Advances in Electronics and Electron Phisics. Vol. 75

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 75

EDITOR-IN-CHIEF

PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche ScientiJique Toulouse, France

ASSOCIATE EDITOR

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

Advances in

Electronics and Electron Physics EDITED BY

PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 75

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto

0

COPYRIGHT1989 BY ACADEMIC PRESS,I NC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED O R TRANSMITTED IN ANY FORM O R BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMlSSiON IN WRITING FROM T H E PUBLISHER.

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LIBRARY OF CONGRESS CATALOG CARDNUMBER:49-7504 ISBN 0-12-014675-4 PRINTED IN THE UNITED STATES OF AMERICA

89 90 91 92

9 8 1 6 5 4 3 2 1

CONTENTS

CONTRIBUTORS TO VOLUME 75 . . . . . . . . . . . . . . . . . . . . . . PREFACE.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii ix

Linear Inverse and Ill-Posed Problems M . BERTERO I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . V . Regularization Theory for Ill-Posed Problems . . . . . . . . . . . VI . Inverse Problems and Information Theory . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 10 36 55 67 96 114

Recent Developments in Energy-Loss Spectroscopy JORGFINK I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Principal Features . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . V . Nearly-Free-Electron Metals . . . . . . . . . . . . . . . . . . . . VI. Rare Gas Bubbles in Metals . . . . . . . . . . . . . . . . . . . . VII . Amorphous Carbon . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Conducting Polymers . . . . . . . . . . . . . . . . . . . . . . . . IX . Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 124 148 157 160 167 181 187 215 226 226

I1. Linear Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . .

111. Linear Inverse Problems with Discrete Data

I. I1.

111.

IV . V.

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

Methods of Calculating the Properties of Electron Lenses E . HAHN Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . Differential Equation of Trajectories . . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . Coupling Between Field and Basis . . . . . . . . . . . . . . . . . V

233 235 237 238 246

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CONTENTS

VI . Representation of the Trajectory . . . . . . . . . . . . . . . . . . 250 VII . Iteration Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 VIII . Electron Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 IX . Conventional Lenses . . . . . . . . . . . . . . . . . . . . . . . . . 262 X . Theory of Micro-Lenses . . . . . . . . . . . . . . . . . . . . . . . 269 XI . Quadrupole Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 294 XI1. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

I. I1. 111.

IV. V. VI .

Derivation of a Focusing Criterion by a System-Theoretic Approach MICHAEL KAISER List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System-Theoretic Approach to Scalar Radiation Problems in Homogeneous Space . . . . . . . . . . . . . . . . . . . . . . . . . . . Focusing by Plane Radiators . . . . . . . . . . . . . . . . . . . . . Focusing in Stratified Media . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329 333 335 348 363 382 383 387

Lightwave Receivers GARETHF . WILLIAMS I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389

INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

461

I1. Receiver and Device Requirements of Lightwave Systems . . . . 393 111. Receiver System and Noise Considerations . . . . . . . . . . . . . 395 IV . First- and Second-Generation Lightwave Receivers . . . . . . . . 422 V. Active-Feedback Lightwave Receiver Circuits . . . . . . . . . . . 431 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

CONTRIBUTORS The numbers in parentheses indicate the pages on which the authors’ contributions begin.

M. BERTERO (l), Dipartimento di Fisica dell’ Universita and lstituto Nazionale di Fisica Nucleare, Via Dodecaneso 33, I-16146 Genova, Italy JORG FINK( 1 21), Kernforschungszentrum Karlsruhe, Institut f u r Nukleare Festkorperphysik, P.O.B. 3640, 0-7500 Karlsruhe, Federal Republic of: Germany E. HAHN(233), Pestalozzi-str. 9,6900 Jena, German Democratic Republic MICHAEL KAISER(329), Dornier System GmbH, P.O. Box 1360, 0-7990 Friedrichshafen, Federal Republic of Germany GARETH F. WILLIAMS (389), N YNEX Science and Technology, 500 Westchester Avenue, White Plains, New York 10604, U S A

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PREFACE The five contributions to this volume reflect many of the themes traditionally covered in this series. The opening chapter by M. Bertero is concerned with a topic that is a widespread preoccupation, namely, the search for satisfactory methods of handling inverse problems and especially ill-posed problems. The author is well known for his many original ideas concerning these questions and the present survey is therefore all the more welcome. The next two chapters, by J. Fink and E. Hahn, are of more specialized interest, although the information obtained by the technique of energy-loss spectroscopy, discussed by J. Fink, is used in many fields. The approach to particle optics developed over the years by E. Hahn is not very well known, probably owing to the relative inaccessibility of the Jenaer Jahrbuch, in which much of his work was published. The present account is devoted mainly to the more recent stages of this research, and I hope that this account in English in these Advances will disseminate more widely Dr. Hahn’s work. Systems theory is a well-developed discipline but there are no doubt problems to which it could be usefully applied but where it is little known. M. Kaiser explains how problems of wave propagation can be clarified with its help. We conclude with a device-orientated account by G. F. Williams of new developments in the field of lightwave receivers. This is a subject in rapid evolution, and the present account should be found very helpful by anyone trying to keep up with the changes. In conclusion, I thank all the contributors most warmly for their efforts and list forthcoming articles in the series. Peter W. Hawkes FORTHCOMING ARTICLES

Parallel Image Processing Methodologies Image Processing with Signal-Dependent Noise Pattern Recognition and Line Drawings Bod0 von Borries, Pioneer of Electron Microscopy Electron Microscopy of Very Fast Processes ix

J. K. Aggarwal H. H. Arsenault H. Bley H. von Borries 0. Bostanjoglo

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PREFACE

Signal Analysis in Seismic Studies Magnetic Reconnection Sampling Theory Finite Algebraic Systems and Trellis Codes Electrons in a Periodic Lattice Potential The Artificial Visual System Concept Corrected Lenses for Charged Particles A Gaseous Detector Device for ESEM The Development of Electron Microscopy in Italy The Study of Dynamic Phenomena in Solids Using Field Emission Amorphous Semiconductors Resonators, Detectors and Piezoelectrics Median Filters Bayesian Image Analysis SEM and the Petroleum Industry Emission Electron Optical System Design Statistical Coulomb Interactions in Particle Beams Number Theoretic Transforms Systems Theory and Electromagnetic Waves Phosphor Materials for CRTs Tomography of Solid Surfaces Modified by Fast Ions The Scanning Tunnelling Microscope Scanning Capacitance Microscopy Applications of Speech Recognition Technology Multi-Colour AC Electroluminescent Thin-Film Devices Spin-Polarized SEM HREM and Geology The Rectangular Patch Microstrip Radiator Active-Matrix TFT Liquid Crystal Displays Electronic Tools in Parapsychology

J. F. Boyce and L. R. Murray A. Bratenahl and P. J. Baum J. L. Brown H. J. Chizeck and M. Trott J. M. Churchill and F. E. Holmstrom J. M. Coggins R. L. Dalglish G. D. Danilatos G. Donelli M. Drechsler W. Fuhs J. J. Gagnepain N. C. Gallagher and E. Coyle S. and D. Geman J. Huggett V. P. Il’in G. H. Jansen

G. A. Jullien M. Kaiser K. Kano et al. S . B. Karmohapatro and D. Ghose H. Van Kempen P. J. King H. R. Kirby

H. Kobayashi and S. Tanaka K. Koike M. Mellini H. Matzner and E. Levine S. Morozumi R. L. Morris

xi

PREFACE

Image Formation in STEM Low-Voltage SEM Languages for Vector Computers Electron Scattering and Nuclear Structure Electrostatic Lenses CAD in Electromagnetics Scientific Work of Reinhold Riidenberg Atom-Probe FIM Metaplectic Methods and Image Processing X-Ray Microscopy Applications of Mathematical Morphology Focus-Deflection Systems and Their Applications Electron Gun Optics Thin-Film Cathodoluminescent Phosphors Electron Microscopy and Helmut Ruska

C. Mory and C. Colliex J. Pawley R. H. Perrott G. A. Peterson F. H. Read and I. W. Drummond K. R. Richter and 0. Biro H. G. Rudenberg T. Sakurai W. Schempp G. Schmahl J. Serra T. Soma et a/.

Y.Uchikawa A. M. Wittenberg C. Wolpers

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 75

Linear Inverse and Ill-Posed Problems M . BERTERO Dipartimento di Fisica dell’liniversita and lstituto Nazionale di Fisica Nucleare Genova Italy

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I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I I . Linear Inverse Problems . . . . . . . . . . . . . . . . . . . . . A . General Properties . . . . . . . . . . . . . . . . . . . . . . B. Inverse Source Problems . . . . . . . . . . . . . . . . . . . . C . Inverse Diffraction Problems . . . . . . . . . . . . . . . . . . . D. Linear Inverse Scattering Problems . . . . . . . . . . . . . . . . E. Radon Transform Inversion and Tomography . . . . . . . . . . . . . F . Fourier Transform Inversion with Limited Data . . . . . . . . . . . . G . Laplace Transform Inversion . . . . . . . . . . . . . . . . . . H. Generalized Moment Problems . . . . . . . . . . . . . . . . . . I11 . Linear Inverse Problems with Discrete Data . . . . . . . . . . . . . . A . General Formulation . . . . . . . . . . . . . . . . . . . . . B. Fourier Transform Inversion with Discrete Data . . . . . . . . . . . . C. Interpolation and Numerical Derivation . . . . . . . . . . . . . . . D . Finite HausdorlT Moment Problem . . . . . . . . . . . . . . . . E. Moment-Discretization of Fredholm Integral Equations of the First Kind . . . IV . Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . A . Moore-Penrose Generalized Inverse . . . . . . . . . . . . . . . . B. C-Generalized Inverses . . . . . . . . . . . . . . . . . . . . . C. The Backus-Gilbert Method for Problems with Discrete Data . . . . . . . V . Regularization Theory for Ill-Posed Problems . . . . . . . . . . . . . . A . Ivanov-Phillips-Tikhonov Regularization Method . . . . . . . . . . . B. General Formulation of Regularization Methods . . . . . . . . . . . C. Spectral Windows . . . . . . . . . . . . . . . . . . . . . . D. Iterative Methods . . . . . . . . . . . . . . . . . . . . . . E. Choice of the Regularization Parameter . . . . . . . . . . . . . . . VI . Inverse Problems and Information Theory . . . . . . . . . . . . . . . A . Modulus of Continuity and Uncertainty of the Solution . . . . . . . . . B. Evaluation of Linear Functionals and Resolution Limits . . . . . . . . . C. Number of Degrees of Freedom . . . . . . . . . . . . . . . . . D . Impulse Response Function: Another Approach to Resolution Limits . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

2 10 11 16 18 21 26 28 31 34 36 37 42 44 49 52

55 56 60 64 61 68 80 84 88 92 96 97 101

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Copyright 0 1989 by Academic Press Inc. All nghts of reproduction in any form reserved . ISBN n-12-014675.4

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M. BERTERO

I. INTRODUCTION In a paper published in 1902 on boundary-value problems for partial differential equations and their physical interpretation, Jacques Hadamard introduced the basic concept of a well-posed problem (Hadamard, 1902). In this first formulation, a problem is called well-posed when its solution is unique and exists for arbitrary (non-analytic) boundary values. From an investigation of several examples related to elliptic and hyperbolic equations, Hadamard concluded that only problems that are motivated by physical reality are well-posed. In particular, he demonstrated that the Dirichlet problem of the Laplace equation and the Cauchy problem of the wave equation with boundary values at t = 0 are well-posed, while the Cauchy problem of the Laplace equation and the Cauchy problem of the 3D wave equation with boundary values at x = 0 are not well-posed. In subsequent work Hadamard emphasized the requirement of continuous dependence of the solution on the data (Hadamard, 1923), claiming that a solution that varies considerably for a small variation of the data is not really a solution in the physical sense. Physical data are never known exactly but only with a certain degree of accuracy, and this should imply that the solution is not known at all. He provided a striking example of this fact using the Cauchy problem of the Laplace equation in two variables,

AU

u,,

+ uYy= 0.

(11

If we consider the following Cauchy data at y = 0 u(x, 0) = 0, uy(x,0)= n-' sin(nx), (2) then there exists a unique solution of Eq. (1) satisfying these conditions, given by u(x, y) = K 2sin(nx)sinh(ny). (3) The factor sin(nx)produces a fluting of the surface that represents the solution of the problem. This fluting, however imperceptible near y = 0, becomes enormous at any finite distance from the x-axis, provided the fluting is made sufficiently small by taking n sufficiently large. Notice that when n -,co, the amplitude of the oscillating data tends to zero while the frequency tends to infinity. This is now a classical example illustrating the effects that arise when the dependence of the solution on the data is not continuous. In honour of his contribution, a problem is now called well-posed in the sense of Hadamard if it has the property of continuous dependence of the solution on the data, though the complete formulation, including the three requirements of uniqueness, existence and continuity, was stated only much later, by Courant and Hilbert (1962, p. 227; cf. Hadamard, 1964, p. 28).

LINEAR INVERSE A N D ILL-POSED PROBLEMS

3

These ideas recall remarks from a biographical sketch of Hadamard. “He liked to work with the rigor of a mathematician and the practical sense of a physicist, and liked to repeat Poincart’s words: ‘La Physique ne nous donne pas seulement l’occasion de rksoudre des problemes, ... ,elle nous fait pressentir la solution”’ (Mandelbrot and Schwartz, 1965).The physics of Hadamard was, however, the physics of the nineteenth century. In fact, the requirements of existence, uniqueness and continuity of the solution were “deeply inherent in the ideal of a unique, complete, and stable determination of physical events.. . Laplace’s vision of the possibility of calculating the whole future of the physical world from complete data of the present state is an extreme expression of this attitude” (Courant and Hilbert, 1962, p. 230). The idea of well-posedness has been extremely useful in the theory of partial differential equations and in functional analysis. A negative consequence was that for many years problems that are not well-posed, now called ill-posed or incorrectly posed problems, were considered mathematical anomalies and were not seriously investigated. Recent developments in physics, and especially applied physics, have shown that ill-posed problems may, however, relate to extremely important physical situations. For example, the solution of the Cauchy problem of elliptic equations has interesting applications in electrocardiography (Colli Franzone et al, 1977),specifically in the reconstruction of the epicardial potential from body surface maps. We would like to emphasize, however, another ill-posed problem, one that has revolutionized diagnostic radiology. In 1971, the first clinical machine for the detection of head tumors, based on a new X-ray technique, called computer-assisted or computerized tomography, was installed at the Atkinson Morley’s Hospital, Wimbledon, Great Britain. (In 1979, Allan M. Cormack and Godfrey N. Hounsfield were awarded the Nobel prize in Medicine for the invention of the technique.) In computerized tomography, images of a cross-section of the human body are created from the attenuation of X-rays along a large number of lines through this cross-section. The processing of the data requires the reconstruction of a function of two variables from knowledge of its line integrals. The solution of this mathematical problem was given many years ago in a paper by Johann Radon (1917). The result of Radon was even more general, since he proved formulas for the reconstruction of a function defined on R” from knowledge of its integrals over all the hyperplanes of R“. In honour of his contribution, a mapping that transforms a function into the set of its integrals over hyperplanes is now called the Radon transform. Therefore, tomography is just a special case of Radon transform inversion. Moreover, Radon transform inversion is an elegant example of an ill-posed problem, since the solution does not exist for arbitrary data and the dependence of the solution on the data, in general, is not continuous. As a consequence, the effect of noise on the solution

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M. BERTERO

is amplified in a way similar to what happens in the Cauchy problem of the Laplace equation. Even if Cormack and Hounsfield had not been aware of the work of Radon (Hounsfield has been quoted as saying “I find I’ve got other tools of thinking than math” (Di Chiro and Brooks, 1979)), there is no doubt that mathematics has been the source of important contributions to the development and refinement of computerized tomography. On the other hand, computerized tomography has stimulated the rapid growth of the theory of ill-posed problems in the past twenty years. Radon transform inversion is an example of an inverse problem. A precise mathematical definition of inverse problems without reference to physics is quite difficult and perhaps impossible. For a mathematician, the distinction between direct and inverse problems is quite arbitrary. “We call two problems inverses of one another if the formulation of each involves all or part of the solution of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other has never been studied and is not so well understood. In such cases, the former is called the direct problem, while the latter the inverse problem’’ (Keller, 1976). Another quote from the same paper shows a route for a characterization of inverse problems based on physics rather than on mathematics.” The main sources of inverse problems are science and engineering. Often, these problems concern the determination of the properties of some inaccessible region from observations on the boundary of the region” (Keller, 1976). In fact, inverse problems are related to indirect measurements, remote sensing and so on. For every domain of physics, it is necessary to specify a definition of the direct problem peculiar to this domain and the definition of its corresponding inverse problem. For example, in classical mechanics, the direct problem is the determination of the trajectories of particles from knowledge of forces. Then the inverse problem is the determination of forces from knowledge of the trajectories. (In this sense, Isaac Newton solved the first inverse problem). In potential theory, the direct problem is the determination of the potential generated by a known mass or charge or current distribution, while the inverse problem is the determination of the mass or charge or current distribution from the measured values of the potential. In the theory of wave propagation, the direct problem is the determination of field distributions in time and space from given constitutions of sources or scatterers. Then the inverse problem is the determination of the characteristics of the sources or scatterers from observations of the fields. (Computerized tomography is an inverse problem in this sense, since it consists in the determination of the space distribution of the X-ray absorption coefficient from the observation of the attenuation of the X-rays passing through the probe). In instrumental physics, the direct problem is the computation of the output

LINEAR INVERSE AND ILL-POSED PROBLEMS

5

of a given instrument with known impulse response function from knowledge of the input. Then the inverse problem is the determination of the input from knowledge of the output, and so on. These examples in mind motivate the following definition of direct and inverse problems. Direct problems are problems oriented along a cause-effect sequence or, in other words, problems that consist in providing the consequences of given causes, while inverse problems are those associated with the reversal of the chain of causally related effects, and therefore consist in finding the unknown causes of known consequences (Turchin et al., 1971). This definition will never be misleading if it is kept in mind that the formulation of any specific problem must be based on well-established physical laws and that physics must specify what is a cause and what is an effect as well as provide the equations relating the effects to the causes. Moreover, a merit of this definition is the attempt of justifying the fact that direct problems are, in general, well-posed while the corresponding inverse problems are, in general, ill-posed. Today, inverse problems are fundamental in several domains of applied science: medical diagnostics, atmospheric sounding, radar and sonar target estimation, seismology, radio astronomy, microscopy and so on. There are routine applications not only in X-ray or MR tomography, but also in seismic data processing for geophysical exploration and in radiometric data processing for meterorological forecasts and monitoring. In recent years, several books have been published on inverse problems in various areas of applied physics, including optics (Baltes, 1978; 1980), astronomy (Craig and Brown, 1986), electromagnetics (Boerner et al., 1983), atmospheric sounding (Zuev and Naats, 1983), computerized tomography (Herman, 1979; 1980; Herman and Natterer, 1981; Natterer, 1986a). Miscellaneous examples and mathematical results may be found in Colin, 1972; Sabatier, 1978; 1987a,b; Talenti, 1986; Cannon and Hornung, 1986. Ill-posed problems for partial differential equations have been investigated by Payne (1975) and Carasso and Stone ( 1 975). Inverse problems, in general, are nonlinear. Two nonlinear problems have been the object of elegant mathematical investigations, the inverse Strum-Liouville problem and the inverse scattering problem. The first one is due most likely to Lord Rayleigh (1877). In describing the vibrations of strings of variable density, Lord Rayleigh briefly discussed the possibility of deriving the density distribution from the frequencies of vibration. A modern version and generalizations are given by Kac (1966). Roughly speaking, the mathematical problem involves the determination of the coefficients of a differential operator from knowledge of its eigenvalue spectrum. Important contributions to this problem have been made by outstanding mathematicians, such as Levinson, Marchenko, and Krein, among others. A short survey is given by Barcilon (1986).

M.BERTERO

6

The inverse scattering problem was originally investigated in connection with the Schroedinger equation and later extended to other equations, such as the Helmholtz equation, impedance equation, and so on. Briefly, the problem is to determine the potential (or refraction index, impedance, etc.) from knowledge of quantities related to the scattering amplitude, e.g. phase shifts, reflection and transmission coefficients, etc. A complete account of the main results is given by Chadan and Sabatier (1977). The above inverse problems are nonlinear. For example, in the case of the vibrating string, the direct problem consists in solving the linear eigenvalue problem -U;(X)

= w ; ~ ( x ) u ~ ( x ) ;k = 0,

1,

(4)

on the interval (0,L), for a given density function p(x) and suitable boundary conditions at the end points 0 and L. But since the eigenvalues W ; are nonlinear functionals of the density function p(x),the inverse problem, consisting in the determination of p ( x )from given values of the of, implies the inversion of a nonlinear mapping. However, when an estimate of p ( x ) , say po(x),is known and the difference f ( x ) = p ( x ) - po(x)is small, it is possible to linearize the functionals w: using, for instance, perturbation theory. In this way, the original nonlinear inverse problem is approximated by means of a linear one for the unknown function f ( x ) . Similar results can be obtained in inverse scattering using Born or Rytov approximation, geometrical optics approximation, and so on. Moreover, some inverse problems related to inverse diffraction are rigorously linear problems. In this paper, we consider only linear inverse problems which have the following general structure. The first step is the definition of the direct problem, which must be linear. Then the solution of the direct problem defines a linear mapping L from the space X of all functions characterizing the properties of the physical sample (such as the density function in the case of a vibrating string or the refraction index in the case of a semi-transparent object, etc.) into the space Y of all corresponding measurable quantities (such as sequences of eigenvalues, scattering amplitudes, and so on). Of course, in the direct problem the data are elements of X , while the solutions are elements of Y. In the corresponding inverse problem, the data and solutions are interchanged. If we assume that the operator L is known (and this implies the need of solving the direct problem), then the inverse problem can be formulated as follows: Given g E Y and a linear operator L: X + Y, find f E X such that g = Lf.

An element of X will be called an object (or a solution) while an element of Y will be called an image (or data function, data vector, etc.). Accordingly, X will

LINEAR INVERSE AND ILL-POSED PROBLEMS

7

be called the object space (or solution space) and Y will be called the image space (or data space). In this paper, as in most mathematical papers on such problems, it is assumed that both X and Y are Hilbert spaces and that the linear operator L: X + Y is continuous. Continuity means that, given any sequence {fn} converging to zero in the sense of the norm of X , the corresponding sequence {LS,) converges to zero in the sense of the norm of Y. In other words, it is assumed that the direct problem is well-posed in the sense of Hadamard. In order to illustrate the difficulties of solving linear inverse problems, we consider a particular example of Eq. (9,the case of a Fredholm integral equation of the Jirst kind of the type K(x,y)f(y)dy,

g(x) =

c

I x I d.

(6)

lab

This can be written in the general form ( 5 ) if we introduce the integral operator (Lf ) ( x )= Jab K ( x , Y ) f ( Y )dY,

c 5 x 5 d,

(7)

which transforms functions on [a, b] into functions on [c, d ] . Moreover, we consider the simple case where the solution and data space are spaces of square-integrable functions, i.e., X = L2(a,b) and Y = L2(c,d ) . Then the operator L is continuous when the integral kernel K ( x , y) is square-integrable. Notice that this is only a sufficient condition for continuity. In the case of a convolution operator, i.e., K ( x , y ) = K ( x - y ) and also a = c = -a,b = d = +a, the operator L is continuous in L2(R) when the impulse response function K ( x ) is integrable and therefore the kernel is not integrable nor square-integrable as a function of the two variables x, y. Assume now, for example, that the integral kernel K ( x , y ) is an analytic function of x for any y E [a, b]. Then, given an arbitrary object f E L2(a,b), the corresponding image g(x) computed by means of Eq. ( 6 )is also analytic. It follows that the inverse problem does not have a solution for an arbitrary g E L2(c,d),but only for functions g in a subset of analytic functions. The problem is ill-posed in the original sense of Hadamard. Also, continuous dependence of the solution on the data does not hold true. If the interval [a, b] is bounded and the kernel is square-integrable, then the kernel is also integrable and, from the Riemann-Lebesgue theorem (Titchmarsh, 1948), it follows that

jab

K ( x ,y) cos(ny) dy -,0,

n + a.

(8)

Thus, we have found a sequence of functions that does not tend to zero

8

M. BERTERO

in L2(a,b), while the sequence of the corresponding images tends to zero. This example is similar to the example of Hadamard for the Cauchy problem of the Laplace equation. The previous example indicates that it is necessary to reconsider the validity of the mathematical model provided by Eq. ( 5 ) for the description of a physical experiment. When we apply the operator L to all the functions of X , we obtain a set of images that may be called the set of exact or noise-free images. In the example discussed above, these functions are a more regular (analytic) than the corresponding functions f and this situation is rather common in most inverse problems, i.e., the operator L that solves the direct problem has rather strong smoothing properties. But measurement errors or noise can destroy the smoothness of the exact image. The measured image is not related to f by Eq. (5), but by the equation g = Lf

+ h,

(9)

where h is a stochastic function that represents random noise. Such a function is, in general, far more irregular than any exact image Lf.For example, Lf is band-limited, while h is not band-limited or has a band much broader than the band of Lf. The remarks above indicate that the space Y must be broad enough to contain both the exact and measured images and that, in general, the set of exact images is a subset of Y. Moreover, if we assume that Y is a Hilbert space, then we must also assume it is equipped with a scalar product such that the norm of the experimental error h is small with respect to the norm of the exact image Lf.(It is for this reason that a space of square-integrable functions is the most convenient one in a number of practical problems.) As a consequence of these properties of the image space Y, the solution of Eq. (5) does not exist for arbitrary g and the problem is ill-posed. In the case of inverse problems with discrete data, however, the solution may exist for arbitrary g (even if it is not unique; see Section 111). It also depends continuously on the data since the space Y is a finite-dimensional Euclidean space and any operator is continuous in a finite-dimensional space. One must never forget, however, that continuous dependence of the solution on the data is a necessary but not a sufficient condition for the numerical stability (robustness) of a solution. Finite-dimensional problems obtained by discretizing ill-posed problems are usually ill-conditioned, even extremely ill-conditioned, so that error propagation from the data to the solution can deprive the solution of any physical meaning. If we return now to discussing linear inverse problems formulated in Hilbert spaces, we find we have a puzzling situation: either we use Eq. (9, but then the solution may not exist, or we use Eq. (9), but then we have only one equation and two unknown functions, f and h. And the search for

LINEAR INVERSE AND ILL-POSED PROBLEMS

9

approximate solutions of Eq. ( 5 ) , i.e., objects f such that Lf is approximately equal to g, will not succeed due to the noncontinuous dependence of the solution on the data. In fact, as clearly illustrated by Hadamard’s example, the set of approximate solutions can contain wildly oscillating and completely physically meaningless functions. More precisely, the set of all functions f, such that the distance between Lf and g is not greater than some prescribed small quantity E in an unbounded set of the object space X. The basic idea in the treatment of ill-posed problem is the use of a priori information about the unknown object to constrict the class of approximate solutions. This means that we need additional information, information that cannot be deduced from Eqs. ( 5 ) or (9), about the properties of the unknown object f.Moreover, this information must be incorporated into the algorithm to produce a physically meanin@ approximate solution. The additional information can consist of upper bounds on the solution and/or its derivatives, regularity properties of the solution (existence of derivatives up to a certain order, analyticity, etc.), localization properties of the solutions (restrictions on its support, behaviour at infinity, etc.), lower bounds on the solution and/or its derivative (positivity of the solution and/or its first derivative), and so on. The idea of using prescribed bounds to produce approximate stable solutions was introduced by Pucci in the case of the Cauchy problem for the Laplace equation (Pucci, 1955),while the constraint of positivity was used by John in the solution of the heat equation for preceding times (John, 1955), another classical example of an ill-posed problem. A general version of similar ideas was formulated independently by Ivanov for Eq. (5) (Ivanov, 1962).This method and the method of Phillips for Fredholm integral equations of the first kind (Phillips, 1962)were the first examples of the regularization theory for illposed problems, formulated and developed by Tikhonov a few years later (Tikhonov, 1963a; 1963b; 1964).Expositions of this theory can now be found in a number of monographs (Tikhonov and Arsenine, 1977; Groetsch, 1984; Bertero, 1982; Morozov, 1984). A summary of this theory will be given in Section V. This will be preceded by a brief section on the theory of generalized solutions, the basis of the general formulation of regularization theory. Regularization theory is essentially deterministic in that it does not make use of statistical properties of the noise or objects. Probabilistic methods for the solution of ill-posed problems have also been developed, but they will not be considered in this paper. Some of them are ad hoc methods and have not yet been formulated in a sound mathematical framework. We need only mention here the method of Wiener $filters,developed as a general method of solving ill-posed problems (Strand and Westwater, 1968; Franklin, 1970; Bertero and Viano, 1978)and which can be considered a probabilistic version of a particular regularization algorithm (Bertero et al., 1980a).This method, of

10

M. BERTERO

course, is very well established and can be used whenever a knowledge of the statistical properties, not only of the noise but also of the object is available. In a sense, the latter type of information plays the same role as prescribed bounds on a solution in the case of regularization theory and therefore in this formulation is the required a priori information (Turchin et al., 1971). The difficulty in the use of Wiener filters is that, in several practical inverse problems, the required correlation functions are not known. When these can be determined, the method can be very useful, as is demonstrated by applications to atmospheric remote sensing (Askne and Westwater, 1986).

11. LINEARINVERSE PROBLEMS

According to the definition introduced by Courant and Hilbert, the problem of solving the functional equation ( 5 ) is well-posed in the sense of Hadamard if the following conditions are satisfied: i) the solution f is unique in X ; ii) the solution f E X exists for any g E Y; iii) the inverse mapping g + f is continuous.

Condition i implies that the operator L: X + Y admits an inverse operator L-': Y + X , while Condition ii means that L-' is defined everywhere on Y. Then, since the continuous operator Lis linear, from a corollary of the Banach open mapping theorem (Yosida, 1966, p. 77) it follows that L-' is also continuous and therefore, in the linear case, Conditions i and ii imply Condition iii. We emphasize that the requirement of continuous dependence of the solution on the data is a necessary, but not a sufficient condition, for the stability (robustness) of the solution against noise. In the case of a well-posed problem, the propagation of relative errors from the data to the solution is controlled by the condition number. If 6 g is a small variation of g and 6f the corresponding variation of f = K ' g , then

I l ~ f l l x / l l f l l x5 cond(~)ll~gllY/llgllY~

(10)

where cond(L) is the condition number given by cond(L) = ~ ~ L ~ 2~ 1.~ ~ L - ' ~ ~

(11)

Here IlLll and llL-'ll denote the norms of the continuous operators Land L-', respectively. When cond(L) is not too large, the problem ( 5 ) is said to be wellconditioned and the solution stable with respect to small variations of the data.

LINEAR INVERSE AND ILL-POSED PROBLEMS

11

On the other hand, when cond(l) is very large the problem is said to be illconditioned and a small variation of the data can produce a completely different solution. It is clear that the distinction between well-conditioned and ill-conditioned problems is not very sharp and that the concept of wellconditioned problem is more vague than the concept of well-posed problem. As we stated in the Introduction, one of the Conditions i-iii may not be satisfied when Eq. (5) is the mathematical model of the given linear inverse problem. In such cases, the problem is said to be ill-posed. According to this definition, since uniqueness never holds inverse problems with discrete data are always ill-posed (see Section 111). In the recent mathematical literature, however, the term ill-posed is used in a more restrictive sense related to the theory of generalized solutions. Since the generalized solution, when it exists, is always unique (see Section IV), the problem of solving Eq. ( 5 ) is called illposed only when the generalized solution does not exist for arbitrary g or, equivalently, does not depend continuously on the data. It follows that only problems formulated in infinite-dimensional spaces can be ill-posed in this sense, while problems with discrete data are always well-posed (but, possibly, ill-conditioned). In succeeding sections we first, investigate some general properties of Eq. (5) in the ill-posed case and subsequently present some of the more significant examples of linear inverse problems. A. General Properties

The ill-posedness of a problem is a property of the triple ( L , X , Y } ; a problem is ill-posed because, for instance, the space Y is too broad. As already discussed in the Introduction, however, the space Y cannot be modified since it must be large enough to contain both exact and measured (noisy) data. Therefore, if we know that in an inverse problem the set of measured data does not coincide with the set of exact data, Condition ii is not satisfied. In many practical circumstances it is quite natural to assume that the object space X is a space of square-integrable functions. It may be convenient however, to adapt the structure of X to the available a priori information about the properties of the solution, such as regularity properties (existence of derivatives of f up to some order, analyticity, etc.) and/or localization properties (approximate size of the support of f,asymptotic behaviour at infinity, etc.). This means that, for functions f defined on some interval [a, b ] , the appropriate Hilbert space can be a weighted Sobolev space whose scalar product is defined by

12

M. BERTERO

Here the star denotes complex conjugation, the pi are given functions which are continuous and positive, and f ( i ) denotes dif/dx'. A more general description of the process of restricting the object space by the use of a priori knowledge is as follows. Let X be the original Hilbert space for which the problem (5) has been formulated (for example, a space of square-integrable functions) and let us assume that the a priori information about the specific solutions we are considering asserts that the solutions of the problem belong to the domain of an operator C from the Hilbert space X into another Hilbert space 2 (the operator C is called the constraint operator and the space 2 the constraint space) with the following properties:

A) the operator C has domain D ( C ) dense in X and is closed (the importance of this property is based on the fact that, as a rule, all differential operators are closed; for a definition see Balakrishnan, 1976); B) the operator C has a continuous (bounded) inverse C - ' . Then it is possible to introduce a new space X , c X , the domain of C equipped with the scalar product ( f 9

4 ) c = (Cf? C4)P

(13)

It is easy to prove that, when Conditions A and B are satisfied, Xc is also a Hilbert space. Moreover, the operator L: X, + Y is also continuous since the topology of X , is stronger than the topology of X . It follows that the inverse problem can be reformulated by taking X c as the new object space. We need only note that the scalar product (12) is a particular case of the scalar product (13). As stated in the Introduction, we will consider Eq. (5) only for a linear and continuous operators L. Therefore, for the sake of completeness, we now summarize a few general properties of these operators which we will frequently use below. We also indicate their physical interpretation. The null space of L , denoted N ( L ) , is the set of all functions f that annihilate L, i.e., N ( L ) = {f E

x ILf

= 0).

(14)

When the linear operator L is continuous, N ( L ) is a closed linear subspace of X . Moreover, N ( L ) is not trivial if and only if the inverse operator does not exist. The null space of the operator L will also be called the subspace of the invisible objects (Rust and Burrus, 1972), since the image of any element of N ( L ) is exactly zero. This also means that the experiment, or the instrument, described by L is unable to detect the objects that belong to N ( L ) .On the other hand, the orthogonal complement of N ( L ) , N(L)', will be called the subspace of visible objects, since the objects that belong to this subspace can be recovered, in principle, from exact data. By the orthogonal projection theorem

LINEAR INVERSE A N D ILL-POSED PROBLEMS

13

(Yosida, 1966; p. 82), an arbitrary object f can be uniquely represented as the sum of a visible plus an invisible component. It is evident that, at most, only the visible component can be recovered from the image in the absence of further a priori information about the object. The range of the operator L, denoted by R(L), is the set into which L maps X R(L)= {

g E

ylg

=

U fE X )

(15)

and therefore R ( L ) is the linear subspace of the exact or noise free images (data). The distinction, discussed in the Introduction, between exact and measured images makes clear that R ( L ) in general does not coincide with Y. Moreover, in the case of inverse problems formulated in infinite-dimensional spaces, R ( L )may not be a closed subspace of Y. In some cases (for example, if L is the Laplace transformation) R ( L ) is dense in the image space Y. The adjoint operator L* is uniquely defined by the following relation ( L f ,g ) v = (f,L*g),,

(16)

which holds for any f E X and g E Y. The operator L* is also linear and continuous and has the same norm as L: IIL*ll = 111.11.The null space and the range of L* will be denoted by N ( L * ) and R(L*),respectively. For example, in the case of the integral operator (7), if we set X = L2(a,b) and Y = Lz(c,d ) , the adjoint operator is given by the equation (L*g)(y)=

jcd

K ( x,y)*g(x)dx,

a I Y I b.

(17)

Finally, we recall two important relations between null spaces and the ranges of the operators L and L*:

R(L) = N(L*)',

R(L*) = N(L)',

(18)

-

where R ( L )denotes the closure of R(L).In other words, by investigating R(L*) we are led to a determination of the subspace of visible objects, while the investigation of N(L*) provides information about the subspace of exact images. These properties will be used in some of the examples of inverse problems discussed in succeeding sections. The relations (1 8) imply the following decompositions of the spaces X and Y X

= N(L)@

R(L*),

Y

= N(L*)@

-

R(L).

We discuss now in brief two important examples of linear operators whose range is not closed. The first is that of a compact operator (Balakrishnan, 1976) whose range is not finite-dimensional (i.e., we exclude the case of finite rank operators). An example of a compact operator is provided by the integral

14

M. BERTERO

operator (7) when the kernel K ( x ,y) is a square-integrable functions of the two variables x, y. More precisely, in this case L is a Hilbert-Schmidt integral operator from L2(a,b) into L2(c,d ) . We consider the general case where the solution and data space do not coincide (for example, in Eq. (7) the intervals [a, b] and [c, d ] do not coincide) and introduce the singular value decomposition of L . We then indicate the relationship with the well-known spectral representation of a compact, selfadjoint operator. The operators L*L and LL* are compact, nonnegative operators in X and Y, respectively. Moreover, N(L*L) = N ( L ) and N(LL*) = N(L*). It follows (Balakrishnan, 1976) that both operators admit a countably infinite set of positive eigenvalues. It is also easy to prove that L*L and LL* have exactly the same positive eigenvalues with the same finite multiplicity (Lanczos, 1961; Kato, 1966).If we denote these eigenvaluesby a$ and count each eigenvalue as many times as required by its multiplicity,the a$ can be ordered in such a way as to form a non-decreasing sequence: a: 2 a: 2 a: 2 . . .. The compactness of L implies that lim 0: = 0. k - +w

Now let uk and v k be the eigenfunctions of L*L and LL*, respectively, associated with the same eigenvalue a: L*Luk = o:uk,

LL*vk = 0 : U k .

(21)

Then the uk form an orthonormal basis in N(L)', i.e., the subspace of visible objects, while the v k form an orthonormal basis in N(L*)*,i.e., the closure of the subspace of the exact images. As is easily verified, it is always possible to choose the pair {uk,q.)in such a way that it is a solution of the following shifted eigenvalue problem Luk = akkvk,

L*uk = akuk.

(22)

The positive numbers are called the singular values of L and the functions U k , the corresponding singular functions. The set of triples {gk; uk, u k } is the singular system of L . Then one can prove the following representation (Kato, 1966) +m

the singular value decomposition of the compact operator L . A similar representation holds true for the operator L* obtained from Eq. (23) by interchanging the role of the singular functions U k and vk. The singular value decomposition (23) implies that the visible components of the object corresponding to small singular values make a small contribution

LINEAR INVERSE AND ILL-POSED PROBLEMS

15

to the corresponding components of the exact image. Therefore, in the case of a noisy image, these components can also be invisible in practice. Moreover, Eq. (23) implies that an image g is an exact image, i.e., g E R(L),if and only if the following conditions are satisfied

These conditions are called Picard's conditions (Nashed, 1976) for the existence of the solution of Eq. (5), since they were first derived by Picard for Fredholm integral equations of the first kind (Picard, 1910). They show that R ( L ) is not closed, since an arbitrary function satisfying the first of the conditions (24) does not necessarily satisfy the second one. It is evident, however, that R ( L ) is dense in N(L*)'. If L is self-adjoint and non-negative (the integral operator (7) is self-adjoint when [a, b] = [c, d ] and also K ( x , y)* = K ( y , x)), we denote by 1, its positive eigenvahes and by Uk the corresponding eigenfunctions. Then we have q,= I , , u k = Uk, and Eq. (23) becomes the spectral representation of the operator L. The second example is that of a convolution operator in Lz(R") &f)(X)

=

J m - Y)f(Y)dY, Rn

x E R",

(25)

where x = {xl,xz,. . .,xn} and d y = dy,dy, . .., d y , . If the impulse response function K ( x ) is integrable, then its Fourier transform k( 0 such that IlLfIlF

+ llCfI13 2 mllf 1:

(192)

for any f E D(C). Such a condition is called by Morozov the completion condition (Morozov, 1984) and is taken as the basic assumption for the solution of problem (187). Inequality (192) was also proved by Bertero (1986) using not only Conditions A and B' (cf. Groetsch, 1986), but also the assumption that N ( C ) is finite-dimensional. By means of inequality (192), it is easy to show that D(C),endowed with the scalar product

( f ,4%

= ( L f ,L+)Y

+ ( C f ,C4).z,

(193)

is a Hilbert space, which we will denote X,. Analogously, the restriction of L to X , will be denoted L,. Assume now that for given 9, there exist least squares solutions u E D(C).It is evident that this is true when Pg E LD(C). Then Eq. (175) implies that for thoseleast-squares solutions, we have Ilullz = IlPgllf IlCulld. Since,for fixed g , IlPgllF is a constant, the solution of problem (187) is equivalent to the solution of the problem

+

Ilfbll,

= inf{llullcl u E %)

"W ) } .

(194)

It follows that the C-generalized inverse of the operator L is just the MoorePenrose generalized inverse of the operator L,. As an example, we consider the case where L: X -i Y is compact. If we notice that, as follows from inequality (192), any bounded set in Xc is also a bounded set in X,we conclude that L , is also compact. We can then introduce its singular system { q - k ; uC,k,v , , ~ ) , the set of solutions of the shifted eigenvalue problem LC'C,k

= %,kvC.k?

LzvC,k

= *C,k'C,k.

(195)

When this problem has been solved, f b is given by Eq. (179), with by ( 0 C . k ; U C , k , V C , k } .

{Ok; uk,vk}

63

LINEAR INVERSE A N D ILL-POSED PROBLEMS

We want to show now that the solution of the shifted eigenvalue problem (195) can be reduced to the determination of the set of solutions { w t ; t)k} of the generalized eigenvalue problem L*Lt)k = o;c*c$k. (196) We notice that this problem is analogous to the problem encountered in the investigation of small oscillations of a mechanical system. In that case, L*L is related to the potential energy of the mechanical system, and C*C is related to the system’s kinetic energy. The first step is the determination of LF in terms of L*. From the relation

(f,LrdC = (Lf, LL?9), + (Cf9 CLc*S)Z = (f,(L*L

which holds true for any f

(197)

+ C*C)LEg), = ( f , L*g)x,

E D(C)which is dense in

X and any g E Y,we obtain

L,* = (L*L+ c*c)-’L*.

( 198)

Therefore, the second of the equations in (195) can be written as follows: L*oC,k = aC,k(L*L + c*c)uC,k.

(199)

Finally, if we apply L* to both sides of the first of the equations in (195) and use Eq. (199) to eliminate +k, we obtain (l

-

“z,k)L*Lk,k

= az,kC*CuCsk,

(200)

and this equation can be equated to Eq. (196) by setting OC,k

=

+ w k2)

-1/2 3

%,k

= t)k.

(201)

Notice that as a byproduct of this procedure, we have found that all the singular values ac,kare less than one. Similar results apply to inverse problems with discrete data. Since in this case L is a finite-rank operator, the C-generalized inverse L: is always continuous, but may be ill-conditioned, the condition number being given again by Eq. (131), with a, and aN- replaced by aC,,and aC,N- respectively. As a concluding remark, we point out that, in the case of the interpolation problem in RKHS, as discussed in Section III,C, if we look for C-generalized solutions associated with the functional (190), then the result is an interpolation in terms of natural splines (Greville, 1969). We merely note that Condition B’ is satisfied whenever k 5 N , where k is the order of the derivative in the functional (190) and N is the number of points. For a discussion of the interpolation and derivation problems in terms of C-generalized solutions, see Bertero et al. (1985a).

64

M. BERTERO

C. The Backus-Gilbert Method for Problems with Discrete Data The Backus-Gilbert method (Backus and Gilbert, 1968; 1970) has been proposed for the solution of the inverse problem consisting in the determination of the structure of the Earth, using data related to properties of the Earth as a whole such as mass, moment of inertia, and frequencies of elasticgravitational normal modes. It has also been applied to Fourier transform inversion (Oldenburg, 1976), the inverse scattering problem (Colton, 1984), and Laplace transform inversion (Haario and Somersalo, 1985). A relationship between the Backus-Gilbert method and the Fejer theory of Fourier series expansions has been discussed by Bertero et al. (1 988a). The Backus-Gilbert method can only be used in the case of an inverse problem with discrete data, when the object space X is a space of squareintegrable functions. In this case, the problem (115) takes the form 9, =

s

f(x)$,,(x)* dx;

II =

1,. . .,N .

(202)

It must also be pointed out that the method does not provide an exact but only an approximate solution of these equations. It is discussed in this section because it shows some analogies with the method of generalized solutions. To introduce the basic idea of the method, let us reconsider for a moment the Moore-Penrose generalized solution, or C-generalized solution, of problem (202). When the 4,, are linearly independent and the data values g, exact, by combining Eq. (202) with Eq. (121) we obtain f'(x) =

1

A+(x,x')f(x') dx',

where

and therefore, at any point x, the generalized solution f '(x) is an average of the true solution f ( x ) . Moreover, this averaged function lies in the subspace spanned by the functions 4". A similar result holds true for any C-generalized solution f g if we bear in mind that f : belongs to the subspace spanned by the functions,,)I satisfying the relations

( f , 4n)X for any f

E

= ( f ,*,,)c;

n = 1,. . ., N

(205)

D(C). When the C-scalar product is defined as in Eq. (13), the

LINEAR INVERSE AND ILL-POSED PROBLEMS

65

functions $, are obtained by solving the equations n = 1,. . .,N ,

C*C$,, = 4,,;

(206) and, when the scalar product is defined as in Eq. (193), the functions $, are obtained by solving the equations (L*L

+ C*C)$,,= q5,,;

n

=

1,. . .,N .

(207)

Then, by introducing the Gram matrix of the functions $, and the dual basis $", one finds for f s an expression similar to Eq. (121) with 4" replaced by $". By combining this equation with Eq. (202), we again find

where

The functions f ' and f,' are solutions of Eq. (202). The Backus-Gilbert method consists in searching for an approximate solution of these equations. Let us denote it by L G ( x ) ,which is also an average of the true solution f ( x ) ,

J

and which depends linearly on the exact values of the functionals (202). This condition implies that the kernel A ( x , x ' ) , called the averaging kernel by Backus and Gilbert, must have the expression N

with functions a,(x) to be determined. By combining Eqs. (210),(21 I), and (202) we have

and therefore ~ B G ( x )is a function in the subspace spanned by the functions a,(x). It is obvious that the generalized solutions discussed in the previous sections have a similar structure. In that case, the functions a,,(x) are determined by requiring that the solution satisfy Eq. (202) and by adding a variational principle for the solution (smallest norm, etc.) in order to ensure uniqueness. Backus and Gilbert follow a different approach and, in particular, do not require that the function (212) satisfy Eq. (202). In fact, they introduce a

66

M. EERTERO

variational principle for the averaging kernal itself, since they require that it must be the sharpest in a sense to be specified. For this purpose, let J(x, x’)be a function that vanishes when x = x’ and increases monotonically with increasing distance between x and x’.An example of such a function is

J(x,x’) = (x - x‘)2.

(213)

Then the unknown functions a,(x) in Eq.(211) are determined by solving the minimization problem

s

S2(x) = J(x,x’)lA(x,~’)1~ dx’ = minimum with the constraint

s

A(x,x’)dx’ = 1.

(214)

(215)

In other words, one looks for a kernel of the form (211) which gives a good approximation of the delta distribution 6(x - x’). If we introduce now the quantities bn =

s

s

+n(X) dx,

snm(x) = J(x,x’)4n(X’)4rn(xr)* dx’,

(216)

we find that the solution of problem (214), (215) implies the minimization of the quadratic functional

S2(x) =

N

1

n.m = 1

Snm(x)am(x)an(x)*

= minimum,

with the linear constraint

This problem can be solved in a standard way using the method of Lagrange multipliers. If S(x) is a non-singular matrix with elements Snm(x)and if we denote by Snm(x)the elements of [S(x)]-’, then the solution of problem (217), (218) is N

a,(x)

= A(x)

1 Snm(x)bm;

m=

1

n = 1,. . .,N ,

(219)

where the Lagrange multiplier A(x) is given by N

A(x)

=

( 1 Snm(x)b,b:) n.m= 1

-1

.

The functions a,(x) depend on the choice of the function J(x, x’).

(220)

LINEAR INVERSE AND ILL-POSED PROBLEMS

67

It should be obvious that for those problems where the generalized for the Backussolution is extremely ill-conditioned, the solution Gilbert method must also be unstable. Numerical results obtained in the case of Fourier series summation (Backus and Gilbert, 1968) indicate however that fBG(x)is more stable than f + ( x ) .This follows from the fact that the kernel (204) is narrower than the kernel (211) and therefore, using f + ( x ) , one requires higher resolution. The connection between resolution and stability will be discussed in Section VI. As a concluding remark, we point out that a convergence result has been recently proved for the Backus-Gilbert method (Schomburg and Berendt, 1987). In this paper, it is assumed that problem (202) is a finite section of a generalized moment problem (Section II,H) satisfying the requirement of is dense uniqueness, i.e., the span of the functions b,,,with n = 1,2,3,. . .,co, in X. Moreover, it is assumed that for a certain set of values gn of the generalized moments, there exists a solution f of the problem which is real-valued and Lipschitz-continuous. Then let &. be the approximate solution provided by the Backus-Gilbert method, using the exact values of the first N generalized moments off, the function J ( x , x ’ ) being given by Eq. (213). The result is that f N converges everywhere to f in the case of functions depending on one or two variables, while f N , does not, in general, converge to f i n the case of functions depending on more than two variables. It is evident that this convergence result can only be proved by assuming that the values gnof the generalized moments are not affected by experimental errors. It is interesting, however, to know that in this case the method can provide an approximation of the exact solution. It should also be important to investigate the convergence (or non-convergence) of the Backus-Gilbert method when applied to problems such as the moment discretization of a Fredholm integral equation of the first kind or the Fourier transform inversion with limited data.

A&)

V. REGULARIZATION THEORYFOR ILL-POSED PROBLEMS

The method of generalized solutions, discussed in the previous Section, provides a satisfactory answer to questions of existence and uniqueness for Eq. (5) only when the generalized inverse is continuous and well-conditioned. As we know, this means that the range of the operator L is closed and the condition number (183) is not much greater than one. The method is not adequate when the generalized inverse is not continuous, or if continuous the condition number (183) is too large. In the first case, the generalized solution

68

M. BERTERO

may not exist because the data are contaminated by experimental errors; in the second case, the generalized solution always exists but it may be deprived of any physical meaning as a consequence of dramatic error propagation from the data to the solution. In both cases, one must introduce methods for obtaining physically meaningful approximations of the generalized solutions. As already discussed in the Introduction, the basic idea is to constrain the solution in some way in order to avoid the wild oscillations generated by noise propagation. Several methods related to various kinds of constraints have been introduced, most of which have been unified in a general theory now known as regularization theory or Tikhonov regularization theory. In this section we sketch the basic ideas of the theory and provide the main references to the mathematical literature, a literature which has grown very fast in recent years. Obviously, our presentation will be strongly biased by our personal experience in this domain. A . The Ivanov-Phillips- Tikhonov Regularization Method

The Tikhonov regularization method was introduced independently by several authors at the beginning of the sixties. The first versions of the method were published in 1962 (Ivanov, 1962; Phillips, 1962) and a more general, unifying formulation-restricted however to the case of Fredholm integral equations of the first kind-was later proposed by Tikhonov (1963a; 1963b). Other important contributions are due to Morozov (1966; 1968) and Miller (1 970). In our presentation, we first sketch the methods of Ivanov, Phillips, and Miller, and successively show their relation to Tikhonov regularization theory. For the purpose of providing a concise outline of these methods, we start by giving a few results concerning the minimization of the functional

IILf - gIIi + aIIfII:, (221) can be any positive number. Let f, denote the function which

@aCfI

=

where LY , is easy minimizes Q a [ f ] .Then, by annihilating the first variation of O a [ f ] it to show that f, must satisfy the orthogonality condition

(Lf- 9, L v ) +~ a ( f a , 4)x = 0

(222)

for any 4 E X.It follows that j i is a solution of the Euler equation

+ crl)fa = L*g.

(2233

+ LYWllx 2 LYllfllxt

(224)

(L*L

Then the inequality

I"*L

LINEAR INVERSE A N D ILL-POSED PROBLEMS

69

which holds true for any a > 0, implies that there exists a unique solution of Eq. (223), which can be written in the form f, = Rag,

(225)

where R,

= (L*L

+ al)-'L*.

(226)

By simple algebraic manipulation, one can also show that R,

= L*(LL*

+ d-'.

(227)

This representation of R, implies that fa belongs to the range of L*, and therefore is orthogonal to the null space of L, as follows from Eq. (18). An important consequence of this property is that f,converges in the limit CI = 0 to the generalized solution 'f associated with g provided that Pg E R(L). Moreover, using the spectral representation of the self-adjoint positive semidefinite operator LL*, it is easy to show that the function E,' =

IILfa - gll:

=

a211(LL* + ~ ~ ) - ' P g I l+? IlQgllP

(228)

is a strictly increasing function of a whose values at a = 0 and a = cc are llQgll; and llgll~,respectively. In a similar way, one can show that the function (229) E,Z = llfalli = IIL*(LL* al)-'gJJ; is a strictly decreasing function of a, whose values at a = 0 and a = cc are llf+l[i (=awhen the generalized solution does not exist) and zero, respectively. For the sake of clarity, we give a more explicit representation of f,and the functions E,' and E,Z in the two particular cases already discussed in Section II,A, namely compact operators and convolution operators. When L is compact, from Eq. (225) and the singular value decomposition of L and L*, one easily derives that fa admits the expansion

+

' (iff' exists, as given by Eq. (179))as a which clearly shows that fa -+ f Moreover, we get

and also

and the properties stated above of the functions ea, Ea are self-evident.

+ 0.

M.BERTERO

70

Analogously,in the case of a convolution operator, the function f , assumes the form

where R is the support of

I?( E , S(g) does not intersect S, and therefore f(,)lies on a circle of radius E and is orthogonal to N ( L ) .

