-5670/98 $25.00 ACADEMIC PRESS 525 B. Street, Suite 1900, San Diego, California 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com (Inired Kingdom Ediiion published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NWI 7DX,UK http://www.hbuk.co.uk/ap

ISBN: 0-12-014745-9 Printed in the United States of America 98 99 00 01 BB 9 8 7 6 5 4 3 2

1

CONTENTS CONTRIBUTORS . . . . PREFACE . . . . . .

I. 11. 111.

IV. V. VI.

. . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . .

Space-Time Representation of Ultra Wideband Signals EHUDHEYMANAND n M O R MELAMED Introduction and Outline . . . . . . . . . . . . . . . . Time-Harmonic Radiation from an Aperture . . . . . . . . . Time-Domain Representation of Radiation from an Aperture . . Illustrative Example . . . . . . . . . . . . . . . . . . WavepacketsandhlsedBeamsinaUnifomMedium . . . . . Phase-Space Pulsed Beam Analysis for Time-Dependent Radiation from Extended Apertures . . . . . . . . . . . . . . Appendix: Asymptotic Evaluation of the Beam Field in (125) . . References . . . . . . . . . . . . . . . . . . . . .

vii ix

3 7 15 24 30

44 59

60

The Structure of Relief JANJ. KOENDERINK AND A. J. VAN Doom I. 11. 111. IV. V. VI.

Introduction . . . . . . . . . . . . . The Differential Structure of Images . . . . Global Description of the Relief . . . . . Contours: Envelopes of the Level Curves . . Discrete Representation . . . . . . Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . .

.

. . . . . . . .

. . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . .

. . . . . . , . . .

66 83 98 138 140 147 147

Dyadic Green’s Function Microstrip Circulator Theory for Inhomogeneous Ferrite With and Without Penetrable Walls CLIFFORDM. KROWNE I. Overall Introduction . . . . . . . . . . . . . . . . . 152 11. Implicit 3D Dyadic Green’s Function with Vertically Layered External Material Using Mode-Matching . . . . . . . . 153 V

vi

CONTENTS

111. Implicit 3D Dyadic Green’s Function with Simple External Material Using Mode-Matching . . . . . . . . , . . . . . . IV. 2DDyadicGreen’sFunctionforPenetrable Walls . . . . . V. 3D Dyadic Green’s Function for Penetrable Walls . . . . . . VI. Limiting Dyadic Green’s Function Forms for Homogeneous Femte VII. Symmetry Considerations for Hard Magnetic Wall Circulators . VIII. Overall Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

.

196 208 213 228 240 273 274

Charged Particle Optics of Systems with Narrow Gaps: A Perturbation Theory Approach M. I. YAVOR I. Introduction . . . . . . . . . . . . . . . . . . . . 278 II. Applicability of Perturbation Methods in Charged Particle Optics 283 111. Calculation of Weakly Distorted Sector Fields and Their Properties withthe AidofaDirectSubstitutionMethod . . . . . . 295 IV. Transformation of Charged Particle Trajectories in the Narrow Transition Regions Between Electron- and Ion-Optical Elements 318 V. Synthesis of Required Field Characteristics in Sector Energy Analyzers and Wien Filters with the Aid of Terminating Electrodes . . . . . . . . . . . . . . . 336 VI. Calculation of the Elements of Spectrometers for Simultaneous Angular and Energy or Mass Analysis of Charged Particles . . . . . . . . . . . . . . . . . 348 VII. Conclusion . . . . . . . . . . . . . . . . . . . . . 384 Acknowledgments . . . . . . . . . . . . . . . . . . 385 References . . . . . . . . . . . . . . . . . . . . . 385 INDEX .

. . . . . . . . . . . . . . . . . . . . . . . .

389

CONTRIBUTORS Numbers in parentheses indicate the pages on which the author’s contribution begins.

EHUDHEYMAN (l), Department of Electrical Engineering, Physical Electronics, Tel Aviv University, Tel Aviv 69978, Israel JANJ. KOENDERINK (65), Buys Ballot Laboratory, University of Utrecht, Faculty of Physics and Astronomy, Princetonplein 5, P.O. Box 8oo00, 3508 TA Utrecht, The Netherlands CLIFFORD M. KROWNE(151), 3810 Maryland Street, Alexandria, VA 22309-2583 TIMOR MELAMED (l), Boston University, College of Engineering, 110 C u d n g ton Street, Boston, MA 02215

A. J. VAN D o o w (65), Buys Ballot Laboratory, University of Utrecht, Faculty of Physics and Astronomy, Princetonplein 5 , P.O. Box 8000,3508 TA Utrecht, The Netherlands M. I. YAVOR(277), Institute for Analytical Instrumentation RAS,Rizhskij pr. 26, 198103 St. Petersburg, Russia

vii

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The four chapters that make up this volume are drawn from very different areas. In the first contribution, E. Heyman and T. Melamed examine a complicated problem that arises in the study of wide-band signals. The familiar frequency-domain approach is not the most appropriate here and the authors describe and explore at length other analytic techniques. This unified presentation of techniques to which the authors have largely contributed should be very welcome to anyone wishing to become familiar with these ideas. The second contribution is really a short monograph on a vexed and important question: how should relief be described and represented? J. J. Koenderink and A. J. van Doorn, who have made numerous contributions to thinking on this problem, give here a systematic and carefully reasoned study of the various aspects of the topic. The themes are defined in the introductory section, after which the authors describe the differential structure of images, the global description of relief, contours, and discrete representations. Readers familiar with Eberly’s book on Ridges in Image and Data Analysis (Kluwer, 1996) will certainly wish to read this chapter. The third chapter needs less introduction from me, being a further contribution by C. M. Krowne on microstrip circulator theory. The appropriate dyadic Green’s functions are explored thoroughly for several designs. Finally, we have a long account by M. I. Yavor on perturbation methods in electron optics. Several practical systems of great practical importance are examined: sector energy analyzers, Wien filters and several conical designs. The theory is set out fully and the examples are analyzed critically, with the result that designers of such systems should find these studies of direct interest. In conclusion, I thank all the authors most warmly for the time and scholarly effort that they have devoted to their manuscripts. A list of contributions to forthcoming volumes is given below. Peter W. Hawkes

ix

PREFACE

X

FORTHCOMING CONTRIBUTIONS Mathematical models for natural images Use of the hypermatrix Image processing with signal-dependent noise Near-sensor image processing The Wigner distribution Modem map methods for particle optics Magneto-transport as a probe of electron dynamics in semiconductor quantum dots Distance transforms ODE methods Microwave tubes in space Effect of proton radiation damage on charge-coupled devices Fuzzy morphology Gabor filters and texture analysis Liquid metal ion sources X-ray optics The critical-voltage effect Stack filtering The development of electron microscopy in Spain Number-theoretic transforms and image processing Contrast transfer and crystal images Conservation laws in electromagnetics External optical feedback effects in semiconductor lasers Numerical methods in particle optics Spin-polarized SEM Sideband imaging Computer-aided design using Green’s functions and finite elements Memoir of J. B. Le Poole Well-composed sets Vector transformation

L. Alvarez Leon and J. M. Morel D. Antzoulatos H. H. Arsenault A. Astrom and R. Forchheimer M. J. Bastiaans M. Ben. and colleagues J. Bird G. Borgefors J. C. Butcher J. A. Dayton J. Deen, T. Hardy and R. Murowinski E. R. Dougherty and D. Sinha J. M. H. Du Buf R. G. Forbes E. Forster and F. N. Chukhovsky A. Fox and M. Saunders M. Gabbouj M. I. Herrera and L. Bni A. G. J. Holt and S. Boussakta K. Ishizuka C. Jeffries M. A. Karim and M. F. Alam E. Kasper K. Koike w. Krakow C. M. Krowne van de Laak-Tijssen, E. Coets and T. Mulvey L. J. Latecki W. Li

xi

PREFACE

Complex wavelets Discrete geometry in image processing Electronic tools in parapsychology Z-contrast in the STEM and its applications Phase-space treatment of photon beams Image processing and the scanning electron microscope Representation of image operators Aharonov-Bohm scattering Fractional Fourier transforms HDTV Scattering and recoil imaging and spectrometry The wave-particle dualism Digital analysis of lattice images (DALI) Electron holography X-ray microscopy Accelerator mass spectroscopy Focus-deflection systems and their applications Hexagonal sampling in image processing Confocal microscopy Electron gun system for color cathode-ray tubes Study of complex fluids by transmission electron microscopy New developments in ferroelectrics Organic electroluminescence-materials and devices Electron gun optics Very high resolution electron microscopy Mathematical morphology and scanned probe microscopy Morphology on graphs Representationtheory and invariant neural networks Magnetic force microscopy Structure, fabrication and performance of color CRTs

J.-M. Lina, B .Goulard and P. Turcotte S. Marchand-Maillet R. L. Moms P. D. Nellist and s. J. Pennycook G. Nemes E. Oho B. Olstad M. Omote and S. Sakoda H. M. Ozaktas E. Petajan J. W. Rabalais H. Rauch A. Rosenauer D. Saldin G. Schmahl J. P. F. Sellschop T. Soma R. Staunton E. Stelzer H. Suzuki I. Talmon

J. Toulouse T. Tsutsui and Z. Dechun Y. Uchikawa D. van Dyck J. S. Villarmbia L. Vincent J. Wood C. D. Wright and E. W. Hill E. Yamazaki


ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 103


ADVANCES IN (MAGINO AND -ON

PHYSICS. VOL . 103

Space-Time Representation of Ultra Wideband Signals EHUD HEYMAN AND TIMOR MELAMED Department of Electrical Engineering-Physical Electronics Tel-Aviv University. Tel-Aviv 69978. Israel

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . I . Introduction and Outline . . . . . . . . . . . . . . . . . . . . . . II. Time-HarmonicRepresentationsofRadiationfromanAperture . . . . . . . . A . Green's Function Representations . . . . . . . . . . . . . . . . . . B. Plane-Wave Representations . . . . . . . . . . . . . . . . . . . . C. Ray Representation . . . . . . . . . . . . . . . . . . . . . . . D . Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . In . Time-Domain Representations of Radiation from an Aperture . . . . . . . . . A . Analytic Signals . . . . . . . . . . . . . . . . . . . . . . . . B. Green's Function Representation . . . . . . . . . . . . . . . . . . C . Time-Dependent Plane-Wave Representation . . . . . . . . . . . . . . D. Ray Representation . . . . . . . . . . . . . . . . . . . . . . . E . Time-Dependent Radiation Pattern . . . . . . . . . . . . . . . . . IV. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . A . The Initial Field . . . . . . . . . . . . . . . . . . . . . . . . B. The Time-Dependent Plane-Wave Spectrum . . . . . . . . . . . . . . C . Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . D . Special Case: Well-Collimated Condition . . . . . . . . . . . . . . . E. Frequency-Domain Interpretation . . . . . . . . . . . . . . . . . . V. Wavepackets and Pulsed Beams in a Uniform Medium . . . . . . . . . . . . A . General Solution . . . . . . . . . . . . . . . . . . . . . . . . B. Properties and Interpretation . . . . . . . . . . . . . . . . . . . . C. Relation to Complex Source Pulsed Beams (CSPB) . . . . . . . . . . . D . Relation to Time-Harmonic Gaussian Beam . . . . . . . . . . . . . . E. Considerations in UWB Synthesis of Collimated Apertures: Isodifiacting versus Isowidth Apertures . . . . . . . . . . . . . . . . . . . . . . . VI. Phase-Space Pulsed Beam Analysis for Time-Dependent Radiation from Extended Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Frequency-Domain Formulation: Beam Summation . . . . . . . . . . . B. Time Domain: Pulsed Beam Summation . . . . . . . . . . . . . . . Appendix: AsymptoticEvahationof theBeamFieldin (125) . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Volume 103 ISBN 0-12-014745-9

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3 7 1 9 11 14 15 15 16 18

22 23 24 24 26 21 28 29

30 32 33 40 40 41 44 46 51 59 61

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ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright @ 1998 by Academic Rws Inc. AU rights of reproductionin any form reserved. ISSN 1076-5670198$25.00

2

EHUD HEYMAN AND TIMOR MELAMED

LIST OF S m B O L S For the wavefields we use a self-consisted symbol system: Time-dependent fields have no special mark; analytic time dependent fields are identified by +; frequencydomain fields are identified by ; plane-wave fields (in either the time or the frequency domain) are identified by -; functions in the initial (data) plane z = 0 are identified by a subscript 0 .

-

Time-dependent field Analytic time-dependent field; see (21) or (22) Frequency-domain field Time-dependent field in the z = 0 plane Frequency-domain field in the z = 0 plane Time-dependent plane-wave spectrum; see slant-stack transform (33) Analytic time-dependent plane-wave spectrum; see (30) Frequency-domain plane-wave spectrum; see (3) Local (phase-space) spectrum in the frequency domain; see (1 10) Local (phase-space) spectrum in the time domain; see (130) Window function in the frequency domain; see (110) Window function in the time domain; see (131) Phase-space beam propagator in the frequency domain; see (115) Phase-space pulsed-beam propagator in the time domain; see (137) 3D coordinate point Time coordinate Transverse coordinates Transverse coordinates in the z = 0 plane Plane-wave spectral coordinates; see (3) and (30) Direction of the spectral plane wave; see ( 5 ) and (7) Spectral time coordinate; see (30) Phase-space coordinates for frequency-domain fields Phase-space coordinates for time-dependent fields Propagation direction of the phase-space beam; see ( 1 17)

SPACE-TIME REPRESENTATIONOF ULTRA WIDEBAND SIGNALS

I.

3

INTRODUCTION AND OUTLINE

Continuing advances in the development of radiators and receivers for electromagnetic, acoustic, and elastic waves have provided the capability to gather data with a high degree of reliability. The recent trends are toward ever wider signal bandwidths because of the enlarged database with high temporal-spatial resolution provided thereby (see examples in [l-31). To extract from received signals the desired information requires data processing methods that link features (observables) in a signal to features in the environment encounteredby the signal during its travel from source to receiver. Such observables-based parameterization requires a thorough understanding of pulsed-fields synthesis, of propagation and scattering of such fields, and of processing short-pulse scattering data. Because of the broad frequencyband of these fields, the conventional route of inversion of frequency-domain(FD) solutions is often less convenient and physically less transparent than direct treatment in the space-time domain where the fields are well localized. Furthermore, direct time-domain (TD) analysis and processing techniques and “TD thinking” make explicit use of the TD observables and can be used to relate distinct TD observables (events) with instantaneous features of the sources, and vice versa. However, it is desirable, whenever possible, to formulate the TD and the FD techniques within a unified analytical framework, thereby emphasizing the differences introduced by the short-pulse excitation. Choosing the proper processing technique, either in the frequency domain or in the time domain, can then be posed as a trade-off that can be expressed in terms of the problem parameters (e.g., pulselength, space-time observation range, etc.). Good engineering, therefore, requires thinking in both domains. Following this motivation we shall consider several alternative field representations in both the frequency domain and the time domain. They involve Green’s function and plane-wave representations, as well as pulsed beams and local spectrum techniques. Such techniques have been utilized for various applications, including antenna characterization,field analysis, inverse scattering, and so on. (See an overview of the applications in [ 1 4 1 as well in the following discussion and references.) The focus of this paper is thus a review of some analytic techniques and wave solutions that are relevant to the analysis of ultra wideband short-pulse fields and data. To convey the ideas in the simplest format we consider the simple problem of short-pulse radiation from an aperture distribution (Fig. 1). The given field in the initial plane may be the physical time-dependent source (forward problem) or it may be a measured field due to a remote sensing or scattering experiment (inverse problem). In the forward problem one is interested in calculating the radiating field, and in its parameterization and optimization. In the inverse case, the time-dependent data should be processed in order to extract

4


FIGURE 1. The-dependent radiation from an aperture in the z = 0 plane. The adopted notations for the Cartesian coordinates are r = (x, e), with x = ( X I , X Z ) . Points in the z = 0 plane are denoted as ro = (m, 0). u(r. 1 ) is the radiated field while the known field distribution in the z = 0 plane may be either the field itself uo(w,t ) = u(r, t)l,=O or its normal derivative U , O ( ~ t, ) = a,u(r, r)l,=o.

the information on the sources (either real or induced sources). ’Qpically, inverse processing involves the same radiation-type integrals used in forward problems. In view with the strategy outlined earlier, we start in Section I1 by reviewing the corresponding FD representations,namely, the Green’s function (or Kirchhoff) integral and the plane-wave integral. The properties of these representations and the physical structure of the field are then discussed by considering explicit approximations, namely, the ray solution and the far-field pattern. Having reviewed the FD formulation, we proceed to discuss the corresponding TD formulations (Section 111). Here, too, we start with the Green’s function and with the plane-wave representations. The first has the well-known form of integration of retardedfields (see, e.g., [ 5 , 61). In the second, the radiating field is expressed as an angular superpositionof time-dependent plane waves. The spectral function (i.e,, the time histories of the plane waves) is found via a Radon transform processing of the space-time data. This operation has the form of a slant stacking of the data; hence, we adopt the notation slant-stack transform (SST). Physically, it extracts the directional information from the space-time data, thereby expressing the field as an angular superposition of time-dependent plane waves. The SST is the TD analog the spatial Fourier transform used in conventional plane-wave representation of time-harmonic fields [5,6]. The TD plane-wave representation has been introduced originally in [7] and [8], and has been extended later on in the spectral theory of transients (STT)in [9] and [lo] as an analytical framework for the analysis of the radiation, propagation and scattering of pulsed fields in various configurations. More recently the SST has been used in the context of near-field scanning of short-pulse antennas [ 111. In [ 121, the TD plane-wave representation has been extended to deal with volume sources, and in [13], this volume formulation has been used as a basis for a theory of TD characterization of antenna systems. The TD plane-wave approach and the SST have also been


5

used for the processing of short-pulsescattering data [14,15]. Finally, in [16], this approach has been combined with a Born-type inverse scattering to formulate a TD difiuction tomography (which is the TD analog of the conventional diffraction tomography used for time-harmonic fields [17, 181). In the present paper we demonstrate the time-dependent plane-wave spectrum approach in the context of the problem depicted in Fig. 1. As in the frequency domain, the TD plane-wave spectrumconsists of both propagating and the evanescent spectra; the latter one describes pulsed fields that decay away from the aperture. Using an analytic signal representation, we derive a unified representation that combines both the propagating and the evanescent spectra. However, for the benefit of the practitioner we also consider the separate role of each spectral constituent and present explicit expressions for each of them directly in terms of the real time-dependent data. To clarify the field structure and the numerical properties of the TD plane-wave integral, we also consider approximate field solutions. In the near zone the pulsed field propagates along space-time rays that emerge from the aperture along directions that are determined by the gradient of the delay function (Section 1II.D). We provide explicit expressions for the ray signal, and show that it is Hilbert transformed when the ray passes through a caustic (a ray envelope that forms a focus surface; see Sections II.C and II1.D). In the far zone (Section 1II.E) the field can be approximated by the pulsed radiation pattern, which is directly related to the time-dependent plane-wave spectrum (i.e., the SST) of the source distribution. The distance where the transition from the near-zone local ray representation to the far-zone global plane-wave representation occurs, depends on the spatial width of the aperture and on the pulselength. Borrowing FD terminology, it is termed here the TD Fresnel distance [see (28) and (47)J.Finally, all these TD characteristics are illustrated for an analytic example (Section IV). In Section V we explore the characteristics of well-collimated ultra wideband wavepackets. Such space-time wavepackets are useful in various applications, including modeling of ultra wideband radar or sonar beams, local interrogationof the propagation environment,transmission of localized energy, secured high-rate communication, and so on. Several classes of such localized space-time wavepackets solutions have been introduced recently [ 19-32]. In the eighties, these wavepackets have been studied primarily in the context of possible synthesis of high-energy, nondiffracting, or weakly diffracting wavepackets (i.e., wavepackets that remain localized up to very large distances), but more recently the emphasis has been placed on other applications (see the following discussion). The possible excitation of such nondiffracting waves by physically realizable sources are discussed at the beginning of Section V, based on the time-dependent radiation concepts introduced in Section III. The rest of Section V is devoted to one class of wavepacket solutions, termed pulsed beams (PB) [30-321. These highly localized wave solutions can be

6


realized by a distributed source in an aperture with jnite support. Unlike other wavepackets, the PBs behave classically in the sense that they remain collimated up to a certain distance (the TD Fresnel distance), and thereafter they diverge along a constant diffraction angle. However, they may be considered as optimal solutions to the collimation problem in the sense that all frequency components remain collimated up to the same distance (the Fresnel distance) [33]. Both the Fresnel distance and the diffraction angle can be controlled by the time-dependent source distribution. The PBs have been used for modeling of wavepackets’ propagation in free space [30,31,33], and for high-resolution probing [34-371. In addition to the foregoing applications, the PBs also furnish a complete basis for general representations of space-time signals that express the field as a phase-space superposition of PBs emerging from all points in the source domain and in all directions. The motivation in these PB summation representations is to obtain local spectral representations that can also be tracked through complicated media. Unlike the more conventional plane-wave representation, each basis function in the PB summation representations accounts only for the local radiation properties of the source near the PB initiation point. Further localization is due to the fact that only those PB basis functions that pass near the space-time observation point need to be considered in the superposition integral. Finally, unlike plane-wave representations, the basis functions in the PB representation can readily be tracked through ambient media, and unlike ray fields, they are insensitive to transition regions (such as caustics or shadow boundaries) [32]. Several alternative schemes for PB summation, which apply for different source configurations have been introduced in [38, 39, 10, 40-421, and will be reviewed in Section VI. As an example for a local PB summation approach, we consider in Section VI the problem of time-dependent radiation from an aperture depicted in Fig. 1. The analysis is based on a local (phase-space) processing of the time-dependent data which extracts the local features of the source and thus expresses the radiated field at a given space-time region only in terms of the relevant local contributions of the source. The processing transform is a local slant-stack transform, which is a windowed version of the SST mentioned earlier and thus extracts the local planewave (directional) information about the window center. Physically, it matches a PB wave propagator that propagates away from the window center along the local radiation direction. This description should be contrasted with the SST that extracts the global directional information of the source distribution, and thus for each direction matches a global time-dependent plane wave to the entire aperture. The local approach is therefore termed a phase-space pulsed beam summation [39, 10, 40, 411. The trade-off in this representation relative to the conventional representationsis an a priori data localization versus more complicated processing integrals and databases. It becomes efficient, therefore, when dealing with very large data objects like those obtained by gathering ultra wideband information over

SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS

7

very large apertures. Specifically,this approach has been utilized in [43] and [44] for local inverse scattering using short-pulse scattering data. This paper is intended to review ideas and techniques associated with spectral representations for time-dependent radiation and propagation, including integral representation, localization techniques, and local wave phenomena. The first half (Sections 11-IV) focuses on “conventional” TD techniques, while the second half (SectionsV-VI) deals with localized waves and local spectrum techniques. Various applications of these techniques have already been mentioned earlier and will be discussed in further detail in the text. This text, however, focuses only on a limited selection of TD formulations and techniques out of those developed recently by these authors and others.

11. TIME-HARMONIC REPRESENTATIONS OF RADIATION FROM AN APERTURE To demonstrate the alternative representationsinvolved in the space-time analysis and processing of time-dependent radiation, we consider the simple model of radiation in a uniform medium with wave speed c (Fig. 1). Specifically we shall be interested in the scalar field u (r, t ) radiated into the half plane z > 0 due a given field distribution in the z = 0 plane. The given field can be either the field itself u o ( q , t ) = u(r, t)lz,O, or its normal derivative U , O ( X O , t ) = a&, t)1,,0, or both. The adopted notations for the Cartesian coordinates are r = (x, z), x = (XI, XZ) while points in the z = 0 are denoted as ro = (xg, 0). This configuration is also relevant for space-time processing wherein uO(r0, t ) is the measured data. We start, however, with the well-known field representations in the frequency domain, which will then be extended to the time domain. Henceforth, we use a circumflex to denote time-harmonic constituents with an assumed exp(-iwt) time dependence (see List of Symbols). In accord with the preceding notations, the known field constituentsin the z = Oplane are either fio(x0) = fi(r)lz=oandor f i z 0 ( x o ) = azfi(r)lz=o. A. Green’s Function Representations

In the Green’s function representation (also termed Kirchhoff or Huygens ’s representation [ 5 ] ) , the known field in the aperture plane z = 0 is replaced by an equivalent source distribution. The radiated field for z > 0 can then be described as a superposition of spherical wave contributions from all points in the aperture (see Fig. 2). In the time-harmonic case, the superposition integral has the form

8


FIGURE2. Green’s function representation of the radiated field: The radiated field for z z 0 is described as a superposition of spherical wave contributions W from all points xo in the aperture. The signals and the notations correspond to the TD representation (25) with (25a-c). The corresponding time-harmonic representation is given in ( I ) with (1 a x ) .

where $(r, ro) is a spherical wave, emitted from each point ro in the z = 0 plane (Fig. 2). It is related to the data on the z = 0 plane via either one of the alternatives

where &kR

w

G(r, ro) = - k = -, 43rR’

C

R = Ir-rol

is the free-space Green’s function and c is the wave velocity. Its normal derivative with respect to the z = 0 plane is given by

Some words of caution should be added here. The three representations in (1a-c) are equivalent only if the initial field at z = 0 is strictlyforward propagating, that is, if all the sources (real or induced) are located at z -= 0. As implied by (1b, c), the field representations in this case require a knowledge of only l i 0 or of its normal derivative ii,~. Equations (lb, c) cannot be used, however, if ii consists of both forward and backward propagating waves. In this case one should use the representation in (la) wherein the kernel $ extracts the forward propagating part in the initial field iio. The integral (I), therefore, does not describe the total field ii but only a+, that is, the forward propagating part of the field for z > 0 [see (9)].

9

SPACE-TIMEREPRESENTATIONOF ULTRA WIDEBAND SIGNALS

B. Plane- Wave Representations Expression (1) describes the field as a superposition of point source fields, excited by the initial field distribution via the alternative relations in (la-c). The radiated field can be described instead as an angular superposition of plane-wave fields, which are matched to the initial field distribution (or sources). The plane-wave spectral information is recovered via the transverse transform pair

The over-tilde notation for spatial spectrum will also be used for TD spatial spectrum [see (29), (33) and List of Symbols]. Thus, ' will be used for the plane-wave spectrum of time-harmonic fields, obtained via the spatial Fourier transform (3a), whereas will be used for the time-dependent plane-wave spectrum as obtained from the transform in (33). Anticipating extension to the time domain we also use here a frequencynormalized spectral variable (transverse wavenumber) = (61,h).Accordingly, has a frequency-independent geometrical interpretation in terms of the planewave angle [see (7)] that can readily be extended to the time domain. To derive the plane-wave representation of the field we use the spectral representation of 6 [45]

-

0 and the ray tube diverges. As follows from the preceding discussion, if the spectral spread of the source is large (i.e., the rays emitted from the aperture are not parallel), then the ray representation can be tracked up to the far (or radiation) zone: If p1,z > 0, the ray representation is tracked continuously all the way up to the far zone, whereas if pl,2 < 0, the ray representation breaks at the caustic, but it can be continued beyond it and up to the far zone using (15). For well-collimated apertures, on the other hand, the rays are nearly parallel (i.e., the phase satisfies VO@ 2i const.); hence, the simple ray representation fails in the far zone. In this case, the far-zone field may be expressed effectively by the spectral representation as described later.

+

14


D. Radiation Pattern For observation points in the far zone, one may approximate

where cos +(P, Po) = f a PO. We use here the vector notations r 3 Irl, P = r / r and the same notations for the source coordinates ro. When substituted in the exponent e i k ( r - r o ( in (2), the second correction term in (I 6) may be neglected if

where L is a measure of the source dimension in the z = 0 plane. In this case we 'tr may use in (2) 6 2: &eikC'ro and thus obtain from (lc)

where the radiation pattern is

g(P) = -2ik cos8&(i$)lc=x/r.

(19)

Here $0 is the spectral distribution of iio as defined in (3b), 8 is the observation angle relative to the z-axis and x is the transverse coordinates of the observation point (i.e., r = (x, z)). For simplicity we used here only the field representation in (lc): A similar expression for is obtained from (lb) if the data in the z = 0 plane is i i , ~rather than i i ~ . modulated by an From (18), the far field has a form of a spherical wave angular radiation pattern g (note that for large apertures the angular variation due The condition i$ = x/r implies to cos 8 is weak in comparison to that of &l+x,r). that the radiation pattern in the direction P is determined by the spectral plane wave t o that propagates in the direction R = P [see (7)l. We may contrast now the general properties of the alternative integral representations in (1) and (6) as applied for observation points in the far zone. The integral (6) has a highly oscillatory phase exp[ik(i$ x (z)] = exp[ikr(P . A)] where P is the observation direction and R is the spectral propagation direction. Thus, the far-field expression in (19) is obtained as a local stationary point contribution about the pertinent spectral direction A = P. The integrand in (l), on the other as implied by (16). Thus, in general hand, is dominated by a linear phase eikP.ro the integration in (1) must be carried over the entire aperture. The result in (19) is thus a global contribution of the aperture.

+


15

REPRESENTATIONS OF RADIATION FROM AN APERTURE 111. TIME-DOMAIN

Having reviewed the basic formulations and concepts associated with the timeharmonic problem, we return now to the subject of time-dependent fields. The more efficient way to handle ultra wideband excitations, at least in the shortpulse case, is to formulate the radiation integral directly in the time domain. The alternative approach is to transform the data into the frequency domain, and then calculate the radiation integrals on a frequency-by-frequency basis, and finally transform the results back into the time domain. In particular for large distances, the frequency axis must be sampled very densely. This is due to the fact that typically the time-harmonic field has the form [cf. (12)] ii(w) E,ii,eiurj where ti are propagation delays along the rays. If tj are large, ii is a rapidly oscillatory function of w that has to be sampled densely in frequency when transformed into the time domain. In certain cases, a common term e'"', where t is an average delay, may be extracted, but the procedure for estimating t is quite complicated, in particular under multipath conditions or if ii is specified numerically. Furthermore, this procedure has to be repeated for every observation point. Thus, for short excitation pulses, direct solutions in the time domain where the field is well localized are preferable. With careful interpretation, though, many of the FD concepts and techniques can be translated to the time domain.

-

A. Analytic Signals

The TD field formulations will be derived by transforming the corresponding frequency-domainformulationsinto the time domain via the inverse Fourier transform

However, in order to gain flexibility in the derivation, in particular in those formulations that involve evanescent spectra, it is convenient to use analytic signal representation defined via the one-sided inverse Fourier transform

+

Here f ( t )is the dual analytic signal correspondingto the real signal f (t) with frequency spectrup f ( o )(see List of Symbols). From the integral definition in (20) it follows that f ( t ) is an analytic function in the lower half of the complex t-plane.

16


This function may also be defined directly from the real data f ( t ) via

+ In most cases one is interested in the limit of f ( t ) on the real t-axis [see (35) for an examele where one needs the function in the complex domain]. From (22), the limit o f f on the real t-axis is related to the real signal f ( t ) via

+

. k t ) = f ( t > iE f(r),

t real

(23)

where 7-f is the Hilbert transform 7-ff = (-nt)-' 8 f , with 8 denoting a cpnvolution. The real signal for real t can therefore be recovered via f ( t ) = Re f(r).

B. Green's Function Representation We start with the Green's function formulation in (1). Utilizing the TD Green's function [cf. (2)]

we obtain the retarded field integral (Fig. 2) d2xo @(r,ro, t - R / C )

(25)

where R = Ir - rol and the retarded kernels are [cf. (la-c), respectively]

+

( Z / R ) [ R - ~ U ~c-'ub] - u,O

2(z/R)(R-'uo

+C ' U ~ )

(25a-c)

where uo(x0, t ) = u lZ=o and U,O(XO, t ) = 8,u l z are ~ known functions and the prime denotes a time derivative. As discussed in connectionwith (1) and (6) [see also (9)], the three representationsin (25a-c) are equivalent only if the initial field at z = 0 is strictlyforward propagating. As implied by (25b,c), the field representation in this case requires a knowledge of either uo or its normal derivative u , ~ Equations . (25b,c) cannot be used, however, if u consists of both forward and backward propagating waves. In this case one should use the representation in (25a) wherein the kernel 9 extracts the forward propagating part in the initial field U O , so that the integral (25a) describes only theforwardpmpagatingpart of the field for z > 0. Numerical implementation of the integral (25) can be made efficient by utilizing the fact that the radiated field is localized in space-time, in particular if the excitation is a short pulse. For a given observation point r, one is interested in calculating the integrals only within a short time window. Furthermore, taking into

17


FIGURE 5. Radiation of collimated pulse. An aperture of width L is driven by a pulsed field with pulselength T .

account the delay R / c from each point ro in the aperture to the observationpoint r, one may limit the integration domain to those points that actually contribute at the specified observation time. For observations in the far zone, on the other hand, in particular under well-collimated conditions (see following example), R does not change much as a function of ro. Hence, all points in the aperturecontribute almost simultaneously. Here, it may be preferable to revert to the spectral formulations (Section 1V.B). To illustrate these considerationswe consider a simple example of a collimated pulsed distribution (Fig. 5 ) . It is assumed that the pulse width T is much shorter than the aperture diameter L , namely, L

>> cT

(26)

and that the pulse delay is uniform across the aperture (Fi 5). For simplicity we only consider points along the z-axis. Using R = 21 z &2z, where po denotes the radial coordinate of the integration point Q, we find that the integration domain in (25)is restricted to

d

t - z / c - T < P , ' / ~ c z< t - Z / C

h

+

(27)

The properties of integral (25) are, therefore, determined by what may be termed the TD collimation (or Fresnel) distance F = L2/cT

(28)

In the collimation zone ( z F), on the other hand, the contributions in the space-time window 0 c t - z/c < T come from the entire aperture. The foregoing discussion provides a TD interpretation to the conventional FD definitions of the near (or Fresnel) zone versus the far (or Fraunhofer) zone. By replacing in (28) T-' w, expression (28) for F reduces to the so-called Fresnel distance k L 2 . Beyond this region [see the Fraunhofercondition (17)], the quadratic phase error in the phase may be neglected, resulting in a global contribution from the aperture in terms of its Fourier transform.

-

C. Time-Dependent Plane- Wave Representation The Green's function representation (25) describes the field as a superposition of spherical wavefronts, which are emitted from each point in the aperture (Fig. 2). In this section we shall consider the dual representation in terms of a spectrum of time-dependent plane waves (see Fig. 3). The waveforms of these plane waves are related to the time-dependent data in the z = 0 plane via what we call the time-dependent spatial spectrum. The time-dependent spatial spectrum can be calculated directly from the timedependent data without recourse to the frequency-domain plane-wave representation (6): The operative relations will be derived by inverting the operations in (3a,b) and (6) to the time domain. To simplify the mathematics, we employ the analytic signal representation, which basically extends the functions into the lower half of the complex t domain where the integral representations converge well. The final formulas will be expressed, of course, for real t . The analytic signal representation also furnishes compact expressions that unify the propagating and the evanescent spectra in a single framework [see discussion after (31)]. We shall use the notational system mentioned after (3a) ( see also List of Symbols): Spatial spectrum constituents are denoted by whereas FD constituents are denoted by a *. Thus, spatial spectrums of time-harmonic and of time-dependent fields are denoted by ' and -,respectively. An over + above any time-dependent field describes its dual analytic signal. With U O ( X O , r) representing the time-dFpendent field in the z = 0 plane, the analytic time-dependent spatial spectrum Go((, t)is defined by

giving, using (3a),


19

FIGURE 6. The slant-stack transform (SST) of the time dependent data [a Radon transform in the three-dimensional (XO, 1 ) plane]. For a given E, the SST extract from the data a transient plane wave that propagates in the R = (t,5 ) (see Fig. 3).