72

M. BERTERO

S , but must lie on the surface of this sphere. Since these points satisfy the condition 11 f (Ix = E, one can use the method of Lagrange multipliers for determining the solution of problem (237). This method consists of the following steps: 1) for any a > 0, minimize the functional (221); 2) since for any OL,there exists a unique minimum point fa of this functional, then search for a value of a such that

(238)

Ilfallx = E*

From the properties of the function E,, Eq. (229), stated above and illustrated in Fig. 7, it follows that there exists a unique value of a, say a(,), which solves Eq. (238).The corresponding solution fa is just Tcaand this is the unique solution of problem (237). We conclude with a few results about the convergence properties of the constrained least squares solution f l E )in the ideal case of experimental errors

0

O((E)

FIG.7. Graph of the function En in the case E mination of a(E).

o(

iIlf'llx

< +a,illustrating the deter-

LINEAR INVERSE AND ILL-POSED PROBLEMS

73

tending to zero. For this purpose, we assume that a family {ge}r,oof noisy data functions is given and that as E -+ 0, g L converges to a noise-free data function g, namely a function in the range of L. Let be the constrained least-squares solution associated with gr and let f = L+g be the generalized solution associated with g. Moreover, let us assume that for any E, case B applies. Then the following results hold true (Bertero, 1986):

rLE)

+

f'r")

i) if IIL+y(lx< E , i.e., the prescribed constant is overestimated, then weakly converges to f + as E -+ 0 (for the definition of weak convergence, see Balakrishnan, 1976); ii) if IIL+glJx= E, i.e., the prescribed constant is precise, then 0; strongly converges to f as E iii) if IILfgl(x> E, i.e., the prescribed constant is underestimated, then strongly converges to the constrained least-squares solution associated with the noise-free data g. +

2)

-+

f'E),

For problems with discrete data, case i must be modified since for sufficiently small E , there is necessarily a transition from case B to case A, and therefore the constrained least-squares solution corresponding to noisy data is not unique. In this case, one can identify, by definition, the constrained leastsquares solution with the generalized solution and therefore strong convergence applies also to this case. This result is also self-evident since weak convergence and strong convergence coincide in finite-dimensional spaces.

2. Phillips Method This method was first proposed for the approximate solution of the Fredholm integral equation of the first kind (Phillips, 1962). A more general formulation was given by Ivanov (1966) and Morozov (1966, 1968) while Reinsch (1967) applied independently the same method to the smoothing problem, a problem which replaces strict interpolation when the values of the function are only approximately given. The starting point is the assumption that an upper bound E on the error is known. We denote by JJg) the set of all elements of X which are compatible with the data g within error E

J€(d= If

E

x 1 IlLf - sllu 4.

(239)

This set is always unbounded when the problem is ill-posed. In the case of a problem with discrete data, for example, it is a cylinder whose basis is an ellipsoid in X , (see Section 111,A). This cylinder is not bounded in the directions orthogonal to X, because the solution of the problem is not unique. On the other hand, in the case of an operator whose inverse is not continuous, J,(y) is unbounded since it is always possible to find a sequence { f n } such that

74

M. BERTERO

llLfnllx + 0 and Ilf,lly-+ 00. Notice that the example of Hadamard discussed in the introduction is just a particular case of this general result. It is easy to prove, however, using the continuity and linearity of L , that JJg) is always a closed and convex set. Since J J g ) contains wildly oscillating and completely unphysical approximate solutions, it is quite natural to look for the smoothest element of J,(g), i.e., the element of minimal norm, which will be denoted by f('). This leads to the problem llj(E)IIX =

inf{Ilf llx 1 llLf - gllr 5

4.

(240)

As remarked by Ivanov (1966) this problem is just the dual of problem (237). Since the set JJg) is closed and convex, from the general theorem of functional analysis (Balakrishnan, 1976)already used for proving the existence and uniqueness of the generalized solution, it follows that there exists a unique solution of problem (240).This solution is not the null element of X provided that the data g satisfy the inequality llglly > E . This inequality is quite reasonable since it implies that the norm of the data is greater then the norm of the noise. If it is not satisfied, it means that the data function (vector) consists only of noise and that it does not contain any information about the unknown object f. Since we exclude this case and since we also assume that if a nonzero component of g orthogonal to the range of L exists, then this component can only be an effect of the noise, we are led to conclude that the following inequalities must hold true: IlQgllr < E

< llgllr

(241)

A representation of the solution of problem (240) analogous to the representation of the constrained least-squares solution can be obtained if we notice thatf") must satisfy the condition IILf") - gllr = E. Then we can again use the method of Lagrange multipliers in order to determine T(').We must minimize the functional (221) for any CI > 0 and then search for a minimum point fa such that IILfa

- gllu = E .

(242)

From the properties of the function E,, Eq. (228), and conditions (241) it follows that there exists a unique value of a, sax d'), which solves Eq. (242) (cf. Fig. 8). The corresponding solution fa is justf('), i.e., the unique solution of problem (240). We see that the difference between the solution of problem (237) and the solution of problem (240)consists only in a different choice of the Lagrange multiplier. In the case where the experimental errors tend to zero, the situation as regards the convergence off(') is far simpler than in the case of the con-

75

LINEAR INVERSE A N D ILL-POSED PROBLEMS

E

Ilasll,

0

o (

FIG.8. Graph of the function c, in the case liQgllv < c < I)glly, illustrating the determination of a"'.

strained least-squares solutions. If { g c } c , o is the family of noisy data functions introduced in Section V,A.l,Tf)the solution of problem (240)with g replaced by g c , and f' E J,(g,) for any E, then it is possible to prove (Ivanov, 1966; Groetsch, 1984; Bertero, 1986) that?:) strongly converges to f ' as E + 0. 3. Miller Method

The method of Section V,A.I requires a prescribed bound on the solution, while the method of Section V,A.2requires a prescribed bound on the error. A paper of Miller (1970) consider the case where both bounds are known. Results similar to those of Miller were also obtained by Franklin (1974). Let us assume that two constants E , E are given and that one wants to find functions f such that IILf - SIIu 5 llfllx _< E. (243) The set K of all functions satisfying these conditions is just the intersection of the set S, given by Eq. (236) and the set Jc(g) given by Eq. (239): K = S, n J,(g). Any function f E K may be called an admissible approximate solution. In such an approach, we must consider two problems: first, how to ensure that K is not empty, in which case we say that the pair { E , E } is permissible; €9

76

M. BERTERO

second, how to extract an element of K in order to produce one specific approximate solution. It is easy to characterize the set of permissible pairs. We first notice that K is not empty if and only if both j ( Eand ) f (') belong to K . The '3f"jar-t is trivial. The "only if" part follows from the variational properties of f ( " ) and j('). For example, the condition that K is not empty implies that there exist elements of J,(g) whose norm is less than E . Since f"") is the element of minimal norm of J,(g), it follows that the norm of ?(')must also be less than E. Moresatisfies, by definition, the first of the conditions in (243),and thereover fore j ( ~ € K). Similar arguments apply to f"'"). The remarks used for proving the previous result also imply that when K is not empty,

Ti')

I17(:(ollxI Il7(")llx

(244)

IILP' - 9 l l Y 5 IILf('"- SllY

(245)

and Moreover, if we recall that E, is a decreasing function of a (or that increasing function of a), we also have

E,

is an

< a(').

(246) Finally, since the previous results imply that when K is not empty, then llf(')ll~I E and IILfl'"' - g/IuI 6,it follows that the set of permissible pairs in the plane { € , E ) is just the set of all pairs to the right and above the curve (see Fig. 9). We also conclude that all functions f,with described by { E , , a between a(E)and f) belong to K . Both f l E )and f(') may be used to determine an element of K . Another possibility, however, is to take the function f, with a = ( E / E ) * .This function will be denoted by 7"").In fact it has been proved by Miller (1970) that if K is satisfies inequalities (243) with ( E , E} replaced by not empty, It is easy to find, however, a sufficient condition that ensures that T(')E K . Let us denote by 0 ( O ' [ f ] the functional (221) with cx = ( E / E )and ~ by the subset of X a(E)

{a,,aE}.

Po)

K'O' =

{f E K

Ic D ' O ' [ f ]

IE'}

(247)

Then since ?(')minimizes the functional @ O ' [ f ] , it is obvious that K'O' is not empty if and only if the following condition is satisfied: @(O)[f'O']

€2

(248)

(Notice that this condition can be easily verified in a numerical application of the method.) Finally, using the inclusion K(O)c K,

(249)

LINEAR INVERSE AND ILL-POSED PROBLEMS

77

/ I

I

E

I

/ II f

OC=O

I

I

/

E/E=const

l

k

0

Ilagll,

11g11,

E

FIG.9. Representation of the set of the permissible pairs { E , E } in the case where the generalized solution f’ exists. Whenf+ does not exist, the line E = llQgll,. is an asymptote of the boundary curve { e s , E m } .

which follows from the remark that any element of K‘O) satisfies conditions (243), we conclude that when condition (248) is satisfied, the set K is not empty and flo’E K . Moreover, the following inequalities are a trivial consequence of inequalities (244)-(246) and the fact that f a E K if and only if tx belongs to the interval [a@),a(‘)]

ll.f~c’llx 1lP)Ilx 5 IIP’llx IIL.f(E’- g u y I IIL.P0’- Sllv II I ~ -PSllY C4-Q

_< ( € / E ) 2 I a(E’

(250) (25 1 ) (252)

78

M. BERTERO

We see therefore that the Miller solution f ( ' ) has a degree of smoothness and f?'). intermediate between that of Finally, as concerns the convergence of flo)to the true generalized solution f+,this approximate solution_has properties similar to those of the constrained least-squares solution f ( E ) (Bertero, 1986) when the error of the data tends to zero.

pE)

4 . Extensions and Comments

The method of constrained least-squares solutions can be considered a generalization of pioneering works on ill-posed problems for partial differential equations (Pucci, 1955;John, 1955; Fox and Pucci, 1958; John, 1960). In these papers, which we referred to in the Introduction one looks for approximate solutions satisfying a prescribed bound. In the method of Section V,A.1, this condition is replaced by a search for approximate solutions in the sphere (236). An extension of this condition is provided in the same paper of Ivanov (1962)and is based on a topological lemma due to Tikhonov (Lavrentiev, 1967;Tikhonov and Arsenine, 1977),which in our context can be formulated as follows: Let H be a compact subset of the Hilbert space X and let us assume that the linear, continuous operator L: X + X restricted to H , has an inverse, Then the inverse operator is continuous. The result of Ivanov can now be formulated as follows: If H is a compact and convex set of X and if the restriction of L to H admits an inverte operator, then for any g E X there exists in H a unique least squares solution f of Eq. ( 5 ) . Moreover the mapping g + f is continuous. It is obvious that in this way one is not obliged to restrict solutions to a sphere (in fact, a sphere is not a compact set). An important example of an application of the theorem is the case where the function to be restored is the distribution function of a random variable, and therefore is an increasing function with values in the interval [O,l]. Then according to the Helly theorem (Titchmarsh, 1958, p. 342), a set of increasingand uniformly bounded functions defined on a bounded interval [a, b] is compact in L2(a,b) and therefore the Ivanov theorem can be used whenever the inverse operator L-' exists. As a second comment we point out that the methods outlined in the previous sections provide approximations of the Moore- Penrose generalized solution. Then one can also look for approximations of the C-generalized solutions introduced in Section IV,B. In fact, the method of Phillips (1962) applies to this case, since it provides an approximate solution of Fredholm integral equation of the first kind such that the L2-norm of its second derivative is as small as possible. The smoothing method of Reinsch (1967) also corresponds to this case.

LINEAR INVERSE AND ILL-POSED PROBLEMS

79

Approximations of the C-generalized solutions can be obtained if we replace the functional (221) with the functional

(D,,,[IfI = IILf- YII: + allcfllk

(253) where the constraint operator C satisfies the properties assumed in Section IV,B and, in particular, property B'. Then one can prove (Groetsch, 1984; Morozov, 1984; Bertero, 1986) that for any a > 0, there exists a unique function X which minimizes the functional (253) and which can be obtained by solving the functional equation (L*L

Moreover, the mapping g

4

+ aC*C)f,,,

= L*g.

(254)

fc,,, given by f ~ ,= u

&,ag

(255)

with RC,a= (L*L + aC*C)-'L* is continuous. This operator can be written in the standard form (226) by introducing the adjoint of the operator L , with respect to the scalar product (193)(Groetsch, 1986).Then, using Eq. (198), by simple algebra, from Eq. (256) we find that RC,Q= b(L:Lc

+ Bal)-'L:,

(257)

where ,f? = (1 - a)-i. In particular, when L is a compact operator (or a finite rank operator as in the case of inverse problems with discrete data), it follows that fC,, has a representation in terms of singular values ac,kgiven by Eq. (195) analogous to the representation of f, in (230). We also notice that when L is an integral operator and C a differential operator such that the scalar product (13) is given by Eq. (12), then the solution of Eq. (254) implies the solution of a boundary-value problem for an integrodifferential equation (Tikhonov, 1963a). The latter is equivalent to the solution of the functional equation (223) in the Sobolev space defined by the scalar product (12) (Groetsch, 1984). If one introduces now the functions EC,a

= 1ILfC.u - glly,

E c , ~= IlCfc,allz,

(258)

one can easily prove that eC,,is an increasing function of a while EC,, is a decreasing function of a. As a consequence, all the results proved in the previous sections concerning the operator (226) can be extended to the present case. More precisely, there exist a unique value of a minimizing under the constraint EC,, < E, and conversely there exists a unique value of a minimizing Ec,a under the constraint eCSaI E .

M.BERTERO

80

A final comment about the Backus-Gilbert method, which can also be affected by numerical instability, is in order. No regularization method has been developed for this algorithm. Backus and Gilbert, however, have introduced two methods in order to improve stability (Backus and Gilbert, 1970), though rigorous results have not yet been proved for these methods. If the covariance matrix C of the noise is known, then from Eq. (212) one easily derives that at any given point x , the variance of the error induced by the noise on L G ( x )is

Then the two methods introduced by Backus and Gilbert are the following: 1) Minimize the functional 0 2 ( x )with the constraint (218) and also the constraint 6’(x) I E2((S2(x)is defined in Eq. (217)). The latter constraint prescribes an upper limit on the desired resolution. 2) Minimize the function S’(x) defined in Eq. (217) with the constraint (218) and also the constraint u’(x) Ic2. The latter constraint prescribes an upper limit on the desired error affecting the reconstructed solution.

Both problems can also be solved by means of the method of Lagrange multipliers and we do not give the details here. We merely wish to remark that each of these two problems is the dual of the other and that they are similar, respectively, to the Ivanov and to the Phillips method for the regularization of the Moore-Penrose generalized solution. B. General Formulation of Regularization Methods

The common feature of the methods presented in Section V,A is that they provide different criteria for selecting a specific element from the same family of approximate solutions, namely fa = Rag, defined by Eqs. (225) and (226) or Eq. (227). This family describes a trajectory in the Hilbert space X . To get a clear picture of this trajectory, we need certain further properties of the operators R,. More precisely, we want to show that: i) for any ct > 0, R,: Y -,X is a linear continuous operator whose norm is bounded by

IIRaII 5 11&;

(260)

ii) if g belongs to the range of L and if f’ is the generalized solution associated with g , then lim((R,g - f + l l x = 0. 010

LINEAR INVERSE A N D ILL-POSED PROBLEMS

81

The proof of i follows from Eqs. (227) and (224). In fact, Eq. (227) implies that IJRagl12= (LL*(LL*

+ aZ)-'g, (LL* + a I ) - l g ) r .

(262)

Then, if we notice that the norm of the operator LL*(LL* + aZ)-' is smaller than 1 and that, by inequality (224), the norm of the operator (LL* + aI)-' is smaller than a-', we obtain inequality (260). As for ii, if g E R(L),then g = Lf' and from Eq. (226) we get I I K g - S'llx = IIRaLf'

- f'llx

= all(LL*

+ aI)-'f'lI,.

(263)

Then Property ii follows from the spectral representation of the self-adjoint positive semi-definite operator L*L and from the dominated convergence theorem (Groetsch, 1984; Bertero, 1986). Properties i and ii can be verified in an elementary way for a compact operator or convolution operator by means of Eq. (230) or Eq. (233), respectively. In the case of noisy data, say g,, where is an estimate of the norm of the error, i.e., of the distance between gp and the exact data g E R(L), 119, - glly

(264)

€9

the approximate solution Rag, may have no limit as ct --t 0 or, as in the case of problems with discrete data, the limit is the generalized solution f: and the distance between f,' and f ' = L'g, i.e., llf: - I ' l l x , may be extremely large. There exists, however, a value of a, say a(oP'),such that the distance bef is minimum. If we write tween Rag, and ' Rage - f'

= (Rag -

f') + Rabe - 9)

(265)

then from Eq. (260) and the triangular inequality, we have IIRagc - J + I 5 I ~IIRag - f +ttx

+ €/&*

(266)

Then since the first term in the r.h.s. is an increasing function of a, as follows from Eq. (263),while the second term is a decreasing one, there exists a unique value of a that minimizes the r.h.s. of Eq. (266). It is obvious that this optimum value of a, a(op'),cannot be determined in practice because its determination requires knowledge of the true solution f'. It is important, however, to know that such an optimum value certainly exists. We can now describe the trajectory of the approximate solutions f a , , = Rag, in the case of a non-empty set K defined by inequalities (243) (see Fig. 10).The trajectory starts at the origin (null element) of X when a = co and for large values of a lies inside the sphere S, defined by Eq. (236). Then for M = d'), the trajectory crosses the surface of the ellipsoid J,(g) defined by Eq. (239), and for values of a between a(') and a(E)passes through the set K .

82

M. BERTERO

\

E

/

FIG.10. Two-dimensional representation of the trajectory described by Rag, as a increases from 0 to to. It is assumed that there exists a generalized solution associated with noisy data. Otherwise the trajectory tends to infinity as a --t 0.

In this part of the trajectory is found the point corresponding to the optimum value of c1 discussed above and also the point corresponding to a=(e/E)’ (at least when condition (248) is satisfied). Finally, for all values of c1 smaller than the trajectory always lies inside the ellipsoid J,(g) and when c( = O its endpoint will be the center of J,(g), i.e., f:, when the generalized solution associated with gc exists; otherwise, the trajectory becomes infinite. These comments on the methods of Section V,A justify the general definition of a regularizing algorithm (in the sense of Tikhonov) which will now be given and discussed. defines a We say that a one-parameter family of operators regularizing algorithm for the approximate determination of the generalized

LINEAR INVERSE AND ILL-POSED PROBLEMS

83

inverse L+ of the linear operator L if: i’) for any a > 0, R,: Y + X is a continuous operator; ii’) for any g E R ( L ) limllR,g - f’llx

= 0.

a10

When the operators R, are linear, then we have a linear regularizing algorithm. It is possible, however, to introduce nonlinear regularizing algorithms €or the solution of linear problems. We will give a few examples of such algorithms in Section V,D. The parameter c1 is usually called the regularization parameter and, in general, is a positive real number. In some cases, however, it may be convenient to introduce a discrete variable that take only integer numbers. In this case, we have a sequence {R‘”’} of regularizing operators and the limit a + 0 is replaced by the limit n + co. If we want a unified notation we may set R, = R(“),where [a-’1 denotes the integer part of a-’ and is equal to n. The meaning of Condition ii’ is obvious. It implies that when g E R(L), it is possible to obtain arbitrarily accurate approximations off’ by means of continuous operators. Moreover, in the case of noisy data g, satisfying condition (264), we still have an inequality analogous to (266), that is ((Rage- f’llx 5 IIRug - f’IIx

+ EIIRaII.

(268) The first term in the r.h.s. can be called the “approximation error,” introduced when the noncontinuous (or ill-conditioned) operator Lf , acting on exact data, is replaced by the continuous (or well-conditioned) operator R,. This “approximation error” tends to zero as a tends to zero. The second term in the r.h.s. represents “error propagation” from the data to the solution and becomes exceedingly large as a tends to zero. Therefore, it is clear that the choice of a will be based, in general, on a compromise between the approximation error and noise propagation. As in the case of the specific example provided by Eq. (226), given a noisy data function g, and a regularizing algorithm {R,}a,o, the family of approximate solutions f,,,= Rag, will describe a trajectory in the Hilbert space X and, in general, there will be a point on this trajectory at a minimum distance from the true solution f’. A similar definition can be introduced for the regularization of C-generalized inverses. Condition i is not modified, while Condition ii is modified as follows: ii”)

for any g E L D ( C ) limllR,g - fdllx = 0, a10

where f: is the C-generalized solution associated with g.

84

M. BERTERO

It is easy to show that the family of linear continuous operators {Rc.a}a,o defined by Eq. (256) is a regularization algorithm for L:. In fact, the representation (257) of these operators coincides with Eq. (226) except for the factor b. Then, since + 1 as a + 0, Eq. (269)can be proved in the same way as Eq. (261). In the case of linear inverse problems with discrete data, the definition of a regularization algorithm needs some modifications (Bertero et al., 1988b). For an ill-posed problem, indeed, a regularization algorithm constitutes a family of continuous (bounded) operators that approximate an unbounded operator. But, for a problem with discrete data, the generalized inverse is always continuous since it is a linear operator on a finite-dimensional space. The problem must be regularized when the norm of L+ is much greater than l/llLll (ill-conditioning),and therefore a regularization algorithm must provide an approximation of L+ with norm smaller than the norm of L'. For these reasons, we say that a one-parameter family of operators {R,},,o is a regularization algorithm for an inverse problems with discrete data when: i) for any a > 0, the range of R, is contained in X,, the subspace spanned by the functions q5,,; ii) for any a > 0, the norm of R, is smaller than the norm of L+, i.e., IlRUll

IIL+II = l/g,-1;

(270)

iii) the following limit holds true in the sense of the norm of bounded operators

A similar definition can be given of a regularization algorithm for a Cgeneralized inverse (Bertero et al., 1988b). In Condition i, the subspace X , is replaced by R(L,*),the orthogonal complement of the null space of Lc which coincides with the subspace spanned by the functions IcI. given by Eq. (205), while in Condition ii the norm of L+ is obviously replaced by the norm of L,f. C. Spectral Windows

For fixed a > 0, consider the function of L given by E(A)= (1 + a)-' and defined on (0, + 00). Then the Tikhonov regularizer (226) can also be written in the form R, = F,(L*L)L*, where the operator F,(L*L) is obtained from Fa@) using the spectral representation (Yosida, 1966) of the self-adjoint, nonnegative operator L*L. Moreover, if the operator R, is applied to a noise-free image g = Lf +,one obtains

LINEAR INVERSE AND ILL-POSED PROBLEMS

85

where W,(A) = A(L + u ) - ' . This function is small in the neighborhood of the spectral point 1= 0 (notice that this point belongs to the spectrum of L*L if and only if the problem is ill-posed), and, therefore, the effect of the regularizing algorithm is a windowing (or filtering) of the spectral components off' related to the ill-posedness of the problem. For example, in the case of a compact operator, using Eq. (230) and the relation (9,u k ) y = (Lf+, u k ) y = ( f + , L * u & = c r k ( f + , u k ) X , we obtain

where we have introduced the notation A k = 0 2 .Analogously, in the case of a convolution operator, we obtain from Eq. (233)

In this case, the spectrum of L*L is the set of values of the function

lm*, < E Q.

A(C) =

The previous remark suggests a search for regularization algorithms of the form

R,

= F,(L*L)L*,

(275)

where now { F,(A)),> denotes a suitable family of functions defined on (0, + 00) and again F,(L*L)is given in terms of &(A), using the spectral representation of L*L (Bakushinskii, 1965; Groetsch, 1980; 1984). The problem is now to find sufficient conditions on {&(A)},>o that would guarantee that { R , ) , > o is a regularization algorithm. These conditions can be given on the window functions Wu(L)= LF,(A). For example, it is not difficult to prove (Groetsch, 1980; Bertero, 1986) that if { W,(A)},, is a family of real-valued, piecewise-continuous functions defined on (0,+ co)satisfying the conditions:

i) for any u > 0,O 5 W,(A) I 1; ii) for any A > 0, lim W,(A)= 1; a10

iii) for any a > 0, there exists a constant c, such that W,(A)Ic,A;

(277)

then the family of operators defined by Eq. (275), with &(A) = 1-'W,(A), is a regularization algorithm. Conditions i-iii are satisfied by the Tikhonov window W,(A) = A(1 + a)-'.

86

M. BERTERO

Another important example is the following:

W,(1) = 0,o I 1 < a;

W,(1) = 1,1 2 a,

(278)

which corresponds to a truncation of the spectral representation of L*L. This regularization algorithm is very important for both compact operators and convolution operators. In the case of compact operators, however, it is more convenient to define spectral windows in terms of singular function expansions as follows: m

and therefore one must introduce a family of window sequences rather than a family of window functions. Conditions i and ii above are unchanged (the variable 1 is replaced by the index k), while Condition iii must be replaced by the requirement that for any a > 0, there exists a constant c, such that Wu,kI cask. Then the use of the window function (278) is equivalent to the use of the window sequence

K,k = 1,

k

Wa,k= 0,

[a-'];

k > [Cr-'1.

(280)

The corresponding regularization algorithm is the well-known method of truncated singular function expansions (Twomey, 1965; Miller, 1970; Groetsch, 1984), also known as numerical jltering. Analogously, in the case of a convolution operator it is convenient to define spectral windows in terms of the Fourier transform as follows:

Again, Conditions i and ii above are unchanged (the variable ;1is replaced by the variable c), while Condition iii is replaced by the requirement that for any c1 > 0, there exists a constant c, such that K( u-l

(282)

or, in other words, equivalent to the use of a cut-off in the Fourier integral. We point out that in the case of a regularization algorithm defined as in Eq. (281), the operator W,(L*L)is a convolution operator given by

CK(L*L)f I@) =

IR"

A,(x - x')f(x')dx',

(283)

where Aa(x)is the inverse Fourier transform of Wa(5).For example, in the case

87

LINEAR INVERSE A N D ILL-POSED PROBLEMS

of functions defined on ( - 03, +a), and the window functions (282) we have (284) A,(x) = (na)-'sinc(x/na). It follows that, in the case of noise-free data, the regularized solution is an average of the true solution over a distance of the order of a. We now present other interesting window functions for the inversion of convolution operators in one dimension. a. The Triangular Window Wa((l)= (1 - aI5I), 151 I a - ' ;

Wa(5)= 0,151 >

(285) The triangular window is related to the approximation of Fourier integrals in the sense of ( C , 1)-summability (Titchmarsh, 1948).In this case, the averaging function AJx) is given by (286) A,(x) = (2na)-l sinc2(x/2na). Notice that this averaging function is positive and therefore in the absence of noise, the corresponding regularization algorithm provides positive approximations of positive functions (Bertero et al., 1988a). b. The Hanning Window

K(t)=+[I

+ cos(na(l)],

151 5 a - ' ;

K(5)= 0,

It1 > a-l.

(287) The Hanning window is well known in the theory of signal processing (Kunt, 1986, p. 136). The corresponding averaging function is Aa(x) = (4na)-'(sinc[(x - na)/na] + 2sinc(x/na)

+ sinc[(x + na)/na]> (288)

which is not positive. The negative parts, however, are quite small and the side-lobes are smaller than the side-lobes of the function (286), so that this window can be very convenient for practical use c. The Gaussian Window

Wa(4;)= exp(-at2/2)

(289)

The corresponding averaging kernel is ~ / ~-x2/2a). A,(x) = ( 2 n ~ ) -exp(

(290) Notice that this kernel is also positive and that side-lobes are absent. The disadvantage is that it can be used only if k( 0 arbitrary.

I AllnEl-"

(335)

LINEAR INVERSE A N D ILL-POSED PROBLEMS

101

When Holder continuity holds true, John calls the ill-posed problem wellbehaved. In this case, only a fixed percentage of the significant digits is lost in determining f from g, and therefore the uncertainty of the solution is not very severe. In contrast, in the case of logarithmic continuity, even an improvement of several orders of magnitude in the noise level does not induce a significant reduction of the uncertainty of the solution. In other words, the information content of the data is practically noise-independent. It is important to realize that the type of continuity does not depend only on the problem, i.e., the operator L , but also on H. For one and the same problem, we can have Holder continuity for certain sets H and logarithmic continuity for others. For certain ill-posed problems, one can have Holder continuity when one prescribes bounds on a finite number of derivatives of the unknown function f . In this case, the problem is said to be mildly ill-posed. Examples are tomography, Abel transform inversion, and numerical differentiation (Bertero et al., 1980; Louis and Natterer, 1983). On the other hand, when prescribed bounds on a finite number of derivatives imply only logarithmic continuity, the problem is said to be severely ill-posed. Examples are Laplace transform inversion (Bertero et al., 1982),the problem of bandwidth extrapolation and, in general, the solution of a Fredholm integral equation of the first kind with analytic kernel (Bertero et a/., 1980a).As already pointed out, however, one can have Holder continuity even in the case of a severely ill-posed problem simply by choosing an appropriate set H of admissible solutions. For example, in Laplace transform inversion, we have Holder continuity if His a bounded set of functions having suitable analyticity properties (Bertero et al., 1982). B. Evaluation of Linear Functionals and Resolution Limits

In several applications one is not directly interested in estimating the solution of a problem, but rather the value of some suitable functionals of the solution. These functionals can be, for example, a moment of given order (the average radius or average occupied volume in problems of particle sizing described in Section II,C.3) or, in general, a linear continuous functional, i.e., a generalized moment. Several examples related to the applications of Abel equation are described by Anderssen (1986). Stated in a rigorous form, the problem is the following: Given an element E X , estimate the value of the functional where f is a solution or generalized solution of Eq. (5). The important feature of these problems is that some of them are far more

102

M. BERTERO

stable than the problem of determining the solution f itself. In fact, for a certain class of these functionals the evaluation problem is well-posed. Since this class can be characterized for any given linear inverse problem, we have here a precise answer to the question succinctly expressed by Sabatier (1984): the need to identify and ask within the framework of indirect measurements well-posed questions about a phenomenon of interest. In general, however, the problem of estimating the functional (336) is not well-posed, and therefore it is necessary to use regularization theory. A modulus of continuity can also be defined for this problem, and by considering special classes of functionals in this approach one can find a rather precise definition of the resolution limits achievable in a given inverse problem. 1 . Evaluation of Well-Posed Functionals

Consider a functional of the form (336) with 4 E R(L*).Then there exists a function $ E Y such that 4 = L*$. By substituting for 4 in Eq. (336), we find Fg(f) = u - 3 L*$h = (Lf, $)Y

*

(337)

In this case, F+,is a continuous linear functional of Lf.Therefore, it is obvious that given the data function g, the estimation of the value of the functional is ag = (s9 $)Y

(338)

*

In other words, these functionals can be estimated directly from the data without any need to solve for the unknown function f.It is also obvious that the dependence of the value of Fg on g is continuous. If 6g is a variation of g and bag the corresponding variation of a+, then, from Eq. (338), using the Schwarz inequality we obtain 16agl 5

Il$llY

' lI&lllY

5

4l$llY.

(339)

It is obvious, however, that the problem can be ill-conditioned because the error ha, can be exceedingly large when Il$lly is too large. It is also important to notice that Eq. (337) characterizes all the functionals that depend continuously on Lf and therefore also characterizes all the functionals that can be directly estimated in terms of g. Assuming again, for the sake of simplicity, that the inverse operator L-' exists, this result can be proved as follows. If the functional F4 has the property

IE$(f)l5 C I I U l l r ,

(340)

where c is a constant independent of f,then, given f~ X , it is always possible to find g E R ( L ) such that f = L-'g. By inequality (340), it follows that IF+,(f)l= I(f> 41x1 = I(L-'g,

41x1

(34 1) Therefore (L-'g, 4)x is a linear and continuous functional on R ( L ) and can CllSllY.

103

LINEAR INVERSE AND ILL-POSED PROBLEMS

-

be extended, by continuity, to a linear and continuous functional on R(L).By the Riesz representation theorem, this implies that there exists an element $ E R(L) such that ( L - l99 4)x = (g*$ ) Y . (342) If we now set g = L f , we have

(f,4)x = (Lf, $)Y

=(

f 3

L*$)x,

(343)

and therefore 4 = L*$. As an example, consider the case of the integral operator (87) whose adjoint is just the band-limiting operator (80) when the values of x are restricted to the interval [- 1, I]. It follows that in the inversion of the integral operator (87), a linear and continuous functional can be estimated directly from the data whenever the function 4 is the restriction of a band-limited function to the interval [ - 1,1]. 2. Evaluation of Ill-Posed Functionals When 4 4 R(L*),the functional (336)cannot be estimated directly in terms of the data g. Therefore, the use of regularization theory is required in this case. It is obvious if ?is some regularized solution of the problem, then the corresponding regularized value of the functional is

-

a+ = (Xcph.

(344) Also for this problem one can define a modulus of continuity as follows: C I H k

4) = S U P { l ( f 3 cpfx 1 .fe ff,llLf IIY 5 €1

1345)

and the analysis runs parallel to that outlined in Section V,A. In particular, in the case where the set H is of the type (321), one can introduce the stability estimate PLO)(EP#J)= SUP{l(fAW

I IILfll: + E211Cfll;

5

.’>

(346)

and inequalities analogous to inequalities (331) hold true in this case. Moreover, it is possible to compute &O)(E;@). The result is (Miller, 1970; Bertero et al., 1980a) p p ( E ; 4) = E([L*L+ &*C]

-14,4);’2.

(347)

From this equation, it is not difficult to prove that $ ) ( E ; 4) + 0 as E + 0 with 4 arbitrary whenever the constraint operator C has a bounded inverse (Bertero, 1982). In conclusion, regularized solutions also provide stable estimations of linear and continuos functionals and the corresponding stability estimates can be easily computed.

104

M. BERTERO

3. Resolution Limits We apply now the analysis of the previous sections to the investigation of resolution limits. We restrict ourselves to the case where X is a space of squareintegrable functions. Then the functional (336) takes the form

(f,4)x =

s

(348)

f(x)4(x)dx

(For the sake of simplicity, we consider only the case of real functions of one variable.) Morever, let us assume that the function 4 is positive and peaked upon the point xo (and, for example, symmetric with respect to x,,), that its integral is 1, and its second central moment is a2,i.e.,

s

$(x)dx = 1,

s

(x

-

xo)24(x)dx = a2.

(349)

Then the value of the functional (348)can be considered a blurred value off at the point x = xo. In other words, we want to estimate a local average off over a resolving length a. It is quite natural to predict that the estimation error (for a fixed noise level E ) will grow for decreasing values of a, i.e., for increasing resolution, and therefore the achievable resolution will be obtained by fixing the acceptable value of the estimation error. Consider the case C = I. Then, as follows from Eq. (347) the absolute error in the estimation of the functional (348) is bounded by

4) = E( [L*L + €'I]

-14, (

by.

(350) It is more interesting, however, to introduce relative errors. This approach is quite natural in the case of stochastic regularization (Wiener filters). Here we can proceed along the lines indicated by Bertero et al. (1980a). The absolute error (350) is the maximum value of I(f,4)xIunder the constraint IILfll: ~ ' I I f l l : I E'. This a posteriori constraint is compatible with the a priori constraint llfllx I 1. Then it is easy to see that the maximum value of [(f, 4)xI under this a priori constraint is just I1411x.Therefore, we can define the ratio as an estimate between the a posteriori and a priori maximum value of [(f, of the relative error, i.e., E,&

+

It is an immediate result that E,&; 4) I 1. We can consider now a family { 4b}b,of functions having different values of the variance a 2 (for instance, Gaussians of variance a'). Then the relative error (351) is a function of E and a, let us say E , , ~ ( E , O ) .Since 4bis an approximation of the Dirac delta function, the L2-norm of c # ~tends ~ to infinity

105

LINEAR INVERSE AND ILL-POSED PROBLEMS

as c -+ 0. Without loss of generality, we may assume that this norm is a decreasing function of c. Then from Eq. (351) one can derive the following properties of the relative error: (a) for fixed c,E,&, a) is an increasing function of E ; (b) for fixed E , &,&,a) is a decreasing function of a and tends to one (100% error) as a + 0. The typical behavior of E,,](E, a) as a function of c is indicated in Fig. 11. Numerical computations of these curves for various inverse problems are given by Bertero et al. (1980a; 1980b) and by Abbiss et al. (1983). Let us make some remarks regarding these results. The plot of E , , ~ ( E , a) as a function of (T represents a trade-off between resolution and error. If we fix the

0.5

1

0

FIG.1 I. Illustration of the trade-off between relative error and resolution. It is assumed that the unit of resolution is some typical length related to the problem (for instance, the Rayleigh resolution distance in the case of an imaging system).

106

M. BERTERO

acceptable error on the averaged solution, for instance lo%, then from the curve we can deduce the corresponding resolution and vice-versa. As indicated in Fig. 11, if we reduce the error on the data, say E' < E, then we have an improvement in resolution. However, if the inverse problem we are considering is affected by logarithmic continuity, the change in the plot of E,,,(E, a) is imperceptible even when there is a change in E of several orders of magnitude. This effect is clearly shown by the computations presented by Bertero et al. (1980b) on the inversion of the Slepian operator (84). In all the problems that exhibit this behaviour, there is a resolution limit that is practically noise independent, representing a fundamental limitation on the possibility of recovering details of the unknown object. In the problem considered by Bertero et al. (1980b), for example, this limit is simply the classical Rayleigh resolution distance. A possibility of going beyond this limit is indicated by Bertero and Pike (1982) for the case when: 1) the full image is measured, so that the data contain more information about the solution; 2) the value of c is small, essentially a limitation on the size of the unknown object (which is important a priori information).

In general, the resolution limit as defined by the previous approach depends on the point xo (see Eq. (349)), and therefore there is no uniform resolution over the domain of the variable x. In the inversion of a convolution operator, however, the resolution does not depend on x due to the translational invariance of the problem. The simplest example of a problem with nonuniform resolution is provided by the inversion of an integral operator of the form (62). Since such an operator can be transformed into a convolution operator by taking as new variables the logarithm of the old ones, it is obvious that in this case we have a uniform resolution in log-variables. This implies that the resolution distance increases for increasing values of the variable r in Eq. (62). In fact it is more appropriate to introduce a resolution ratio 6, rather than a resolution distance (McWhirter and Pike, 1978).The meaning of this resolution ratio is the following: given two delta pulses at positions r l and r 2 , it is impossible to resolve these pulses unless r2 2 d0r1. A general method for the estimation of b0 is discussed by McWhirter and Pike (1978) C. Number of Degress of Freedom

The concept of number of degress of freedom was first introduced in optics in terms of the sampling expansion and successively clarified in terms of the basic properties of the PSWF (Toraldo di Francia, 1969a). The typical step

LINEAR INVERSE A N D ILL-POSED PROBLEMS

107

wise behaviour of the eigenvalues of prolate spheroidal functions indicates that while the object can have an arbitrary number of degress of freedom, i.e., an arbitrary number of large components with respect to PSWF, the image always has a finite number of components. This number, called by Toraldo di Francia the Shannon number, is given by S = C / I C and is proportional to the space-bandwidth product. In fact the Shannon number is approximately equal to the number of sampling points interior to the geometric image and is essentially a characteristic parameter of the optical instrument, giving a measure of the information transmitted by the instrument itself. In subsequent work (Toraldo di Francia, 1969b), it was recognized that this number is,in fact, noise-dependent. The dependence is so weak, however, that the original conclusion remained valid for all practical purposes. An interpretation of the result was later given in terms of the theory of ill-posed problems by showing that the problem of inverting the Slepian operator is affected by logarithmic continuity (Bertero et a/., 1980a). The concept of number of degress of freedom or, more precisely, the concept of a noise-dependent number of degress of freedom, can be generalized to all inverse problems that can be treated in terms of singular systems, i.e., problems corresponding to the inversion of a compact operator and inverse problems with discrete data (Twomey, 1974; Bertero and De Mol, 1981a; Pike et al., 1984). As discussed in Section V, an acceptable regularized solution of these problems can be provided by a truncated singular function expansion. When the noise level E is known and a priori information on the solution is represented by a prescribed bound E on the norm of the solution, the truncation rule is given by Eq. (308). This condition has also a nice statistical interpretation. Assume that the data is represented by Eq. (9) and that both f and h are representatives of zero-mean, uncorrelated random processes. Moreover, assume that the signal f is from a white noise process with power spectrum E 2 and that h is also from a white noise process with power spectrum e2.Then the variance of any given component of f with respect to the basis ( U k f is: ( I ( f , u , J X I 2 ) = E 2 . Analogously, the variance of any given component of h with respect to the basis ( u k } is: ( ~ ( hUk)yl2) , = E’. From the relation ( g , uk)y = (Lf, 0k)Y + ( k uk)Y = ak(f, uk)X + (h,uk)Y we get 9

This equation shows that the variance of a given component of the data consists of two term, the first being the contribution of the object and the second the contribution of the noise. The first term tends to zero as k + 00, or becomes very small for large k in the case of ill-conditioned inverse problems with discrete data; the second term is constant. Moreover, the first term is a

108

M. BERTERO

decreasing function of k. Therefore, for k greater than some critical value, the variance of the noise is greater than the variance of the object contribution and the corresponding data components do not contain any information about the object. Equation (308) simply states the requirement that the variance of the object contribution must be greater than the variance of the noise. In the case of the Slepian operator, the quantity

N ( E / E )= max{k

+ Ilok 2 E / E } ,

(353)

is approximately equal to the Shannon number. Therefore, it is quite natural to call it the number of degrees offreedom (NDF) in the case of the more general problem. The N D F is a function of the signal-to-noise ratio E / E and gives a measure of the information content of the data. It is a very useful parameter for estimating, in general, how many distinct elements one can resolve with the available data. From the examples discussed in Section 111, it follows that for some important inverse problems with discrete data, such as the moment problem or Laplace transform inversion, the N D F can be quite small (on the order of 3 or 5, and in any case less than 10).For other problems, such as tomography, the N D F can be very large. Here is another way of introducing a distinction between mildly and severely ill-posed problems. In general, in the case of a mildly ill-posed problem where it is possible to restore Holder continuity by means of prescribed bounds on a small number of derivatives of the solution, the N D F depends rather strongly on the signalto-noise ratio. It is possible to obtain a significant increase in the N D F by increasing E / E .In contrast, in the case of a severely ill-posed problem affected by logarithmic continuity, the N D F is nearly independent of the signal-tonoise ratio, at least in the case of reasonable values of this quantity. This usually happens if the singular values 4 tend to zero exponentially fast as k -+ co (this is the case, for example, of the Slepian operator). Then if No is the value of N ( E / E )corresponding to some preassigned value of the signal, can easily deduce that to-noise ratio, say E O / e Oone N ( € / E ) = ( N o - 1)

+ clog,,(E€,/€E,)

(354)

where c is a constant (Bertero and De Mol, 1981a). Therefore, if No is very large, even when improving the signal-to-noise ratio by many orders of magnitude there is no significant improvement in the NDF. This is what happens in the problem discussed by Toraldo di Francia mentioned at the beginning of this section, where the N D F can be considered as practically noise-independent and equal to No. It is important to point out that the N D F can always be introduced (and computed) in the case of an inverse problem with discrete data, since in this

LINEAR INVERSE AND ILL-POSED PROBLEMS

109

case one can always use singular function expansions, as indicated in Section II1,A. Then the N D F can depend not only on the signal-to-noise ratio E / E ,but also on the number N of data points. This is true, for example, in the case of the finite Hausdorff moment problem (Section 111,D). If we define the N D F as the number of singular values greater than (recall that the singular values are just the square roots of the eigenvalues of the Hilbert matrix), then we have N D F = 4 for N = 4,N D F = 6 for N = 10, N D F = 8 for N = 50, and N D F = 9 for N = 100. Clearly, the N D F depends on the number N of given moments, even if the dependence is rather weak. This dependence, however, is related to the fact that the (infinite-dimensional) Hausdorff moment problem does not correspond to the inversion of a compact operator, and therefore the singular values and singular functions of the finite Hausdorff moment problem do not have a limit as N -i 00. . The situation is different when a problem with discrete data is a finite version of an infinite-dimensional problem correspopding to the inversion of a compact operator. In this case, the N D F can be defined for the infinitedimensional problem and the N D F of any finite version of it cannot be greater than the limiting NDF. But when the number of points of the finite version is sufficiently large, its first singular values are good approximations of the corresponding singular values of the compact operator. It follows that for N greater than a suitable number No of data points, the N D F is independent of N and equal to the N D F of the compact operator. An illustration of this behavior for Poisson transform inversion is given by Bertero and Pike (1986). We have also some indications that in the inversion of a compact operator, satisfactory results can be obtained using a finite version of the problem with a number of (suitably placed) data points, which just coincides with the N D F of the infinite-dimensional problem. We give an example taken from the inversion of the Laplace transformation in a weighted space, using as a weight the gamma distribution (Bertero et al., 1985~).In this case the Laplace transformation is compact and the NDF, defined as the number of singular values greater than lop2,is 4.In Fig. 12 we give the reconstruction of two delta pulses having the same mass, i.e., 0.5. Clearly, using four geometrically spaced data points, it is possible to obtain a reconstruction that practically coincides with the reconstruction obtained using 32 uniformly spaced data points. Of course, the position of the four points must be optimized (in this case, the criterion is minimization of the condition number). As a conclusion, we find that this example provides a strong indication of the fact that the N D F can coincide with the optimum number of data points. In other words, one needs as many data points as pieces of information that can be extracted from the complete (infinite-dimensional) noisy data (i.e., the maximum number of these pieces of information).

110

M. BERTERO

-0.5 FIG. 12. Reconstruction of two delta pulses in the case of Laplace transform inversion in a weighted space using 32 equidistant points (dotted line) and 4 geometrically spaced points (full line).

D . Impulse Response Function: Another Approach to Resolution Limits

Given a regularizing operator R , with a fixed and an exact (noise-free) data g = L f , the corresponding regularized solution is f, = R , L f . Since f, converges to f when the operator L has an inverse L-' and to f ' , when L has a generalized inverse L', the operator

T,= R,L

(355)

is an approximation of the identity operator in the first case and of the projection operator over N(L)' in the second case. This is the mathematical interpretation of the operator T,,emphasizing the fact that, in the noise-free case, the regularized solution f, is just a (stable) approximation of the true solution f (or generalized solution f '). There exists, however, an interesting physical interpretation of T,. Since L describes the transmission of the signal by the instrument in the absence of noise, while R, describes the processing of the data by the computer in the absence of round-off errors, T, describes the total effect of both the transmitting instrument and the processing computer (in the absence of any kind of error).

LINEAR INVERSE AND ILL-POSED PROBLEMS

111

If T, is an integral operator,

fh) = ( T , f ) ( x )=

s

A,(& x ’ ) f ( x ’ ) dx,

(356)

the averaging kernel A,(x, x ’ ) is the impulse response function of the system consisting of instrument plus computer. Moreover, if for fixed x , A,(x, x ‘ ) as a function of x‘ has the form of a central lobe flanked by decreasing side-lobes, the width of the central lobe may be used as a measure of the resolution achievable at the point x by means of the algorithm R,. Notice that the form (356) for the approximate solution is just the starting point of the Backus-Gilbert method. Moreover, in the case of inverse problems with discrete data, if we assume that X is a space of squareintegrable functions and that the regularized solutions are defined by means of windowed singular function expansions, the kernel A,(x, x ’ ) is given by N- 1

In this approach, it is obvious that the resolution is determined by the choice of the regularization parameter. This choice, however, can be an important point in the case of mildly ill-posed problems, while in the case of severely illposed problems the choice of the regularization parameter depends very weakly on the noise and we again find a resolution limit that is practically noise-independent. It is obvious that the form and the width of the central lobe depend, in general, on the regularization algorithm. We think, however, that the width does not depend strongly on the algorithm, since it is a measure of the achievable resolution and, therefore, a measure of the information that can be extracted from the data. We must never forget the principle formulated by Lanczos and emphasized at the beginning of Section V. Though this question must still be investigated carefully, it is useful to examine an example in support of this claim. Let us consider inversion of Fraunhofer diffraction data. This corresponds . to inversion of an integral operator of the form (62) with K ( x ) = J 1 ( x ) 2 / xThe corresponding Fredholm integral equation of the first kind can be approximated conveniently by means of the exponential sampling method (Ostrowsky et al., 1981). When the problem has been discretized in this way, one can compute singular values and singular functions. For example, using 32 geometrically spaced data points and assuming that the support of the unknown function is in the interval [7,130] (the wavelength of the incident radiation is taken as a unit of the radius of the particles), one finds 18 singular values greater than

112

M. BERTERO

In Fig. 13 we give various reconstructions of a delta pulse of unit mass obtained using various regularization algorithms. The radius of the particles is represented on a logarithmic scale in units of the wavelength of the incident radiation. In all the reconstructions the maximum number of singular functions is 18. In the figure we have superimposed 10 different reconstructions corresponding to 10 different contaminations of the exact data by means of random errors of the order of 1%. In this way a clear picture of the robustness of the solution is obtained. In Fig. 13(a) we plot the reconstruction obtained by means of truncated singular function expansion. This provides the maximum resolution

a)

FIG.13. Reconstruction of a delta pulse in the case of the inversion of Fraunhofer digraction data. (a) Truncated singular function (TSF) expansion. (b) Tikhonov regularization with c( = (c) TSF expansion with triangular window. (d) TSF expansion with Hanning window.

113

LINEAR INVERSE AND ILL-POSED PROBLEMS

1

10

100

1000

FIG. 13. (Continued)

achievable by means of 18 singular functions, but this resolution is obtained at the cost of very large side-lobes. In the other parts of the figure, we plot the reconstructions obtained by means of various filterings of the previous truncated singular function expansion. In particular, Fig. 13(b) corresponds t o the Tikhonov window, Fig. 13(c) corresponds to the triangular window, and Fig. 13(d) corresponds to the Hanning window. In all these cases, we have a loss in resolution of approximately a factor of 2 with respect to the reconstruction of Fig. 13(a) but the side-lobes are always much smaller than in Fig. 13(a) and the reconstructions are nearly positive. The previous example seems to indicate that we can obtain positivity at the cost of a loss in resolution, a result which is in conflict with a rather commonly held opinion about the beneficial effect of the constraint of positivity in the solution of inverse problems. This question, however, is beyond the scope of

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this paper, which is essentially devoted to linear methods for linear problems and is mentioned only as an example of the many questions that are still open.

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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS VOL. 75

Recent Developments in Energy-Loss Spectroscopy* JORG FINK Kernforschungszeutrum Karlsruhe Institut f u r Nukleare Festkorperphysik Karlsruhe Federal Republic of Germany

I. Introduction . . . . . . . . I1. Principal Features . . . . . . A . Fundamentals of Electron Scattering . . . . . . . . B. Models of Dielectric Functions . . . . . . . . . . C. Interface Effects. . . . . . . . . . . . . . . . . D. Core-Level Excitations . . . . . . . . . . . . . . E. Data Evaluation . . . . . . . . . . . . . . . . I11. Instrumentation . . . . . . . . . . . . . . . . . A . Principle of Operation . . . . . . . . . . . . . . B. Electron Source, Monochromator. Analyzer. and Detector . C. Zoom Lenses, Accelerator. and Decelerator . . . . . . D . Scattering, Characterization and Preparation Chamber . . E. Vacuum System and Electronics . . . . . . . . . . F . Spectrometer Performance . . . . . . . . . . . . IV. Sample Preparation . . . . . . . . . . . . . . . . A . Thin Film Deposition . . . . . . . . . . . . . . B. Preparation from Macroscopic Solids . . . . . . . . C . Ion Implantation and Doping . . . . . . . . . . . V . Nearly-Free-Electron Metals . . . . . . . . . . . . VI . Rare Gas Bubbles in Metals . . . . . . . . . . . . A. Pressure and Density in Bubbles . . . . . . . . . . B. Surface Plasmons on Bubbles . . . . . . . . . . . VII . Amorphous Carbon . . . . . . . . . . . . . . . . VIII . Conducting Polymers . . . . . . . . . . . . . . . . IX. Superconductors . . . . . . . . . . . . . . . . . A . Transition Metal Carbides and Nitrides . . . . . . . B. A-IS Compounds . . . . . . . . . . . . . . . . C . Ceramic Superconductors . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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122 124 124 129 142 144 147 148 148 . 150 . 152 . 153 . 153

. 155 157 157 . 158 . 159 . 160 . 167 . 167 . 177 181 187

215 . . . . 216 . . . 220 . . . . 221 . . . 226 . . . 226

* This article has been accepted as Habilitation Thesis by the Fakultat fur Physik. Universitat Karlsruhe 121 Copyright 0 1989 by Academlc Press Inc All nghts of reproduction in any form reserved

ISBN 0-1 2-014675-4

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JORG FINK

I. INTRODUCTION Scattering experiments with fast particles have become an important tool in various fields of physics. Compared to absorption experiments, they make it possible to investigate not only the frequency but also the wave length of possible excitations of matter, by means of angular resolved measurements. In solid state physics, most of our experimental knowledge about the structure of solids has been obtained from elastic scattering experiments with X-rays, electrons, and neutrons. Inelastic scattering experiments with neutrons have provided considerable information on the dynamic properties of solids in the low-energy range ( E < 100 mev), i.e., on phonons and magnons. Inelastic scattering of light (Raman scattering) has been a powerful means of probing the vibrational motions of atoms and molecules in bulk material. Recently, methods of scattering thermal neutral atoms (mainly He) on the surface of solids have been developed to gather information on surface phonons (Toennies, 1984). The advent of powerful synchrotron radiation sources has led to the first pioneering inelastic X-ray experiments for the study of phonons (Burke1et al., 1987) and collective excitations of electrons (Schulke et al., 1984, 1986). Inelastic scattering of electrons has provided important information on the vibrational modes of adsorbates (Ibach and Mills, 1982) and on surface phonons of metals (Rocca et al., 1986) in the low energy range ( E < 100 mev). At higher energies, inelastic electron scattering or electron energy-loss spectroscopy (EELS) in transmission has been the classical method for investigating the collective excitations of electrons, i.e., plasmons. The method has been further improved during the years to provide information on interband transitions between valence bands and conduction bands and on core-level excitations in order to learn about the unoccupied density-of-states (near-edge fine structure) and on the lattice structure (extended-edge fine structure). EELS at medium energy ( E < 20 eV) is in competition with optical spectroscopy and at higher energies (20 < E < 2500 eV) with X-ray absorption spectroscopy using synchrotron radiation. In general, the resolution offered by optical spectroscopy is superior to that encountered in EELS. However, this does not hold for soft X-rays in the energy range of 250 to 800 eV where it is still not an easy matter to build high-resolution monochromators. Moreover, as mentioned above, the scattering methods have the great advantage that an additional degree of freedom-momentum transfer-can be varied, which is not possible in optical spectroscopy. Furthermore, the ease with which one can cover the excitation spectrum from infrared to soft X-ray (2500 eV) is still a tremendous advantage even with powerful synchrotron radiation sources. Inelastic X-ray scattering is developing to a tool that

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

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extends EELS to higher momentum transfer. However, at lower momentum transfer the cross-section for inelastic X-ray scattering becomes very small; as yet, EELS is without competition in this region. In comparing EELS with other methods, we should not forget that, experimentally, EELS suffers from a serious problem that is not found in neutron and X-ray scattering; that is, the interation of the electrons with the solid is far stronger than one would like so that only samples with a thickness of about 1000 A can be probed. This is, in some cases, an advantage, but in many cases it entails a considerable effort in sample preparation. This hurdle and the complicated instrumentation may be why EELS has not become as widespread a method as, for example, photoelectron spectroscopy. The Franck-Hertz phenomenon in gases, which has been the source of information on electron energy levels in gaseous atoms and molecules, was first applied to solids by Rudberg (1929). In this doctoral thesis, he reported measurements of the kinetic energy of electrons that had been reflected from the surface of metals (Cu and Ag). The first electron energy-loss measurements in transmission with primary energies from 2 to 8 keV were performed by Ruthemann (1941) and showed discrete excitations in Be and A1 that were later explained by free-electron plasmons of the metals (Pines and Bohm, 1952). The same author reported (1942) for the first time the excitation of core-level electrons in collodium by EELS. A first review on these early-day EELS experiments was given by Marton et al. (1955). The first systematic applications of EELS to solid-state physics (mainly collective excitations and interband transitions) were made by Raether’s group at Hamburg (Raether, 1965; Daniels et al., 1970; Raether, 1977; Raether, 1980). This work was continued and extended to core-level spectroscopy by Schnatterly’s group at Princeton (Schnatterly, 1979) and other groups. Alongside these investigations, electron microscopists became interested in EELS as an extremely powerful tool for microanalysis using core-level excitations. Excellent reviews of the field were given by Colliex (1984), Colliex an Mory (1984), and Egerton (1986). With time the energy resolution of these spectrometers using a transmission electron microscope (TEM) together with an energy analyser below the microscope column was improved and interesting near-edge structure investigations were performed (Colliex et aZ., 1985; Leapman et al., 1982;Grunes et al., 1982; Lindner et al., 1986). However, most of these modified TEM had excellent spatial resolution but very poor momentum transfer resolution. Therefore, TEM were used in only a few investigations on valence band excitations and, in particular, momentum transfer depending measurements. This work reviews EELS studies by my colleagues and myself during the last five years using a dedicated electron energy-loss spectrometer. Following a theoretical introduction; and a description of the Karlsruhe spectrometer, it describes work on nearly-free-electron metals, rare gas bubbles in metals,

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JORG FINK

amorphous carbon, conducting polymers, and superconductors. A short review on these activities has been given previously (Fink, 1985a). This contribution should not be regarded as a systematic discussion of the method of EELS. Rather, it is an illustration of the present state of the technique applied to solid-state physics and to material science. 11. PRINCIPAL FEATURES

A . Fundamentals of Electron Scattering

The geometry of the electron scattering experiment is shown in Fig. 1. An electron specified by a momentum hk, is scattered into a state with momentum hk, . The energy loss and momentum transfer (momentum transfer a function of the energy loss and scattering angle 6 ) are given by hw

=

E , - El = hz(kz - kT)/2rn

and

hq = hko

- hk,.

The basic quantity that is measured is the partial differential cross-section d20/di2dE.It is the fraction of electrons of incident energy Eo scattered at an angle 8 into an element of solid angle dQ with energy between El and El dE. The cross-section has the dimension (area/energy). In the transmission electron energy loss experiments, the energy and momentum of the incident electrons are very large compared to the energy and momentum of the electrons in the solids. In the Karlsruhe spectrometer, the primary energy is at present

+

SCATTERED ELECTRON

FIG. 1. Definition of energy loss and momentum transfer in a transmission inelastic electronscattering experiment.

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170 KeV, which corresponds to a wave number k, of 228.4 k', or a wavelength 1 of 0.0275 A. In solid-state physics, interesting wave vectors are around 1 A-' and, therefore, the scattering angles are extremely small. For the values given above, the scattering angle 0 is about 4 mrad or 0.25".The wave vector q can be decomposed into two vectors, one parallel to and one perpendicular to, the incoming electron beam. For 141 ,

(4)

where O ( t ) = exp(iH,t)O exp( - iH,t), H , (w)is the Fourier transform of H,(t), and 6 is a very small positive real number. In connection with electronscattering experiments, it is interesting to look at a particular response function, namely the response of the charge density p(r,t) = en(r,t) to an external, scalar, electrostatic potential @)ex,, i.e., a longitudinal electrostatic perturbation. The perturbing Hamilton operator can be given by

and the density-density response function x is then defined by (Pind(r,w))

s

( 1 / 2 ~ ) dleio'(pind(r, t)> = jd3rt

x(r, rf,w)Qext(r',0).

Using Eq. (4) the response function x is then given by

For a homogeneous system, such as the jellium model for free electrons where the electron charge is compensated by a homogeneous positive background due to the ions, the response function depends only on r - r' and the Fourier transforms of Eqs. (6) and (7) are given by

There is a close similarity between x(q, w ) and the dynamic structure factor S(q, w ) given in Eq. (4). A detailed comparison shows that the structure factor can be expressed in terms of the imaginary part of the density response function:

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JORG FINK

where exp( - hw/k, T ) - 1 = - I for electron excitations with hw >> kBT.Thus the calculation of the differential cross-section reduces to a calculation of the imaginary part of the density-density response function. It is interesting to connect this result to another response function the dielectric function. The dielectric function may serve as a simple unifying concept for the theories of the electron gas. In its most general form it is defined by E(r, t ) =

ss

d3r' dt'E^-'(r, r', t - t')D(r', t).

(1 1)

For a homogeneous system, the dielectric tensor depends only on r - r'. The same holds for the macroscopic viewpoint in which an average is taken over the system, therefore, it is a homogeneous system which is studied. The macroscopic dielectric function eM is measured by an external probe that transfers momentum corresponding to a wavelength that is large compared to the dimensions of the Brillouin zone. Then the Fourier transform of Eq. (1 1)is given by E(q, 4 = EM1(%w)D(q, w). Now, with D =

and E

=

(12)

-Val,,,we may write

'/EM(q, 0) = E(q, w)/D(q, w)

= @tot(q,w)/@ext(qy0).

(13)

Using @lol(q,0)= OeXl(q,0)+ @in&, w), Eq. (8), and the Fourier transform of the Poisson equation @)ind(q,m)= uqpind(q,co)(uq = 4 n / q 2 is the Fourier transform of the Coulomb potential), we obtain

By combining Eqs. (2), (6),and (ll), we obtain an important result for the relation between the differential cross-section and the macroscopic dielectric function :

where Im[ - l/a,(q, w)] is the macroscopic loss function. In many cases, the loss function is almost independent of q, in which case the differential crosssection for inelastic electron scattering decreases approximately as l/q2. According to Eq. (lc), q 2 = q i + 4:. For a given energy loss, q I 1can be calculated from Eq. (lb); q L (or angular) distribution of the inelastically scattered electrons is then a Lorentzian with full width of 2 q I I .For incident electrons of 170 keV kinetic energy, q , , = 0.0067 k' for a 10-eV loss and q I 1=

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

129

0.67 A-' for a 1000-eV energy loss. Therefore, valence band excitations show a rather narrow q,, (angular) distribution, while high-energy core excitations tend to have larger ones. First, from In the following, we summarize some useful relations for E(q, o). the fact that the response of the system is causal, one obtains the KramersKronig relations

For the calibration of the loss functions and also for a consistency check of the data, sum rules are useful, the most important of which are as follows: f m

Jo

do o Im( - 1/&(q,o))= (n/Z)w;

:j

d o w Im E(q, o)= (n/2)w;,

where the plasmon frequency is given by w,, = ( 4 ~ n e ~ / r n ) " ~

and n is the density of the valence electrons. Sum rules are frequently used to define an effective number of electrons, n e f f ,contributing to excitations up to a finite frequency range oc:

jT

d o o Im E(q, o)= ( n / 2 ) o ; ( n e f f / n ) .

B. Models of Dielectric Functions 1 . Drude-Lorentz Model

The Drude-Lorentz model is based on a phenomenological view. Nevertheless, it is still extremely helpful for a rough understanding of the dielectric functions and, in particular, the loss function for zero momentum transfer. It is assumed that excitations of the electrons can be expressed by a sum of oscillators satisfying the equation of motion

+

rni(d2r/dt2 yidr/dt

+ o ? r ) = qiE(t),

where m i , yi, mi, and qi are the masses, damping constants, eigenfrequencies,

130

JORG FINK

and charges, respectively, of the oscillators. By solving the differential equation the polarization qir and thus the dielectric functions can be derived to be 1

&(W) =

+ 471ci mi(w’

niq? - o2- iyiw)’

where n, are the oscillator strength of the oscillators. In order to illustrate typical dielectric properties of solids, we show in Fig. 2 the loss function, = Re&,and c2 = ImE as calculated by Eq. (22) for various combinations of oscillators, the parameters of which are listed in Table I. For simple sp metals, the charges qi = e, are almost free, i.e., the energy of the one oscillator that is needed is zero and the damping y is given by the scattering of the charges by phonons or impurities. This “Drude-model” leads to the dielectric functions shown in Fig. 2a. Close to the zero-crossing of cl, where e2 is small, the loss function Im( - I/&)= E ~ / ( E ; + E : ) has a strong maximum due to collective excitations of the electrons, i.e., the plasmon. The plasmon energy is given by Eq. (20) and is proportional to the square root of the density of the electrons. In the noble metals, an additional oscillator appears due to transitions from the filled d bands to states just above the Fermi energy. The influence of these interband transitions on the dielectric functions is shown in Fig. 2b. If the oscillator is strong enough, it causes a second zerocrossing of c1. This leads to a second maximum in the loss function below the free-electron plasmon, called an interband plasmon in contrast to the freeelectron intraband plasmon. Negative polarization due to interband transitions near the free-electron plasma energy leads to a shift of the zero-crossing of and therefore shifts the intraband plasmon to higher energies. A typical example for this situation is Ag or TiN. In transitionmetals, various additional oscillators appear that strongly distort the free-electron loss function (see, e.g., Wehenkel and Gauthe (1974)). Excitations from inner shells are simulated in TABLE I PARAMETERS ( I N eV) FOR

THE

DRUDE-LORENTZ DIELECTRIC FUNCTIONS SHOWN IN FIG.2

Drude

Lorentz

nl

”0

hwo

yo

(a)

1.0

0

4

-

(b)

0.83 0.95 -

0 0

4

0.17 0.05

(c)

(d) (el (f)

-

4 -

-

_

-

_

1 .o 0.83 0.80

hol

4

35 8 8 8

yI

_ 0.5 0.5 4 4 4

n2

w2

_

_

-

ha3

n3

y2

_

-

y3

-

-

-

-

-

-

-

-

-

-

-

-

-

3 3

0.5

-

0.17

0.16

0.5

-

-

-

-

0.4

1.5

0.5

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

13 1

21 20

FIG.2. Calculations of the loss functions and the dielectric functions E , and c2 of solids using the Drude-Lorentz model. The parameters are given in Table 1. (a) Free-electron metal. (b) Free-electron metal with a strong interband transition at low energy leading to an interband plasmon. (c) Free-electron metal with a core-level excitation. (d) Insulator. (e) Semiconductor with a strong interband transition across the fundamental gap. (f) Semiconductor with transitions related to defect states in the gap.

132

JORG FINK

Fig. 2c by an additional oscillator far above the plasmon energy. In this case, e2 of the oscillator is rather small (c2 0.7 A-', it is difficult to detect these observed for q > 0.25 kl. excitations, since there is considerable broadening. The maxima of these excitations are shown in the insert of Fig. 14 and are close to the upper border of the gap between intraband and interband transitions in agreement with the calculations of Foo and Hopfield (1968). This indicates that these zoneboundary collective states in Na for small q are dominantly caused by interband transitions. Similar zone-boundary collective states have been observed recently in Li and Be with much higher intensity as in these metals, the pseudo-potential is much stronger than in Na metal. The latter experi-

0.5

qrA

0.45

I

I

1

2

I

I

I

3 4 5 ENERGY ( e V )

I

6

FIG. 14. Momentum dependence of single-particle excitations in single-crystalline Na for q 11 [I lo] (solid line) and 911[lo01 (dashed line).The difference between the two directions is due to zone-boundary collective excitations. In the insert, the dispersion of the maximum of these excitations is shown.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

167

ments were performed by inelastic X-ray scattering using synchrotron radiation (Schulke et al., 1984, 1986, 1987). In summary, the recent experimental studies on heavy alkaline metals that form a more strongly-coupled plasma than, e.g., A1 or the light alkaline metals, have raised a number of new questions. First of all, is the extension of the RPA dielectric function by a dynamic local field correction sufficient or are more refined theories necessary? What is the influence of the low-lying unoccupied d states close to the Fermi energy on the dielectric functions of the heavy alkaline metals? Is it possible to prepare dilute alkaline metals to study even more strongly coupled plasmas? There are many new experimental and theoretical problems.

VI. RAREGASBUBBLESIN METALS A . Pressure and Density in Bubbles

Rare gases (RG) are insoluble in metals. For example, the heat of interstitial solution for He in A1 is negative and as large as -2.7 eV per atom, thus making the equilibrium He concentration negligibly small at room temperature. The microscopic origin for this fact is the net pseudo-potential from the RG atoms that the conduction electrons of the metal experience is repulsive as it entails orthogonality of the wave functions to the RG closed shell system. When RG atoms are somehow introduced, a formation of clusters is observed, and at still higher concentrations RG atoms precipitate into bubbles. The growth mechanism of the bubbles, which is strongly correlated with the pressure inside the bubbles, is still a matter of debate. EELS has provided a method of measuring the density and pressure in RG bubbles in metals and during the last five years various studies have been performed in this field. While RG bubbles are interesting per se, they are also of enormous technical importance. In fission reactors, such fission products as Kr and Xe are produced in the fuel elements, and He by (n,a) reactions. Still higher doses of He will be produced by ( n , a ) reactions in the first wall of future fusion reactors. The RG bubbles lead to strong embrittlement of materials due to bubbles in grain boundaries. At the surface, RG implantation produces blistering and exfoliation. In addition, RG bubbles are important for ion beam milling of layer structures as well as in microelectronics in which metallic films are produced by sputter deposition. Ion-beam mixing is used to glue protective layers on metal surfaces to improve tribological properties of metals. EELS has led to significant results in many of these fields. In this section we describe some typical investigations related to the fundamental properties of RG bubbles in metals.

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JORG FINK

1. He in Metals

In an early theoretical work, Ohtaka and Lucas (1978) had already suggested studying the thermodynamic state of He in bubbles by measuretransition. The earliest experimental studies were ments of the He 1 'S0-21P1 realized by measuring this transition by means of ultraviolet absorption using synchrotron radiation (Rife et al., 1981). In this work the first EELS measurements were reported. In this section we review similar systematic studies of RG bubbles in metals. Figure 15 shows a representative loss spectrum of a 1000-A thick single-crystalline A1 film in which 13 at.% He was homogeneously implanted. Besides the single- and double-volume plasmon at 15 and 30 eV, respectively, and a surface plasmon of the Al,O,-coated, flat surface of the A1 metal film at 6.7 eV, two additional peaks are observed that are related to the He bubbles. At 24 eV, there is a strongly broadened He 1S2P transition that shows a tremendous pressure shift compared to the freeatom value 21.23 eV. The peak at 10.6eV can be assigned to a surface plasmon on the He bubble interface. These surface plasmons will be discussed in more detail in Section VI,B. The origin of the strong blue-shift of the He 1s-2P transition is the strong interaction of the excited-state 2P wave function, which has a mean diameter roughly four times greater than the 1s ground state, with the wave functions of the neighbouring He atoms in the ground state. The Pauli exclusion principle excluding the occupation of the 1s' closed shell by a third electron leads to a strong blue shift with increasing He density. Other mechanisms that may lead to a shift of the He line have been discussed (Ohtaka and Lucas, 1978; Lucas et al., 1983). The effect of a resonant transfer of the optical excitation to neighboring atoms via dipole-dipole interaction can be calculated by the

-

He 1'5, -2'P,

Jzp iq& Ln

VP

2 W

2xVP

5 vl

+ II) t W z

z

0

5

10

15 20 25 ENERGY l e v 1

30

35

40

FIG.15. Electron energy-loss spectrum of He-implanted A1 film (cHe= 13 at.%).SP: surface plasmon; BSP: bubble surface plasmon; VP: volume plasmon. The He I1S,-2'P, transition is strongly shifted compared to the free-atom value of 21.23 eV.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

169

Clausius -Mosotti formula for the dielectric function and leads to a small red-shift. The calculation of the effective dielectric functions within the framework of the Maxwell-Garnett theory leads to a small blue shift. The latter two effects are important at low densities such as that of liquid He, whose number density n = 0.022 k3. At higher densities ( n 0.1 k 3 these ) , effects are negligible compared to the effect of Pauli repulsion. In order to obtain the pressure and density inside He bubbles, it is necessary to correlate the energy shift with He density. While there is agreement that the energy shift is proportional to density in a given energy range, there is still considerable disagreement as to the proportionality constant. One approach assumes pairwise additivity of the interactions, which are taken from ab initio quantum mechanical calculations of the 2p excimer potential energy curves (Lucas et al., 1983); many-body perturbations are neglected. The radial pair distribution function required for these calculations is computed from theoretical models of dense fluids. The blue shift of the resonance line is found to depend nearly linearly on the density: A E = Cn, with C = 31 A3eV. Another approach uses a multiconfigurational selfconsistent field method for calculations of the ground and excited states of a He atom surrounded by a nearest-neighbor shell of 12 He atoms (Taylor, 1985). The He,, cluster geometry was taken to be fcc. These calculations predict a linear relationship between energy shift and density for number densities 0.01 I n I0.05 k 3with a proportionality factor C = 30 A3eV. However, above n = 0.06 k 3a red-shift is predicted. The third approach (Manzke et al., 1982; Jager et al., 1983) uses ab initio self-consistent electron band-structure calculations for solid (fcc) He for densities n in the range 0.10-0.25 A-3. While the use of density functional theory for ground-state properties is well established, the calculation for excited energies poses serious problems. Therefore, the energy shift was obtained by a calculation of the difference of two total energies, the ground state and the excited state described by He atoms with 1 electron in the 2p shell located on Au sites in a Cu,Au structure where each excited He is surrounded by 12 non-excited ones on the Cu sites. The calculations employ the augmented spherical-wave method. The procedure, called ASCF (self-consistent field), was applied with great success to determine the excitation energies in LiF (Zunger and Feeman, 1977). In the density range mentioned above, the calculations yield good proportionality between energy shift and density, with a proportionality factor C = 22 A3eV. The three approaches result in relationships between line shift and density that differ considerably and, therefore, due to the non-linear relationship between density and pressure, lead to highly different pressure values. The first approach of Lucas et ul. (1983) starts from the low-density limit of an excited He, molecule. At higher densities, pairwise addition of

-

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JORG FINK

the effects of all surrounding He atoms is assumed. This assumption, which does not take into account many-body perturbations, may be the reason for the high C value. The second approach predicts a maximum line shift of 1.3 eV, in clear contrast to experimentally observed line shifts of up to 4 eV. Therefore, the result for lower densities also seems to be questionable. These problems are probably due to the fact that the cluster chosen was too small. The third approach describes the high-density, many-electron system quite well, but assumes the excited atoms do not greatly influence the next nearest neighbor sites. However, estimates of this effect indicate that even smaller values for C would be expected if the concentration of excited He atoms is reduced. Therefore, according to the third approach, the value C = 22 A3eV is a maximum value. Since this value of C is based on reasonable assumptions and is supported by various experimental findings as outlined below, it serves as the basis for the analysis of EELS data. It should, however, not be forgotten that more sophisticated calculations are necessary to improve the accuracy of the proportionality factor in order to obtain reliable values for the pressure in He bubbles. Once the density is calculated by the E(n) relationship, the pressure inside the bubbles can be derived by an equation of state (EOS). A discussion of EOS of He was given recently in a review on density and pressure of He bubbles by Donnelly (1985).In this field, the data are less controversial with respect to a pressure evaluation from densities. The analysis of EELS data was based on an EOS for He at extremely high densities, including solid fcc He worked out by Trinkaus (1983). Using these tools based on theoretical calculations, it was possible to derive the density and pressure in bubbles as a function of implanted He concentration or as a function of annealing temperature. These results gave important information on the growth mechanism of He bubbles. The pressure shift decreases with increasing He concentration as shown in Fig. 16 for A1 implanted with 3, 13, and 26 at.% He (see also the values of nAE and pAEin Table 111). A transmission electron microscopy (TEM) study of the mean bubble radius R as a function of implanted He concentration showed that the mean radius increased with increasing cHe.Figure 17 shows He pressure inside the bubble plotted against the inverse mean radius. Despite the fact that the pressure (due to uncertainties in C ) and the radius may be connected with errors up to 50% and 30%, respectively, the data show a linear relationship between the pressure and the inverse radius with a proportionality factor of about 900 KbarA. For temperatures above one-half the melting temperature ( T , ) ,the bubbles are expected to grow by absorption of thermal vacancies and the pressure relaxes to the thermal equilibrium pressure p = 2y/R, where y is the surface free energy (y = 1 N/m for Al). These pressures are indicated by a broken line in Fig. 17. The experimental findings of a much higher pressure clearly demonstrate that the bubbles are overpressurized and not in thermal equilibrium. Even a 50% error in the pressure due to uncertainties in C would

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

171

ia t 'A 1i°C '

a L

l

Al

26

-

Al

13

-

Al

13

43C

Al

13

49C

I

ENERGY i e V )

FIG.16. He 1'S0-2'P, linesafter subtractionof the backgrounddue to thematrix for Al,Ni, and amorphous NiP samples implanted with various He concentrations. The labels h and I indicate high- and low-dose implantation. TAindicates the annealing temperature over a time period of 2 h.

TABLE 111 DENSITY AND PRESSURES IN RG BUBBLES WITH MEANRADIUS R FROM ENERGY SHIFT( A E ) AND BRAGGREFLECTION SHIFT (Ak).pE A N D pL ARE THE CALCULATED EQUILIBRIUM PREssURE AND THE THRESHOLD h E s s U R E FOR LOOP PUNCHING, RFSPECTIVELY RG Matrix He He He He He He He Ne Ar

Kr Xe

Al A1

Al Al Ni Ni

NIP Al A1 Al AI

CRG TA (at%) ("C)

3.3 13 26 7.7 low high 2

-

475 -

2

-

3 3

-

(A) 6.5 10 20 25 -

15

13 15 17.5 13

AE (eV) 3.32 2.74 1.94 1.0 -4.7 3.7 2.75 1.4 1.0

-

-

~ A E

(A-3) 0.151 0.125 0.088 0.0455

0.214 0.168 0.125 0.073 0.04 -

PIE (Kbar)

nAk

Pd*

-

-

32 20

-

10

(k')(Kbar)

PE PL (Kbar) (Kbar)

150 85 31 5

-

-

7.5

500

-

-

-

250 80 40 60

-

-

-

-

31 15

-

-

15

0.035 0.026 0.023

30 20 30

13 11 15

-

-

123 80 40 32 280 -

62 53 46 61

172

JORG FINK

RIA)

0.1

0.05

i/R

0.15

[R-7

FIG. 17. Pressure in He bubbles in an A1 matrix as a function of the inverse mean bubble radius 1/R.Closed circles: Not annealed samples. Open circle: Annealed sample.

show these discrepancies. Some 20 years ago, Greenwood et al. (1959) predicted that for T < Tm/2, growth of bubbles, and thus pressure, is controlled by the emission of dislocation loops. For the bubble radii investigated in this study, the threshold pressure for loop punching is given [according to Trinkaus (1983)l by p = 2y/R pb/R, where p is the shear modulus (260 kbar for Al) and bis the length of the Burgers vector of the loops (for Al, b = 2.3 A I( [1 1 11). This gives a proportionality factor between pressure and inverse mean radius of 800 KbarA (chain line in Fig. 17), which is very close to the experimental value of 900 K-barA.Therefore, the results on the pressure derived by EELS with the combination of TEM studies clearly support the model of an athermal growth mechanism via loop punching at low temperatures. Annealing the samples increases the diameter of the bubbles and the pressure shift decreases. This is shown for a 13 at.% AlHe sample in Fig. 16. As mentioned above, for T > Tm/2 the pressure is released to the thermal equilibrium pressure p = 2y/R by the absorption of thermal vacancies. For a sample containing 7 at.% He and annealed at 475"C, a pressure shift A E = 1 eV and bubble radius of 25 A were observed. From the energy shift, the pressure is 5 Kbar, which is not far from the equilibrium pressure of 7.5 Kbar (see open circle in Fig. 17). This again supports the present understanding of bubble growth at high temperature. It also gives some confidence to the

+

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

173

theoretically derived proportionality factor between energy shift and density. When annealing the sample at even higher temperatures, most of the bubbles move to the surface of the A1 film and He emerges from the sample. There probably remains some adsorbed He on the surface or some bubbles with almost zero pressure. This small amount of He gives a spectrum very close to that of atomic He with 1'S,-2"P1 transitions with n = 1,2, and 3. This is shown in Fig. 16 for a 13 at.% AlHe sample annealed at TA= 490°C. Even higher pressures for He bubbles were observed in Ni, which has a higher shear modulus. As shown in Fig. 16, a high-dose sample showed a line shift AE = 3.7 eV, while for a low-dose sample AE 4.7 eV could be extracted from the spectrum. For the low-dose sample, the data evaluation is somewhat uncertain because of features due to Ni interband transitions in the vicinity of the He line. Extremely high densities and pressures (250 Kbar and 500 Kbar; see Table 111) could be derived which are again close to the non-equilibrium pressure pL,which for small bubble radii is close to the threshold pressure for Ni self-interstitial atom emission. We also note that the He density in these bubbles is about ten times as large as that of liquid He at normal conditions and exceeds by far the density of the fluid-solid phase transition at room )is. interesting to note that recently evidence for a temperature(n = 0.14 k 3 It solid-fluid transition near 250 K for 3He bubbles in palladium titrite has also obtained (Abell and Attalla, 1987).Recently, He implanted in amorphous NiP samples has been investigated by Fink and Jager (1987b).In these amorphous materials, large pressure shifts were also observed (see Fig. 16) and a pressure of 80 Kbar is derived for He bubbles having a mean radius of about 15 A. As shown in Table 111, these values indicate overpressurized bubbles also for this amorphous material. Thus, a new mechanism for bubble growth in amorphous systems must be developed based on other than the loop punching mechanism adequate only for crystalline materials.

-

-

2. Heavier Rare Gas Bubbles Because of controversial results on the relationship between line shift and density for the He 1s-2P transition, the existence of high pressures is still called into question (Donnelly, 1985). Therefore, we extended our studies to the heavier RG bubbles in metals (vom Felde et al., 1984).The great advantage of these systems is that the proportionality factor can be directly extracted from experiments, since EELS and optical measurements on solid RG are available. Then the line shift between atomic and solid RG can be easily extrapolated to the densities in the bubbles, which are only slightly larger than the densities of solid RG at normal pressure. The disadvantage with respect to pressure determinations by line shift measurements is the fact that these line shifts become smaller with increasing Z due to the smaller mean radius of the wave functions of the excited state. Moreover, for heavier RG such as Kr

174

JORG FINK

and Xe, the bubble surface plasmons and the 1s-2P transitions appear at the same energy and lead to a coupling between the two excitations. Therefore, the line shift cannot be extracted from the spectra. This is illustrated in Fig. 18 where typical loss-spectra of homogeneously RG-implanted A1 films are shown. For Ne and Ar, the 1s-2P transitions can be clearly realized at 18.3 and 12.8 eV, respectively. For Kr and Xe, the spectra are more complicated and will be discussed in the next section. For Ne and Ar, the pressure shift and the density (nAE)and pressure (paE) as derived by the method discussed above are given in Table 111. The pressures are again considerably higher than the equilibrium pressures and close to the threshold pressure for loop punching. Moreover, by comparing these results with those evaluated for He bubbles, we clearly see that for similar bubble radii almost the same pressure values were detected for the AlHe, AlNe, and AlAr systems. These findings support the evaluation method used for the He bubbles and the observed high pressures for the very small He bubbles. Pressure evaluation from the energy shift of excitations is not restricted to the valence electrons. For example, the L2,3 edge of Ar in A1 was recently measured by vom Felde (1986). This spectrum is compared in Fig. 19 with those of atomic and solid bulk Ar under normal pressure measured by synchrotron radiation (Haensel et al., 1971). With increasing density, a considerable broadening of all lines is observed. In addition, the lowest 2p-4s

I

SP

BSP

10

VP

ENERGY lev)

20

-

FIG. 18. Loss spectra (q = 0) of Al films implanted with 3 at.%Ne, Ar, Kr, and Xe. SP: Al surface plasmon; BSP: bubble surface plasmon; VP: volume plasmon of Al; arrows indicate valence electron transitions (Ne, Ar) or coupled oscillations of valence excitations excitations and bubble surface plasmons (Kr, Xe).

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ENERGY (eV1

FIG.19. Ar 2p edges. Gas and solid Ar were measured by X-ray absorption spectroscopy using synchrotron radiation (Haensel et al., 1971). Upper curve: Ar bubbles in A1 measured by EELS (vom Felde et al., 1986).

transition, which is probably excitonic in nature, shows a blue-shift with increasing density. Correlating the energy shift between atomic Ar and solid Ar under normal pressure with the density of solid Ar and assuming a linear relationship between energy shift and density, it is possible to derive the density of Ar inside the bubbles and the pressure can be calculated by an EOS. This yields a pressure of -40 Kbar which is close to the value derived from the 1S-2P transition. Recently, similar measurements on core-level shifts of the Ne K shell by soft X-ray emission techniques were performed by Lefebvre and Deconninck (1986) presenting a further tool for the study of pressure in RG bubbles in metals. Another aim of the EELS core-level measurements was to get information on the density via evaluation of the EXAFS oscillations. While these oscillations can be clearly recognized in solid bulk Ar, they are almost completely washed out in the AlAr spectrum. This is probably caused by strong contributions of those electrons backscattered from the A1 host. The mean free path of the electrons (-20 A) is comparable to the diameter of the bubbles. According to the above EELS studies of RG bubbles in metals, the measured pressures indicate that Ne should be liquid and Ar solid at room temperature. Calculating the pressure with the values of the mean radii derived by TEM, the Kr and Xe bubbles should also be solid. Therefore, during this study it was realized for the first time that solid RG bubbles at room temperature should be detectable by electron diffraction (vom Felde et al., 1984). Independently, Templier et al. (1984) obtained similar results by

176

JORG FINK

TEM investigations. Typical elastic electron diffraction patterns on RGimplanted A1 films are shown in Fig. 20. These spectra were taken with the EELS spectrometer, setting the energy loss to zero. Besides the (1 11)and (200) reflections of Al, the (111)(and for Kr, the (200))reflections are observed for Ar, Kr, and Xe, while for Ne only a diffuse intensity appears at about 2.4 k', as is typical of a liquid. The diffraction peaks of the solid RG systems are slightly broadened due to the finite bubble volume. In addition, a correlation of the intensity of the (111) A1 reflection with that of the (1 1 1) RG reflection upon rotating the scattering plane around the electron beam is observed. This finding is explained by an epitaxial growth of the RG bubbles in the matrix. From the diffraction peaks, the lattice constants, and thus the density, and, using an EOS, the pressures as listed in Table 111 as well, could be derived. These pressures are again above the equilibrium pressures but fall below the threshold pressures for loop punching by a factor of two. However, in view of the large errors expected for these values, the results of the two methods (energy shift and diffraction shift) are still compatible. Finally, we mention that electron diffraction on solid RG bubbles has now become a domain of electron microscopy, since sample preparation of singlecrystalline films is much easier thanks to the small diameter of the electron beam in TEM. However, for He, the latter method cannot be applied because

I

I ,

RG A1 RG (1111 (111)(200) (220)

,

,

, , 3 4 MOMENTUM TRANSFER(k'1

1

,

,

,

2

-

FIG.20. Electron diffraction pattern of A1 films implanted with 3 at.% Ne, Ar, Kr, and Xe. Besides the A1 (11 1 ) and (200) reflections, the Ar, Kr, and Xe (1 11) and for Kr the (220)reflections can be recognized.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

177

of the small scattering amplitude of the He atoms. Therefore, there is as yet no observation of crystalline He in bubbles by diffraction methods. Thus, EELS (and for some metals, VUV spectroscopy) is a powerful method of obtaining information on the thermodynamic properties of He in bubbles. In summary, EELS investigations on RG bubbles yield a consistent picture on overpressurized bubbles for T < TM/2and equilibrium pressures for T > TM/2.In this review, we have not mentioned similar studies as a function of annealing temperature (Manzke et al., 1983a),which give information on bubble growth via loop punching, vacancy absorption, and coalescence at higher temperatures. Open problems are at present the exact determination of the proportionality factor between energy shift and density for He, the energy shift due to the interaction with the bubble wall, and the excitation spectra of RG dissolved in metals (if there exist any at higher implantation concentrations).

3. Surface Plasmons on Bubbles There are numerous studies of collective electronic excitations in the case of planar surfaces like those of films and layers (Raether, 1977,1980), whereas surface plasmons on inner surfaces in metal cavities, i.e., in voids or RG-filled bubbles, have rarely been investigated. The pioneering experimental work was done by Henoc and Henry (1970), who observed the images of He bubbles in an A l matrix in an energy-selective electron microscope. There are also theoretical studies on bubble-surface plasmons (BSP) (Natta, 1969; Ashley and Ferrell, 1976; Lucas, 1973).A short introduction to this theoretical work was given in Section 11,B. In the following we review recent experimental studies on surface plasmons in RG-filled voids (Manzke et al., 1983b; vom Felde and Fink, 1985). In Fig. 15, the BSP at 10.7 eV for a 13 at.% AlHe sample can be recognized between the surface plasmon of the planar Al/AI,O, interface at 7.0 eV and the volume plasmon of bulk Al at 15 eV. The energy of the bubble surface plasmon and the width of the volume plasmon as a function of the concentration of implanted He is shown in Fig. 21. According to Eq. (44), the monopole, dipole, quadrupole, and the 1 = 00 mode should appear at o, = 15 eV, J2/3 up= 12.25 eV, J3/5 o, = 11.62 eV and J1/2 o, = 10.61 eV for E = 1 (unfilled void). The monopole mode, which is actually a breathing mode (see Fig. 7), superimposes on the volume plasmon mode. The increasing width of the line at 15 eV (see Fig. 21) with increasing cHemay be caused by an increasing coupling of the monopole modes of the individual bubbles with increasing filling factor. This phenomenon, which leads in a regular array of bubbles to a band structure of breathing plasmons, was described by Lucas (1973).For 1 > 0, according to the results of Ashley and Ferrell(1976), only the

178

JORC FINK 1.0

-I

0.9

- 0.8 >

+BSP I=O

E 0.7 N

~'0.6 - 0.54

* 2-0

'

C H (at.%l ~

3'0

FIG.21. Bubble surface plasmons measured in He-implanted Al films. Upper part: Line width BE,,, of the plasmon at 15 eV, which is composed of the volume plasmon and the I = 0 BSP mode. Lower part: Energy of the I = 1 BSP dipole mode. Calculation for a single bubble: dashed line. Effective medium theory using a constant He polarizability: Chain line. Effective medium theory using density-dependent He polarizability : Solid line.

dipole mode should have considerable intensity for small momentum transfer [see Eq. (45)]. As the energies shown in Fig. 21 were measured at zero momentum transfer, the results should be compared with theoretical models for the dipole mode. For these calculations, the dielectric constant of He is evaluated in terms of the Clausius-Mosotti formula for atoms in cubic coordination E(O) =

1

[

: I-'

+ 4zncr(o) 1 - -nncc(o)

,

where a(w)is the He polarizability and n the number density. a ( o ) is simulated by an oscillator wo at the 1s-2P transition using Eq. (22). Neglecting yo, the oscillator strength should be no = (rn/e2)a(0)og,where a(0) = 0.205 A3 is the quasistatic polarizability of He as measured in the visible for the high-density gas. Using the He densities as derived in the last section, which decrease with increasing cHe,we obtain an increasing broken line in Fig. 21, which is in contradiction to the decreasing experimental results due to the weaker coupling of the BSP to the dielectric medium. Thus Natta's equation for a single bubble is not adequate for the description of the experimental results. An improved theoretical description has to take into account long-range dipole interactions between the bubbles. This is realized in the theory of for small Garnett (1904) for a long-range effective dielectric function E,'~(o)

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

179

particles in cubic coordination given by Eq. (46). The filling factor f is the = A1 atomic volume fraction of the spherical particles AV/V= ~~~/(nG~,)(i2~, volume). cM= cAl is well described by the Drude model. For cB = cHe, the above Clausius- Mosotti formula was taken. From the effective dielectric function cerf(w),the energy-loss function Im( - l/cerr(w))was calculated. It has poles given by Eq. 47 that describe the excitation energies of the A1 volume plasmon, the He 1s-2P transition, and the BSP dipole mode. Using the density-independent atomic polarizability of He, a(0) = 0.205 A3, the calculated BSP energy is now decreasing with increasing cHe(see chain line in Fig. 21). But still the slope deviates from that of the experimental points. The following effects may explain the remaining discrepancy: The excitation energy w, may depend on the bubble size; this was discussed by Aers et al. (1979) who found that for small bubble radii, the BSP dipole mode is in fact shifted to higher energies. Recently, density-functional calculations on surface plasmon energies in voids by Wu and Beck (1987)have also predicted a strong blue shift for small radii due to the fact that electrons diffuse into the void and the density within the void increases as the void radius decreases. Surface plasmon energies up to about 14 eV, i.e., far above the experimental values, were calculated for radii close to the experimental situation. However, the comparison should be regarded as very ambitious, since there are many important differences between the theoretical model and the experimental system. First of all, in the experiment the voids were filled with high-density He which, due to the high pseudopotential, probably strongly pushes the electrons back into the metal. Besides these microscopic effects, there is some evidence from optical measurements on superdense He (Vidal and Lallemand, 1976; Loubeyre et al., 1982) that the atomic polarizability is decreasing with increasing density. Using a linear dependence a = 0.286 - 1.348n(n in k3), the solid line in Fig. 21 was calculated and is now close to the experimental points. However, it should be mentioned that a linear relationship between a and n is speculative and other mechanisms may be responsible for the discrepancies between the chain curve in Fig. 21 and the experimental points. As shown in Fig. 18, for Ne and Ar bubbles in Al, BSP are observed at 11.7 and 10.2 eV, respectively. The shift, going from Ne to Ar, is explained by the higher atomic polarizability of Ar compared to that of Ne. For Kr and Xe, the spectra shown in Fig. 18 are much more complicated in the energy range where BSP should appear. The peaks cannot be explained by the interband transitions of bulk RG that were measured by Keil (1966). Obviously, for Kr and Xe, the 1s-2P transitions appear at the same energy as the BSP, and therefore there is a coupling of the two interactions. As in the case of He bubbles, the effective dielearic constant cerf and the loss function were calculated. The filling factors (f 3 - 8%) were derived Im( - 1/ceff(~)) from the RG densities discussed in the last section or from measurements of N

180

JORG FINK

the RG content by Rutherford backscattering experiments and from TEMdetermined bubble radii. The RG dielectric functions were reproduced by means of several Lorentzian oscillators. For AlNe, the BSP appeared at 11.4 eV, which is close to the experimental value of 11.7 eV. A similar calculation for AlAr yielded 10.0 eV, whereas 10.2 eV was measured. The calculations also reproduced the RG excitations and the A1 volume plasmon quite well. Of course, the effective medium theory could not reproduce the large pressure shift of the 1S-2P transitions, since the dielectric function for RG was adjusted to the atomic values. The observed half-width of the BSP considerably exceeded the calculated ones, which are nearly insensitive to any reasonable variation of the parameters in the Lorentz model. A random (not cubic) distribution of RG bubbles was given as an explanation for the discrepancy. According to the calculations of Persson and Liebsch (1982), considerable broadening of the BSP should be expected in this case. For AIXe, the agreement between theory and experiment is less satisfactory, as is shown in Fig. 22. The calculated peaks are close to the experimental ones, but the intensities are not well reproduced. The differences were explained by an imperfect simulation of the Xe oscillators. Figure 22 also shows the momentum dependence of the AlXe excitations in this energy range. For low momentum transfer, the measurements provide information on the longwavelength properties of the solid and, therefore, the effective medium theory is adequate. At higher momentum transfer (shorter wavelength) the spatial variation of the electronic structure is detected and for wavelengths small compared to the dimensions of the bubbles, the individual excitations of the RG in the bubble and that of the A1 host should appear. This decoupling of the excitations at higher momentum transfer is indeed observed in the experiment as shown in Fig. 22. There is a strong variation of the spectra with increasing q and at q = 0.2 k', the spectrum shows maxima at energy positions where maxima were observed for bulk solid Xe (Keil, 1966). The decoupled BSP is not observed at higher q since it is strongly damped at higher momentum transfer. No energy shift was observed for Xe excitations, as these spectra were taken on annealed samples containing only large bubbles. Moreover, as mentioned above, the pressure shift decreases with increasing atomic number. The spectra in the intermediate momentum transfer range are not understood at present. There is no effective medium theory available to account for frequency and momentum-dependent dielectric functions. Together with the experimental results, such theories could give valuable results on the microstructure of precipitations in metals. It is remarkable that all the above experiments on surface plasmons could be rather well explained by classical theories. Even at a bubble radius R = 6 A, which is close to the Thomas-Fermi shielding length of electrons (-2 A), there are at least no strong deviations from classical macroscopic theories for

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

18 1

0.2 0.15 0.1

0

I

7

I

8

I

9

/

I

10 11

ENERGY lev)

FIG.22. Loss spectra of a Xe-implanted A1 sample taken at q = 0, 0.1, 0.15, and 0.2 k’. Arrows on the uppermost curve mark Xe valence electron transitions. The background is subtracted. The dashed curves show the loss function calculatedon the basis of Maxwell-Garnett theory.

the dielectric constant. To see quantum effects, even smaller voids or empty voids should probably be studied. Another interesting field are the “inverse bubbles,” i.e., metallic spheres in dielectrica. Currently, such studies on alkaline spheres in MgO are being performed by vom Felde et al. (1987). These studies indicate deviations from classical models due to finite size effects. VII. AMORPHOUS CARBON There exist various modifications of semiconducting amorphous carbon. Evaporated amorphous carbon (a-C) has a gap of about 0.4-0.7eV. In amorphous carbon produced by the ion beam technique (i-C) the gap varies from 0.3 to 3 eV. Recently, hydrogenated amorphous carbon (a-C:H) produced by plasma deposition from hydrocarbons has attracted considerable interest. These films have an attractive combination of properties: extreme hardness (between sapphire and diamond), high electrical resistivity, optical transparency in the infrared, variable high refractive index, and resistance to chemical attack. They are, therefore, of particular interest for coatings for optical components, for example. Although amorphous carbon modifications have been investigated for many years, the microstructure is a matter of continuing debate. The problem is rather complicated since carbon has three different bonding configurations while other four-valent amorphous

182

JORG FINK

materials, such as a-Si, have only the tetrahedral bonding configuration. The microscopic origin for this difference between C and Si is that C has no 2p core levels and, therefore, the 2p level in C is lowered to the 2s level, thus allowing three different hybridizations of the four valence electrons. These are shown in Fig. 23a. The sp3 configuration, which appears, for example, in diamond, has four tetrahedrally directed cr bonds. In the sp2 configuration typical of graphite, there are three trigonally directed cr bonds. The fourth electron is a a orbital normal to the cr bonding plane. Finally, in the sp' hybridization there are two cr bonds and two a bonds. The a electrons are weakly bound and therefore lie closer to the Fermi level than the cr states. The density of electronic states for the amorphous carbon modifications is schematically shown in Fig. 23b. The size of the gap is determined by the separation of the filled a band (valence band) and empty a* band (conduction band), both originating from the sp' carbons. One of the most interesting questions related to the short-range order of amorphous carbon modifications is the relative concentration of sp3 carbons (tetrahedral bonding, without a electrons), sp2 carbons (trigonal bonding, with one K electron), and sp' carbons (linear bonding with two a electrons). Another important question is whether the A electrons are localized in small islands of sp2 carbon or delocalized into broad A bands. In this section,we review EELS studies (Fink et al., 1983a; Fink et al., 1984a; Fink et al., 1987c)on these questions.

SP

(a)

(b)

FIG.23. (a) Three different hybridizations of the four valence. electrons of a carbon atom. (b) Typical electronic density-of-states of amorphous carbon materials.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

183

The near-edge structures of the carbon 1s level (see Fig. 24) give direct evidence on the existence or non-existence of n electrons.Neglecting core-hole effects,these structures reflect the density of unoccupied states, i.e., the n* and (r* states. Thus, graphite shows a transition at 285 eV into the n* states and above 291 eV into the (r* states. In diamond, there are no n electrons and therefore the edge starts at 290 eV by transitions into (r* states. The sharp structure at 289 eV has been demonstrated to be a core exciton (Morar et al., 1985). The spectra for a-C and a-C:H show is-n* transitions at 285 eV, indicating the presence of 5~ electrons. Therefore, these spectra clearly demonstrate that the name "diamond-like" carbon for a-C: H is misleading. There must be a considerableamount of sp2carbon atoms in the a-C: H films. A quantitative analysis of the concentration of sp2carbons, however, was not possible as the cross-sections of the ls-n* and the 1s-a* transitions are dependent on the detailed band structure of these materials. On heating the a-C:H samples, the features in the spectra resemble more and more those in the graphite spectrum, indicating the transition into a graphite-like structure due to hydrogen release. The measurements of the extended structure of the edges (not shown) revealed strongly damped oscillations for both a-C:H and a-C. From this, an amorphous structure with variations of the radius of the first neighbour shell of -0.1 -0.2 A could be derived using Eq. (50). Quantitative information on the concentration of sp2 carbons could be derived from an analysis of the loss spectra in the energy range 0-40 eV. The

1

'

1

'

'

I

"

"

I

"

GR a-C H T= 1000°C

a-C H T= 650T a-C H T=20"C a-C

DIA

280

300 ENERGY lev)

FIG.24. Near-edge structures of the carbon 1s level excitations for graphite, a-C:H (annealed and as-grown),evaporated carbon (a-C), and diamond.

184

JORG FINK

loss functions and the real and imaginary part of the dielectric functions as calculated by a Kramers-Kronig analysis are shown in Fig. 25. The dominant peak in the loss function is the collectiveexcitation of all valence electrons, i.e., the n + B plasmon, which appears at 27,23,21,22,25,and 34 eV for graphite, as-grown a-C:H, for two annealed a-C:H samples (T = 600 and loOO°C), for a-C, and for diamond, respectively. As the energy of the plasmon is proportional to the square root of the density of the total number of valence electrons, if we neglect the H atoms, it is, a measure of the density since each carbon atom contributes four valence electrons. Thus, the density increases when going from a-C:H via a-C and graphite to diamond. Upon annealing a-C:H up to 6o0°C, the number of valence electrons decreases due to partial hydrogen release, and, therefore, the plasmon energy decreases by about 2 eV. At higher temperatures, the density increases due to a transition from an amorphous to a dense graphite-like structure, and, therefore, the plasmon energy increases again. This demonstrates nicely the usefulness of measurements of the n + B plasmon for density determinations, a technique proposed by Leder and Suddeth as long ago as 1960. In the loss functions shown in Fig. 25, a second feature is seen near 6 eV (except for diamond) originating from collectiveexcitation of n electrons, i.e., a n plasmon, which we found to be more pronounced in the spectra of graphite and annealed a-C:H than in those of as-grown a-C:H and a-C. This n plasmon is related to a n-z* transition showing up in c2 near 4 eV. The n-z* oscillator causes a zero-crossing of c1 near 6eV, where t 2 is small and, therefore, the loss function Im( - 1/c) = c2/(c: + E : ) shows a maximum, i.e., the n plasmon (see also Section 11,BJ). The second maximum in tt near 12 eV is related to B - B * transitions. There is a clear separation of n-n* transitions and B-B* transitions in graphite and in annealed a-C:H. The minimum at -8 eV between the two transitions is less pronounced in the other two systems. The fact that strong n-n* transitions are found also in as-grown a-C:H implies the presence of sp2 hybridized carbon atoms. The concentration of sp2 carbons was derived using the sum rule on me2 [see Eq. (21)]. Integration up to o,= 8 eV gives the number of n electrons, and integrating to infinity yields the number of all the valence electrons, i.e., the n + B electrons of carbon and the electrons of the H atoms forming the B bond to the C atoms. The ratio between those two numbers combined with the known H concentration allows the sp2 carbon concentration to be calculated. For asgrown a-C:H, this analysis showed that about one-third of the C atoms is in the trigonal sp2 configuration (graphite-like carbon) while two-thirds are in the tetragonal sp3configuration (“diamond-like”carbon). This result is almost independent on rather large variations of preparation conditions such as pressure or bias voltage on the plasma. The same result was recently obtained for a-C:H, F, where the H atoms were partially or totally replaced by F

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

185

Eti€RGV(sV)

FIG.25. Loss function and dielectric functions E , and E~ for graphite, a-C:H (as-grown and annealed), evaporated carbon (a-C), and diamond.

186

JORG FINK

atoms. Annealing the a-C :H samples caused a transformation of sp3carbons into sp2 carbons due to a release of H atoms. At an annealing temperature of 600"C, the concentration of sp2 carbon atoms reached 50%. For the amorphous evaporated carbon a-C, a sp2 carbon concentration of almost 100% was derived. Information on the localization of the n electrons could be obtained by measuring the dispersion in momentum transfer of the K plasmon. For localized n electrons, the n and the n* bands should be narrow and, therefore, the n plasmon should show no dispersion. In contrast, for delocalized 7c electrons, dispersion of the K plasmon should be observed. Analyzing the n plasmon dispersion data in terms of the quadratic momentum dependence E = E , + a(h2/m)qz,no dispersion, i.e., a = 0, was found for as-grown a-C:H and a-C. Upon annealing a-C:H, a non-zero value of a was observed above 500°C. For TA= 600°C and T A = 1000"C, the dispersion coefficients a = 0.12 and CI = 0.25, respectively, were derived. This indicates that in as-grown a-C:H and in a-C, the n electrons are localized, i.e., what is known as the n band is composed of narrow non-dispersive levels as, e.g., the band gap tails in amorphous Si or molecular levels in organic molecules. One possible model for a-C:H in which this band structure might occur would assume the existence of larger nonpercolating n-bonded clusters in a matrix of sp3 carbons. The assumption of larger sp2 molecules instead of statistically distributed sp2 carbons is necessary to obtain the low gap of about 1.8 eV. The important fact that the gap of a-C:H is dependent on the size of the sp2 clusters was first pointed out by Bredas and Street (1985) and by Robertson (1986). The reason for localization of n electrons in a-C may be that clusters with n bonds align in a plane and are separated from each other by non-overlapping n orbitals at a dihedral angle along the bond of 90". For a-C:H, the model of sp2 clusters in a sp3matrix is also supported by recent measurements of the carbon 1s near-edge spectra of a-C:H, F, as shown in Fig. 26. For fully fluorinated a-C:F, the F/C concentration was 0.6, while for the Is-n* transitions the ratio of the intensity of the line for C bonded to one F atom at 287 eV to that of the line of C bonded to no F atom at 285 eV is only -0.3. From this fact, a non-equilibrium distribution of F atoms was derived with -70% of F atoms bound to sp3 C atoms and only 30% of the F atoms bound to sp2 C atoms. This result indicates again the presence of sp2 clusters, since it is only the border of the sp2 clusters that can be decorated with F(or H) atoms. The delocalization of the n electrons in a-C:H upon annealing can be explained by percolation of the growing sp2 carbon islands (sp3 carbons are transformed to sp2 carbons by H release). This causes a closing of the gap as observed by EELS or optical spectroscopy due to broadening of the K bands. At the annealing temperature of 600°C where the dispersion coefficient a starts to deviate from zero, conductivity also strongly increases.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

I 280

’,

1

1

187

I

290 300 ENERGY lev)

FIG.26. Near-edge structuresof the carbon 1s level excitations for a-C: H, F produced from fluorinated benzenes.

Investigations of a-C: H, F by EELS have provided detailed information on the microstructure of these systems. In particular, the concentration of carbon atoms in sp2 (graphite-like) and those in sp3 (diamond-like) configuration could be determined. Moreover, measurements of the dispersion of the n-n* transitions gave strong evidence for sp2 carbon clusters in a matrix of sp3 carbon atoms. This microstructure is probably the reason why up to now these systems could not be transformed to highly conducting materials by means of doping with acceptors or donors as, e.g., the conjugated polymers. Finally, changes of the microstructure as a function of preparation condition and annealing temperature have been studied.