Note that an important feature of the spectral formulation in (3a) is that the spatial wavenumber is normalized with respect to w and, thus, has afrequencyindependent geometrical meaning in terms of the plane-wave angle [see (7)]. This permits an inversion in the order of integration (legitimate when Im t 5 0 and in the limit of real r ) and a closed-form evaluation of the w-integration, giving

Repeating the same procedure for the field representation in (6) with (6c), one obtains

&r, t ) = -(21rc)-*

s

d26a:Ho[ 0 but a1 may be any number; if IYI = 0, then (62) is a pure Hilbert transform of (48)]. In order to understand the waveforms involved in (62), we spall express them explicitly in a standard form. Noting that the argument of the 6 function in (62) has an imaginary part, we shall express it symbolically as 6(r - i c ) where E = TO ; C - ' ~ R sin 8 > 0. Expression (62) involves, therefore, the following standard waveforms:

i +

+ 1 & ( t ) = Re6(t - i c ) = -

1

l7€ ( t / € ) 2

t 1 'Hl&(t)= Im6(t - i c ) = -

l7E

+1

-t/c

(f/€)*

+1

The amplitude of+the radiation pattern (62) is controlled by the imaginary part of the argument of 6: c = ;To ; c - ' ~ Rsin8 > 0. Noting from (45) that the far-field pattern is related to the spectral pattern via 161 + sine, we may express the beamwidth of the radiation pattern in terms of the spectral width of (60)

+

sin@ = D

(64)

Under the well-collimated conditions (54)-(53, we have a narrow spectral spread, that is,

@zD 1, the integral can be evaluated in closed form by using 5' 2: 1 - 1(12 as in (66), giving

This is the well-known expression for a Gaussian beam. Thus, the pulsed beam field in (66) is the TD counterpart of an ultrawideband spectrum of the Gaussian beams (70). These Gaussian beams, however, have the special property that they are characterized by a frequency-independent parameter a!, that is, they have a frequency-independent collimation (Fresnel) distance ( Y R but a frequencydependent width as described by (69). More on these pulsed beams and Gaussian beams will be said in the next section.

30


v. WAVEPACKETS AND PULSED BEAMSIN A UNIFORM MEDIUM This section deals with the characteristics of well-collimated, short-pulse (ultra wideband) wavepackets. Such space-time wavepackets are useful in various applications, including modeling of ultra wideband radar or sonar beams, local interrogation of the propagation environment (medical probing and imaging), transmission of localized energy, secured high-rate communication, and so on. During the 1980's, the research emphasized the possible synthesis of high-energy, nondiffracting, or weakly diffracting wavepackets, that is, wavepackets that remain localized up to very large distances, but more recently the emphasis has shifted to other applications. Several classes of localized space-time wavepacket solutions have been introduced to address these applications, in particular in connection with the problem of reducing the far-field diffraction [ 19-32] (see also a critical review paper in [53]). In the case of the focus wave mode (FWM) and its relatives [ 19-23], the nondiffracting mechanism is caused by an interplay of forward and backward propagating fields [24]. If the backward propagating part (which cannot be launched by any physical antenna) is removed, then the collimated wavepacket starts to defocus approximately at the TD collimation (Fresnel) distance mentioned earlier. (To explorethis phenomenon analytically, one should substitute the known analytic expression for the FWM in the Kirchhoff radiation integrals (25c), or equivalently (lc) or (6c) or (35c). As discussed there, the Kirchhoff kernel which involves both uo(x0, t) and U,O(XO,t ) extracts the forward propagating part from the initial field [54]). The weakly diffracting Bessel beams [27, 281 are free-space mode functions (i.e., plane waves in cylindrical coordinates; see [45]) whose JO cross section is concentratednear the propagation axis. Ideally, such beams require infinitely wide source distributions, in which case they remain collimated up to infinity, but for finite apertures, they eventually diffract. Supercollimation is therefore achieved by using apertures (source distribution) that are much wider than the beamwidth (i.e., the width of Jo), in which case the wavepacket stays collimated for longer distances than the collimation distance associated with the beamwidth. On the other hand, this wavepacket defocuses much earlier than the optimal collimation distance that can be achieved with the entire aperture [see (47) and (95)]. Thus, the Bessel beam does not provide an efficient solution for applications involving longrange propagation, yet it can be used as a superresolving wavepacket for near-field applications (e.g., medical interrogation and imaging) where the aperture width can be much wider than the beamwidth. The bullets [25, 261 are wavepacket solutions that behave classically in the sense that they diverge at infinity along diffraction cones and decay like r - ' . The intriguing property of the bullets is that they vanish outside the radiation


31

cone (the rge). These solutions have been synthesized by using the fact that TD solutions can be expressed directly in terms of the TD radiation pattern. This is done by first associating the TD radiation pattern with the TD plane-wave spectrum [see (44)-(45)], and then synthesizing the exact solution as a plane-wave superposition [see (32)] [55]. (Note, however, that such representation involves only the propagating spectrum and is valid only after the source has been turned off [56, 121.) Thus, the bullets are formally synthesized by specifying a radiation pattern that vanishes outside the given far-field cone. In the near zone, however, these solutions lose their wavepacket structure and in fact require sources with infinite support. The class of wavepacket solutions to be considered here is termedpulsed beam (PB). Unlike other wavepackets, the PBs behave classically in the sense that they remain collimated up to a certain distance (the TD Fresnel distance), and thereafter they diverge along a constant diffraction angle 0 whose width can be reduced by using shorter pulses or wider apertures. Although the PBs diffract in the far zone, they may be considered to be optimal solutions to the collimation problem in the sense that all frequency components remain collimated up to a given distance (the Fresnel distance) (Section V.D) [33]. These PBs are strictly causal and are generated by a finite aperture. Exact solutions for this class of PB can be obtained in an elegant form by extending the source coordinates into the complex space. These globally exact solutions are, therefore, termed complex source pulsed beams (CSPB) [29-3 11. Here, however, we shall only be interested with the solution in the limited domain that brackets the wavepacket. In this domain one obtains approximate expressions which, however, have a simpler analytical form. Such approximate solutions may be found from the exact solutions (see Section V.C), but here we shall derive them directly from the differential wave equation. Via this approach we shall also derive in Section V.A a generalized class of PB solutions that can be extended to a nonhomogeneous medium. These solutions maintain their general structure even through propagation in such a medium or reflections at curved boundaries [32], and thus can be regarded as eigen wavepacket solutions of the time-dependent wave equation. These solutions may, therefore, be used to model local diffraction phenomena and for local interrogation [35-371. Alternatively, it is also possible to derive these PB solutions by synthesizing an appropriate aperture distribution and evaluating the radiation integral [in fact, expression (66), which has been derived as an example for the time-dependent plane-wave synthesis of pulsed aperture distributions, is a special case of the PBs considered later]. An important application of these PBs is their use as basis functions for spectral expansions of general time-dependent fields [38,10,41,42]. This last subject will be considered in Section VI.

32


A

x1,2

I1

FIGURE10. Pulsed beam in a uniform medium. The drawing depicts a cross section in the principal plane ( x j , z). T. D j . R j and (YR, denote the pulselength, beamwidth, wavefront curvature, and collimation distance, respectively. Note that D j >> cT [see (96)l. The lines y = const. are the propagation lines [see discussion in connection with (83)l. The thick line in the z = 0 plane represents the rigorous source distribution for the globally exact complex source pulsed beam [30].

A. General Solution We consider pulsed beam (PB) solutions u(r, t ) of the time-dependent wave equation

(a:, + a2: +a:

- c-*a:)u(r, t ) = o

(71)

in a medium with uniform wave velocity c. It is assumed that the PBs propagate along the z-axis in the coordinate frame r = (x,z), x = ( X I , x 2 ) (Fig. 10). From reasons which will be clarified soon [see (76)] we utilize the analytic signal representation. Thus, if A(r, t) is an analytic wave solution, then both UR(r,

t ) = ReA(r, r )

and

ul(r, t ) = ImA(r, t ) = 71uR

(72)

are real wave solutions. Here we shall consider only U R since it also defines U I via (72). Furtheyore, U I or any linear combination of U I and U R may be obtained by multiplying u by an appropriate complex constant and taking the real part. Since the PB is localized in space-time we shall express in a moving coordinate frame

A

+

A(r, t ) = U(r, t),

t =t

- z/c.

(73)

The only approximation in the analysis will be to assume that the shape of the wavepacket changes slowly along the propagation path, that is,

la$l 0 are the reciprocg of the eigenvalues of I', (recall that I', is positive definite). Recalling that f in (81) decays as y increases, it follows that the wavepacket contour lines in the transverse plane are described by y = const. Thus, the wavepacket is elliptical in the transverse plane and its principal semi-axes along ( X I , , x12) y e proportional to I1,2. The wavepacket shape also depends on the decay of f (t - i y ) as a function of y [see a specific example in (93)]. As a function of z, the lines y = const. have the same field magnitude and are, therefore, termed


35

propagation lines (see Fig. 10). The wavepacket waist in the x ~ ,direction ,~ is, therefore, located at z that minimizes 11.2(z). 2. Special Case: An Isoaxial Astigmatic Pulsed Beam

In the general case, the two real symmetrical matrices r R and I?! cannot be diagonalized simultaneously;hence, Eq.(80) is an astigmatic PB whose amplitude and curvature axes are nonaligned and their orientation also changes with z [3 11. We shall consider here the simpler case of an isoaxiul PB where the principal axes of r R and of I?/ coincide and, thus, their orientation remains constant with z. The resulting PB is still astigmatic but it has a much simpler structure. In this case, the matrix I' has the diagonal form

+

r ( z ) = diag(c-'(z - iaj)-'],

aj = L Y R ~ ia~,( Y R ~> 0

(84)

where the complex constants a j , j = 1,2, are found from the initial value I'(0). Note that (84) complies with (79) and the condition a R j > 0 guarantees positive definiteness of rl. Equation (80) becomes

[Note thyt (66)js a rotationally symmetric special case of ( 8 5 ) with a1 = 4 2 = a, and f(r) = s(t - ;TO). Recall that (66) has been derived from the timedependent plane-wave integral, in the narrow angular-spectrum limit, whereas the present solution is a direct solution of the differential equation (75)]. To parameterize the properties of the PB we note that in this case Rj and Ij of (82)-(83) are found from 1

z-ia,

-

i +R j 1; 1

giving

As discussed in (82), R, are the wavefront radii of curvature, while from (83) 1, controlsthe amplitude decay along XI,. In aplane z = const., the amplitudecontour lines are ellipses with principal axes xi. In the plane (xi, z ) the amplitude contour lines are described by the condition x ~ / I ~ ( = z )const. The waist occurs at z where I, is minimal, that is, at z = - a l j . Near the waist, for Iz ' Y I ~I > ( Y R ~ Ij , 2: (z a ~ ~ ) / f i ~ ~ ;

+

+

+

36


hence, the beam contour line satisfies y = x;/Zf(z) 2 a ~x;/z2 , = const. and the PB opens up along a constant diffraction angle 0,. The foregoing discussion identifies a~~as the collimation length F, of the PB in the (xi,z ) plane. In fact, a key feature in this PB solution is that all its frequency components have the same collimation distance (see discussion in Sections V.E.1 and V.E.2). As has been discussed+after(83), the beamwidth and diffraction angle also depend on the decay rate of f ( r - iy) as y increases. An example is given in (93) later.

3. The Real Field The real PB field is given by taking the real part of (80) [see (71)]. To clarify the structure of the real field we shall express it in terms of the real waveform f y (t) defined via (see (23))

Clearly, f,,( t ) decays as the parameter y increases: On the beam axis y = 0 and fy = f is strongest, but as the distance from the axis increases, y increases and the waveform f,,weakens. Substituting (87) in (81) and taking the real part, using also A(z) = AR i A I , the real PB field has the form

+

where y ( x , z ) is given in (83). Thus, the real PB field is a sum of two real waveforms: f,,and its Hilbert transform 'Flf,. Their relative amplitudes depend on z via A R ( z )and A / ( z ) where A(z) is given in (80). At z = 0, A R = 1 and A/ = 0 and the waveform is f,,,but for z > 0, the waveform is gradually Hilbert transformed as the proportion between AR and A / changes. Note that for z + 00, A(z) z-' as follows from (80). As an example, consider the isoaxial case where A is now given in (85). Here we may obtain explicit expressions for the amplitude functions A R ( z )and A / (z). Considering, for example, the special case of a rotationally symmetric PB with a1 = a2 = a. Here A(z) = - i a / ( z - ia)so that AR = [al(a~ + z ) +cr:]/[(a~ z ) ~ a:] and A / = - ~ R z / [ ( u I z)* a:]. Substituting into (88) we find that the waveforms of U R change from f,,in the z = 0 plane to z-'(a/ ( Y R X ) ~as, , z++ 00.JNote that the astigmatic PB in (66) is a special case of this example with f ( t ) + S(r - TO)]. Another simple example is Q R , # a~~but a/,= a/,= 0 (i.e., the waists in both principal directions are at z = 0). Here the waveforms change from f,,in the z = o plane to z-',/-'Flf, as z + 00.

-

+

+ +

+

+

37


4. The Aperture Field

The PB distribution in the aperture plane is found from (80)

(89) For simplicitylet us assume that Im I'(0) = 0, so that the initial wavefront is planar. ~ , that In this case the PB is isoaxial-astigmatic with real a!j = c ~ R so

Thus, in order to synthesizethe PB, the pulsed sources at the center of the aperture should be strong and short, but as the distance from the center increases, the sources should be weaker with a longer pulse (i.e., with lower-frequency content). Illustrative examples for specific pulse distributions have been given in [30],[38],and [32].Specifically,for the analytic 6 pulse in (93) we obtain the initial distribution in (48) with a! = Q R . From the general properties of analytic signals, the following qualitative characteristics are found: If the real signal f ( t ) is characterized by a pulselength TO, then the pulselength T, and the amplitude A, of the distribution in (90) are

+ h12/2ca!R1-'* (91) Thus, the energy of the off-center pulsed sources also decay like (TO+ 2 y ) - ' . Tp(m)rv TO -k l%12/2ca!R,

Ap(%)

[TO

5 . The Axial Energy It is also instructive to see how the wavepacket energy changes along the propagation axis. Here we shall only consider the axial energy density defined as Ilu(r, f)1121x=o. The off-axis energy distribution as well as the total energy have been calculated in [37, Appendix B ] . In order to evaluate the energy integral it is convenient to use, again, the analytic signal formulation. According to this formu!ation, the+energy llg1I2 = g2(t)dt of areal signal g ( t ) is one halfof the energy llg1I2 = Ig(t)12dr of its dual analytic signal i ( t ) . Applying (85) for x = 0, and assuming for simplicity that the parameters a!j = a ! are ~ real, ~ we obtain

s

s

where llf1I2 is the energy of the real signal f ( r ) . Thus, the energy density remains essentially constant up to the diffraction points a!Rj where it starts to decay like 2-2.

38


6. Analytic Delta Pulsed Beam Finally, we consider the characteristics of the PB (80) for a specific excitation pulse. A simple example is provided by the analytic delta pulse [see ( 4 9 ) )

The analytic PB field is given now by using (93)in (80) [or in (85)for the isoaxialastigmatic case]. The real field solutionsare given by (88) where the real signals fy (t) and 'Hf, (t) are given now by S,(r) and 'HSc(t)in (63), with 6 = ;TO y and y is given in (83). From the discussion in (63), the half-amplitude pulsewidth and the peak value of the wavepacket are TO 2 y and n-l (; TO y ) - ' , respectively. Thus, the waveform is shortest and strongest on the beam axis ( y = 0) and it decays as y increases away from the axis. The peak reduces to half of the axial value when To y = TO.Solving for y and substituting in (83) using (86a), we find for the half-amplitude beamwidth in the principal direction xi,

+

+

+

+

D j ( z ) = DO,

d

m

,

DO,

=

2dZZ,

(94)

where Doj is the width at the waist z = -arj. From (94),the far-field diffraction angle is 0, = D0,/aRj = 2 4 c T o / u ~ ~

(944

We therefore observe that the physical characteristics of the wavepacket, namely the waist beamwidth Doj,the collimation length Fj = a~~and the axial pulselength TOare related by what we may call the TD diyraction (or Rayleigh) limit: Fj = Dij/4cTo

(95)

Recall that it is assumed in this section that the PBs are well collimated, implyR70 ~ 0 is chosen to satisfy

with wmaxdenoting the upper frequency in the data U O ( X O ,t). The reason for this choice will be discussed after (146). Thus, replacing the window in (120) by

and by applying (21) we obtain, instead of (142), the window function

where from (49), &2)(t) = $ " ( i ) = 2/nit3. This window has essentially the same form as (48) (it is raised to the power 3 and a! is replaced by B); hence, its properties are as discussed in (50)-(53). Finally, it should be noted thpt for j?real and very large, the window in (142) behaves like a planar window S " ( t ) so that the local spectrum integral reduces to the SST limit as in (132). The phase-space pulsed-beam propagators B due to the window (145) are obtained by substituting in (139) the spectral window [see (56)]

as obtained by the slant-stacktransform (30). For well-collimatedwindows, closedform expressions for B can be obtained via an asymptotic evaluation of the resulting integral (139) directly in the time domain [cf. the evaluation of (66)]. For simplicity, however, we shall derive these expressions from the asymptotic FD beams B, in (125). Noting that the TD window in (145) corresponds to the FD window in (144), we find that the corresponding FD propagators are obtained by multiplying the propagators B, in (138) by (-iw)2e-(1/2)0Tg.Furthermore, g the using (122) we have for this FD window f i 2 = ( - i ~ ) ~ : g e - ~so~ that integral in (136) for N t diverges like eWTp. However, if TB is chosen according to (143), then it has no effect over the entire frequency band of the data and we may use Tp = 0 in the expression for f l 2 [see also discussion in (136)l. Applying


57

(138) and, recalling that all the parameters in (125) are w-independent, we obtain B =ReB,

(147) where the beam coordinates ( x b l , nb,Z b ) are defined in (123) and r is given by (126). This expression has the PB form in (80)(subject to a few differences, which will be discussed in the next paragraph). Thus, it is identified as a PB that emanates from the point xo = t, in the z = 0 plane at a time t = i , and propagates in the i% direction alpng the zb-axis. Its pfopyrties have been discussed in (81)-(92), for a general f and in (93)-(96) for f = 8 . As discussed after (126) in connection with the Gaussian beam window, the elements of r in (126) depend only on zf-' = Zb - x b , tan8, whereas in the PB (80) they depend only on the location Zb along the beam axis. Thus, as in (125), expression (147)conforms smoothly with the initial field distribution W(m,t ; Y) at the z = 0 plane. For large Zb, on the other hand, the elements of I? behave as in (127) and, thus, (147) changes gradually into+the conventional PB form (80). Specifically, since in our case I? is diagonal, B in (147) obtains the form i? (8:) with the replacements: z + Z b ; x j +-X b j ; a1 + piz; a2 + p; and f + 8'(t - ~ p as) in (93). The interpretation of this expression is discussed after (93): It is astigmatic PB; in the (z, X b j ) plane the waist is at Zb = -a!,,the collimation distance is F, = c ~ R ~ the beamwidth D, is given in (94),and the diffraction angle at (94a).

4 . Illustrative Example for the Local Spectrum In this example we calculate the time-dependent local spectrum of the initial distribution (48) via the local SST (130), using the Gaussian 8-window (145). The resulting integral can be evaluated in closed form, giving

-2ilsc Uo(Y) = Re a-l + p*-1

(1

i - -To - -Tp i 2

+ c-'(a + /?*I-'

It should be noted that for large and real /? the window in (142) behaves like a plane window 8 ( t ) and the local spectrum reduces to the SST limit in (132) [see discussion in connection with (142)l. Indeed one observes from (148) that if

,

58


IBI >> la[,I%l and Ts 0: Every two integrals are dependent, that is, a(w1, w~)/a(x,y) = 0. When w(x, y) is a complete integral (i.e., not constant over any area), then all integrals are the functions with continuous first derivatives that are dependent of w . There exists a complete integral that is k-times continuously differentiable. The integral can be built from the characteristics. If an integral is required to pass through a given curve (Cauchy’s problem), a unique solution exists when the curve is everywhere transverse to the characteristics. If there are isolated singular points and if the characteristics in neigborhoods of the singular points are closed curves, then there exists a complete integral with w: w; > 0 except at the singular points [l, 191.

+

The final item applies to the level curves but not to the creep equation, of course. The creep equation is generically not exact; it fails because p x # -qy, that is, zxx zyy = Az # 0. Thus, the Laplacean of the height measures “how nonexact” the creep equation is. The partial differential equation for an integrating divisor is

+

Itscharacteristicsaretheintegralcurvesofthevectorfield( p ( x , y), q ( x , y), dAz), where p ( x , y), q ( x , y) and Az(x, y) are known functions of ( x , y). Thus, the projection of the characteristics on the xy-plane are the projections of the fall curves, whereas the vertical component is the Laplaceanof the height that measures the degree of nonexactness of the creep equation. Notice that the degree of nonexactness Az equals az/as, that is, the derivative of the height function with respect to its scale. Thus, the integrating divisor d ( x , y)

104

JAN J . KOENDERINK AND A. J. VAN DOORN

has an intimate relation to the structure of the scale space of the height function z(x, y ; s). 1. Nature and Existence of Complete Integrals

The creep equation p dy - q dx = 0 is guaranteed (though not likely globally) to have an integrating divisor 6 ( x , y ) such that O ( x , y ) dw(x, y ) = p d y

- 4 dx,

(59)

where w ( x , y ) denotes a complete integral. Neither the integral nor the integrating divisor are unique, of course. If F ( 6 ) denotes an arbitrary function, then O’(X, Y ) = O ( X 3 y ) F ( w ( x ,Y ) )

(60)

denotes another integrating divisor of the creep equation. Notice that solutions of the creep equation must simultaneously be solutions of

m,y ) F ( w ( x ,Y ) ) = 0.

(61)

We may assume the integrating divisor and the general solution to be independent functions; that is,

We will assume this to be the case throughout the paper. When the integrating divisor and the general solution are dependent, the curves 6 ( x , y ) = c’ would also satisfy w ( x ,y ) = c ; that is, the level curves of the integrating divisor would coincide with the fall curves defined by the general solution.

2. Singular Solutions The curves 6 ( x , y ) = 0 satisfy the creep equation. When 6 and w are independent, these curves cannot be members of the complete integral. Such curves are singular fall cuwes; though they satisfy the creep equation, the complete integral doesn’t capture them [55, 301. We will call the fall curves that can be represented as w ( n , y ) = c as regularfall a w e s .

3. Consistent Labeling of Fall Curves The congruence of level curves has a natural parameterization: We simply use the value of z as a label for the level curve. The only ambiguity that might arise is that the level curves often consist of a number of disconnected components. But at least in a sufficiently limited neighborhood the height serves as a convenient way

THE STRUCTURE OF RELIEF

105

to label the level curves. The classical terminology (German: kofierte Projekfion) stresses especially this aspect. The situation is quite different with the fall curves: On the face of it there appears to be no principled way to label them. One might of course single out a fall curve and give it a label (a,say), but there is no way to go from one to the next in “even steps.” Indeed, in the literature the fall curves are usually left without a system of labeling and one is satisfied to simply name a few ones (a,B, y , . . .) that figure in the discussion. There is a solution to this problem if one is able to specify a complete integral of the creep equation: The complete integral of the creep equation itself provides us with a principled way to parameterize the congruence of fall curves. Because the equation w ( x , y ) = c is the implicit equation for a fall curve, the parameter c - o r the value of w-is a label for the fall curve. Thus, w tells us whichfull curve, just as z tells us which level curve. (See Figs. 21 and 23.) Notice that affine transformations of the z-axis merely relabel the fall curves and level curves but do not change them as mutually orthotomic curvilinear congruences. A simple (but nongeneric!) example is

Here the labeling of the fall curves is uniform on all level curves because it has been taken uniform on one and propagated through the characteristicsof the creep equation (Cauchy’s problem). The integrating divisor has an isolated zero at the origin: This corresponds to the degenerated rut (all water will flow into the origin).

FIGURE 21.

An isotropic bowl. Left: The surface. Right: The integral of the creep equation.

106

JAN J. KOENDERINK AND A. J. VAN DOORN

(See Fig. 21.) Notice that the fall curve labeling is simply the azimuth, which is clearly the natural choice given the symmetry of the immit. In general we can assign a smooth labeling to the fall curves on any curve that meets all fall curves to be labeled transversely. (For instance, we may use the arclength along some fiducial level curve.) Then we can propagate the labeling along the fall curves to other regions. The problem here is that at some places the labeling is arbitrarily compressed because the fall curves approach each other asymptotically. This happens exactly at the singular solutions of the creep equation, the curves where the integrating divisor vanishes. At such loci the labeling breaks down. They are the singular solutions of the creep equation; that is, the curves 0 = 0. A simple generic example is the parabolic gutter 2

=x

+ y2/2,

(66)

here a consistent labeling is given by the integral

with integrating divisor

(68)

0 =y.

The various surfaces and the flow are very intuitive (Figs. 22 and 23). The differential equation of the integrating divisor is a0

- +y-

ax

a0 = 0; ay

its characteristics are illustrated in Fig. 24.

Y

I

.U

8

hS

I

1.1

FIGURE22. A parabolic gutter. Left: The surface z(x, y). Right: The level curves z ( x , y) = c and the fall curves w ( x , y) = c.

THE STRUCTUREOF RELIEF

107

1

I

FIGURE 23. A parabolic gutter. Left: The integral surface w ( x , y) of the creep equation. Right: The integrating divisor B(x, y ) .

D. Description in Terms of Intrinsic Properties Since we have consistent labelings for the orthotomic curvilinear congruences of level curves and fall curves, we may free ourselves completely from the arbitrary choice of coordinates (of the plane) and express all relations in terms of the labels. Thus, we arrive at an elegant, intrinsic description of the scalar field.

108


-1 X

FIGURE 24. The field of directions of the characteristics for the integrating factor. Notice that the plane y = 0 is an asymptote and will enforce a zero of the integrating divisor +(x, y ) at y = 0.

1. Basic Differential Relations By far the fastest way to construct the required framework is to use differential forms [12,60]. A basis of differential forms (or covectors) {wx,my} in the plane are the differentials of the coordinates; that is, wx = d x , oy= dy. Thus, we have the basic contractions

a

w x ( e x )= --dx

ax

a

wx(ey)= --dx

aY

a

w y ( e x )= --dy

ax

a

oy(ey)= --dy aY

= 1,

(70)

= 0, = 0,

(72)

= 1.

(73)

We use the Hodge star operator "*" that rotates the first-order forms over 7r/2 and toggles scalars and two forms: *1 = W,

A

my,

*w, = my,

*ay= -w,, *(wx A w , ) = 1.

It is easy enough to translate expressions in terms of the w , d, A, and in terms of classical differential calculus. For convenience we notice the relations that are most generally useful in the present context: The operator *d *d = A, that is, the Laplacean operator. The scalar product of two gradients such as V f Vg


109

appears as *(df A *dg), whereas the exteriorproduct of two gradients Vf x Vg = fxgy - fygx appears as *(df A dg). The defining equation for the level curves is simply dz = 0, whereas the creep equation is the dual expression d z = 0. The functions w and S are defined through the defining relation bdw = *dz. The function w is assumed to be a complete integral of the creep equation whereas 6 is an integrating divisor. We assume w and 6 to be independent; that is, *(dw A do) f 0. Starting from this small set of equations we proceed to apply Cartan's theorem (i.e., d2 = 0) to every differential form in sight until there is nothing left to differentiate. The result of this brute force approach is the following list of basic differential relations

(78)

*dz = 6 dw, *dw = -dz/6, dS A dw = (d6/6)

A

d(6 A dz)/S2 = -(dS/6)

*dz = d * dz, A

*dw = d * dw,

*(dz A dw) = a2/S,

(79) (80) (81) (82)

dw A *dz = dz A *dw = 0, *(dz A *dz) = (T 2 , *(dw A *dw) = a 2 / O 2 . There are quite a few remarkable symmetries and analogies in these relations. For instance, Eqs. (78) and (79) reveal an interesting duality between the parameters w and z. Of course, this is more like window dressing though because these equations merely repeat the defining equations for the fall curves (78) and level curves (79). Eqs. (80) and (81) specify the Laplaceans of w and z and are reminiscent of the Cauchy-Riemann equations from the theory of complex functions. Eq. (80) is a partial differential equation that characterizes the integrating divisor 19. Eq. (82) defines the element of area of the xy-plane in terms of the parameters w and z, likewise Eqs. (84) and (85) define the element of arc in the ny-plane in terms of the intrinsic parameterization. Equation (83) is trivial and merely reflects the fact that the level curves and fall curves are orthotomic curvilinear congruences. Since the reader might find it useful, we list here the conventional expressions for these basic results: BAz = VZ ' VS, S A W= -VW * V6,

vz x v w = (T2/6, v z . v w = 0, IIVZ1l2 = u2, IJVW112= 2 / 0 2 .

110


These equations have immediate geometrical significance. We will study this in more detail in the next section.

E. The Geometrical Meaning of the Diaerential Relations 1. The Line EEement

The metric in the xy-plane in terms of the intrinsic parameterization (z, w) is given by the classical line element ds2=dx2+dy2=dsi+dsl=

dz2

+ O2 dw2 0 2

The line element has to be understood in the sense ds@ds=dx@dx+dy@dy,

(93)

that is, as a symmetric tensor [60]. In &.92 ds, denotes the arclength along the level curves, whereas ds, denotes the arclength along the fall curves. Apparently the fall curves crowd injnitely close together when the integrating divisor 19 vanishes, that is, along the singular fall curves. Clearly this is one intuitively very important characteristicof the ridges and courses and it will indeed prove to be an important one. Apparently the complete integral w(x, y ) is singular when the integrating divisor 0(x, y) vanishes. Indeed, starting from the relation qdx - p d y = 0dw we may write

thus, w(x, y) fails to be an entire function. This will be the case no matter which complete integral andor integrating divisor we accept. 2. The Gauss-Weingarten Equations of Classical Suface Theory

The material in this section applies primarily to the Euclidean 3-dimensional setting, and not to relief as such (in which we disregard arbitrary monotonic transformations of the z-axis). However, the material will find frequent application and does throw some light on the structure of topographic relief, hence, its inclusion here. In the classical theory of curves and surfacesone expresses all quantities in terms of adaptedframes. In this way the differential invariants appear as by magic. In the theory of space curves one uses the Serret-Frenet frame (tangent, normal and binormal),in the theory of surfacesone uses the Gauss-Weingarten frame (tangents along the parameter curves and the surface normal). The trick is to express the


111

movements of the frame as one progresses from one point on the manifold to the next in terms of the adapted frame itself. The second fundamental form of the surfaces in three dimensions is [we use the notation r = { x ( z , w), y ( z , w), z), N denotes the surface normal] Il(dz, dw) = L dz2

+ 2M dz dw + N dw2

- Rdz2 - 2BSdzdw - B2T dw2

a24iT7

(95)

(96)

The coefficients of the connection (the Christoffel symbols; “symbols” because the rfj are not the components of a tensor r, although they certainly look that way) are:

r:,

S

= ---j-p’

These quantities occur in the classical theory of surfaces in the ( z , w}-parameterization. The Gauss equations of classical surface theory express the second-order differential of the position in terms of the Gauss-Weingarten frame; that is,

+ + +

+ + +

rzz = rlllrU rtlrU LN, rzw= rf2ru rf2rU MN, rww= r;2r,, rl2rU NN. The Weingarten equations of classical surface theory express the change of the normal in terms of infinitesimal changes in the parameters. In the present case the normal (which characterizes the attitude of the local contact element or tangent plane) is represented by the slant and tilt; thus, we have to look for expressions that relate the changes in slant and tilt to infinitesimal changes of the values of the

112


intrinsic parameters z and w . These relations are: dz 6dw Udt = S- - T = Sds, - T d s , , U

U

dz ddw dU = R- - S= Rds, - Sds,. U

U

Evidently the differential invariant S measures the tilt change in the direction of the fall curves and the slope change along the level curves, whereas the differential invariant T measures the tilt change along the level curves, and the differential invariant R measures the slope change along the fall curves. An important invariant relation between directions immediately follows from the Weingarten equations concerning the so-called conjugation relations between directions on the surface. These relations are equally valid for directions in the xy-plane. (We have met them before.) Given two directions in natural coordinates ( d z ,d w ) and (St,Sw), we call them (mutually) conjugated directions when

R d z S z - d S ( d z S w + 6 z d w ) - 8 2 T d w 6 w =0,

R ds, SS,

-S

(ds, SS,

+ ds, SS,)

or,

- T ds, SS, = 0.

(108) (109)

The latter form of this relation is nothing but the definition of the invariants R, S, and T , of course. Notice that d s , = 0, ds, # 0 leads to SSs, TSs, = 0; thus, the conjugated direction of the creep is Ss, : As, = -S : T . Likewise, the conjugated direction of the isohypses is Ss, : Ss, = - R : S . A self-conjugate direction is an asymptotic direction. Thus, the creep direction is self-conjugated at R = 0 (i.e., when the fall curves inflect) and the isohypses are self-conjugated when T = 0 (i.e., when the isohypses inflect). Notice that the curvilinear congruence that is conjugated to the creep is invariant against afinities ofthe z-axis. These are the level curves of the slope squared function, which is itself not an invariant. Closely related is the so-called indicatrix of Dupin, which is defined as the curve

+

Rdz2

+2dSdzdw + d2Tdw2 = f e 2 .

(1 10)

The indicatrix is either an ellipse or a pair of hyperbolae. Geometricallythe indicatrix is obtained as the intersection of the surface with planes parallel to the tangent plane but infinitesimally displaced from the tangent plane. Thus, the indicatrices yield an immediate and very vivid impression of local surface shape. Conjugated directions are simply conjugated diameters of the indicatrix; this explains immediately why the relation is preserved in the projection on the xy-plane. The principal tangents (asymptotic directions) are self-conjugated. (See Fig. 25.) When the Hessian vanishes, the indicatrix degenerates into a pair of parallel lines. This indicates that the surface is locally cylindrical. The principal tangents merge into the direction of the generators of the cylinder. It must, clearly be the


113

RGURE 25. The field of indicatrices of Dupin for the example landscape. The graylevel indicates whether the surface lies above or below the cutting plane. When the indicatrix is a hyperbola only the part inside a disk about the fiducial point is drawn. The indicatrices have been sampled for points on a Cartesian lattice. Notice the variation in size (thus curvature) and shape (elliptic and hyperbolic).

case that if a ridge crosses the parabolic curve, it runs along the cylinder axis; thus, these geometrical entities are important features of the topography. Notice that most of the material in this section applies only to the Euclidean 3-dimensionalcase, and not to relief proper. Some conceptscarry over to the case of R2 x A though; for instance, the concepts of conjugacy and asymptotic directions. 3. The Mainardi-CodauiEquations of Sur$ace Consistency The relations discussed in this section also apply to the Euclidean 3-dimensional setting rather than relief proper. The classical conditionsof surfaceintegrity are obtained when we apply Cartan’s operator d to the Weingarten equations and use Cartan’s theorem, that is to say d2 = 0. We obtain (setting d2a = 0 and d2r = 0) Sw

6 + BT, = --(2S2 + T 2+ R T ) ,

(111)

Rw

+ 6 S z = - >fJ6( R + T ) S .