VIII. CONDUCTING POLYMERS Since the pioneering work of Chiang et al. (1977), conducting polymers have emerged as a new class of materials. It was realized that certain polymers can be “doped” with electron acceptors (oxidation by AsF;, J;, ClO,, ... counterions) or electron donors (reduction with Li’, Na’, NH;, .. . counterions) to conductivity levels approaching those of metals. The highest conductivity reported up to now (Naarmann, 1987)is D = 1.5 lo5 S/cm, which is not far from the room temperature conductivity D = 6 l o 5 S/cm of Cu. The conducting polymers have interesting technological application, such as replacement of conventional metals in electronic shielding and antistatic equipment, rechargable batteries, photovoltaic cells, and perhaps in the next century electronic circuits on the basis of conducting polymers and molecular crystals. From the more fundamental point of view, conducting polymers are

188

JORG FINK

of great interest as these systems are quasi-one-dimensional semiconductors and, after doping, quasi-one-dimensional metals. The polymers that can be doped up to metallic conductivities are the conjugated polymers, i.e., the carbon atoms are in the sp2 configuration (see Section VII). The most important conjugated polymers in this field are shown in Fig. 27. Besides the strong bonding by (T electrons to the next carbon atoms and to the H atom, there is a weaker bonding due to n electrons indicated by a double bond. The density of the electronic states is similar to that of the hydrogenated amorphous carbon systems illustrated in Fig. 23b. The prototype of the conducting polymers is polyacetylene (PA), the trans modification of which is shown in Fig. 27. A schematic one-dimensional band structure of trans-PA is shown in Fig. 28. The detailed curvature of the bands depends on the geometry and on interchain interactions (AndrC and Leroy, 1971; Grant and Batra, 1979; Mintmire and White, 1983; Ashkenazi et al., 1985; Springborg, 1986). Due to the strong (T bonds, the splitting into bonding 0 and antibonding (T*bands is about 7 eV. As there are 6 C B electrons and 2 H (T electrons per unit cell (2 H and 2 C atoms per unit cell), the 4 (T bands are completely filled. As to the n electrons, there are 2 n electrons per unit cell and, therefore, the x band is completely filled, too. In a single-particle band-structure calculation for trans-PA with equal C - C distances, there is no gap between the n and n* bands and, therefore, PA should be a metal. There are two effects in onedimensional systems (e.g., platinum chain salts, organic charge-transfer salts, or conducting polymers), that lead to a breakdown of single-particle band-

-

trans-POLYACETYLENE (PA) POLY PARAPHENYLENE (PPP) POLYPHENYLENEVINYLENE (PPV)

*

POLYPYRROLE (PPY)

POLY THIOPHENE (PT)

&+

POLY ANILINE (PAN11

FIG.27. Chemical structures of conjugated polymers that play an important role in the field of conducting polymers. Carbon and hydrogen atoms are not shown explicitly.

wo lH

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

189

___----,-----==

> -

1

w

z w

”,/---____-__--

--__ -- . -. ...___*--

--....

0 k n /a FIG. 28. Schematic band structure of trans-polyacetylene.

structure calculations. First, the electron-electron Coulomb interaction U can be sufficiently strong compared to the total band width W,resulting in a Mott-Hubbard insulator with finite gap. Second, the electron-phonon coupling may lead to a Peierls-Frohlich transition with a distortion of the lattice structure with wave vector 2kF. We then obtain a gap at k = 2kF as shown in Fig. 28 with partial localization of the n: electron density on every second bond as indicated by a double bond in Fig. 27. This non-equal 7c electron distribution is termed bond-order wave. It leads to a difference in length between single and double bonds of about 0.06 A. The theory for the electron-phonon model for PA has been worked out by Su, Schrieffer, and Heeger (SSH) (1980), Rice (1979), and Brazovskii (1978, 1980). The model, which is essentially a tight-binding model with an electron-phonon term, has been extremely useful in describing many of the experimental observations. On the other hand, Horsch (1981) has shown that correlation effects cannot be ignored in these systems. Recently, Baeriswyl and Maki (1985) and Baeriswyl (1987) have postulated that electron-electron correlations may be the dominant factor for dimerization in PA and that the size of the electronphonon coupling constant should be much smaller than previously assumed. At present, there is a lively debate on the size of the two effects and experimental results are extremely helpful to clarify the situation. Upon doping a t low levels, electrons are not removed from the n: band (acceptors) or added to the 7c* band (donors) as would be expected in a rigidband model. Rather, the dopants produce defects on the polymer chain that lead to defect levels in the gap. The most important defects that appear on doping with donors (n-type doping) are illustrated in Fig. 29 for PA and polyparaphenylene (PPP). In PA, the most important defect is a conjugational

190

JORG FINK

SOLITON

a=-e

S=O

rn

FIG.29. Defects induced by n-type doping in trans-polyacetyleneand in polyparaphenylene. On the right side, the correspondingdefect levels in the gap and their occupationfor the case of ntype doping is illustrated.

defect in which the double bonds in the zig-zag structure switch from the right-hand slopes to the left-hand slopes. This defect has in the positive, neutral, and negative form zero, one, or two dangling bonds, respectively. In Fig. 29 we show the negative defect with two dangling TC bonds caused by the fact that one electron has been transferred from the donor to the chain. The defect may be also viewed as a domain wall or as a phase slip of 180" in the order parameter of the bond-order wave. In the SSH model, the defect can be described by a solitary wave, and the defects are therefore often termed solitons. The formation of a soliton implies local suppression of the Peierls gap with a defect level in the gap, the so-called mid-gap state. This state is empty, half filled, or doubly filled for positive, neutral, or negative solitons, respectively. The existence of the mid-gap state was first proved by Suzuki et al. (1980) using optical spectroscopy. Charged solitons that have no spin may lead to the spinless conductivity observed at low doping concentrations. There exist many review theoretical and experimental articles concerning solitons in PA (Schrieffer, 1985; Baeriswyl, 1985; Etemad et al., 1982; Streitwolf, 1985; Roth and Bleier, 1987; Heeger et al., 1987). While PA is the prototype of the conducting polymers, on the other hand it has the rather exceptional property that the energy of the right-hand slope doublebond structure is the same as that of the left-hand slope double-bond structure. Thus, PA has a degenerate ground state. The majority of the other

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

191

conducting polymers have the common feature of a non-degenerate ground state; that is, when the positions of the double and single bonds are reversed, an inequivalent structure with different energy results. For PPP, the groundstate structure is the aromatic form shown in Fig. 27. The other structure in PPP is the “quinoid structure,” where n electron charge is transferred from the rings into the bonds between the rings. In non-degenerate ground-state polymers, single defects such as solitons cannot be stable. Upon oxidation or reduction, the quinoid structure is locally more stable. At low doping concentration, singly charged defects, i.e., a charge with lattice distortion, appears which in solid-state physics are called polarons (see Fig. 29). At higher concentrations, doubly charged bipolarons are more favourable. Polarons and bipolarons are related to two defect levels in the gap. The occupation of these levels is illustrated for n-type doping in Fig. 29. Theoretically, polarons and bipolarons were predicted by Brazovskii and Kirova (198 l), Bredas et al. (1982), and Fesser et al. (1983). Experimentally, these defects were first observed in PPP by Crecelius et al. (1983a) using EELS and in polypyrrole (PPY) by Scott et al. (1983) using ESR. Polarons and bipolarons may, in principle, also appear in PA as a combination of two solitons, a soliton and an antisoliton. However, this is energetically less favourable than the formation of a single soliton. With increasing doping concentration, there is an overlap of the bipolarons, which in reality extend over about four rings. This leads to a broadening of the bipolaron levels in the gap. As shown in Fig. 30 for n-type doping, this may lead to an overlap with the conduction band and the valence band at the highest doping concentrations. Then a partially filled conduction band, and thus the metallic state, is formed. Similarily, for PA, with increasing doping concentration the mid-gap state is broadened by the formation of a soliton glass and at sufficiently high concentrations the metallic state is reached due to a disorder-induced quenching of the Peierls distortion (Mele and Rice, 1981). Recently, a first-order transition from a soliton lattice to a

FIG.30. Evolution of the band structure of n-type doped non-degenerate ground-state polymers (e.g.,polyparaphenylene)as a function of dopant concentration.

192

IORG FINK

polaron lattice has been discussed (Kivelson and Heeger, 1985). The detailed electronic structure of the metallic state and the transition from the semiconducting to the metallic state is still under debate. In the last five years, we have studied the electronic structure of undoped and doped conjugated polymers by EELS. Previous reviews on these studies have been given by Fink (1985b), Crecelius (1986), Fink (1987), and by Fink et al. (1987d). Microscopic investigations may lead to a better understanding of the interesting macroscopic transport properties such as the high conductivities. We begin with a review of some EELS investigations on undoped systems, in particular highly oriented PA (Fink and Leising, 1986; Fink et al. 1987e). In Fig. 31, electron energy-loss spectra are shown as a function of the angle 8 between the chain axis c and the momentum transfer q. The spectra which is small compared to the extension of were taken at q = 0.1 kl, the Brillouin zone. For uniaxial crystals, the loss function is given by Im[- 1 / ( q cosz 8 sin28)] [see Eq. (35)], where ell and are the principal components of the dielectric tensor. Therefore, for highly oriented samples, the two components can be determined separately. For q 11 c (8 = 0)

+

QY-K-TO

' 3b ENERGY (eV)

Lo

'

FIG.3 I . Electron energy-loss spectra of highly oriented trans-polyacetylene as a function of the angle between the chain axis and the momentum transfer. The spectra were taken at q = 0.1 A-' which is small in comparison with the dimensions of the BriIlouin zone (after Fink and Leising, 1986).

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

+

193

there is a strong n o plasmon at 22.5 eV due to all valence electrons and a strong n plasmon at 4.9 eV due to a n-n* interband transition. In addition, there are various shoulders above 8 eV that indicate less intense oscillators due to P O * transitions. Turning the momentum transfer perpendicular to the chain axis causes the n plasmon to become strongly reduced and also changes the shoulders in energy and intensity. In Fig. 32, the loss function, E ~ and , c2 are shown for q = 0.1 A-' parallel and perpendicular to the chain axis. These data were derived by a Kramers-Kronig analysis from the loss spectra as described in Section I1,E. For the calibration of the loss function, the optical refractive indices nll = 3.25 and n, = 1.33 were used. For ql(c, the loss function and the dielectric functions are very close to those shown in Fig. 2e, which were formed from two oscillators at 3 and 8 eV. In the onedimensional system PA, the lower oscillator, i.e., the n-n* transition across the fundamental gap at the zone boundary, is very pronounced. The joint density-of-states of the n and n* band shows in this case a square-root singularity at the gap energy 1.8 eV. The maximum of &* is, however, broadened due to interchain interaction and local field effects. The n-n* oscillator leads to a zero-crossing in cl at 4.9 eV and, therefore, the loss function shows a strong maximum at this energy. As this maximum is closely related to the n-n* transition, the n plasmon is not a free-electron plasmon but an interband plasmon. For q l c , the n oscillator is strongly reduced in c2 by a factor of 170, indicating a very small polarizability of the n electrons perpendicular to the chain axis. The observed anisotropy is in qualitative agreement with calculations in the framework of the SSH model by Baeriswyl et al. (1983). At higher energies, there are several oscillators due to ~-r-r* transitions at 12, 14.6, and 19 eV (for q 11 c) and at 16.5 and -23 eV (for q l c ) . The transitions can be partially assigned to transitions in the zone centre and at the zone boundary in the band structure shown in Fig. 28. For q l c , the strong shoulder near 10 eV should be assigned to a n-O* transition. According to the calculations of Mintmire and White (1983), a large joint density-of-states should appear for this normally forbidden transition, because the n and the lowest O* band are effectively parallel in a large portion of the Brillouin zone (see Fig. 28). In the following, we discuss the momentum dependence of the dielectric functions of trans-PA. The n plasmon shows a linear dispersion in momentum transfer in the range q = 0.07 to 1 k'with a slope of 3.3 eVA as can be seen in Fig. 33. This was already recognized by Ritsko (1982) on non-oriented samples. A linear dispersion of the n plasmon was also observed by Ritsko et al. (1983a) for polydiacetylenes. The n plasmon in PA decays close to the The linear dispersion of the n plasmon zone boundary at q = n/a = 1.28 k'. is, of course, related to a linear dispersion of the n-n* oscillator and should not be confused with the dispersion relation of a free-electron plasmon. The

-

194

JORG FINK

16

PPY

I

(5

PT

I

FIG.32. Loss functionsand dielectricfunctions of undoped conjugatedpolymers.PA:highly oriented trans-plyacetylene; PPP: partially oriented polyparaphenylene;PPV: highly oriented polyphenylenevinylene;PPY: non-oriented polypyrrole; PT:non-oriented polythiophene. For oriented samples, the solid (dashed)line gives data for q parallel (perpendicular) to the chain axis.

195

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY 10

PA THEORY

05

-

0 0 0 0

--1 0 tO

-

€ 0 0 0 0 0

0 0

6

L

2

8

x3

12

2

4

ENERGY l e v )

6 8 1012 ENERGY IeVI

0

2

L

6

8

1 PT

PPV

12

10

E K R G Y lev1

q

01 02 03 04

05

06

T

,

I

2

,

I

4

,

.

,

l

l

l

6 8 x) ENERGY lev1

,

12

2

4

6 8 1 0 1 2 ENERGY (eV1

O M T T - K T T ENERGY l e v 1

FIG.33. Momentum-dependent loss functions in the range of I electron excitations. PA: highly oriented polyacetylene; PPP: partially oriented polyparaphenylene; PPV: highly oriented polyphenylenevinylene; PPY: non-oriented polypyrrole; PT: non-oriented polythiophene. For the oriented samples, 9 is parallel to the chain axis. For comparison, calculations of the I plasmon in polyacetylene in the framework of the SSH model, including local field corrections, are shown.

196

JORG FINK

i i b r -

(a1

30

30

20

20

N

W

W"

10

10

0

0

1

0

0

5 10 15 ENERGY lev1

:; 30

r

J 15

7

5 10 15 ENERGY I eV)

10

-

I

(el 30

~

20-

10

00.5

\

qbi-')

\

1.1 0

ENERGY (eV1

5 10 15 ENERGY 1 e V )

FIG.34. Momentum dependence of the H-Z* interband transition in polyacetylene. (a) E 2 ( q , w ) as determined by a Kramers-Kronig analysis from the experimental loss function. (b) Calculation of the joint density-of-states of the x and I[* band. (c) Calculation of E2(q,w) without local field corrections. (d) Calculation of E 2 ( q , a ) with local field corrections for pII > 1. (e) Calculation of E2(q,a)with local field corrections for pII = 0.3 A and pL = 1 A.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

197

q dependence of the n-n* oscillator is shown in Fig. 34a in a momentumdependent plot of c2 as derived by a Kramers-Kronig analysis of the loss spectra. For comparison, we show in Fig. 34b the q dependent joint densityof-states of the n and n* band as calculated in a tight-binding model using a simple sampling method. We used E, = 1.8 eV for the gap and W = 11 eV for the total width of the n electron system. In Fig. 34c, the imaginary part of the dielectric function calculated in the SSH model using Eq. (36) is shown with off-diagonal elements omitted (Drechsler and Bobeth, 1985). Finally, in Figs. 34d and 34e we show c2, calculated again from Eq. (36) but now with the inclusion of local field corrections (Neumann, 1986; Neumann and von Baltz, 1987). In Fig. 34d the extension of the Wannier function was chosen pII > 1 A, while in Fig. 34e the calculation is based on more realistic values pI, = 0.3 A and pL = 1 A. Both in the joint density-of-states and in c2 without local field corrections, Van Hove singularities that are square-root singularities in the one-dimensional system are seen. In the (co,q) diagram given in Fig. 35, these singularities are given by full lines. Due to the small matrix element of the n-n* transitions in the zone center, the high energy singularity is strongly reduced in the calculations for c2. When local field corrections are taken into account, the singularities are rounded and the maximum of c2 is shifted to higher energies. This effect increases

AI 5

O O

4a n

FIG.35. Momentum-dependent experimental data on the z plasmon (D) and the maximum of the n-n* interband transition in E* (U) in trans-polyacetylene. Full curves: calculated Van Hove singularities of the n-z* transitions that limit the range of possible interband transitions. Dashed and dot-and-dash curve: calculations of the z plasmon and the maximum of e2 by Neumann and von Baltz (1987).

198

JORG FINK

particularly when more extended Wannier functions are used for the calculation. At higher momentum transfer, the splitting of the lower singularity is strongly washed out. Figure 35 gives a comparison of the experimental data for the plasmon energy and the maximum of c2 with theoretical data by means of the model also used for Fig. 34e. The best agreement could be achieved using the above parameters for the more extended Wannier functions and EB = 1.8 eV and W = 11 eV (Neumann and von Baltz, 1987). A comparison with the simple calculation of the joint density-of-states yielded a slightly higher value W = 12.5 eV (Fink and Leising, 1986). The derived width W = 11 eV is in excellent agreement with estimates of W in the range 10- 12 eV from band-structure calculations, while the gap energy is close to that determined by optical spectroscopy. At this point, it should be noted that an approximation without local field corrections, but with the introduction of a next-nearest-neighbor hopping integral also gave good agreement between theory and experiment (Drechsler et d., 1986). As can be seen in Fig. 35, rather good agreement between theory and experiment can be achieved for the energy position of the collective excitations and the interband transitions in the 71 electron system. However, the shape of the e2 curves in Figs. 34a and 34e is rather different at higher momentum transfer. In particular, the onset in c2 remains in the experiment at the gap energy for zero momentum transfer, while the theoretical calculations predict an increasing gap at higher q. It is interesting to note that this q-independent onset also appears in the loss function shown in Fig. 33 and is therefore not an artifact of the Kramers-Kronig analysis. For comparison, in Fig. 33 the calculated loss function is plotted and clearly shows a momentum-dependent onset. The tail below the single-particle gap has been explained either in terms of excitonic transitions due to high electron-electron correlation energy or by a direct generation of charged soliton-antisoliton pairs assisted by quantum fluctuations in the ground state (Fink, 1987). The latter excitations were also observed in measurements of the photoconductivity (Etemad et al., 1981)and in measurements of the photoinduced absorption (Blanchet et al., 1983). Above q 0.7 kl, there is considerable intensity in the single-particle gap due to multiple scattering processes. On the other hand, the shape of the calculated c2 is rather narrow, while in the experiment there is a splitting that is significant in the calculations only when local field effects are neglected. The origin of this difference has not been clear until now. The calculations also predict a strong narrowing of the plasmon at higher q and, contrary to the postulation of Neumann and von Baltz (1987), the plasmon should be completely undamped when leaving the region of interband transitions at q > 1.1 A-'. In the experiment, strong damping of the n plasmon is observed near that momentum transfer. This probably happens because c2 is strongly increased above -8 eV due to 6-0* transitions not taken into account in

-

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

199

the calculations, (see Figs. 32 and 34a), which leads to a damping of the n plasmon. In discussing the momentum-dependent dielectric functions of PA, we should also mention that the n c~ plasmon shows a linear dispersion for 0.1 I q I 1 k'with a slop of 3.5 eVA. For q > 1 A-' there is a positive deviation from the linear dispersion. An explanation for this linear dispersion may be the fact that the energy of the c ~ - c ~ *transition near 12 eV (probably at the r point) increases linearly with increasing momentum transfer similar to the case of the n-n* transition. This may then cause a linear dispersion of the c~ + n plasmon when using Eq. (23). Let us now return to the n electron system. There are deviations between theory and experiment, given the exact shape of the dielectric functions at higher q, yet the overall agreement shown in Fig. 35 may lead to the conclusion that for PA the Peierls- Frohlich model is adequate and that electron-electron correlations seem to be less important. On the other hand, momentum-dependent calculations of the dielectric functions for non-zero correlation energies would be highly desirable. There is an indication from the calculation of neff for the n electron system for q )Ic up to 6 eV using Eq. (21) that correlations are important. From the experimental data, we obtain for the optical effective mass m * / m = 1.7, while a calculation on the basis of the SSH model, including local field corrections, yields m * / m = 1.44 (Neumann and von Baltz, 1987). The difference between these values may be explained by electron-electron correlations not taken into account in the SSH model. According to calculations of the sum rule in the framework of the onedimensional Hubbard model (Baeriswyl et al., 1986), the experimental value can be explained by high on-site correlation energy U 7 eV or UIW- 0.6. This high values would indicate that PA is an intermediate case in which neither electron-phonon coupling nor electron-electron correlations can be neglected, a situation that is very difficult to treat theoretically. It has been claimed that the small difference in the optical mass of only about 20% may be also caused by other effects such as next-nearest-neighbor hopping, etc. However, it should be noted that very similar values for U have been derived from photoinduced absorption measurements (Vardeny et al., 1985) and from ENDOR experiments (Grupp et al., 1987). Moreover, theoretical estimates of U by Baeriswyl and Maki (1985) are in excellent agreement with the above experimental values. The discussion of the results for PA have illustrated the possibility of obtaining information on the electronic structure of conjugated polymers. Although the dispersion of the n plasmon is not exactly parallel to that of the n-n* transition, some semi-quantitative information on the width of n bands can be obtained directly from plasmon dispersion without performing the Kramers-Kronig analysis. Various undoped conjugated polymers such as

+

N

200

JORG FINK

PPP (Crecelius et al., 1983b; Fink et al., 1983b; polypyrrole (PPY) (Ritsko et a!., 1983b; Fink et al., 1986; Fink et al., 1987f), polythiophene (PT) (Fink et al., 1987a),and polyphenylenevinylene (PPV) (Fink et al., 1987b)have been studied. The loss functions of these polymers are shown in Fig. 32. For highly oriented samples, the data are given for q parallel to and perpendicular to the chain axis. The PPP sample was partially oriented, and, therefore, data taken for q perpendicular to the chain axis are obscured by the projected, much stronger contributions of transitions for q 11 c. Therefore, only data for q 11 c are given. For PPY and PT, up to now no oriented samples could be obtained. On all the conjugated polymers, a n r~ plasmon is realized near -22.5 eV, indicating the same concentration of valence electrons and the same energy of the G - Q * oscillator [see Eq. (23)]. In P P P and PPV there are now two n plasmons related to two n-n* transitions. Strong CT--(T* transitions are always observed between 10 to 20 eV. In Fig. 33, the momentum dependence of the n plasmons is shown. While some of these show a strong dispersion, others are almost without any dispersion. The lowest n plasmon in P P P at 4 eV shows a strong linear dispersion for 0.07 I q I0.3 A-’ with a slope of 2.9 eVA, which is slightly less than that of PA (3.5 eVA). This result suggests a band structure for the n and n* band forming the fundamental gap of PPP very close to that of PA. On the other hand, then plasmon at 7 eV shows no dispersion typical of a transition in which at least the n or the n* band or both are flat (see Fig. 4). Since from symmetry arguments, the n* bands look like the n bands with the Fermi level acting as a mirror plane, both the n and the n* band related to the 7-eV A plasmon must be flat, as shown in Fig. 4a. These flat bands, with a gap of about 6 eV, are related to the n electrons that are localized in the benzene rings, the wave functions of which have nodes at the carbon atoms connecting the rings. The wide R bands with a gap of about 3.2 eV related to the 4-eV plasmon are then n electrons that are delocalized over the polymer chain. Thus, from the loss data a n band structure combining Figs. 4a and 4c with gap energies of 6 and 3.2eV, respectively, can be derived. This is in excellent agreement with band-structure calculations for the n electron system in PPP by Bredas et al. (1984). The situation in PPV is very similar but there is an indication of a third transition between the two dominant n plasmons that may actually be a “mixed” transition between the flat bands and the wide bands, apparently not allowed in PPP. In both PPY and PT, the lowest n plasmons show strong dispersion, indicating that the transition across the fundamental gap is related to a wide n band and a wide n* band. The maxima in the loss functions at higher energies show almost no dispersion, meaning that at least the R band or the n* band or both related to the transition must be flat. The investigation of the n band structure may give important information on the formation of conjugated polymers. This is illustrated here for the

+

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201

example of PPV (Fink et al., 1987b). This polymer is formed by thermal transformation from the precursor polymer poly(p-xylene-x-dimethylsulphoniumchloride) in which the benzene rings are connected by a saturated CH [S'(CH,),] - CH2 bridge. By thermal elimination of dimethylsulphide, the saturated bridge is transformed into the non-saturated -CH=CHbridge. The ratio of the non-conjugated to the conjugated segments, and, therefore, the average conjugation length, depends on the annealing temperature and time. In Fig. 36a we show the loss function of the precursor polymer at various stages of thermal conversion to PPV, and in Fig. 36b the dispersion of the lowest maximum of the loss function near 4eV. For the as-cast precursor, the 4 eV transition is small and shows almost no dispersion typical of a molecular transition in short conjugated systems. With increasing temperature, the length of the unsaturated segments increases, which leads to an increasing n-z* transition at 4eV with increasing dispersion in q. Therefore, these measurements provide information on the transformation mechanism when passing from the precursor polymer to PPV. In the following, we turn to the doped conjugated polymers and review investigations of the evolution of the n electron structure as a function of doping concentration. For highly oriented PA, the experiments at higher momentum transfer are rather difficult to perform because the effective thickness of the samples strongly increases upon doping with heavy counterions and samples below 2000 A could not be prepared. Nevertheless, loss

20ooc 4.6 13OOC

RT.

-

f

lb)

-

>, - 4.4 &

LL w

24h 30OOC 2h 200OC 2h 130°C

R.T.

4.2 -

4.0 -

3.8 0

2

4

6 8 1 0 ENERGY ( e V I

0

0.1

0.2

0.3

0.4

MOMENTUM TRANSFER

FIG.36. Formation of polyphenylenevinylene from the prepolymer poly(p-xy1ene-xdimethylsulfoniumchloride) upon heat treatment. (a) Loss spectra. (b) Dispersion of the first maximum in the loss spectra.

202

JORG FINK

measurements on doped samples in a limited q range are also possible (Fink et al., 19878).In Fig. 37a the change of the n plasmon of AsF; doped PA as a function of the conductivity of the sample is shown for q 11 c. The highest conductivity corresponds probably to an AsF; concentration of 15% per C atom. The transition into a metallic state with finite Pauli susceptibility is reached near 1000 S/cm. At low doping concentration, intensity appears in the gap of undoped PA due to the fqrmation of a soliton or midgap level. With increasing dopant concentration, the n plasmon is broadened and shifted to lower energy. At the highest dopant concentration, the onset of the loss function is close to zero and the shift of the plasmon is 2.1 eV. For q h , the dielectric function due to the n electrons is probably small compared to the background dielectric function cB because of the cr electrons and therefore the loss function is given in this case by Im( - 1 / ~ )= ct/(.$ E : ) = &&/:.; However, contributions from cannot be excluded, in particular at the highest dopant concentrations where the samples become amorphous, while there still remains some electronic anisotropy. The data in Fig. 37a and 37b clearly show a transformation from a semiconductor via a system with a defect level in the gap to a system where the gap is closed or at least less than some tenth of an eV. In Fig. 38, the evolution of the plasmon dispersion upon

+

I

I

Ib)

ql

=o.taa,,

(S/cm

3300 2700 1000

300 3

0

I 2 4 6 8 1 0 ENERGY ( e V )

ENERGY lev)

FIG.37. Evolution of the loss function in the z electron range of highly oriented PA upon doping with AsF; for various conductivities parallel to the chain axis. (a) q parallel to the chain axis. (b) q perpendicular to the chain axis.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

203

2L

10

02

0.4

06

08

10

MOMENTUM TRANSFER

(k')

FIG.38. Dispersion of the R plasmon in AsFi-doped, highly oriented polyacetylene for various conductivitiesparallel to the chain axis.

doping with AsF, is shown. Up to oI1= 300 S/cm, the dispersion of the plasmon is not changed compared to that of undoped PA. That means that the overall n band structure is not changed and that the conductivity must be connected with defect levels in the gap. Between oII= 300 and 1800 S/cm there is a rapid transition to a new dispersion curve with a steeper slope indicating a stronger dispersion of the n and n* band near the Fermi level, which is to be expected for a closed gap. In addition, for oII 2 1800 S/cm a step of -0.3 eV is realized in the dispersion curves which moves to q = 0.25 A-' for the highest dopant concentration. The step is explained in the following way. At low q there are only transitions from the valence band into a broad soliton band that has filled the gap almost completely. At higher momentum transfer, there are more and more transitions from the valence band to the conduction band. The step in the dispersion would then indicate a gap between the soliton band and the conduction band of about 0.3 eV. The explanation is supported by the fact that the step appears at lower q for a lower dopant concentration at which the unit cell, and thus the length of the soliton band in q space, is smaller due to the larger distance of the counterions in real space. A gap of the same size between valence band and soliton band is to be expected but could not be detected because of the limited energy

204

JORG FINK

resolution. The results support a model of a semiconductor-metal transition via a soliton band. No indication of a narrow half-filled polaron band is observed. Therefore, the above model of a polaron metal is not supported. Finally, we show in Fig. 39 a first attempt to perform a Kramers-Kronig analysis of the loss function of AsF; doped PA. For the calibration of the loss function, E ! = 40 and E: = 8 at the energy of -0.25 eV (Leising et al., 1987) was used. The dielectric functions reveal the shift of the n-n* transition to very low energy (E, I0.2 eV) and the remaining strong anisotropy of the n-n * oscillator. This indicates a highly anisotropic one-dimensional metal for the p-doped PA which was also derived from conductivity measurements and optical studies (Leising et al., 1987). Similar studies on n-doped, highly oriented PA have been performed recently (Fink et al., 1987e; Fink et al., 19878; Fink and Leising, 1987a). In Fig. 40,the loss function and the dielectric functions of K-doped highly oriented P A is shown. A smaller shift to lower energy is observed in all the n-type doped systems. For K, the shift is about 1 eV. For the more electropositive metals Rb and Cs, the shifts are slightly larger. Like the p-type doped PA, there are changes in the o-o* and n-o* transitions. In addition, a strong K 3p core excitation near 20 eV is observed that is very similar to that observed in K-intercalated graphite (Grunes and Ritsko, 1983). Since the effective thickness of the samples was

1.5

PA ( AsF;

1

FIG.39. Loss function and dielectric functions of fully AsFi-doped, highly oriented polyacetylene. Solid lines: q parallel to the chain axis; dashed lines: q perpendicular to the chain axis.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY 15

205

PA (K')

&-

__.--. --- '-.__ x5

---___-.

FIG.40. Loss function and dielectric functions of fully K+-doped, highly oriented polyacetylene. Solid lines: q parallel to the chain axis; dashed lines: q perpendicular to the chain axis.

smaller than that of the p-type doped ones, the Kramers-Kronig analysis could be performed also for higher momentum transfer. In Fig. 41, E 2 ( q , w ) for K-doped PA for q c is shown. Compared to the undoped case, the x-n* oscillator is shifted to lower energies. It shows within error bars a linear dispersion. The extrapolation to zero q gives almost zero energy, indicating the closing of the gap at the highest dopant concentrations. This indicates a similar excitation spectrum as predicted by Williams and Bloch (1974) for a one-dimensional metal. However, looking closer to the data, there are deviations. For example, there are again steplike deviations from a linear plasmon dispersion that at present are not understood in detail. The situation is slightly different for the non-degenerate ground state polymers. In Fig. 42, we show a typical loss function of K-doped PPP at Iow dopant concentration. There are two maxima near 1 and 2 eV in the gap that can be assigned to transitions from the filled bipolaron levels to the conduction band (B*-n* and €3-n* transition). According to the calculation of Fesser et al. (1983), the lower maximum should be much stronger in intensity than the higher maximum, with a ratio of about 20, in rough agreement with the data shown in Fig. 42. It is interesting to note that the observed ratios are not always 20 but are in some cases considerably higher, depending on the method of doping, the counterion, etc. No systematic variation, however, is

206

JORG FINK ___

7--

0

5 10 15 ENERGY ( e V )

FIG.41. Momentum-dependentimaginary part of the dielectricfunctionfor q parallel to the chain axis for fully K -doped, highly oriented polyacetylene. +

observed. On the other hand, various symmetry-breaking mechanisms have been predicted, such as high correlation energy or interchain interaction, that change the oscillator strength of the two transitions (Sum et al., 1987). The shape of the two transitions are also in good agreement with theoretical predictions. It indicates a transition from a narrow defect-level into a

0

1

2 ENERGY (eV1

3

FIG.42. Transitions from occupied defect levels in the gap into the ~band l * for K+-doped polyparaphenyleneat low dopant concentration.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

207

continuum of n* states. Attempts to measure the form factor of these excitations, ie., the momentum dependence of the intensity, have so far not been successful due to the low cross-section and increasing background at higher momentum transfer. Such measurements could, in principle, provide information on the size of the defects. The evolution of the loss function upon doping is illustrated in Fig. 43 for the case of Cs-doped PPP. At low concentrations, again two bipolaron transitions are realized. With increasing dopant concentration, the B*-n* transition increases at the expense of the lowest n-n* transition. At about 25 at.% per monomer, the n-n* transition at 4 eV has disappeared and a new shoulder is realized on the high energy side of the B*-n* transition that disappears at the highest dopant concentrations ( - 50 at.% per monomer). The n-n* plasmon at 7 eV remains almost unchanged, indicating that the benzene rings are not destroyed upon doping. This finding is supported by the fact that the doping is completely reversible, i.e., upon heating the sample Cs can be evaporated and the undoped loss spectrum is obtained again. In the

-

0 2

4 6 8 1012 ENERGY ( e V )

FIG.43. Evolution of the loss function of polyparaphenylene (PPP) upon doping with Cs+. Uppermost curve: fully Cs+-doped PPP ,.( 50 at.% per monomer).

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JORG FINK

fully doped sample, the strong plasmon at 2.5 eV may be explained by a n plasmon related to a n band that is closed upon doping, i.e., a free-carrier plasmon. However, by comparison with the momentum dependence of this excitation (see Fig. 44) here the situation is completely different from that of PA. The maximum shows no dispersion, indicating a relation t o a still existing, narrow bipolaron band. At higher momentum transfers, a second maximum, showing a strong dispersion, appears which may be assigned to a n-n* transition. At the highest dopant concentrations, the lowest n* band is probably partially filled already, a fact that allows n-n* transitions only at higher momentum transfers or at lower dopant concentrations in agreement with the experiment. The data further suggest that there is a narrowing of the n-n* gap near -25 at.% by about 1 eV, since the energy of the z-n* transition is reduced by this amount. In summary, the present data support an evolution of the n electron system as shown in Fig. 30. However, complete overlap with the valence and conduction band could not be observed. Similar data on PPP have been reported for AsF; doping (Crecelius et al., 1983a), Li doping (Fink et al., 1984b, 1985b), and Na doping (Fark et al., 1987). The existence of a narrow bipolaron band at even the highest dopant concentrations has been also observed in EELS studies on other non-degenerate ground state polymers, such as PPY (Fink et al., 1986; 1987f), PT (Fink et al., 1987a) and PPV (Fink et al., 1987b). It is probably a general law that these non-

I

0

1

2

3 L 5 ENERGY ( e V )

6

FIG.44. Momentum dependence of the loss function of fully Cs+-dopedpolyparaphenylene.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

209

degenerate ground state polymers cannot be doped to the full overlap of the bipolaron bands with the conduction bands and that narrow bipolaron bands remain in the gap up to the highest dopant concentrations. At the end of this section on conducting polymers, we briefly report on recent results on core-level excitations in undoped and doped conjugated polymers. Only some of these results have been published in the references given above. In Fig. 45, the C 1s absorption edges of conjugated carbonhydrogen polymers are shown. The spectrum of PPP is almost identical to that of gaseous benzene (Horsley et al., 1987). For benzene, the peaks at 285.2 and 288.9 eV were ascribed to the 1s-n*(ezu) and 1s-n*(b,,) transitions, since benzene has two unoccupied n* states. The peaks at 293.5 and 300.2 eV, the latter being less pronounced in PPP, were assigned to two 1s-o* shape resonances. The shoulder at 287.2 eV was assigned to a 3p Rydberg state, although, since it remains in the solid, the assignment is probably not correct. It is remarkable that the spectrum of the molecule and that of the polymer is almost the same. This indicates that in these cases the core-level excitations only sample a very local part of the density of the unoccupied states. The

280

290

300 ENERGY l e v )

30

FIG.45. Carbon 1s edges of hydrogen-carbon conjugated polymers such as polyacetylene (PA), polyparaphenylene (PPP), polyphenylenevinylene (PPV), and polynaphtalinevinylene (PNV).

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JORG FINK

delocalized n electrons have no influence on the spectrum. In PPV, the spectrum is almost the same; only the first shoulder above the first 1s-n* resonance is shifted slightly to higher energies. For PA, the first two peaks, which are probably again 1s-n* transitions, also appear in butene molecules. For q1 = 0, the momentum transfer q is given by q = qll = 0.19 k'for an energy loss of 280 eV [see Eq. (lb)]. When the polymer chain is perpendicular to the beam, there is strong overlap of the n* lobes with the momentum transfer, and therefore the matrix element in Eq. (48) is large, yielding strong 1s-n* transitions. Upon increasing ql, the total momentum transfer turns the matrix element more and more parallel to the chain axis. At q1 = 0.7 kl, in Eq. (48) for the ls-a* transition is strongly reduced, leading to a much weaker peak in the C 1s edge as shown in Fig. 45. For q l c and less pronounced for q IIc, there are two Is-o* shape resonances. As described in Section II,D, the energy positions of these resonances can be used to determine bond lengths. Using the empirical relation A E = 54 Ar, an energy difference of about 4 eV leads to Ar = 0.075 A for the difference between single and double bond. This is in excellent agreement with X-ray (Fincher et al., 1982) and electron (Chien et ai., 1982) scattering data, yielding Ar = 0.06 A, and with NMR data by Yannoni and Clark (1983), yielding 0.08 A. However, it should be kept in mind, as outlined in Section II,D, that the model we have used is based on empirical data and that various exceptions to this model exist. In principle, such splitting of the ls-o* resonances should also appear in PPP, PPV, and polynaphtalinevinylene (PNV), but in this case the ratio of the number of double bonds to that of single bonds is much less than one. Finally, we note that the origin of the splitting of the 1s-n* transition in PNV is not clear at present. In Fig. 46, we show C 1s edges for other conjugated undoped polymers. Polymethineimine (PMI), which is similar to trans-PA but with every second C-H group replaced by a N atom, has a strong 1s-n* resonance at 286.6 eV, which is about 2 eV above that of PA. This is probably caused by chemical shift due to the more electronegative N atoms. From XPS measurements, it is well known that one N atom bonded to a C atom causes a chemical shift of the C 1s level of about 1 eV. In polyacrylonitrile (PAN) at low temperatures, which is a polyethylene chain where every second H atom is replaced by a C G N group, this group causes the ls-x* resonance at 286.8 eV. Upon heating PAN at 3OO0C,PAN transforms into polypyridinopyridine, which is composed of trans-PA bonded on each second C atom by a r~ bond to a PMI chain. Therefore, there appear two 1s-n* transitions at -285 eV and 287 eV corresponding to those of trans-PA and PMI (Ritsko et al., 1983c; Fink et al., 1983~).In PPY, there are two n resonances separated by 1 eV because the so-called /%carbons (two per monomer) are bonded to C and H atoms only, while the a-carbons (two per monomer) are bonded to one N neighbor each. In PT, the 1s-n* transition at about 285 eV also shows a

-

-

-

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

21 1

PAN

200Y PAN

300°C PPY

PT

280

290

300

0

ENERGY ( e V )

FIG.46. Carbon Is edges of N and S containing conjugated polymers such as polymethineimine (PMI), polyacrylonitrile (PAN), polypyrrole (PPY), and polythiophene (PT).

splitting, which is, however, rather small because S and C have almost the same electronegativity. In context of an EELS investigation of the monomer, the peak at 287.2 eV was ascribed to a C-S 1s-a* resonance (Hitchcock et al., 1986). In summary, the 1st-n* transitions in the conjugated undoped polymer are very close to those of short molecules. Therefore, the spectra are not related to delocalized n electrons but to n electrons localized in small molecules. The first 1s-n* transition, which appears close to 285 eV for all nbonded carbon compounds, including graphite, indicates that this transition reaches even more localized, probably atomic-like states. The N 1s spectra of these materials have the same character (see Fig. 47). There is a 1s-n* transition near 401 eV for N bonded to an H atom. In PMI, radiation-damaged PPY, and in a polyaniline sample (PANI), 1s-n* transitions appear at lower energy due to N atoms not bonded to a H atom, which therefore have more charge. In n-type doped metallic polymers, the core holes are stronger shielded and the core-level edges are characterized by an extremely sharp rise at threshold as in metals. For Na- and K-doped PA, the molecular C 1s-n* resonance is strongly reduced, as shown in Fig. 48. A similar C Is edge is found by Ritsko (1981) for AsFS-doped PA. The very near-edge structure of PA(K+)is close to

212

JORG FINK

PPY - 6 s - R I

390

400 410 420 ENERGY ( e V )

FIG.47. Nitrogen 1s edges of polymethineimine (PMI), (210;-doped polypyrrole (PPY-

Cloy), radiation-damaged butanesulfonate-doped polypyrrole (PPY-BS-RD), and Cl0;doped polyaniline (PANI).

that of K-doped graphite (Grunes and Ritsko, 1983). The two maxima may be explained by two C sites, one of them close to a counterion, the other not. However, the intensity of the second peak is much too small to support this explanation. A splitting may also occur to metal carbon hybridization. Furthermore, the second peak may be caused by metal s states above the Fermi level. It is very interesting to note that in the fully doped PA, there is now only one 1s-o* resonance, indicating an almost equal C-C bond-length as predicted for a Peierls system with a closed gap. The 1s-n* transition of Na+, K', Rb+, and Cs+-doped PPP are strongly reduced compared to those in undoped PPP. There is a sharp edge with a tail at higher energies. It is as yet unclear why Li+-doped PPP is an exception. Perhaps there is a partial ionic binding of Li to PPP. It should be noted that there was no evidence of such a bonding in the valence excitations. In Fig. 49, C 1s edges of p-type doped conjugated polymers are shown. In all cases, a shoulder or peak below the threshold appears that may be explained by an excitation into empty defect levels in the gap or by a negative chemical shift. From the charge extraction out of the carbon atoms, a shift to higher energy should be expected [see Eq. (49)and Section II,D]. However, since the extracted charge per counterion is distributed over several carbon atoms in the polymer chain while the charge on the counterions is concentrated on one site, the Madelung term may

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213

I Cls "a')

IK')

PPP ILI+ 1

PPP INa') PPP IK')

PPPIRb* I

280

290 300 ENERGY ( e V )

310

FIG.48. Carbon Is edges of n-type doped conjugated polymers such as N a + -and Kf-doped polyacetylene (PA)and Lif-, Na+-,K+-, Rb'-, and (2s'-doped polyparaphenylene (PPP).

overcompensate the first term in Eq. (49). The carbon atoms close to the counterions may then cause spectral weight at lower energy. Finally, we note that for PANI, the peak at 287 eV can be assigned to a ls-.n* transition of those carbon atoms that are bonded to a N atom. It is interesting to investigate also the core-level excitations on the counterions in the conducting polymers. As an example, in Fig. 50 we show core edges of Na' in PPP. For comparison, the same edges are shown for Na metal. Since charge is transfered from the Na atom to the polymer, the core levels are shifted to lower energies and, therefore, the threshold is shifted to higher energies. This is observed in both the Na 1s and Na 2p excitations. In NaOH, the Na 2p peak appears at 33.3 eV (Kunz, 1966), which is 0.5 eV above that of PPP (Na') shown in Fig. 50. This indicates a slightly weaker oxidation of Na in PPP than in NaOH. Since in both spectra of Na' in PPP there are no pronounced thresholds, it is believed that most of the 3s and 3p states are well above the Fermi level, which indicates a rather complete charge transfer to the polymer. In Fig. 48 the 2p edges of K + in PA and in P P P are recognized. The difference in spectral shape is not clear at present. K in PA shows a shape very similar to that of K metal while K in PPP shows a shape that is closer to that of oxidized K.

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PPY ( C I O ~ )

PT (CIO,)

PAN1 (ClO,

l " " 1 " ~ ' 1 " " 1 ' ' " I ' " ' I " '

290

280

300 ENERGY (eV 1

0

FIG.49. Carbon 1s edges of p-type doped polymers such as polypyrrole (PPY), polythiophen (PT) and polyaniline (PANI).

1 Nals

Na-METAI

Na-METAL

30

32 34 36 ENERGY l e v )

FIG.50. Sodium 1s and 2p edges of Na metal and as counterion in polyparaphenylene

(PPP).

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

2 15

In summary, important information on conducting polymers can be obtained from core-level spectroscopy using the spectra as finger prints similar to XPS as used in chemical analysis. It should be emphasized that the energy resolution in present day EELS spectra is better by a factor of about 3- 10 than conventional XPS spectroscopy and, therefore, smaller shifts can be detected. On the other hand, it should not be forgotten that in EELS, changes of both the core level and the conduction bands have to be considered. More theoretical calculations on core-level excitations in these systems are highly desirable, since only then will detailed information on the electronic structure be obtained.

IX. SUPERCONDUCTORS The recent discovery of high- T, superconductivity in certain metallic ceramics (Bednorz and Muller, 1986) has strongly revived interest in the field of superconductivity. Generally, any microscopic understanding of superconductivity requires detailed knowledge of the normal-conducting properties, of which electronic properties are central. Within the conventional Eliashberg theory, which is believed to describe essentially all known “conventional” superconductors (e.g., Pb, Nb, TIN, Nb,Sn,. . ..), the electronic properties occur in the “Eliashberg function” a2F(o),which is the main ingredient in this theory. It is defined by

Here E k is the energy of an electron of momentum k, w k - k .a phonon frequency belonging to momentum k- k’, G k k . the electron-phonon coupling constant, and N ( E F ) the electronic DOS at E F (Scalapino, 1969). For simplicity, band and branch indices have been omitted. Equation ( 5 5 ) represents a Fermi-surface average over electron-phonon coupling. A crucial point for the validity of Eliashberg’s theory is the applicability of Migdal’s theorem, which implies a slowly varying electronic DOS around E F on the scale of typical phonon energies hwDebye.So spectroscopic investigation of electronic states around EF may not only serve as an experimental check of the electronic data in a2F(w),but also tell us whether N ( E ) is indeed slowly varying around EF, and thus whether Eliashberg’s theory is applicable at all. For the conventional refractory compound superconductors and the A 15 superconductors, this will be illustrated below. Similar studies on the organic superconductor /?(BEDT-TTF),I, have been performed recently by Nucker et al. (1986). Within the context of unconventional superconductors (heavy fermions, new high-T, ceramics, etc.), spectroscopic data may still provide valuable help

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for discriminating between different coupling mechanisms. So, if instead of phonons, low-lying excitations of electronic origin (excitons, plasmons) play the role of virtually exchanged bosons, those excitations should also be seen in the real spectral DOS in high resolution energy-loss spectroscopy. Moreover, these superconductors are expected to be highly correlated systems, which implies that the local density-functional theory may not provide a good description of the electronic structure. This breakdown can be recognized by electron spectroscopy studies and furthermore, important parameters can be determined at least roughly to guide theory in the choice of a model. A. Transition Metal Carbides and Nitrides

First, we review investigations of the electronic structure of 3d and 4d transition metal carbides and nitrides (Pfliiger et al., 1982; Pfliiger et al., 1984; Pfliiger et al., 1985a; Pfluger et al., 1985b). Some of these refractory materials have a rather high T, on a before-1986 scale (e.g., for NbN, T, = 16 K). The band structure in the vicinity of the Fermi level is composed of bonding nonmetal p orbitals and of antibonding metal d orbitals. In the carbides, there is strong hybridization of the p states with the d states, which decreases as we approach the nitrides and oxides. The position of EF is determined by the number of available valence electrons. In the case of Tic, EFis situated in the very minimum between the p and the d states, while for TiN, EF is shifted upward into the d bands and N(EF) increases. In Fig. 51, we show the loss

Frc. 51. Loss functions and dielectric functions of Tic and TiN.

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2 17

function and the dielectric functions of T i c and TIN, as derived from the loss spectra. In Fig. 52 we show a comparison of the reflectivity derived from EELS measurements for TIC with optical data obtained using synchrotron radiation (Lynch et al., 1980). In the lower part, the loss function calculated from the optical data is compared with the measured loss function. The overall agreement between these results obtained by the two techniques is rather good. In Fig. 53, the optical joint density-of-states for T i c and TiN as derived from the loss spectra is compared with calculations of the joint density-ofstates using the self-consistent Gaussian-LCAO method of Applebaum and Hamann (1978). Matrix elements and energy-dependent life-time effects were not taken into account. In both compounds, there is, besides a Drude part, a strong interband transition at about 5-7 eV from the bondingp bands into the

1

0.2

0

2.0 Z

E Z

i?

1.0

0

ENERGY IeV 1 FIG.52. Comparison of EELS data (solid line) and optical data (dashed line) for T i c (after Pfliiger et a/., 1984). Upper part: measured reflectivity (after Lynch et al., 1980) and reflectivity derived from EELS data. Lower part: measured loss function and calculated loss function from reflectivity data.

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JORG FINK

FIG.53. Optical joint density-of-states (solid line) of Tic and TiN as derived by a KramersKronig analysis of the loss functions. For comparison, the calculated joint density-of-states is shown (dashed line). Note that in the calculations, matrix elements and energy-dependent life-time effects are not included.

antibonding d bands. The oscillator causes a zero-crossing of el below the oscillator energy (see also Section II,B and Fig. 2b). In Tic, EF is in the very minimum between the p and d bands and there are strong interband transitions also at lower energy which strongly damp the plasmon. For TiN, where EF is within the d bands, intraband transitions between pure d states localized at the metal atoms are dipole-forbidden and e2 and the optical joint density-of-states have a minimum close to the energy at which el has a zerocrossing. Therefore, the interband plasmon is highly pronounced. A similar highly pronounced plasmon was also observed in ZrN, while in VC, VN, NbC, and NbN it is more or less broadened due to intraband transitions. Structures in the loss function at higher energies can be attributed to transitions from non-metal 2s states into the unoccupied d states. For the carbides and nitrides they appear at 12 eV and 19 eV, respectively. The volume plasmons of all valence electrons are close to the expected energies of the freeelectron plasmons. In Fig. 54, the non-metal 1s absorption edges of Tic, TIN, VC, VN are shown, the binding energies of the 1s level being 281.5, 397.0, 282.6, and

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219

P

A

vc

i

FIG.54. Nonmetal 1s core-level edges of Tic, TiN, VC, and VN (solid lines). For comparison, the total unoccupied DOS (dashed line) and the calculated transition probability (dotted line) is shown. In the calculations an energy-dependentlife-time broadening is included.

396.8 eV, respectively. For comparison, we also show in Fig. 54 the broadened total DOS calculated by a self-consistent LCAO program. For VN, a singleparticle calculation of the broadened transition probability using a cluster program based on the multiscattering formalism is also shown (Winter, 1984). The measured peak positions are well reproduced by the DOS calculation. This indicates that the transitions from the non-metal 1s states into the rather delocalized 2p states can be well described by single-particlecalculations, and that in this case interaction with the core-hole is rather weak. The situation is completely different in the case of metal 2p excitations into the 3d states, where due to the large overlap between the 3d states and the 2p states there is strong interaction with the core-hole. Now the edges cannot be explained any longer by a simple DOS calculation (Pfliiger, 1983).In the non-metal 1s edges, the first two peakscan be assigned to metal 3d-t,,and 3d-e, states hybridized with non-metal 2p states. Thus, the crystal splitting of the 3d states can be

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clearly recognized in all four compounds. The strong peak about 12 eV above the Fermi level was ascribed essentially to a transition into a non-metal 2p state. These core-level measurements indicate that the non-superconducting materials T i c and VC have a lower DOS at EF compared to that of the superconducting materials VN and TiN. With an energy resolution of 0.2 eV, no strongly varying DOS close to EF could be detected. In general, besides investigations with other spectroscopic techniques, the above EELS investigations have served as a check of our present understanding of the electronic structure of these materials. B. A-15 Compounds

Up to 1986 the A15 compounds were the class of materials with the highest superconducting transition temperatures. Moreover, these alloys are also characterized by other unusual physical properties, such as phonon anomalies, temperature-dependent Knight shifts, and others. This has been attributed to a high and rapidly varying DOS with peaks having a width of about 50 meV close to EF. Band structure calculations indeed predict such features in the DOS. A direct spectroscopic verification of the existence of these peaks has been achieved by core-level spectroscopy using EELS (MullerHeinzerling et al., 1985a). In Fig. 55, we show the N b 3d5,, edges of Nb,Sn, Nb,Ge, and Nb,Al, which appear at about 202.5eV. Spectra on nearstochiometric, high-T,, and on off-stochiometric, low-T, samples are shown. For comparison, the partial p-symmetry DOS and the unoccupied part broadened by the energy resolution (0.2 eV) obtained from a tight-binding fit to self-consistent augmented-plane wave calculations (Mattheiss and Weber, 1982) are also shown. It is believed that the measured edge is close to the unoccupied DOS because the s and p final states in this case have only a small overlap with the core-hole. Indeed, the measured high-T, edges are close to the calculated ones, indicating the existence of a peaked DOS close to E F . Moreover, the high- T, samples exhibit much more pronounced peaks than the corresponding low-T, samples. This is in agreement with specific-heat data. Finally, there is a clear trend of increasing peak heights from Nb,Sn to Nb,Ge and, in particular, to Nb,Al, which originates from the shift of EF from the top to the bottom part of the high-DOS region, leaving more and more of the high-DOS region unoccupied and therefore contributing to the nearedge peak. These core-level measurements have revealed directly the existence of the high and strongly peaked DOS at EF in the A15 compounds which is responsible for many anomalies, in particular for the high superconducting transition temperatures. Further information on the electron structure has been obtained from valence band excitations, producing dielectric functions

22 1

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-05

00 05

10

15

Energy above EF (eV) FIG.55. Nb,,, core-level edges of Nb,Sn, Nb,Ge, and Nb,AI on off-stoichiometric, low-T, and stoichiometric, high-T, samples. The thin line through the data is a guide to the eye. All curves are normalized to the same height at 1 eV above E,. For comparison, the calculated broadened unoccupied density-of-states and the unbroadened density-of-states for the three compounds are shown (after Muller-Heinzerling et al., 1985a)

over a broad energy range that can be compared with band-structure calculations (Miiller-Heinzerling et al., 1985b; Miiller-Heinzerling, 1985). C . Ceramic Superconductors

The last chapter of this section on superconductors is devoted to new ceramic systems that possess superior superconducting properties such as high in the 30-95 K range, high critical fields, and high critical currents. Since this is presently an extremely rapidly developing field, the presentation of EELS and other electron spectroscopy results can reflect only a current viewpoint whose correctness remains to be proven. At present, it is still far from clear what type of interaction leads to the high superconducting transition temperatures. Many theoretical models have been proposed so far based on a pairing by phonons, excitons, plasmons, or on electron-electron Coulomb interaction (Rice, 1987; Fulde, 1988). In particular, the problem of providing the relevant parameters for the latter interaction poses a challenge to theorists and experimentalists. At present (October 1987), there are basically two systems under consideration, La,-,M,CuO, with M = Sr, Ca,Ba . .. and MBazCu,O,-,

rs

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with M = Y and most of the rare earths. It is generally believed that in these compounds, two-dimensional C u - 0 planes and, in YBa2Cu307-, , besides those also one-dimensional C u - 0 chains, form the electronic states close to the Fermi level. These low-dimensional structures are therefore important for the transport properties of these systems. For La,CuO,, band-structure calculations predict a splitting of the Cu 3d egstates due to a non-cubic crystal field, and therefore the antibonding Cu 3dX+* states hybridize with the 0 2p states forming a half-filled wide band (Mattheiss, 1987; Yu et al., 1987).The Fermi surface of this half-filled two-dimensionalband has pronounced nesting properties, and therefore a charge-density wave or a spin-density wave may occur depending on the size of electron-phonon or the electron-electron Coulomb interaction. Electron spectroscopy studies may help to clarify whether the single-particle calculations are still valid, or whether the above mechanisms lead to a breakdown of these calculations. Preliminary information on electronic structure can be obtained from the loss function of these materials (Nucker et al., 1987a) shown in Fig. 56. The free-electron plasmon of all valence electrons should appear in both compounds close to 26 eV, in excellent agreement with the experimental result of 25.5 eV for YBa2Cu,O7. In La,CuO,, the plasmon is shifted to 29.5 eV due to a very strong oscillator at 13.7 eV (see Section 11,BJ and Fig. 2). According to the XPS and BIS spectra for these materials (Nucker et al.,

I

0

SrO.ZSCuo~

10

20 30 40 ENERGY ( e V 1

50

FIG.56. Loss spectra of La,,,,Sr,.,,CuO, and YBa,Cu,O,.

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

223

1987b),this oscillator should be assigned to a transition of the occupied La 5d states to the narrow unoccupied La 4f band. There is an additional shoulder near 21.5 eV that can probably be attributed to a Sr 4p excitation. In YBa,Cu,O,, shoulders at about 17,19 and 29 eV can be assigned to Ba 5p,,, , 5p1,2,and Y 4p core-level excitations. The origin of the shoulders at 13.6 and 35 eV is not clear at present. Unfortunately, the samples available at present, which were prepared by cutting with an ultra-microtome (see Section IV), are small and have numerous pin-holes. Therefore, strong contributions from the direct beam produce a large background at low energies. Thus, the data below about 2 eV are still connected with some uncertainty and Kramers-Kronig transformation could not be performed. However, momentum-dependent data in this energy range would be extremely interesting and could decide whether there exist low-energy excitations that may lead to a pairing of electrons. Up to now, no pronounced plasmon could be detected that would be expected from optical data on these materials. In Fig. 57, the 0 1s edges of pure La2Cu0, and La,8,Sro,,,Cu0, are shown (Nucker et al., 1987b, 1987c, 1988). The binding energy of the 0 Is level as determined by XPS is indicated by the dashed line. If we interpret the edges as the local DOS of the unoccupied states at the oxygen atoms, this line will denote the Fermi level. In the Sr-doped system, there is a peak close to E F , the intensity of which increases roughly proportional to x. Beyond 5 eV above EF, strong spectral density is observed due to 0 2p states hybridized with La 5d and 4f states. This is in agreement with band-structure calculations within the local densityfunctional approximation using the LMTO-ASA method (Temmerman et al.,

x = 0.15

0

ENERGY (eV1

FIG.57. Oxygen 1s edges of La,-,Sr,CuO, for x

= 0 and 0.15.

224

JORG FINK

1987). It is very interesting to note that in pure La,CuO,, there is within error bars no DOS at EF,although the band-structure calculation and calculations that include the matrix elements predict almost the same local 0 DOS or the same spectral weight near E F , respectively, for the pure and for the doped material. Moreover, a peak or a shoulder at about 2 eV above EF is not predicted by the calculations. This indicates clearly a breakdown of the local density-functional band-structure calculations for these materials. Taking into account the energy resolution of 0.4 eV chosen for the registration of these spectra, a gap of about 1 eV can be derived for La,CuO,. An interpretation of the gap in terms of a Peierls-Frohlich model due to electron-phonon coupling can be excluded because no charge density wave, i.e., no distortion of the C u - 0 distances in the planes, has been observed experimentally. On the other hand, the finding of an antiferromagnetic transition in La,CuO, with a Nee1 temperature of TN= 230 K strongly suggests that the observed gap is due to strong correlations of d electrons on the Cu sites, i.e., a MottHubbard gap. The next important question is: How strong are the correlawould lead to a 3d count close to 9. This would then tions. A rather large U..,, lead to a closed 0 2p band between the lower and the upper Hubbard band and thus an insulating state with a gap would be formed (Emery, 1987). The peak at the Fermi level in La,,8,Sr,.,,Cu0, could then be explained as holes on the oxygen sites. This peak has never been found in photoelectron spectra where the Cu 3d states have a much larger cross-section and the 0 2p states have a smaller one. Thus we conclude that the holes created upon doping have predominantly 0 2p character. The hole-like character of the charge carriers was also derived from Hall-effect studies (Hundley et al., 1987; Ong et al., 1987) and by optical spectroscopy (Geserich et al., 1987). The indication of a large correlation among d electrons is supported by XPS measurements on the density-of-states of the valence electrons (Fujimori et d., 1987; Fuggle et al., 1988), where a peaking of the spectra dominated by Cu 3d states should occur at 2 eV below $;,while in the experiment it is observed at -4 eV. Furthermore, strong correlations would prevent a Cu 3d8 (Cu3+) configuration in agreement with the experimental results of XPS studies on Cu 2p core-level excitations (Niicker et al., 1987b). Finally, from Auger electron spectroscopy it is clear that U,,, for the Cu 3d electrons is not less then 4- 5 eV (Fuggle et al., 1988). The situation in the YBa,Cu,O,-, system is very similar as shown in Fig. 58. There may be at present some ambiguities on the threshold in YBaZCu307-,, since it is not clear whether the 0 1s binding energies for the four 0 sites have the same value. The density-functional band-structure calculations again predict a similar density-of-statesat EF for y = 0 (superconducting, = 92 K) and for y = 1 (insulating), while in the experiment, for y = 1, at which the one-dimensional C u - 0 chains are destroyed, the DOS is

-

RECENT DEVELOPMENTS IN ENERGY-LOSS SPECTROSCOPY

I

01s

~

y

225

- 0.2

- 0.3 - 0.5 - 0.8

1 . 4 .%a%-.

5 k E~ 5;O

5% '

510 '

ENERGY ( e V )

FIG.58. Oxygen 1s edges of YBa,Cu30,-, for y about 0.2,0.3,0.5,and 0.8.

-

close to zero. This indicates again a gap due to the strong correlations among d electrons. For lower y, the DOS at EF increases and the peak at 3 eV, which appears in the band-structure calculations at 2.5 eV above E,, disappears. It is interesting to note again that the states at E, are not seen by those spectroscopies in which the Cu 3d cross-section is high (Fujimori et al., 1987; Bianconi et al., 1987). As with La,-,Sr,CuO,, we conclude that the states near EF and also the holes have a predominantly 0 2p character. The recent studies by electron spectroscopy have revealed many details on the very complicated electronic structure of cuperoxide-based superconductors. While from UPS, XPS, and Auger electron spectroscopy important information on the parameters U and A (the charge transfer energy for Culigand hopping) could be derived, the 0 1s core-level excitation by EELS directly shows the breakdown of single-particle band-structure calculations in the insulating materials and possibly also in the superconducting systems. It should also be noted that this spectroscopy is at present the only method capable of detecting pronounced changes of the electronic states close to EF upon variation of x and y. The DOS of the holes on 0 sites, which are created upon doping or upon changing the stoichiometry, could be measured directly by this local probe. Finally, it should be emphasized that, contrary to almost all the other electron spectroscopies, EELS is not a surface-sensitive method and thus avoids all the ambiguities entailed in surface preparation, surface contamination, and oxygen diffusion.

-

226

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ACKNOWLEDGEMENTS The author gratefully acknowledges the initiation of the construction of the electron energyloss spectrometer by W. Schmatz and M. Campagna. My colleagues G. Crecelius, W. Czerwinski, H. Fark, A. vom Felde, R. Hott, P. Johnen, A. Litzelmann, R. Manzke, Th. Miiller-Heinzerling, N. Niicker, J. Pfluger, B. Scheerer, and J. Sprosser have contributed significantly to the work presented in this review. The author is grateful to D. Baeriswyl, R. von Baltz, H.-J. Freund, J. Fuggle, J. Heinze, D. Kaletta, P. Koidl, H. Kuzmany, W. Jager, G. Leising, H. Lindenberger, N. Neugebauer, H. Rietschel, J. J. Ritsko, S. Roth, G. A. Sawatzky, M. Stamm, D. Schweitzer, K. Sturm, H. Trinkaus, W. Weber, W. Wernet, G. Wegner, J. Zaanen, and R. Zeller for fruitful collaboration and stimulating discussions.

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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 75

Methods of Calculating the Properties of Electron Lenses E. HAHN Jena German Democratic Republic

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

233

11. Equation of Motion . . . . . . 111. Differential Equation of Trajectories.

. . . . . . . . . . . . . . . . 235 . . . . . . . . . . . . . . . . 237 IV. Methods of Solution . . . . . . . . . . . . . . . . . . . . . . 238 V. Coupling between Field and Basis

V1. Representation of the Trajectory . VII. Iteration Cycle . . . . . . . VIII. Electron Mirrors . . . . . . IX. Conventional Lenses . . . . . X. Theory of Micro-Lenses . . . . XI. Quadrupole Optics. . . . . .

. . . . . . . . . . . . . . . . . 246

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

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. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . 250 . . 255 . . 258 . . 262 . . 269 . . 294

XII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

325 328

I. INTRODUCTION

The concept of electron optics first brings to mind the imaging properties of electrostatic magnetic fields, in which a ray bundle can be deflected and focused. Provided that wave optical properties play no role, the Lorentz equations of motion form a starting point for the calculation of trajectories as constituent parts of a bundle. If one considers the integration of trajectories, i.e., the solution of the corresponding differential equation, for which there are standard procedures (for example, the Runge-Kutta method), the optical point of view can be introduced only afterwards in terms of choice of initial values. The interaction between field, trajectory, and optics, which finds expression in the Eikonal theory, based on Fermat’s principle, is lost in the calculation of individual trajectories. We consider here the integration of the equation of motion as an optical problem: The optics is coupled with the field, and this coupling is realized in the form of an iteration process by which a linearization of the equation of 233 Copyright 0 1989 by Academic Press, Inc All rights of reproduction in any form reserved. ISBN 0-12-014675-4

234

E. HAHN

motion is attained. The solutions of this linearized equation of motion are represented as a linear combination of a fundamental system forming the “basis” that incorporates the optics. Three differential expressions, related to field and trajectory, which are associated with the round-lens, quadrupole-lens, and deflection-system components of the field, become field functions related by an (n - l)lh order approximation to a particular trajectory. Its optical equivalent arises through a local field-basis coupling in the form of step matrices (lens matrices) and a deflection vector. The four-vector of the trajectory is thus represented in an nth order approximation as a function of its initial values. The field-basiscoupling consists of a homogeneous first-order differential equation for the step matrices, whose solution is given explicitly, so that the introduction of “extraneous” integration procedures is not necessary. This concept not only renders a very precise trajectory calculation possible, but also has a number of advantages in the treatment of problems connected with optical conjugation arising from the cross-sectional properties of ray bundles, especially with quadrupole optical conjugation. In the equation of motion or, alternatively, the differential equation of the trajectories, the time t or the abscissa z along the optic axis is usually chosen as the independent variable. The solution, accordingly, appears as a sequence of points (motion) or as a line of points (trajectory). Regarded mathematically, threre is no essential difference between the two representations, since motion can also be regarded as a representation of the trajectory in parametric form. We use here as independent variable, the basis variable w, which is defined in terms of the fundamental system associated with the round-lens component of the field. From an optical point of view, motion can be regarded as a process of successive mappings. To each process step (integration step) corresponds an operator (step matrix) that produces the mapping. The mapping can be spacelikeor timelike. The objects of the mapping are firstly the straight lines of a ray space, which touch a trajectory in the vertical planes of a “partial-lens” corresponding to the integration step, and, secondly, certain points or lines that are defined as intersection points or lines of a ray bundle and that characterize a state of the ray space. The spacelike mapping is a collineation determining which straight line of the object-side ray space of the partial lens is identical with which straight line of its image-side ray space, i.e., belongs to the same trajectory as a tangent. The optical action of the process step can be understood by comparing identical objects in two different ray spaces. The two ray spaces exist simultaneously and are spatially separated by the integration step along the taxis or the z axis. The timelike mapping is produced by a “micro-lens.’’ The “micro-lens” only appears in the form of its optical action (lens matrix) and may be

CALCULATING THE PROPERTIES OF ELECTRON LENSES

235

compared to an “event” separating two states. The ray space is present always only in one state. The identity of its rays cannot be determined, since a comparison analogous to the “partial-lens” concept is not possible. In the “micro-lens’’concept, the intersection configurations of rays are determined, especially vertices of incident and exiting rays, whose identity is secured by means of optical conjugation. The basis variable has a double function: as a substitute for t or z, it serves to describe the progress of the ray, and as the argument of periodic solution functions, it serves to describe a state and the conjugation. In this double function resides an important advantage of the theory set out below. OF MOTION 11. EQUATION

The relevant equation of motion (Lorentz equation) for the geometrical optics of an electron with negative charge (-e), rest mass m,, and velocity v in a time-invariant, electromagnetic field (cp, A ) may be written

d dt

-4

m,v

= egradcp

+ ecurlA x v.

Here cp denotes the electrostatic potential with -grad cp = E as the electro-static field strength. A is the magnetic vector potential with curl A = B as the magnetic induction and cis the velocity of light. It is useful to introduce the proper time r of the moving electrons instead of the time t

dt d? =

-

1

= 1 + - ecp m,,c2’

(2)

The electric potential cp is normalized in such a way that erp is equal to the kinetic energy. With v = ds/dt, the equation of motion (1) takes the form

d2s ds mo2 = egrad@+ ecurlA x -, dz dr which differs formally from the nonrelativistic equation (c -P should be replaced by T and cp by

@=(I+&)’.

(3) 00)

only in that t

(4)

The coordinates x, y, z form a rectangular (right-handed) Cartesian coordinate system with z as the abscissa of the optic axis. We write the coordinates x, y in complex form r=x+iy

r*=x-iy.

(5)

236

E. HAHN

In the vortex-free region of the vector field B = curlA the magnetic field intensity (induction) can be represented as the gradient of a scalar potential B = curl A = -grad II/.

(6)

If A,, A,, A , are the components of A, the system of equations (four real equations)

a aZ

-($

-

a

iA,) = 2i-(A, ar

+ iA,) f7)

is equivalent to ( 6 )together with the equation div A = 0 (Lorentz convention for cp independent of time), and the Laplace equation, which has the following appearance for rl/,

+

(4&

$)*

= 0%

holds for all four functions ( $ , A , , A,, and A,) in which the freedom from sources (divB = 0) and from vortices (curlB = 0) of B = curlA has been incorporated. By the use of (7) one obtains the following equations of motion for the coordinates (r, z ) of the moving electrons: + j

e d2z dt2 -m,

-+

j

*

(a$

az {a$

dt

dr dr d t

(9)

2 az d t

a+ d r * ) } ar* d t

‘

Scalar multiplication of ( 3 ) with dsldt and subsequent integration with respect to z yields the energy relationship

Between the arc length ds and its projection dz on the z axis, there exists the relation (dashes mean differentiation with respect to z ) (ds)’ = (1

+ r’r*’)(dz)2,

(12)

and one obtains for the derivative of the z coordinate of the moving electron with respect to the proper time t

CALCULATING THE PROPERTIES OF ELECTRON LENSES

237

Repeated differentiation and comparison with (10) yields the relation

111. DIFFERENTIAL EQUATION OF TRAJECTORIES

If we regard the equations of motion (9), (10) as equations determining the trajectories r(z), z(r) in parametric form, a constant in the modified electrostatic potential @ has no influence, since only its gradient is effective. In order to obtain a trajectory it is necessary to set the initial values. In the case of the differential equation systems of (9) and (lo), these are the initial coordinates and velocity. In the trajectory bundle, emerging from a point, the initial velocities may be different. Fixing the constant in the potential @ via relation (1 1) would have the effect that the electrostatic potential cp would depend on a particular trajectory. On the other hand, in the electrostatic potential cp(r, r*, z), the solution of the Laplace equation (8), with I) replaced by cp, is fixed by its boundary values at the given geometrical electrode arrangement. In these electrode potentials, the potential of the cathode is implicit; it is set at cp = 0. The various emission velocities of the electrons emitted by the cathode are taken into consideration by adding a quantity q (chromatic parameter) to the potential cp. For the modified potential @, we must then write

instead of (4). If we bear in mind that in the equations of motion, (9),(lo), and (13), the electric potential arises only in its modified form (15), we can subsequently neglect the relativistic correction term 1 + e(cp + q)/2m,c2 and of further simplification of the simply replace cp by (cp + 7). adopt the convention that I) notation, we replace the denotes (e/8m)'/2 +. We write for short

and have in the normal case of the ray propagation in the direction of the positive z axis according to (1 3) the following relation between z and t dz

-= dt

Jq.

238

E. HAHN

For a trajectory in an electron mirror, the sign of the ray propagation in the z direction reverses at the point of reflection, determined by the zero value of (c + (p). In this case, the square root in (17) is ambiguous. Using the relation (17) we can eliminate the variable t on the right-hand side of the equations of motion (9), (10) and obtain, in terms of the agreed simplifications,

8q

d2z

Likewise, we can eliminate the variable t on the left side of (18) and obtain the differential equation for the trajectory r(z). 1 D(r) = (c + @)r"+ -(c 2

84) + @'r' - __ + i44dr*

Multiplication of (20) with r*' and addition of the complex conjugate leads to the relation

which is equivalent to relation (14), taking into account the agreed simplifications. If r(z) is a solution of the differential equation (20) with the initial values r(z=), r'(z,) in the object plane z, for a given value of the chromatic parameter q, and if one inserts this in expression (16), then integration of (17) vields

The function z(t) arising from the solution of this equation, together with r(t) = r(z(t)) then satisfies the differential equations (18), (19) of the equations of motion. For the lower intergration limit zo in (22) we adopt the following conventions: In the normal case (conventional lenses) the quantity (c + 4) has no zeros in the lens space, and zo is the abscissa of some point on the axis representing the center of the lens; the choice of zo is independent of the trajectory.

Iv. METHODSOF SOLUTION The differential equation (20) is a complex composite of two coupled, ordinary second-order differential equations and hence can be integrated stepwise, for given initial conditions, by means of well-known numerical

CALCULATING THE PROPERTIES OF ELECTRON LENSES

239

methods, e.g., the Runge-Kutta-method (Zurmiihl, 1963). Since this procedure is universally applicable, it cannot reflect the specific structure of the differential equation (20).Such a specific characteristic results, inter aka, from the existence of a paraxial region, the possibility of expanding the field as a power series in r and r*, and the classification of these fields as multipoles of various orders. The procedure developed here is an iterative process. This enables the differential equation (20) to be linearized in such a way that, stepwise, differential equation systems of the paraxial type are formed. The corresponding coefficient functions arise out of the r, r’ and z-dependent “field expressions” governed by the differential equation structure and field distribution; these expressions become “field functions” (functions of an axial variable) by the insertion of the trajectories obtained in a cyclic operation (as an approximation for the prescribed initial conditions). These “field functions” change in successive iteration cycles until these trajectories become the solution of the differential equation (20). The solution of the differential equations takes place by the construction of fundamental matrices, which are products of step matrices. The coupling between the elements of the step matrix and the corresponding field functions is derived by a specially developed formula system of order h6, where h is the step length in the z raster (Hahn, 1985). A . Fundamental Matrix

Consider a differential equation of the paraxial type d2r dt2

-

+ f ( t ) r = 0,

in which the field function f ( t )for a given field expression f ( r ,r’, z) is obtained by using the functions r(t), r’(t),z ( t )from the above-mentioned cycle instead of the free variables r, r’, z. Each solution of (23) is a linear combination of two linearly independent solutions rl(t), r2(t);these form a fundamental system. The matrix

is referred to as a fundamental matrix. The initial values of the fundamental system at t = 0 are defined by the relation S ( t = 0) = 1 (unit matrix). The Wronski determinant of the fundamental system is the determinant of S and det S ( t ) = 1 irrespective of t .

240

E. HAHN

The differential equation for the fundamental matrix

is on the basis of definition (24) equivalent to (23) and has the advantage of being linear and first order. If we replace the field function f(t) by f ( t ) + g(t), the fundamental matrix then goes over into the product S P, where P(t) is determined by the differential equation,

-

-+g(t)s-l(; dP dt

i)S.P=O

and P(t = 0) = 1. We put p(t) = S 2

+ T2,

S R(t) = arctan T

and call p the amplitude and R the phase of the basis defined by the fundamental system S(t), T ( t ) or simply the amplitude and phase of the fundamental matrix S(t). It can be seen that

and we can use R as a new axis variable (basis variable). We express the differential equation (27) in terms of R. In terms of R, S has the product form (basis form)

The dot notation indicates differentiation with respect to R, and C(R) denotes the rotation matrix

C(R) = We obtain

dP dR

cosR sinR

-sinR cosR

CALCULATING THE PROPERTIES OF ELECTRON LENSES

24 1

B. Quadrupole Matrix When the additional field function is complex, that is, it has the form g + ih with g and h real, we have a system of differential equations of the paraxial type to consider: d2x dt2 + ff + g 1 . x + h y = 0,

-

d2Y + (f - g) ' y dt2

+h

(33)

x = 0.

The fundamental matrix X = SQ

(34)

now has four rows. Its four column vectors

j = 1,..,4,

(35)

are formed analogously to (24). In (34) the 2 x 2 matrix S(t), defined with S(t = 0) = 1 according to (26), is to be replaced by a four-row matrix according to the relation (38). The quadrupole matrix Q(t), with Q(t = 0) = 1, is determined by the differential equation

/o

0

1

o\

In this the complex-valued field function, (g + ih) is a four-row matrix. First g + ih is written as a two-row matrix according to the rule g+ih=

g

+ ih

(37)

and then into a four-row matrix with real elements according to the rule

+

ac + iu, iy,

+

db + ip ia)=[

1; p 1;). c

d

d

(38)

242

E. HAHN

The product of the two quantities is independent of the form of presentation, the multiplication being carried out according to the usual rules. On introducing the transformation (28),we obtain the following equation, which is the analog of (32):

\o

0

0

o/

In the case h = 0 , Q has the “separated” form

\

913

and we have

as the solution matrix of the differential equation (32) and

as the solution matrix of the differential equation (32) with - g instead of g. The following theorems hold: 1) The determinant of the solution matrix Q of (39) is constant, independent of Q, and has, in particular, the value 1 when (e.g., at Q = 0) Q = 1. 2) The inverse Q-’(n) is obtained directly from Q(Q) (with det Q = 1) by a simple rearrangement of its elements. If A, 6, C,D are two-row matrices and A’, B’, C’, D’ are their transposes, then:

C. The Step Matrix

If Q, are the nodes of a raster, where n is an integer, then for the step matrix of Q we have

-

Q(Qt + 1)Q - ‘@L) = C - ‘(On+ i)Qm,,,

+ 1 C(QJ.

(44)

243

CALCULATING THE PROPERTIES OF ELECTRON LENSES

On,,+ is the “phase-rotated” step matrix of the interval (R,, R,+

\Y, + uy, - x y + uy, whose real elements are determined by the value of the function q = p2(g + ih) at the boundary points of the step interval and whose first and second integral are determined over the step interval. For the reciprocal phase-rotated step = Q,;+,, rule (43) holds matrix

Qn+

1.n

=

i

xa

+ ua,

y, - 08, - uy, - Y , + u,,

-Xy

Ya -xa

+ ua, + Uar

-xp - ug,

-Yp

-Yy - u y , Xy

Xa

- uy,

+

+

-Yg - up xp -

ua,

Ya - u a ,

Ya -Xa

For the matrix Q at R,, with n a positive integer, we have

Q(QJ= C-1(Q,)Q,,-1,,,Q,,-2,,,-

1

+ Va + ua

.

(46)

.60,1C(WQ(QJ

(47)

Q(W= C-1(R,)Q,+i,,Q,+2,a+i... Qo,-iC(%)Q(%).

(48)

1..

and for Q at R, with n a negative integer,

The formulae for the elements of the phase-rotated step matrix (45) were developed on the basis of the differential equation for the four-row quadrupole matrix Q(n) 10

0

1

o\

using the rotation matrix C(R) (31) for a prescribed complex-valued function q(R). Here C and q are to be understood as four-row matrices with real elements, according to the rules (37) and (38). For each step interval (a,,a,,+ we form the following quantities, which will be needed as “building blocks” in the formation of the matrix elements of Q , , n + l :

a+ia=

qdR,

b

+ ip = - ] z + ’ ( O

-

*‘ +2*“+‘)qdR

(50)

244

E. HAHN

-

The following formulae for the elements of the phase-rotated step matrix On,,+ (45) reproduce the exact solution n-tI

Q(R)Q-’(Q,) = 1

n-

+ ‘6 +

Dn6 + D 0%

+ ...

(53)

of the differential equation dQ -=DQ dR

(54)

with an accuracy of the order (fin+ - QJ6. The terms of the series (53) consist of an intricate series of integrations (the symbol n-means integration with respect to R with the lower integration and matrix multiplications, which require a solution. limit 0,) We note the results: 1 2

X, = - U S

- b - d-

s4

+ 120

1 S4 y , =-as - p - 6-2 120 1 2

+ y(c2 +

1 y, =-as

s4 + p + 6-120 + y(c2 + r’,,,

s3 xp = as+ a- - c - s 2

5

SZ

yp = as+ a-

5

60 3

-yS-S2 60

3

S2 x = a - - c - s s -2 5 60

s2

s3

5

60

y =cr--y-s

- (d2

5760

s4 S6 + b + d- 120 + c(c2 + y2)- 5760

s2

= S + (UC

S6 72)-

~6 = - U S

2

U,

+ 7’)- 5760 S6

C(C’

-2

s3 + ~ 7 24 ) - - (c’ +

+ 8’)-

s4

1440

s3

U, = (U? - w)-

120

s4 + ( ~ -6 7 d ) -120

S6

s5

+

c(c2

+

+

y(c2

s5 + y2)- 960

y2)-

960

(55)

CALCULATING THE PROPERTIES OF ELECTRON LENSES

245

“4

= u, - (cd

+ y6)”120 s3

vg = v,

- (ay - ac)-

60

up = -s

+ (ac + cry)-?203

- (c2

+

”5

up = -(up - crb) - (c6 - yd) 2-

180

uy = s

U?

=(

+ (CZ +

s5 72)-

480

s5

~6 yd)-.

720

D. Local Basis The fundamental matrix X in (34) is the product of successive step matrices X(tn+l)X-’(cn) = S(tn+l)C-’(Qn+ l)Qn,n+lC(Qn)S-’(tn).

(57) Each step matrix (n = 0, k 1,. . .) is calculated independently of the others. This constitutes an essential difference between this method and the usual integration methods with continuous calculation of the solution curve. The relation of the step matrix with the field functions (coefficients of the differential equation) is valid for a discrete interval of the t axis and z axis, respectively. We may say, therefore, that the step matrix (57) is a “discrete solution” of the differential equation (33). It is invariant under a basistransformation of the form S(t) = S‘”’(t)A,,

(58)

in which A , is a two-row matrix with constant elements: X(t,+

l)x-yl,)= s ~ ” ~ ( t . + , ) c - ’ ( ~ $:!,c(Q$))s~)-’(tn). ~ : ! , ) Q ~ ! , + (59)

@)and p(”)arethe phase and amplitude of S(”).The related building blocks of the transformed basis, (50) to (52), with q ( n ) = - p*( g + ih)

(60)

yield, when inserted into (45), the phase-rotated step matrix &,+ 1. The invariance of the step matrix (57) under a basis transformation (58)tells us that

246

E. HAHN

for each step t,, t,+ the basis S‘”)[as a solution matrix of (26)] may be chosen freely. If we take A , = S(t,), we have been careful to ensure that at t = t,, the amplitude p(”)has the value unity and the phase a(”) has the value zero.

V. COUPLING BETWEEN FIELD AND BASIS With a suitable substitution of the spatial variable (r,z) the differential equation (20) can be separated into simply constructed differential equations, in terms of the substituted variables. In the substitutions

p , y, R, and 0 are real, w is complex.

A. Field Representation The field expression x(r, r*, z ) differs from the magnetic scalar potential $ only by a rotationally symmetrical round-lens component of the magnetic field: $- =

(-- 1)”

22”(p - l)!(p + l)!

Yb2”l(z)r”r*”.

Y,(z) is the axial function describing the magnetic round-lens potential, and Y hzN1is its 2pIh derivative with respect to z. Of the components of the vector potential in cylindrical coordinates (r, a, z ) that, according to (6), describe the round-lens component, only the a Component A , = A(r, z ) is nonzero. For the z component - a$o/az of the magnetic induction B = curl A of the round lens we have

a

1 -__ -- -(r,

aZ

r ar

A).

For the derivative of (62) with respect to z it follows that

-a$- _ ax = -rz(h)

aZ aZ

ar

r ’

(44)

CALCULATING THE PROPERTIES OF ELECTRON LENSES

247

and for the sum of (63) and (64) one finds

$x(r,r*, z ) is the magnetic scalar potential minus the magnetic round lens potential $o(r, r*, z). A(r, z ) has the series expansion

The electric potential q ( r , r*, z ) is composed of the round-lens component qo(r,r*, z), the deflection potential qx,l(r,r*, z ) and the quadrupole potential q x , 2 ( r , r * , z )all ; higher multipole potentials are included in qx = qx,l+ qx,z. Similar relations hold for the magnetic potential $(r, r*, z). In the series expansion

Vi2”](z) denotes the 2pthderivative of the axis function of the k-fold magnetic multipole field. The vertical bar notation(. .. .) is introduced as a short-

I..

hand for a frequently occurring operation on two arbitrary complex quantities, A = a + ib and X = x + iy: We write ( A X )= + ( A X * + A * X ) = ax by. The electric potential q ( r , r * , z ) has an analogous representation in terms of the axial function Ok(z)of the k-fold electrostatic multipole fields.

I

+

B. Field Expression The pair rl(t) = h s i n n ,

r2(t) = & c o s ~

forms a fundamental system of the differential equation

The expressions sin R R,(z) = -, Y

cos R R ~ ( z=) Y

likewise form a fundamental system of the differential equations

d2R 7 dz + y 3 ( j + y ) R

d ’R 7+ y4Fy * R = 0. dz

(68)

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E. HAHN

The differential equation (69) is of the paraxial type (23), if the differential expression F,(p) is understood to be the result of a basis-field coupling giving rise to a field expression FJr, r', z) and the trajectory in parametric form r(t),z(t) is inserted so that Fp appears as a function Fp(t) of the axial variable t . The differential equation (71) is of the paraxial type if the differential expression y4Fy(y) is specified as a field expression y4F(r, r', z) and the trajectory r(z) inserted so that y4Fyappears as a function of the axial variable z. In view of the substitution (61), the expression F J p ) in the differential equation is related to y4Fy in the following way:

(c + 4 c+cp @)I'

=-

($GT$y

YET$

(72) *

This relationship defines either y4Fy or Fp as a field expression, when either Fp or y4Fy is specified as a field expression. For this, we insert in the differential equation (20) for r the substitution (61) and obtain

L) = 0,

(73)

together with (the suffix 1 indicates that the separation is not yet definitive),

(73 ax

The separation of the field expression in the substituted differential equation (73) into H, K,and L is unambiguous only in the paraxial region, since H must be real and both K and L must be continuous for r = 0. In the higher order region, the ordering is not unambiguous, as one can see from the identities

which serve to transfer components of higher order from H I to K , or L,. It is

CALCULATING THE PROPERTIES OF ELECTRON LENSES

249

even possible to change components of higher order from K , to L, and vice versa, as can be seen from the following identities.

The above separation (74) to (76) is unsatisfactory for the following reasons: a) Magnetic deflectors and quadrupole fields can have an effect on H, through the a x / a z term, unlike electric multipole fields, for which this is not possible through the term dpo,/rdr*. In K, and L, total differentiation with respect to z is called for. For abbreviation we put

and form the identity

When this identity is added t o the differential expression (73), the field expressions H, K, and L take the following (definitive) forms

The differential expression D(r) in (73) goes to zero if we require in the first place that Fp,regarded as a field expression Fp(r, r’, z), be specified through the

250

E. HAHN

basis-field coupling

F,=H and secondly that w(Q) satisfies the differential equation W

+ w - Kw* - L = 0.

(85)

The first requirement means that in the differential equation (26) for the fundamental matrix, the field function f(t) may be written f ( t ) = H(r(0, r w , do).

(86)

The second requirement is fulfilled by the expression

in which Q(Q) satisfies Eq. (39) with

VI. REPRESENTATION OF THE TRAJECTORY A. Four-Vector We wish to transform the representation (87) into a representation for the four-vector of the trajectory

p and Rare the amplitude and phase of the fundamental matrix S(t),which is related to the field function f = H through the differential equation (26). S(t) possesses the basis representation (30). On the basis of the substitutions (61) one has

Furthermore, the following identity holds for an arbitrary complex quantity v(t) independent of the field functions H(t) and K(t), to which S and Q are

CALCULATING THE PROPERTIES OF ELECTRON LENSES

251

related:

If we insert (87) into (90) and add (92), we obtain

If we now take for v the quantities defined in (79), we obtain after making the substitution (61),

Finally we apply (65) and (83) and obtain the following expression for the four-vector (89) of the trajectory as a function of its given initial values in the plane-z,

252

E. HAHN

In this representation the four-vector that is relevant for beam deflection has the meaning

If we denote the electric and magnetic field strengths of the deflector fields qx,l and t,bx,l by E(D)and B(D),respectively,

E(D)= -grad cpx, B(D)= -grad $x, (97) and the z components of the B field that remain after removal of the round lens field by

then (96) takes the form

The interpretation as beam deflection arises from the fact that we have freed the quadrupole expression in the original structure (75) from the term

By this means the field expression H determining the fundamental matrix S is freed from components of deflector and quadrupole fields and the field expression K (82) determining the quadrupole matrix Q is reduced to that of

the effective quadrupole field expression in the paraxial region. In the representation (95), depending on a particular trajectory, the variable t is a substitute for z and vice versa, and the relationship established in (22) holds. The calculation of the fundamental matrix S and the quadrupole matrix Q is performed in that t or z raster in which the corresponding field expression is given. In the usual case, the appropriate raster is an equally spaced z raster. In the case of an electron mirror, it is necessary to change over to a t raster in the neighborhood of the reflection plane (mirror zone), since the reversal point zo depends on the individual trajectory. The transformation of the field expression from a given z raster to the corresponding expression for a t raster is certainly possible if the reflecting field and the modulated field of the mirror surface to be imaged can be described analytically.

CALCULATING THE PROPERTIES OF ELECTRON LENSES

253

B. Initial Matrix

The field function (86) H(t) is related through (26) to the fundamental matrix S(t). If we replace the field function H(t) by 0 + H(t), the initial matrix

corresponds to the identically vanishing field function, from which one originally began. The initial matrix with S(t = 0) = 1 is also defined through the differential equation

""+(-; dt

gso=o.

The amplitude and phase of the initial matrix Soare denoted by p and w t = tano.

Now S is the product of So and P:

(103)

S ( t ) = S,(t)P(t).

P(t), with P(t = 0) = 1, is coupled with the field function (86) H through the differential equation

The discrete solution of this differential equation is the step matrix formed in an analogous way to (44) I

P(wn+I)P-l(wn) = c-'(wn+ 1)Pn,n+lC(wn)

(105)

in which P,,,, is the phase-rotated step matrix. Since q = - H p2 is real, this can be reduced to the two-row matrix

The formulae ( 5 5 ) and (56)serve to calculate its elements, whose building blocks can be formed according to formulae (50)to (52)and transformed to the z raster (on the assumption that in the integration interval of the expression

254

E. HAHN

(16) c + @ is appreciably different from zero) as follows: Zn+1

H

The quantity So(z) denotes S,(t(z)) and likewise for To. The amplitude p = 1 + t 2 and the phase o = arctan t of the fundamental matrix (90) S,(t) are related to the abscissa z through Eq. (22) for t(z). The discrete solution of the differential equation

is the step matrix S(t,+ l)S-l(t,,),and this comes in a formal way from (57), if we regard S there as the initial matrix Soand Q as the phase o S(tn+1)S-'(tn)

=

SO(tn+l)C-'(0n+l)~n,n+l

c(on)so(tn)*

(109)

It is invariant under a basis transformation of the initial matrix So(t)of type (58), and we can assign to each step interval ( t n , t n + l ) , or ( Z ~ , Z , + ~ ) , a particular (local) initial matrix Sg)(t),for example, with the normalization SF'(t,)= 1 Sgyt) = S,(t)S,'(t,) =

(t

: t,

a);

The phase u'") and amplitude p(") of Sg' tano'"'

=

t - t,,

p(n)

=1

+ ( t - t,)Z

=

l/cos2(w("))

111)

should be used for forming the building blocks of the elements of the phaserotated step matrix P,,,,, instead of o and p in (107). The elements of the phase-rotated step matrix Q,,,, of Q in (95) rely on the fundamental matrix S (or alternatively S'"))as a basis. With p and R as amplitude and phase, the formulae for the building blocks (50)to (52) in the z raster (q = K/p2 = K)run

CALCULATING THE PROPERTIES OF ELECTRON LENSES

255

as follows:

d

+ ih = (p2K),,+, - (P'K),,

In the usual case, i.e., for an ordinary electrostatic/magnetic lens, the field expression (16) c + (P = (p(r,r*,z) + q - (JZ$rr)(J-rr>*

(1 13)

is appreciably different from zero in the optical channel. The z raster, in which the explicitly z-dependent functions in the field expressions H, K, and L are given, need not be abandoned when performing iterative trajectory calculation. The integration over o or R needed for the calculation of the elements of the phase-rotated step matrix (45) according to (50) can be carried out immediately by integration in a prearranged, preferably equidistant, z raster. Amplitude and phase can be extracted from the fundamental system of the corresponding fundamental matrix according to (28), and the differential d o or dR can be eliminated, based on (62), by means of dz.

CYCLE VII. ITERATION Given the initial values (trajectory parameters) r(za), r'(za), q, the fundamental matrix S, the quadrupole matrix Q and the four-vector (89) of the trajectory are calculated in each iteration cycle, according to the given formulae, at the nodes of the prescribed z raster. By insertion of the four-vector into the field expressions, these are turned into field functions. In the subsequent iteration cycle these lead to new values of S, 9, and the four-vector (89) at the nodes of the z raster. The series of iteration cycles is either broken off when a prescribed number of cycles has been worked through or proceeded with further if the results of successive cycles are not in sufficiently good agreement, or if the field expressions need to be supplemented by the inclusion of higher order terms.

256

E. HAHN

A. First Cycle

In the first cycle, the four-vector (89) is set identically equal to zero, whereupon the field expression is reduced to its axial distribution.

H

1 4

= -@z(z)

+ Il/z(z),

+

= K = e'2e{@2(z) i 4 a Y 2 ( z ) ] , (114)

m = @ , ( z )+ i 4 J o o ~ , ( z ) , n = 0. The initial matrix Soof each cycle always has the form of (100) or (1 10). The fundamental matrix Soof the first cycle is S(1)= s0 pel). (115)

PI')is determined via the field function ( H ( r = 0, r' = 0, z ) and S ( ' )contains the optical properties of the round-lens field in the paraxial approximation. The field function K(r = 0, r' = 0, z ) determines the quadrupole matrix Q('), in which p") and 51") are the amplitude and phase of S ( l ) . After carrying out the remaining integration over the deflector part and the matrix multiplication according to (95), we obtain the four-vector ( a r t ,r)(') as a result of the first cycle. Since a change of trajectory parameter (apart from = 0) has hardly any influence on the field function, no new integrations are necessary for other trajectories in this paraxial region. Between the (real) components of the four-vector of the trajectory in the plane z and the plane z,, there exists a linear transformation that depends solely on z and z,. B. Second Cycle and Aberrations of the Third Order

In the course of the second cycle, the coefficients of the transformation are found to be functions of the trajectory parameters, since the field function of a field expression now depends on the trajectory. Deviations (aberrations) arise with respect to the path of the corresponding trajectory in the first cycle, and these can be interpreted as image aberrations, when z and z, are optical conjugates. It is chiefly the third-order aberrations that are involved, since in the second cycle a field function is formed by the insertion of a trajectory of the paraxial region in the field expression. The polynomial of the aberrations of third order is not restricted to the geometrical parameters, beam position r(z,), and beam direction (r'(z,) in the object plane z,, but also includes a variation of the parameter q of the initial energy (chromatic parameter), as well as the ray position in the image plane due to deflection.

CALCULATING THE PROPERTIES OF ELECTRON LENSES

257

One may recall that in electron-beam lithography, the beam deflection is program controlled according to the illumination pattern to be reproduced. The number of (real) third-order aberration coefficients is considerable, of the order of magnitude of hundreds. The formal expression of the complete aberration coefficients analogous to the 10 or so third-order image aberration coefficients of conventional round-lens optics is not necessary in our procedure. The aberration polynomial of third order, for example, of a deflection objective in which only the beam direction (beam position in the pupil plane) and the beam deflection (ray position in the target plane) are varied contains a total of 35 complex polynomial coefficients. These can be calculated since 35 trajectories with different ray parameters can be computed in perhaps two cycles. The aberrations of the trajectories are related to the polynomial coefficients by a linear system of equations. The formal solution of the system of equations poses no difficulties, given a suitable choice of ray parameters. The (complex-valued) elements of the reciprocal matrix of the system of equations are either zero or roots of the equation X 8 = 1. The values of the polynomial coefficients obtained in this way do not agree exactly with the values of the formal image aberration coefficients of third order, since in our procedure the solution of the differential equation of the trajectory itself is not made from the standpoint of an approximate procedure in the sense of a development of the differential equation expression in powers of r and r’. This less-than-exact agreement has no influence on the final result, since the aberration coefficients of third order form only an intermediate stage on the way to a higher accuracy of trajectory integration.

C. Aberrations of Higher Order In the results of the third iteration cycle we obtain conclusive information as to what extent the limitation to third-order aberrations is permissible. An extension of the aberration polynomials to the relevant region of fifth order and higher by taking into account the appropriate terms in the field expressions is in principle possible. The description of the properties of an optical system by a multiplicity of aberration coefficients is, however, unsuitable for the evaluation of optical quality, which can be much better demonstrated by a reconstruction of the intensity profile. The “objects” of electron-beam lithography are deflection fields and apertures; its images are intensity patterns. Intensity profiles can be calculated by starting from a particular surface element in the target plane, calculating the representative trajectories backwards to the source, applying there the weighting of the directional brightness, and distinguishing the passage

258

E. HAHN

through aperture planes between transmission or blanking. This basic approach thus constitutes an accurate trajectory calculation. With the iteration procedure here described, the question of accuracy is reduced to the question of the required number of cycles and to the question of how accurately the electric and magnetic fields can be described or included in the array plane of the z raster. VIII. ELECTRON MIRRORS

+

It is a characteristic of the electron mirror that the expression c (p (113) has a (simple) zero position zo for each trajectory. This arises since the denominator in (16) increases indefinitely. The vanishing of the numerator for finite values of the denominator is not excluded and indicates that the trajectory crosses the potential surface (mirror plane) (cp q = 0) at normal incidence and is turned back on itself. In the application of the method previously described for trajectory calculation, two difficulties arise. The first is that in (22) the integrand for the matrix element t(z) of (100) So,is singular at z o , and secondly zo is not fixed but depends on the particular trajectory. These difficulties may be overcome if in the vicinity of the mirror field region (mirror zone), a t raster is employed instead of the z raster. Thereby, in the formulae (107) and (112), which serve for the calculation of the elements of the step matrix of P or 9, the integrand is finite in zo because of Eq. (17).

+

A. Determination of the Zero Position zo In the t raster, the zero position has the fixed value t = 0, and zo must be recalculated for each cycle. When the four-vector Eq. (95)has been calculated, its components for t = 0, which we designate by uo, v o , x o , y o , are inserted into Eq. (113) and the equation c + i$= 0 is solved for z = zo. For this an iterative procedure (regula falsi) can be used:

The iteration can be included in the course of the iteration cycle ( j = 1,. ..), as is described in the following formula:

CALCULATING THE PROPERTIES OF ELECTRON LENSES

259

ut), I#), x f ) , y $ ) are the components of the four-vector for t = 0 as a result of the jthcycle. zko) is the zero position of @ ( z )+ q = 0. zbo)(t)is the inverse function of

so that the field functions formed in the first cycle appear as functions of t. If, in the field expression of Eq. (19) (right-hand side), we replace (J?@ r', r) by {u(')(t),~ ( ' ) ( t ) , x ( ' ) ( t ) , ~ (obtained ' ) ( t ) } in the first cycle, Eq. (19) goes over to an ordinary differential equation for z(t). Its solution z("(t), with the initial values z")(t = 0) = zb'), dz")/dt(t = 0) = 0, forms approximately a solution to the differential equation system (18), (19). Thereby, the initial value zbl) is determined by Eq. (1 17) with j = 1. B. Calculation of the Abscissa Function z ( t )

1. Iterative Determination

As a result of the

jIh

iteration cycle, the values of the abscissa function

zci)(t)can be calculated stepwise and iteratively in the points t , of the t raster

by means of the formula

If we consider the components (u(j),d j ) , x(j),y(j))of the four-vector as known functions of t, then Eq. (119) is an exact step representation of the solution z(j)(t)of the differential equation (19). It is, however, a matter of an implicit representation, since the value z(j)(tv+ to be calculated first with the help of Eq. (I 19) is already required on the right-hand side in the calculation of the field values q(r, r*, z ) , d q / a z , d q / d r , and acp/ar* in the field expressions (113) (c + @) and (21) (c + @)'. The integral equation (119) can be solved iteratively if on the right-hand side the value z ( j - I ) ( t , + calculated on the previous cycle is used in place of z ( j ) ( t , +'). 2. Numerical Integration of the Diyerential Equation for z(t) One might possibly consider determining the abscissa function z(j)(t)by solving the appropriate differential equation. As a result of the j I h iteration

260

E. HAHN

the four-vector ,IT%/,( r) is given by (95) at the points of the t raster. Inserted in Eq. (19), this yields an ordinary differential equation of the second order for z(j)(t), which can be integrated numerically after setting the initial values z(j)(t = 0) = zbj), d z ( j ) / d t ( t= 0) = 0, by, for example, a Runge-Kutta procedure. Here use is made of the assumption-which is usually necessary in an iterative solution procedure of equation (ll9)-that the electric and magnetic fields together with their derivatives in the mirror zone are given analytically in r and z. For in each iteration cycle the mapping of the fixed t raster is different, and the mapping points z ( j ) ( t )of the t raster do not, in general, match those of a given z raster. Outside the mirror zone, in order to facilitate the transition from the t raster to the z raster, a transition zone is called for in which the t raster is flexible and accommodates itself with respect to the given (equidistant) z raster, in whose nodal points the values of the axial functions of the fields are defined. In the application of the Runge-Kutta procedure, this requirement may have the consequence that the step length t , + - t , has to be adjusted to a prescribed increase z,+ - z , of the solution; numerical integration methods for the solution of differential equations are not in general adapted to this task.

3. The Transition Zone On the basis of the given connections (119) between the conjugated step distances of the t and z raster, the solution of the equation

is easier in terms of t!,ji - tl;" than the solution in terms of the step distance in the z raster, since the coefficient of the linear term has in each case a prescribed fixed value. The contribution of the integral is small in third order and yields, by integration by Simpson's rule,

Since Eq. (120) takes the form of a quadratic equation in t;il - tl;" with coefficients whose values in the equidistant nodal points in the z raster of the

CALCULATING THE PROPERTIES OF ELECTRON LENSES

26 1

transition zone can be calculated by the use of the four-vector (95) of the trajectory resulting from the j I h iteration cycle,

In general, the coefficient of the quadratic terms is small compared with that of the linear term (in a uniform field it is zero), and one solves the equation iteratively. With Eq. (122) an adjustment is made of the abscissa differences of the (equidistant) underlying z raster of the transition zone to the (nonequidistant) abscissa differences of the particular overlaid t ( j ) raster of the transition zone. In order to be able to assign values of the abscissae t?) itself, one needs a knowledge of an abscissa pair ( t ( j ) ,z) whose partners t ( j ) and z are correlated. In the mirror zone the equidistant t raster predominates. To its zero point t = 0 is correlated, according to Eq. (1 17), the abscissa z$) of the reversible point. The abscissae z(J)(t,) correlated with the points t , of the (equidistant) t raster of the mirror zone are calculated by means of Eq. (1 19). The (equidistant) t raster has such a length that at least one mapping point z(j)(t,) comes to lie in the transition zone. Its distance to the nearest neighbor point z, of the (equidistant) z raster is known. Then Eq. (122) yields the distance t , - t, and thereby the abscissa t , correlated with this raster point z,. The requirement concerning the overlapping of the two rasters is based on the following reasoning. For the calculation of the elements of the phaserotated step matrix (%), (56), quadrature (50)is needed over the relevant step interval. In the region of the (equidistant) z raster, the values of the integrand are available only in the nodal points. In order to carry out the quadrature with sufficient accuracy, nodal-position values outside the actual integration interval are also needed. When using symmetrically arranged quadrature formulae (with constant step lengths), for reasons of accuracy, at the start of the integration in the range in which the z raster dominates, nodal point values from the transition zone are also needed. The width of the zone is determined by the number of iterative cycles. The nodal position values in the transition zone are predetermined by the integration process in the t raster. This is carried out in equidistant t rasters and the result, i.e., the matrices, S and Q as well as the four-vector interpolated at the image points t(j)(z,) of the underlying z raster of the transition zone. This has the advantage that all quadratures (and also interpolations) can be carried out uniformly with symmetrically arranged formulae at constant step length.

262

E. HAHN

We understand by “field call” the evaluation of a field expression by the insertion of coordinates (r,r’,z), so we can say: In the region in which the z raster predominates “field calls” exist only in grating planes of the (equidistant) z raster. For this, field representations of type (67) are suitable. The axial functions are primarily given at the nodal positions of the z raster and must not be interpolated in z. In the region in which the t raster predominates, i.e., in the region of the mirror and transition zone, “field calls” take place in the grating planes of the (equidistant) t raster. Since the z(j) raster (as a mapping) that is correlated to the t raster (as original) is in general not equidistant, it is necessary in the field representations of the type (67) with given axial functions in equidisThis tant z points to interpolate from the nonequidistant mapping points di). requirement on the field representation in the mirror and transition zone means that the fields must be expressed analytically not only in r but also in z. For an electron-mirror objective, as has, for example, been proposed in Hahn (1979) for the program-controlled modulation of a multiray bundle, simple field models are available in closed form, which is suitable for answering questions of contrast production.

LENSES IX. CONVENTIONAL A . Fundamental Matrix R(z)

The expression (1 13) for c + is everywhere (i.e., in the channel under consideration) different from zero, and it is usual to eliminate the variable t from the equations of motion (18), (19). This is based on optical considerations, i.e., the interest in the properties of the beam complex is of primary importance and not the mechanical considerations, which are concerned with individual trajectories and the temporal relations of the motion. Tacitly, thereby, the usual space-time considerations are used as an auxiliary aid, in which the abscissa z of the optical axis is used as the space variable and t as the time variable. This approach is a hindrance to the decision, based on mathematical and optical principles, to give to the variable t or the variable z or the basis variable R precedence of independence from a problem-orientated point of view. For us, t and z are substitutes of a basis variable, whereby the substitution of a particular optical state can depend on local or global circumstances. This becomes clear from the basis-field coupling that we have set up in Eqs. (73) and (84), with t as an independent variable and have to carry it out analogously with z instead of t.

CALCULATING THE PROPERTIES OF ELECTRON LENSES

263

I. S(t)as Basis For S(t) as a fundamental matrix of the fundamental system (68) the differential equation becomes

Fp is an abbreviation for the differential equation expression formed with the amplitude p

and takes on, as a result of the basis-field coupling(84), the meaning of the field expression (81) I$

= H(r,r’,z).

(125)

This is transformed by the insertion of the inverse function (22) z(t) and the four-vector (95) calculated in the process of iteration cycles, into a field function F,(t). A discrete solution of the differential equation (123) is the step matrix S(t,+ JSW1(tn), whose elements can be calculated with the help of the procedure developed in the section, Methods of Solution. 2. R(z) as Basis

For R(z) as fundamental matrix of the fundamental system (70),

the differential equation takes the form

F, is an abbreviation for the differential expression formed with the eigenfunction y(Q)

L’ + 1

F,(Y) = Y

and takes on, as a result of the basis-field coupling, for the purpose of

264

E. HAHN

satisfying the differential equation (20) for the trajectories,

(I=)" H 1 D(r) - -y4Fy c+@ r c+cp

+-

y4 +(W + w - K w * - L) = 0, w

the meaning of a field expression

( 129)

In order to rearrange these in a field function dependent only on z, one has to insert the four-vector (95) calculated in the process of iteration cycles in the nodal points of the z raster. On the basis of the relation (60)existing between p and y, we can transform S(t)into R(z).

and eliminate S(t)in Eq. (95). A discrete solution of the differential equation (127) is the step matrix R ( Z " + ~ )R-'(zn), . whose elements can be calculated with the help of the method developed in Methods of Solution. In the field expression (130) a problem arises with the additional term, as opposed to the field expression (125), since on it a double total differentiation is required. Clearly, this tells against the use of the fundamental matrix R(z), if this is directly calculated analogously to the fundamental matrix S(t). This disadvantage is not compensated by the advantage that the substitution (22) can be omitted.