(1 12)

U2

To this we may add the differential equation for the fall curves; that is, (setting d2w = 0) 6z =

6L 7’

Eqs. (111) and (1 12) are the so-called Mainardi-Codazzi equations of classical surface theory. They are satisfied by virtue of the fact that the surface is indeed a surface.

114

JAN 1. KOENDERINK AND A. J. VAN DOORN

These equalities are often of some utility if we need to simplify expressions for various differential geometric properties. F: The Topographic Curves

At this point we have developed sufficient theory in order to be able to start the study of the so-called topographic curves [MI. We will start with the simplest examples (i.e., review the level curves and fall curves again) and gradually move to topics where the recent literature has not yet managed to reach a concensus. 1. The Congruence of Fall Tangents and Its Second Caustic Surface

+

Consider the collection [in (2 1)-dimensions] of the tangent planes of the topographic surface @ (say) that possess some fiducial slope, slope angle 01 (say). This one-parameterfamily of planes envelopes a developable surface S (say) that might be called the “slope developable of slope angle (Y.” It is a surface of constant slope, and all its fall curves are straight lines. Its level curves are parallel curves, that is, solutions of the Eikonal equation p 2 q2 = const (wide supra). This developable surface 3 has a certain edge ofregression 6 (say), such that the tangent planes of this curve envelope the developable. The curve 6 must also have constant inclination (Y.The developable S touches the surface @ along a certain curve (say). This is the basic geometry that characterizes a curve of constant inclination (q). (See Fig. 26.) When the topographic surface @ is illuminated by the sun in the zenith the curve appears as a curve of constant illumination or isophote. The generators of the

+

FIGURE26. Left: Surface of constant slope and its crest curve. This is the surface generated earlier, see Fig. 15. Right: The same surface with the generators of the developable surface extended in order to show the edge of regression of the surface. All generators are tangent to the edge of regression. Notice that the surface self-intersectsat the crest curve and is seen to form a swallowtail-likestructure (apparently the edge of regression itself may display singularities, in this case a cusp). It is important to distinguish between the edge of regression 6 and the crest x .

115


developable 3 are the tangents to the fall curves at the points of JJ; let’s call them the fall tangents of v . When we vary the inclination a,we obtain all slope developables that together envelope the topographic surface 0. At the same time the edges of regression of these slope developables (the curves c ) describe a certain surface W (say). All the fall tangents are also tangent to the surface W; thus, they are bitangents of 0 and q. Thus, we conclude that the original surface 0 and the surface of edges of regression of the fall tangents of a certain inclination are the caustics of the congruence of all fall tangents of the topographic surface. We will denote the original surface theJirst caustic surjace and the surface @ the second caustic surjace of the congruence of fall tangents. If we consider the tangents of any given fall curve, these form a developable surface of which the fall curve is the edge of regression. They touch the second caustic surface along a curve y (say) that is the image of the fall curve on the surface W. In the (singular) case that all fall curves of @ are planar curves the second caustic surface is the envelope of a one-parameter family of planes and thus has to be developable. This simply characterizes all topographic surfaces with planar fall curves, an observation that was first made by Wunderlich [66]. In general the second caustic surface will not be a developable surface, of course. If the topographical surface 0 itself is a developable surface, the second caustic surface degenerates into a curve, for the case of planar fall curves even into a straight line. In the generic case both surfaces @ and W will be nondegenerated and nondevelopable. That the second caustic surface will be a developable surface if all the fall curves of the topographical surface @ are planar curves (not necessarily in vertical planes) is a very special case, which is of historical interest because for some time ridges and courses of generic topographical surfaces were believed to be planar [26]. Wunderlich [67] has extensively researched this issue and has also characterized all surfaces for which the fall curves are especially simple (i.e., quadrics). Such surfaces can often serve as convenient illustration, though one should be aware of their nongenericity (Fig. 27). The notion of the second caustic surface W for any topographic surface 0 is indeed a very useful one and we treat it in some detail in the next subsection. 2. Description of a Topographical Surjace in Terms of Its Support Function

If we regard the topographical surface @ as the envelope of its tangent planes, we can write it as x cos u

+ y sin u + 4V = h(u, v),

(1 14)

where the surface parameters (u, u ) have a simple geometrical significance: The parameter u is the tilt ( 5 ) and the parameter v the tangent of the slope angle (tan q).

FIGURE 27. A surface with confocal hyperbolas and ellipses as level curves and fall curves respectively. The parametric representation of this surface is x = cos u cosh v , y = sin u sinh v , z = sin u. Left The surface. Right: The relief and creep.


117

The function h ( u , u ) may be called the supportfunction of the surface: It is the distance from the origin to the line of intersection of the tangent plane at the point { x , y, z} and the xy-plane. This is seen immediately when we note that the surface normal n is given by the expression

n = {cosssin6o,sintsincp,cos6p}= { u c o s u , u s i n u , l ] / ~+l u 2 ,

(115)

thus (d is the distance of the tangent plane at { x , y. z] from the origin)

and we have that h ( u , u ) = d / sin p, which is indeed the distance from the origin of the intersection of tangent plane at the point { x , y, z} with the xy-plane. The support function h(u, u ) describes the surface completely, thus, it can take the place of the height function z(x, y). The height function describes the position of points on the surface whereas the support function describes the position of its tangent planes: These are dual descriptions. An explicit parameterization of the surface is given by the expressions x ( u , u ) = (h

+ uh,)cosu

- h,

sinu,

y(u,u) = (h+uh,)sinu+h,cosu, z(u, u ) = -u 2 h".

(117) (118) (119)

This representation is frequently advantageous since it describes the surface directly in terms of the slant and tilt of its surface elements. We essentially treat the surface as the envelope of its tangent planes. Since we have obtained an explicit parametric expression of the surface (in terms of the support function) we can immediately find all of its differential geometric properties via the standard algorithms. Considerthe developablesurfacesof constantinclinationthat touches the surface 0 along curves: These curves have to be the u-parameter curves. Such a surface will have an edge of regression, which is a curve. When we vary the parameter u these edge curves will sweep out a surface Q (say). The generators of this developable surface are tangent to Q, (they are rays of constant inclination since they are common to two infinitesimallyclose tangent planes of constant inclination) but they are also tangent to the edge curves, thus to Q: Hence, they are the common tangents of our original surface 0 and the surface W. When we consider the line congruence of fall curve tangents, then the original surface Q, and the surface Q are the caustic surjaces of this congruence. The surface Q is the second caustic surface defined by the surface 0. It is not difficultto find an explicitrepresentation of the second caustic surface Q: The developable surface of constant slope E: is characterized by u = const. Here the parameter u is the parameter of the curve arctan u. The edge of regression of

118


the developable surface thus can immediately be obtained from Eqs. (1 17)-( 1 19). When we then vary both the parameter u (position on an edge of regression) and the parameter u (tells which edge of regression) we obtain the explicit parametric representation of the second caustic surface 9. The parametric representation of the second caustic surface 9 of the fall tangents is: x ( u , u ) = -h, sinu y ( u , u ) = h, cosu z(u, u ) = ( h

- h,, cosu,

+ h,, sinu,

(120)

+h u b .

It is a straightforward exercise to express the differentialgeometry of the surfaces or 9 in terms of the parameters { u , u ) , that is, in terms of the slant and tilt of the local surface. Most importantly, for the surface @ the metric is Q,

ds2 = ( h

+ h,u)2du2 + 2h,,(h + h,,) du du + (4h: + h:,) d u 2 ,

(123)

whereas the second fundamental form is

II(du, du) = L dU2

+ 2M du dv + N du2,

( 124)

with L=

M=

N =

+ h,,,) sin u - (h,, + uh) cos u ) JG-7 u(h,, cos u + (2h, + h,,,) sin u ) diT7 -2h, + u(3h,, cos u + h,,, sin u ) u((h,

dG-7

( 125) ( 126)

(127)

We present an example in Fig. 28.

G. Generic Structure 1. Morse Critical Points At a Morse critical point the second-order germ of the height is Z(X,

1 Y ) = - ( K 1 X 2 -k K 2 Y 2 ) , 2

(128)

where K I , ~2 denote the principal curvatures in the three-dimensional setting. The complete integral is w ( x , y) = x-KZyK',

(129)


119

FIGURE28. The surface with support function h ( u , u ) = exp(u)u2/2 and its second caustic. A few rays of constant inclination have also been drawn.

and the integrating divisor

Possible ridges and ruts are the coordinate axes x = 0, y = 0. In case the point is elliptic ( ~ 1 ~> 20 ) the level curves are concentric ellipses. The fall curves approach the critical pointfrom a single direction only; thus, only one of the directions x = 0, y = 0 is a candidate ridge or rut. (See Fig. 29.)

FIGURE 29. An anisotropic bowl. Left: The immit surface with level curves. Right: The level curves and creep near the immit.

120


I

U

e

0s

1

FIGURE30. A nonsymmetric saddle. Left: The saddle surface with level curves. Right: The level curves and creep near the saddle.

In case the point is hyperbolic ( K I ~2 < 0) both the level curves and the fall curves are (mutually orthogonal) hyperbolas. (See Fig. 30.) The special fall curves that meet at the saddle are also curves of steepest ascent or descent. Moreover, the other fall curves approach them asymptotically, reason why these were singled out by Cayley, Maxwell, and Jordan as ridges and ruts. However, these curves may fail certain other intuitively reasonable properties of ridges and ruts (vide infra). These curves are generally termed separatrices because they separate the families of hyperbolic arcs. In general, there seems to be no local way to recognize a separatrix (i.e., except from following it from the saddle point). Thus, Jordan indeed denies that the special fall curves are in any way to be distinguished from the regular ones. This seems to contradict common sense though: One has the impression that it is pretty clear where the courses should run even if we have no occasion to go to the mountains and locate their defining saddles. Even more strongly, it seems counterintuitiveto deny that local courses depend only on local relief, for then local changes high up in the distant mountains could influence their course over here. Such influence should intuitively be limited to the water supply, not to the local course. Indeed, Rothe’s criterium 6 = 0 (vide infra) defines the course locally; there is no need to follow separatrices to the saddles. The Morse critical points of the example landscape are illustrated in Fig. 3 1.

2. Parabolic Curves The parabolic curves are intrinsic geometrical entities; that is, if you tilt the entire landscape, they remain well defined. We may assume that in the generic landscape the parabolic curves are smooth, closed curves (they bound elliptic regions) and that they are typically inclined, that is, that they osculate level curves (or, for that matter, fall curves) at isolated points that are distinct from the critical points. The


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FIGURE31. The Morse critical points of the example landscape. The immits are denoted 0 , the summits 0, and the saddles Q.

parabolic curves contain certain higher-order singularities, namely the Gaussian cusps (where the cylinder axis coincides with the tangent to the parabolic curve, O 0; also termed rufles) and the inflection points. The ruffles are defined by C ~ = we may assume that these typically lie on inclined parts of the curve. In neighborhoods of a parabolic point each point has a mate (at the other side of the parabolic curve) with a parallel tangent plane (same tilt and slope). In neighborhoods of a ruffle we find triples of such points. These properties characterize the parabolic points and ruffles [4]. When the parabolic curve osculates a level curve, this will be a critical point of the slope squared function, which can be either a saddle or an extremum. Such isolated points of locally steepest slope in the landscape are obviously remarkable. If the landscape is illuminated by the sun in the zenith, these points appear as critical points of the illuminance. They have been studied in this context [41] and are of interest in the context of computer vision and human visual psychophysics. (See Fig. 14.)

3. Ridges and Ruts The intuitive notion of a ridge is perhaps most clearly illustrated by the example of the ridge of a conventional western roof the horizontalintersectionof two inclined planes that is the highest part overall. Ridge derives from the Anglo Saxon hrycg or hricg, the Scottish rig or rigg. Principal meanings according to Webster’s are “an animal’s back, a range of hills, or the horizontal line found by the meeting of two sloping surfaces; as in the ridge of a roof.” Ridges are sometimes called “divides” since they act as watersheds. If we invert the height the ridge turns into what may be called a “rut” or a “course” since it is the likely locus for a river. In German the ridge is known as Kummweg (ridge path) or Riickenlinie [back (as an

122


FIGURE32. Intuitive definition of a ridge as the junction of two roof planes.

animal’s back) line], the rut as Rinnelinie (rut line) or Thalweg (or Talweg, valley path). This latter term (Thalweg) was taken over by the French geometers, who refer to the ridge as the ligne de faite (ridge line). (See Fig. 32.) The definition of these topographic curves has been the subject of long and heated debate [44,551. Even today the issue is not generally been considered settled. One issue is whether there exists a local criterion or whether ridges and courses can only be defined on the global landscape. Jordan [34] flatly denied that there could be such a local criterion. He, as Cayley [ 151and Maxwell [46], defined courses and ridges as the fall curves through the saddles and running into summits and immits to define ridges as strings of summits and courses as strings of immits. This closely approximates the intuitive ideas of the cartographers (Fig. 33). Rather early a concensus arose that the fall curves through the saddles are courses. Jordan grants that courses may also be fall curves that start from the highest point of a concave slope though. If you consider an overall convex hill slope which contains a concave intrusion, the level curves are seen to develop an undulation (where they touch the curve T = 0, the condition is Co3 = 0). At one side of the level of the undulation the level curves are convex throughout; at the other side they have a pair of inflections (where they cut the curve T = 0) with a concave segment. Here we can immediately draw horizontal bitangents to the surface; the undulation is the limiting point of these bitangents. Such points have been implicated by Jordan as points where springs issue forth from the slope: The initial points of courses running down the slope (Fig. 34). In Cayley’sdescription a ridge can be followed from a saddle to a summit; then at the summit we can continue in the same direction (the direction of minimal normal curvature) to end up at another (sometimesthe same!) saddle. (See Fig. 35.) There we can again continue (keeping on a straight course), and so on. Thus, the ridge becomes a string of saddles interspersed with summits and a smooth space curve throughout. The same can be done for the courses of course. Perhaps a slight flaw in this contmction is the fact that though this curve indeed runs through the

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FIGURE33. “Caterpillars” in early maps: These signify strings of hills connecting to form mountain ridges.

0

4.5

-1 -1.5 -2

-w -3

-15

-1 -05

0

0.5

1

1.5

2

FIGURE34. A convex surface with concave intrusion. Left: The surface. The black area is elliptically curved, the gray (and also the black) area is concave. The white part of the surface is convex. At the top of the concave intrusion we have the point where a spring might be expected. Right: The structure. The parabolic curve and the convex-concave boundary (T = 0) osculate at their common top.

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FIGURE 35. A chain of Cayley ridges. It consists of asymptotic fall curves from the saddles. Since all fall curves generically reach the extrema from one of to opposite directions, the ridges (or courses) N n smoothly (i.e., without change in direction) through the summits (or immits). Near an extremum the (extended) ridge osculates the special fall curve through the extremum, but does not coincide with

it.

summits in the direction of the special fall curve (the one all other fall curves asymptotically approach) it fails to coincide with the special fall curve but only osculates it. Thus, the fall curve congruence slightly deviates from Cayley’s (and Jordan’s, Maxwell’s) ridge at the extrema. (See Fig. 36.) Another choice would have been to follow the fall curves that extend the directions of minimum normal curvature at the summits and immits. However, empirical studies indicate (and a simple analysis verifies this) that these curves are numerically badly behaved. Moreover, the fact that they may entirely miss the saddles counts against them. These curves for the example landscape are illustrated in Fig. 37. Clearly they are not likely candidates for course or ridge-hood. If one considers the relation between a saddle and one of its immits there exist two limiting (nongeneric)cases. In one case an asymptote of the saddle meets the immit along its direction of minimum normal curvature (we imply absolute value here); in the other it meets it along the direction of largest normal curvature. But notice that a slight perturbation will always let it meet the immit along the direction

FIGURE 36. Cayley’s ridges and courses in the example landscape. Left: The ridges. The faces are the dales. Right: The courses. The faces are the hills.


125

FIGURE37. The fall curves passing through the extrema along the direction of the special direction: Clearly the result is less than fortunate when such curves are thought of as candidates for course or ridge-hood!

of least normal curvature because this is the fall curve to which all others (except the direction of maximum normal curvature) asymptotically approach. The other one has probability zero. In the generic case the asymptote of the saddle will not coincide with the special fall curve from the immit. This latter curve does not meet the saddle at all but proceeds to a summit, which it generically meets along its direction of minimum normal curvature. So what is the course? Empirically(!) perhaps the most reasonable (and numerically stable) definition is to ascend at an immit along the fall curve that connects the immit with the saddle and follow it upstream until the surface fails to be elliptically concave. At that point a course can be said to “spring forth” from the slope (basically Jordan’s insight). A similar (fully symmetric) procedure can be applied near a summit (although it is awkward to speak of a “ridge springing forth,” however, if we invert the landscape we see it as an “inverted” spring). In this picture there will be no courses near a saddle at all, which makes formal sense because there is no true (i.e., only asymptotic) confluence of the creep (vide infra) and also accounts for one’s experience in the actual landscape: Near a saddle one has marshy conditions at most, but no streams. This definition yields very convincing results, but we grant that there is a certain amount of arbitrariness (Fig. 38). However, all definitions we know from the literature suffer from definitely worse defects. That this is not necessarily the case is clear from the example of the (Jordan-type) spring on a slope example. No matter how exactly one defines the ridges and courses, they must necessarily be of Rothe’s type (a = 0), and we fail to understand Rieger [54]’s arguments to the contrary. Rothe’s definition is indeed the only viable one available (since other local definitions like De Saint-Venant’s and others violate Boussinesq’s condition) if the region of interest doesn’t contain (the relevant) critical points.

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FIGURE 38. Ridges and courses in the example landscape. The parts of the fall curves connecting the summits and immits with the saddles are drawn only in the dome and bowl areas. Notice that the courses in the landscape run into the immits and end there (the immits act as drainholes) instead of running towards the “ocean” (that is to say: the edge of the picture). Real landscapes don’t contain immits except for some rare cases (lakes).

The first one to suggest a purely local criterion was De Saint-Venant [18]. According to De Saint-Venant a course is the locus of extreme slope along the level curves. (“Linge de pente maxima”.) This is certainly an intuitively reasonable idea. (See Fig. 40.) Thus, the condition for a course is simply (of course it works in the same way for ridges)

(g

= 0)

A

(dz = 0).

This immediately leads to the condition 19s= 0. However, here De Saint-Venant only considers the possibility S = 0 and thus arrives at a differential equation for the courses (named after him):

--(--$) az az

a2z

ax ay

ax2

-

((g)2-($)2)-=0. a2z (132) axay

The alternativepossibility 19 = 0 doesn’t figure at all in De Saint-Venant’s account. According to a novel definition due to Eberly [22,23] ridges should be regarded as Ridges are loci of maximum height along the direction of maximum value of the second order directional derivative. Although this certainly sounds like a novel idea, it nevertheless turns out to lead to De Saint-Venant’s equation again. Thus, perhaps the definition can be said to throw some new light on the meaning of De Saint-Venant’s equation. The De Saint-Venant’s curves for the example landscape are shown in Fig. 39.


127

RCURE 39. De Saint-Venant’s curves (the curves S = 0) of the example landscape. Compare these with the Cayley ridges and courses (Fig. 36) and the ridges and courses in our definition (Fig. 38). Notice that there is a superficialsimilaritybut that these curves are differentin most details. Notice also how at many points the De Saint-Venant ridges are tangentto the isohypses,thus violating Boussinesq’s condition that water should run downhill!

A problem with De Saint-Venant’s equationis that the condition S = 0 is highly nongeneric because it is highly unlikely that the condition will coincide with the independent condition dw = 0, that is, that the course is also a special fall curve. This latter condition is equally intuitive and is due to Boussinesq [8,9]: Bluntly stated it means that water tends to run downhill (see Fig. 40). The two conditions (S= dw = 0) taken together imply that d t = 0. This is Breton de Champ’s [lo] memorable observation: The implication is that the courses are planar curves running in vertical planes! De Saint-Venant’s locus only coincides with a fall curve if the surface happens to be a member of a certain class of surfacesthat satisfy a certain third-order differential equation (easy to derive from the representation in terms of the support function) see [66]. Evidently most surfaces are not of this

FIGURE40. Left: De Saint-Venant’s principle: The fall curve a that issues forth from the point P on the isohypse i is steeper than any of the fall curves in its immediate neighborhood. Thus, all the water will tend to collect in a course along this ligne a2 pente murim as Lk Saint-Venant calls it. Right: Boussinesq’s principle: When a is a fall curve, then we don’t expect water to run along the direction p. (Although intuitively obvious most ridge and course finders in common use today don’t recognize this simple observation!)

128


FIGURE41. A helicoidal gutter surface. Left Piece of the surface. Right: Relief map of a much larger part of this surface. In practice we study only part of this landscape in order to avoid the origin or branch cuts, like the part depicted on the left.

type. This clearly poses a problem: For instance, it forbids rivers to do what they usually do, which is to meander. As a historical curiosity, some topographers actually held the opinion that this is how water runs downhill! These geographers applied this nongeneric solution to geomorphology and drew some weird (rather counterintuitive)conclusions from it (e.g., [26])! That the De Saint-Venant ridges are located at intuitively absurd places can be illustrated rather easily. In Fig. 41 we show a helicoydal gutter surface. Here the De Saint-Venant course is curved, but it is evidently located at the wrong place: This is immediately seen when we numerically integrate the creep equation (Fig. 42). Rieger [54]objects against examples like this that “the heightjumps along

FIGURE 42. Numerically integrated creep field of the helicoYda1 gutter surface. Here the region considered is a hemi-annulus. The circular arc is De Saint-Venant’s course: Notice that the fall curves have inflections on this curve. Evidently the numerically computed fall curves cross this curve and instead asympotically approach a quite different circle, which is the true course. This circle is Rothe’s special fall curve (19 = 0, that is a singular solution of the creep equation). Although some authors still cling to the De Saint-Venant definition of ridges and courses, or at least seriously doubt the validity of Rothe’s explanation of ridges and courses, it is hard to understand why given examples like this.


129

the course” (he eventually discards the Rothe-type solution 6 = 0 altogether), but he is evidently mistaken since we may simply define a nice region of interest (e.g., an open disk) such as to avoid this trivial problem. (Another simple “solution” is to solve the problem on the infinitely covered plane, taking the origin as a branch point. The surface then becomes a helicoydal gutter with infinitely many turns.) The possibility ignored (or rather: missed) by De Saint-Venant is 6 = 0. Clearly this is the generic case though! These curves also satisfy the creep equation; thus, they are the singularfall curves. It was Rothe’s [55] seminal insight that De Saint-Venant’s and Boussinesq’s conditions are not at odds at all, but serve to single out the singular fall curves as candidate courses and ridges. In our opinion Rothe’s solution definitively solves the problem and also very nicely shows why De Saint-Venant’s reasoning-which is certainly very reasonable-typically leads to nonsensical results. Here we shall take the singular solutions of the creep equation as Candidate ridges and courses: Some additional requirements have to be fulfilled for them to qualify (vide infra). Notice that the creep equation is exact whenever qy = - p x , that is, whenever L = R T = 0, which again is the case for the so-called minimal surfaces. In such cases we don’t need an integrating divisor; 6 can be taken a constant. Thus, minimal surfaces can have no ruts or ridges in Rothe’s sense. An example is the saddle z(x, y) = ( x 2 - y 2 ) / 2 . However, the generic hyperbolic quadric is

+

1 2

z(x, y > = - ( a x 2

- by2),

(133)

and the complete integral of the creep equation is w ( x , y) = xby-‘, the integrating divisor O(x, y) = xb-’y-’-’. Thus, the symmetric saddle is a very special case. Since it is arbitrarily close to saddles with the x - and y-axes as singular solutions, it makes sense to simply define these as singular solution for the symmetric case. Whether a singular fall curve will be a candidate ridge or course can be very simply decided: It depends solely on concavity or convexity of the surface and can thus be decided purely locally. The sign of the differential invariant T is decisive. For the saddles two of the branches are (candidate)courses and two are (candidate) ridges. In case the zero locus of T transversely meets the singularfall curve, we have the changeover from a candidate course to a candidate ridge. This is in fact Jordan’s observation that a course can start on a slope where the surface turns concave. A singular fall curve can indeed be ridge in some parts and course in another. In very special (nongeneric)cases the singular fall curve may actually be both ridge and course [ 5 5 ] ! Needless to say that one needn’t bother with such cases. (see Fig. 34.)

130


H. Transport of Stuff In the real landscape the rivers transport all water raining down on them toward the ocean. If we assume a constant, uniform rainfall, a certain amount of stuff per area element has to be carried along by the creep. After a sufficiently long period an equilibrium state will have been reached in which all streams have reached a given size such that the stuff is transported to the ocean without local accumulations or depletions. In the true landscape there are hardly immits; in general the water will tend to flow into the immits (rather than the ocean) of course. In the landscape the immits will fill and spill over, thus forming lakes with an outlet. We will ignore such effect here and consider immits as drain holes where the inflow simply vanishes. Clearly, conceptually it makes but little difference to the concept whether the flow is into the ocean (infinity) or into the immits. The equilibrium size of the streams and immits is obviously an interesting geometrical entity that depends on the structure of the landscape and lets us judge from the size of the streams how extensive their basin at any point along their course is. This measure might well prove important in image processing where it will yield (global) a measure of the relative importance (river or spring) of the courses. We know of no attempt to compute such data, nor any formal attempts to characterize the information contained in these types of data. It remains an important open problem. A numerical calculation via a simulation is easy enough, of course: One simply “rains” points (from a uniform distribution) on the regions and follows the flow through numerical integration. Density can be monitored by counting how many flow lines traverse each pixel. 1. Confluence

In his article on topographic curves in the Encyklopudie der mathemarischen Wissenschaften mit EinschluJ ihrer Anwendungen Liebmann [44] remarks that “in general the fall curves approach the courses,” the kind of insipid statement that appears strangely out of place in the Encyclopedia of Mathematical Sciences. The observation is quite apt though. Rothe [55] correctly notices that in order to form a river the river bed has to be in a position to collect water from an extended region (the so-called basin of the river). Thus, we need an additional condition of confluence: Somehow fall curves should approach and merge the course (and similarly for ridges, of course). In the topographical parlance one might say that the singular fall curve needs tributaries, which actually merge it at junctions before we will consider to call it a course. The singular fall curves are in a good position here because we have already seen that the fall curves tend to crowd closely together near a singular fall curve. The candidate course is given by B(n, y ) = 0, with a ( w , B ) / a ( x , y ) f 0. The direction of a regular fall curve is given by q / p = --w,/wy. In order for the

131


FIGURE 43. Left: A “false course” z = x + y 2 / 2 - x 3 / 3 . Though there exists a candidate course that runs from the saddle to the irnrnit, there fails to be confluence;thus, this is in no way a true course. It can be seen that the water actually runs into the irnrnit from a direction orthogonal to the course. Right: The relief for this case.

regular fall curve to meet the singular curve, one thus has to check the condition O x P y

= wxlwy = - q / P .

( 134)

Because of the condition a(w,O)/a(x, y) f 0 the issue can certainly be decided. The junction can either be at some definite point on the candidate course (e.g., an extremum) or it can be at infinity (e.g., at the asymptote from a saddle). This condition immediately discards many candidate courses as viable (see Fig. 43). (A small perturbation completely changes the qualitative structure; see Fig. 44.)In the case of the saddles thejunction is ar injinity. Thus, there will be no true course near the saddle, and whether the candidate course is an actual one depends on the shape

.I

u

.

u

I

u

-1.5

-I

u

0

0.5

I

IS

FIGURE 44. Left: The previous case being nongeneric we here show the effect of a not too large perturbation z = x + y 2 / 2 - x 3 / 3 + x y / 2 . Now the water suddenly runs into the immit from the orthogonal direction (according to the perturbation from either side)! This really doesn’t change the final conclusion that this candidate course should be discarded. Right: The relief for this case.

132

JAN J. KOENDERLNK AND A. J. VAN DOORN

FIGURE 45. Left: Cylindrical gutter with circular isohypses. Right: Isohypses and flow lines for the gutter. The isohypses are circular arcs; thus, the vertex locus is ill defined: Yet there evidently exists a well-defined course.

of the landscape far away from the saddle: In many cases the saddle is essentially irrelevant in determining the course, in contradistinction with Jordan, Cayley, and Maxwell’s course definition. The most reasonable definition of courses seems to be the special fall curves that connect the saddles with the immits, taken to the points where the surface changes from concave to convex. Such courses are thus curvilinear line segments. An analog definition can be framed for the ridges. In cases where there are no saddles (e.g., in image processing with small images, or in our “gutter” examples) the only way to proceed is to consider Rothe’s special fall curves. Other definitions (e.g., De Saint-Venant’s or the loci of level curve vertices) need not even be considered seriously because they (generically) violate Boussinesq’s condition. That the locus of isohypse vertices is not a particularly fortunate ridge definition is clear from such simple obervations as that a gutter with circular isohy ses (every point is a vertex point! See Fig. 45, the height function z =x - 1 - y2) has a well-defined course and that a gutter with elliptical isohypses has a course that fails to coincide with the vertex locus (Fig. 46, height function z = x - 3y/5 - 2d-15).

e

\

\

I

\

\

\

\

\

\

\

\

\

\

-

-

\

FIGURE 46. Left: The isohypses of this gutter are elliptical. The upper line is the vertex locus, the lower line is Rothe’s special fall curve (8= 0). Right: Numerically integrated flow. Clearly the flow converges on a course that does not coincide with the vertex locus, which is indicated by the horizontal line.


1

15

2

25

3

35

133

4

FIGURE 47. Left: An infinitely extended river bed like the FloridaEverglades. The height function is z = x + y2/2 - x 3 / 3 . The flow lines are asymptotically horizontal, but at different location. Each fall curve has its own asymptote (which is parallel to the singular fall curve) thus, it will never have a junction with the course, not even at infinity. There is no confluence. Right: The same flow with a perturbation term x y / 2 . Notice that the perturbation doesn’t really change much. Small (but finite!) perturbations really don’t affect the conclusion that one doesn’t have a well-defined river bed here.

In cases where it remains undecided whether the singular fall curve would be a ridge or a course ( T = 0) additional complications occur. Of course this case is nongeneric since the condition T = 0 typically defines a curve. However, in practice it may happen that the value of T is small over an extended area and that due to experimental uncertainties the sign of T must remain in doubt. In such areas both the level curves and the fall curves must be straight fines, and since they are orthotomic both must be congruences of parallel lines. In such cases there can be no confluence, hence no course, but the river bed assumes a large area like the Everglades in Florida. (See Fig. 47.) This case is, of course, not generic. However, slight perturbations really don’t destroy this state of affairs, thus, one may easily encounter it in practice. An interesting problem concerns thejunction of two courses. It is easy to build simple models for such a case, an example is Z(X,

y ) = x4/4

+yx2 +

EX.

(135)

For E = 0 we have a simple, symmetrical situation. If one computes the flow, it is seen that the course splits into two subcourses, the junction being of the pitchfork variety. Indeed, the configuration of ridges and courses must be like a trident, the junction being at an osculation of the concave/convexboundary with a level curve. If one slightly “tilts the landscape” (by letting E # 0), we have a “flip-flop” where the water either takes one subcourse or the other, the minor course will run dry. This apparently simulates the behavior of actual streams quite well. (See Figs. 48 and 49.)

134


RGURE 48. Left: The symmetrical junction relief. Right: The symmetrical junction Row. The ridgekourse configuration has a trident structure.

If one follows a river upstream, one meets with countless junctions of the preceding type. The tributaries grow smaller and smaller. At some distance upstream it becomes rather arbitrary to distinguish between the main stream and the tributaries (one has a fractal behavior). According to the resolution, the branches become arbitrarily small. Eventually they end at points where they spring forth from the slopes. Higher upstream the flow is as seepage through the soil. We end this section with the discussionof the most remarkable generic properties of the loci considered in this paper. We will not prove these relations in exrenso. A simple way to derive all these relations is to write the height as a truncated Taylor expansion (a fourth-order binary polynomial suffices) and then to use a symbolic package like Mathematica to compute all relations of interest (be sure not to display these relations since they become very complicated!). Then study the various specializations at the origin: These are simple expressions and the results follow without any hardship. In the cases treated here order four is sufficient. In

FIGURE 49. Left: The perturbed junction relief. Right: The perturbed junction Row. This is the generic case. All the water takes one of the branches, the other runs dry. The configuration of ridges and courses now has two branches.

135


general one will pick an order such that the final result is still general; that is, at all stages in the computation one should carry superfluous terms. By specializing the expressions to the origin, one rids oneself from this fluff automatically and arrives at the general expression.

I. De Saint-Venant’s Curves Though the condition of De Saint-Venant fails as a local criterion for ridges or courses, these curves are of some interest in their own right. When a fall curve intersects such a curve it has an inflection of the horizontal curvature: That is essentially De Saint-Venant’s condition. The curves may intersect the concavekonvexboundary ( T = 0) transversely. At such points the Hessian ( R T - S 2 ) must also vanish; thus, we have an intersection of De Saint-Venant’s curve, the convexkoncaveboundary, and the parabolic curve. Both the level curves and the fall curves inflect. (See Fig. 14.) The curves have transverse self-intersectionsat the Morse critical points. At the extrema one branch is locally steepest, the other locally shallowest. At points where the curve is tangent to the fall curves it changes from locally steepest to locally shallowest. J. CliffCurves

The cliff and plateau curves are characterized through the condition R = 0. Cliffs are often marked on topographic maps since they are evidently important for human traffic. Their properties simply follow from the defining condition. First of all, since H = RT - S2 we have H = -S2 5 0 on these curves: They are apparently restricted to hyperbolical areas. The curves S = 0 are loci of inflection of the fall curves in the horizontal plane. When a cliff curve meets such a curve, it has to be on a parabolic point since H necessarily vanishes. The cliff curves have osculations with the parabolic curve. Near such a point they run completely in the hyperbolic area. The fall curve intersects both the cliff curve and the parabolic curve transversely at such a point (see Figs. 50 and 5 1). The example illustrates the height function z(x, y) = x y2/2 x2y/2 xy2/2 y3/6. Cliff curves change into plateau curves when they are tangent to a fall curve. If you are only interested in cliffs, you may say that the cliff curves end at such points. At a saddle the cliff curves self-intersect transversely. We have four halfbranches of cliff and plateau curves. The cliff curves for the example landscape are illustrated in Fig. 52. Cliff curves often occur in working drawings of sculptors since they vividly reveal the relief as seen from a given vantage point. The cliff curves for a sculptor’s drawing of a male face en projil are illustrated in Fig. 53.

+

+

+

+

136

JAN J. KOENDERINK AND A. J. VAN DOORN 1 0.7s

0.S

O U 0

-

.

a= 4.1

4.75 I

a75

as

QZ(

o

0.u

u

0.7s

I

FIGURE50. The level curves and fall curves for a case of osculation of a cliff curve with the parabolic curve (in the center of the figure).

FIGURE51. The case of osculation of a cliff curve with the parabolic curve. Left: The local surface habitus. Right: The cliff curve, fall curve, and parabolic cuwe (the black curve).

FIGURE 52. The cliff curves of the example landscape. Here the area has been tinted according to the sign of the invariant R; the cliff curves are thus the boundaries of the tinted areas.


137

FIGURE 53. The cliff curves in a technical sculptor’s drawing. Cliff curves are quite similarly indicated in cartography.