3. Initial Matrix Ro(z) The substitution (22) can be regarded as a component of a method for calculating a special initial matrix Ro(z) that is based on the transformation (13 1). Belonging to the initial matrix

r i together with S,(t) and (22) for t(z), is the fundamental system

CALCULATING THE PROPERTIES OF ELECTRON LENSES

265

which is a solution of the differential equation

With the special initial matrix (1 32), the disturbing term in the field expression is removed, and there remains for the matrix P R ( z ) Pg(z) = R i ' ( z ) R ( z ) ,

(135)

the differential equation

R, has the basis representation

1/Y is the amplitude and o is the phase of R, and the differential equation for P R ( m )has the form

Comparison with (104) shows that by considering the existing transformation (61) between p and Y ( Y instead of y, o instead of Q), P R ( z )and P ( t ) coincide, whereby the existing substitution between z and t is described by (22). Analogously to (109),there holds for the step matrix of R. R(zn+l)R-l(zn)= R ~ ( Z n + ~ ) C - ' ( o n + l ) e n , n + l C ( o " ) R ~ ' ( Z n ) .

q

The phase-rotated step matrix -H/(c + @)Y4 is real.

(139)

en,,+ is a two-row matrix as in (106) since

=

B. Paraxial Region In the paraxial region the additional term in the field expression (13 1) does not cause trouble but ensures that the second derivative of the electrical potential in the field expression (1 30) vanishes.

266

E. HAHN

The study of the paraxial optical properties of a particular lens defined by means of the axial functions Oo(z) and Yh(z) usually begins with the calculation of two independent linear solutions of the differential equation. R“

+ f(z)R

= 0.

(141)

This differential equation is of the paraxial type and can be solved by means of the procedures given in Methods of Solution. A discrete solution of the differential equation (141) is the step matrix R(zn+l)R-l(zn). For, by supplying its initial value R(zo) = 1in the axial point z , of the “lens centre,” the fundamental matrix R(z) is the sequential product of step matrices. A disturbing term need not be eliminated, and we take as initial matrix Ro in the product (135) a special solution of the differential equation

By supplying at the particular raster interval ( z n , zn+ values of the initial matrix

the relevant initial

there holds for the initial matrix R,, generating the corresponding local basis,

R, has the basis form (137) with tanw

=z -

Zn

+ Zn+ 1 2

’

Y(w)= cosw

(145)

For P,(o) the differential equation has the form

this can be solved by means of the step matrix of P,

-

P,(zn + ~ ) P‘(Zn) R = C - ‘(wn+ 1)Pn.n+ 1C(wn).

(147)

By means of the local basis defined by Eq. (143),the rotation matrix has the following form in the interval delineated by the limiting nodal points zn and

267

CALCULATING THE PROPERTIES OF ELECTRON LENSES Zn+1:

c - l ( w n + l )= c(mn) =

il + (

1

Zn+ 1

- Zn

-Zn+1 - Z n

1

[

Zn+1 - 2 ,

y y / 2

1

2

1.

(148) The building blocks (50) to (52), out of which the elements of the phaserotated step matrix Pn,n+ (it is, as in (106),a two-row matrix since q = -fly4 is real), are formed, according to formulae (55) and (56), are: a = -J;;+'{l

+

(. -

"+'

+

2

'")'}. f d z ,

h = S : : + ' { l + ( z - zn+ 12+ zn)2}arctan(

c

=

-{

1

+ (zn+1;

z.)')'

- zn+ 12+

zn

).fdz,

- ( f ( z n + l+) f(z,)},

C. Checking and Reconstruction We now consider the procedure for the solution of the paraxial differential equation (141) as a development process of the step matrix (139) and as a solution of the differential equation (127)for the fundamental matrix R(z).The field function y4Fymay not at first have the special form (140)but consists as in (1 30) of two summands F and G. We therefore write instead of (127) dz

0

and designate R,+,(z) as the optical equivalent of F + G. A slice of field enclosed within two vertical planes with abscissae znand zn+ may be regarded as a partial lens, whose optical action can be described in the sense of a tangent

268

E. HAHN

mapping by the step matrix RF+G(z,+l)Ri;G(zn). For this, we also call the step matrix a partial-lens matrix and introduce into the paraxial optical case F + G = f the notation

If G is excluded (G = 0), the fundamental matrix RF is the equivalent of F. If G is included, RF is an initial matrix in the transition from RF to RF+, = RFP,. The coupling between P, and G subsists in the differential equation (146) (with G instead of f ) , whose solution is represented by the step matrix. On the amplitude 1/Y2and phase o of the initial matrix RF is supported the "processing" of G into the building blocks (50) and (52) that enter into the elements of the phase-rotated step matrix k,,,+ according to formulae (55) and (56). (The actual basis field coupling is described by the differential equation (54), for which (53) is an exact solution). In order to set the process in motion, the provision of an initial matrix is necessary. In the case of the field function (130), the initial matrix was the optical equivalent of F'=

-(.ycsCp)"/.yzq.

In the case of the paraxial field function (140), f,the initial matrix is the optical equivalent of F = 0, and the process yields, with G = f,the fundamental matrix R,. This is now the optical equivalent off, which in the sense of the development process means that F goes over into f and G into 0. In order that the process may once more assume an initial position, we have once more to extract from the basis formulation (137) of the initial matrix RF the nodal position values of amplitude and phase. O n these is once more based the processing of a newly compounded field function G , which is represented by its nodal point values at the points of the z raster. If we take G = -f, there arises as a result of this (second) process step, the step matrix of the initial matrix (144) R,

from which we began in the first process step. In this way, the partial lens thicknesses z, + - zn are reproduced. Since the abscissae of the nodal points of the z raster are prescribed in advance and, in the second-process step, not used explicitly, their comparison with the result (152) enables a check to be made on the accuracy of the calculation. If we consider the partial-lens matrix (151)(n = 1,2,3, N ) as primary data, we can reconstruct, with the aid of the above second-process step, the abscissae of the splicing positions (vertical planes) in which two successive

CALCULATING THE PROPERTIES OF ELECTRON LENSES

269

partial-lenses touch. Here the supplying of the splicing position values f, = of the field function is again required. It is now worth noticing that we can also reconstruct, without prior knowledge of the splicing position values J,, i.e., with only a knowledge of a sequence of partial-lens matrices, the partial1. This is a result lens thicknesses and the partial-lens excitation n T n of the theory of micro-lenses developed below. This opens up the possibility of operating, in a corrective manner, on the step matrices calculated according to Methods of Solution so that these are self-consistent in relation to the z raster and to the field, and also form a consistent solution of the differential equation (141). f(z,)

+

x. THEORY OF MICRO-LENSES A . Macro-, Partial- and Micro-Lenses

The optical properties of a conventional lens (macro-lens)-its axial field function (140) may first of all fall to practically zero outside the interval (z,,z,)-derive from the fact that each trajectory is a linear addition of two linear independent solutions of the paraxial ray equation. Each object side incident line (object ray) is optically conjugate to an exiting straight line (image ray). This collineation rests on the premise of continuous and steadily progressing trajectories that smoothly bind the object and image rays over the lens interval (zl, z,). I . Partial Lens Image

If the interval (zl,zr) lies inside the field, the object and image rays are represented by the trajectory tangents in the vertical planes .z = z1and z = zrr respectively, of this partial lens. The cardinal elements determined by the partial lens matrix (15 1) R(z,)R-'(z,)

=

(

J!

zFB represent the abscissae of the object and image side focal points, fA, the focal lengths, and and QB the values of the electric potentials in the object and image side vertical planes z, and z,, respectively)

(zFA,

fB

270

E. HAHN

describe the collineation, caused by the partial lens, of the object rays and the image rays in the sense of a tangent mapping. If the lens interval ( z , , z r ) of a macro lens is covered by N gap-free and non overlapping partial lens intervals ( z ~ ,z~ ~, , (n ~ = ) 1,2,. ..N ) , the lens matrix of the macro-lens is the product of the partial-lens matrices

(1; -@)=( T

-qN

- 2sN

(

-41

irN). tN .. -2s1

31.1 -t1

).

(155)

The concept of partial-lens arises in connection with the splitting up of a macro-lens according to its fields. The lens matrices are thereby the results of the solution of the differential equation of the trajectories for prescribed axial functions of the field and division of the axial step intervals. The arrangement of the partial-lenses along the optical axis ( z axis) is thought of spatially and we speak of a “partial-lens image,” when we are looking at the situation chiefly from a spatial point of view. Consequently, the so-called first process step in IX.C, in which the partial-lens matrices are calculated, is strongly determined by the characteristics of partial-lens imagery. 2. Micro-Lens image If we regard the lens matrix as an expression of the optical action and think of this, as in the designated second-process step, as given beforehand, we can then interpret the product representation (155) as a splitting up of the macrolens according to its optical action. The component part in this kind of splitting we call a micro-lens. The micro-lens can be considered as an “event” whose action is characterized by the change from one state to another; the first state characterizes the situation before the micro-lens action, the other the situation after. By “state” we mean the structure of a ray complex in an overall field-free space (ray space), which comprises the image-side field-free space z 2 zr,nof the n”‘ partial-lens and the object-side field-free space z I z ~ , of~ +the~ (n + l)Ih partial-lens, where zrPn = z ~ , ~is+ the common “dotted” splicing position. We speak of the “micro-lens image,” when we mainly have it in mind as a time concept. In the micro-lens image, we are concerned with a time series (n = 1,2,. . . N ) of those ray spaces that touch the space of the partial-lens image in its “dotted” splicing position. In the partial-lens image, the “dotted” splicing positions are the nodal points of the z raster. In the role of a fixed-space reference system, the z raster exists as a whole, i.e., it is always at hand. The optical equivalent of the field function appears in the form of trajectories that are represented by their ordinates at the nodal points of the z raster. In the micro-lens image, from the whole of the z raster only one nodal position is present, i.e., of the nodal points of the z raster, there exists (momentarily) in the

27 1

CALCULATING THE PROPERTIES OF ELECTRON LENSES

ray space only one dotted splicing position. The metrics in the ray space are thus decoupled from that of the z raster. The spatial separation of two consecutive dotted splicing positions, i.e., the partial-lens thickness, is not “measurable” in the traditional sense. For this (to be possible) the two ray spaces, correlated with both raster points as dotted splicing positions, must be present simultaneously. That is, however, not possible since the existence of the two ray spaces, by the action of the micro-lens (as “event”) do not coincide in time but are separated. Moreover, in the step matrix (151) representing the micro-lens, no information concerning the step length z,+~ - z, is “stored,” which was in fact required during its formation phase in the first process step previously mentioned. In the micro-lens image we consider the step matrix as given and reconstruct thickness and excitation of partial lenses in such a way that these-carried over into the partial-lens image-have their optical equivalent in precisely these step matrices.

3. Reconstruction of Trajectories We consider the differential equation of paraxial type (141). Let R,(z), R,(z) be a fundamental system and let R be the fundamental matrix according to (126). We specify two neighboring nodal points of the z raster as abscissae z = z1 and z = z, of two vertical planes of a partial lens. This posseses, for z < zI, an object-side field-free space and for z > z,, an image-side field-free space. In the region between z1and z,, the field function f(z) is optically active. The fundamental matrix goes over, on the object side, into R,(z)

and on the image side into R,(.T)

(-

R,(F) = z

-

2,

(157)

:)R(z,).

The axial points with the abscissae z and Tare optically conjugate with respect to the partial-lens (zl, z,), if in Rr(z)R;’(z) = (?!

=(;

z,

y ) ( ~ : ~ I:)( ;;;), -z+z,

O) 1 (158)

the matrix element a,, is zero. R,(z) describes a state of R in the object space of the partial-lens and, correspondingly, for R,(T) in the image space. Both the object space and the

272

E. HAHN

image space extend from z = -LO to z = +a.A solution y(z) of the differential equation (141) may have the value y;, y, in z1and the value y:, y , in z,. Between these two line elements there exists the transformation:

In the partial-lens image, both line elements exist in both vertical planes of the partial-lens simultaneously, next to each other. This idea is applicable to all partial-lenses of the macro-lens. A solution curve is shown as an envelope of the line elements defined in the nodal points of the z raster. In the partial-lens image, the reconstruction of trajectories is relatively simple on production of the partial-lens matrices, since the step lengths remain unchanged and so may be regarded as given in advance. This method of reconstruction of trajectories is not transferable to the micro-lens image. 4. Identity and Optical Conjugation a. Basis Function

The function (basis function) P(z)

defined by the above quadratic form, satisfies the differential equation 1 PI1 2 P

- --

(1

;)2

-

)(;

2

+ f(z)

= 0.

The quadratic form with i12

= D,,D2, - D f 2

>0

is positive definite. With a partial-lens for which f ( z ) = 0 outside its lens interval zI 4 z I z,, P(z)goes over, in object space, into the object side tangent parabola

and in the image space into the image-side tangent parabola 12

The geometrical defining components of both tangent parabolae are the abscissae zpASand zpeSof their apex points, their vertical heights PASand PBs,as well as their curvature parameters

CALCULATING THE PROPERTIES OF ELECTRON LENSES

273

b. Basis Variable When we define an axial point (object) in relation to the mapping of a partial lens in such a way that we place it at the starting point (vertex) or, in general, at the intersection point of a ray bundle, for the purposes of an unique determination of its position on the axis, we must add whether the relevant intersection point is meant to be asymptotic or real on the object side or asymptotic on the image side. In the micro-lens image, the identity of a ray bundle that makes its appearance in successive ray spaces is ensured by optical conjugation. To this end, the relation between the intersection point of a ray bundle in the ray space and the variable that localizes the intersection point must be unique and the description of the optical conjugate metrically simple. Since the optical state of the z axis in relation to the ray bundle is three-valued, we replace the axial variable z by the basis variable o according to the following definitions: n n a. Object-space condition - -- - < w < LIP 2

2

dz

n

n + + w ( z ) = jl vjl 2

~

71 --

-

z - ZpAS

=-

cot

2

1 (? + ->. j l 2

n n b. Lens-space condition - - < o I 2 2

c. Image-space condition

n n <w m, and the fixed basis of the micro-lens ( m - 1) is an initial basis for a progressive determination of the coherent basis elements of all micro-lenses with n m - 1. Both initial bases are fixed by

-=

With these initial conditions, the coherence conditions (226) and (227) are fullfilled. 1 is a length, that with 1 = 1 represents a unit length from which all other sizes of dimensions can be related to a given length (for example, the focal length of a macro-lens). a. Recursion Formulae for n 2 m

290

E. HAHN

($[; +

-5+

cpi"'] = (;)n-l[

I+.,

40, 0 - 1)

(244)

1351

tan[($+[(;)n-= -tan[(:).

-

1171

+

%.n%.

cos[(;)n

n

-

-{

-2sn++cot(;)n-l[-;+ t

IIn ..-1)]}

uv,n

b. Recursion Formulae for n I m -- 1

cot[(;)n

- 1171 =

g& t

+ .I.+"].

- cot 1

Uv,n

=

{-tn+$cot(;)n+l[;+

uwl

-

COS[($

cpi.+'i]},

lIn

($[-; + cpj"i] + [($

+ .y+')]

= (;)n+l[;

tan{(;)$)

.-

]

=tan[(:).

-l

- 71,

1]5}

=+.qJq Uq,nVv.n

If the refractive power of a micro-lens (n) vanishes, r,

= 0 and (c/z),

4 . Consistency

In the N

(245)

+ 1 nodal points cpn on the cp-scale of the macro-lens

= 1.

CALCULATING THE PROPERTIES OF ELECTRON LENSES

29 1

the values of the functions u ( q ) and 6(q) are determined: (n = 1 . .. N )

The nodal position values of the eigenfunction V ( q )are

and these define the eigenfunction V ( q )of the macro-lens in the framework of the limited accuracy due to the discrete dividing up of the optical action. The thickness of the nlh partial lens is reconstructed from the relation (202)

By numerical integration, nodal point values in the vicinity of the integration interval can be included. The situation is thus similar to that which arose previously in the determination of the building blocks (50) with a prescribed field function in a discrete raster. Since here, however, the values of the derivatives of the integrand are known from (240), advantageous integration formulae can be used, based on Hermitian interpolation. We assume that each micro-lens has a refractive power (- r / 2 ) that is greater than zero. In that case, the phase function 6(q) increases monotonically in the interval -n/2 to +n/2 and the inverse function q ( 6 ) is single valued. With (239), V ( q )follows from (240)

[f

]+

-cos - q - 6 ( q )

(250)

dduq Integration yields the consistency condition

If V ( q ) is to be an eigenfunction of the macro-lens, it must satisfy the differential equation (1 75)

Since V ( q )is reconstructed, we can consider this equation in connection with (177) as a determining equation for the reconstructed field function f(z)

(253)

f(z) = V4((P)H(d= u4(v)c0s2 z

292

E. HAHN

and obtain for the reconstructed excitation of the micro-lens

r+’

f ( z )dz =

[’“

&, + 1

(254)

U2C(P(6)l d6.

Also here, the differential of the integrand in the nodal positions 6, is known, in view of (250), so that numerical integration based on Hermitian interpolation may be used. In general, the reconstructed magnitudes (249) and (254) (right-hand side) will differ by small amounts from the values on the lefthand side, which are to be considered as given in advance.

-12rn =

[Ii1

f ( z ) d z - [6:t’u2[p(b)]d6,

(255)

d

s,. v20’ 9”tl

-, ;2

= (z, + 1 - z,)

--

-

d(P

(256)

By applying the nodal position values of the field function f(z), the derivative of the integrand of (251) is also known, in view of (253), and numerical integration based on Hermitian interpolation can be applied. In general, the consistency condition (251) will not be fullfilled exactly, and a correction 6/C is needed.

,.

log, un = log-u n + 1 un

Un

+

j”’ril [:

tan -q@)

]

-6

dh.

(257)

As will be shown below, the correction magnitudes set out in (255), (256), and (257) can be regarded as elements of a “phantom” matrix whose optical action arises as a result of the testing for consistency and through the formation of

i.e., through the transition of the micro-lens matrix from the “uncorrected” (no superscript bars on quantities) to the corrected state (with superscript bars), whereupon the “phantom” matrix disappears. If we, in fact, develop the elements of the partial-lens matrix (153) in a series of ascending powers of thickness h = z, - z l , there arises, since we develop the fundamental system R,(z), R 2 ( z )at each of the two vertex points in a Taylor series and eliminate the

CALCULATING THE PROPERTIES OF ELECTRON LENSES

293

higher derivatives by means of the differential equation (141), 1

2'"

=

L+l + f f l h

= 1J

-4

+

2

+

l

+fn

2

O(h3),

h2 2!

1

h3 3!

- - - f:, -

2

+ O(h4),

A comparison of (259) with (255) and (256)justifies the interpretation of the corrections ?,/2 and -2S;, as elements of the phantom matrix. For the ratio q / t of the partial-lens matrix we have

-4 = I t

f:+1 + f:,h" 2

+

O(h4),

3!

For a phantom matrix, this ratio is, however, a free choice. Even so, it is permissible for s^ and -?I2 to be also negative. Externally, i.e., as component parts of the splitting up of the macro-lens according to its optical action, the phantom matrices do not make their appearance, except as correction matrices, differing only slightly from the unit matrix. They allow a splitting up of the optical action in the micro-lens image, which is consistent with the primary data of the partial-lens image. From the representation (184) of the elements of a micro-lens matrix, it appears that in the limiting case A/p = 1, the matrix elements -4 and --t have the limiting values v/u or u/v, respectively. Since the determinant of the phantom matrix has the value 1 we determine that:

-

V

-ij, = 1 ( 1 - ?,$")1'2, un

-t,

u, = ,(1

- ?,Q1'?

un

The elements of the phantom matrix (n = 1,2,. . . N ) are determined by (255), (256), (262), and (257). Inserted in (258) they give rise to a new dismantling (155) of the optical action of the macro-lens whose parts (factors) as micro-lens matrices reproduce more exactly the unchanged prescribed partial-lens thickness and field excitation and fullfill better the consistency conditions (251). O n this sequence of lens matrices, altered in this way, the correction procedures by means of phantom matrices can be applied iteratively. As a result of the procedure described, there arises a series of step matrices that can be designated as a consistent solution of the differential equation (141) in terms of the prescribed raster.

294

E. HAHN

XI. QUADRUPOLE OPTICS A . Osculating Micro-Lenses

1. The Basis Variables of Macro- and Micro-Lenses In the development of the theory of micro-lenses we have taken as a starting point that the solution of the differential equation of the paraxial type (141)is given in the form of a field function, defined in the nodal positions of a z raster, in the form of a sequence (155) of step matrices. In the partial lens image, this sequence corresponds to a fundamental matrix (123), in terms of which a fundamental system R,(z), R2(z) is defined. Each positive definite quadratic form (160) P ( z ) of a fundamental system can serve here as a basis function of the macro-lens. In accordance with (166)-(171), R is set up as basis variable and, in accordance with (174), Y(0)is set up as an eigenfunction. The eigenvalue A / p is determined in such a way that the points f c o of the independent variables in the differential equation of the paraxial type correspond to the boundary points f n / 2 on the R scale of the macro-lens. The independent variable in the case of the differential equation (141) is the abscissa z with the fundamental matrix R(z), and in the case of the differential equation (23), the independent variable is t with the fundamental matrix S ( t ) . The optical conjugation of axial points in the object, lens, and image space caused by the action of the macro-lens is described by an equation of the form (173). In the micro-lens image, to each step matrix (139) or (109), respectively, as well as an eigenvalue (L/p),,and there is related an eigenfunction Y(")(o)(") w(")scale. The step matrix (micro-lens matrix) is an invariant of the basis transformation that in its partial-lens image corresponds to a change of the fundamental system. The five basis elements u,,", u,,,, (A/p),,, a?),cop' define a basis. They determine the elements of the lens matrix (184) and describe a basis-related eigenfunction of the macro-lens in the form of osculating eigenfunctions (1 78) in the object and image space of a particular micro-lens. If in a sequence of micro-lenses the bases are coherent, then the envelope of the osculating eigenfunctions of the micro-lenses is an eigenfunction of the macro-lens. In the S cop', the basis variable 0 of the macro-lens is micro-lens interval 01") Io(") a linear substitute of the coherent basis variable of the particular micro-lens.

&=nqi)v- 5 [(;)"v= 1

114-

o=n+l

11;.

(264)

CALCULATING THE PROPERTIES OF ELECTRON LENSES

295

2. Step Matrix S(t)as Lens Matrix We now take as a starting point that the solution of the dimerential eqaution of the paraxial type (23) is given in the form of a sequence (155) of partial lens matrices

that have been prepared as step matrices (109). In the partial lens image, this sequence corresponds to a fundamental matrix (24) S(t)by means of which a fundamental system rl(t), r2(t)is defined. Any positive definite quadratic form p ( t ) of a fundamental system can serve here as a basis function of the macro-lens; in accordance with (166) to (171), with formal t instead of z and p(t) instead of P(z), as a basis variable R instead of o and, in accordance with (174), Y(Q) as an eigenfunction. Analogous to (185), S(t)has the basis representation

which agrees with (30),if p is replaced by p / A and R by AQlp. In the lens space of the micro-lens there exists between R and t , analogous to (168), the transformation

3. Optical Conjugation with a Lens Section

The optical action of a component part of a macro-lens (lens section) may be represented by a sequence of micro-lenses with indices n to m. By the iterative application of (220) one obtains the transport basis transformation in a particular lens space of the micro-lens between the variables w(")and dm) within the limits of the lens section.

If the bases of the micro-lens series of the lens section are coherent, then

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E. HAHN

p,,, = 1 and M,,, is the product of rotation matrices (228) Mm,n

=

-

(269)

C(6, - 6,).

In the partial-lens image t n + l - t , and t m + l - t , are the thicknesses of the two partial-lenses, corresponding to the micro-lenses at the beginning and end of the lens section. For vanishingly small thicknesses, the eigenvalues ( A / P ) ~and ( A / P ) ~take the value 1, and we denote the basis variables of the osculating micro-lenses at the point R, by o and at the point R, by 0. The lens section extending from t, to t,,, in the t raster, on the particular w scale of the two micro-lenses abutting the macro-lens section, gives rise to an optical conjugation c5 = w -

13.. q(;)” -

v=n

B. Conjugation of the R a y Coordinates

1 . Nodal Position Matrix of the R a y Coordinates

If, along the component path (t,, t,), quadrupole fields are operative, the question of optical conjugation between the basis variables o and 0 of the touching micro-lenses arises once more. As well as a differential equation of the paraxial type (23), we now have to deal with a differential equation of the type (33), whose solution in the form of a fundamental matrix (34) SO came into the formulation (95) of the four-vector of the trajectory (89). The field expression (75) or (82), respectively, is different from zero and optically active in the paraxial region, as is revealed by the differential equation (85). From the coefficient of w*, another characteristic of the intersection properties of the ray bundle is determined that establishes an optical conjugation on the basis of a line-grating imaging. The differential equation ( 8 5 ) for the “ray” w(R) has the solution

s stands for the nodal position matrix

s-1 =

cosp

-cosa (272)

w, = w1 + iw,, wg = w 3 + iw, are the ray coordinates at the nodal positions R, and Rp.

297

CALCULATING THE PROPERTIES OF ELECTRON LENSES

2. Transformations of the Ray Coordinates The four-vector (w,,w,) of the ray coordinates, as a result of (87), has the form

The quantities fi and fi indicate the beginning and the end of the active length on the Q scale of the macro-lens, inside which the field expressions K and L, defined in (82) and (83) are effective. Before the occurrence of this activity, the ray w(Q) is in the state A

sin -(Q - Q,) P w(Q) = A sin - (Q, - Q,) P

A

"'a

-

sin -(Q P

-

Q,)

A

WB 7

(274)

sin -(a, - 0,) P

where w, and w, are the values of w in arbitrary nodal positions Q, and 0,. After the occurrence of activity, i.e., as the active path is being traversed, w takes up the state 2 A sin -(Q - aZe) sin -(Q P P W ( 0 ) = (275) w, WB A sin - ( Q ~- *), sin '(*, - *,) P P

a,)

2

where W, and W, are the values of

W

in arbitrary nodal positions

a, and

3. State of the Ray in the Space of the Touching Micro-Lens

The beginning and the end of the active path (fi,h) is limited by a corresponding micro-lens. Since we may assume coherence, in each lens interval ( w , , ~ , )or (O,,Wr), respectively, the basis variables w and W are linear substitutes of the basis variable Q of the macro-lens. In the lens interval of the micro-lens at the beginning of the active path we place the nodal points Q, and Q,, and in the lens interval of the micro-lens at the end of the active By means of (263) we can go over from the path, the nodal points a, and basis variable of the macro-lens to that of particular micro-lenses and obtain for the touching ray, before the activity event

a,.

w(0) =

sin(o - a) sin(w - p) wfl(Q), sin(a - B) w ~ ( Q )- sin(a - p)

(274)

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E. HAHN

and for the state of the touching ray after that,

We have passed from the micro-lens with nonvanishing lens interval (wl, w,) to the touching micro-lens and have tacitly ignored- the condition that the nodal positions o, = u, us= p, and 0,= CC, GB= p, respectively, lie in the lens interval of particular micro-lenses. This is allowable since already in (273) the above-named transition could be carried out.

In this representation Rand are the beginning and the end of the action path on the Q scale of the macro-lens. w,(R), w,(Q) are the coordinates of the touching ray at the nodal points u, p in the lens space - n/2 w < n/2 of the touching micro-lens at the point R, and W u ( a ) , WB@) are the coordinates of the touching ray at the nodal points CC, Bin the lens space - 4 2 < 0 < 4 2 of the touching micro-lens at the point fi.

-=

4 . Relation Between Ray and Trajectory

The ray (274) w(R) is according to (61) a substitute for the trajectory r(t) with p ( t ) as the basis function of the macro-lens and @(t)as the rotation-angle function. In the object or, alternatively, image space of the lens section ( R , a ) of the macro-lens, the trajectory tangent is a transform of (276) w(R) or, alternatively, (277) w(a), with p as a basis function (tangent parabola) of the particular touching micro-lens. The trajectory tangent in object or image space of the lens section has the nodal position value r., rB or, alternatively, Fa,Fo,which arises by a rotational stretching (61) of the ray coordinates w,(R), wa(R) or, alternatively, W,(Q), W,(n). The conjugation (278) of the beam coordinates contains the identity of the trajectory before and after the occurrence of activity of the field expressions K and L. In the investigation of the astigmatic conjugation that the fieid function gives rise to between the w and the KI scale, the rotational stretching plays no part in it. It has significance only in connection with the supplying of imaging magnifications and position angles of the line-grating imaging in real three-dimensional space and has, therefore, not been taken into consideration. Likewise, we dispense with a consideration of a deflection term caused by the field expressions L in (278).

CALCULATING THE PROPERTIES OF ELECTRON LENSES

299

C . Collineation I . Collineation Matrix

For the study of the astigmatic conjugation between the basis variables w and W of the micro-lens touching the action path (Qfi) of the quadrupole field, the collineation matrix K = ( k i , j ) K

= s(C1,

p)Q(fii)Q-’(Sz)~-’(~, p)

(279) forms a starting point. It is an expression of the quadrupole action and describes a transformation existing between the ray coordinates in object and image space of the lens section (Q, fi)

By object and image space, respectively, of the lens section, we understand the lens space of the micro-lens touching the lens section on the object or image side, respectively. The choice of the nodal positions w = a,fl, or, alternatively, 6 = CC, in the object and image space, respectively, is at our disposal. However, the difference between a and p and CC and respectively, must not be a multiple of 7r for then, the determinant of the collineation matrix sin2(g- 8) det K = sin2(@- /?)

p

p,

is not zero and the transformation described by (280 ) is invertible. The ray coordinates correlated with each other through the collineation matrix represent one and the same object. In the partial lens image, the lens spaces (ray spaces) of both touching micro-lenses exist simultaneously. By “object” should be understood the trajectory that binds smoothly the object side ray with the image side ray. In the micro-lens image, both ray spaces are considered sequentially in time. The conjugation described by (280) is an expression of the identity of single rays and not of points (axial points), if these have their origin in the vertex of one and the same ray bundle. For the property of a ray bundle, in any ray space, to possess an intersection, at which all rays of the bundle intersect, does not hold good under the action of a quadrupole. In order to solve the identity problem of ray bundles in the microlens image, even in quadrupole systems, we require, analogous to XI.A.4.b, the unambiguous and metrical simplicity of the quadrupole conjugation.

300

E. HAHN

2. Line-Grating Imaging The first two equations of (280) set out a relationship between the ray coordinates w l , w2 in an a plane (a plane whose axial point is the nodal position w, = a) and the beam coordinates W1, W2 in an &plane, together with the ray coordinates w3, w4 in a plane, as parameters. By linear combination, we can eliminate any one of these two parameters

a

+ [ki2kz31w2

k23Wi

- k ~ 3 W 2= [ k i i k 2 3 l w i

k24wi

- k14$z = [ k i i k z 4 l w 1 f

Ck12k241W2

[ki3kz4IW4,

(282)

+ Cki3k241W3.

(283)

-

In the formulation of astigmatic imaging, two-row minor determinants of the collineation matrix K must be formed, which we abbreviate as follows =

i <j,

[k,kj,],

k 0. There is no stigmatically conjugate pair of points. b. Parubolic, if q’ = 0, q > 0. There is one stigmatically conjugate pair of points. c. Hyperbolic, if q‘ = q = 0. There are two pairs of stigmatically conjugate points. In this classification, the special case, in which the elements of region I1 and

+

111 of the system matrix (318) are jointly zero, i.e., (348) D2(w)= C(w)’ H ( O ) ~and = C(W)’ H(uS)2 vanish identically in w or 0, respectively,

o2

+

is not included. In this case the quadrupole optical imaging is fully stigmatic. 4 . Second Kind of Classijication

The state of a quadrupole system is classified by criteria fullfillment of the two quadrupole discriminants according to a scale a to f. a. b:2

+ b:3 - b223 # 0,

and the state is elliptic.

then q q’ > 0,

CALCULATING THE PROPERTIES OF ELECTRON LENSES

b. b4,

+ b:3 - b i 3 = 0,

b23

then q‘

# 0,

317

= 0,q > 0

and the state is parabolic. c. bjk= 0,

then qq’ > 0

a , , - a22 - a33 > 0,

and the state is elliptic. d. bjk = 0,

- a33 c

a,, -

0,

then q’ = q = 0

- a33 = 0,

then q’ = q = 0

and the state is hyperbolic. e. bjk = 0,

a,, - a,,

with c = 0

and the two stigmatically conjugate point pairs coincide.

f. bjk

0,

Ujk

= 0,

and the conjugation is fully stigmatic. H . Constants of the Quadrupole Optical Conjugation

1. Elliptic and Parabolic State Here q > 0, and with the substitution x = -cot(w - wo) = p

+ q tan(cp - cpo)

(370)

and the abbreviations ( I , s both positive) r2 = ( p - p’)’

+ ( q + q’)’,

s2 = ( p - p’)’

+ (q - q‘)’,

(371)

one obtains the integrand (358)of the elliptic integral in the normal form dcp (372) 2 C(w), + H ( w ) ~ J1 - k 2 sin2 cp The constant cpo(modn),which depends on w o , is tied to the condition

r q E G F T w z dw = du =

rscos2cpo = ( p - p ’ ) ,

+ 4”

rssin2cp0 = 2q(p’ - p). (373)

- q2,

k is the modulus of the elliptic integral u

= F(k,cp) =

10

dcp

k 2 =-

4rs

+

(374) (r s)2 ‘ ,/I - k2sin2cp ’ It is independent of wo and coincides with the image-side value and is a function of the two discriminants of the quadrupole optical state. This is brought out in the following representation from a consideration of (366)and (367): 1 r2 s2 1 --k2 = (375) 2 r2 + s2 + J ( r 2 + s212- (r2 - s212’

+

318

E. HAHN

r 2 + s2 -- 29; 2

+ c 2 cos 2y + 3c2

(yy + = (2q;

c2

(3761

cos 27 - c 2 ) 2

(377)

Furthermore, the quantity g;

is independent of coo or is,,respectively. 2. Substitution Matrix The substitution (370) may be written in the form p cos(0 - 0 0 ) = P cos(cp - cpo)

+ 4 Wcp - cpo),

(379)

psin(w - w,) = -cos(cp - cp,),

where p2 is defined by the sum of the squares of the two right-hand sides. By multiplying both equations together or, respectively, with themselves, we form

and obtain for the substitution matrix

I

1 R,= 0 (0

0 cos 20, - sin 20,

o \ sin 20, cos 20,

I

1+p2+q2 ~2q

P

P

1

+ p2 - 42

\ +p2-q21 2q

~

-

4

-1 +p2+q2 P

-l

2q

*i 1

0 c0s2cp0 -sin2cp0 . sin2q0 c0s2cpO o ,

The image-side substitution matrix R, is obtained by placing bars on all relevant quantities.

CALCULATING THE PROPERTIES OF ELECTRON LENSES

319

3. Conjugation Matrix

The astigmatic conjugation in the variables o and (5 is indicated by the condition (333) and this becomes, on substituting cp and Cp, sin2F [c0s12J

- A,.

with the conjugation matrix A,

[c0s12J sin2cp = 0,

RYSF’R, = ( ~ j k ) . (383) The inverse function of (374) cp(u) is denoted as amplitude and sin cp = snu, cos cp = cnu are the Jacobi elliptical functions. The astigmatic conjugation in the variables ii and u is represented as A,

=

(384)

On the other hand, we know that the relation (356) holds between astigmatically conjugate points. This means that by replacing U by u + A, equation (384) must be satisfied identically in u. The addition theorem of the elliptical functions permits this if the conjugation matrix A, has the welldetermined structure

A,

= 033

l

cn ’A 1 + dn2A

0

1 -dn2A 1 + dn2A

0

2 dnil 1 + dn2A

0

1 - dn2A 1 +dn2A

0

1

1-4

(385)

4. Transformed System Matrix

The conjugation matrix (383) A, is a three-row matrix of the region I of a transformed system matrix S(”‘)

320

E. HAHN

with the quadrupole matrix H‘ as generator sin@, cos@,

:)(

-cos@,)L(ij sin@o & p

sino, -cosW,

-cosoo

COSO,

sin(5, sinw,

For the transformed system matrix S(”’),the orthogonality relation (316) holds with H’ instead of H. From this we obtain, from a consideration of the structure (385) of A,, the two determining equations for A (3 - (4 + k2)sn2A+ (4 - 2k2)sn41}

(388)

These are supplemented by the determinant relation of the matrix equation (383)

5. Eigenvalue Determination

We write for brevity u = 4(1

2b

=a

+ a2)+ (4 - 2k2)g2,

+ 3k2gi, 12312

2b’ = (sign cr22)a“Z

(391)

-A

and test with these quantities, depending only on the two discriminants of the quadrupole optical state, the content of the determining equations (388) to (390). By a linear combination of (388) and (390), we obtain, firstly, an expression for ui2 12a,2, = a and secondly, a quadratic equation for sn2A

(392)

asn4A - 2bsn21 + 39; = 0.

(393)

Further, by substituting (390)into (389)by the use of (392), a second quadratic equation for sn2L is obtained asn4A - 2b sn2A - 39; 9

= 0.

(394)

CALCULATING THE PROPERTIES OF ELECTRON LENSES

32 1

From this one obtains, on the one hand, by addition, the determining equation for the eigenvalue. b + b' sn21 = (395) a and, on the other hand, by subtraction, and application of (399, the relation

b2 - b',

=

3 q 02'

1396) Depending on the choice of the sign of a,,, which with (359) ymod R , and ?mod n is adjustable, b' is positive or negative (391). If it is agreed among ourselves that q should not be smaller than q'(positive), one then has to decide that 6' should be negative. sign a,, = -sign A

(397) With 0 < 1< F(k, cp = n/2), 1is unambiguously determined via (399, and 1, = 1,1, = - 2 are the eigenvalues of the advanced and retarded quadrupole imaging that depend only on the two discriminants of the quadrupole state. 6. Hyperbolic State

Here q' = q = 0 and the substitution (370) is not applicable. In(360) y takes on the value 0 mod n and the modulus k of the elliptical integral (374) has, due to (371) r = s = 2c, the value k = 1. According to (369) a' > 2 and A' = a2 - 1. The determining equation (395) for the eigenvalue 1goes over into coshZ1=

-

1 = A,.

(398)

With bjk = 0, comparison of (366) and (367) shows that qo = 0 and H ( o ) = const G(w) or A(W)= const G(c5). It is sensible to determine oOmodx/2 in such a way that (359) p o = 0. a23 aZ2 tan20, = = -.

'33

(399)

a23

c ( o o )is determined from (365) and both zeroes w, = wl, w 2 ,of the radicand (358) are fixed by cot(@, - wo) = c(og), (40) with -n/2 d wl < o2< 4 2 on the w-scale. The zeros w1 and w2 are stigmatically conjugate to the zeros W,, Wz of D 2 ( ( 5 )= C(O),+ H ( 6 ) , , in accordance with (339)

322

E. HAHN

Instead of substitution (370) there now arises with dq

= du/2

s w c p + cpo) = c(wo)tan@ - coo) cosh(cp + cpo)

if wo “internal” to o-partial scale

(402)

sinh(cp - cpo) cosh(cp - cpo) -

if wo “external” to w-partial scale.

(403)

1

cot(a, - wo)

c(00)

The w scale is broken up by the two zeros wl, ozmod IC into two partial scales. If w runs from w1 to w2 and thereby passes w,, coo lies “internally” with respect to the partial scale, otherwise “externally.” On the image side, 2W0 = w1 + Oz mod 7~ must be chosen such that with wo internal or external, respectively, Wo must also be internal or external, respectively. The substitution (403)(aoexternal to the partial scale) can be written in the form pcos(w - wo) = c(w,)sinh(cp - cpo)

(404)

p sin(w - wo) = -cosh(cp - qo).

This agrees formally with (379)if we put p = 0, q = c and replace sin(q - qo), cos(q - cpo) by s i n W - q0X cosh(cp - q0). The substitution matrix R, in cosh2q (405)

C

with wo external to the partial scale, has the form 0

0

-c+-

C

1 c

c

cosh2qo - sinh2qo cosh2qo 0 The form with wo internal to the partial scale (substitution 402) comes out of this by the transformation 0,+ 0 0

+ n/2,

c + l/c,

500 +

-cpo.

(407)

The condition (333) for the quadrupole optical conjugation has, with the

CALCULATING THE PROPERTIES OF ELECTRON LENSES

(

323

substitutions cp and Cp, the form

c0s;2q)

[c0s;2v)

sinh2Cp A, sinh2cp = 0,

(408)

with the conjugation matrix A,, formed analogously to (383). O n the other hand, there exists between cp and Cp, because of (356), the relation

2(4i - cp) = 1. (409) With the replacement of 24i by (2Cp + A), equation (408) must be identically fullfilled by cp. The addition theorem of the hyperbolic functions permits this if the conjugation matrix A, has the well-determined structure A, = RT. Sl("). R, = (sign A)

[ -'

-coshA)'

(410)

With this equation the quantity q0 - pois also determined. The substitution existing between @ and 6 can be achieved via (402) and (403),respectively, by a formal placing of bars over all relevant quantities. 7. Cardinal Elements If the biquadratic form (333) in = tan 6,c = tan w turns out as the product of two bilinear forms, the setting to zero of one of the two factors,

gives rise to a conjugation, which in the usual form

can be described with the cardinal elements

The substitution (370),which is valid for the elliptical and parabolic state, has the form

-'("i"')=(: P coscp

:)(-;:)=(-;:)(:

Ey

(414)

324

E. HAHN

with sincp,

By multiplication by

-coscpo)(;

A)

(-:

=(-;ow)(;

:)(

sinw, -cosw,

cosw, sino, 1415)

there results

3'c -0.

(416)

We form the scalar product of the object- and image-side vectors (414) and (416)

"I(;

and obtain by linear combination the form (411) with the cardinal matrix

(; ;)=(;

;y[

);

sin(@- cp), - C M @ . (419) cos(@- q), sin(@- cp) When this matrix has the same value for all point pairs w, W of the advanced line-grating image, i.e., it is constant, and if this prbperty also holds for all point pairs of the retarded astigmatic conjugation, the cardinal elements are fixed by (413). The astigmatic conjugation (356) by translation of the variable u, shows us that this case occurs when the modulus k of the elliptical integral has the value k = 0. Then the difference @ - cp is constant and has for the advanced conjugation the value A, = 2, and for the retarded conjugation the value A, = -A. In the general elliptical state (k > 0) and in the parabolic state (k = l), the cardinal elements are not fixed, since the cardinal matrix is not constant. In the hyperbolic state, we obtain for the cardinal matrix constructed for the analog to (419), if we start out from the substitution (402) with o,internal to the o

325

CALCULATING THE PROPERTIES OF ELECTRON LENSES

partial scale -

(: i)

=

-sinh(cp-cp)

(g ): (

L3)

-cosh(rp-cp)(A

cosh(cp - cp)

sinh(@- cp)

C (420)

with

(: ):

sinhq, = (coshg,

-coshqo)( -sinhq,

Ol ~

](

sinw, -coswo

coswo sinw,

& (421)

The difference (p - cp is constant for the conjugate points w, W and has for the advanced conjugation the value &I2 = 112, and for the retarded conjugation the value &/2 = -11/2. The cardinal matrix is invariant with respect to a transformation (407) of the object- and image-side quantities if thereby the sign of (p - cp is reversed. This means that the sign of the eigenvalue does not characterize the state of the cardinal matrix. That is, however, not necessary, since a conjugation described by (412) is already unambiguous.

XII. CONCLUDING REMARKS To conclude, we would like to answer the question, how is the method set out here to be classified in relation to existing methods? If one formulates the laws of motion in the form of a differential equation (Lorentz equation of motion), the solutions reveal the trajectories. If one wishes to reduce to a minimum the necessary expenditure of effort needed for the numerical integration of the differential equation and to find closed forms for the optical properties of the field configurations, one can develop the differential expression in terms of the powers of the ray position and direction and so construct the trajectory by means of perturbation methods. 0. Scherzer (1936) went this way and set out formulae for the image aberrations of third order. If one formulates the laws of motion as a variational problem, one can then, as W. Glaser (1952) did, from the outset formulate the electron motion as an optical problem and treat the electron optical imaging in the region of the third order by analogy with the Seidel image aberration theory. Of basic importance here is the representation of the action function (Eikonal) as an integral over the refractive index along a particular trajectory of the paraxial region. The refractive index (Lagrange’s function) is so constructed that the Euler equations of motion associated

326

E. HAHN

with the variational problem agree with the Lorentz equations. Since the trajectories, expressed in beam parameter variables, can be expressed in terms of two fundamental solutions of the paraxial differential equation, these ray parameters are explicitly present, up to the fourth order, in the action function. Partial differentiation with respect to them leads to formulae for the image aberrations of third order. They agree with those obtained from formulae based on a perturbation method. Both these methods were developed for electron optics at a time when the implementation of numerical calculations was tedious and time-consuming. It is therefore understandable, with the advent of large computers, that the methods of solving electron optical problems might also change, especially in those cases in which the degree of difficulty and complexity has greatly increased, in particular in the area of microlithograph y. If the field configuration is described sufficiently accurately, and if one uses an integration method with sufficiently small step widths, one can without any further analysis, introduce direct trajectory integration (DTI methods). By a choice of an array of initial values and numerical integration of the differential equation for each initial value, one can build up a discrete trajectory bundle and, from a system of equations, calculate the aberrations (Kasper, 1985). With the DTI method, the trajectories of a discrete bundle are calculated independently of each other. A single trajectory gives no information about the structure of the surrounding ray bundle, in particular, about defocusing and astigmatism. Since the image aberrations are determined from the differences between the trajectories evaluated, oscillations or discontinuities in the higher order coefficients of the field representations can have a very harmful effect on the final result (image aberrations). A combination of the DTI method and the perturbation method avoids these difficulties. The differences between the trajectories are then represented in the form of integrals and the higher order coefficients may be eliminated by partial integration. In electron lithography, the electron beam serves to process a target (resist layer) and the sharpness of the shaped ray bundle (electron probe) is influenced by local image aberrations. These cannot, in general, be derived from the global image aberrations, since the deflection objective contains correcting fields that are altered in a controlled manner, according to the position of the electron probe on the target. In Hahn (1980) a method was described in which the intensity inside a shaped electron probe (line probe) may be structured in a controlled manner over many parallel electron channels in order to attain a higher fabrication productivity (writing speed). With this kind of modulation, a quadrupole lens is used to produce an astigmatically structured ray bundle. In this example, the advantage of the method described here is particularly clear, since the

CALCULATING THE PROPERTIES OF ELECTRON LENSES

327

quadrupole matrix is calculated as a whole and need not be constructed from an assortment of single trajectories. The method described here can be applied in such a way that each trajectory can be precisely calculated by providing its initial values. Thus, the actual trajectory integration is carried out on the principle of the field-basis coupling, i.e., the coupling between the electromagnetic field and the rayoptical totality of the ray bundle surrounding a particular trajectory. The field-basis coupling is effected not pointwise but intervalwise in the form of a step matrix (lens matrix) whose representation (53)within the framework of an iteration process, and in connection with the representation (95),is a solution of the equation of motion. The trajectory is consistent with the primary data (raster and field values) and with the secondary data (lens matrix), since from the secondary data the primary data can be reconstituted (correction by phantom matrix). In the development of this theory, the starting point made use of concepts that are directly related to space and time considerations and can be recognized as thought categories. Its application finds both these categories in the concepts of the partial-lens image and the micro-lens image. In contrast to the partial-lens image, in which a partial-lens can be characterized in the usual way, with vertical planes as limiting surfaces and entering the exiting rays, the micro-lens image is unusual. A micro-lens is described only in terms of its action, in that one postulates that in a ray space a first (object-side) state is transformed into a second (image-side) state so that one cannot compare the two states simultaneously. Such a comparison serves, in the partial-lens image, to determine the action (lens matrix), since the identity of the object and image rays, i.e., the common membership of one and the same trajectory, is postulated. In the micro-lens image, such a comparison is not possible, and even not necessary, since the action is known. What must be determined in the micro-lens image is the identity. This problem is solved in the framework of a theory of micro-lenses by showing that the optical conjugation is unambiguous and metrically simple. The basis variable is a substitute of the axial abscissae and in this substitution, which varies from iteration to iteration, are the paraxial optics of a micro-lens with the particular trajectory included as principal ray of a bundle. In a further development of the method, not included here, the particular trajectory is the axis of a curved line coordinate system, on which the field has been transformed. Here, the paraxial optical properties of the ray bundle, enveloping a particular ray as principal ray, are directly represented. For the quadrupole optics, in a similar way, the unambiguity and metric simplicity of the optical conjugation have been built up. The substitution of the basis variable leads to an elliptical integral and enables a classification to be made of the state of a quadrupole optical system.

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REFERENCES Glaser, W. (1952). “Grundlagen der Elektronenoptik.” Springer, Wien. Hahn, E. (1985). Optik 69,45. Hahn, E. (1979). DD 147018, HOlj 37/30. Hahn, E. (1980). DD 158 197 HOlj 37/30; US 4.472.636 HOlj 37/302. Kasper, E. (1985). Optik 69, 117. Scherzer, 0.(1936). 2.Phys. 101,693. Zurmuhl, R. (1963). “Praktische Mathematik fur lngenieure und Physiker.” Springer, Berlin, Heidelberg, New York.

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VCL 75

Derivation of a Focusing Criterion by a System-Theoretic Approach MICHAEL KAISER Dornier System GmbH Friedrichshafen. Federal Republic of Germany

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1. System-Theoretic Approach to Scalar Radiation Problems in Homogeneous Space . A . The System-Theoretic Concept . . . . . . . . . . . . . . . . . . B . Scalar Formulation of the Radiation Problem . . . . . . . . . . . . . C. The Electromagnetic Radiation Problem . . . . . . . . . . . . . . . 111. Focusing by Plane Radiators . . . . . . . . . . . . . . . . . . . . A . Illumination of the Infinitely Extended Radiator, Scalar Treatment . . . . . B. The Optimum Focusing Illumination and Its Radiated Field . . . . . . . . C. The Effect of Limiting the Optimum Focusing Illumination . . . . . . . . D . Comparison with the Conventional Focusing Illumination . . . . . . . . E. The Electromagnetic Case . . . . . . . . . . . . . . . . . . . . IV . Focusing in Stratified Media . . . . . . . . . . . . . . . . . . . . A . Propagation of Plane Waves . . . . . . . . . . . . . . . . . . . B. The Transfer Function Tensor . . . . . . . . . . . . . . . . . . C. Examples . . . . . . . . . . . . . . . . . . . . . . . . . V . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Theorems of Fourier and Hankel Transform Used in This Contribution . . . . B. Derivation of the Optimum Focusing Input Illumination (Eq. 59) . . . . . . C. Derivation of the Optimum Focusing Input Illumination with the Constraint of Finite Aperture Dimensions (Eq. 73) . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

329 333 335 335 339 344 348 348 350 352 358 358 363 363 361 372 382 383 383 385 386 387

LISTOF SYMBOLS a

aperture radius

a(k.. k. )

spectral function of the electric field

a.(k.. k. )

transverse part of a@.. k. )

a.(k.. k. )

longitudinal part of a(k.. k. ) 3 29 Copyright CI 1989 by Academic Press Inc. All righrs o f reproduction in any form reserved. ISBN 0-12-014675-4

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MICHAEL KAISER

a',,"(K),a;,.(K) spectral function of the parallel polarized part of the incident and reflected field, respectively, in the n-th layer for an electric field strength impressed in the input plane ai,(K), a;,(K)

spectral function of the perpendicular polarized part of the incident and reflected field, respectively, in the n-th layer for an electric field strength impressed in the input plane

Af,, A',

amplitude of an incident or reflected plane wave, respectively, in the n-th layer

Ailn, Ailn

parallel polarized part of Af, or A:, respectively

A ; , , A;,

perpendicular polarized part of Af, or A;, respectively

bf,.(K),bil,(K) spectral function of the parallel polarized part of the incident and reflected field, respectively, in the n-th layer for a magnetic field strength impressed in the input plane bi,(K), b;,(K)

spectral function of the perpendicular polarized part of the incident and reflected field, respectively,in the n-th layer for a magnetic field strength impressed in the input plane diameter thickness of the n-th layer distance two-dimensional Fourier-transformed vector of the electric field amplitude factor components of e spectral function of the electric field transverse part of eo components of eo vector of the electric field tangential electric field in the input plane components of E amplitude factor of the electric field strength electric input field strength components of the electric input field strength amplitude of an incident homogeneous plane wave frequency Fourier transform of the function F ( x ) two-dimensional Fourier transform of the function F ( x ,y) Hankel transform of the function F ( p ) input function of a system

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

33 1

input function of a two-dimensional system circular symmetric function of p = operator of the one- and two-dimensional Fourier transform, respectively Fourier transform of the function G ( x ) two-dimensional Fourier transform of the function G(x, y) output function of a system output function of a two-dimensional system vector of the magnetic field strength tangential magnetic field in the input plane amplitude factor of the magnetic field magnetic input field strength components of the magnetic input field strength imaginary unit

J-l

Fourier-transformed surface current density electric surface current density wave number propagation vector wave number of the n-th layer transverse components of the propagation vector or variables of the Fourier transform, respectively z-component of the propagation vector of the n-th layer radial component of the propagation vector or variable of the Hankel transform, respectively

k:,k;

propagation vector of an incident or reflected wave, respectively, in the n-th layer

ko

free space wave number

K

magnitude of the transverse part of the propagation vector

K

transverse part of the propagation vector

n,n'

normal vectors

r, r'

field point vector, source point vector

rllnr T i n

reflection coefficient at the n-th layer for parallel and perpendicular polarization, respectively 3-dB radius distance between an aperture element and the focus surface transfer function of a one-dimensional system

MICHAEL KAISER transfer function of a two-dimensional system transfer function of a two-dimensional circular symmetric system impulse response of a one-dimensional system impulse response of a two-dimensional system impulse response of a two-dimensional circular symmetric system time variable Fourier-transformed scalar wave function U, 6, U, unit vectors in Cartesian coordinates unit vectors in cylindrical coordinates Fourier-transformed optimum focusing illumination time-dependent voltage amplitude factors scalar wave function unit vectors of the parallel polarized incident and reflected field, respectively, in the n-th layer unit vectors of the perpendicular polarized incident and reflected field, respectively, in the n-th layer volume Cartesian coordinates of field point input plane coordinates focal plane coordinates depth of the n-th boundary intrinsic impedance of the n-th layer attenuation transfer function tensor of the system electric input field strength-radiated electric field transfer function tensor of the system magnetic input field strength-radiated electric field elements of y ( k , , k,; z) elements of y"(k,, k,; z) impulse response tensor of the system electric input field strength-radiated electric field impulse response tensor of the system magnetic input field strength -radiated electric field elements of r ( x , y , z) elements of

r H ( ~y,, Z )

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

6

loss tangent

6(x), 6(x, y)

one- and two-dimensional Dirac delta function, respectively

333

delta-operator relative permittivity permittivity of vacuum wavelength free space wavelength wavelength in a dielectric integration variable cylindrical coordinates of the field point cylindrical coordinates of the input plane reflection coefficient at the n-th layer for parallel and perpendicular polarization, respectively (assuming no further boundaries are present) Fourier-transformed correlation function correlation function phase angle angular frequency one- and two-dimensional convolution operators, respectively, e.g., U(x, Y)

** S(X. Y)

correlation operator complex conjugate of S

I. INTRODUCTION

System theory has become a widespread tool in electrical engineering and is well known to the engineer from communication and control theory. System theory has yet to find very many applications in the solution of wave propagation problems, however. In the present paper a focusing criterion for electromagnetic waves will be derived by means of a system-theoretic approach. Emphasis is placed neither on an extensive representation of all aspects of system theory nor on mathematical rigour in deriving some of the theorems of computational methods commonly used in system theory. Rather, we shall be concerned with the applicability of system theory to the treatment of wave propagation problems. For an extensive treatment of system theory, the reader is referred to the books by Papoulis (1968) and Gaskill (1978); the reader more interested

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MICHAEL KAISER

in the mathematics of system theory is referred to Papoulis (1962) or Bracewell (1965). There is an increasing number of applications of electromagnetic waves outside their classical domain (the transmission of information), for example, in industrial measurements, remote-sensing tasks, imaging purposes, and, in the medical field, diagnosis and therapy. In many of these applications, it is desirable to focus radiated electromagnetic energy to a well-defined region or to have a preferred reception of the radiation emanating from a certain region; it is essential that the transmitting or receiving device be focused on that region. As an example, in localized microwave hyperthermia in cancer treatment (Melek and Anderson, 1980) focusing is required in order to concentrate the electromagnetic energy effectively on the tumor so as to reduce damage to healthy tissue. Focusing is also essential in receiving antennas of microwave thermographic measurement devices used in the detection of warmer areas (i.e., areas with enhanced levels of emitted radiation) within the body, e.g. a tumor, (Myers et al., 1979). In this case focusing is essential to obtain good spatial resolution. In non-destructive material testing, inhomogeneities, e.g. flaws, in an otherwise homogeneous material can be detected by means of scattered radiation (Holler, 1983). In these applications, the medium in which focusing is required is usually not homogeneous. In microwave hyperthermia applications, skin and fat layers must be penetrated before the microwave radiation reaches the target area in the tissue below. As in many other applications, a planar stratified medium may serve as a good model, where skin, fat and muscle tissue are approximated by layers with different (complex) permittivities (Guy, 1971). Under certain assumptions, the same model may be used to approximate the different geological layers of the earth in the application of electromagnetic methods in geophysical probing (Wait, 1979; Hansen, 1983). The goals of the present paper are the development of a method of synthesizing an illumination focusing electromagnetic radiation in stratified structures and the determination of the radiated field. The classical focusing criterion given in the literature (e.g., Wehner, 1949) is a heuristic one and assumes a distribution of elementary sources in the aperture, the contributions of which have to interfere constructively at the focus. The computation of the field radiated from a focusing antenna into a stratified medium is a rather difficult field-theoretic problem. Here we shall treat radiation problems by a system-theoretic approach. Like the classical methods, such an approach allows us to carry out a field computation from a given source distribution. A synthesis of an aperture illumination for producing a desired radiation field is also possible. The input plane/focal plane system can be completely described by its impulse response in the spatial domain or by its transfer function in the spatial

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

335

frequency domain, respectively. They are interrelated by the Fourier transform. The transfer function can be given analytically for a homogeneous medium as well as for a structure composed of planar layers between input and focal plane. The impulse response, however, can be represented by a closed expression for the homogeneous medium only, whereas the stratified medium requires a numerical solution. The desired focusing aperture illumination is obtained from the impulse response.