K. The ConvedConcave Boundary The convedconcave boundary runs fully in the hyperbolic area. The level curves inflect on this boundary. These curves are very important because they decide on the ridginess/courseness of the candidate ridges/courses (19= 0 loci). When the convedconcave boundary meets a cliff curve the Laplacean must vanish; thus, these three curves meet transversely at special points. Both the level curve and the fall curve inflect there. When a ridge or course curve meets the convedconcave boundary transversely, the type toggles; that is, ridge becomes course and vice versa. Hence, such points have been implicated (by Jordan) as the likely origins of springs and we may well call them spring points. The convedconcave boundary of the example landscape is illustrated in Fig. 54.

FIGURE 54. The convedconcave boundary of the example landscape. The critical points are also indicated: Notice the relation to the saddles.

138


L. The Loci of Vertices of the Isohypses As a short calculation will reveal, the vertices of the level curves are characterized through the (cubic) condition

2s2- C12a = 0.

(136)

These curves pass both through the extrema and through the saddles, and they have a self-intersection at both. It appears evident that the vertex loci run all the way from the extrema to the saddle points, though they can certainly bifurcate and reunite again. The generic pattern has (to the best of our knowledge) not been studied yet.

OF THE LEVELCURVES Iv. CONTOURS: ENVELOPES

In the natural landscape overhangs essentially don’t occur since gravity would soon remove them. Thus, the level curves will simply foliate the plane and have no envelopes. In images we can only have one image intensity per pixel; thus, envelopes likewise don’t occur. In the case of relief envelopes are very common though. The reason is simply that most reliefs are due to the surfaces of objects of limited extent. From any vantage point one sees only the front of the object, that is, only part of its surface. The level curves of the distance have singularities at the locations where the lines of sight are tangent to the object: In such cases the projection of the tangent plane degenerates into a line (Figs. 55 and 56). These curves are evident in the topographic rendering as envelopes of the curvilinear congruence of level curves. At an envelope the level curve is generically tangent to the envelope and one branch (the part of the level curve at one side of the point

FIGURE55. Generation of the contour. The bean-shaped object is projected upon the plane n from the point P.The grazing visual rays form a cone C that meets the plane ll in the curve C. The preimage of C on the object (the curve p, called the rim) is smooth, whereas the curve C has a cusp due to the fact that the rim may be tangent to the visual ray at isolated points.


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FIGURE 56. Generic behavior of level curves at the contour. Left: A generic contour or fold. Middle: A near point on the contour. Right: A far point on the contour.

of tangency) will be visible, whereas the other branch will not be visible due to the fact that it belongs to points on the backside of the object. Since the sign of the normal curvature of the surface at a point of the preimage of the contour is well determined (because the object has to lie at one side of the contour), the sign of the curvature of the contour in the plane of projection is simply the sign of the Gaussian curvature of the surface. Thus, the contour is concave when the surface is (locally hyperbolic) and convex if it is (locally) elliptic (Fig. 57). The envelopes will typically be smooth curves, except for self-intersections(socalled T-junctions) and so-called cusp points. At a cusp the (visible) branch of the envelope ends. Cusps occur when the line of sight runs along the preimage of the envelope on the surface of the object (it can be shown that the line of sight then touches the surface along a principal tangent or asymptotic direction, hence the surface will locally be hyperbolic). In that case the projection of the preimage degenerates into a point (Fig. 58). The fall curves cusp at the envelope, although only one of the branches will be visible. Indeed, when the fall curve meets the envelope the line of sight has to be tangent to the preimage of the fall curve. At isolated points of the contour the level curves may meet the contour in such a way that either only an isolated point, or both branches of the level curve are visible. This happens when the line of

FIGURE 57. Convex and concave contour segments. The surface part marked H is hyperbolic, that marked E is elliptic. The parabolic curve meets the contour at P and Q. Thus, we obtain the concave segments 02, u3, and the convex segments 01and u4.

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FIGURE58. Generic singularities of the contour. P is the typical case: The point is not on a contour. Q denotes the fold: The point is on the contour. R denotes two transversely intersecting folds: The point is a 7'-junction. S is the cusp: The point is the endpoint of a contour.

sight touches the surface along a principal direction, the preimage of the envelope then runs along the other principal direction. At such points the preimage of the contour has a local extremum of depth. These extrema have the character of a depth maximum or a depth saddle. (See Fig. 59.) These special points complement the Morse critical points in the interior of the contour.

V. DISCRETEREPRESENTATION In applications we rarely meet with scalar fields as such: We typically have access to observations of samples at a (possibly very large but certainly finite) number of sample points. This introduces a number of important and difficult problems. d

d

FIGURE59. Near and far points of the contour. Here o denotes the contour and (p the fall curve through the singular point. Notice that point P is like a saddle, whereas point Q is like a depth maximum.


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A. Triangulations

In order to deal with surface properties we certainly need more than just a collection of sample points. Some data structure is needed that allows us to glue these completely distinct items together. The way to handle this is to introduce a triangulation of which the sample points are the vertices. Since we have samples of the values on the vertices we then have a polyhedral approximation of the surface. In order to define the triangulation we have to define its edges andfaces. The edges will be ordered pairs of distinct vertices such that any pair of vertices defines at most a single edge. Not all possible pairs will be used to define edges because we will certainly require that no two edges will intersect each other. We need a sufficient number of edges to ensure that they divide the territory into a number of triangular faces. Aface is an ordered number of vertices such that when (vl , u2, ug) is a facet, then (u1, 1121, (u2, ug), and (2)3,01) are edges, possibly in the wrong order (e.g., (u2, vl] is the edge ( u ~u2} , in reverse order). We require that the faces tesselate a simply connected area. This still leaves an enormous amount of ambiguity in the choice of triangulation. We describe some common choices next. The required data structures are then first of all a list of sampled values ( z l , . . . , z ~ )where , V denotes the number of sample points. The order in this list (index {vl , . . . , u v ) = ( 1, . . . , V}) is used as a label for the vertex. Then we need a listofedges((ui,u j ) ,...1, wheretheorderinthelist((e1, ...,eE) = (1, ...,E)) is used as a label for the edge. Finally, we need a list of faces ( ( u i , vj, u k } , . . .), where the order in the list ((f~, . . . , f ~ =) (1, . . . , F } ) is used as a label for the face. This type of data structure is the most basic one and turns out to be very convenient in most algorithms when implemented in mathematics packages such as Mathematica or Maple. For many algorithms one desires more explicit relations between the vertices, edges, and faces (although in principle everything can be found from the elementary data structures). It usually pays off to precompute a number of additional data structures, depending on the task. These data structures mainly serve to be able to find neighbors or boundary elements and to proceed from one item to the next. Examples are the edges that form the boundaries of a face, the neighbor of a face for a given edge, the ordered list of edges that meet a given vertex, and so on. 1. Regular Lattices

Regular lattices are either Cartesian grids with added diagonals or lattices with hexagonal symmetry. In practice the differences tend to be slight, either in quality of the results or the complexity of the algorithms. This is the case because the Cartesian lattices with additional diagonals are simply hexagonal lattices, at least, if all diagonals are drawn in the same direction. The hexagonal lattices have the

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FIGURE60. Left: Cartesian lattice. The edges list is ((1.2). (2,3), (4,5), (5.6). (7.8). (8.9). (1,4), (4,7), (2.5). (5,8), (3.6). (6.9). (I.5), (2.61, (4,8), (5,911 and the faces list is ((1,2,5),(1,5,4), (2,3,6), (2,6,5),(4.53). (43.7). (5,6,9), (5,9,8)). Right: Hexagonal lattice. The edges list is ((1.2). (1,3), (l,4), (1,5). (1,6). (1,7). (2,3), (3,4), (4.5). (5.6). (6.7). ((7,211, and the faces list is ((1.2.3). (1,3,4), (1,4,5), (lS.6). (1.6.7). (1.72)).

theoretical advantage because they are more nearly isotropic; the Cartesian lattices are typically more convenient to handle. The choice is essentially arbitrary. (See Fig. 60.) 2. Delaunay Triangulations

When the sample points are irregular (or actually random) the Delaunay triangulation is the optimal choice. This type of triangulation leads to faces that are as closely equilateral as can be. This is very important because very “thin” triangles lead to numerically unstable behavior. Nowadays computational geometry has provided us with optimal algorithms for actually computing Delaunay triangulations and-although costly-they typically are a viable option. (See Fig. 61.) Mathematics packages such as Mathematica or Maple readily perform the task.


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3. TriangulationEditing In most cases of interest the triangulation will be a very large data structure and it will be next to impossible to make changes manually without assistencefrom some smart program. A triangulation editor lets one graphically pick vertices, edges, or faces and allows one to, for example, delete an element (the editor should automatically “mend” the hole and update the data structure), add an element (the editor should automatically “splice it in” and update the data structure), barycentrically subdivide a face (the editor should automatically subdivide neighboring faces so as to add up with a valid triangulation and update the data structure), and so on. It should be possible to indicate an element and retrieve various properties of it, or to obtain pertinent data concerning the triangulation as a whole.

B. Isohypses and Slope Field The level curves and fall curves on the triangulated surface degenerate into piecewise linear polygonal arcs. This is because we have only toconsider planar surfaces: Any point on the triangulation is generically in a face, and thus part of a unique planar surface. The level curves and fall curves for planes are simply straight lines which are easy to compute. In fact, most math packages such as Mathematica or Maple support contour maps, which simply plot the level curves for a Cartesian-basedtriangulation. (See Fig. 62.) The fall curves are more problematic, not because they are particularly

FIGURE62. Level curves in the discrete case. Left: The triangulation. Right: The contour curves. Over the interior of the faces the level curves are straight-line segments.

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difficult to compute, but because there is no principled way to space them. In order to do that one would need a complete integral of the creep equation. In practice one has to be satisfied with an essentially arbitrary selection of drawn fall curves. One useful method is to pick a fiducial level curve. Then the fall curves may be launched from points of the fiducial level curve spaced at equal arclength intervals.

C. Morse Critical Points To find Morse extrema (summits and immits) on the triangulated surface is an easy task. A vertex is a summit if all neighboring vertices are below it. Thus, one simply has to find all neighbors of a vertex and check them. The neighbor vertices are the vertices that are part of edges that also contain the fiducial vertex. For the immits the procedure is similar. In order to find all summits and immits we simply visit all vertices and check them for the required properties. The only problem that might occur is that some vertices might turn out to be at the same height. This is actually a rather likely event if the samples have been discretized as is most often the case in image processing. The simplest way to avoid the problem is toforce genericity. One sorts the samples and finds the equal samples. Then one perturbs the heights by small amounts (less than a discretization step) in order to make all heights distinct. This introduces some arbitrariness, but this is irrelevant as the perturbations are in the noise level anyway. (In practice one simply replaces the heights with the rank order, where the order is assigned arbitrarily in the case of ties.) More principled methods tend to be (much) more complicated because a great number of possible exceptions has to be handled and the chunks of data that appear as single entities can become arbitrarily large (when all values turn out to be equal). Such methods are often more of a pain in the neck than a real boon. It is much more difficult to find the saddles. In order to find saddles we need to order the neighbors of a vertex something that is quite unnecessary if we look only for extrema. When we visit all neighbors in order and notice whether the sampled values are higher or lower than the fiducial values we end up with the following possibilities (of course one should mind the fact that the order is periodic): All values are lower: Then the fiducial vertex is a summit. All values are higher: Then the fiducial value is an immit. There is a run of lower and a run of higher values: Then the fiducial value is a generic slope point. In all other cases the fiducial vertex is a saddle. There are severalcomplicationsthat might occur here. First of all, we have assumed that the vertex is not on the boundary of the triangulation. These boundary vertices have to be handled separately, we will not go into that here. Second, the saddle may be either of the Morse type in which case we should have an alternation of runs


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of lower, higher, lower and again higher neighbor vertices, or it may be a “higherorder” saddle. For instance, for the (most common) vertex with six neighbors we may find the sequence - which indicates a “monkey saddle.” There are several ways to handle this complication. Since we can find the maximum number of neighbors for any given vertex we can figure out beforehand what types of higher-order saddles are at all possible and simply include these in our classification. Another method is to introduce dummy vertices and edges and split the higher-order saddle into generic Morse ones. The first possibility is typically better if one uses a Cartesian-based triangulation, since there exist no reasonable ways to handle the messed-up lattice (which counts as a disadvantage). The latter method is to be preferred if it is not prohibitive to complicate the triangulation slightly because then we have the advantage of Morse critical points throughout without (awkward) exceptions.

+ + +-,

D. Ridges and Ruts

Finding ridges and ruts on the triangulated surface turns out to be a simple task. In fact, one might say that ridges and ruts are more easily defined on triangulated surfaces than on smooth ones. At least everyone seems to agree on the definition in the discrete case whereas there is as yet no concensus on the definition in the continuous case! Ridges and ruts will definitely have to be sought among edge progressions because there can be no such entities on the faces since these are planar. The surface at an edge is simply the butterfly structure of the two faces that meet at the edge (so-called winged edge representation). When we consider the height gradient on both faces we meet with the following possibilities (Fig. 63): There are paths of steepest descent that run from one face, over the edge, onto the other face. If this is the case the edge is certainly not part of a rut or ridge but part of a generic slope region.

FIGURE 63. The creep field at discrete ridges and ruts. Left: A regular edge: The flow simply runs over the edge from one face to another. Middle: An edge that is a course: From both adjacent faces the flow runs into the edge. The flow can only continue its descent via the edge, which acts as a true course. Right: An edge that is a ridge: In both adjacent faces the flow runs from the edge into the face. The flow can only start its descent at the edge which acts as a true watershed.

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FIGURE 64. Morse critical points, ridges and ruts in the discrete case.

There exist descending paths on each of the two faces that meet at a point of the edge. Then the edge must be a rut since the creep will run from the faces to the edge and continue by way of the edge. There exist ascending paths on each of the two faces that meet at a point of the edge. Then the edge must be a ridge since the edge will act as a watershed and the creep will run from the edge into each of the faces. We can simply check these conditions and do an exhaustive search for ridge and rut elements. Afterward we can concatenate the individually found elements into edge progressions of maximum length. Slight complications arise here because bifurcations do occur and ruts and ridges may meet, but in principle the procedure is indeed straightforward. (See Fig. 64.)We have used such methods routinely for triangulations of up to about a thousand faces. For larger triangulations one should search for optimum algorithms. As far as we know such work is not available at present though. In order to find the ridges and ruts as indicated one really needs the heights at the vertices and the layout in the xy-plane, the mere height order and triangulation topology doesn’t suffice (as it does for the computation of the Morse critical points). To see what the problem is consider the situation depicted schematicallyin Fig. 63. Suppose that B is above A , and Q above P. Then if Q is above both A and B and P below both A and B we certainly have a regular edge. If the depth order of the points is APQB we cannot decide, for consider any situation (regular, ridge,


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or course), then you can always place points P, Q on the two faces in the given order, thus-conversely-the order is consisent with any of these interpretations.

VI. CONCLUSION We have extensively discussed the theory of topographic maps (structure of scalar fields in two dimensions). This is what is availabletoday. It will be evident from our exposition that there are still quite a few “white areas on the map.” Indeed, though the subject has a venerable history (serious mathematics started [20] at about the beginning of the nineteenth century with much activity continuing to the present day) it cannot be regarded as a closed subject. Such important and intriguing topics as the proper definition of ridges and courses still remains a matter of hot debate. A major lack of knowledge is in the area of generic properties of the topographic features. Most of the early work focused on nongeneric, specific examples, which where often generalized wrongly. Since many of the interesting properties are of a rather high order it will be a nontrivial task to fill in the required understanding. If we-as is almost mandatory-study the topography with scale (or resolution) as a parameter the task becomes even more daunting. However, even on the level of fairly simplistic descriptions there are still surprises in store; in this paper we have several illustrations of generic facts that have apparently completely escaped earlier authors. An almost completely novel area is that of the discrete representation. Certainly real observations were used in the past (most of the developments were done with very practical issues in mind), but one typically used techniques of descriptive geometry [49, 561, which are quite alien to modem computer analysis. With the advent of modem symbolic algebra and novel geometrical algorithms,possibilities have been created that were completely out of reach even a decade ago. In the area of the structure of scalar field there are still extensive opportunities for progress in these fields.

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6y )

ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL. 103

Dyadic Green’s Function Microstrip Circulator Theory for Inhomogeneous Ferrite with and without Penetrable Walls CLIFFORD M. KROWNE Microwave Technology Branch. Electronics Science and Technology Division. Naval Research Laboratory. Washington. DC 230375

I . Overall Introduction . . . . . . . . . . . . . . . . . . . . . . . I1. Implicit 3D Dyadic Green’s Function with Vertically Layered External Material Using Mode-Matching . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Self-Adjoint Operators for Vertically Layered External Material . . . . . . C . Implicit Dyadic Green’s Function Construction . . . . . . . . . . . . D. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . I11. Implicit 3D Dyadic Green’s Function with Simple External Material Using ModeMatching . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Fields. Mode-Matching Technique, Nonsource and Source Equations . . . . . C. Implicit Dyadic Green’s Function . . . . . . . . . . . . . . . . . D. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . IV. 2D Dyadic Green’s Function for Penetrable Walls . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . C . Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . D . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . V. 3D Dyadic Green’s Function for Penetrable Walls . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . C . Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . D . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . VI . Limiting Dyadic Green’s Function Forms for Homogeneous Ferrite . . . . . . A. 2D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . B . 3D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . VII . Symmetry Considerations for Hard Magnetic Wall Circulators . . . . . . . . A . 2D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . B . 3D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . VIII . Overall Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

151 Volume 103 ISBN 0-12-014745-9

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ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright @ 1998 by Academic Ress Inc. All rights of reproduction in any form reserved. ISSN 1076-5670/98$25.00

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CLIFFORD M. KROWNE

I. OVERALLINTRODUCTION

Our focus here is on canonically shaped circulators, namely, those with circular symmetry. Great advantage may be taken of mathematical physics tools when restricting the problems examined to canonically shaped objects, such as those with circular symmetry. For electromagnetic circulators, many actual devices possess some degree of circular symmetry, so symmetry restrictions are not unrealistic. As one moves further and further away from simple objects which are symmetrical, like a circle in a two-dimensional model or a circular cylinder in a threedimensionalmodel, the ability to derive explicit dyadic Green’s functions becomes more difficult. Finally, when the complexity is increased to the extent when the circulator puck (composed of ferrite anisotropic material) is surrounded by layers of substrate material and layers of superstrate material, with radially changing composition, all that is reasonable to seek is an implicit Green’s function, which retains some of the features originally employedto find the simpler explicit Green’s function, and new aspects found in the mode-matching method. Adding in the mode-matching technique (Sections I1 and 111) with its variable number of matching modes seems to irrevocably eliminate the ability to solve for explicit dyadic Green’s function elements. Systems of equations, or equivalently, matrices in matrix equations contain the properties of the environment external to the puck and the manner in which the outlying areas interface with the puck itself. But what is lost in acquiring the field solutions through explicit dyadic Green’s functions is gained in treating much more complicated surrounding environments, which still retain circular symmetry. The radially changing composition mentioned above may be thought of as zones of specific radial width, reaching out toward infinite radial size. There may be a finite number of zones, and the last zone can be either terminated at infinity or at an electric or magnetic wall. Another area where an attempt to obtain explicit dyadic Green’s functions may shed more light on the electromagnetics is that of field extension beyond the circulator perimeter (Sections IV-VI). It is often a reasonable approximation, substantiated by comparison to experiment, to model the contour between microstrip ports as an impenetrable magnetic wall. But it might be very instructive to obtain expressions giving the dependence on the external material outside of the inhomogeneous puck. This may be done under specific assumptions for both two- and three-dimensional approaches regarding the field behavior and coupling between the internal puck fields and the outside fields on either side of the nonport boundaries. Significant simplifications occur when the inhomogeneity of the ferrite puck is taken to limit to a single uniform disk (two-dimensionalmodel) or cylinder (three-dimensionalmodel) of material (Section VI). Although the magnetic sources (driving functions) which represent the microstrip feeding lines can be placed at arbitrary azimuthal angular locations along

THEORY FOR INHOMOGENEOUS FERRITE

153

the circulator perimeter, for both two- and three-dimensional models, tremendous simplifications occur in the various dyadic Green’s functions used to obtain the s-parameters and electromagneticfields when symmetricplacement of the ports is imposed (Section VII). For threefold, fourfold, or sixfold symmetry of the ports, where the mathematics is still manageable, drastic reduction of the required number of Green’s function elements used in building up s-parameter solutions occurs in the two-dimensional treatment.

11. IMPLICIT 3D DYADICGREEN’SFUNCTION WITHVERTICALLYLAYERED EXTERNAL MATERIAL USING MODE-MATCHING A . Introduction

Developing Green’s function approaches for canonical structures can be particularly advantageous when solving inhomogeneousboundary-valued problems, as is the case for planar circulator problems of the microstrip variety. The driving force occurs on the r = R surface at the point 4 = 4’ and z = z’ or on a strip at 4 = 4’. Obtaining explicit dyadic Green’s function expressions is known to be very convenient and allows extremely rapid numerical computation of electromagneticfields and s-parameters [ 1, 21. In that work, the circulator was a circular femte puck, but with completely arbitrary radial variation of the descriptive parameters of the problem. The puck itself was made up of a number of annular rings, each with different widths, and with different material properties for the magnetization M, and demagnetization factor N,,. The magnetic biasing field Happwas also allowed to vary in an arbitrary radial manner. Two- and three-dimensional dyadic Green’s functions were obtained, which depended upon recursive relations to find final expressions. Although these expressions are compact and explicit, the recursive nature of the development necessarily contains embedded information, making the actual algebraic dyadic Green’s functions immensely complicated. Therefore, computer techniques are essential in studying the behavior of the dyadic Green’s functions. But because of the canonical nature of the structure geometry, and the theoretical techniques employed in the derivations, these dyadic Green’s functions lead to field evaluationsbetween 1,000 and 10,000 times faster than intensive numerical techniques like finite difference or finite element methods. What we are desirous of doing in this section is dropping this complex inhomogeneous puck into a medium which consists of radial zones beyond the circulator puck perimeter. Each zone is made up of an arbitrary number of horizontal layers, stacked vertically in the z-direction. This arrangement outside of the puck will constitute yet another inhomogeneous problem, in addition to that of the puck

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CLIFFORD M.KROWNE

itself. In principle, the region above the puck, bounded on the lower side by an electric wall formed by a microstrip conductor, and on the top by a metal limiting wall, can also be viewed as a zone. Such a structure as outlined above can be used to treat the case where a circulator ferrite puck is dropped into a hole in a substrate, and possibly covered by a superstrate. Both the substrate and superstrate may be broken up into layers. All of the material external to the puck will be considered to be isotropic, but with the possibility that the cylinder above the puck and the layers in the radial zones can have permittivity properties, permeability properties (unbiased), or both simultaneous permittivity and permeability properties. Each radial zone, stretching vertically from the lower ground plane to the top horizontal wall, made up of many different layered regions, is viewed as a waveguide section, with a collective radial waveguide propagation constant. At a cylindrical wall r = r , , mode-matching is applied. The j index increases in value from j = 1 at the the puck-external medium interface rl = R, to jN at the last interface. The last interface may be chosen as open in which case a radiation condition could be applied or as an electric or magnetic wall requiring explicit but simple mode-matching conditions for the last zone’s vertically stacked regions. Here we will treat a specific case of the general situation outlined in the last paragraph. The puck will be placed inside a substrate like that found in microwave monolithic integrated circuits (MMICs), with a ground plane bounding it from below. An electric wall, representing microstrip metal, will constrain the fields within the femte puck material from above, and this electric wall will be flush with the substrate surface. Immediately above the puck will be isotropic material, not necessarily the same as that for the medium beyond the circulator puck perimeter. One zone exists beyond the perimeter, and it consists of the substrate on the bottom and another material region on top, not necessarily the same as the inner zone above the puck. The top layer, consisting of an inner and outer radial ordered set, constitutes the superstrate, which could be chosen by default in the simplest situation to be air. For the substrate being part of an MMIC, it could be one of a number of semiconductor materials like Si, GaAs, or even heterostructure material. For the case where a more hybrid-like circuit is used, it could be an unbiased magnetic material, even the same or related to that used for the puck itself. Furthermore, depending upon how the biasing magnetic field is obtained for the puck, the electric wall above the puck may be a microstrip-keeper metal combination to allow self-biasing of the femte material in the puck. With the use of a conventional biasing magnet, the origin of this field is considered to come from outside the whole structure shown schematically in Fig. 1. Allowing for a magnet to be placed in a layered arrangement above the puck (as in Fig. 2 with no space shown here, for this particular diagram, between the magnet and the microstrip disk) constitutes a greater complication to the problem which won’t be addressed in this paper, although

-

155

THEORY FOR INHOMOGENEOUS FERRITE 2

multl-layered vertical zone

Top Cover

/ / / / / / / / /I / / / / / / / / / /

OT EdT P d T source wall

OB Ed0 P d 0

I

hl

c e.w.

OT hO

TI

rL

C

hc

lnhomo Ferrite

OB 7

Ground Plane R

RGURE1. The ferrite circulator structure including the regions above and surrounding the device puck. This figure is formed by taking a cut plane at d, = const (in 3D).

the theoretical principles for accomplishing such inclusion will be treated in this work. Self-adjoint operators are found for the differential equations describing the z-dependent field variation in the medium zone external to the circulator puck. The external medium is in general inhomogeneously layered, consisting of media with permittivity properties, magnetic properties, or both. For the simplest case in which each zone has regions of only one trait (i.e., not mixed), and that trait is

R

FIGURE2. The femte circulator sttllcture including the regions above and surrounding the device puck with the presence of an external biasing magnet. This figure is formed by taking a cut plane at d, = const (in 3D).

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CLIFFORD M. KROWNE

dielectric, information is available on the TE axisymmetriceigenvalue equation [3], eigenvector forms [4,5], or scalar potential governing equations and the eigenvalues and eigenvectors using self-adjointness properties [6]. Eigenvalue equations characterizingthe radially sectioned medium outside the puck are found, as are the eigenvectors. When the z-dependent parts are multiplied with the radial and azimuthal dependences,the complete field expressions are determined. Source constraint equations driving the circulator are then combined with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements. Mode orthogonality is employed to encourage sparsity in matrix system development where appropriate or convenient. The self-adjoint operators lead to testing functions which may be used to test field continuity equations, thereby reducing some infinite summations to single-term contributions. The implicit Green’s function is particularly useful because field information and s-paramaters may be found in real space, completely avoiding typical inverse transformations. Consideration of field extension into the surrounding medium, beyond the circulator perimeter, including fringing such that fields may extend out and then above the height of the circulator nonreciprocal puck, is an essentially physical motivation for this theoretical work. The approach is a good approximation to a very complicated geometric and inhomogeneous problem, given the irregular effects arising from application of the dc biasing magnetic field and the actual finite-width microstrip input and output lines. For narrow microstrip lines the expectation that the fields extend beyond the device perimeter, with azimuthal symmetry, is very good, and essential to this canonical treatment. When some of the microstrip lines attain widths which are a noticeable fraction of the puck radius, the error introduced by the symmetry assumption for r > R for the fields will be directly related to the fraction of the circumference occluded by the presence of the line itself. B. Self-Adjoint Operatorsfor Vertically Layered External Material 1. Introduction

Consider the situation where the electromagnetic field occupies three areas (see Fig. 1). The first area C is that filled by the femte nonreciprocal material for r < R and 0 p z p hc. The second area or zone 0 has two regions for r > R, a bottom region O B with0 Iz p hc andatopregion OT withhc 5 z Iho. Andthethird region I has r < R and hc Iz Iho. Conducting walls are assumed at z = 0, h (all r ) and z = hc ( r < R ) , and the radiation condition in effect as r + 00. Fig. 3 shows a cross section through a z = const plane, with microstrip lines coming into (or out of) the ports located at $1, &, . . . ,4i, . . . ,4~~on the r = R puck surface. In order to maintain the same field structure formulas and parallel construction techniques inside and outside of the puck, governing equations are developed in the zone r > R with the stacked 0 B and 0 T regions utilizing field components


157

Port - femte region

7/ /

interface

,

Port 1

[E

Inhomo. Wall ( non - port)

\\

; (UPo, K)I

Multi-layered external vertical zone [Ed,

dl

Port i

FIGURE3. This sketch represents a cut plane at z = const (in 3D) for the femte. circulator structure.

from the puck in the isotropic limit. Specifically, TM, (e mode) and TE, (h mode) equations are sought. Maxwell’s sourceless equations assuming eiordependence are

V x E = -iwpH

V x H = ioEE

( 1ab)

2. TM, Operator Properties Take the curl of (la), noting that both p and E depend upon spatial location, namely, that p = p(r) = p ( r , 4, z ) and E = E(r) = ~ ( r4,, z ) [71.

V x V x E = -iwV x (pH)

+

= o2p&E vp(r) x V x E LL

using both curl expressions (1) to remove any H field dependence. Using an identity to eliminate the curlcurl term,

Realizing that the divergence of a curl in the left-hand side of (lb) is zero, the divergence of the electric field in (3) may be replaced by

V .E =

--a

Vdr) E E

Equation (4)has been obtained from the general spatial variation being reduced to that in only the z-direction because of regional changes within a zone (i.e., for

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CLIFFORD M.KROWNE

a two-region zone as being considered here, E ( Z ) = &fB for 0 5 z 5 hc and for hc 5 z 5 ho with a discontinuity at z = hc). E(Z) = Inserting (4) into (3), the vector electric field governing equation is found:

k2 = W'PE = 02p(r)E(r)= W * P ( Z ) E ( Z )

(6)

If the TM to z is selected as the basic mode electric type (out of the two required for a complete field description) with only the electric field existing perpendicular to the transverse plane, this being precisely the same directional field setup inside the puck, then (5) vastly simplifies to +k2Ez = O

(7)

+k2E, = 0

(8)

Since all of the other field components (transverse) depend on formulas written in terms of E,, only its expansion need be considered first. m

h

n=-m

k,=b

bo

n=-mm,=l

Here it is seen that the electric field solution is the sum over the radial variation, z directed harmonic, and azimuthal harmonic products. The A coefficient provides


159

the term by term weighting. The infinite sum over k, propagation constants in forward ( f ) and backward (b) directions may be changed into only a backward wave summation,but now requiring explicit forward and backward A coefficients. added to the A coefficients store that wave direction Superscripts "-"and information, letting the subscript propagation index being simplified to merely k, = b. This process is collected together in the next line. Recognizing that in each zone only the radial propagation constant n is definable (and therefore capable of being indexed over the entire zone), but that k, varies from region to region within a zone, the fifth line is obtained. Finally, the n solutions to be determined later, can be assigned for the solutions, index numbers ordered as me = 1 , 2 , . . . , 00. The associations for Z i e( z ) in (9) can be summarized as

"+"

acting as the separation constant,

Select the first two derivative terms of (13), plus the first of the last bracketed sum, as the inhomogeneous linear operator L T M ,

Invoking ( 14) enables ( 13) to be recast as

L TM z ; < ( z ) = -A;< Zi,(Z)

(15)

Operator equation (1 5) is in eigenvalue form, the eigenvalue operator on the righthand side of the equation merely the eigenvalue constant. Requiring the radiation condition to hold as r += 00 makes

R ; p ) = Kl (0:r)

(16)

Next let us find the adjoint of L to better understand the behavior of the inhomogeneously loaded waveguide zone. For ease of mathematical argument and brevity, abbreviate two different Z i e( z ) solutions as u and u and define the inner

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CLIFFORD M.KROWNE

product on the interval (0,h o ) as

with weight w(z). To find the adjoint, study

1

ho

(u, Lu) =

w(z)u(Lu}d z

Sometimes it is convenient to explicitly place the weight in the bracketed expression when we wish to be reminded of its presence, as in ( w u , Lu). Anyway, we seek to convert (18) into the form (L'u, u) by repeated application of integration by parts and thereby identify the adjoint form La of the operator 15. The weight we choose here is

Therefore,


=-loe$du

161

+ (qu,u )

From (20) the adjoint form of the operator can be identified as L$M = L T M

(21)

Thus, it is seen that the operator is self-adjoint [8,9,10]. Equation (21) was obtained by using a number of boundary conditions, which will be covered in this section briefly. Electric wall conditions at z = 0, h~ require transverse field components E, and E4 to be zero. From the Ed field component expression [ 111, for example,

one observes that pure Neumann conditions hold at the ends of the inner product domain. This conclusion follows because (22) originates when exponential (harmonic, plane waves) are chosen for the z-functionalbehavior. Thus, the coefficient p,, from the second piece of E4 generating the TM mode, indexed for j z-directed

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CLIFFORD M. KROWNE

modes, contains a ik, factor, implying the presence of a partial derivative a/az operator.

The same conditions, of course, hold for u. Because within a region of a zone E(Z) is stipulated to vary continuously, and for the zone to vary, at most, piecewise continuously, because the discontinuitiesonly occur at the region interfaces, ds/dz will be well behaved at the domain ends. Furthermore, requiring E ( Z ) to be constant within a region makes a Neumann condition also hold for it. Thus,

Neumann conditions (24) on d&(z)/dz are not required for obtaining the selfadjoint relation (20). This is also the case for TE modes, as will be demonstrated later, It is emphasized here that although L T M is self-adjoint, it is not representing lossless media. The media can be dielectrically lossy, magnetically lossy, or lossy in both regards simultaneously. Thus, the eigenvalue in (15) may be complex. In fact, for ordinary media, we expect it to be complex. Now let us review in light of this fact, the short derivation of orthogonality as implied by self-adjointness. Consider that any self-adjoint operator L obeys ( L u , u ) = (u,L u )

(25)

Let

L T M U= -AhCu; L T M U= -A',ume

(26ab)

associating u and u with, respectively, the eigenvalue indices me and m:. Placing (26) into (25) yields

(-A;/,

u) = (U,-A;p)

or

(A;; - A;,) (u,u ) = 0 or

(u,u ) = 0

(27)

Orthogonality relation (27) holds precisely because the z-eigenfunctionsare associated with different eigenvalues. In our case, the eigenvalue difference is between different complex eigenvalues. Relation (27) says that the z-eigenfunctions are orthogonal no matter how many different regions are stacked in a zone, and this


163

is true regardless whether we use only dielectric regions, magnetic regions (unbiased), or intermix these two types of regions, or even if we further complicate the situation by using regions with both dielectric and magnetic characteristics. As a final point of discussion, proof of L T M operator self-adjointness can be found in a more general manner than by (20) by associating L T M with a general second-order derivative operator, showing the connection between this secondorder derivative operator and another standard operator form Ls, and then finally demonstrating that L S is indeed self-adjoint. A general second-order derivative operator can be expressed as

d2 d L, = 4 z ) - b(z)dz2 dz Next define the L S operator as

+

1 d w(z)dz In L S make the following associations:

Substituting ~

( z )p ,( z ) , and q ( z ) into L s ,

1

= L,

+ c(z)

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CLIFFORD M.KROWNE

We can readily show that L S is self-adjoint by studying


165

Self-adjointness of L S was assured because evaluation of d i n i t e integrals n two steps of the derivation relied upon either Dirichlet or Neumann boundary conditions holding, and this is known to be true for many classes of operators acting on variables, as it is for true for the LTM operator. In fact, for the LTM operator ( 14)Neumann conditionsare valid for the variables u and u of the problem. Now let us associate the LTM operator with a general second-order derivative operator by finding the coefficients a ( z ) ,b ( z ) , and c ( z ) . Expand the second operator term in (14),

{

-l d_s } + k 2 L T M = -d+2 - - -I+ d& -d dz2 s d z d z dz s d z Comparison to L , allows the identifications 1d s = 1; b ( z ) = --; E dz C ( Z ) may also be written as U(Z)

C(Z)

=k

At last, the coefficients in the L S operator can be determined.

+

q(z)= c(z)=w2s(z)p(z)

3. TE, Operator Properties Following the philosophy of the TM, development in Section B.2, take the curl of (lb).

V

x

V

x H = iwV x

= u2psH

(EE)

+VE(r) x V x H &

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CLIFFORD M.KROWNE

using both curl expressions (1) to remove any E field dependence. Using an identity to eliminate the curlcurl term,

+

V2H- V ( V H) W'PEH

+V E ( r )x V x H = 0 E

(29)

Realizing that the divergence of a curl in the left-hand side of (la) is zero, the divergence of the magnetic field in (29) may be replaced by

Equation (30) has been obtained from the general spatial variation being reduced to that in only the z-direction because of regional changes within a zone (i.e., for a two-region zone as being considered here, p(z) = pdOB for 0 5 z 5 hc and p(z) = p y T for hc Iz 5 ho with a discontinuity at z = h c ) . Inserting (30) into (29), the vector magnetic field governing equation is found:

':)A

a:(

. ( ( l a -H- -z - q ) r a4

a2)i}=0

(31)

If the TE to z is selected as the basic mode magnetic type with only the magnetic field existing perpendicular to the transverse plane, then (3 1) greatly simplifies to

or

Since all of the other field components (transverse) depend on formulas written in terms of H,, only its expansion need be found. c

o

b

167


=

1

Rnhk,(r)[e

ik z

hO- + e-ikzzAhO+ Ank, nk,

Ie'

in@

n=-ca k,=b

n=-camh=l

Here it is seen that the magnetic field solution is the sum over the radial variation, z-directed harmonic, and azimuthal harmonic products. The a solutions to be determined later, more correctly denoted as am,,are assigned for the solutions are different than urn< index numbers ordered as m h = 1,2, . , 00. Solutions amh for the T M , case. The associations for Zkh(I)in (34) can be summarized as

..

h Ahozh mh (z) = z:(z)= z k z ( z ) = eikzz nk,

+ e-ik,zAhO+

nk,

= eik,zAhO

nk2

+ e-ik,zAhO

n,-k,

(35)

Now inserting (34) into (33) and applying separation of variables, with 2

= -(amok>

(36)

acting as the separation constant,

Select the first two derivative terms of (38), plus the first of the last bracketed sum, as the inhomogeneous linear operator L T E ,

(39) Utilizing (39) enables (38) to be recast as

Requiring that the radiation condition to hold again as r + 00 makes

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CLIFFORD M.KROWNE

Next let us find the adjoint of L to better understand the behavior of the inhomogeneously loaded waveguide zone. For ease of mathematical argument and brevity, again abbreviate two different ZLh(z)solutions as u and u and use the inner product definitions (17) and (18) on the interval (0, h o ) . We seek to convert ( 1 8) into the form ( L a u ,u ) in order to identify the adjoint form L" of the operator L = L T E . The weight we choose here is w(z) = W ( Z )

Therefore,

(42)

THEORY FOR IMIOMOGENEOUS FERRITE

= ( L $ ~ v u, )

169

(43)

From (43) the adjoint form of the operator can be identified as L$E = L T E

(44)

Thus, it is seen that the operator is self-adjoint. Equation (44)was obtained by using a number of boundary conditions, which will be covered in this section briefly. Electric wall conditions at z = 0, ho require transverse field components E, and E$ to be zero. From the E$ field component expression in (22), one observes that pure Dirichlet conditionshold at the ends of the inner product domain. Coefficient s,, from the first piece of E+ generating the TE mode, indexed for j z-directed modes, has no ik, factor, implying u ( 0 ) = 0; u(ho) = 0

(45ab)

The same conditionshold for u. Because within aregion of a zone p(z)is stipulated to vary continuously, and for the zone to vary, at most, piecewise continuously, because the discontinuitiesonly occur at the region interfaces, dp/dz will be well behaved at the domain ends. Furthermore, requiring p(z) to be constant within a region makes a Neumann condition also hold for it. Thus,

Although (46) holds, conditions on d p ( z ) / d z are not needed for obtaining the self-adjoint relation (43). This is like the case for the TM mode, where (24) was not required.

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CLIFFORD M.KROWNE

Finishing the presentation of TE, operator properties, again something may be said about the Ls operator as was done for the TM, operator properties. L s may be found for the TE modes merely by referring to the previous TM presentation and replacing E by p everywhere and noting that Dirichlet boundary conditions replace Neumann boundary conditions. 4. Eigenvalues for TM, Modes Eigenvalues umeof the TM, modes, and the consequent values of A&, can be found by applying the boundary conditions on the electric field component

E;'(r, 4, z ) = 0; z = ho

(47)

Ezo(r, 4, z ) = 0; z = 0

(48)

and continuityconditionson the azimuthalcomponents of the electric and magnetic fields

E;o(r, 49 z)lhc+ = E;o(r, 4, z)lhc-

(49)

H;o(r, 4, z)lh,+ = H;o(r, 47 z)lhc-

(50)

EZo can be written using the third line form of (9), and (22).

n=-w

k,=b

ikzj

.

'

where P'- - k i - kZj'

i W d

s.---

ki

- kzj

.

, u'--

-

iWEd

ki

- kZj

(52abc)

The comparable H: expression to (22) is [ 111

Electric wall boundary conditions (47) and (48) yield with the help of (5 1) eikPThoAeOT- - e-ikPThoAeOT+ -0 nm, eikpn.O

e O B - - e-ik;n.O

nm,

eOB+

=0

(55)

(56) Anmc Anmc These conditionshave resulted from the global nature of the radial om,propagation constant or me index, allowing the radial function and factors depending only on this index to drop out of the equations. Superscripts T and B denote, respectively, the top and bottom regions in the outside zone. It should be realized here that kZ has an implicit dependence on me so that when we see the perpendicular propagation


171

constant, it is understood that it constitutes an abbreviation for :k (for the top region, for example). Continuity conditions (49) and (50) become, using respectively (51) and (54), [ e i k p r h c A rnm, O T - - e-ik;ThcAeOT+ nm,

3 k z o T - [eik;Bhc A enm,O B - - e-ikpBhc A:::']

kpB (57)

Adding and subtracting these two equations from each other, and utilizing (56) gives the top region amplitude coefficients.

It is also helpful to define a ratio of the two amplitudes, AeOT+

so that

Similarly, refemng back to (56),

Insert amplitude relations (59) and (60) into the electric wall condition (55) to find the characteristic eigenvalue equation:

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CLIFFORD M. KROWNE

+

(a),

Keeping the bracketed groupings in identifying 81 = k,OThl kPBhc and 82 = kfTh, - kpBhc, the exponential transcendental eigenvalue equation can be converted to a trigonometric form.

+

I-

sin[kfThl k,OBhC sin[kpThl - kpBhc] -

5-g d

+

$ d

Here h , = ho - hc. Note that for the limiting case where the top and bottom regions become identical, the numerator becomes zero and (65) reduces to the familiar form sin[kpho] = 0. Another useful form may be obtained by grouping according to the ratios kpB/k,OT and & f B / & f T . Again, the exponential transcendental eigenvalue equation is changed into a trigonometric expression, which can be set down in two ways. E fB k: O T tan[k,O T h,] = --

koT

kp koB

‘d

Ed

‘d

tan[k,OBhC]

5 tan[kPTh,] = -+tan[kpBhc] The first form of this tangent relationship is written in terms of the propagation constant and dielectric ratios. The second form relates the top region quantities to the bottom region quantities. Equation (66b) can be shown to have the same form as indicated for a stacked inhomogeneous dielectric zone [6]. But we must be aware that now the material of the regions can be dielectric, magnetic, or of mixed permittivity and permeability character. The transcendental eigenvalue equation for the TE, will be seen later to differ explicitly from the pure dielectric form even though the TM, has not. That (66), or (65), constitute eigenvalue equations for , ; a can be understood if the separation equations are found from the differential governing equation ( 13)for Z i Cwith the help of (10) providing the exponential Z& form. Inserting (10) into (13) for the top and bottom regions in the zone, realizing that the inhomogeneous dielectric constant term drops out inside each region, yields

Invoking (1 1) for the separation constant, and taking the positive branches in (67), gives


173

Inserting (68) into (66b) gives a single transcendental equation in terms of the unknown Material region propagation constants are delineatable once (6) is examined.

02.

OT 2 (kd

)

OT O T

2 =@

pd

Ed

;

OB 2 (kd

)

2 OB O B

= w pd

Ed

(69)

Return to the amplitude ratio R f T . Following the same reasoning in finding (66) using grouping by k,OB/k,OTand

Eliminating the bottom region information in (70) by using the eigenvalue equation form (66b), the compact formula

results. Using ratio grouping and (66), the backward and forward amplitude coefficients in (59) and (60) for the top region are now

5 . Eigenvalues f o r TE, Modes Eigenvalues a,,,,,of the TE, modes, and the consequent values of A;*, can be found by applying the boundary conditions on the electric field component E j o ( r , 4, z) = 0; E;'(r,

4, z) = 0;

z = ho

(74)

t =0

(75)

and continuity conditionson the azimuthal components of the electric and magnetic fields E;O(r,

4 7

Z)Ihc+

= E j 0 ( r 9 4, Z ) l h c -

H;O(r, #, z)lhc+ = ~

i O ( r4, , z)lhc-

E j o can be written using the third line form of (34), and (22).

(76) (77)

174 H:'

CLIFFORD M. KROWNE

can be expressed, using the first part of (53), as

Electric wall boundary conditions (74) and (75) yield with the help of (78) eikp'hoAhOTnm, eikpB.O

hOBAnmt

+e-ikpThoAhOT+

=0

nm, + e-ikpB.O

hOB+ Anme

=0

(80) (81)

These conditions have resulted from the global nature of the radial am,propagation constant or m h index, allowing the radial function and factors depending only on this index to drop out of the equations. It should be realized here that k, has an implicit dependence on m h so that when we see the perpendicular propagation constant, it is understood that it constitutes an abbreviation for kg: (for the top region, for example). Continuity conditions (76) and (77) become, using respectively (78) and (79).

Adding and subtracting these two equations from each other, and utilizing (81) gives the top region amplitude coefficients.

Define a ratio of the backward and forward wave amplitudes,

175


so that

Similarly, referring back to (81),

Insert amplitude relations (84) and (85) into the electric wall condition (80) to find the characteristiceigenvalue equation:

(89)

+

Keeping the bracketed groupings in (89), identifying 81 = k,OThr k,OBhCand 0, = k P T h l - kPBhc ,the exponential transcendental eigenvalue equation can be converted to a trigonometric form.

For the limiting case where the top and bottom regions become identical, the numerator becomes zero and (90) reduces to the familiar form sin[k?ho] = 0. Another useful form may be obtained by grouping according to the perpendicular propagation constant and permeability ratios k,OB/k,OTand pfB/pdOT.Again, the exponential transcendental eigenvalue equation is changed into a trigonometric expression, which can be set down in two ways. OT

5tan[kfThr] = Pd

k: tan[kfBhC] k,OB

PJoT

PdOB

--

= -- tan[kfBhC] k,O k,OB The first form of this tangent relationship is written in terms of the propagation constant and dielectric ratios. The second form relates the top region quantities to the bottom region quantities. Equation (91b) shows that the magnetic differences between regions in a zone appear explicitly in the eigenvalue equation for the TE, -tan[kfThl]

176

CLIFFORD M. KROWNE

modes, whereas (66b) demonstratedthat the dielectric differences between regions in a zone appear explicitly in the eigenvalue equation for the T M , modes. ~ , be understood if That (91), or (90), constitute eigenvalue equations for C T ~ can the separation equations are found from the differential governing equation (38) for ZLh with the help of (35) providing the exponential Zkh form. Inserting (35) into (38) for the top and bottom regions in the zone, realizing that the inhomogeneous permeability term drops out inside each region, yields

Invoking (36) for the separation constant, and taking the positive branches in (92), gives

Inserting (93) into (91b) gives a single transcendental equation in terms of the unknown ath. Looking at the amplitude ratio R f T in (87) again, using the same procedure in finding (91) using grouping by k p B / k p T and p f B / p f T ,

Eliminating the bottom region informationin (94) by using the eigenvalue equation form (91b), the compact formula

results. Using ratio grouping and (91), the backward and forward amplitude coefficients in (84) and (85) for the top region are now


177

6. Eigenvectors of the TM, Modes Return to (lo), extracting out the third exponential wave form. Factor out the backward wave,

Writing (98) in terms of the different regions (two here), using the definitions (61) and (63,

The fourth and sixth forms of (99) created the scaled formulas for the eigenfunctions, separated from the unknown coefficients found in ZZJz). Known ratios

available from (59), enable the unknown fields to be written in terms of one unknown amplitude for each zone (one external zone beyond the circulator in the current problem). The scaled forms are related to one another by

178

CLIFFORD M.KROWNE

Retrieving (71) and (72), the first scaled eigenfunction (the generator for the eigenvector) form in the top region can be evaluated, completing its explicit formula:

(2cos(kgz);

Oszihc

One can easily show that (102) corresponds directly to a continuity condition at the z = hc interface if it is multiplied by E(z), thereby satisfying the normal component of the displacement field continuity. D,e0(C 4, Z ) l h C + = D,e% 49 Z ) I h c -

(103)

Enlisting (99), the orthogonal property for Zip(z) given by (27) carries over to TmP(z).That is,

(z:* (z), 2;; ( z ) ) = (z:< (z),

z; (2))

=0

( 104)

It is instructive to verify that (102) indeed satisfies (104). Proceeding,

cos (k,O,h, ) cos(k;! h / )

Referring to a compilation of integrals [ 121, the first definite integral in (105) is evaluated to be %hCc0s(kgz)c o s ( k 2 z ) dz

Referencing the integration variable in the second integral to the z = h o top plane, and making appropriate constant changes, allows the use of (106) again. The difference between perpendicular propagation constant eigenvaluesin the different regions, which act as a prefactors in both integrals, are found to be equal by (67)

179


to

( 0 2 )-~(cJ:)~, reducing the inner product expression in (105) to

(Qz)7

.Qz))

invoking the eigenvalue equation (66). Thus, we have verified (104)directly. The orthogonality relation may now be stated as

( Z i e(z), T m L (z)) = C:,Sm,,m:

(107)

Let us borrow the right-hand side of (103, and particularize to me = mi,to obtain : , . the square of the normalization constant, which is equal to C

where [ 12J

1 lc

"

hr

11, =

cos2(k&fz)dz = - - sin(2kghc) k%: 4

ho

12e

=

2

OB

cos (kZm,[ho- z])dz =

+

*] 2

sin(2kghl)

+ k'Th']2

( 109a)

(109b)

7. Eigenvectors of the TE, Modes

Go back to ( 3 3 , extracting out the third exponential wave form. Factor out the backward wave,

180

CLIFFORD M. KROWNE

Writing (1 10) in terms of the different regions (two here), using the definitions (86) and (88), [eikpTZ + e - i k p ' z R O T h +,-ikpnzR0B h e ~ k P T z+ e - i k p T z R O T h

3

AhOT-

;

]

AhOB-

; OPzshc

nmh

nmh

OT-. ] A hnmh

2i sin(kpBz ) A:::-; [eikpTz + e - i k p ' z ~ f T ] R T B mh

9

hcIZ5ho

hc 5 Z l h 0 OPzshc

; hc P z 5 h o

}

OSzihc - Z -

h

AhOBmh

=

{

nmh

+

[eikpTz e-ikp'zRfT]; hc 5 z 5 h o 2i sin(kpBz)R:T-; 0 5 z 5 hc

- zh A h O T mh

nmh

(111)

The fourth and sixth forms of (1 11) created the scaled formulas for the eigenfunctions, separated from the unknown coefficients found in ZLh( z ) . Known ratios

available from (84), enable the unknown fields to be written in terms of one unknown amplitude for each zone. The scaled forms are related to one another by

Retrieving (95) and (96), the first scaled eigenfunction (the generator for the eigenvector) form in the top region can be evaluated, completing its explicit formula:

Equation (1 14) can be shown to correspond directly to a continuity condition at the z = hc interface if it is multiplied by p ( z ) , thereby satisfying the normal component of the B field continuity.

181


Utilizing (1 1 l), the orthogonal property for Zkh(z) given by (27) carries over to (z) when using the proper weight function in the inner product construction. That is,

zkh

z:h(z), 2;:

(Zkh(z), z;

(z)) = (

(z)) = 0

(1 16)

It is also informative to verify that (1 14) does satisfy (1 16). Proceeding,

6"

sin(kg[ho - zl) sin(kz![ho

- z1)dz

(117)

Refemng to a compilation of integrals [12], the first definite integral in (1 17) is evaluated to be

Referencing the integration variable in the second integral to the z = ho top plane, and making appropriate constant changes, allows the use of (1 18) again. The difference between perpendicular propagation constant eigenvalues in the different regions, which act as prefactors in both integrals, are found to be equal by (92) to (a? )2 reducing the inner product expression in (1 17) to mh

invoking the eigenvalue equation (91). Thus, we have verified (1 16) directly.

182

CLIFFORD M. KROWNE

The orthogonality relation may now be stated as

( Z : h ( ~ )Z:;C~)> ' = CLh6mn,rn;

(1 19)

Let us borrow the right-hand side of (1 17), and particularize to mh = mi,to obtain the square of the normalization constant, which is equal to Clh. CLh= -4pfB

1

hc

sin2 (kzm,z) O B dz - 4

[s]= OB

where [ 121

1

hc

Ilh

=

sin2(k:2z)dz

=

k%?

4

sin(2kghc)

1

k: +hC:2

(121a)

(121b)

C. Implicit Dyadic Green's Function Construction 1. Fieldsfor Puck, Interiol; and External Zones

Electromagnetic fields to be used in the following continuity conditions employing the mode-matching method are summarized here from the previous section in the 0 zone, plus those found also in the I and C regions:

m=l n=-m


183

m=O n = - w

w

w

m,=l n = - w 0 0 0 0

mh=l n = - w

w

w

m=l n = - w c a m

m=O n=-w

Field expressions in (130)-( 133) for the circulator puck region are available in [2] and [l I].

184

CLIFFORD M. KROWNE

2. Mode-Matching Theory at Intelfaces

Use of tangential field continuity at the r = R interface will be used to connect the various regions with z as the variable. That is, {Ezv E,, Hz, H#)lr=R- = {Ez, Ed, Hz, H&)lr=R+

(134)

This expression must be applied at the CO interface and at the I 0 interface. It will not hold at the CO interface for the magnetic field components because the r = R surface contains singular forcing functions (delta functions). In the spirit of applying rigorous mode-matching theory by projecting testing functions on the continuity equations and integrating, we find for the I 0 interface

and for the CO interface,

Finally, for the O(C

+ I ) interface,

Here $r>,m are z-dependent testing functions of the particular region i = I, 0,C as applied to the f type of field continuity equation for component j type, with the mode index m. Projection of these functions on the field continuity equations in the fashion of ( 1 3 9 4 142) allows their orthogonal properties to be used, with proper attention paid to weighting functions, thereby reducing the complexity of the eventual system of equations describing the problem. Wherever orthogonality is exploited, matrix sparsity is enhanced. For the exterior zone problem, the work required to determine the orthogonal properties of the scalar generating potentials is balanced by their convenience in use and sparse matrix behavior. We will use such potential properties here, noting that it is possible to use an external unloaded

THEORY FOR INHOMOGENEOUSFERRITE

185

zone cavity testing function (with no orthogonal features, but easy to identify). These testing functions are

+Lzm = +&,, = sin(k:,[z

- hc]);

I mn k,, = -, hl

m = 1,2,. . . , Mf"

m = 0, 1, . . . , M C

$5,= sin(kFmZ);

(144)

(145)

m = 1,2, ..., MC

Notice that the infinite summations contained in the field representationshave been truncated to Mi, i = I, 0, C in the respective regions, with mode-type differentiation in the external zones. Mi must be chosen with some care. M Iand MC may be chosen relative to M o (ignoring, for the moment, mode-type differentiation) as

There is some flexibility in the specific ratio converting M o into the other summation limits [ 131. These issues are referred to as the relative convergence behavior. How large to choose M o is an issue of absolute convergence, specifically how accurate an answer we desire for the problem solution.

3. Nonsource Governing Equations Selection of the subset (if a full set is not needed) of continuity equations is not unique [ 141, the choice being dependent upon individual inclination, sometimes numerical advantage, and the requirement that the number of equations equal the number of unknowns when the missing source equations are included (to be addressed later). Thus, we choose (133, (136), (141), and (142), which become, once the testing function projections and integrations are performed,

186

CLIFFORD M.KROWNE

m=l

m=l

m=l

mh = 1,2,.

. . , M:

(153)

In these previous relationships, for the internal I region,

(ki)2 = ( D : ) ~

+ (k:m)2

Also, the normalized S:h constant used in (153) is

The overlap integrals found in (150)-(153) are given by [ 121

(154)


187

188

CLIFFORD M. KROWNE

(165) Orthogonality relationships relevant to these equations are found in (104) and (1 16) for respectively and [used in (152) and (153)], and below for (150) and (151).

ce

zkh

m,m'> 0 where Em

{

112; m = O = 1; m>O

Also

was used in (158), and a similar formula exists for the h mode.

(167)

189


+

+

+

+

Four system equation set (150H153) constitute Mf M: M$ M: 1 individual equations [2(M, M o ) 1 if the e and h mode index limits are the same]. But the unknowns are

+

+

m anmO; 1

= 0, 1, . . . ,Mc

( 170a)

1,2,. . . ,MC

( 170b)

2

anmO; m =

,

me

= 1,2, . . . , M ;

(170c)

AhoT-; nmh

mh

= 1 , 2 , . . . , M:

( 170d)

A:;;

m = 0, 1 , . . . , MI"

( 170e)

Ail;

m = 1 , 2 , . . . , M:

(170f)

+

+

AeOT-.

+

+

+

+

making a total of M ; M: M; M: ~ ( M C 1) of them MI MO+ MC 1) if the e and h mode index limits are the same]. Notice that the m = 0 case has been left off of the second radial mode coefficient index listing (170b) because it corresponds to the zero perpendicular thickness situation and we expect the first radial mode to completely dominate. Therefore, we see that exactly 2Mc 1 more equations are required to describe the structure.

+

+

4. Source Governing Equations The missing equations come from two source equations governing the exchange of energy between the magnetic delta forcing functions acting on the r = R surface through a point aperture, and the puck structure. The components of the surface magnetic field are chosen to couple the external environment to the circulator puck structure, thereby defining a Green's function construction. The two source equations are H@A(#, Z) = H$'S(Z - z')S(#

- #')A#'

H z ~ ( #Z> . = H;'S(Z - z%(# - #')A#'+

+ H t ( @ ,Z)

(171)

H,O(#,

(172)

Z)

Both H ~ and A H z may ~ be expressed by a Fourier series constructed from the same expansion functions as used to represent the circulator puck field components on the interval (0, hc) by simply using the extended field on (-hc, h c ) ~ 5 1 .

190

CLIFFORD M.KROWNE

H$Am

(177)

= HiAm =0

Here ex indicates extended field and (177) results from the cos or sin nature of the field component variation within the circulator puck. It is convenient to retain the exponential Fourier series on (-n,n) for an additional representation of the # dependent coefficients found in the z-expansions (173) and (174). Thus,

n=-co m

n=-w

With the foregoing information, namely (171)-( 18 l), the proper constraints on the sources, can be imposed. It is done through the requirements that the tangential magnetic field components to the r = R surface be continuous in a limiting process just to the inside and outside of the device perimeter. Hi

Ir=~-

= H @ AIr=R+

Hf

Ir=~-

= H z A Ir=R+

(182ab)

Use of the azimuthal orthogonality property and recognizing the similarity of the perpendicular coordinate expansions in both the puck and aperture (source) surface, Hcir @mn

= HiAmn

cir

Hzmn

- HfArnn

(183ab)

where we have written out cir to emphasize the difference between the puck expansion coefficients and the aperture mnth coefficient, which happens to be a cosine type. Placing (175) into (180) gives, considering the left-hand side of


191

(183a) first, H;Amn

=Em f x H m , ( a ,

nhc

r ) c o s (m F n) ez - i n 4 d + d z

0

(184)

Next insert the source equation (171) into (184),

-

(

rnnz )H$' A+'e-i"@' + 0 integral 5 cos F nhC

Retrieving H: in (129), the second integral in (185) can be evaluated for the 0B and 0 T regions being identical, demonstrating the reduction of one infinite summation:

At this stage, constraint (183a) can be imposed, invoking (133) for the righthand side, using the previous (185) and (186) results. The fifth system equation, containing MC 1 subequations, can now be stated as

+

192

CLIFFORD M. KROWNE

Overlap integrals on the right-hand side of (187) are determined to be

The sixth system equation containing Mc subequations is found by treating (182b). Place (176) into (18 I), obtaining

Inserting the aperture source expression (172) for the perpendicular magnetic field component into (190) gives

(F )

1 mnz HzCAmn= -sin Hfr' Aq5ie-in@ 0 integral XhC

+

(191)

Using (127), the outside integral second term is expressed as .

b

o

o

0

With the help of (132) for the puck field on the left-hand side of (182b), and employing (191) and (192), the final perpendicular source equation is written as

m = l , 2 , ...,Mc (193)


193

The overlap integral in (193) is given by

1

hc

I$.mh =

sin(kFmz)zt,(z) dz = 2i

sin (kFmz) s i n ( k g z ) dz

Orthogonality relationships implicitly contained in (187) and (193) for the puck test eigenfunctions are

1

hc

LcI,b&,I,b&,dz =

hC m, m' > 0 sin(kFmz)sin(kFm,z)dz= -L3mm~; 2 (195)

5. Complete System Equation and Dyadic Green 's Function

The six system equations, four sourceless as seen in (150)-(153), and two with magnetic sources in (187) and (193), may be stated in compact form in a single

A representativenmth element for each class of unknowns is shown in the left-hand side column vector. The matrix entries are as follows: zero for no contribution of the subscript type of unknown, one for a single entry contribution, and X for a sum of all that particular class of unknowns indicated by the subscript. On the right-hand side of the equation are the source forcing terms. It is possible to reduce

194

CLIFFORD M.KROWNE

the size by analytical effort of the system matrix (197) due to the appearance of null and unity entries, and solve a smaller inhomogeneous linear matrix problem, albeit with fewer unknowns determined initially. The remaining unknowns are captured by solving subsidiary matrix equations of generally smaller size than the reduced system matrix. Setting H:'A#' = 1 and H$'A# = 0 (or the reverse) allows for the solution of the suite of unknown coefficients in the column vector. When these solutions (or the reverse) are placed in the electromagnetic field formulas, the dyadic Green's function elements are generated. This may be put down symbolically as

S represents system (197) and S( 1,O) corresponds to the azimuthal magnetic source turned off. F is the field equation operator and produces the correct component (first superscript) for the desired field (electric or magnetic field, indicated by the first subscript). It is clear that if we had only one term for each summation, then the compressed form in (197) would represent a 6 x 6 system, and it would be possible but extremely tedious to pull the forcing terms through the determinant solution for each unknown and obtain explicit real space dyadic Green's function elements. But for our problem here with incommensurateheights (or other geometrical dimensions), the problem is in practice impossible to solve for an explicit dyadic Green's function. One would look naturally to reciprocal space to obtain by analytical means compact explicit dypdic Green's function expressions. But there is a tremendous advantage in not going to reciprocal space, and that is that the implicit dyadic Green's function can be used to obtain the field behavior directly without any transformations. Furthermore, these Green's functions can be used to obtain the s-parameters for the circulator structure too. Mention is made here that the diagonal nature of the relation between different region amplitudes in the outer zone allows one amplitude (per mode class) to characterize the entire zone (the T amplitude was chosen). Once they are determined by (197), the subsidiary diagonal equations (100) and (1 12), stated explicitly below, may be employed to capture the B amplitudes.

RBT-

0

...

0

0

R2BT-

...

0

lo

0

. .. R L i -


195

D. Conclusion Source constraint equations have been combined with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements for a very general canonical circulator geometry. The approach allows the inclusion of layered surrounding material beyond the radius of the ferrite puck, as well as a covering material above the puck, enabling a more realistic or complete description of the circulator structure. New self-adjoint operators are found consistent with the surroundingmaterial having dielectric, magnetic, or simultaneouscharacter. Inclusion of substrate and superstrate effects are possible with this treatment. Assessment of vertical field fringing is a natural consequence of this analysis approach. Obtaining the dyadic Green’s function as described in the paper is particularly useful because field information and s-paramaters may be found in real space, completely avoiding typical inverse transformations. In this approach are included the inhomogeneous properties of the circulator puck due to chosen radial variation of the ferrite material parameters, nonuniform applied magnetic dc biasing field, or finite puck geometric effects on the bias field. All of these inhomogeneousproperties in the puck region are naturallyincorporated into the dyadic Green’s function. Numerical evaluation should be efficient and only limited by the well-understood features of the mode-matching technique. The geometry considered correspondsto that found when one of two self-biasing configurations are employed using hexagonal ferrite films [16-181: (1). The puck itself is a hexagonal material with the exterior zone a layered combination of materials or (2) the puck is spinel or garnet material and the exterior material a hexagonal material (it may be anisotropic, a situation not treated here). In the first structure the electric wall condition above the puck is maintained by microstrip metal. For the second structure, it is maintained by a combination of the microstrip metal and low-coercivity“keeper” plate (or cover) permalloy. Hexagonalmaterials include Ba, Pb, or Sr, iron oxide compounds. The garnets are the usual Y, iron oxide compounds, and the spinels are the Ni, Li, or MgMn, iron oxide compounds. If we had not insisted on self-biasing,but used an external magnet lying immediately above the circulator puck, configuration Fig. 2 would have resulted. (This structure was not be addressed here.) The methods covered in this section still apply, with extra allowance made for the magnet excluding fields in the I region if

196

CLIFFORD M. KROWNE

it is viewed as a perfect conductor forming electric walls at its boundaries. Field exclusion occurs for a volume encompassing the magnet’s thickness and diameter 2R. Operator methods developed in this section are also still required. One may wonder what happens if the external radial zones can be constructed out of simple uniform material for each zone. For such a situation, the operator work here may be avoided and considerableease is found in Section I11 in obtaining a formulation compared to the labor required in this section.

111. IMPLICIT 3D DYADICGREEN’S F U ” I O N WITH SIMPLE EXTERNAL MATERIAL USING MODE-MATCHING A. Introduction Recently, consideration of field extension into the surrounding medium, beyond the circulator perimeter, has been examined with the idea of constructing a more descriptive dyadic Green’s function, which can allow assessment of external dielectric effects (actually both isotropic permittivity and permeability). This has been done for both 2D [ 191 and 3D models [20]. But neither model has the ability to enable field extension and fringing such that fields may extend out and then above the height of the circulator nonreciprocal puck. It is the intent here to show that a 3D approach may correct this deficiency by combining source constraint equations with a rigorous mode-matching treatment where the outlying radial zones are single cylindrical regions with one material throughout each zone. Restriction of the structure radial zones vastly simplifies the analysis compared to Section I1 where considerable work was required in an operator theory development. None of that is needed here. Mode orthogonality is utilized to encourage sparsity in matrix system development, eventually resulting in implicit dyadic Green’s function elements. When the Green’s function elements are obtained this way, they are in direct space, and consequently field information and s-parameters arise immediately, completely avoiding typical inverse transformations. The approach allows the inclusion of surrounding dielectric material as well as a cover plate above the ferrite puck. The approach is a good approximation to a very complicated geometric and inhomogeneous problem, given the irregular effects arising from application of the dc biasing magnetic field and the actual finite-width microstrip input and output lines. For narrow microstrip lines the expectation that the fields extend beyond the device perimeter, with azimuthal symmetry, is very good, and the dyadic Green’s function elements will be nearly exact, only limited by numerical precision. When some of the microstrip lines attain widths which are a noticeable fraction of the puck radius, the error introduced by the symmetry assumption for r > R for the


197

R

FIGURE 4. The ferrite circulator structurewith a uniform radial zone and a region above the device puck. This figure is formed by taking a cut plane at I$= const (in 3D).

fields will be directly related to the fraction of the circumference occluded by the presence of the line itself.

B. Fields, Mode-Matching Technique,Nonsource and Source Equations Consider the situation where the electromagneticfield occupies three regions (see Fig. 4). The first region C is that filled by the ferrite nonreciprocal material for r < R andO 5 z 5 h c . The second region 0 has r > R andO 5 z 5 h o . And the third region I has r < R and hc 5 z 5 h o . Conducting walls are assumed at z = 0, ho and the radiation condition in effect as r + 00. Such a geometry could result from magnetless, self-biased circulators.

m=O n=-m m

m

m = l n=-m m

m

L

m = l n=-m

J

- uLA:!,,airnJ,' (uimr)eing

198

CLIFFORD M.KROWNE m o o

m=On=-m o o m

m=l n=-m

+ Li pRoAn ;

L

: K, (afmr)

c a m

m=O n=-m

M

M

Field expressions in (210)-(211) for the circulator puck region are available in [2, 111. Use of tangential field continuity at the r = R interface will be used to connect the various regions at the appropriate z value. That is,

This expression must be applied at the CO interface and at the I0 interface. It will not hold at the CO interface for the magnetic field components because the r = R surface contains the singular forcing function (a delta function). In the spirit of applying rigorous mode-matching theory by projecting testing functions on the


199

continuity equations and integrating, we find for the I0 interface

and for the CO interface,

Lastly for the 0 (I

+ C) interface,

+ij,

are z-dependent eigenfunction factors of the particular region i = Here I, 0, C as applied to the f type of field continuity equation for component j type. Projection of these functions on the field continuity equations in the fashion of (215)-(219) allows their orthogonal properties to be used, thereby reducing the complexity of the eventual system of equations describing the problem. These testing functions are

+,:

=

= cos(kim[z- h c l ) ;

+Lzm = $i#,= ~ i n ( k ; ~ [-z h c l ) ;

,+:

= cos(k;,z);

+$,

= sin(kF,z);

+zm= cos(kgz); +&,

mlr

k,,c - -, hC mlt , :k = -, hC 0 mlt k,, = -, h0

I mlt kzm = -, hl I

mn

k,, = -, hI

m = 0, 1, ..., M I (220) m = 1 , 2 , . . . , M I (221)

m = 0, 1 , . . . , M C

(222)

m = 1 , 2 , . .. , M c

(223)

m = 0, 1 , . .. , M o

(224)

m = 1,2,. .. , Mo k,,0 = -, m= (225) h0 Notice that the infinite summations contained in the field representations have i = I, 0, C in the respective regions. Mi must be chosen been truncated to M i , = sin(kgz);

200

CLIFFORD M. KROWNE

carefully. M Iand Mc may be chosen relative to M o as

There is some flexibility in the specific ratio converting M o into the other summation limits, and discussions regarding this limit management is found elsewhere [ 131. These issues are referred to as the relative convergence behavior. How large to choose M o is an issue of absolute convergence, specifically how accurate an answer we desire for the problem solution. Selection of the subset (if a full set is not needed) of continuity equations is not unique [ 141, the choice being dependent upon individual inclination, sometimes numerical advantage, and the requirement that the number of equations equal the number of unknowns when the missing source equations are included (to be addressed later). Thus, we choose (215a), (215b). (218), and (219). The orthogonality and projection relationships relevant to these equations are [12]

m, m' > 0

(229)


m,m’ > 0

=ilosin(Ez

) sin (T-:[ z

20 1

(234)

)

- h] dz = Ismtm I0 (235)

where

Placing (227)-(236) into (215a), (215b), (218), and (219), where all summations in the field components in the inner products [i.e., as in (215)] are switched to m’ indexing, yields the four system equations

202

CLIFFORD M.KROWNE

m'=O

m'=O

m = 0, I , . . . , M o

(240)

m'= 1

m = 1 , 2 , . ..,M o These four system equations constitute the unknowns are

(241)

MI + M o + 1) individual equations. But

1 anmO; m = 0, 1,. . . , MC

2 anmO; m = 1 , 2 , ...,M c

A;:;

m = 0 , 1 , ..., M O

hO Anm; m = l , 2 , ..., M o

A$;

m = 0, 1 , . . . , M I

hl Anm; m = 1 , 2 , .. . , M I

+

+

+

making a total of 2 ( M , M o Mc) 3 of them. Notice that the m = 0 case has been left off of the second radial mode coefficient index listing because it corresponds to the zero perpendicular thickness situation and we expect the first

203

THEORY FOR INHOMOGENEOUS F E R R m

+

radial mode to completely dominate. Therefore, we see that exactly 2Mc 1 more equations are required to describe the structure. The missing equations come from two source equations governing the exchange of energy between the magnetic delta forcing functions acting on the r = R surface through a point aperture and the structure. The componentsof the surface magnetic field are chosen to couple the external environment to the circulator structure, thereby defining a Green's function construction. The two source equations are H @ A ( Z~) , = H$'S(Z - z')S(# - #')A#' H z ~ ( 4Z), = H;'S(Z - z ' ) S ( ~ - #')A#

+ H:(#,

+ HP(#,

Z)

(248)

Z)

(249)

Both H+A and H z may ~ be expressed by a Fourier series constructed from the same expansion functions as used to represent the circulator puck field components on the interval (0, h c ) by simply using the extended field on ( - h c , h c ) [15].