APPROACH TO SCALAR RADIATION 11. SYSTEM-THEORETIC PROBLEMS IN HOMOGENEOUS SPACE A. The System- Theoretic Concept In this chapter, the application of system-theoretic methods to wave propagation problems shall be introduced. For this purpose some basic system-theoretic concepts will be presented. Below, the term, “system,” shall apply to a configuration having at least one input, while its output or outputs exhibit a reaction due to a stimulus at the input. Thereby, the reaction depends on the properties of the configuration and on the stimulus itself (Gaskill, 1978). For example, the stimulus of an electric network may be a voltage pulse at the input terminals which results in a voltage pulse at the output terminals that depends on the shape of the input pulse and on the physical realization of the network (Fig. la). Another example for a system is the configuration of two (fictitious) parallel planes in homogeneous space between which electromagnetic waves propagate. If the tangential components E,, and Eoy of the electric field E,(x’,y’,O)= E, are given in the input plane z = 0, the radiated field E(x, y , z ) = E, is completely determined in any output plane z # 0 due to the uniqueness theorem valid for electromagnetic fields (Honl et al., 1961). If one assigns one input and one output to each component of the electric field strength, then the two parallel planes represent a system with two inputs and three outputs for the propagation of electromagnetic waves (Fig. lb). To analyze a system, a suitable mathematical model that describes the behavior of the system has to be developed. Usually this requires certain simplifications for otherwise the mathematics involved would be considerably more complicated. For example, when the wave propagation between two parallel planes is treated by system-theoretic means, one assumes that it is possible to impress an arbitrary field distribution in the input plane. The realization of this field distribution and any possible feed-back of the system

336

MICHAEL KAISER

(b)

FIG.1. Examples of systems. (a) Electric network. (b) Parallel planes.

to the input are neglected. Despite these restrictions, the model may be a useful tool for system analysis as long as one is aware of the simplifications. A system can be described by operators. Such an operator L assigns to a function &(x) which is a member of one class of admissible input functions a function G,(x) of a second class of output functions: L{&(x)} = Gi(x),

i = 1,2,3,.. . , n .

(1)

Such a rule of assignment may be given in many different ways: by a differential or integral equation, by a table, or by a graphical representation. In optical diffraction for example, Kirchhoff’s formula

is commonly used. In the region bounded by the closed surface S , it is a solution of the scalar Helmholtz equation AV(r)

+ kzU(r) = 0

(3)

from the boundary values U(r’) and dU(r’)/an‘ on S . Here r = xu,

+ yu, + zu,

is the field point vector,

r’ = x’ux + y’u, the source point vector, and region V.

+ z’u,

a / a d the normal derivative directed into the

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

337

If applied to the system of parallel planes in Fig. lb, Kirchhoff’s formula represents a method of computing the field in an output plane z # 0 from the boundary values in the input plane z = 0, assuming that contributions from the surface at infinity can be neglected. The assignment rule (2) has the properties of linearity and shift invariance. The analysis of a general system is practically impossible; only systems with very special properties allow a detailed analysis. The most important properties are those of linearity and shift invariance. For linear systems, the superposition principle is valid; i.e., if a system reacts to two stimuli F,(x) and F2(x)with outputs G , ( x ) = L { F , ( x ) }and G2(x) = L { F 2 ( x ) } respectively, , it will react to the sum of the input signals with L{aF,(x)

+ bF2(x))= aG,(x) + bG2(x).

(4)

A system is called shift invariant if as a result of a shift of the inputs by a certain quantity, the outputs are shifted by the same amount. If

L ( W } = G(x), (5) then a shift-invariant system reacts to the input function shifted by x o with L { F ( x - xO)}

G(x - xO),

(6) i.e., with an output signal which is shifted by x o relative to the original signal but still having the same shape. Systems possessing the properties of linearity and shift invariance (linear shift-invariant (LSZ) systems) can be analyzed particularly easily. The eigenfunctions of the operators describing LSI systems are complex exponential functions (Gaskill, 1978). If an LSI system is stimulated with F ( x ) = exp(jk,x), then the system reacts with =

s(kx)ejkxx= G ( 4 ,

(7 1

where s(k,) is the eigenvalue associated with the eigenfunction exp(jk,x). s(k,) is called the transfer function of the system; it is computed by Fourier transforming the impulse response S(x) of the system according to

s(k,) =

rm

S(x)e-jkx”dx = F, { S ( x ) } ,

(8)

where S(x) is the reaction of the system when stimulated by a delta function: S(x)= L{S(x)}.

(9)

The impulse response describes a system completely. Since an arbitrary input function can be represented by

338

MICHAEL. KAISER

as a linear combination of delta pulses, the reaction of an LSI system can be computed by convolving the input function with the system's impulse response or, in k,-space, by multiplying the Fourier-transformed input function by the transfer function: F ( O S ( x - Od5

G(x) = m: J

= F(x)

* S(x)

(1 1 4

OFT A subsequent inverse transform to the x-space yields the output signal G(x). For our purpose, the Fourier transform and its inverse have been defined slightly differently from the usual formulation (Papoulis, 1962), for the onedimensional case by

f(k,) =

Spyrn

F(x)e+jk-xci, = F, { F ( x ) )

(1 2 4

OFT

F(x) = 2n

Sm -m

f ( k , ) e - j k x xdk,

= F;'

{f(k,)},

(12b)

= F,

{ F(x, y ) }

(13 4

and for the two-dimensional case by

f(k,, k,,) =

F ( x , y)e+j(kxx+k yY) d x d y

OFT

If applied to radiation problems, x, y in Eq. (13a) are transverse coordinates (Fig. 1b), and k,, k, are the transverse components of the propagation vector; in Eq. ( 1 3b), k,, k, are the variables after performing the Fourier transform, i.e., spatial frequency domain variables. The sign of the transformation kernel has been chosen to represent the radiated field as a superposition of plane waves a(&,,k,,)exp( -jk r) propagating into the half-space z > 0 when a time dependence like exp( + j o t ) is assumed. The quantities transformed to the spatial frequency domain are denoted by lower-case function symbols, whereas quantities in the spatial domain are denoted by upper-case function symbols. The Hankel transform is an appropriate tool in the treatment of circularly symmetrical problems (Papoulis, 1968). To distinguish it from the Fourier transform f(k,, k,,), the Hankel transform f ( k , ) of the circular symmetric

.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

339

function F ( p ) is denoted by an overbar: f(k,) =

1;

F(P)Jo(k,P)P &J

fHT f m

with k , = , / W

and p = J m ,

(15aM

where Jo denotes the Bessel function of order zero. The Fourier transform of the circular symmetric function F ( p ) is obtained by multiplying its Hankel transform by 2n: Further theorems of Fourier and Hankel transform used in this contribution are given in the appendix (V1.A). B. Scalar Formulation of the Radiation Problem As noticed in the previous section, it is possible to treat the problem of wave propagation between two parallel planes by a system-theoretic approach. In this section a scalar formulation of radiation problems will be given. A scalar treatment is need not be limited to acoustic problems; it is also a useful approximation in optics where aparture dimensions are usually much larger than the wavelength (Born and Wolf, 1975). A scalar treatment is also very accurate in the design and analysis of large reflector antennas (Silver, 1949).An extension of the technique to electromagnetic problems will be given in the next section. The starting point of the following considerations is the linear shiftinvariant system of two parallel, infinitely extended planes depicted in Fig. 2. Here the plane z = 0 represents the input plane with the impressed sources, while a parallel plane at a fixed out arbitrary distance z from the input plane represents the output plane. The radiated field in the region z > 0 must solve the Helmholtz’ equation Eq. (3) and the Sommerfeld radiation condition, and it must take the values of the field distribution U(x’,y’,O) given in the plane z = 0. Such a solution was given by Rayleigh (1877, 1878) and more recently by Sommerfeld (1964):

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MICHAEL KAISER

FIG.2. System of two parallel planes in homogeneous space.

a

e-jk"(*-x')z+(y-y')Z+zz

--

az 2nJ(x

- x ' ) 2 + ( y - y')2 + z2 x'=y'=O

-_

2x x 2

+y2

+z

xz

+ y2 + z2

= S(X, Y , Z )

(19)

represents the impulse response of the system of two parallel planes in homogeneous space separeted by an arbitrary distance z = const. So the field in the plane z # 0 (output), which is generated by the field impressed in the plane z = 0 (input),can be computed by a two-dimensional convolution of the input distribution with the impulse response of the system:

w, Y, z) = w,y, 0) ** w, Y, 4.

(20)

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

341

As in the system theory of electric networks, where the response of a filter to an input function can be computed either in the time domain or frequency domain, the evaluation of the radiated field is not only possible in the spatial domain, but also in the spatial frequency domain. For that purpose the Helmholtz’ equation Eq. ( 3 ) will be Fourier-transformed according to Eq. (1 3a) with respect to x and y :

Taking into account the given source distribution u(k,, k,; 0) in the plane z = 0, the solution of Eq. (21b) for z > 0 is given by u(k,, k,; z ) = u(k,, k,;

O)e-jkz”

with k,

= J k Z - k: - k ; .

Here the positive real root is chosen when k , is real, and the root with a negative imaginary part is chosen when k, is imaginary so as to satisfy Sommerfeld’s radiation condition. In eq (22), e

-j.fk2-k2

-kZz r

y

= s(k,,k,; Z)

(24)

is the transfer function for a system of two parallel planes in homogeneous space. Thus, in the spatial frequency domain multiplying the Fouriertransformed input field by the transfer function in accordance with Eq. (22) is equivalent to the field computation by convolving the input distribution with the impulse response in the spatial domain: V(x7 Y , 4 =

Sw,

U(X’,y’, O)S(X - x’, y - y’, Z) dx’ dy‘

(254

S I m

=

Y , 0) ** S(X, Y , 4

OFT u(k,, k,; z ) = 4 k , , k,; O)s(k,, k,;

4.

(25W

In order to find the relation between impulse response and transfer function, it is advantageous to make use of the Hankel transform pair (26)

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MICHAEL KAISER

It can be shown (cf. Appendix V1.A) that the relation

exists, according to which the transfer function is the Fourier transform of the impulse response and vice-versa, corresponding to the one-dimensional case in Eqs. (1 1a,b). At the end of this section, two commonly used approximations in diffraction theory shall be introduced, the Fresnel and the Fraunhofer approximation. The first one, which is valid if the following assumptions are satisfied: distances z larger than several wavelengths, small transverse wave numbers kx and k,, small wavelength I, can be obtained by expanding the root in the exponent of the exact transfer function of Eq. (24) in a binominal series, neglecting terms of order hi her than the quadratic, and a subsequent low-pass filtering by c i r c ( J d / k ) (Blume, 1976):

With the ideal low-pass function in the spatial frequency domain

we associate a b-function approximation in the spatial domain (Bracewell, 1965), where J1 denotes a Bessel function of order one. The approximating function approaches a &function the smaller the wavelength I becomes, i.e., the larger the radius k = 2 4 1 of the ideal low-pass function. The consequence for the approximated impulse response Eq. (28b) is that the convolution of the first term with a second term that is similar to a &function for all practical purposes leaves only the first term as long as the wavelength 1 is small compared with all other dimensions.

FOCUSING CRITERION BY A SYSTEM-THEORETICAPPROACH

343

The field UFres(X3Y , 2) = SFres(X,Y , z,

** u(x,Y , O)

(30)

resulting from a convolution of the input distribution with the approximated impulse response (28b) is called the Fresnel diffraction field. However, one usually obtains the above result by quite a different approach, i.e., by solving an inhomogeneous Helmholtz’ equation (in the spatial domain) using a Green’s function of the unbounded space (Kirchhof’s formula) or the halfspace (Sommerfeld, 1964; Goodman, 1968). If the field at greater distances z from the input plane is to be evaluated, one enters the Fraunhofer zone. In order to obtain the Fraunhofer field, one replaces the exact impulse response in Eq. (25a) by the Fresnel approximation of Eq. (28b) and takes the constant factors out of the integral:

Furthermore, if the relation k

z >> - ( X I 2 2

+ y’2),ar

is valid, i.e., the distance z is much larger than, essentially, (aperture dimensions)2/wavelength, the phase factor in the integrand can be replaced by 1. The result

(33) is usually called the Fraunhofer jield. It is obvious that the Fraunhofer field is directly proportional to an integral of the two-dimensional Fourier type , Thus the Fraunhofer field is directly over the aperture distribution U ( x ’ , y ’ 0). proportional to the spatial frequency spectrum u(kx,k,; 0) of the input distribution with the spatial frequencies X

kx = k -Z,

Y and k, = k -Z.

(34)

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MICHAEL KAISER

C. The Electromagnetic Radiation Problem

The vector character of electromagnetic fields has to be taken into account if the linear dimensions of the radiating aperture are comparable to the wavelength or if polarization effects are to be studied, as, for example, in crosspolarization discrimination. Therefore, the propagation of electromagnetic waves in a system of two parallel planes in homogeneous space (Fig. 3), taking into account the vector character, will be studied below by a system-theoretic approach. In the input plane z = 0, the tangential electric field strength E,(x’,y‘,O)= E, shall be impressed. In order to compute the radiated field in a plane parallel to and at an arbitrary distance from the plane z = 0, the impulse response and transfer function of the system of two parallel planes in homogeneous space-now in the case of electromagnetic wave propagation-will be evaluated. The starting point of the following investigations will be Huygens’ principle in its vector form. In the case of plane source distributions as given here, the electromagnetic field E(r), H(r) may be computed from the tangential components Et(r’)or Ht(r’)in the aperture S (Severin, 1951):

ss

E(r) = -rot 271

e-jklr-r‘l

[n’ x E(r’)]

~

1 1 H(r) = -rot rot 271 J’wopu, ~

Ir - r’l

lj[n’

ds‘,

(35)

e-jklr-r‘l

x E(r‘)]

~

Ir

- r’l

ds‘,

or e - j k l r -r’1 1 1 E(r) = - -rot rot sJ[d x H(r‘)] ds’, 271 j m O E , Ir - r‘1 ~

H(r) = -rot 271

ss

(36)

e - j k l r -r’I

[n‘ x H(r’)]

~

Ir

-

r’l

ds’.

As in the scalar case, the radiated field will take the values assumed in the aperture if the field point is moved into the aperture. Applied to our system of parallel planes (Fig. 3), Eq. (35) yields a e- j k l r - 1’1 EJr) = -EX(r’)% (374 271 Ir - r’I ds‘ E,(r) = -27L

ss ss

a

e - j k l r -r‘1

EY(r’)%

Ir

- r’l

a

+ E7(r’)-ay

ds’ e-jklr-r‘l

Ir - r’]

]

ds‘

(37c)

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

345

FIG.3. System of two parallel planes in homogeneous space for the propagation of electromagnetic waves.

for the electric field. The integrals in Eqs. (37a-c) are of the convolution integral type Eq. (lla). Hence, from a system-theoretic point of view the components of the radiated field (“output field”) originate from a convolution of the aperture field components (“input field”) with impulse responses r i k ( X , y, z) as in Eq. (20): a e -jk & rxx = - EJX, y , z) = E J X , y, 0) ** rx,(x, y , z), az 271 Jx’ + y’ + z’ ’ (384

346

MICHAEL KAISER

The scalar impulse responses

rik(r) can be combined to form the tensor

ie., r:;i ij 0

rX,(r)

r(r)=

(39)

which represents the impulse response tensor r(r)of the system of two parallel planes (aperture plane/field point plane). Using the impulse response tensor, the electromagnetic field in the output plane Z) **

w,Y, 0)

(40) can be computed as a two-dimensional convolution of the impulse response with the aperture field. The equivalent circuit of the system aperture/field point plane is depicted in Fig. 4; to an input E , the system reacts with E,. As in the case of scalar waves, the computation of the radiated electromagnetic field can be carried out in the spatial frequency domain. For that purpose, the Helmholtz equation for the electric field ~ ( xY, ,Z) = r(x, Y,

AE(x, y, Z)

a2

-e(k,,

aZ

k,; z ) + ( k 2 - k:

+ k 2 E ( x ,y ,

- k;)e(k,,

Z)

=0

0

(4 1a)

FT

k,; z ) = 0

(41W

is Fourier-transformed with respect to x and y (Collin and Zucker, 1969). The solution of this ordinary differential equation is, as in Eq. (22),a plane wave

e(k,, k,; z) = e(k,, k,;

O)e-jkZ"

(42)

with the spectral coefficient e(k,, k,; 0) = e,. The sign of

k,

= J k 2 - kf - k;

is determined according to the rules set up in Section ILB for scalar waves. By means of an inverse two-dimensional Fourier transform of Eq. (42), one

FIG.4. Equivalent circuit of the aperture plane/field point plane system,

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

347

obtains the electric field in the half-space z > 0 E(x, y, z ) = F;'{e(k,, k,; O)e-jkz')

=$JJ:a

(43)

e(k k,; O)e-j(kxx+kyy+kz~)dk dk X

X'

Y

as a superposition of plane waves with propagation vectors k

= u,k,

+ u,k, + u,k,

=K

+ u,k,.

(44) This form had been introduced by WeyI (1919) to treat the propagation of electromagnetic waves along a boundary between two media with different electromagnetic properties. The spectral function e, can be computed from the tangential field in the plane z = 0; one obtains

eor =m:JJ

E,(x', y', O)e+j(kxx'+kyy') dx' dy' = F2{Eo(x',y', 0)}

(45)

for the transverse component, while the normal component is given by

Substituting Eqs. (45) and (46) in Eq. (43), one obtains the components of the radiated field and their Fourier transforms, respectively: E,(x,y,z)

= F;'{e,(k,,

0

k,; O)e-jkz"}= F;'{F2{E,(x',y',0)}e-~k*'}

FT ex(k,, k,; z) = e,(k,, k,; O)y,,(k,, k,;

u

E , ( x , y , z ) = F;'{e,(k,,

4,

k,; O)e-jkz'} = F;1{F2{Ey(~',y',0)}e-jkzz}

FT ey(kx,ky;z ) = ey(kx,k,; O)r,y(k,,k,;

4,

(47a) (47b) (48a) (48b)

348

MICHAEL KAISER

Equations (47b)-(49b) may be combined to yield the Fourier-transformed electric field

-

e(k,, k,; z ) = ~ ( k ,k,; , 4 e(k,, k,; 0) in the output plane z as a product of the transfer function tensor

lo

Yx,(k, k, ; 4 7

y ( k , k,; 4 =

0

(50)

“i

?,,(k, k, ;4 Y Z , ( L k,; z) Yzy(kxrk,; 4 0 and the Fourier-transformed field vector in the input plane z = 0. If one considers the transform (27), it becomes obvious that Eqs. (40) and (50) are linked by a two-dimensional Fourier transform. So the transfer function tensor y(k,,k,; z ) in Eq. (51) is the Fourier transform of the impulse response tensor r (x,y , z ) in Eq. (39): 9

r ( X , y , z )= F I ~ { Y ( ~z)}. ,,~,;

(52)

This result corresponds to the scalar case, where the transfer function is also obtained by a Fourier transform of the impulse response. BY PLANERADIATORS 111. FOCUSING

A. Illumination of the Injinitely Extended Radiator, Scalar Treatment

Now we apply the system-theoretic approach to scalar radiation problems in order to derive a focusing illumination. Consider the linear shiftinvariant (“isoplanatic”) system of two parallel planes shown in Fig. 2, the plane z = 0 being the input plane and the plane z = zf the output plane containing the focus. In the plane z = 0, a field distribution U(x’,y’,O) shall be derived that will produce a spatially limited field U ( x ,y , zf)in the output plane having its maximum amplitude (focus) at the point ( x f , y , , z f )A. singular field distribution of the type 6 ( x - x f , y - y f ) in the output plane would fulfil the above requirement ideally; however, such a field includes infinite energy, as can be shown from Parseval’s formula (Papoulis, 1968), and, consequently, the input field producing this field distribution would also have infinite energy. Therefore, the input illumination maximizing the relation

between field intensity at the focus and available input power shall be derived

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

349

(Kaiser and Hetsch, 1984).Using Eqs. (25a,b), the value of U at the focus can be determined by multiplying the Fourier transform of the input field distribution u(k,, k,; 0) by the transfer function s(k,, k,; z ) of the input plane/focal plane system and performing a subsequent inverse transform in the spatial domain:

(54)

Making use of Parseval’s formula, the integral in the denominator of Eq. (53) can be expressed in the spatial frequency domain:

Squaring both sides of Eq. (54), one obtains Schwartz’ inequality

x

$JjIals(kx, k,;

z,-)e-j(kxx+kyy)

I2dkdk,,

(56)

and thus the upper limit for the intensity at the focus. The first integral on the right side is identical to the one in Eq. ( 5 5 ) , so that Eq. (56) can be written as

The value of the integral is determined by the transfer function s(k,, k,; z,-), i.e., by the geometry of the input plane/focal plane system. The upper limit for the intensity at the focus is achieved if uo = u(k,, k,; 0) = ii,[s(k,,

k,; ~ , - ) e - j ( ~ ~ ” J + ~ y ~ J ) ] *

(58)

is chosen (cf. Appendix VI.B), where ii, is the amplitude of the input illumination. Transforming Eq. (58) to the spatial domian yields ~op,(x’,Y’,o) = iioS*C-(x‘

- X f ) , - ( Y ’ - YJ), Z f l ,

(59)

which is the optimum focusing illumination, proportional to the complex conjugate of the inverted impulse response shifted by (x,-, y,-). This result is true for any linear shift invariant system, i.e., where there are no restrictions on the medium between the input plane and the focal plane besides linearity and independence of the transverse coordinates.

350

MICHAEL KAISER

Finally it should be mentioned that the derivation of the optimum focusing illumination sketched in this section is similar to the derivation of the matched filter principle in communications applications. Here it is the intensity at the focus relative to the input power which is to be maximized, whereas in communications applications we wish to maximize the signal-to-noise ratio of a signal disturbed by white noise at the time of sampling. B. The Optimum Focusing Illumination and Its Radiated Field

In this section, the case of a homogeneous lossless medium between an input plane and a focal plane will be treated. Since a system of two infinitely extended planes in homogeneous space is circular symmetric with respect to the z-axis, it is advantageous to give the impulse response of the input plane/focal plane system in cylindrical coordinates. Equation (19) yields, with (x2 + y2) = p2:

Jw,

Making use of Eq. (16) with k, = the transfer function can be given as the Hankel transform of the impulse response multiplied by 2n:

From this one obtains-assuming, without any loss in generality, the focus to be located on the z-axis-the optimum focusing illumination according to Eq. (59) and its Fourier spectrum:

With the illumination given by Eq. (62a)the input power is proportional to

Because of the circular symmetry of the functions to be convolved, the field in the focal plane U(P,Zf) = UOP,(P,O)** S(PJf) = ~,S*(P,Zf)** S(P,Z,) (64) resulting from Eqs. (20) and (59) with xf = y, = 0 is circular symmetric too.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

35 1

Since it is not possible to evaluate the convolution integral in cylindrical coordinates (Gaskill, 1978)

analytically, the field will be computed in the spatial frequency domain. The two-dimensional Fourier transform of Eq. (64), 4k,;

Zf) = Uop&

O)s(k,; Zf) = %14kp; Zf)I2,

(66)

shows that the spectrum of the field distribution in the focal plane is equal to the magnitude of the transfer function squared and multiplied by ii,. Using Eq. (61), one obtains:

The evanescent waves ( k , > k ) can be neglected for sufficiently large values of z f / l (cf. Fig. 5). With this “low-pass’’ approximation, the field in the focal plane becomes

In this approximate solution, only waves with transverse wave numbers k , k all having the same amplitude iio contribute to the field in the focal plane. The inverse transform of Eq. (68) to the spatial domain, which is possible analytically, yields the field distribution

-=

250

t

1.o

a‘dB

/

z, = 5d

1.5

-

2.0 k,,/k

FIG.5. Attenuation a of the evanescent waves ( k , > k).

352

MICHAEL KAISER

in the focal plane. The optimum focusing illumination in the input plane according to Eq. (62a) and the resulting field distribution in the focal plane with and without taking the evanescent waves into account are shown in Fig. 6 for various input plane/focal plane distances. Only in the extreme near field (e.g., zf = 0.1 A) do the two field distributions differ from each other, whereas they differ only slightly for zf > ,Ias expected. Using Eq. (69) yields

-

k 27rp

U ( p = 0 , ~ =~ limu^,-J,(kp) ) p-0

=

k2 uo471 A

as an approximate value for the field strength at the focus. For 2-, >> A, the exact value resulting from Eq. (64) approaches this value for the given input illumination:

In Fig. 6a the width of the focus resulting from the approximate solution is essentially broader than the one resulting from the exact solution. According to the uncertainty principle, which is valid for the Fourier transform (Papoulis, 1968), using the band-limited spectrum always yields a larger value for the 3-dB radius than the complete spectrum, i.e., taking into account the evanescent waves. The 3-dB radius is the distance from the z-axis where the amplitude has dropped to l/& times its maximum value (cf. Fig. 6a). This quantity can be considered a measure of the “quality” of the focus. The approximation according to Eq. (69) yields 1.62 ,I A r3dB= -~ 32 -. 271 4 So the field in the focal plane is concentrated in an area with diameter of approximately half a wavelength. The axial resolution has been investigated 10 A by computing for a distance between input plane and focal plane of z f. = the field by means of Eqs. (18) and (62a). It is less distinct than along the transverse coordinates (cf. Figs. 7 and 6c).

C. The EfSect of Limiting the Optimum Focusing Illumination

The optimum focusing illumination derived in the previous section by matching it to the impulse response is of infinite extent, which, of course, cannot be realized. Therefore, the effect on the shape of the focus of limiting the optimum illumination in accordance with Eq. (62a) by an aperture of

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

353

t

1.C

1

lp5

0

0.5

0 0 0.5 1.o 1.5 2.0 PlA FIG.6. Optimum focusing illumination in the input plane and resulting field distribution in the focal plane for different input plane/focal plane distances (with and without taking the evanescent waves into account). (a) Input plane/focal plane distance zf = 0.1 1.(b) Input plane/focal plane distance 2, = 1.(c) Input plane/focal plane distance z, = 10 1.

3 54

MICHAEL KAISER

0

2

0

0.5

6

4

8

1.o

0.5

0

1.o

FIG. 6. (Continued)

1.5

2.0

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

w

0

10

30

20

40

P’ll

t 1 .o

0.5

0

0

0.5

1.o

FIG.6 . (Continued)

1.5

2.0

355

356

MICHAEL KAISER

1

1 -

fi

0

z, = 101

0

z

FIG.7. Field distribution along the z-axis for the optimum focusing illumination and a distance between input plane and focal plane of 2, = 10 1.

radius a will be examined. This yields the illumination (cf. Fig. 8) q,opt(P’,0) = U0*,1(P’,O)circ(p’/a)

(73)

with circ(p’/a) =

1,

0,

OIp‘la p’ > a

(74)

and UJp’, 0) according to Eq. (62a). Among all possible illuminations with U,(p’,O) non-zero only if p‘ < u and where the integral

takes the same value, this illumination produces the maximum field strength at the focus as shown in the Appendix (V1.C). The analytical determination of the field distribution in the focal plane U,,op,(p,2,)

= W S * ( p , zf)circ(Pl41

I

HT

** S(P9 Zf)

(764

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

357

1.0

0

---------_____ 0

a

P'

FIG.8. Finite focusing illumination in the input plane.

resulting from Eq. (64) is not possible either by evaluating the convolution integral or by a transformation to the spatial frequency domain. Therefore Eq. (76a) is evaluated numerically. The 3-dB radius of an aperture focused at a distance zf = 10 2 as a function of aperture radius relative to fixed focal distance is shown in Fig. 9. As expected, the 3-dB radius becomes broader as u/zf decreases. However, a significant deterioration does occur if the aperture radius approaches the focal distance or becomes smaller. With p = 0, the amplitude at the focus can be determined analytically from Eq. (76a):

z,= 101

optimum focu ing illuminati n ( a - 0 0 )

0.5 0

0

1

2

3

4

5

alzr

FIG.9. 3-dB radius as a function of the aperture radius

358

MICHAEL KAISER

As expected, for finite values of a the amplitude is smaller than the value from Eq. (71). As a + 03, it approaches this value.

D. Comparison with the Conventional Focusing Illumination The focusing criterion given in the literature (Wehner, 1949; Bickmore, 1957; Sherman, 1962) requires a phase distribution that compensates for the different distances between the aperture elements and the required location of the focus. The contributions of all elements then add up in-phase at the focus (cf. Fig. 10).Until now, conditions for the amplitude distribution have not been derived. The latter has been taken constant over the aperture. The assumed relative phase distribution $(p’) = kAR(p’) = k ( J m - zf)

(78) in the input plane causes a radiation of spherical waves converging to the focus. Rewriting Eq. (62a) yields

with =k

J m -a r c t a n ( k J m ) (80) and allows a comparison of the phase distribution according to Eq. (78) with the focusing criterion derived previously. In both cases the phase variation with p‘ is very similar for zf > 1.

A

I

\

\/I’ = const.

FIG.LO. Spherical waves converging to the focus.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

359

Now let us compare the field radiated from an aperture of radius a with an optimum illumination according to Eq. (73) to the field radiated from a 0) of the same radius. In order to conventional focusing illumination Ua,con(p', minimize the effect of limiting the illumination, which theoretically extends to infinity, the aperture radius a is chosen so that the amplitude for p' = a is only 10% the amplitude at the center of the aperture (p' = 0), ie., With zJ 2 1 this occurs at a = 3 zJ (cf. Eq. (79). The amplitude of the conventional focusing illumination is chosen so that the integral

takes the same value for both illuminations. Figure 11 shows the fields along the aperture axis and in the focal plane for the optimum and conventional illumination. The focal distance is one wavelength. With the optimum focusing illumination field strength is maximum at the focus, whereas the conventional illumination produces the maximum field at a distance shorter than the desired focal distance. Because of the term a r c t a n ( k J m ) in the equation of the optimum illumination, phase variation with p' is less rapid than with the conventional illumination. So the in-phase superposition of the contributions of all aperture elements occurs at a distance which is slightly larger than expected from the conventional method according to Fig. 10. Furthermore, the field at the focus radiated by the optimum focusing aperture illumination is higher than the one radiated by the conventional illumination. The radial extension of the focus radiated by the latter is smaller, whereas the axial extension is nearly equal in both cases. The side lobes in the focal plane are lower for the optimum illumination. Figure 12 shows a comparison of both illuminations for a focal distance z - 10 A. The maximum of the radiated field occurs at z = z, in both cases, as f: the influence of the term arctan(k,/-;) has decreased compared with the previous example. E . The Electromagnetic Case

Suppose we wish to consider the vector character of the electromagnetic field, for example, in determining the radiated field. According to Eq. (35) this can be done using either the tangential components of the electric field in the input plane or, according to Eq. (36),the tangential components of the magnetic field. Which of the two methods is best, or even which is the only one

360

MICHAEL KAISER

1.o

.\0

31

P

0.5

z, = r?

0

n-

Z

t

1.o 1.5 2.0 P/A FIG.1 I. Fields of optimum focusing aperture and conventional focusing illumination for an aperture radius a = 3 iwith an input plane/focal plane distance of 1 1.( 1 ) Along the z-axis. (2) In the focal plane. 0

0.5

applicable, depends on the technical realization of a focusing radiator. In this section focusing by an impressed electric field strength will be treated. The vector of the electric field strength in the input plane z = 0 shall be parallel to the x-direction (Fig. 3). As in the scalar problem, the input distribution that maximizes the radiated power density at the focus shall be determined. According to Eqs. (37a-c),and input field strength oriented along

361

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

1.o

t

0.5

0 2, = 102

0

t

\

0. 0

0

/ - - -

0.5

1.o

1.5

2.0

.

c

PIJ

FIG. 12. Fields of optimum focusing aperture and conventional focusing illumination for an aperture radius a = 30 I with an input plane/focal plane distance of 10 I . (1) Along the z-axis. (2) In the focal plane.

the x-axis produces the components Ex and E, of the radiated field. In order to maximize the x-component, the input field strength has to be matched to the component r,, of the impulse response tensor. With Eq. (59), one obtains where go is the amplitude of the illumination of the input plane and the focus,

362

MICHAEL KAISER

1l k l

z

FIG.13. Magnitude of Ex in the xz-plane (2, = 10 A).

Z

FIG. 14. Magnitude of E, in the xz-plane (z, = 10 A).

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

363

dB

0 -

10

- 20 -

30

- 40 - 50 - 60

0 FIG.

1 2 XI1 15. Relative magnitude of Ex and E, in the focal plane (zf = 10 A).

without any loss in generality, is assumed to be on the z-axis (xr = yf = 0). Since the element r,., of the impulse response tensor in Eq. (38a) is of the same form as the scalar impulse response in Eq. (19), the input field distribution has the same shape as the optimum focusing scalar illumination (Eq. (62a)). The field components Ex and E, are computed by means of Eq. (83), applying Eqs. (47a,b)and (49a,b). Figure 13 depicts [ E x for [ a focal distance zr = 101and the circular symmetric input illumination given in Fig. 6; like the aperture illumination, IE,I is circular-symmetric with respect to the z-axis. As expected, one obtains a field strength that greatly exceeds the one generated by a homogeneously illuminated aperture. The result for the z-component is different. Because of symmetry properties, it vanishes along the z-axis for the chosen input illumination. The magnitude of E, in the xz-plane is depicted in Fig. 14, where the maximum is about 50 dB below IExI at the focus (Fig. 15). IV. FOCUSING IN STRATIFIED MEDIA A . Propagation of Plane Waves A structure composed of planar dielectric layers with different permittivities and conductivities (Fig. 16) may be a good model (Guy, 1971) for the treatment of wave propagation in stratified media (e.g. biological tissue, geophysical structures), as long as the wavelength is short by comparison with

364

MICHAEL KAISER

'///// FIG. 16. Structure composed of planar layers (r, is the reflection coefficient at the (n + l)th boundary).

the radii of curvature of the layers. In this chapter the field produced by an electric or magnetic field strength impressed in the input plane z = 0 in such a layered structure will be determined via the spatial frequency domain. For this purpose the transfer function of the input plane/output plane system within the layered structure must be determined. According to Chapter 11, a field computation in the spatial frequency domain is equivalent to a field representation by a spectrum of propagating and evanescent plane waves. The solution of the problem of reflection and refraction of plane waves at parallel boundaries can be found in several standard textbooks (e.g. Wait, 1970; Kong, 1975).Here, however, special emphasis is placed on a representation suited to the given task of field computation via the spatial frequency domain. According to Fig. 16, the structure shall fill the half space z > 0. The n-th layer consists of a (in general) lossy dielectric with complex permittivity E,, and thickness d,,. All layers shall be non-magnetic (ppn= 1). The field in the n-th layer can be expressed as the superposition of an incident and a reflected wave according to

the propagation vectors of which can be described as

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

365

and

since the components u,k,

+ u,kY = K,

K

=

J

~

parallel to the stratification are equal at adjacent sides of the boundaries due to the boundary conditions for the electromagnetic field. With k:r. kir

=

k:

= kie,,

(89)

for the longitudinal part of the propagation vectors in the n-th layer. The positive real root is chosen when k>; is real, and the root with a negative imaginary part when k:; is imaginary so as to satisfy the radiation condition. With the unit vectors u;, = u;, = Ul

=

u, x K K ' ~

and

introduced in Fig. 17, the field according to Eq. (84) can be divided into components parallel and perpendicular to the plane of incidence, respectively:

This representation is valid not only for plane homogeneous waves ( K < k,), but also for evanescent waves ( K k,,). Then uflnand u i l nare complex even for k, real, and thus d o not have the geometrically intuitive meaning of direction vectors. However, for a field representation in the form of a superposition of plane waves as presented in the next section, Eqs. (91)-(93) can still be used (Kerns, 1981). The reflections at the boundaries between two layers can be described starting from the reflection coefficients of two adjacent semi-infinite media with appropriate material properties. For parallel polarization, the reflection coefficient at the boundary to the (n + 1)th layer (cf. Fig. 16) is given by (Kong,

=-

366

MICHAEL KAISER

and for perpendicular polarization by

Starting from the last layer n = N, the reflection coefficientsIland l, rln at the boundaries of the layered structure are obtained by the ~ecursionformula

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

367

with

w, = e-J' 2 d n & 5 ?

(98) The amplitudes A\,, and A:, in Eq. (94)can be obtained from the amplitudes Ail, and A i 0 of the incident wave known at the first boundary z = 0 utilizing the relation

with the recursion formulae

and

following from the boundary conditions. In analogy to Eq. (99), the terms

+1 k,(,+l)k,

1 + rl/n

e-jk.mzn

kznkn

e-jk~(.+r)zn

1+

II(n+ 1 )

e-jkr(n+i)dn+i

and e-jksnzn e-ikz(n+

112,

1

1 + rLn + T l ( n + l F -jk=c,

+ l ) d n+ I

represent the transmission coefficientsbetween the n-th and the (n + 1)th layer at the boundary z = z,. Thus Eq. (94)for the field in the n-th layer can also be written in the form E, = j?JAi ne-jkk.r + e-j2krnzne-Jk; .r) I1

+

( II

Ai ( e - j k k . r I I n

n Iln

e-j2k.,z,e-jkL.r

+ I n

(102)

)I.

B. The Transfer Function Tensor 1. Electric Field Impressed in the Input Plane

In the plane z = 0 of the stratified medium of Fig. 16, a tangential electric field strength E, is impressed. The field that radiates into the layered halfspace z > 0 shall be computed via the spatial frequency domain. Therefore, the transfer function y,(k,, k,; z) of the system input plane z = 0 and output plane z > 0 within an arbitrary layer of the structure of Fig. 16 has to be evaluated. In what follows, this will be termed the transfer function of the stratified medium.

368

MICHAEL KAISER

According to Eq. (43), the field within the n-th layer can be represented as a continuous spectrum of incident and reflected waves, as in Eq. (102) (Clemmow, 1966):

+

,i

I

I n

(K)(e-jkL.r

+

I n

e-j2kznz,e-jkL.r

11d k x dk, .

With the definition of the two-dimensional Fourier transform (13a,b), one obtains the spectral functions of the field in the first layer from the field Eo:

and

(107a)

369

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH I ' y y n ( k x j k y ; Z)

k: k2 ~ i , , ( K Z) ; + S S I I ~ ( Kz), ;

=z

( 107c)

(1 07d) and

Yzyn(kx,k,;

4

k -Z t n ( K ;4

=

( 107e)

kzn

are the elements of the transfer function tensor y,,(kx,k,; z) from the plane z = 0 to the n-th layer of the structure. The factors slln(K;z ) and s,,,(K; z ) are computed as follows:

n (1 +

n-1

SIl,Ln(R

4=

n n

r II , L i ) e - j 2 k z l d i

i= 1

+ r l l, L n e - j 2 k z n z n e + j k z n z

(e-jkznz

(1 + rll,lie-i2k.,di)e-jk."i"~

1.

L

i= 1

( 108)

for both polarizations, whereas n- 1

JJ (1 + r

tAK; 4 =

n

i)e-j2kz~d~

i= 1

n

(1 +

(e-jkznz

II i e - i 2 k z L d i ) e - j k z n z n -

-

rllne

-j2kznzne+jkz,z

). (109)

I

i= 1

In Eqs. (108) and (109), the term

n

(1 + r I I, l i e - J 2 k r i d i ) e - j k , " z ,

-

1

i=l

represents the transmission coefficient from the boundary z = 0 of the stratified medium to the boundary of the n-th layer. With the elements of the transfer function tensor, the field in the stratified medium can be computed usings Eqs. (50) or (40) derived in Section 1I.C. 2. Magnetic Field Impressed in the Input Plane

If one assumes that a tangential magnetic field strength H, is impressed in the plane z = 0 of the layered structure in Fig. 16, the magnetic field in the n-th layer can be represented as a superposition of plane waves in an

370

MICHAEL KAISER

analogous manner as the electric field in the preceding section. Applying Maxwell's equations to the electric field (103) in the n-th layer yields the associated magnetic field

+ bi,(K)[(k;

+ (k;

x

+ (k:

x uI)r I n e-i2kznzne-jk~"])dk, dk,,

Uiln)rllne-j2f~~z~e-ik:.r]

x u,)e-Jk;"

(111)

JX

where Z, = is the intrinsic impedance of the n-th layer. As in Section B.l, one finds for the spectral function within the first layer:

and

The spectral functions bf,,(K) and b\,(K) for the other layers are computed by making use of the recursion formulae (100) and (101). In analogy to Eqs. (107a-e), the transfer function tensor H

Y3kx,k,;z)

H

=p :; i: Yzxn

(114)

Yzyn

can be determined. The electric field in the spatial frequency domain en = y f ( k x ,

0

k y ; Z) *

FT En = I';(k,, k,;

Z)

(uz x ho)

(115a)

** (u,

(115b)

x H,)

is obtained by multiplying the transfer function tensor by the Fouriertransformed input illumination. The electric field (in the spatial domain) can be determined alternatively by an inverse two-dimensional Fourier transform or according to Eq. (115b) by convolving the impulse response tensor

with the vector u, x Ho.

FOCUSING CRITERION BY A SYSTEM-THEORETICAPPROACH

371

The elements of the impulse response tensor are: (117a)

and (117e) with n- 1

and "-1

(e-jkznz

-

-j2k.,zne+jkz.z rllne

1.

If the field radiated by a dipole antenna located in the input plane is to be determined,it is more advantageous to start from the dipole current. For that purpose, the relation U, x

H, = J,

( 120)

372

MICHAEL KAISER

is introduced, where Js denotes the surface current density in the input plane. The current distribution is obtained via the current element of an infinitely thin dipole (Borgnis and Papas, 1965): Js(xb, yb) = 1, dl 6 ( ~ ’ xb, y’ - yb).

(121)

Here I , denotes the dipole current and dl the length of the dipole. Thus the relations derived above are also applicable to the determination of the field of a distribution of discrete currents. C . Exa.mples

1. Perpendicular Incidence of a Plane Homogeneous Wave on a Structure of Two Parallel Plane Layers (Biological Tissue) As an introductory example, we will discuss perpendicular incidence of a plane homogeneous wave on a structure (Fig. 18) to illustrate field computations in a stratified medium via the spatial frequency domain. This example may serve as a model for biological tissue. Superimposed on muscle tissue, which is assumed to be of infinite extension in the positive z-direction, there is a layer of fat tissue, typically 2 cm thick (Guy, 1971). A skin layer above the fat layer can be neglected in the model (Guy, 1971), as it is only about 0.4 mm in thickness, which is small by comparison with the wavelength of the radiation used (& ‘v 45 mm at 915 MHz and & ‘v 18 mm at 2,450 MHz; Johnson and Guy, 1972). The assumption of infinite extension of muscle tissue can be justified by the fact that reflections at deeper boundaries can be neglected due to high losses in muscle tissue (tan 6 N 0,6). It is obvious that this boundary-value problem can also be solved in the classical manner by superposing incident and reflected waves in both layers,

L FIG. 18. Structure of two layers as a model for biological tissue.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

373

the amplitudes of which can be determined by observing the boundary conditions. This simple example, however, shall be used to demonstrate the applicability of system-theoretic methods to the solution of wave-propagation problems within stratified media. The incident wave is assumed to be linearly polarized in the x-direction, thus

E' = u x E o e - j k O z , The field in the plane z = 0 is given by Eo = uxEo(l

D

( 122)

+ ro)

(123a)

FT (123b) eo = uxEo(1 + r 0 ) W x ) W y ) 3 where ro is the reflection coefficient at the air/fat tissue boundary. The field within the layered structure can be determined by substituting Eq. (123b) in Eq. (50),where the transfer function tensor is given by Eqs. (107a-e) and by a subsequent transformation into the spatial domain. With a reflection coefficient rl at thefat/muscle tissue boundary, the elements of the transfer function are given by (cf. Eqs. (107a-e):) -jk.

Yxxl

= Yyyl =

I

+

- j k . id1 e j k z t z

1 + rle-J2kz~dt

( 124)

and (1 + r l ) e - j k z l d t e j k z z z Yxx2

= Y y y 2 = (1 + r l e - j 2 k z i d t ) e - j k z z z '

(125)

The remaining elements are zero on both regions. As expected, the electric field-like the incident wave-has only an x-component in both layers given by

and

314

MICHAEL KAISER

The field distribution in the fat/muscle tissue structure is shown for both frequencies in Fig. 19. One can see that in addition to the propagating wave for f = 2,450 MHz there also is a standing wave, which leads to a maximum of the field strength in the fat tissue producing undesired high-energy dissipation in the fat tissue. In addition, the field within the muscle tissue decays somewhat faster than at 915 MHz, which means that input power has to be increased proportionately to generate the same field strength at the same depth. This results in a further increase of energy dissipation within the fat tissue. 2. Focusing Illumination on a Biological Tissue In order to focus electromagnetic radiation, it is necessary to match the aperture illumination to the impulse response of the input plane/focal plane system, as discussed in Chapter 111. The layered structure depicted in Fig. 19 with the material parameters given in the tissue shall again serve as a model of body tissue. The “target area” (focal plane) of the microwave radiation shall be at an overall depth z = zr = 4 cm. In the plane z = 0, the focusing illumina-

/

/

I

I

I I

1-

/

I

I

\i;

2450 MHz

1.0

\

L915MHz

0.5

0 0 2 4 6 z/cm FIG.19. Field within biological tissue (material parameters taken from Johnson and Guy, 1972) at 915 (2,450)MHz for the perpendicular incidence of a plane homogeneous wave.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

375

tion, by Eq. (83), given as

(128b) e(k,,k,; 0)= u,~Oy:x;z(L k,; Zf) shall be impressed. The field radiated into the tissue is obtained either by Eq. (40), by convolving the input illumination with the impulse response tensor given by Eqs. (107a-e) and (52), or in the spatial frequency domain by Eq. (50) by multiplying the Fourier-transformed input illumination by the transfer function tensor given by Eqs. (107a-e). The transfer function and, thus, the focusing input illumination in the spatial frequency domain as well can be given by analytical expressions, whereas numerical solutions are required for the corresponding quantities in the spatial domain. For that reason, the radiated field has been first computed in the spatial frequency domain according to Eq. (50). Then it has been tranformed to the spatial domain using an FFT algorithm, where the difficulties arising from the numerical evaluation of Fourier integrals have to be taken into account (Bergland, 1969; Howard, 1975). The focusing input illumination and its spectrum are shown in Fig. 20 as described by Eqs. (128a,b). The resulting field in the focal plane z = z, is depicted in Fig. 21, the field along the z-axis in Fig. 22. The y- and zcomponents of the radiated field are the same as in the case of focusing in a homogeneous lossless medium again much smaller than the x-component. Contrary to expectations and unlike the case of a homogeneous lossless

f = 915 MHz

(4

(b)

Fic. 20. Focusing illumination for biological tissue according to Fig. 19 (focus in a depth of 4 cm). (a) Spatial domain. (b) Spatial frequency domain.

376

MICHAEL KAISER X

0

dB

t

5

10

15

20

crn

25

X

f = 915 MHz

5 10 15 20 crn 25 Y FIG.21. Field in the focal plane for the illumination given in Fig. 20. (a) Along the x-axis. (b) Along the y-axis. 0

medium, there is no local field maximum with respect to the axial direction at the desired focus. This is due to several effects. As a comparison of Fig. 11 and Fig. 12 reveals, the local maximum at the focus becomes less prominent as the focus approaches the aperture due to the diminishing influence of edge zones of the aperture, and in the given case the focal distance is in the order of half a wavelength in the medium. Furthermore, the high attenuation of biological tissue causes rapid decay of the field in the axial direction. Thus, the spatial

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

377

0 -8

-16 -24

-32

.

-40

I

0

4

_I

I

!

2

4

6

a

cm

10

c z

FIG.22. Field along the z-axis for the illumination given in Fig. 20.

resolution in the radial direction, which can also be observed for a homogeneous illumination, can be attributed primarily to the finite extension of the radiating aperture. Of course, the effect is amplified by in-phase superposition of the contributions of the aperture elements due to the focusing criterion derived in Chapter 111. In general, one cannot speak of significant focusing effect within biological tissue, though for the example we have chosen the field strength at the focus is about 6 dB higher than that generated by a comparable homogeneous illumination. For comparison, the field generated in the layered structure by a circular homogeneous illumination of radius a in the input plane, E, = uxiocirc($),

has been included in Fig. 23. The amplitude of the homogeneous illumination is, by design, equal to the amplitude of the focusing illumination; because of the choice of the radius a equal power is radiated from both illuminations. As expected, the focusing illumination generates a higher field strength at the focus than the homogeneous illumination (Fig. 23a); in addition, spatial resolution is better. In the axial direction, the field decay of the focusing illumination occurs more slowly than that of the homogeneous illumination. 3. Nondestructive Testing A further possible application of focused electromagnetic waves lies in non-destructive testing of nonconducting materials for flaws and other

378

MICHAEL KAISER

- - --

focusing illumination (4 = 4 cm)

. . ,

-10 '. -20

--

homogeneous illumination

__c

O

a

X

-30 *

-40.. .

..

-50 .-

f = 915 MHz

c

5

0

10

dB

f I €,(On 0, z)/% I

O

L

20

15

cm

25

X

f = 915 MHz

focusing illurnination homogeneous iIIumi nation

.'.. .. 0

2

4

cm

6

Z

FIG.23. Field for the focusing illuminatian given in Fig. 20 and for a homogeneous illumination.(a) In the focal plane along the x-axis. (b) Along the z-axis.

inhomogeneities that cause a regional change of electric properties. If the object to be investigated is illuminated by an electromagnetic wave, the energy will be partly scattered back (Holler, 1983).A receiving antenna will deliver a particularly strong output signal if focused at the location of the scatterer. If a criterion for focusing at the scatterer is known, information as to its location can be derived from this criterion. By comparison with reflectivity measurements using non-focused antennas, in this method the signal re-radiated from the inhomogeneity is more strongly weighted than are contributions from other points of the object to be investigated. The result in an increase in sensitivity.