(254)

H$Am = H;Am = 0

Here ex indicates extended field and (252) results from the cos or sin nature of the field component variation within the circulator puck. It is convenient to retain the exponential Fourier series on (-n,n) for an additional representation of the 4 dependent coefficients found in the z-expansions (250) and (25 1). Thus,

n=-w 60

n=-w

204

CLIFFORD M.KROWNE

With the foregoing information, namely (248)-(258), the proper constraints on the sources can be imposed. It is done through the requirements that the tangential magnetic field componentsto the r = R surface be continuous in a limiting process just to the inside and outside of the device perimeter. Hilr=R-= H & A I r = R +

Hflr=R-

= HzAIr=R+

(259ab)

Use of the azimuthal orthogonality property and recognizing the similarity of the perpendicular coordinate expansions in both the puck and aperture (source) surface, Hcir

6mn = H i A m n

cir Hzmn

- HfAmn

(260ab)

where we have written out cir to emphasize the difference between the puck expansion coefficients and the aperture mnth coefficient which happens to be a cosine type. Placing (252) into (257) gives, considering the left-hand side of (260a) first,

HiAmn= 5 J h x H m , ( ~z)cos( ,

nhc 0 Next insert the source equation (248) into (261),

Retrieving

m )zez- i n 6 d 4 d z F

H t in (209), the second integral in (262) can be evaluated:

(261)

THEORY FOR INHOMOGENEOUSF E m

205

At this stage, constraint (260a) can be imposed, invoking (213) for the righthand side, using the previous (262) and (263) results. The fifth system equation, containing Mc 1 subequations, can now be stated as

+

:AAmNaLmO

+ :A2mNanm02

- U 0, , eOA ~0 ~ , ~ ~0 ~ , K ~ ( ~ m~ =~0,, R1, ..., ) ] IMC ~ ? ~ ;

(264)

The sixth system equation containing MC subequations is found by treating (219b). Place (253) into (258). obtaining

Inserting the aperture source expression for the perpendicular magnetic field component into (265) gives

Using (207), the outside integral second term is expressed as

With the help of (212) for the puck field on the left-hand side of (259b), and employing (266) and (267), the final perpendicular source equation is written as I

i [2AArnNanrnO

+ hz ~ n2m N a n 2 m OHZCA,. I m = 1 , 2 , . .. , MC (268)

C. Implicit Dyadic Green's Function The six system equations, four sourceless as seen in (238)-(241), and two with magnetic sources in (262) and (264), may be stated in compact form in a single

206

CLIFFORD M.KROWNE

matrix formula: o h0 ChO

Oh 0 IhO

ChO ChO

A representative nmth element for each class of unknowns is shown in the lefthand-side column vector. The matrix entries are as follows: zero for no contribution of the subscript type of unknown, one for a single entry contribution, and C for a sum of all that particular class of unknowns indicated by the subscript. On the right-hand side of the equation are the source forcing terms. It is possible to reduce the size by analytical effort of the system matrix (269) due to the appearance of null and unity entries, and solve a smaller inhomogeneous linear matrix problem, albeit with fewer unknowns determined initially. The remaining unknowns are captured by solving subsidiary matrix equations of generally smaller size than the reduced system matrix. Setting H:rA@' = 1 and H$A@ = 0 (or the reverse) allows for the solution of the suite of unknown coefficients in the column vector. When these solutions (or the reverse) are placed in the electromagnetic field formulas, the dyadic Green's function elements are generated. This may be put down symbolically as Gr3@,z;z E,H;H(r,

4, z) = Fi,Y[S(I?0)l G'&3;H(rr @ z.@ 4, z ) = Fi,Y[S(O,111

(270) (27 1)

S representssystem (269) and S( 1,O) correspondsto the azimuthal magnetic source turned off. F is the field equation operator and produces the correct component (first superscript) for the desired field (electric or magnetic field, indicated by the first subscript). It is clear that if we had only one term for each summation, then the compressed form in (269) would represent a 6 x 6 system, and it would be possible but extremely tedious to pull the forcing terms through the determinant solution for each unknown


207

and obtain explicit real space dyadic Green’s function elements. But for our problem here with incommensurate height (or other geometrical dimensions), the problem is in practice impossible to solve for an explicit dyadic Green’s function. One would look naturally to reciprocal space to obtain by analytical means compact explicit dyadic Green’s function expressions. But there is a tremendous advantage in not going to reciprocal space, and that is that the implicit dyadic Green’s function can be used to obtain the field behavior directly without any transformations. Furthermore, these Green’s functions can be used to obtain the s-parameters for the circulator structure too.

D. Conclusion Here we have combined source constraint equations with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements for a more general circulator geometry. The approach allows the inclusion of surrounding dielectric material as well as a cover plate above the ferrite puck, enabling a more realistic or complete description of the circulator structure. An assessment of vertical field fringing is possible with this treatment. Obtaining the dyadic Green’s function in this manner is particularly useful because field information and s-paramatersmay be found in real space, completelyavoiding typical inverse transformations. But the theory avoids many of the complicationsinherent in the more complicated radial zones seen in the geometries considered in Section 11. In this approach are included the inhomogeneous properties of the circulator puck due to chosen radial variation of the ferrite material parameters, nonuniform applied magnetic dc biasing field, or finite puck geometric effects on the bias field. All of these inhomogeneous properties in the puck region are naturally incorporated into the dyadic Green’s function. Numerical evaluation should be efficient and only limited by the well-understood features of the mode-matching technique. The geometry considered correspondsto that found when one of two self-biasing configurations are employed using hexagonal ferrite films [16-181: (1) the puck itself is a hexagonal material with the exterior material a dielectric or (2) the puck is spinel or garnet material and the exterior material is a hexagonal material. In the first structure the electric wall condition above the puck is maintained by microstrip metal. For the second structure, it is maintained by a combination of the microstrip metal and low-coercivity “keeper” plate (or cover) permalloy. Hexagonal materials include Ba, Pb, or Sr, iron oxide compounds. The garnets are the usual Y, iron oxide compounds, and the spinels are the Ni, Li, or MgMn, iron oxide compounds. If we had not insisted on self-biasing but used an external magnet lying immediately above the circulator puck, configuration Fig. 5 would have resulted. Again

208

CLIFFORD M. KROWNE / / /// / / / / / / / / / / / / / / /

I t

I

,

i R

FIGURE 5. The femte circulator structure with a uniform radial zone and a region above the device

puck with the presence of an external biasing magnet. This figure is formed by taking a cut plane at

4 = const (in 3D).

the methods covered in this section apply, with extra allowance made for the magnet excluding fields in the I region if it is viewed as a perfect conductor forming electric walls at its boundaries. Field exclusion occurs for a volume encompassing the magnet’s thickness and diameter 2R. This structure will not be addressed here.

N.2D DYADICGREEN’SFUNCTION

FOR PENETRABLE WALLS

A. Introduction In the previous inhomogeneous 2D model for microstrip circulators [l], it is assumed that the non port boundaries are magnetic walls, confining energy exchange to only the ports. However, in a theoretical treatment which goes into great lengths to correctly model the detailed internal circulator field behavior, and allow extraction of the device s-parameters, by viewing the circulator itself as made up of a sufficient number of annuli, it may be both reasonable and perhaps necessary to understand the effects of electromagnetic field extension into the non-ferrite region. This extension is accomplished by still having the ports interface with the external circuit, but now replacing all magnetic walls on contour lines between ports with penetrable boundaries obeying the proper electromagnetic interfacial boundary conditions. We develop here a new dyadic Green’s function [21], which allows us to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with permittivity &d and permeability pd (see Fig. 6). The three common cases of air as the external medium, a dielectric as the external


Port - femte region

a

& -\

; (CL, PO,

DIRECT

209

ISOLATED

Inhomo. Wall ( non - port)

b

FIGURE6. Cross section through a z = const circulator plane parallel to the ground plane. The diagram shows the special case of only three ports, labelled counterclockwise. External material exists beyond the r = R puck radius.

medium, and an unmagnetized ferrite as the external medium are all easy to treat with the new dyadic Green's function. Obtaining an explicit dyadic Green's function is a result of reducing the problem down from three dimensions examined in Sections Il and III to two dimensions here.

B. Formulation Applying the radiation condition as r + 00, leads to the selection of the modified Bessel function of the second kind K,,(kdr) for use in the external field construction, r > R . It is assumed that the same field modes are maintained in the device for radii exceeding the circulator radius so that a consistent 2D modeling procedure holds internal and external to the device. Additionally, the microstrip edge effect and fringing field contribution provide the correct field available from the circulator puck for coupling to the external environment when multiplied by the factor f. With these constraints, the internal TM, persists for r > R , = [ l / ( i W p d ) ] a E , d / a rand ,

Hi

Requiring continuity of the perpendicular electric field at r = R , f E f ( R ,4) = E,d(R9 $ 1 9

210

CLIFFORD M. KROWNE

Factor f is not a simple quantity, but one multiplicative component making it up may be thought of as the ratio of the difference between the total integrated field over the entire open surface to that contained in the puck, divided by that in the puck. It may be computed by capacitance calculations, and is estimated to fall between a few percent and 10% for typical device pucks. Factor f in an approximate way must both allow for a consistent fringing in the 2D model and for the fact that the 2D model is only approximately reflective of the 3D nature of the actual physical device. More discussion on these points is found in Section VI covering limiting dyadic Green's function forms for homogeneous ferrite devices. The forcing function for the Green's function is applied at (r', 4'). r = R , through the equality

H p ( R , 4) = H y ~ a ( 4 4'1A4'

+ H $ ( R , 9 # 4')

(274)

The perimeter azimuthal magnetic field can be represented by a ID Fourier expansion,

Azimuthal magnetic field in the circulator, as r + R from the inside within the last annulus i = N, is given by 00

H i ( R , 4) =

an0 [anN(reCuf)C:haN(r)+b.~(recur)C:~~,,,(r)]e'"@

(276)

n=-co

Here the coefficients anN(recur) and b,N(recur) come from a recursive transfer matrix process. Equating the fields in (275) and (276),

H i ( R , 4) = H F ( R , 4) an0 [anN(recur)c$,N(r)

(277)

+ bniv(recur)~$bN(r)]

Inserting (272) into (2781, r + R from the outside, utilizing azimuthal orthogonality, yields ano{ [w(recur)C:haN(r) +bnN(reC~r)~:hbN(r)~


21 1

C. Dyudic Green's Function With the solution of u,o from (279) in terms of the forcing field H ~ and A using the development in [ 1 11 as a guide, the elements of the dyadic Green's function in the i th annulus may be derived as l b O ani (recu)Cneai ( r ) GLdH;(r,4; R , 4') = 2n n = - w ynN- fkd

+ bni (recu>cnebi( r )e;n(,#,-,#,') yzc

i o p d K , &:R)

KA (kd R ,

(280a)

Dyadic Green's function element subscripts are interpreted as follows. The first refers to the resulting field type, the second to the forcing field type, and the third to the annulus under consideration. For the superscripts,the first is the resulting field component and the second the forcing field component. It is evident from (280) that the new dyadic Green's function elements are those of a circulator device with hard walls (namely, magnetic walls), but with a modification to the form of the denominator term. This modification is in the form of a subtraction from , depends on the properties of the external the original circulator divisor y n ~and medium, on the internal circulator field behavior through y i i , and on the factor f . The elements of the dyadic Green's function external to the circulator, in the nonport regions, are

(281a)

(281b)

(281c)

212

CLIFFORD M.KROWNE

These r > R dyadic Green’s function elements are completely new and not only contain the denominatorcorrection term but also functionalforms which assure that any fields constructedfrom them will properly decay outside of the device. Results in both (280) and (281) correspond to a Green’s function which requires a radian azimuthal contour integration. That is, the differential element is d 4 . To make the Green’s function correspond to a line integral of differential element length ds, each dyadic element like G$Hd(r,4; R, 4’) should be multiplied by l/R. Besides being able to treat the case of an external dielectric with Ed and the case of the external medium being air is accomplished by setting &d = and pd = po, the free-space values. Finally, the case of an unmagnetized ferrite is done by setting &d = &f and pd = pf (the unmagnetized 2D diagonal value) which are available. In order to see that coupling the external medium to the circulator puck affects the field behavior, it is merely necessary to show that the Green’s function (GF) elements are modified. This may be done by examining the lower-order leading azimuthal terms. All GF elements have their denominators modified by the multiplier (1 - me;)where the ratio part is

Now from running both 2D Green’s function and FEM simulationsof a MnMg ferrite puck device of radius R = 0.270 cm and thickness h = 0.0675 cm [ h / ( 2 R )= h/D =0.125], we find the center frequency to be about 9.0 GHz [22]. For this device w,,, = -yMs = (2.8MHz/Oe) x (2300G) = 6.44 GHz (magnetization frequency). The off-diagonal relative permeability tensor element K M w m / w = 0.716 when wo x 0 (internal dc bias field small) and a,,,M 0 (losses neglected), and the diagonal relative permeability tensor element p x 1. The effective permeability will be pe = 0.488. The relative dielectric constant of the ferrite material is & f r = 13.3. The propagation constant in a surrounding air environment kd = 1.89 m = 4.80 radianskm. Electrical length radianskm and in the puck k, = w angles are defined as 6d = kdR = 0.509 rad and 0, = k,R = 1.30 rad. In terms of electrical wavelengths, D/Ad = 0.162 and D/Ae = 0.413. To simplify matters, let us assume enough uniformity to use the limiting form of (282) as the required number of annuli plus the disk N’ + 1 (N’ = N 1). Then we can write for the ratio

+

Evaluating (283) for the axisymmetric mode n = 0 and the first few asymmetric modes with nonreciprocal contributions n = f l and n = f2,we find that they are all a few tenths or smaller. The negative asymmetric modes are about ten times


213

smaller than the positive asymmetric modes. And for n = f 1 0 , the value is down by another factor of a thousand. Thus, we see from this example that the inclusion of the external medium modestly affects the Green’s function, and so we expect both modest changes in the field behavior and s-parameters.

D. Conclusion Dyadic Green’s function applicable to the case where the inhomogeneous ferrite circulator walls are no longer perfect magnetic conductors, a good but imperfect assumption at best, has been derived. The 2D dyadic Green’s function elements are seen to contain a modification from the impenetrable wall case, which is easy to recognize. The modification may easily be turned off, returning the device to the original hard wall situation. External permittivity and permeability effects on device electromagnetic fields and s-parameters may be found from the new dyadic Green’s function. This Green’s function is expected to be particularly relevant to new work proceeding on developing microstrip thin film ferrite circulators [16]. We note in passing here that the present approach will yield numerical results in a few secondsper frequency point versus times exceeding a thousandfold increase using intensive numerical 2D and 3D finite element method (FEM) or finite difference time domain (FDTD)method. The next section to come follows the philosophy in this section in that an explicit dyadic Green’s function is sought, but in three dimensions. A number of assumptions and approximations must be made to accomplish that, and these are covered in Section V.

V. 3D DYADIC GREEN’SFUNCTION FOR PENETRABLE WALLS A. Introduction In the previous inhomogeneous3D microstripmodel [2], it is assumed that the nonport boundaries are magnetic walls, confining energy flow to only the ports. This may be a reasonable assumption in view of the thin dimensions in the z-direction compared to the extent on the lateral surface. But by its very nature, a 3D model could allow accurate description of field extension beyond the circulator perimeter and fringing fields, depending upon how general and complex the formulation. Here we will develop a treatment which extends the perfect electric wall at the ground plane beyond the perimeter and assumes that the fields exit the device with no vertical fringing, essentially maintaining a distributionin the 0 < z < h region. This approach will enable a manageable derivation of an explicit dyadic Green’s function and the ability to model the field extension into the outlying dielectric

214

CLIFFORD M. KROWNE

region while also maintaining the radial mode splitting inside the puck due to finite device thickness of the planar structure. Issues such as the real fringing near the microstrip line-circulatorinterface or the field fringing above the circulator height can’t be addressed. There are procedures to address the field fringing above the circulator height and they will be partly treated in B and C in this section, but for more general structures the procedures entail substantial complications because of the different electric wall heights beyond the perimeter, and this height may even vary depending upon the environment external to the actual device region. Some attempt to address these complications was already presented in Sections I1 and 111. Field extension is accomplished by replacing all magnetic walls on surfaces (in a cross section, a contour line) between ports with penetrable boundaries obeying the proper electromagnetic interfacial boundary conditions. A new 3D dyadic Green’s function is found, which allows one to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with permittivity &d and permeability pd. The three common cases of air as the external medium, a dielectric as the external medium, and an unmagnetized ferrite as the external medium are all easy to treat with the new dyadic Green’s function.

B. Formulation Figure 7 shows a three-dimensional sketch of the circulator structure. It is highly exaggerated in the axial direction z so as to make viewing easier. The circulator thickness is h. There are a finite number of ports placed at azimuthal locations 4; with angular widths A& and having their microstrip lines with physical widths w ; where i = 1,2, . . . , N,. For r < R, there is a solid cylinder of magnetized ferrite material, which is radially inhomogeneous. For r =- R, there is a homogeneous filling material, except of course where the ports exit and enter the circulator puck. At r = R is located the ferrite-external region inhomogeneous boundary, which constitutes an imperfect wall, that is, where field coupling is allowed. The ground plane is at z = 0, and the microstrip circulator metal restricting the puck fields is at z = h and r < R. Conductive metal is also found on the incoming and exiting microstrip lines at z = h. Microstrip line widths are required to be in total angular width A ~ =T X ( i = 1,2, . . ., N , ) A4; R. It is assumed that field conditions are maintained external to the puck in such a manner for radii exceeding the circulator radius that the same z-indexing modal set can be used internal and external to the device. With this assumption,

j=O n = - w w o o

He(r, 4, z) =

iufhj Sin(k,jZ)Kn(adjr)ein9

(293)

j=1 n=--w

Here the characteristic equation for the radial separation constant a d j is given in the outside region by adj

=

dm,kd

=W

m

(294ab)

where the perpendicular indexing for the discrete spectrum of allowed values is done according to

jn k,j = -; h

j = (0 or l), 2,

...

(295)

with the first j index choice determined by the first nontrivial field component. The azimuthal magnetic field component H$j (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the coefficient factors q and t of the partial differential operators a / & and 8 / 8 4 equal to zero in (289d) (see [ 111 where exponential form is associated with the perpendicular propagation index j ) : (296) with (297ab) Similarly, the azimuthal electric field component E,di (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the coefficient factors F and q of the partial differential operators a/a# and a / a r equal to zero in (289b):


219

with iwuA

(299)

Requiring continuity of the perpendicular electric field at r = R ,

f q 4)~= E,, ~ ( R4),

(300)

Factor f is chosen to correct for the actual vertical fringing seen in a 3D circulator and not treated by the model adopted here for simplicity reasons. The approach here does account for external field extension by assuming to first approximation that the jth perpendicular mode spectrum inside the puck holds outside also. f used here modifies the inhomogeneous formulas found in [24], and enables the effect of the external environment to be turned off (set f = 0), or left on (f # 0). Null f value must be done with some thought in the formulas to follow, but it can be demonstrated that the limit exactly returns the problem to the hard wall 3D case. Circulator field ES in the i = N t h ring at its most extreme position r = R is expressed as M

M

Here the modal coefficients aANj and b f N j ,i = 1 or 2, of the radial functions C,$,jNR, g = a or b , can be related by the recursive nature of the problem to the modal coefficients in the disk (i = 0; containing an implicit singularity at r = 0 which has been removed)

The "recur" indicates that the 4 x 2 matrix is determinedrecursively. Putting (292) and (301) into (300), and using longitudinal and azimuthal orthogonality, yields

f [:AAj&j

+ :A$&,]

making the coefficient in the external region

= a,djKn@djR)

(303)

220

CLIFFORD M.KROWNE

with e 1 zAnj

- II - ‘nNj

eA2. z nJ = 2I.

czln e a j N R + bt%jCiibjNR

n N J. c ; l z : a j ~ ~

+

b:ijci;bjNR

+aikjCr$ajNR

+ b%jCi2bjNR

(305)

+

+

(306)

a n22N j C i 2 a j N R

b:’,iCi;bjbjNR

Requiring continuity of the azimuthal electric field at r = R, (307) Circulator field EG in the i = Nth ring at its most extreme position r = R is expressed as

+

aiNjc::ajNR

+

bi~jcf:bjNR]

ein@

(308)

The dielectric region field E$ as r + R from the outside is obtained by (298) with the help of (292) and (293).

(309) Inserting (308) and (309) into (307), and using longitudinal and azimuthal orthogonality, yields


22 1

Here g(z) is the functional behavior of the forcing azimuthal magnetic field in the z-direction. The perimeter azimuthal magnetic field can be represented by a 1D Fourier expansion, 00

H,P"'(R,#) =

1 A,,e'"@, n=-m

A,, = - / r H F ( R , $)e-'"@d+ 2rc

Azimuthal magnetic field in the circulator, as r last annulus i = N, is given by

(315ab)

R from the inside within the

3

(3 18) with h 1 h

2

11

@1

+ b!thjctibjNR +a i i j c % j N R +b;fijc:ibjNR

(319)

12

@1

12 $1 + bnNjCnhbjNR + a%jc!tajNR

(320)

= 'nNjcnhajNR = 'nNj'nhajNR

+b;fijc:lbjNR

and H @ t ~ being j the overlap integral specifying the amount of the jth mode behavior contained in the forcing azimuthal magnetic field. It is

(321) H$j is found from m o o

$(r, 9 , Z) =

1 C cos(kzjz)[

h d 1 1 +Anja,oj

j=O

n=--60

where (296) has been used to find

+ ~Anjanoi]e'"@ h d 2 2

(322)

222

CLIFFORD M. KROWNE

Inserting (322) into (3 18), letting r + R from the outside, and utilizing azimuthal orthogonality, yields

The perpendicular magnetic field forcing function for the Green's function is applied at (r', 4'), r = R , through the equality H p ( R , 4) = H ~ A ~ ( z )-S 4')A4' (~

+ H e ( R , 4 # 4')

(326)

Here h(z) is the functional behavior of the forcing perpendicular magnetic field in the z-direction. The perimeter perpendicularmagnetic field can also be represented by a 1D Fourier expansion,

Perpendicular magnetic field in the circulator, as r + R from the inside within the last annulus i = N, is given by

Equating the fields in (327) and (328),

gives

with

j the overlap integral specifying the amount of the jth mode behavior and H z ~being contained in the forcing perpendicular magnetic field. It is

1

h 2 HzAj = h H z ~ sin(kzjz)h(z)d z

(333)


223

H i is found from (293) and (3 10). Inserting (293) into (330), letting r -+ R from the outside, and utilizing azimuthal orthogonality, yields

C. Dyadic Green's Function

Let us introduce some new definitions in order to make transparent the forms of the disk modal coefficients in (325) and (334).

Introducingthese formulas into the two equations forming a simultaneous solution set for the disk modal coefficientsin terms of the driving field magnitudes,

A'.a' . + A2.a2 . = -HqAjA4'e-'"@', 1 nJ no/

nJ n0J

2rr (337ab)

-I

+

-2

2

1

Bnja;oj BnjanOj= -H ,A#re-in@' 2rr zAJ

enables a solution to be found

These equation forms are identical to those derived for the 3D circulator problem with hard walls between the ports. The right-hand sides are the driving terms. From (338) and (339) the dyadic Green's function elements for the ith ring are

224

CLIFFORD M. KROWNE


225

Here z-directed weights K: and K $ are given by

.:=;I

h

21

h

COs(kzjz)g(z) d z ,

Kij =

sin(k,jz)h(z) dz

(353ab)

by invoking (321) and (333). It should be noted that the T ( r ) radial indexed functions are related when r = R to the definitions in (305), (306), (31 l), (312), (319), (320), (331), and (332), and their specific forms may be found in [lo]. The dyadic elements for r > R are to be discussed next. Consider (293), (310), (338), and (339) to obtain

Employing (292), (304), (338), and (339), one obtains

226

CLIFFORD M.KROWNE

Generalizing (322) for r # R (r > R ) , and using (338) and (339) yields

ein(@-4') x [h$Aij(r)A:j- hd@Anj(r)AAi] 2

(359)

where the radial functions are

(361) Setting coefficient factors r: = 0 and q = 0 of the partial differential operators a/ar and 8/84 equal to zero in (289),

From (362). the next two dyadic Green's function elements are identified as

(363ab) remembering the definition of dyadic elements [25], and writing it for this situation as

Thus, using (354)-(357), and (363),

227


Enlisting (298), with the help of (364),

(367ab) Thus, the corresponding dyadic Green's function elements are

Setting coefficient factors q = 0 and r = 0 of the partial differential operators 8 / 3 4 and a/& equal to zero in (289c),

The two associated dyadic Green's function elements are identified as G$Hdj = pj-

ac$Hj u j

+--r

ar

a4

9

aGzHj GzHdj = p j - ar

u j acgHj + -r a+

(37 1ab)

Therefore, using (354)-(357), and (371), ,

0

0

0

0

228

CLIFFORD M. KROWNE

D. Conclusion For the case where the inhomogeneousferrite circulator walls are no longer perfect magnetic conductors, a revised and extended 3D dyadic Green’s function can be obtained, which describes both the fields internal to (within the circulator perimeter) and external to the nonreciprocal medium. It is seen that updated disk modal coefficients are found enabling a transparent extension of the dyadic Green’s function within the inhomogeneous annuli regions. Dyadic Green’s function element expressions external to the puck are also provided, and of course they are entirely new in relation to the old perfect wall case. External permittivity and permeability effects on device electromagnetic fields and s-parameters may be found from the new 3D circulator dyadic Green’s function. The next section, Section VI, will treat the much simpler case of a homogeneous circulator puck. Insights gained in this section and in the previous one on 2D aspects of the dyadic Green’s function should be helpful.

VI. LIMITING DYADICGREEN’SFUNCTION FORMSFOR HOMOGENEOUS FERRITE A. 2 0 Dyadic Green’s Function 1. Introduction

Ferrite planar circulators are often built with circular symmetry, and they are electrically thin enough to warrant a model based upon a 2D approach [ 191. Furthermore, the use of a canonical structure can provide guidance on design for structures with noncanonical geometry. It has already been shown that s-parameter and field results may be obtained numerically with great efficiency using a 2D microstrip dyadic Green’s function, which is based upon judicious treatment of the source point singularity and mode-matching [23]. Such an approach avoids explicit use of the completenesstheorem for the homogeneous part of the problem, a tremendous advantage since the final dyadic Green’s function has one less infinite summation.


229

But in the previous 2D model [l], it is assumed that the nonport boundaries are magnetic walls, confining energy exchange to only the ports. This seems like a reasonable assumption, given that the actual device may only have magnetized material in the circular region, and that the surface current perpendicular to the perimeter on the microstrip goes to zero at the boundary. (This particle current being zero does not mean displacement current is null, thereby allowing a finite H$ value.) Convincing experimental evidence for this supposition is certainly found in the literature, and this is also indicated by recent theoretical and experimental data reviewed [22]. Nevertheless, in order to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with dielectric constant ~ d and , also thereby achieving the capability of varying the permittivity of the external dielectric, we develop here a new dyadic Green’s function satisfactory to accomplish the task. The permeability pd is also allowed to differ from the free space value. We restrict ourselves to the 2D homogeneous circulator case because it approximates many actual operating devices, readily allows explicitly developed compact Green’s functions, and enables the modificationsto the hard wall device to become evident. Thus, more transparent formulas will arise in this section compared to Section IV.

2. Fields and Constraints Applying the radiation condition as r +. 00 leads to the selection of the modified Bessel function of the secondkind K,, (kdr)for use in the external field construction, r > R: M

(374a) (374b) Requiring continuity of the perpendicular electric field at r = R,

we find (376) where

230

CLIFFORD M. KROWNE

f attempts in an approximate way to allow for a consistent fringing in the 2D model, which has some inherent degree of 3D nature. The forcing function for the Green's function is applied at (r', 4'), r = R , through the equality

H r ( R , 4) = H @ A J (-~ #')A$'

+ H $ ( R , 4 # 4')

(378)

The perimeter azimuthal magnetic field can be represented by a 1D Fourier expansion, W

H,p"(R, 4) =

A,ein@

(379)

n=-W

Azimuthal magnetic field in the circulator disk, as r + R from the inside, is given by


or

Inserting (374) into (380), r + R from the outside, utilizing azimuthal orthogonality, yields

3. Dyadic Green's Function With the solution of ano from (384) in terms of the forcing field H ~ Athe , elements of the dyadic Green's function for r < R (within the circulator puck) may be written


23 1

down as

4. Factor f Factor f is estimated as f = f w f,. f w weights the parameter dependence expressed in f,. Closed-form formulas, based upon self-consistent static solutions, exist for microstrip capacitive (electric) end effect [26]. Stretching the microstrip end so as to connect one corner to the other constructs the circulator perimeter, and allows us to roughly obtain f,. f

CT - C - _ Cf h - -cf c c AEd

,-

(39 1)

232

CLIFFORD M. KROWNE

Assign A = nR2 and W = 2nR, and place them in (391) and in the equivalent additional radial length Alf expression, which relates to the fringing capacitance

c,:

c is the speed of light in vacuum, h the substrate thickness, Z, the microstrip impedance based upon &d dielectric loading causing an effective dielectric constant &de. We replace &de by &d under the W / h > > 1 limit. The left-hand side of (391) is given by [27,28]:

Using (392) and (393) in (391), the final formula for f is (noting that the r subscript denotes relative value) 0.824h &,d fp

=

&,d

+ 0.300 R/h + 0.042 - 0.258 R/h + 0.127 0.2217 +0.1061n

Cover location h’ >> h in deriving (394). One would expect the f w prefactor to contain information on the azimuthal mode structure, and this will be displayed as a dependence on the azimuthal mode index n. The prefactor f w may be very complicated, and the best that one can do is to obtain some reasonable degree of approximation. But that will not be done here.

B. 3 0 Dyadic Green’s Function 1. Introduction Although the theoretical work has been completed for a 3D model of the ferrite circulator, it has been done for the relatively general case of an inhomogeneous device puck. Here we would like to present the considerably simplified case for only a disk which is homogeneous [20]. The motivation for this is twofold. One is a historical parallel with the developments for the original planar work, which in the earlier years found sufficient accuracy using a uniform ferrite region in a 2D model. The other is the more streamlined formulas and more transparent formulation. The other part of what we want to do here is to allow for either perfect magnetic walls as before, containing the fields within the nonreciprocal region, or for


233

imperfect walls leading to field extension into the area external to the device ferrite material. In the previous inhomogeneous 3D microstrip model [2], it is assumed that the nonport boundaries are magnetic walls, confining energy flow to only the ports. This may be a reasonable assumption in view of the thin dimensions in the z-direction compared to the extent on the lateral surface. But by its very nature, a 3D model could allow accurate description of field extension beyond the circulator perimeter and fringing fields, depending upon how general and complex the formulation. Here we will develop a treatment which extends the perfect electric wall at the ground plane beyond the perimeter, and the microstrip conductor covering the ferrite puck into the region r > R but excluding all port areas where microstrip lines enter and exit the device.

2. Fields and Constraints Applying the radiation condition as r + 00 leads to the selection of the modified Bessel function of the second kind K,(adjr) for use in the external field construction, r > R. It is assumed that the electric wall conditions are maintained in the manner described previously for radii exceeding the circulator radius so that the same z-indexing modal set can be used internal and external to the device. With this assumption,

Here the characteristic equation for the radial separation constant udj is given in the outside region by

where the perpendicular indexing for the discrete spectrum of allowed values is done according to jn kzj = -; h

j = (0 or l), 2,. . .

(398)

with the first j index choice determined by the first nontrivial field component. Azimuthal magnetic field component H$ (only the transverse part) may be made to retain a form congruent with the puck field constructionby setting the coefficient factors q and t of the partial differential operators a/ar and a/a# equal to zero

234

CLIFFORD M. KROWNE

(see [l 11):

with

p=-

ikzj

ki - kZj

, u=-

iW&d k i - kZj

Similarly, the azimuthal electric field component E$ (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the and a/ar equal coefficient factors F and q of the partial differential operators to zero:

with

Requiring continuity of the perpendicular electric field at r = R ,

f q R , 4) = E,d(R,4)

(403)

Circulatorfield ES (in the i = Nthring, N = 0) at its most extreme position r = R is expressed as 0 3 0 3

EE =

C C c o s ( k ~ j Z ) [ a ~ o j J n ( a l j+a~OjJn(azjR)]e'"@ R) j=O

(404)

n=-m

Putting (395) and (402) into (401), and using longitudinal and azimuthal orthogonality, yields

f [Jn(aljR)anoj 1 Jn(a2jR)a;oj] = a,djKn(adjR)

(405)

making the coefficient in the external region

Requiring continuity of the azimuthal electric field at r = R, (407) f E ; ( R 4 ) = E,d(R. 4 ) Circulatorfield E; (in the i = Nth ring, N = 0) at its most extreme position r = R is

235


expressed as

Q2j + -(iwpo + sj)cl,)J;(a2jR)

(408)

bj

The dielectric region field E$ as r -+ R from the outside is obtained by (401) with the help of (395) and (396). 0 0 0 0

$!I

=

C

j=l n=-m

i sin(kzjz)

iPjn

d

4--an,Kn(ad,R) R

1

e'"'

(409) Inserting (408) and (409) into (407), and using longitudinal and azimuthal orthogonality, yields

The azimuthal magnetic field forcing function for the Green's function is applied at (r', #), r = R, through the equality

H p R , 4) = Ht$'Ag(z)a($

- #')A#' 4-H:(R, # # 9')

(413)

Here g(z) is the functional behavior of the forcing azimuthal magnetic field in the z-direction. The perimeter azimuthal magnetic field can be represented by a 1D

236

CLIFFORD M. KROWNE

Fourier expansion, 00

Hp(R,#) =

1 A,e'"$, n=-m

An =

2n

H r ( R , #)e-'"@d#

(414)

--11

Azimuthal magnetic field in the circulator, as I + R from the inside within the last annulus i = N = 0, is given by


H;(R, 4 ) = H r ( R , 9 ) gives

and H@fAjbeing the overlap integral specifying the amount of the jth mode behavior contained in the forcing azimuthal magnetic field. It is

Htj is found from

;=O n=-w

where (398) has been used to find

237


where 2AAj and

are given by

Inserting (419) into (4 17), letting r -+ R from the outside, and utilizing azimuthal orthogonality, yields

= - 1H t ~ j A 4 I e -in@'

2n

The perpendicular magnetic field forcing function for the Green's function is applied at (r', + I > , r = R, through the equality

Hp"'(Rt4) = H z A ~ ( Z ) S (&A# ~

-k H,d(R, 4

# 4')

(425)

Here h(z) is the functional behavior of the forcing perpendicular magnetic field in the z-direction. The perimeter perpendicular magnetic field can alsobe represented by a 1D Fourier expansion,

c oi)

Hp"'(R,#) =

n=-w

Bnein@,

Bn = 2n

/" -"

H:a(R, #)e-'"@d 4

(426)

Perpendicular magnetic field in the circulator, as r + R from the inside within the last annulus i = N = 0, is given by


H;(R, 4) = H p ( R , 4 )

(428)

238

CLIFFORD M.KROWNE

gives

and H , A ~being the overlapintegral specifyingthe amount of the jth mode behavior contained in the forcing perpendicular magnetic field. It is

H S is found from (396) and (410). Inserting (396) into (429), letting r + R from the outside, and utilizing azimuthal orthogonality, yields

3. Dyadic Green's Function Let us introduce some new definitions in order to make transparent the forms of the disk modal coefficients in (428) and (437).

-1 c.-)12j B n1.=I Jn(a2jR)- f h:Aij, bj

-2 cj - A l j Jn(a2jR) - f ':A:j Bnj = bj (434ab) Introducingthese formulas into the two equations forming a simultaneous solution set for the disk modal coefficients in terms of the driving field magnitudes,

1 + Aijui0 = H ~ I AA4~e 2n I

Atja;,

whose solution is

-in@'

,

B,!,ja;,

1

+ Bijaio = Hz~jA@ e 2n I

-in@'

(435ab)


239

These equation forms are identical to those derived for the 3D circulator problem with hard walls between the ports. The right-hand sides are the driving terms. From (436) and (437) the dyadic Green’s function elements are determined to be (c subscript on the left-hand side of the equations denotes circulator disk since the ith ring degenerates to i = 0)

(439)

c

1 ” 0 3 1 i K$ sin(kzjz)[L?ijCif,jO(r) G z H c= 2n j = I n=-w DABj

240

CLIFFORD M. KROWNE

. " "

(447)

(449) Here z-directed weights K$ and Kij are given by (450)

by invoking (419) and (431). The dyadic elements for r > R possess the same form as previously provided in Section V and so will not be repeated here.

VII. SYMMETRY CONSIDERATIONS FOR HARD MAGNETIC WALLCIRCULATORS A. 2 0 Dyadic Green$ Function 1. Introducrion

Green's function methods are widely employed for analyzing circulators. Most circulators are designed with symmetry [29], and this fact may be used to significantly reduce the number of Green's function elements which must be calculated when determining the s-parameters and electric field. Although we have


24 1

$b = 2n/3

b

Port - ferrite region

-

( non port)

C $C =

- 2d3

FIGURE9. Two-dimensional (2D) circulator diagram showing the special case of only three ports, labeled counterclockwise. Port interfaces act as source driving functions, with the rest of the perimeter acting as a magnetic wall. Inside the perimeter located at r = R , an inhomogeneous ferrite material exists, which is broken up into N annuli, each annulus being considered uniform in material.

demonstrated how such simplified elements may be obtained from the general expressions which are valid for arbitrarily placed ports for the case of a three-port inhomogeneous device [2], here derivations for this device as well as the fourport and six-port cases will be given to provide a complete picture for multiport symmetric inhomogeneous circulators. The value of the Green’s function approach, besides its great mathematical elegance at times, is that it allows fast calculation for canonical structures such as the circular circulator. Thus, the utilization of the formulas provided here greatly assists the effort in simulating the s-parameters and z-component of the electric field E,. Focus is on finding the s-parameters because from them inhomogeneous circulator performance can be assessed. However, discussion is given on getting E,(r, 4) so that field plots are available. Presentation to follow will use a counterclockwiselabeling of the ports, shown in Fig. 9 for the case of a symmetric three port device. Note that the circulator puck is composed of concentric rings with differing ferrite parameters, creating inhomogeneous internal loading. 2. Three-Port Symmetric Circulator

There are a tremendous amount of simplifications which result by constraining the inhomogeneous circulator to a symmetric disposition of the port locations. Ea

= GaaHa

GabHb

+ GacHc

(45 la)

242

CLIFFORD M.KROWNE

= G b a H a -k

-k

GbcHc

Ec = G c a H a -k G c b H b 4-

GccHc

Eb

GbbHb

(45 1b) (45 1c)

In the general case for the 2D model, (45 1) holds for an inhomogeneousthree-port device. Examination of the EH dyadic Green's function element for n = N, the last ring, allows full advantage to be taken of the inherent threefold symmetry [lo]. .

m

Let us make this expression more transparent by defining the source azimuthal location to be 4j = 4: and the field location to be 4 = $i . Furthermore, abbreviate

G"EQ;IN (R,4; R,4:)

I

= G(4iT 4j)

(453)

@: =@ j

On the right-hand side of (452), collect the azimuthal exponents into one factor, while noting that the radial variation, here with r = R, is stored in the prefactor 7;; = 7;; (R). Then (451) becomes

(454) Right away, we notice from (454) that the Green's function only depends upon the difference between the source and field locations. That is,

G(4iv 4 j ) = G(4i - 4 j )

(455)

This does not mean, however, that G(@i,4,) = G($,, 4i). In fact, because the material is a nonreciprocatingmedium, we know this can't hold. Certainly, circulating action would cease if this type of Green's function dyadic element symmetry existed. Another clue that this type of symmetry doesn't exist is seen by reexamining (454) again. Let us find the G($,, 4i) dyadic element from (454) by switching azimuthal angles. . o o

Now define a new summation index as n' = -n. Substitutingthis into (455) yields

.

-02

(457)

But because notation is arbitrary, change the n' into n for the index, and recognize that the order of summation doesn't matter, especially as the same integers are


used as in the original G(&,

243

Green’s function. .

w

Comparing this to the expression for G(qji, 4j) in (454) enables us to see that except for the prefactor, the dyadic Green’s function formulas are identical. It is this prefactor, nevertheless, which is all important here and maintains the device nonreciprocity! pZNis not equal to p;i, or

(459) Because of the all important relationships in (459), G ( @ j 4i) , # G(4ii,@ j )

(460)

is true and we are assured of our nonreciprocity. However, property (455) and the symmetric angular distribution of the port locations will reduce the number of actual Green’s function elements required to be calculated. Note that the notation Gij

= G(4iV 4j)

(461)

is used in (45 1). Thus, by evaluating the Green’s dyadic for @i = # j , the self-terms, it is found that .

w

All self-term Green’s function dyadic elements are equal, and we denote this fact by assigning

Equation (462) obviously also means that G(4i, 4i) = G ( + j , 9j)

(464)

For the off-diagonal Green’s function dyadic elements, it is found that the number of unique terms is less than the total number of off-diagonal elements. Denote the total number of radially located ports as N n p . Then the total number of diagonal elements is NT,diag

= Nm

(465)

but the number of unique diagonal elements is one by the argumentsin the previous

244

CLIFFORD M. KROWNE

paragraph. The total number of off-diagonal elements is

but the number of unique off-diagonal elements is considerably less,

for an even number of ports and

for an odd number of ports. A device with three ports would have Nu,+d = 2. This is a much smaller number than the result in (466), giving NT,~-,,= 6. The savings for larger Nnp rapidly goes up because of the quadratic term in (466). The determinant of the system to solve for the radial azimuthal &-fields also displays all the individual Green's function dyadic elements.

Let us define

- 4 .J

A#.. '1- 4 .I

(470)

Recognizing that a right-handed system with (r, 4) or ( x , y) in the plane of the paper requires counterclockwise labeled ports (i.e., i = a, b, c or i = 1, 2, 3) to have progressively more positive valued azimuthal angles, if the input port angle is set to 0 radians, then $a

= 0;

4b

= 2x13; 4~= 4x13

(47 1)

In order to see that there are only two unique dyadic Green's function elements requiring calculation,it is best to begin to evaluate the particular off-diagonalterms in (469). It will become apparent what the trend is once this examination process is started. By (454), (461), and (469),

Moving down the first column, excluding the diagonal, the ij = 21 or ba element is .

w

(473)


245

Invoking (47 l), %a

= #b - #a = 2x13

(474)

and inserting this into (474) yields one of the unique dyadic Green’s function elements.

(475) with Gba = G+

(476)

Moving down to the next element in the system matrix, the 31 or ca element, we find the angular argument

and put it into

obtaining the other unique dyadic Green’s function element.

with Gca = G -

(480)

The dyadic Green’s function elements in the first row in (469), are found from those already determined by noting that the azimuthal angle differences have their signs reversed. Finally, the cb (and bc) element is determined once it is noted that A#& = #c - #b = 4x13 - 2x13 = 2 ~ / 3

(481)

By (472), this implies that the cb element is G - . Therefore, in summary we have found that

(482a) (482b)

(482c)

246

CLIFFORD M. KROWNE

Placing (482) into (469) gives

Now the three azimuthal H-fields can be simplified as follows:

(484a)

(484b)

(484c)

+

Note that the and - indices on the Green’s functions are mislabeled in [2] for the Hb and Hc results (they have been reversed inadvertently). The s-parameters are found in the usual manner, (485a) (485b) (48%)

247


3. Four-Port Symmetric Circulator Equation (451) comes from the more general construction to be covered here. The z-component of the electric field at the circulator perimeter, r = R, is given by

where Hq = H$. Evaluating this at 4 = 4q,p = 1,2, 3 , 4 for

=4 gives

or written out explicitly for all p indices, dropping the understood component index on the electric field,

In matrix form this reads as

[ [zi: E4

G12 G13 = cZ2G23 G31 G32 G33 G41 G42 G43

[21

G14 G ~ ~ ] G34 G44 H4

Making port 1 the input port,

The remaining three ports have the impedance conditions

or put in a more compact form,

(489)

248

CLIFFORD M. KROWNE

Placing (490) and (492) into (489) produces the matrix equation for finding the azimuthal H-fields.

The H-field solutions are

(494a)

(494b)

(494c)

H4

1 =D4x4

2 - -D4x4

G2I

(G22

+ Hout(l) hi.=&

+ G (r, 4; 4iso ( 4 i n ) ) Hiso( 1 ) ) Einc(#in)

(538) Numerical results based upon (538), with a theoretical technique employed to derelative to Ei"C(4a), has been done for specific circulator parameters rive EinC(+in) ~311.

7. Conclusion Sets of Green's function elements pertinent to the perimeter of the inhomogeneous circulator have been determined for various multiport symmetric circulators. It has been demonstrated how these Green's function elements are employed for finding the device s-parameters. Utilization of these formulas leads to an economy of effort spent in numerical evaluations. Also, derivation of an expression suitable for electric field contour plotting within the inhomogeneouscirculator boundary has been provided. B. 3 0 Dyadic Green's Function 1. Introduction In part A of this section, we have presented the symmetric considerations for 2D circulators, and have found a great reduction in the number of specific Green's fucntions which must be evaluated to determine the s-parameters. And it was also seen that only one type of Green's function is required to act as a generator to construct the rest of the Green's functions needed to completely represent the internal electric field. Here we present the 3D dyadic Green's function results.

2. Three-Port Symmetric Circulator There are a tremendous amount of simplifications which result by constraining the circulator to a symmetric disposition of the port locations. For a three-port circulator [ 1 11 Ea = TiaHa Eb = T;a Ha

+ T:b Hb + TfCHc + T;b Hb + TlC Hc

(539a) (539b)

26 1 (539c) (540a) (540b) (540c) (541a) (541b) (541c) (542a) (542b) (542c) (543a) (543b) (543c) (544a) (544b) (544c)

(545a) (545b) Gfh

(cc)MCc

(54%) (546a)

(546b)

(546c)

(547a)

(547b)

262


263

The dyadic Green's functions elements at the circulator perimeter are given by the following formulas, which are then used to obtain the azimuthal dyadic Green's functions elements employed in the preceding equations.

(556a)

(556b)

(556~)

(556d)

(556e)

(5560

(557a)

264

CLIFFORD M.KROWNE

x [AAjT'ijN(R) - AijTL~jN(R)]e-i"4e'"0

7".( R )

x [AAj

- A i j T'),

( R ) ]e-i"@': ein@

(557d)

(5570

To get the analysis started for the 3D case, we must try to compact these complicated expressions. With this desire in view, consider how factorizable non-4dependent parts of the 2D Green's function elements were created earlier. The same thing may be done here by setting the z-coordinate to z = zs, and defining

Psy = Kzj+ COS(kzNj+Zs)-DABj 1 [Bn,TnejN(R)- B,!jTi;N(R)] 2

p:$jh

1

= Kzj+ COS(kzNj+Zs)-

DABj

= iKzj+ sin(k,Nj+z,>-

1

DABj

Pnr$jh = iKzj+ Sin(kZ,vj+zS)-

1

DAEj

zl

[AL,T,Z,:.N(R)- AijTi:jN(R)]

(558b)

[BijT,',:.,(R) - B,!jTLiN(R)] ( 5 5 8 ~ ) [AAjT':,(R)

- AijTL:jN(R)] (558d)

[BnjTnejN(R) 2 4J1 - BjjT2,(R)] DABj @ezh 1 pnNj = iKd+ sin(kzNj+zs)[ A A j T S N ( R) Aij7"jN(R)] DABj = iKzj+ sin(kZNj+zs)-

(558a)

(558e) (5580


265

The Green’s function elements may then be written in the much abbreviated forms (560a) .

w

w

(560b)

(560d)

(561a)

266

CLIFFORD M.JSROWNE .

w

w

(561b) .

w

o

o

(561c) .

w

w

(561d)

(561e) .

w

w

By constructingazimuthal summationsover the new coefficients j&iS ‘sf where ,f2 are electric or magnetic fields, removing the extra j-index arising from the z-variation, allows the 3D Green’s function elements to take on forms identical to the earlier 2D forms studied. The modified coefficients are defined as

(,,tSare cylindricalcoordinatesand fl

w

(562a) j-0 w

(562b) j=O

(562c) j=O

(562d) j=O

(562e) m

(563a) j=O


267 (563b)

(563c) j=O w

(563d)

(563e)

j=O

The Green's function element expressions can now be set down using these f p They are modified ~ ~ ~ " scoefficients.

.

w

(565a) (565b)

268

CLIFFORD M.KROWNE

(565c) .

w

(565d) (565e)

Because our interest is in only the z-directed component of the electric field E,, and a subsidiary z-component magnetic field H,is employed, the first superscripts may be removed since they are understood and the compressed notation used for the only four Green’s function elements required.

(567a) Gih(i?j ) = Gih($iy 4,) = G i h ( R , $ i ; R , 4 j ) = %

C ;7; W

zhein(@,-4,)

n=-w

(567b) Following the logic in the 2D theory for a three-port device, unique azimuthal symmetry dyadic Green’s function elements are defined for the four basic types of dyadic Green’sfunctionelements. Each basic dyadic Green’s function element will have three unique azimuthal symmetry dyadic Green’s function elements. Thus, a total of twelve are needed and provided here. For the E, H4 symmetry elements *

w

(568a)

THEORY FOR INHOMOGENEOUS FERRITE .

269

w

(568b)

(568c) For the Ez Hzsymmetry elements (569a) , 1

‘?,eh

=

c w

= ze. zh 2inlr/3 YnN

(569b)

n=-w

(569c) For the HzHb symmetry elements (570a) .

W

(570b) .

m

(570c) And finally for the HzHzsymmetry elements .

W

(571a) I G$,hh

=

c

= zh, zh 2 i n n / 3

YnN

(571b)

n=-w

(571c) The simplification of the azimuthal dyadic Green’s function elements includes the following rules for the EzHb symmetry elementsbased upon counterclockwise

270

CLIFFORD M.KROWNE

port labeling

For the H, Hd symmetry elements

And finally for the H,H, symmetry elements


27 1 (579a) (579b) (579c) (580a) (580b) (58oc)

(581a) (581b) (581c)

(582a)

(582b)

(582c)

(583a)

(583b)

272

CLIFFORD M.KROWNE

=

['f,

hh

-

'1

2

- 'k,

hh':,

(584c)

hh

The Mi, expressions are related to one another because of the symmetry simplifications.

-

Maa

= Mbb = Mcc =

Mab

= Mbc = GI, hh ['f, hh = -Mcb =

('5,

[G:,hh

2 hh)

3'

2

-'t.hhGt3hh

'3 - ('t, - 't, :,[' hh

(585b)

hh) hh

-

(585a)

'1

(585c)

The T: relationships may be put into matrix form again for compactness and ease of viewing their properties.

,!'

eh

,!'

eh

'$,

eh

1


273

Once all the preceding symmetry relations are found, the electromagneticfields and, as a consequence, the s-parameters, can be found.

3. Conclusion Sets of Green’s function elements pertinent to the perimeter of the inhomogeneous 3D circulator have been determined for a single multiport symmetric circulator case, namely, the most common three-port case. It has been demonstrated how these Green’s function elements are employed for finding the device s-parameters. Compared to the 2D situation, these formulas are much more involved. But a similar economy in the number of Green’s function elements occurs here and is passed down throughout the expressions required to find the s-parameters. The same process may be undertaken for four or six symmetrically displaced ports, with the provision, however, that the algebra involved in working with the Green’s functions increases tremendously, as we would expect.

VIII. OVERALLCONCLUSION Electromagnetic circulators, which are commonly utilized at several hundred MHz, at a few GHz, in the microwave regime where the free space wavelength is on the order of centimeters, to the millimeter wave region which exists beyond 30 GHz up to 300 GHz, are a key component of high-frequency electronics and will probably be around in the foreseeable future. Circulators are employed in very high-power applications as well as in moderate- and low-power electronics. The lower-power applications involve hybrid and monolithic circuits, which are integrated with solid state devices, and it is for these higher-frequency, lowerpower applications where much of the interest and utility of microstrip circulators arises.

274

CLIFFORD M. KROWNE

We have tried in this contribution to bring together several approximation possibilities or economizing techniques which may be applicable toward fast computeraided design (CAD) using rigorous mathematical physical methods for microstrip circulators, and to address or handle real issues encountered by devices in circuit environments, which have not appeared in the literature or are of very recent origin. Consequently, the effect of placing the circulator puck in an external medium affecting its behavior has been examined from several aspects. The puck has been placed in a vertically layered, radially zoned medium (Sections I1 and 111). The puck has been placed in a much simpler medium where only a single outlying radial region exists (Sections IV through VI). Finally, ways to employ symmetry to reduce significantly the number of Green’s function elements and vastly speed up numerical computation have been presented (Section VII). Our focus has been on techniques that are more important for their mathematical physics content and potential for user-friendly engineering design capability than for their ability to obtain nearly exact solutionswith numerically intense simulators, which use extensive computer code, memory, and time.

REFERENCES [I] Krowne, C. M., and Neidert, R. E. (1996). Theory and numerical calculations for radially inhomogeneous circular ferrite circulators, IEEE Trans. Microwave Theory & Tech. 44(3), 419-431, March. [2] Krowne, C. M. (1996). 3D dyadic green’s function for radially inhomogeneous circular ferrite circulator, IEEE Microwave Theory Tech. Symposium Digest, San Francisco, CA, June 18, pp. 121-124. [3] Maystre, D., Vincent, P., and Mage, J. C. (1983). Theoretical and experimental study of the resonant frequency of a cylindrical dielectric resonator, IEEE Trans. Microwave Theory & Tech. 31(10), 844-848, October. [4] Hui, W. K., and Wolff, I. (1991). Dielectric ring-gap resonator for application in MMIC’s, IEEE Trans. Microwave Theory & Tech. 39(12), 2061-2068, December. [5] Hui, W. K.,and Wolff, I. (1994). A multicomposite, multilayered cylindrical dielectric resonator for application in MMIC’s, IEEE Trans. Microwave Theory & Tech. 42(3). 415423, March. [6] Michalski, K. A. (1986). Rigorous analysis methods, In Dielectric Resonators (D. Kajfez and P. Guillon Eds.). Dedham, MA: Artech House, pp. 185-258. [7] Krowne, C.M. (1998). Implicit 3D dyadic green’s function using self-adjoint operators for inhomogeneous planar ferrite circular with vertically layered external material employing modematching, IEEE Trans. Microwave Theory Tech. M I T 46(4), April. [8] Morse, P. M., and Feshbach, H. (1953). Purr I, Methods of Theoretical Physics, New York McGraw-Hill. [9] Friedman, B.(1956). Principles and Techniques of Applied Mathematics, New York: Wiley. [lo] Courant, R. (1962). Partial differential equations, in Purr 11, Methods of Mathematical Physics, by R. Courant and D. Hilbert. New York: Interscience. [ 1 I] Krowne, C. M. (1996).Theory of the recursive dyadic green’s function for inhomogeneous ferrite canonically shaped microstrip circulators, in Advances in Zmuging and Electron Physics, Vol. 98 (P. W. Hawkes, Ed.). Academic Press,pp. 77-321.

THEORY FOR INHOMOGENEOUS FERRlTE

275

[12] Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table oflntegrals, Series, and Products, Academic Press, San Diego. [13] Wang, I. J. H. (1996). Generalized Moment Methods in Electromagnetics. Wiley, New York. [I41 Chu, T.S., Itoh, T.,and Shh, Y.-C. (1985). Comparative study of modematching formulations for microstrip discontinuityproblems, IEEE Trans. Microwave Theory Tech. MlT 33(10), 10181023, October. [ 151 Mautz, J. R. (1995). On the electromagneticfield in a cavity fed by a tangential electric field in an aperture in its wall, IEEE Trans. Microwave Theory Tech. M l T 43(3), 620-626, March. [I61 Carosella, C. A., Chrisey, D. B., Lubitz, P.,Honvitz, J. S., Dorsey, P., Seed, R., and Vittoria, C. (1992). Pulsed laser deposition of epitaxial BaFelzO19 thin films, J. Appl. Physics 71(10), 5107-51 10, May 15. [I71 Karim, R., Ball, S. D., Truedson, J. R., and Patton, C. E. (1993). Frequency dependence of the ferromagnetic resonance linewidth and effective linewidth in manganese substitutedsingle crystal barium ferrite, J. Appl. Physics 73(9), 45 1 2 4 515, 1 May. [ 181 Morisako, A., Nakanishi, H., Matsumoto, M., and Naoe. M. (1994). Low temperature deposition of hexagonal ferrite films by sputtering,J. Appl. Physics 75(10), 5969-5971, May 15. [ 191 Krowne, C. M. (1997). 2D dyadic green’s function for homogeneousferrite microstrip circulator with soft walls, The 22nd Intern. Conf. Infrared MillimeterWaves Dig., pp. 168-169, Wintergreen Resort, VA, July 19-25. [20] Krowne, C. M. (1997). Homogeneous ferrite microstrip circulator 3D dyadic green’s function with and without perimeter interfacial walls, The 22nd Intern. Conf. Infrared Millimeter Waves Dig., pp. 170-171, Wintergreen Resort, VA, July 19-25. [21] Krowne, C. M. (1998). Inhomogeneous ferrite microstrip circulator 2D dyadic green’s function for penetrable walls. Intern. J. Electronics 84(6), June. [22] Newman, H. S., Webb, D. C., and Krowne, C. M. (1996). Design and realization of millimeterwave microstrip circulators, Intern. Con$ Millimeter Submillimeter Waves Appl. Ill Dig., SPIE Proceedings 2842, 181-191. [23] Krowne, C. M., and Neidert, R. E. ( 1995).Inhomogeneous ferrite microstrip circulator:Theory and numerical calculationsusing a recursive green’s function, 25th European Microwave Conference Dig., pp. 414420, September 4-7, Bologna, Italy. [24] Krowne, C. M. (1997). Femte microstrip circulator 3D dyadic green’s function with perimeter interfacial walls and internal inhomogeneity, Microwave Optical Tech. Lens. 15(4), 235-242, July. [25] Tai, C.-T. (1993).Dyadic Green’s Functions in Electromagnetic Theory, IEEEPress, Piscataway, NJ. [26] Gupta, K. C., Garg, R., and Bahl, I. J. (1979). Microstrip Lines and Slotlines, Artech House: Dedham, MA. [27] Hammerstad, E. 0.. and Bekkadal, F. (1975).Microstrip Handbook, ELAB report STF44 A74169, The University of Trondheim, The Norwegian Institute of Technology. [28] Garg, R., and Bahl, I. J. (1978). Microstripdiscontinuities,Intern. J. Electronics 45.81-87, July. [29] Krowne, C. M. (1997). Symmetry considerations based upon 2D EH dyadic green’s functions for inhomogeneous microstrip ferrite circulators, Microwave Optical Technology Letts. 16(5), 176-186, October 20. (301 Newman, H. S., Neidert, R. E., Krowne, C. M., Vaugh, J. T., and Popelka, D. J. (1996). CAD tools for planar ferrite Circulators:Development, verification, and utilization, Femte CAD Workshop, IEEE Microwave Th.Tech. Symposium, San Francisco, CA, June 17. [31] Newman, H. S., and Krowne, C. M. (1998). Analysis of ferrite circulators by 2D finite element and recursive green’s function techniques, IEEE T m s . Microwave Theory Tech. M7T 46(2), 167-177, February.


ADVANCES IN IMAGING AND ELECTRON PHYSICS,VOL. 103

Charged Particle Optics of Systems with Narrow Gaps: A Perturbation Theory Approach M. I. YAVOR Institute for Analytical Instrumentation RAS, Rizhskij p r 26, 198103 St. Petersburg, Russia Tel. +7 (812) 251-86-63. Fax +7 (812) 251-70-38, Email: [email protected]

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . II. Applicability of Perturbation Methods in Charged Particle Optics . . . . . . . A. Electromagnetic Field Structures Suitable for Application of Perturbation Methods B. How to Apply Perturbation Methods Correctly . . . . . . . . . . . . . 111. Calculation of Weakly Distorted Sector Fields and Their Propertieswith the Aid of a Direct Substitution Method . . . . . . . . . . . . . . . . . . . . . . . A. Electrostatic Field and Charged Particle Trajectories in an Imperfectly Manufactured Sector Energy Analyzer . . . . . . . . . . . . . . . . . . . . . B. Magnetostatic Field and Charged Particle Trajectoriesin an Imperfectly Manufactured Sector Magnet . . . . . . . . . . . . . . . . . . . . . . . . C. Electromagnetic Field and Charged Particle Trajectories in an Imperfectly Manufactured Wien Filter . . . . . . . . . . . . . . . . . . . . . . . D. Parasitic Beam Distortions in Charged Particle Analyzers Based on Sector Fields and Wien Filters and Their Correction . . . . . . . . . . . . . . . . N. Transformation of Charged Particle Trajectoriesin the Narrow Transition Regionsbetween Electron- and Ion-Optical Elements . . . . . . . . . . . . . . . . . . A. Charged Particle Beam Transport through the Gaps of Multiple Magnetic Prisms B. Transformation of Charged Particle Trajectories in the Gaps between Lenses of Closely Packed Quadrupole Multiplets . . . . . . . . . . . . . . . . V. Synthesis of Required Field Characteristicsin Sector Energy Analyzers and Wien Filters with the Aid of Terminating Electrodes . . . . . . . . . . . , . . . . A. Electrostatic Field in the Gap between Two CurvilinearElectrodes Terminated by Split Shielding Plates . . . . . . . . . . . . . . . . . . . . . . . B. Influence of Split Matsuda Plates on the Field and Optical Propertiesof Sector Energy Analyzers and Wien Filters . . . . . . . . . . . . . . . . . C. Synthesis of a Required Field Distribution in a Vicinity of the Beam Main Path in a Sector Analyzer with the Aid of Split Shielding Plates . . . . . . . . . . VI. Calculation of the Elements of Spectrometers for Simultaneous Angular and Energy or Mass Analysis of Charged Particles . . . . , . . . . . . . . . . . . . A. Electrostatic Field of a Poloidal Analyzer . . . . . . . . . . . . B. Calculation of Particle Trajectories in a Poloidal Analyzer . . . . . . . . C. Focusing Properties of a Poloidal Analyzer . . . . . . . . . . . . . . D. Electrostatic Field of a Conical Mirror . . . . . . . . . . . . . . . E. Electrostatic Field of a Conical Lens with Longitudinal Electrodes . . . . . . F. Electrostatic Field of a Slit Conical Lens . . . . . . . . . , . . . . . G. Elimination of a Beam Deflection in Conical Lenses . . . . . . . . . . H. Focusing of Hollow Charged Particle Beams by Conical Lenses . . . . . . VII. Conclusion . . . . . . . . . , . . . . . . . . . . . . . . . .

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218 283 283 294 295 295 302 306 311 318 318 329 336 336 344 346 348 348 352 361 363 361 312 314 378 384

211 Volume 103 ISBN 0-12-014745-9

ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright@ 1998 by Academic Press. Inc. All righu of repduction in any form reserved. ISSN l076-5670/98525.00

278

M. I. YAVOR

. . . . . . . . . . . . . . . . . . . . . . . .

385

. . . . . . . . . . . . . . . . . . . . . . . . . .

385

Acknowledgments References

I. INTRODUCTION

A design of any charged particle optical system (electron or ion microscope, energy or mass analyzer, mass separator,particle accelerator,etc.) is always based on simulation of electron or ion trajectories through an electromagnetic field created by a certain set of electrodes, currents, or magnetic pole faces. At the early stage of development of charged particle optics, simple model field distributions were used for this simulation such as bell-shaped lens fields or sector and multipole fields in a sharp-cutoff approximation; these distributions allowed analytical representation that made trajectory analysis relatively simple. The simulation usually resulted in analytical formulas, which clearly revealed physical properties of the system under investigation and thus enabled the achievement of optimal solutions easily. For practical purposes, however, a framework of analytical field models appeared to be too narrow. Even a small deviation from an analytical model could be treated only with the aid of a numerical solution of the partial differential equations for electromagnetic field distributions followed by either a subsequent numerical ray tracing of trajectories or, at best, by a calculation of aberration integrals. A progress in computer engineering has led during the last decade to development of a variety of sophisticated modem numerical methods for electromagnetic field calculation (a survey of these methods can be found, for example, in [ 11) and corresponding software for charged particle optical simulation [2]. However, practical optimization of electron- and ion-optical devices with a help of such software usually is a complicatedproblem, and the reason is not only that numerical computations are time-consuming and wasteful of computer resources. With a numerical simulation, the main advantage of analytical models is lost: a clearness and visuality of the physics involved. Instead of general physical properties of the system and trends in its behavior with changes of parameters, a designer sees only a result of a particular calculation for a given set of the system parameters, and this result as a rule does not tell much about a way or even a possibility to optimize a system in a desired direction. Thus, a question arises: How would it be possible to enlarge the area of application of analytical calculation methods in charged particle optics? A positive answer to this question may be a development of approximate analytical methods, provided that their accuracy in practical cases would be generally comparable to the accuracy of conventional numerical simulation.

CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS

279

In many areas of physical knowledge where complex processes are studied (such as hydrodynamics, propagation and diffraction of electromagnetic or acoustical waves), a perturbation theory techniqueis widely used, which enables us to enlarge considerably the area of application of analytical calculation methods with all their advantages being preserved. The perturbation theory methods may be applied to a problem if one or several small physical parameters are inherent in it; this allows one to obtain a mathematical solution of the problem by the successive approximation method in a form of power serii in these small parameters. The goal of the present chapter is to demonstrate that perturbation methods can be successfully and fruitfully used also in the optics of charged particles. It should be clearly emphasized that the perturbation methods just mentioned have nothing to do with the approach of aberration expansions that has remained the base of the charged particle optics for a long time. We imply perturbation theory expansions to be the serii in powers of small parameters that are characteristic for a physical system and thus are present in coefficients of the differential equations describing this system. In optics of charged particles the corresponding parameters characterize specifics of the electromagneticfield distribution (for example, a small length of the interval of the field inhomogeniety, like that observed in a sector magnet fringing field). In the theory of charged particle beam aberrations the situation is different: Studied here are not the expansions of the particle trajectory coordinates with respect to the small parameters inherent in the electromagnetic field structures (usually it is not assumed at all that such parameters are present in the system) but the expansions in power serii with respect to small initial conditions for the charged particles (these conditions can be defined either explicitly or through the beam aperture parameters). Actually the aberration coefficients are just the Taylor serii coefficients, and successiveapproximationmethod is only a tool for obtaining these coefficients in case where explicit solutions of the trajectory equations are unknown. Note that aberration Coefficientscan be also calculated without using any successive approximation methods, for example, by differential algebraic methods [3]. A presence in electromagnetic field structures of charged particle optic systems of features described by small parameters is very rarely used for calculation of both the fields themselves and the particle trajectories in these fields. Actually, in a trajectory simulation the only example of the application of a perturbation method is a development of the so-called fringing field integral method to describe a charged particle beam transformation in the fringing fields of sector electrostatic and magnetic analyzers as well as of quadrupole lenses and multipoles (the essence of the method is summarized in [4]; see Section II.A.2 for a more detailed history of the problem). Among the electromagnetic field calculation methods, the only application of a perturbation approach is an investigation of field deviations caused by weak distortionsof the electrodes,magnetic pole pieces, or coils; however, even this investigation usually requires a numerical simulation of the field deviation

280

M.I. YAVOR

distribution, if performed with the aid of the general perturbation methods by F. Bertain, M. A. Monastyrskij,and other authors, which are surveyed in [ 5 ] . There were only few attempts to obtain analyticalrepresentations of electromagneticfield deviations, which are mentioned in Section 1I.A. 1. It should be noted that even these rare attempts to apply perturbation methods in charged particle optics in many cases finished with incorrect results, which can be found in the publications devoted to both the development of the fringing field integral method and the investigation of electromagnetic field deviations due to the electrode or magnetic pole distortions. The reason for the corresponding mistakes is due to neglecting an important feature of the perturbation theory. As a matter of fact, from the mathemathical point of view, perturbation expansions are specific cases of the so-called formal asymptotic expansions. A characteristic feature of the asymptotic serii is that these serii are, as a rule, diverging. One may only state with a certain confidence that, provided that the expansion parameter is small and that there are no any other “side” large values (coefficients or functions) in the equation under consideration, the main contribution to the solution of the problem is represented by the first term of the asymptotic expansion while the succeeding terms describe smaller corrections. However, a presence of the side large parameters is typical for many physical problems, including the problems of charged particle optics. For example, a small parameter used (explicitly or implicitly) in the fringing field theory considerations is usually a small length of the fringing field region as compared with the length of the main field. But at the same time, the smaller the fringing field length is the larger the field gradient is in the fringing field region. If this fact is not taken into account properly, the fringing field transfer matrix may be calculated partially or completely incorrectly. Generally, an arbitrary specification of an “evident” small parameter without a thorough study of a problem to be solved may lead to confusion. A very good and widely known example is deducing a formula for the focal length of a thin round electrostatic lens. It is well known [6] that, being obtained in two different ways (starting with an ordinary paraxial trajectory equation and with an equation relative to a transformed coordinate function), the corresponding formulas are different! It is not easy to understand directly which of the resulting expressions is correct and which one is wrong; an explanation given in [6] seems to be rather unclear. A real difference between the two ways discussed is a different specification of the small parameter in the problem. Working with the ordinary paraxial trajectory equation, one implicitly assumes that a small parameter in the problem is a small deviation of the charged particle coordinate inside the lens from its initial value; however, an explicit attempt to build the asymptotic expansion based on this small parameter fails. Following the second way, a small parameter is assumed to a small deviation of the ratio of the particle coordinate to the fourth root of the electrostatic field potential from the initial value of this ratio. Such a definition of


28 1

the small parameter leads to a correct final analytical expression for the thin lens focal length. Thus, a development of the perturbation methods in charged particle optics requires, first, a specification of the circle of the problems where application of these methods may be promising and, second, an attentive and thorough analysis of such problems in order to provide a correct use of perturbation algorithms. In the present chapter we will mainly concentrate on the optics of charged particle spectrometers. The reason is that the relations inherent in geometrical dimensions of the elements of such spectrometersvery often allow one to introduce some typical small parameters. For example, energy or mass dispersive elements like electrostatic or magnetic sectors or electrostatic cylindrical or spherical mirrors usually have gaps between the electrodesor pole faces that are small compared to their curvature radii. Typically lengths of the particle trajectories inside these elements as well as inside quadrupole lenses or multipoles are considerably larger than the sizes of the fringing fields. In some types of electrostatic lenses often used in electron and ion spectrometers, like so-calledtransaxial and conical lenses, the electrostatic field is concentrated in the regions whose sizes are small as compared with the curvature radii of the electodes. Generally, all these elements may be called optical elements with narrow gaps, though explicit mathematical expressions for the corresponding small parameters may be different. In spite of the similar nature of all these small parameters, they allow the perturbation methods to be applied to very different kinds of problems in charged particle optics. The three main applications, which will be considered in the present chapter, are calculation of effects of weak distortions of the electrode and magnetic pole surfaces; investigation of a transformation of charged particle beams in the fringing fields and short transition regions between adjacent elements; and, finally, calculation of electromagneticfields and electron optical properties of the elements with narrow gaps. The state of the development of the corresponding problems and possible directions of the use of perturbation methods to their solution are discussed in Section 11, which contains also some general rules for a correct application of perturbation expansions. The subsequent sections are devoted to specific problems in charged particle optics solved by perturbation techniques. Section I11 presents a way of analytical calculation of the electromagnetic field disturbance and its influence on charged particle beam distortions in sector field analyzers and Wien filters. A method used in this section is based on the idea of the substitution of the Taylor expansion for the field potential written on the optic axis to the boundary conditions defined at distorted electrode or magnet pole surfaces. Section IV describes an extension of the fringing field integral method for sector fields and quadrupole lenses to the case of closely packed arrays of such elements where the fringing fields of the adjacent elements overlap. Section V demonstrates an original and elegant analytical method for calculation and synthesis of the electrostaticfield distribution

282

M.I. YAVOR

1

Systems with narrow gaps: (air gap)/(curvaiure radiw) (‘(4)= MI+ h+ t2(4I9

(12) (13)

where i j and f describethe trajectoriesin the nondisturbed field of the ideal toroidal condenser, 41 and f~ are small displacements of the beam as a whole in the radial and vertical directions, and 4 2 and f 2 describe other trajectory distortions. In the


301

first aberration order the corresponding trajectory equations read

+ Afj = C , f ” + BS = 0,

(14)

~ =I bo,

(16)

= co,

(17)

fj”

4;’ + A

l; + BtI

(15)

+ = blfj + b2fj’+ b3f + b4c’ + b 5 +~ bby, t; + B f 2 = clfj + + cgf + c4f’ + c5a + c6y.

6;’

A42

c2fj’

(18) (19)

Here the parameter y = ( m / q - mo/qO)/(mo/qo)denotes a relative deviation of the mass-to-charge ratio for an arbitrary particle with respect to the reference particle possessing a nominal mass. This parameter evidently does not contribute to the equations for charged particle trajectories in the toroidal analyzer (i.e., b6 = C6 = 0); it is introduced in 9 s . (18) and (19) to unify the forms of these equations with the forms of equations in cases of a sector magnet and a Wien filter, considered in Sections 1II.B and 1II.C. The coefficients in Eqs. (14)-( 19) are A

=3+h20,

B=h02,

C=U,

bo = -2xOO - XI07 co = bl = - ( l O h 2 0 h30 IS>% - 2 h 2 0 x 0 0 - 12x00- 6x10- x 2 0 , b2 = 24; x&, b3 = -(%2 h 1 2 1 f 1 - 2x01- X I I , 64 = -2fi, -KO19

+

b5 = (hzo CI

=-

c2

= 2i;,

+ +

+

+ 6)41 + 4x00+ XIO, +

~ o 2 h12)l1

- 4x01- x11.

+ + c5 = h 0 2 h + XOl,

= -(4ho2 h i d 4 1 - 2ho2XOO - X02, c4 = 24; XLO, C3

In these coefficientsretained are only the terms linear with respect to the parameters characterizing the field disturbances. Since Eqs. (14) and (15) can be easily solved analytically, the solutions of Eqs. (16)-( 19), which describe beam distortions, can be expressed in a form of abrerration integrals. For some particular types of manufacturing defects, as, for example, electrode shifts or ellipticity, these integralscan be evaluated analytically. The beam distortions are represented by Eqs. (16H19) in the first aberration order. However, the equations for higher-order parasitic aberrations caused by manufacturing imperfections can be also obtained using the procedure just described.

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M. I. YAVOR

B. Magnetostatic Field and Charged Particle Trajectories in an Imperfectly Manufactured Sector Magnet

In this section we consider a general case of an inhomogeneous sector magnet, whose pole surfaces are rotationally symmetric with respect to the z-axis of a cylindricalcoordinateframe {r,4, z) and also symmetricwith respect to the median plane z = 0 (see Fig. 6). Generally an equation describing the pole surfaces of such a magnet is

where the subscript j = 1 corresponds to a lower pole, j = 2 to the upper one, ro is a main beam path radius inside the magnet, and b, is half a gap between the pole faces, measured at the radial distance r = ro from the sector symmetry axis z. The function Q can be represented as a power series

Here the coefficients pm are defined by the shapes of the pole surfaces. For a homogeneous magent P m = 0 for all m z 1 ;in the case of a conical magnet with half a cone opening angle (Y (see Fig. 6) pm = 0 form z 2 and p1 = (rotan a)/b,; finally, for a toroidal magnet with a radius of the pole face curvature R one has p2 = r;/(2bmR),p3 = 0, p4 = r$/(8bmR3),and so on. Note that we assume all the coefficients pm not to be large; in particular, this means that for a toroidal magnet a condition R >> ro should hold. In the presence of inaccuracies in machining or assembling the magnet poles, Eq.(20) transforms to

where the functions 6qj describe small smooth distortions of the magnet pole surfaces. As in Section IILA, we proceed now to the dimensionless coordinates q = (r ro)/ro and = z/ro and introduce a small dimensionless parameter E

= bm/ro.

Then Eq. (22) takes the following form:

< = (-l)’e[g(~) + Sqj(q9 411.

(23)

As examples of manufacturing defects, we can consider the following two simple cases:


303

1. A variation of the air gap in the magnet. Consider the lower and upper pole faces to be displaced in the vertical direction so that the vertical distances between their real and nominal positions are A j (4);then Sqj (4) = (-l)jAj(4)/bm; that is, the functions q j represent the ratio of the magnet pole face displacements to half of the air gap (note the opposite signs adopted for displacements in the same direction of the lower and upper poles). 2. A mutual inclination of the magnet pole faces in the radial direction. Consider the pole faces of a homogeneous magnet to be manufactured inaccurrately so that they are not parallel to the median plane z = 0 but slightly inclined as in case of a conical magnet; let O j (4) = (-l)jc(t36qj/t3q)(4=0 be small angles in the radial direction between the pole surfaces and the median plane, measured at the radial distance r = ro. Then the derivatives of the functions Sqj with respect to q represent the ratio of the angles of the inclinations of the magnet pole surfaces in the radial direction to the small parameter e . Note that these angles are thus assumed to be small enough as compared with 6 . Note also that the opposite signs are adopted for inclinations in the same direction of the lower and upper poles. Similar to what was done previously in Section III.A, we describe the magnetic field in the inhomogeneous magnet by its scalar magnetic potential w; we represent the latter in the form of an expansion in the vicinity of the main beam path: M

.

The coefficients A i k can be represented as the sums A i k ( 4 ) = aik 4- ( Y i k ( 4 ) where the constants aik are the expansion coefficientsfor the nondistorted magnet and the terms cxik ( 4 )describe a small deviation of the scalar magnetic potential caused by pole face distortions. To obtain analytical formulas for a i k and (Yik, we represent them as asymptotic power serii 00

00

n=O

n=O

and substitute Eq. (24) together with expansions of Eq.(25) into a Laplace equation L(w)= 0 for the scalar magnetic potential, where the operator L is expressed by Eq. (7), as well as into the boundary conditions at the pole faces (we assume here an infinite magnetic permeability of the poles, so that the scalar potential values at the pole surfaces are constants fWO):

~ [ v (-1)jc{q(~) , + Sqj(q9

41= ( - 1 ) j ~ o .

(26)

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M.I. YAVOR

As a result of such a substitution, we obtain a set of algebraic equations for coefficients a::) and a!:’. The procedure of deriving these equations, which is similar to the procedure used in the case of an electrostatic sector field, is described in detail in [5] and is not reproduced here. From the corresponding equations, analytical formulas for the coefficients with arbitrary numbers i , k, and n can be extracted. We list next the coefficients a i k for i k 5 3 and a i k for i k 5 2, assuming the magnetic scalar potential WO to be chosen so that the condition aol = I is satisfied:

+

+

= a10 = a12 = a20 = a02 = a30 = 0,

+ E 2 [ . .I, + PI + 2P2 + e2[’

a21 = 2(P: - P 2 ) a03

= -2p: €

am(@>= --(Sq2

2

€

a10(4J)= p

q

*

*

* I 7

- 6%) +€3[*-1,

+ P l ( S q 2 - 6ql)l+ 63[*-1, - h),, + PI(Sq2 - w,

2 - Sqd,

E

azo(@) = $-(&I2

- 2 ( P ? -P2) ( 8 q 2 - 6 q l ) l + ~ 3 [ . * * 1 . 1 UOl(@) = --(h +Sq2) + € 2[...l, 2 1 all(@) = Pl(6ql 6q2) - -(6q1 8 q 2 ) , E 2 [ * *I, 2

+

E

402(@) = p

% 2

- Sql)dd + (&I2

+

+ - b),, + (1 - PI) (Sqz - Sql),

+ (2P: - p2 - P I ) (6q2 - h ) l + E 3 [ . . .I. Here the subscripts r] and 4 denote the derivatives with respect to the corresponding variables, calculated at = 0. In the coefficients aik retained are only the terms linear with respect to the functions S q j ; these functions are also assumed to be calculated at r] = 0. The formulas obtained for the coefficients a i k of the magnetic field distribution in an ideal inhomogeneous magnet generalize to the case of a toroidal magnet earlier results reported in [25] for a conical magnet. Most interesting, however, are the analytical expressionsfor the coefficients ajk which characterize a magnetic field disturbance. Suppose that, similarly to the case of a toroidal electrostatic analyzer, the nominal kinetic energy KOand the nominal mass rno of the charged particles in the beam


305

is such that a particle possessing this energy and mass moves along a circular path of the radius ro inside the ideal magnet (where a magnetic flux density is chosen so that the coefficient aol = 1). Then the trajectory equations in the inhomogeneous sector magnet take the following form:

where u and y are relative deviations of the energy-to-charge and mass-to-charge ratios for an arbitrary charged particle with respect to a reference particle possessing the nominal kinetic energy and mass. A substitution of the field distribution in the imperfectly manufactured magnet into Eqs. (27) and (28) leads to the equations that describe beam distortions caused by inaccuracies of the pole surface shapes. If we represent the particle coordinates as sums of Eqs. (12) and (13), then in the first aberration order the resulting equations for the terms of these sums take the form of Eqs. (14)-( 19), where the coefficients are

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M.I. YAVOR

c3 = (2all Al c4 = 91,

+ azdii1 + a l l ,

l A 1 c5 = c(j = --allao (x I x h o

+ (x I c-JT)cr + (x I Y)Y

+ (x I aa).,” + (x I acr)aocr +

Y = (Y

* * *

,

I BIB0 + (Y I Y>Yo + (Y I aB)aoSo + -

* *

(38) (39)

with respect to the powers of the initial parameters ao, xo, SO,yo, cr, and y. A position of a final x-image is characterizedby the condition (x I a) = 0. In doublefocusing mass spectrometers the condition (x I 6) = 0 also holds. As a rule, optical schemes of high-performance analyzers are optimized so that some higher-order aberrations are eliminated in the final image plane, for example, (x I aa) = 0. In the presence of machining or assembling inaccuracies in the analyzer, a dependence of the final trajectory coordinates on the initial parameters becomes more complicated: x = AXm

+ [(x I a)+ Axalao + [(x I x) + Axx1x0

+ A x u b + [(x I v>+ AX,IY + AxpBo + AXyYO + [(x I aa)+ Axaalai + [(x I + A - ~ ~ ~ l a o ~ + [(x

ID)

(YO)

+ AxapaoBo + Y = AYm

* * *

,

+ [(YI B ) + A Y ~ I B +o [(Y I Y) + A Y ~ I Y+O

(40) * *

(41)

where the coefficients Ax and Ay with various subscripts are small values depending on the type of defects. In an image plane each of these coefficients correspond to a certain type of parasitic aberrations and beam distortions. The terms AXm and Aymare responsible for a lateral shift of a main beam path and the image as a whole. The term Axa is responsible for displacement of a Gaussian image plane along the optical axis and, thus, for an image defocusing at the ideal position of the exit analyzer slit.

3 12

M. I. YAVOR

To the same effect leads a displacement along the optical axis of an energy focus (achromatic plane) in double-focusing mass analyzers in case of a nonvanishing coefficient Axo. The coefficient Axx is responsible for a change of a linear magnification of the image and the coefficient Ax,, for a change of the mass dispersion in a mass analyzer. The aberrationsjust listed are caused by defects that do not violate a symmetry of the system with respect to its median plane. If this symmetry is violated as a result of some manufacturing defects, two more parasitic aberrations appear, which are characterized by the coefficients Ax, and A x y . Consider these aberrations in more detail. Assume for the moment that a charged particle analyzer is triple focusing [ ( x I a)= ( x I 6) = (y I /?) = 01 and its entrance slit is infinitely narrow. Then particles leaving one point of this slit are focused to the first order in a point with the zero x-coordinate in the Gaussian image plane, whereas the image of all the slit is a vertical line (whose height Hi = (y I y)&, where HOis the entrance slit height). If only the parasitic aberration An, is nonvanishing in Eq. (40), then charged particles leaving one point of the entrance slit will still be focused in one point, but the x-coordinate of this point will depend on the initial y-coordinate of the trajectory. This means that the image of the whole entrance slit is still a thin line; it is, however, no longer vertical but inclined by some angle with respect to the y-axis. If, as it is most often the case, the parasitic aberration AxB is also nonvanishing, the particles leaving one point of the entrance slit under different vertical angles form a linear image parallel to the x-axis; the length of the line is proportional to the coefficient Ax, and to the vertical angular spread in the beam. Thus, the image of the whole entrance slit is defocused and has a parallelogram-like shape. In case the analyzer is not focusing in the y-direction, the shape of the image in the situation just considered generally remains (in the approximation of the first aberration order) a parallelogram, whose upper and lower sides are, however, not parallel to the x-axis. An example of the parallelogram-like defocusing of the image, simulated by a computer program ISIOS [ 1071, is shown in Fig. 8. This type of defocusing can be also observed experimentally 11081. The described effects exhaust the types of the first-order parasitic beam distortions in the x-direction, observed in the image plane. Thus, these types are 1. lateral image shift, 2. shift of the Gaussian image plane along the optical axis and the corresponding image defocusing in the nominal Gaussian plane, 3. change of the linear magnification of the image, 4. change of the energy dispersion or a longitudinal shift of the energy focus (achromatic plane) and a corresponding image defocusing in the Gaussian plane of a double-focusingmass analyzer, 5 . change of the mass dispersion,


313

RGURE 8. Computer simulation of images, produced by the mass analyzer of [108], which are created by the ions of three masses 999.96, 1o00, and 1ooO.04 a m u emerging from the 0.008-mm-wide and 4-mm-high entrance slit with a 1.6" horizontal angular spread and a 0.4" vertical angular spread: in a perfectly manufactured analyzer (a), in the case of an imperfectly manufactured inner electrode of a cylindrical electrostatic deflector, which has a conical shape with a cone opening angle of 0.5 mrad (b), and in the case where the effect of this imperfection is reduced by an excitation of a rotated quadrupole corrector that eliminates the parasitic aberration Axa (c).

6. tilt of the image, 7. parallelogram-like image defocusing. First-order parasitic aberrations in the vertical direction y, except the vertical shift of the main beam path and the image as a whole mentioned above, are not of a practical interest for charged particle analyzers because they do not influence their energy or mass resolution and can only slightly decrease the transmission of the analyzer. Note that the quality of the image in charged particle analyzersis deterioratednot only by the first-order parasitic effects but also because of higher-order parasitic beam distortions, such as, for example, a parasitic second-order angular aberration Axaa (which can be important in the systems where the angular aberration (x I aa) is eliminated) or a mixed angular aberration AX,^, which appears if the symmetry of a system with respect to its median plane is violated. The higher-order parasitic aberrations can be estimated based on the same approach that was used in Sections 1II.A-1II.C for deducing equations which describe first-order parasitic effects. Actually, the second-order parasitic beam distortions in practice determine tolerances for the optical elements of charged particle analyzers, since the first-order parasitic aberrations can be more or less successfully compensated for by correcting elements. Consider now what types of the electrode or magnet pole distortions are responsible for different kinds of parasitic beam distortions. A shift of the optical axis in the x-direction is described by Eq. (16). In an electrostatic sector analyzer this shift is caused by nonvanishing values of the coefficients x~ and XIO of the electrostatic field expansion, that is, by changes of the field potential and strength on the ideal circular main beam path. The formulas

3 14

M.I. YAVOR

obtained for these coefficients in Section 1II.A show that such changes are mainly contributed by the variation of the interelectrode gap, characterized by the difference ( S f 2 - Sfl), and the variation of the position of the middle of the gap relative to the ideal circular optical axis, characterized by the sum (Sf:! Sfi). The latter variation, which is the contribution to the beam axis shift of the pure potential change (i.e., of the variation of the beam kinetic energy), is considerably smaller than the former variation, which is the contributionof the pure field strength change (i.e., of the variation of the beam deflecting force), for comparable values of the distortions Sfj: indeed, in the formulas for the coefficients xw and x l o the sum (Sf2 Sfi) is multiplied by the small parameter E . In a sector magnet the shift of the optical axis in the x-direction is caused by a nonvanishing value of the coefficient sol, that is, by a change of the magnetic flux density on the ideal optical axis. This coefficient (see Section 1II.B) is proportional to the variation (Sql 6q2) of the air gap between the pole surfaces. In a Wien filter the shift of the optical axis in the x-direction, as follows from the results obtained in Section III.C, is caused by the summary action of all the listed factors. A shift of the optical axis in the y-direction is described by Eq. (17). In an electrostatic sector analyzer this deflection is caused by a nonvanishing value of the coefficient x o l , that is, by a sum ~ ( S f l Sf.)(. characterizing the summary inclination of the electrodes with respect to the analyzer median plane. In a sector magnet this shift is determined by the coefficient ( ~ 1 0 . This coefficient (see Section 1II.B) is proportional to the summary inclination of the magnet poles with respect to the median plane; in case of a conical magnet the value a10 is contributed also by a deviation of the middle of the air gap from this plane. In a Wien filter the shift under consideration is caused by the action of all the listed factors. Anyway, a sensitivity of the vertical shift of the optical axis to the electrode and magnet pole face distortions is smaller than the sensitivity of the horizontal shift of the optical axis, since both coefficients xol and a10are proportional to the small parameter E . The effects of the image defocusing in the x-direction are described by Eq. (18). In particular, a longitudinal shift along the optical axis of a Gaussian image plane and of the achromatic plane in the double-focusing mass spectrometer is determined by the coefficients bl ,b2 and b5 in this equation. In an electrostatic sector analyzer (considered by itself or as a stage of a mass analyzer) these effects are caused by the same factors as the shift of the optical axis in the x-direction and depend on the coefficient x20 which, in turn, is dependent on the sums (Sf1 Sf2)cc and (Sfl+ Sf2)## characterizing the deviation of the electrode curvatures from their nominal values. In a sector magnet the effects in question, apart from the factors which cause the horizontal shift of the optical axis, are determined also by the coefficient all which, in turn, is dependent on the sum (Sgl 8 q ~ )that ~ , is, on the rate of the mutual inclination of the homogeneous magnet pole faces (or

+

+

+

+

+

+


3 15

on the inaccuracy of the mutual inclination of the conical magnet pole faces). In a Wien filter parasitic effects under consideration are influenced by all the listed factors. Finally, a defocusing of the x-image caused by a violation of the symmetry of the system’s geometry with respect to the median plane is determined by the coefficients b3 and b4 of Eq. (18). In an electrostatic sector analyzer this defocusing is influenced by the same factor as a vertical shift of the optical axis and by a nonvanishing value of the coefficient ~ 1 1 . This coefficient, in turn, depends on the rate of a mutual inclination of the electrodes in the vertical direction, characterized by the expression (Sf2 - Sfl)(. In a sector magnet the parallelogram-like defocusing is also caused by the same factor as a vertical shift of the optical axis and by a nonvanishing value of the coefficient 4 0 2 and a longitudinal variation of the coefficient 400,that is, as shown by the analysis of the corresponding formulas given in Section III.B, by parasitic pole face curvatures and inclinations of the pole faces with respect to the median plane. Note, however, that both these coefficients are proportional to the small parameter E . Thus, the parallelogramlike defocusing in a sector magnet is considerably smaller than in an electrostatic sector analyzer for comparable magnitudes of the electrode and magnet pole face distortions. In a Wien filter the defocusing in question is caused by all the listed factors. The conclusions made in the previous paragraph are important for a practical design of charged particle analyzers based on sector fields. They mean that a most significant source of the parallelogram-like defocusing (which is most difficult to correct) in sector field mass analyzers is an electrostatic sector analyzer. Calculations show (see [109]) that a typical mutual inclination of the electodes of a cylindrical sector analyzer of 1 mrad (such an inclination can be caused not only by inaccurate assembling of the electrodes but also by occasional wavelike distortions of the electrodes in the vertical direction) can considerably decrease the resolving power of a double-focusing mass spectrometer with compensated second-order aberrations. Moreover, such an inclination creates not only large first-order parasitic beam distortions but also considerable second-order parasitic aberrations like Ax,p, which prevent restoring a nominal mass resolving power of the analyzer by aberration correctors. Thus, much attention should be given to an accurate machining and assembling of the electrostatic sector condenser electrodes. In some cases, where a precise manufacturing is difficult (in particular, in mass separators with large electrode sizes), it can be advantageous to avoid using not only toroidal electrodes that are difficult to assemble properly but electrostatic sector analyzers as well, replacing them by achromatic magnetic systems whose stages are floated at different electrostatic potentials [ 1101. Consider possibilities of compensation for first-order parasitic effects. A shift of the optical axis in the x-direction can be compensated for by a change of the

316

M.I. YAVOR

electrostatic field strength in a sector condenser and Wien filter or of the magnetic flux density in a sector magnet and Wien filter. Since this compensation provides only a spatial correction in a given profile plane but not an angular compensation, in multistage systems it is advantageous to place additional deflecting elements along the optical axis. A vertical shift of the optical axis cannot be compensated for by a change of the excitations of sector field elements or a Wien filter. In this case different ways of correction are possible. In a sector magnet an efficient means of correcting a vertical beam deflection can be a small inclination of the magnet as a whole in the radial direction. In an electrostatic analyzer or Wien filter supplied by Matsuda plates, compensation may be achieved by applying additional electrostatic potentials of the opposite signs to the upper and lower Matsuda plates. Similar to the case of the horizontal deflection, it is advantageous to place additional deflecting elements in the system. A longitudinal shift of the image plane can also be easily compensated for in different ways: by changing the lens excitations, if any lenses are present in the optical scheme, by applying voltages to the Matsuda plates in electrostatic sector analyzers or Wien filters, or just by mechanical shift of the entrance or exit slits. Similarly, one can correct the position of the achromatic plane in a double-focusingmass spectrometer;however, to restore nominal positions of both shifted Gaussian image plane and achromatic plane require combination of these methods. Most difficultis to compensatefor a parallelogram-likeimage defocusing caused by a violation of the system's symmetry with respect to the median plane. Obviously such compensation requires that additional correcting elements be present in a system that can induce fields not symmetrical to the median plane. A most efficient element of this type is a quadrupole corrector, which is a quadrupole lens (electrostatic or magnetic) rotated about its axis to 45". Note that though rotated quadrupoles have been routinely used as stigmators in electron microscopes for many decades, in mass spectrometersthey are still not used commonly. Instead of a rotated quadrupole corrector one can employ electrostatic hexapole or octopole elements in the system to which additional voltages are applied in a special manner [ 1111. One can also apply voltages to electrically isolated poles of magnetic quadrupole lenses [ 1121. Nevertheless, the correction of the parallelogram-likedefocusing is rather complicated. The problem is that there are two first-order parasitic aberrations, Axp due to the vertical aperture angle and Ax,, due to the vertical size of the entrance slit, in a system whose symmetry with respect to the median plane is violated. One rotated quadrupole corrector can eliminate only one of these aberrations; both aberrations can be compensated for only by two rotated quadrupoles. Sometimes it is possible to use one corrector and additionally tilt the exit slit for the full correction (see Fig. 8) but this way is not suitable for multicollection devices.

CHARGED PARTICLE OpIlCS OF SYSTEMS WITH NARROW GAPS

317

The effectiveness of the correction is rather sensitive to the position of the rotated quadrupole corrector with respect to the imperfect stage of the system because an improperly located corrector may increase one of the parasitic aberrations Axp or Axy while correcting another [ 1071. Besides Matsuda plates, whose action is consideredin detail in Section V, we can mention two more special correctors of parasitic beam distortions. A widely used device for tuning magnetic mass spectrometers and especially mass separators is the correction coils [ 1131. The coils are mainly produced as printed circuit boards [114, 1151 though they can be also made of explicit wires or produced by cutting sheets of copper with thicknesses of several millimeters. Correction coils are very effective in compensating for a shift of the image plane (nl-coils) and the correction of the second-order angular aberration (nz-coils), which can become nonvanishing in the Gaussian image plane after its shift [1161. In principle n 1 -coils can also compensate for a shift of the energy focus plane; however, it is difficult to achieve coincidence of the Gaussian image and energy focus planes by using only one correcting element, since it shifts both these planes. Thus, at least two correctors are required. The lateral displacementof the beam and longitudinal shifts of the image and energy focus planes can be achieved by conventional quadrupolecorrectors. Though the quadrupole lenses used for the beam transport and focusing can be in principle applied for these purposes, it is adviseable to include in the system more flexible adjustable multipole elements [117]. At least one such element should be placed before the dispersive elements to allow the image plane position to be shifted independently of the energy focus plane. It should be noted here that such multipoles can be also used for correction of the parallelogram-like image defocusing, since they can provide quadrupole field components rotated by 45" with respect to the field of conventional quadrupole lenses. An example of the use of adjustable multipole correctors can be a set of three such elements implemented in the ISOLDE-3 mass separator [ 1181, which allows considerable improvement in its mass resolving power. We emphasize again that all the listed methods and elements are efficient for correction of the first-order parasitic effects. It should be noted that in case of large enough manufacturing inaccuracies such a correction can lead to an increase of second-order parasitic effects. For example, a correction of a longitudinal shift of the Gaussian image plane in analyzers with eliminated second-order angular aberration inevitably leads to a violation of the condition of elimination, which prevents achievement of a nominal performance of the analyzer. Secondorder parasitic aberrations present in imperfectly manufactured systems or induced by correction elements restrict a possibility of compensation for parasitic beam distortions and determine finally technological machining and assembling tolerances.

318

M. I. YAVOR

Iv.

TFtANSFORMATION OF CHARGED PARTICLE 'rkAJECTORIES IN THE NARROW TRANSITION REGIONS BETWEEN ELECTRONAND ION-OFTICAL ELEMENTS

This section presents an extension of a fringing field integral method to a situation where effects of overlapped fringing fields between two adjacent electron-optical elements, placed close by each other, are considered. However, a technique of scaling with explicit separating out a small parameter, proposed in this section, is a useful tool which provides a correct and efficient application of a fringing field integral method to various problems, including investigation of effects of single fringing fields. A. Charged Particle Beam Transport through the Gaps of Multiple Magnetic Prisms

In this subsection we study a transformation of charged particle trajectories in a narrow transition region between two homogeneousdeflecting magnets, characterized by different magnetic flux densities in their air gaps. Such a region is formed, for example, in multiple magnetic prisms used for separation of illuminating and reflected electron beams as shown in Fig. 2. First of all, we will obtain a power series expansion for the magnetic field in the transition region based on the magnetic field distribution in the direction normal to the magnet boundaries; a magnetic field is characterized by its scalar magnetic potential W and we suppose the magnetic permeability of the poles to be infinite. We consider a general case where two homogeneous magnets with the same median plane Y = O are separated by a gap whose width G is comparable with air gaps 2Kl and 2K2 between the poles of these magnets. Generally a gap between the magnets may be circular with the radius of curvature R and inclined by an angle h with respect to the beam optical axis T (see Fig. 9). To be precise, R is the radius of some circle arc whose center coincides with the centers of the magnet boundary curvaturesand which intersects the beam optical axis at the same point as a so-called effective boundary between the magnets; the concept of the effective boundary will be specified later. The curvature of the gap is assumed to be positive if the second magnet boundary is concave. We introduce a Cartesian coordinate frame (X, Y, Z) whose origin C lies at the point of the intersection of the beam optical axis with the effective boundary between the magnets. The Z-axis is tangent to the beam optical axis; the XY-plane is the effective boundary plane. A condition specifying the position of the point C will is given later by E q . (59). We also introduce another Cartesian coordinate frame Y, 2) with the same origin C, which is tilted with respect to the coordinate frame ( X ,Y, Z) by the angle h about the Y-axis.

{x,


319

FIGURE9. A particle trajectory T passing through a narrow gap of a width G and a curvature radius R between two magnets. Shown are coordinate systems used in calculation.

We make now a first important step: namely, we proceed to dimensionless coordinates x = X / K 2 , y = Y / K 2 , z = Z / K z , t = x / K z , = z / K 2 and introduce a dimensionless cylindrical coordinate p = (r - R ) / K z (the axis of the cylindrical coordinate frame passes through the center of the curvatures of the magnet boundaries). Thus, in the new coordinates the air gap of the second magnet is formed by the surfaces described by equations y = f 1, and the dimensionless width G / K2 of the transition region between the magnets in the new coordinate frames is comparable to 1. A scalarmagnetic potential W in the gap between the magnets can be represented in a form of an expansion

w ; ,

where prime denotes a derivative with respect to p.

(43)

320

M. I. YAVOR

If we assume the distribution of the flux density B ( p ) = -K;’aW/ayl,,o in the median plane Y = O to be known from calculations or measurements, then all the coefficients wi can be evaluated from Eq. (43). Let us suppose wo(p) = W(0, p ) = 0. Then w2 = w4 = 0, w3 = K 2 [ B ” ( p ) ( p - p2p .)B’ ( p ) ] ,205 = Kz[-B””(p) - 2pB”’(p) . .]. With small 6 we obtain the following relations between the coordinates p , 6 , and 2‘:

+.

+

+

a

Using these relations, we can finally represent the magnetic field distribution in the vicinity of the 1,O) = V for the middle electrode. For the two-electrode lens we assume that one of the electrodes has the zero potential and the other one, with half the slit width being b, has the potential V; we suppose that the {-coordinate of the latter electrode is zero. Then the boundary conditions at the electrodes are @(')( 16 I > (2, (2) = 0 and @(')( It I > 1,O) = V. In both cases the function @ ( I ) satisfies the zero boundary conditions at the electrodes. The solution of Eq. (141) can be obtained with the aid of the Schwarz-Christoffel transformation that gives a conformal mapping of the plane z = { it to the upper half-plane of the variable w = u iu. This conformal mapping can be found in [119, 1201. Having obtained this solution, the solution of Eq. (142) is represented in the following form:

+

+

where the functions $1 and +2 satisfy the equations

Besides, both these functions satisfy the zero boundary conditions at the electrodes and are to be finite at the infinity. The function +I that satisfies Eq.(156) and the boundary conditionsjust listed is

where x = 0 for a three-electrode einzel lens. For a two-electrode immersion lens this function is x ( t , { )= u ( t , {) with u being the imaginary part u = 3 w ( z ) of the Schwarz-Christoffeltransformation mentioned previously. The function $2 can be represented as

A substitution of Eq. (159) into Eq. (157) shows that the function h ( t , {) satisfies the two-dimensional Laplace equation and the following boundary conditions at

314

M. I. YAVOR

the electrodes:

h(( > ( j ,

Tj)

= H j l h ( t c - t j , < j > = -Hj$

(160)

with j = 0, 1 , 2 for a three-electrode lens, j = 0 , 2 for a two-electrode lens, and the constant parameters H , being

The constants H j for some electrode geometries are tabulated in [ 1041. The function h ( 6 , C) is easily calculated with the aid of the Schwarz-Christoffel transformation discussed earlier. Note that this function is antisymmetric with respect to the (-coordinate and that

To estimate the accuracy of the perturbation method, we give a comparison of the results of a calculation performed by the proposed analytical formulas and by a high-precision numerical integration. Figure 26 shows the distributions of the potential q and the field strength component En normal to the lens optical axis, along this axis in an einzel three-electrode conical slit lens with f3= 30” and a typical value of the small parameter E =0.25. Calculations were performed for the lens geometry where all the slit gaps were the same and the distances between the central and outer electrodes equal b. The constants H , in this case are H1 = H2 = -0.097, HO = 0.244. It is seen that the accuracy of the potential calculated by the perturbation method inside the lens is about 2% and the accuracy of the calculation of En is about 15%; this accuracy becomes somewhat lower at the tails of the field. G. Elimination of a Beam Deflection in Conical Lenses The electrostatic field component En considered in Sections V1.E and V1.F causes a deflection of the charged particle beam from the lens optical axis. This effect is due to a “shift” of the potential distribution in a conical lens in the direction of the axis of rotational symmetry, as compared with the potential distribution in a two-dimensional reference lens. Such a deflection is usually small but still can achieve several degrees in case of strong lenses. Thus, in case a variation of the lens excitation is required, it is desirable to reduce the field strength component normal to the optical axis. Consideralens with longitudinalelectrodes. A field strength component En(6, C) = -aQ((,

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