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

379

To better understand the underlying principles, the two-dimensional model of Fig. 24 shall be considered in the subsequent mathematical treatment. In this case the vector problem reduces to a scalar one. The principal results may be transferred to the three-dimensional case. We start with a line inhomogeneity at x = xo, z = zo in a lossless dielectric plate which is assumed to have an infinite extension in the x- and y-directions. The radiation emanating from the electric line source S is linearly polarized in the y-direction. It induces a secondary line source iio6(x - xo) at the location of the disturbance, radiation from which will also be linearly polarized in the ydirection. The radiation generated by this secondary line source produces the field UAX, 0) =

r,(x

-

( 130)

x0,z 0 )

in the plane of the receiving antenna z = 0, where T,,(x,z,) is obtained by performing an inverse (one-dimensional) Fourier transform of yyy(kx;z ) by Eq. (107c). This signal will be received optimally by an antenna focused optimally at the location of the inhomogeneity (xo,zo), which, by Eq. (59), must be illuminated by U,,(X’,O)

(131)

= Gor,*,c-(x’ - xo),zol,

as it is a transmitting antenna. Because of certain reciprocity relations satisfied by any electromagnetic field, a receiving antenna focused at this location will weight the radiation at that point in accordance with Eq. (131). To determine the position of the scatterer (inhomogeneity), the receiving antenna has to be focused successively to any line x, z of the object to be investigated; the received signal will take its maximum value if the antenna is focused at the location of the inhomogeneity. For the procedure described in this contribution, focusing is performed “synthetically”. To that end, the scattered signal Us that has been recorded by the antenna R along the x-axis and stored in a computer must be weighted by the illumination U,,, corresponding to a

Er3

=1

FIG.24. Dielectric plate with an inhomogeneity acting as a scatterer.

380

MICHAEL KAISER

receiving antenna successively focused to all lines within the object to be investigated. Mathematically, the scattered signal Usin the plane z = 0 has to be correlated with the “illumination” of the receiving antenna. This correlation has to be performed successively for all depths z within the thickness of the plate (for practical purposes, in appropriately chosen steps A z ) in order to obtain information on the depth zo of the inhomogeneity: @(x,Z ) = Us(x,0) Uopi(x,Z ) =

s_a &(t,

O)Kpt(t

r(t - x o ) T * ( x - l , z ) d < ,

- X , 2)d t

d, < z < d ,

+ d,.

(132)

The correlation function @ ( x ,z ) will reach its maximum if the receiving antenna has been focused according to Eq. (131) to the line x = x o at a depth z = z o . Then @ ( x , z ) is an autocorrelation function that reaches its maximum value for x = x o . After a Fourier transform, the correlation integral (132) can also be evaluated in the spatial frequency domain. For correlation and convolution, the relation @ ( x , z )= U,(x,O) U o p { ( - x , z )= US(x,O)* iior;y(x,z)

q ( k , ; z ) = u,(k,; 0 ) COY*(-k,;

Z)

(133a) (133b)

is valid (Luke, 1975), which turns the process of computing an integral to a multiplication in the spatial frequency domain. An inverse transform to the spatial domain yields the correlation function. Evaluating the correlation function in the spatial frequency domain has a further advantage, in that it is no longer necessary to compute the impulse response of the structure to be investigated by Fourier transforming the transfer function. Below, we will study PVC-plates with thickness of d = 150 mm for inhomogeneities, e.g. flaws, by the method described above as simulated by computer. The radiation scattered by the flaws is simulated by line sources at the assumed locations of the disturbances. In order to test the transverse resolution, it is assumed that there are two line sources located side by side at a depth of 85 mm and separated by a distance D. The emitted radiation is computed for the plane z = 0, Fourier-transformed and multiplied by the spectrum of the optimum illumination at the depth d = 85 mm in accordance with Eq. (133b). The signals obtained by inverse Fourier transforms for two distances D are depicted in Fig. 25. One can see that the two line sources can be resolved even at a distance of one and a half wavelengths (A, = 11.6 mm). With a separation of five wavelengths, the result is even more distinct (Fig. 25b).

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

381

5 10 15 20 25 cm 30 X 0 FIG.25. Simulated imaging of two flaws side by side at a depth of 85 mm for two different separations. (a) Separation of line sources D = 1.5 I , = 17.4 mm. (b) Separation of line sources D = 5 & = 58 mm.

(W

In a second simulation, it is supposed that there are two line sources below each other, separated by 80 mm. The signal in the plane z = 0 generated by the two line sources is correlated with the optimum illumination computed for various depths z. The correlation function O(x,,, z) for the different values of z obtained at the coordinate xo of the flaw is depicted in Fig. 26. One can see that

382

0

MICHAEL KAISER

5

10 15 crn 20 FIG.26. Simulated imaging of flaws below each other.

z

the method’s resolution capability in the vertical direction is considerably worse than in the horizontal direction. Maxima of the function O(xo,z) do not occur at the true locations of the line sources, the separation does not agree with that of the line sources, and, furthermore, the sources are indicated by very broad maxima. In addition, the performance of the method is degraded by reflections at the boundaries between the media, which results in the superposition of a standing wave. To obtain a better resolution in the axial direction, a combination of focusing techniques with time-delay measurements seems to be a promising approach.

V. CONCLUSIONS In this contribution, the applicability of system-theoretic methods to the treatment of wave propagation problems has been demonstrated. Starting from methods known from the literature for the computation of the field radiated from a plane source distribution, a system-theoretic description-at first in scalar, but then also in vector form-has been presented. For the scalar treatment, the input plane/output plane system can be characterized by its impulse response and by its transfer function in the spatial domain and the

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

383

spatial frequency domain, respectively. In the electromagnetic case, the impulse response and the transfer function are tensors. The spatial domain and the spatial frequency domain are interrelated by the two-dimensional Fourier transform. Using this system-theoretic representation, a focusing source distribution is given by matching the input illumination to the impulse response of the input plane/focal plane system. The illumination found in this way is optimum in the sense that it produces the maximum power density possible with the available input power. This criterion has been used for focusing electromagnetic radiation in homogeneous space as well as in a stratified structure. To compute the field in such a structure and determine the optimum focusing input illumination, the associated transfer function and impulse response are determined. The impulse response is obtained by an inverse twodimensional Fourier transform of a transfer function that describes the propagation of plane waves from the input plane to the focal plane. We discussed the perpendicular incidence of a plane homogeneous wave on a structure that serves as a model for biological tissue as an initial example to illustrate the application of system-theoretic methods to field-theoretic problems. Furthermore, the optimum focusing illumination for this structure and the resulting field distribution are determined. As a third example, the focusing criterion derived initially is applied to non-destructive testing of dielectric plates. These examples show that focusing in the transverse direction can be achieved very well, whereas resolution in the axial direction is less prominent. However, if compared to the field of a homogeneous illumination radiating the same power, the focusing illumination generates a field at the focus which is about 6 dB higher. If the focusingcriterion given in this contribution is applied to non-destructive testing, good agreement with the true locations of the disturbances is only found if they are located side-by-side. Further investigations are required if we wish to improve the focusing effect in the axial direction.

VI. APPENDIX A . Theorems of Fourier atad Hankel Transform Used in This Contribution 1. List of Theorems

From the two-dimensional Fourier transform and its inverse defined by Eqs. (13a,b), the following theorems used in this contribution can be derived.

384

MICHAEL, KAISER

With f ( k x ,k,) as the two-dimensional Fourier transform of F ( x , y ) , one obtains: 1. Differentiation in the spatial domain

2. Differentiation in the spatial frequency domain

3. Shifting in the spatial domain F(x - xo, y - yo)

f ( k , , k,)e+j'kx"o+kyyo)

(136)

4. Shifting in the spatial frequency domain F ( ~y ) ,e - i ( k x o x + L y o ~ )

* " f ( k , - kxo, k, - k y d

(137)

5. Complex conjugate function in the spatial domain

F*(X,Y)

::f * ( - k , ,

-k,)

(138)

6. Complex conjugate function in the spatial frequency domain F*(-x, -y)

f *(k,,k,)

(139)

2. Correlation of Impulse Response and Transfer Function b y the Two-Dimensional Fourier Transform ( E q . ( 2 7 ) ) Making use of the relation between the two-dimensional Fourier transform of circular symmetric functions and the Hankel transform given by Eq. (16), one can derive the following transforms used in the present chapter: 1. Impulse response S ( x , y , z ) and transfer function s(k,, k,; z ) for the scalar radiation problem (cf. Section I1.B) Taking into account the relation between the Fourier- and Hankel transforms given by Eq. (16) and making use of the Hankel transform given by Eq. (26), one can derive the Fourier transform pair 1 e - j k & c 2 + y 2 + z 2 FT -jdk2 - k i - k : z o -j 271 J x 2 + y 2 + z 2 J k 2 - k,2 - k; ' Differentiating both sides with respect to z yields the relation between the impulse response and the transfer function given by Eq. (27). 2. Impulse response tensor r ( x , y , z ) and transfer function y(k,, k,; z ) for the electromagnetic case (cf. Section 1I.C)

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

385

The elements Txx(x,y, z ) and Tyy(x,y, z ) are of the same form as the impulse response S(x, y, z ) for the scalar radiation problem. Therefore, the transform for the scalar case given by Eq. (27) is also valid for Txx(x,y,z) and yxx(k,, k y ; z )in Eqs. (39) and (51) respectively. y , z) and rzy(x,y, z ) can be obtained Using Eq. (38c) the elements Tzx(x, by differentiating the left side of the transform (140) with respect to x and y , respectively: (141a) (141b) According to Theorem (134),the right side of Eq. (140)has to be multiplied by ( - j k x ) and ( -jky), respectively. This yields the transforms ( 142a)

and (142b) Comparison with Eqs. (49a,b) reveals that the right sides of Eqs. (142a,b) are identical to the elements y z x ( k x , k y ; z )and yzy(kx,ky;z)of the transfer function tensor as given by Eq. (51). Thus, it has been demonstrated that the spatial domain and the spatial frequency domain are interrelated by the twodimensional Fourier transform in the electromagnetic case as well. B. Derivation of the Optimum Focusing Input Illumination (Eq. ( 5 9 ) )

m

u(kx,k,; O)s(k,, k,; zf)e-j(kxxf+ky”f)dk x dk Y ’ This delivers the following power density at the focus:

(143)

386

MICHAEL KAISER

Application of Schwarz’ inequality yields

Is(kx,k,; zf)e-j(kxxf+kyyf)12 dk, dk,.

IU(x’,y‘,0)12dx’dy’42:

(145) Here the right side is constant, as the value of the integral is determined by I U(x’,y’, 0)l and s(k,, k,; zJ). Therefore, the ratio in Eq. (53) is maximum if equality holds. This is true for:

’

u(k,, k,; 0) = iio[s(k,, k,,; zf)e-j(kxxf+kyyf) 1*

- Cos*(k k,;

f )e+j(kxxf+k,Yf).

X)

(146)

Using Theorems (136) and (1 39) of the Fourier transform, one obtains: U(x’,y’,O) = CoS*[-(x’

-

XI),

-(y’ - Y J ) , Z f ] .

(147)

With this input illumination, the ratio in Eq. (53)takes its maximum value:

C. Derivation of the Optimum Focusing Input Illumination with the Constraint of Finite Aperture Dimensions (Eq. (7 3 ))

If the input illumination U(x’,y’, 0) is non-zero only within a finite region S, the value of U at the focus can be computed thus: W X f ,YJ,Z f ) =

w, 0) ** S(X9Y , Y 7

ZJ)lx=x,, y = y f

u(x’,y’,o)s(xf - x’, yf - Y‘,Zf)dX’dY’. Application of Schwarz’ inequality yields r r

( 149)

FOCUSING CRITERION BY A SYSTEM-THEORETIC APPROACH

387

With the arguments from the previous section, the ratio

is maximum if equality holds. This is true if U(X’,y’, 0) = iioS*(x, - x’, y, - y’, zs)

holds within the region S.

REFERENCES Bergland, G. D. (1969). “A guided tour of the fast Fourier transform,” IEEE Spectrum, 6, July, 41-52 Bickmore, R. W. (1957). “On focusing electromagnetic radiators,” Can. J . Phys., 1292-1298 Blume, S. (1976).“Analogie zwischen elektrischen-und wellenphysikalischen Systemen,” Optik, 46, 333-335 Borgnis, F., and Papas, C. (1965). Randwertprobleme der Mikrowellenphysik. Springer, Berlin. Born, M., and Wolf, E. (1975). Principles of optics. Pergamon Press, Oxford. Bracewell, R. (1965). The Fourier TransJorm and its Applications. McGraw-Hill, New York. Clemmow, P. C. (1966). The Plane Waoe Spectrum Representation of Electromagnetic Fields. Pergamon Press, Oxford. Collin, R. E., and Zucker, F. (1969). Antenna Theory, pt. 1. McGraw-Hill, New York. Gaskill, J. D. (1978). Linear Systems, Fourier Transforms, and Optics. Wiley, New York. Goodman, J. W. (1968). Introduction to Fourier optics. McGraw-Hill, New York. Guy, A. W. (1971). “Electromagnetic fields and relative heating patterns due to a rectangular aperture source in direct contact with a bilayered biological tissue,” IEEE Trans. MTT-19, 214-223 Hansen, V. (1983). “Electromagnetic methods in geophysical probing,” Proc. URSI Symp. 1983, Belgium, 399-404 Holler, P. (edJ(1983).“New procedures in nondestructive testing,” Proc. Germany-U.S. Workshop, Fraunhofer-Institut, Saarbriicken, Aug. 30-Sept. 3, 1982, Springer, Berlin. Honl, H., Maue, A,, and Westphal, K. (1961).“Theorie der Beugung,” In S. Fliigge (ed.), Handbuch der Physik, Vol. 25, Springer, Berlin. Howard, A. Q. (1975). “On approximating Fourier integral transforms by their discrete counterparts in certain geophysical applications,” IEEE Trans. AP-23,264-266 Johnson, C. C., and Guy, A. W. (1972). “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE, 692-718 Kaiser, M., and Hetsch, J. (1984). “Derivation of an optimum focusing aperture illumination by a system-theoretic approach.” Optica Actn, 31, 225-232. Kerns, D. M. (1981). Plane wave scattering-matrix theory of antennas and antenna-antenna interactions. NBS monograph 162, Washington, U. S. Government Printing Office. Kong, J. A. (1975). Theory of electromagnetic waves, Wiley, New York. Luke, H. D. (1975). Signalibertragung. Springer, Berlin. Melek, M., and Anderson A. P. (1980). “Theoretical studies of localized tumor heating using focused microwave arrays,” IEEE Proc. (F), 319-321

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Myers, P. C., Sadowsky, N. L., and Barret, A. H. (1979). “Microwave thermography: Principles, methods, and clinical applications,” J. Microwave Power, 14(2). Papoulis, A. (1962). The Fourier Integral and its ,4pplications. McGraw-Hill, New York. Papoulis, A. (1968).Systems and Transforms with ,4pplications in Optics. Mdjraw-Hill, New York. Lord Rayleigh. (1877,78). The Theory of Sound. The Macmillan Company, New York. Severin, H. (1951).“Zur Theorie der BeugungelektromagnetischerWellen,” 2.Phys. 129,426-439 Sherman, J. W. (1962).“Properties of focused apertures in the Fresnel region.’’ IRE Trans. AP-10, 399-408 Silvers, S. (1949). Microwave Antenna Theory and Design. McGraw-Hill, New York. Sommerfeld, A. (1964). Optik (Optics).Akademische Verlagsgesellschaft, Leipzig. Wait, J. R. (1970). Electromagnetic Waves in Stratified Media (2nd ed.). Pergamon Press, Oxford. Wait, J. R. (Guest Editor). (1979). Special Issue on Applications of Electromagnetic Theory to Geophysical Exploration. Proc. IEEE, 979-1015 Wehner, R. S. (1949). Limitations of Focused Aperture Antennas. Rand Corp., RM-262, Santa Monica, Cal. Weyl, H. (1919). “Ausbreitung elektromagnetischer Wellen iiber einem ebenen Leiter,” Ann. d. Phys., 4. Folge, Vol. 60,481-500.

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 75

LIGHTWAVE RECEIVERS GARETH F. WILLIAMS* NYNEX SCIENCE AND TECHNOLOGY WHITE PLAINS.NEWYORK

I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Receiver and Device Requirements of Lightwave Systems . . . . . . . I11. Receiver System and Noise Considerations . . . . . . . . . . . . A . Digital Receiver System Considerations . . . . . . . . . . . . B. Pin-Photodiode Receiver Noise and Sensitivity Calculations . . . . . C. Avalanche Photodiode Receiver Noise and Sensitivity Calculations . . IV . First andsecond-GenerationLightwave Receivers . . . . . . . . . A . Voltage-Amplifier Optical Receivers . . . . . . . . . . . . . B. Integrating Optical Receivers . . . . . . . . . . . . . . . C. Transimpedance Optical Receivers. . . . . . . . . . . . . . D. Integrating Transimpedance Optical Receivers . . . . . . . . . V. Active-Feedback Lightwave Receiver Circuits . . . . . . . . . . . A . Introduction . . . . . . . . . . . . B. Micro-FET Feedback Receivers. . . . . . . . . . . . . . . C. Capacitive Feedback Receivers . . . . . . . . . . . . . . . D . Dynamic Range Extenders . . . . . . . . . . . . . . . . E. Hybrid IC Active-Feedback Receivers . . . F . IC Active-Feedback Receivers . . . . . . G . Design Scaling Laws for IC Receivers . . . . . . . . . . . . . H. Sensitivity Calculations for Present- and Future.-Technology IC Receivers References . . . . . . . . . . . . . . . . . . . . . .

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I . INTRODUCTION This chapter covers the development of optical receiver circuits as influenced by photodetector and transistor noise physics and by lightwave system requirements. These considerations led directly to the new highsensitivity. wide.dynamic.range. active-feedback receiver ICs. This chapter also covers photodetector and transistor technology choices and some probably future device developments. in light of the receiver design advances. * Chapter prepared at AT&T Bell Laboratories. Holmdel. New Jersey 389 Copyright 1989 AT&T Bell Laboratories. reprinted by permission .

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the noise physics, and the system requirements. This chapter is written for both receiver designers and device physicists. The earliest optical receiver circuits were simple voltage amplifiers and were very noisy. These receivers were followed by high-sensitivity integrating front ends (Personick, 1973a, 1973b; Goell, 1974). Because these receivers integrate the signal, they require subsequent equalization (differentiation) with its attendant problems and have limited dynamic range. Thus, although these early integrating receivers showed promise in laboratory experiments, neither they nor the voltage-amplifier receivers were ever used in a commercial system. The transimpedance receiver was developed next and was the first commercially viable lightwave receiver. Its advantage is that it avoids signal integration; unfortunately, it is less sensitive than integrating receivers. The first commercial transimpedance receivers were developed for 0.8-pm wavelength, 45-Mb/s applications and used silicon avalanche photodetectors (APDs) to achieve high sensitivity (R. G. Smith et al., 1978). More recently, a silicon bipolar IC transimpedance receiver for the TAT-8 transatlantic cable was pioneered by Paski (1980) and perfected by Snodgrass and Klinman (1984); it operates at 296 MB/s and a 1.3-pm optical wavelength. The superior sensitivity of integrating receivers has continued to attract interest; recently, integrating receivers have come back in long-wavelength (1.3- 1.6 pm) receivers using low-capacitance conventional (nonmultiplying) pin photodiodes and fine-line microwave FETs (Hooper et a!., 1980; D. R. Smith et al., 1980;Gloge et al., 1980; Ogawa et al., 1983).These improved integrating receivers offer sensitivities comparable to present 1.3- to 1.6-pm APDs below 100 Mb/s and good sensitivities to at least 1 Gb/s (Linke et al., 1983). However, they still require equalization and have limited dynamic range. As a result, these improved integrating receivers have primarily been used for laboratory demonstrations, where these difficulties do not matter, and in a very few leading-edge systems. Nearly all commercial systems have continued to use conventional non-integrating transimpedance receivers. A new type of lightwave receiver, the active-feedback receiver, offers high sensitivity with a wide-band nonintegrating response, plus the widest dynamic ranges yet achieved (Williams, 1982, 1985; Fraser et al., 1983; Williams and LeBlanc, 1986). Active feedback receivers avoid the signal integration of previous high-sensitivity designs, yet do not compromise the sensitivity; in fact, these new receivers are somewhat more sensitive than corresponding integrating receivers. The dynamic-range-extender circuitry incorporated in these receivers provides the wide dynamic range; in fact, most implementations of these new receivers cannot be saturated by present lightwave transmitters. Active-feedback receivers can be realized inexpensively in IC form using any fine-line FET IC technology and are the first step towards the single-chip lightwave regenerators of the future. Although specific commercial

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designs are beyond the scope of this chapter, note that IC versions for 1- to 50-Mb/s datalinks and local-area networks are in production by AT&T Technologies (Morrison, 1984; Steininger and Swanson, 1986); high-speed versions for transmission systems operating at bit rates to 1.7 Gb/s have also been announced (Gloge and Ogawa, 1985; Dorman et al., 1987). Because APDs were the only way to achieve high sensitivities with early receiver circuits, lightwave device physicists focused on APDs. With today’s high-sensitivity receiver circuits, sensitivity improvements can come from advances in either photodetector or transistor technology. The lightwave device physicist must now be aware of both. For example, present long-wavelength (1.3- 1.6 pm) APDs offer little sensitivity advantage over nonmultiplying pin photodiodes below 100 Mb/s when used with either the integrating receiver circuit or the new highsensitivity, active-feedback receiver circuits. Where this dividing line will be in the future depends on the rate of long-wavelength APD development versus that of the fine-line FET technology used in the receiver. In addition, at bit rates up to a few hundred Mb/s, reducing the longwavelength APD dark current to that of pin photodiodes is presently more important than achieving a higher ionization coefficient ratio, k , for lower excess multiplication noise; because receiver amplifiers are now so sensitive, the optimal APD gain is small and k is less critical. This makes high-sensitivity, long-wavelength APDs easier to develop. These receiver advances and their device implications can also influence research into new types of devices. For example, these receiver advances and their implications led directly to the staircase APD proposal of Williams et al. (1982). Only one type of carrier should impact ionize in this structure, giving very low noise multiplication. However, as mentioned in the previous paragraph, a low leakage current is just as important as low-noise multiplication. The low operating voltage and large average band gap of the staircase APD should allow even smaller leakage currents than in long-wavelength nonmultiplying pin photodiodes, where the entire depletion region is narrow gap. Finally, the staircase APD might not have been proposed without the new receiver circuits, because the maximum multiplication of the detector may typically be only 30; the high sensitivities of the receiver circuits mean that higher multiplications will not be needed in order to realize the full sensitivity of the detector. The circuit and device developments to date have been driven by systems requirements, especially the move from the 0.8-pm to the 1.3-pm wavelength. The early first-generation 0.8-pm transmission systems used high-gain, lownoise silicon APDs, which solved both the sensitivity and dynamic range problems of the early receiver circuits. However, second-generation transmission systems operate in the 1.3- to 1.6-pm region, where the lower fiber loss N

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allows wider repeater spacings. Unfortunately long-wavelength APDs comparable to 0.8-pm silicon APDs do not as yet exist. These receivers must presently use either pin photodiodes or APDs with lower gains and higher noise than the 0.8-pm silicon APDs. What is lost in the detector has had to be made up in the following preamplifier. In addition, APDs and their associated variable high-voltage supplies are presently too expensive for datalinks and local-area networks and not reliable enough for transoceanic submarine cables. In these systems, the low noise amplification and wide dynamic range must at present be achieved in the preamplifier rather than in the detector. Section I1 briefly discusses lightwave systems requirements and assesses their impact on present and future optical receivers and devices. This area is sometimes ignored, but it will continue to determine which new device and receiver developments will be used. Section I11 lays the foundation for the receiver circuit discussions of Sections IV and V. Almost all lightwave systems are digital; Section I11 begins by reviewing how optical-fiber digital regenerators are designed to achieve the best sensitivity of which the optical receiver is capable and reviews the derivation of the digital bit-error rate as a function of the receiver noise. It then reviews the classic Smith and Personick (1980)calculations of the receiver input device noise, presents FET and photodetector technology figures-ofmerit, and discusses sensitivity calculations for the new active-feedback IC receivers of Section V. These calculations are then compared with sensitivity results reported in the literature. This section then covers technology choices between GaAs FETs and silicon FETs and between pin photodiodes, InGaAs/InP APD detectors, and Ge APD detectors. Section IV reviews and discusses the first- and second-generation lightwave receiver circuit designs. The early first-generation commercial 0.8-pm transimpedance receiver amplifiers traded sensitivity for bandwidth and used a silicon APD to make up the sensitivity loss; in the design of R. G. Smith et al. (1978),the APD improved the sensitivity by 15 dB over the same receiver with a pin photodiode. The later silicon bipolar transimpedance receiver ICs (Paski, 1980; Snodgrass and Klinman, 1984) were designed for 1.3-pm applications and use nonmultiplying pin photodiodes. They have achieved good sensitivities at bit rates of several hundred megabits per second. These receivers are a genuine tour de force and probably represent the end of their line of development. The second-generation 1.3- to 1.6-pm pin-FET receiver circuits use integrating or “high-impedance’’ receiver designs. Section IV discusses both the simple integrating receivers, which have been introduced commercially in a few leading-edge systems by the British Post Office (Hooper et al., 1980) and the integrating transimpedance receivers favored by Gloge et al. (1980) and Ogawa et al. (1983). Both types achieve very high sensitivities; however, they integrate the signal and have little dynamic range.

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Section V describes the new active-feedback receivers. This section first reviews the basic features of these designs, including both the novel feedback techniques and the dynamic-range-extender circuitry. Response, stability, and sensitivity considerations are covered in detail. This section then briefly discusses an illustrative design for a complete hybrid IC active-feedback receiver. This is followed by an extended discussion of active-feedback receiver ICs. Two generic IC designs are presented; these can be realized in any of the standard fine-line FET IC technologies, including NMOS, CMOS, and GaAs. Section V also presents scaling laws, which are used to generate different versions of these IC receiver designs for different bit-rate applications, then presents calculations of IC receiver sensitivities for bit rates between 10 Mb/s and 4 Gb/s. These high-sensitivity active-feedback IC receivers will achieve further sensitivity advances as the pin-photodiode and FET technologies develop. The development of submicron gate FETs is driven by very-high-speed GaAs and silicon digital IC programs; the new optical receiver ICs are compatible and will piggyback on these efforts. In addition, the pin-photodetector capacitance appears to be halving every two to three years. Section V concludes with theoretical calculations of the probable sensitivities to be expected from future IC receivers using these improved pin and FET technologies. In long-haul terrestrial systems at several hundred megabits per second or above, these IC receivers often use InGaAs/InP APD detectors instead of pin photodiodes. Future IC receivers for these applications will require APDs with lower dark currents and lower junction capacitances than present devices. The importance of a small ionization coefficient ratio, k, will depend on whether the dark currents are decreased or the commonly used bit rates are increased faster than the FET technologists are able to achieve shorter channel lengths. Smaller APD junction capacitances would also reduce the importance of a small k ratio.

11. RECEIVER AND DEVICE REQUIREMENTS OF LIGHTWAVE SYSTEMS

This section first looks at the requirements of present and future systems and then at the likely impact of these requirements on present and future receiver and device technologies. The important receiver design goals are high sensitivity, wide dynamic range, adaptability to high bit rates, and low cost. Most of the lightwave literature to date has focused on technology for long-haul transmission systems, which comprise the bulk of the present optical-fiber applications. The key goals in transmission systems are long repeater spacings, to minimize the

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number of repeaters, and high channel capacity. Accordingly, most of the receivers in the literature are optimized for high sensitivity and high bit rate; wide dynamic range and low cost are less important. However, most future optical-fiber applications will be in the loop plant (between the central office and the subscribers), in local-area networks (e.g., on-premises office-of-the-future networks), in optical-fiber cable TV, and in data links (Miller, 1979). All require high sensitivity, wide dynamic range, and low cost; the bit rate varies with the application. The new high-sensitivity, wide-dynamic-range active-feedback receivers discussed in Section V were originally invented for these applications (Williams, 1982, 1985). The IC versions can be realized economically in fineline digital VLSI technology and are adaptable to bit rates in excess of 1 Gb/s. They therefore are a superior alternative for high-bit-rate transmission systems as well. These loop, local-area-network, cable-TV and datalink applications require high sensitivity because they all frequently include some type of veryhigh-loss optical data path. They all require wide dynamic range because the minimum loss path in each is a short, almost connectorless, fiber run with a loss of a few decibels or less; essentially the entire transmitter power can appear at a receiver. Field-installed optical attenuators are not a viable means to reduce the dynamic range requirement because of the expense of installing them and of changing them if the system is reconfigured, plus the need to keep a record of what value attenuator was installed where. Low cost is imperative because the total number of receivers in these applications is so large; each customer, work station, or computer may have its own receiver. In loop feeder, the highest loss path is between the central office and the telephone-call concentrator of the furthest neighborhood. The higher the sensitivity, the wider the radius that a single central office can serve without repeaters; the number of customers increases as the square of the radius and every decibel is an additional 1-2 kilometers. On-premises local-area networks also need high-sensitivity receivers, especially in star or tree configurations in which each transmitter’s power can be divided among many receivers and several fiber branches. The same applies to fiber cable-TV systems. Typical local-area-network signal paths can also have many fiber connectors, each with a signal loss; in addition, a local-area network should be extendable to several buildings, e.g., on a campus or in an industrial park; link lengths can be up to a few kilometers. The implications for circuit and transistor technology designers are clear. From now on, lightwave receivers will be IC receivers; the transistor technologies used will be IC technologies (with the possible exception of integrated pin-FETs). Most future applications are in loop, local-area networks, cable-TV, and datalink systems, which require relatively high per-

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formance and very low cost; those receivers absolutely must be realized as ICs. As mentioned in the introduction, GaAs and silicon FET IC technologies are presently preferred; any new transistor technology, e.g., high-electronmobility transistors or heterojunction bipolar transistors, should also be realizable in IC form for these applications. In addition, the new nonintegrating high-sensitivity wide-dynamic-range receiver circuits developed for those applications are best realized in IC form (Section 5). Furthermore, the big development efforts that can be expected for loop-feeder receivers will ensure that IC receivers will always be at or near the sensitivity limit of the detector and transistor technologies, and hence superior for transmission systems as well. New detectors will first be adopted in long-haul transmission systems, where sensitivity is most important and cost is secondary. They will be adopted in loop, LANs, cable-TV and datalink applications only if their cost comes down; these systems are cost critical, and the projected advances in pinFET sensitivities should adequately meet the future needs of these systems. Most of the applications will run on 5 volts; few can justify the expense of a high-voltage supply for an APD. Therefore, low-voltage detectors, such as heterojunction phototransistors, integrated pin-FETs, or staircase APDs are preferred possibilities; high-voltage devices like conventional APDs will probably be restricted to transmission systems only.

111. RECEIVER SYSTEM AND NOISE CONSIDERATIONS This section covers the systems background and the device noise theory for the receiver amplifier circuit discussions of Sections IV and V. It begins with a tutorial on receiver systems, then reviews the classic Smith and Personick (1980) expression for the input device noise of pin-FET receivers. It then extends these expressions to include the noise of the rest of the receiver circuit and derives figure-of-merit expressions for pin, FET, and FET IC technologies. It then calculates theoretical sensitivities for FET IC receivers and compares present silicon and GaAs FET IC technologies on the basis of these noise calculations, experimental results from the literature, and circuit considerations. This section concludes by reviewing the APD receiver noise theory of Smith and Personick (1980) and calculating sensitivities for present-technology APD-FET lightwave receivers. Lightwave receivers using bipolar transistors will not be considered because they are presently less sensitive than FET lightwave receivers at all bit rates. However, new bipolar technologies, such as heterojunction bipolar transistors (Kroemer, 1982,1983) may someday change this.

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This section focuses on digital receiver systems because most optical fiber systems are digital. Analog video is the only major exception, but will likely be replaced by digital video once economical video codec ICs become available. These codecs are realizable in the silicon or GaAs FET technologies favored for the receivers; ultimately, the receiver and codec will probably be on the same IC. Section 1II.A is a brief tutorial for device physicists on the filtering and digital signal recovery circuits that extract the digital bit stream from the analog output of the optical receiver circuit and on how these circuits affect the digital-receiver sensitivity. This section contains just enough detail to place the optical receiver in the system and to motivate the noise treatment of Section 1II.B; circuit engineers may wish to consult Maione et al. (1978) to see a complete digital-signal recovery circuit. Section 1II.B covers sensitivity and noise calculations for pin-FET receivers, including the active-feedback IC receivers of Section v. This section first reviews the connection between the signal-to-noise ratio at the output of the receiver and the digital bit-error rate at the output of the digital signalrecovery circuit, and the assumptions involved. Subsection III.B.l then presents a semi-intuitive derivation of the Smith and Personick (1980)expression for the receiver front-end noise due to the input FET, the pin photodiode, and the input resistor. These devices account for the majority, but not all, of the receiver noise. Subsection III.B.2 extends this expression to include the noise contributions of the first-stage load device and of later stages; the resultant new expression can be used to calculate sensitivities for the complete IC receivers of Section V.F. Subsection III.B.2 also presents lightwave receiver figure-of-merit expressions for pin, FET, and FET IC technologies and then discusses the device implications. For example, microwave or VHSIC FETs are preferred even for 10- to lOO-Mb/s lightwave receivers, but for noise, not speed, reasons. Subsection III.B.3 calculates theoretical sensitivities of present-technology silicon and GaAs FET IC receivers as a function of bit rate. Sensitivity results from the literature are also included. In theory, the two technologies should give essentially the same receiver sensitivities; GaAs FETs have a higher saturation velocity, but silicon FETs have a lower channel-noise factor.' In practice, GaAs is presently a few decibels superior at most bit rates, probably because fine-line GaAs FETs began as an analog microwave technology. Therefore, GaAs noise problems were important and were solved. Fine-line silicon FETs began as a digital IC technology in which analog noise was unimportant and was almost ignored. The extra channel noise in fine-line GaAs FETs is largely due to high-field scattering from the high-mobility central valley to low-mobility satellite valleys (Baechtold, 1972). This effect is used in G u m diodes.

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Section 1II.C discusses noise and sensitivity considerations for avalanche photodiode (APD) receivers. This section includes an APD receiver noise tutorial following the APD receiver noise theory of Smith and Personick (1980),which is based on the APD device noise theory of McIntyre (1966). It then shows optical receiver sensitivity calculations for present-technology APDs used with present-technology IC receivers. It considers only 1.3- to 1.6-pm wavelength APDs; 0.8-pm APD systems are obsolete because their repeater spacing is less than that of 1.3-pm pin-photodiode systems due to the much greater fiber loss at 0.8pm. These calculations indicate that receivers using present-technology low-leakage-current InGaAs/InP APDs are more sensitive at all bit rates than receivers using present germanium APDs and that at a typical maximum operating temperature of 85”C,present-technology pinFET receivers can be more sensitive than present germanium APD receivers at bit rates below 200 Mb/s. In fact, these calculations indicate that even present InGaAs/InP APDs have little sensitivity advantage (at 85OC) over presenttechnology pin-FETs below 100 Mb/s and are less sensitive below -50 Mb/s due to their higher leakage currents. A . Digital Receiver System Considerations

A typical digital receiver system block diagram is shown in Fig. la; associated waveforms are shown in Fig. 1b. For now, assume a non-return-tozero (NRZ) signal format; if a bit is a “one,” the signal is high during the entire bit interval; if a bit is zero, the signal is low during the entire bit interval. The receiver amplifier takes the photocurrent signal input, ,i and gives an output voltage, v,, of the same waveshape; however, this output voltage also includes noise due to the detector and the receiver amplifier. The average noise shown on v, is much less than the signal; however, occasionally a random fluctuation is bigger than the signal and can turn a 1 into a 0 or a 0 into a 1, causing an or one error per error. Typical systems require a bit-error rate (BER) of billion bits. The digital-receiver sensitivity is then the input optical signal power required for a signal-to-noise ratio high enough to get a BER of The receiver output noise can be reduced by filtering; the receiver bandwidth is usually wider than necessary to pass the photocurrent signal; the extra bandwidth contains extra noise. This extra noise is removed by the channel filter, improving the signal-to-noise ratio at the input to the digital decision circuit. The total noise power on the channel filter output signal, uf is the spectral noise power density S(w) of the amplifier plus that of the photodetector integrated over the filter bandwidth; the lower the filter cutoff frequency, f,,the lower the noise bandwidth and the less noise at the input to the decision circuit.

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CHANNEL FILTER

-

DlGlTA L

vf

L

BIT STREAM

OUTPUT

CIRCUIT

1 0 1 0 0

1 0 0 1'1 1

(b) I

>voh t

!

:

:.TIME

TIME

T I ME DIGITAL 1 OUTPUT 0

h-•

TIME

FIG. 1. Optical receiver system: (a) block diagram, (b) typical waveforms.

The digital decision circuit recovers the digital bit stream from uf . There are two types of decision circuits. The asynchronous decision circuit is presently cheaper and has been used in datalinks. The synchronous decision circuit, in which a clock is recovered from the signal and is used to sample of at the center of each zero or one (Maione et al., 1978), is more sensitive because it allows the use of a narrower channel-filter bandwidth, which removes more noise. In time, the synchronous circuit will be included in the receiver IC and will thus cost essentially nothing. Note that the receiver amplifier typically includes automatic gain control (AGC) so that the peak-to-peak signal at the decision circuit input is held constant, independent of the input optical signal level. This ensures proper decision circuit operation. This AGC function has typically been implemented in a postamplifier (not shown) after the input receiver amplifier (Maione et al., 1978); in the new high-sensitivity, wide-dynamic-range amplifiers of Section V, AGC is provided in the input amplifier as well. The asynchronous decision circuit (Fig. 2) is just a discriminator. It gives a logic zero output when the filter output, uf, is less than the decision threshold voltage, V,; it gives a logic one output when uf is greater than V,. (Typically, V,

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vT

FIG.2. Asynchronous decision circuit optical receiver system.

is midway between the “zero” and “one” analog signal levels.) Each digital transition (zero to one or one to zero) in the output bit stream occurs at the moment the signal uf passes through V,. Any noise as the signal transition passes through V, will cause a transition timing error. These timing errors are minimized by maintaining fast rise and fall times on the signal u,; this means a wide channel-filter bandwidth, hence high noise, thus reducing the achievable sensitivity. The channel-filter bandwidth for an NRZ asynchronous receiver would typically be about twice the bit rare B,giving a risetime of about 0.17 times the bit interval. (The 10% to 90% risetime is 5, x 0.35/BW). The synchronous decision circuit (Fig. 3a) samples the filtered signal, u,, at the center of each bit interval, as indicated by the arrows in Fig. 3b, and decides whether it represents a zero (u, < V,) or a one (of > V,). The circuit then puts out a standard length one or zero pulse exactly one bit-interval long. The sampling circuit is triggered at the center of every bit by the clock recovery circuit,* which reconstructs the transmitter’s digital clock signal (timing) from the received signal. The decision circuit output bit intervals are each one cycle of the recovered clock. Since the signal is sampled only in the center of each bit interval, the rise and fall times can be very slow; this means a narrow channelfilter bandwidth and hence low noise. Thus, the synchronous decision circuit gives the best digital-receiver sensitivities. The channel-filter bandwidth for a synchronous NRZ receiver is typically set at 0.56 times the bit rate (Maione et al., 1978; Smith and Personick, 1980), thus removing the high-frequency part of the NRZ signal spectrum. The resultant waveform, u,, is shown in Fig. 3b; the risetime is now about 0.6 times the bit interval; the waveform is still satisfactory because the time between sampling points is one bit interval. The noise bandwidth is now only 28% of

* Typically, the clock recovery circuit is a phase-locked loop in which a voltage-controlled oscillator is synchronized on the bit transitions in the analog signal (Maione et al., 1978). This circuit effectively averages over many bit periods and many transitions; noise on any one transition is averaged out.

400

CHANNEL FILTER

+

DIGITAL DECISION

OUTPUT

FIG.3. Synchronous decision circuit optical receiver system: (a) block diagram, (b) typical waveforms.

that of the asynchronous receiver example; the receiver sensitivity is increased accordingly. Typical optical receiver noise power spectra (Section 1II.B) contain a term independent of frequency, which dominates at low frequencies and a term proportional to the frequency squared, which dominates at high frequencies. The frequency-independent noise term is due to the device leakage currents and the circuit; the frequency-squared term is primarily due to the input FET. In high-sensitivity designs (Sections IV and V), the frequencyindependent term can essentially be eliminated over most of the bandwidth. Therefore, the frequency-squared noise term is dominant; integrating this term over the signal bandwidth gives a total electrical noise power proportional to the receiver bandwidth cubed. Since the synchronous receiver example has only 28% of the noise bandwidth of the asynchronous receiver, for example, the electrical noise power is only 0.283 = 0.022 times as much; the noise volt= 0.1 5 times as much. Since the signal voltage is proportional to age is the optical power, the minimum optical signal is 0.15 times smaller. This is an 8.3-dB optical sensitivity advantage for the synchronous detection example. Until now, transmission systems have used synchronous decision circuits for maximum sensitivity; asynchronous circuits are used in some very-short-

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haul, low-sensitivity datalinks. However, since a linear channel and synchronous detector circuit can ultimately be very economically integrated on the same IC as the receiver amplifier, asynchronous systems will no longer be cheaper. This means that almost all transmission, loop, and local-areanetwork receivers plus many datalink receivers will use synchronous detection for better sensitivity. Therefore, the noise treatment of Section 3.2 assumes synchronous detection. So far, this discussion has assumed an NRZ optical data transmission format. Other formats such as return-to-zero (RZ), biphase or Manchester, and block codes have been considered (Takasaki et al., 1976). Both RZ and biphase approximately double the receiver noise bandwidth, for an approximately 4.5-dB optical sensitivity penalty; they therefore are not preferred. Block-coding schemes (Rousseau, 1976; Brooks, 1980) in NRZ format have been used with the integrating receivers of Hooper et al. (1980) discussed in Section 1V.B. These involve a smaller sensitivity penalty. The biphase and block-coding schemes reduce the low-frequency content of the signal, which can be helpful in ac-coupled systems or in integrating receivers; in addition, they provide more frequent signal transitions to help maintain synchronization of the clock recovery circuit. At present, the NRZ format is preferred for sensitivity; however, it may be used either with self-synchronizing scrambling or with block coding; both can be realized in IC form on the same chip. B. pin-Photodiode Receiver Noise and Sensitivity Calculations

This section discusses noise and sensitivity calculations for lightwave receivers using (nonmultiplying) pin photodiodes and FET input stages, and lays the noise-theory foundation for the receiver circuit designs of Sections IV and V. Subsection III.B.l is a tutorial review of the classic Smith and Personick (1980) expressions for the receiver front-end noise due to the input FET, the photodetector, and the input resistor. Subsection III.B.2 first extracts pin- and FET-technology figures of merit from the Smith-andPersonick front-end figure of merit, then discusses the device implications. It then extends the noise expressions to include noise from the first-stage load device and from later stages; the resultant expression gives accurate sensitivities for IC receivers. Subsection III.B.2 also defines a simple expression for an IC technology figure of merit. Note that the figures of merit help dictate device design; the noise expressions as a whole help dictate the receiver circuit designs of Section V. Subsection III.B.2 calculates theoretical sensitivities of present-technology GaAs and silicon FET ICs versus bit rate, and compares the two technologies on the basis of the calculations, some results from the literature, and circuit considerations.

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The receiver-circuit noise expressions are for equivalent RMS input noise currents; the real noise sources in the amplifier and detector are replaced by a single equivalent noise current source at the input to a noise-free equivalent amplifier. This is convenient because the signal-to-noise ratio at the amplifier output is then just the photocurrent divided by the equivalent input noise current. In addition, these results are not affected by the feedback techniques used in the new receivers of Section V; the noise sources can be referred to the input before mentally closing the feedback loop; the loop is thus closed around the noiseless equivalent amplifier; by inspection, the equivalent input current noise source, which is outside the loop, is unchanged. Thus these open-loop noise calculations apply to all the circuits of Section V. For digital receiver systems, the signal-to-noise ratio at the digital decision circuit input is given by the photocurrent signal divided by the receiveramplifier-equivalent input-noise expression, provided that the noise bandwidth is taken as the channel-filter bandwidth. As mentioned in Section III.A, the channel-filter bandwidth for a synchronous NRZ receiver system is typically 0.56 times the bit rate B. The photocurrent signal can be had from the optical signal power by remembering that a I-eV photon has a wavelength of 1.240 pm; at that wavelength, 1 watt of detected optical power gives 1 ampere of photocurrent. This then gives -

-

A

I = qP1.240'

where I i s the average photocurrent in amperes, q is the photodiode quantum efficiency, i' is the average optical power in watts, and A is the optical wavelength in micrometers. The problem now is to turn this signal-to-root-mean-square noise ratio at the decision circuit into a digital bit-error rate (BER). Consider a synchronous detection digital receiver system, as discussed in Section 1II.A. The digital decision circuit samples the signal once at the center of each bit interval; if the signal, s, is below the decision threshold D,the bit is read as a zero; if s is greater than D it is read as a one. The probability of error, i.e., the BER, is the probability that the noise at the sampling instant will bring the zero signal above D or the one signal below D. By the central limit theorem, the noise amplitude probability distribution is Gaussian if the noise amplitude is the sum of many small independent physical process (Davenport and Root, 1958). The probability distribution for the zero-level signal is then, following Smith and Personick (1980).

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LIGHTWAVE RECEIVERS

where s(0) is the noise-free, zero-level signal, oo is the signal variance, or root-mean-square zero-level analog noise. The probability E,, of mistakenly identifying a zero bit as a one is then (Fig. 4) Eo, = P(s > D) = ___ &To

Jw

ds e - [ s

-s(o)Iz/2~i

0

The derivation of Elo, the probability of mistakenly identifying a one-bit as a zero, is similar. Changing variables, the general error probability is P(E) = -

d x e-xz/2 = 1 erfc

2

(g) ,

,DECISION LEVEL

PROEAElLl TY

FIG.4. Error probabilities for two-level digital system. E,, is the probability of mistakenly identifying a zero bit as a one bit; E,, is the probability of mistakenly identifying a one bit as a zero bit. [After Smith and Personick (19801.1

404

GARETH F. WILLIAMS

where

and for j = 0, P ( E ) = Eel; for j = 1, P ( E ) = El,. For pin receivers, the zero-signal-level RMS noise, go,and the one-signallevel RMS noise, a,, are essentially equal. Assuming a random bit stream (maximum information content), ones and zeros are equally frequent. The optimum decision level D is then midway between s(0) and s(l), and Eol = Elo. The bit-error rate (BER) is now

where Q

=

41) - s(0) 2(,*,2)1/2 ’

where ( s : ) ~ /=~ a is the RMS noise. Q is then just half the peak-to-peak signal to RMS noise ratio at the digital decider input. However, as mentioned, this SNR is just the photocurrent signal to equivalent input RMS noise current ratio. Assuming that the zero-level photocurrent is zero, the average photo current is half the peak (one-level) photocurrent I p k and

where (i,2)1/2 is the RMS equivalent input noise current. Equations (4) and ( 5 ) give the pin-receiver bit-error rate (BER) in terms of the photocurrent signal to RMS equivalent input-noise current ratio. The BER as a function of Q is shown in Fig. 5. A typical system requirement is for a BER of lop9;this corresponds to Q = 6 or an average photocurrent of six times the RMS equivalent input-noise current. This corresponds to a peakto-peak signal at the digital decider input that is 12 times the RMS noise. The optical power to get that photocurrent is then the digital optical receiver sensitivity. The actual digital sensitivity of some early experimental receivers was less than that predicted from the measured RMS analog noise on the basis of the above Gaussian noise discussion (Williams 1982). The reason was nonGaussian device noise due to surface leakage or bulk defects; later receives with better devices did not have this problem. The fundamental device noise sources (Sections 1II.B.1 and III.B.2) are almost always Gaussian because they are the sums of many small independent physical processes, as required by the central limit theorem (Davenport and Root, 1958). For example, a photodiode bulk-leakage current of 1 nA corresponds to 63 independent

405

LIGHTWAVE RECEIVERS 10-5

I

10-6

-w

-a

10-7 10-8

L e

B (L

10-9

w

&

1o-io

t

L 10-li

-

J

m

20

10-’2

U

a

10-43

I 0-44 10-15

4.5

5

5.5

6

6.5

7

7.5

8

Q FIG.5. Probability of error versus the average signal-to-RMS-noiseratio, Q. [After Smith and Personick (1980).]

carrier generations per bit at 100 Mb/s; similarly, the channel Johnson noise current of a typical input FET is produced by the Brownian motion of about lo6 electrons. The result is the Gaussian noise amplitude probability distribution, which falls off exponentially in the square of the noise amplitude; this is why an average photocurrent that is only six times the RMS equivalent input-noise current can give a BER of On the other hand, device defeat or surface leakage noise can be the result of only a few physical processes at a few sites and therefore non-Gaussian; this means more bit errors for a given noise level. Thus, both the RMS noise spectrum and the noise-amplitude distribution must be measured for both devices and receivers. In addition, early devices with defect and surface noise problems may have non-Gaussian noise, leading to lower digital receiver sensitivities. For example, even though the ideal shot noise of a 20-nA photodiode leakage current may be acceptable in theory, a real diode with 2GnA leakage is unacceptable if the leakage is a non-Gaussian process due to a few device defects or surface leakage paths. Thus, a highleakage device may be acceptable in theory but unusable in practice if the

406

GARETH F. WILLIAMS

leakage is a non-Gaussian process. However, a mature device technology, in which surface and defect noise sources are under control, usually has Gaussian noise. I . Input Device Noise Theory for pin-FET Receivers

Figure 6 shows a generalized pin-FET receiver amplifier. This subsection presents a tutorial review of the classic Smith and Personick (1980)theory for the noise due to the input devices shown in Fig. 6, i.e., the channel noise of the input FET, the Johnson noise of the bias/feedback or input resistor, and the shot noise due to the input FET gate and pin-photodioide leakage currents. This theory gives much of the information about FET- and pin-technology requirements and amplifier design constraints. Subsection III.B.2 derives pin, FET, and FET IC technology figures of merit, and extends the Smith and Personick expression to include the noise of the input FET's load device and the noise of subsequent stages. This is useful for sensitivity calculations for the circuits of Sections IV and V. As mentioned, the equivalent input noise current is the same whether or not the feedback techniques of Section V are used; for simplicity, the discussion below assumes no feedback. The input bias or feedback resistor R i contributes a mean Johnson thermal input noise current squared per frequency bandwidth df of 4kT d ( i i ) R = -df. Ri The different frequency components of the noise are both independent and orthogonal. Therefore, the total mean-square input noise ( i i ) due to the input resistor is simply the mean-square spectral noise density of Eq. ( 6 )integrated over the channel frequency bandwidth. If IF( f )I is the magnitude of the normalized channel-filter frequency response, then

FIG.6. General pin-FET receiver amplifier.

LIGHTWAVE RECEIVERS

407

the independent noise currents squared are multiplied by the filter response magnitude squared, then integrated. By inspection, this resistor-noise integral is proportional to the bandwidth or to the bit rate B times a numerical factor that depends on the shape of the filter frequency response function F(f).(The same filter function is scaled up or down in frequency for different bit rates.) Changing variables from the frequency f to y = f / B (the frequency normalized to the bit rate) gives (i:)R

4kT

= -I,B, Ri

where

where F * ( y ) is the filter frequency response shape; the numerical factor I , is called the second Personick integral. The next question is how to normalize the channel frequency response F(f). F(f) must give the fraction of the input-noise-current frequency component that appears at the linear channel output to the digital decider. This is best done by first normalizing in the time domain so that a unit input photocurrent pulse maps to a unit output pulse to the decider. The corresponding filter frequency function normalization then applies to the noise as well. The receiver input is a photocurrent; the input pulse shape h,(t) is normalized to correspond to a unit photocurrent over a bit interval T = 1/B:

i T

Jm

h,(t)dt

=

1.

-m

These receivers are used with synchronous decision circuits, which sample the waveform at the center of each bit interval when the pulse is a maximum (logic one bit) or a minimum (logic zero bit). Thus, a unit filtered pulse is one which is of unit amplitude at the sampling instant, t o ; hOlJI(t0)

=

1.

(11 )

Defining H,(f) and H,,,,(f) as the fourier transforms of h,(t) and ho,,(t) then gives the normalized filter function:

Note that, for a given filter design, changing the input pulse shape changes the magnitude but not the frequency dependence of F(f);F(f)is normalized such that a unit average photocurrent pulse of whatever pulse shape chosen gives a

408

GARETH F. WILLIAMS

unit amplitude input at the sampling instant to the decision circuit, but the shape of F( f ) depends only on the channel filter design. The remaining question is how to arrive at the frequency dependence of the channel filter function F( f ); in other words, how to design the channel filter. By inspection, a narrow channel-filter bandwidth means smaller Personick integrals and less receiver noise; however, a narrow filter bandwidth also means slow rise and fall times, which cause the filtered bit-pulse houl(t) to spread into the neighboring bit intervals. Thus, choosing the filter function is a tradeoff between noise considerations in the frequency domain and pulseshape considerations in the time domain. Since the decision circuit samples the waveform at the center of each bit interval, the bit-pulse houl(t) should peak at the center of its own bit interval and should be zero at the centers of the neighboring bit-intervals to avoid inter-symbol interference (ISI). The filter typically is designed by iterating between the time and frequency domains, alternately minimizing the IS1 and the noise bandwidth, respectively. The result usually is a filter bandwidth of 0.5-0.6 times the bit rate for a NRZ signal. For such a typical filter design, I , worked out to be 0.56 (Maione ef al., 1978). The FET gate-leakage current and the pin-leakage current contribute an input shot noise current squared per unit frequency bandwidth of d < i i > , = 2qlidf,

(13)

where the total leakage current I , is the sum of the pin-photodiode leakage current legin plus the FET leakage current I,,,,, and q is the electron charge. Integrating over frequency, as before, gives the total input-leakagecurrent noise in the channel filter bandwidth: (ii>/ =

s.'

(ii),

2yI,I, R,

0s:

=

2 q 1 / l F ( /')I'd/'

(14)

(15)

where I , is the second Personick integral as before. 'The input FET channel noise is csscntially the Johnson noise of the unpinched-off portion of the channlel ncxt to the source. This channel conductance is simply the transconcluctanct: of the FET; the drain current noise per unit bandwidth is then

d ( i , ~=)4 k~ 7 ~: y ,~, w~7 ~ whew

(16)

r is a factor to account lor high-lield effects in short-channel transistors

(Bacchtold, 1972). For I-pm gate length silicon MOSFETs, is typically 1 to 1.2 (Ogawa er ul., 1983);for 1-pm GaAs FETs, I- is typically 1.4 to 1.8 (Ogawa, 198 1 ) .

LIGHTWAVE RECEIVERS

409

This mean-square FET drain noise current must be turned into an equivalent input mean-square noise current. The drain noise current squared can be turned into an equivalent gate (input) noise voltage squared by dividing by the transconductance squared (i,” = gie;):

The corresponding mean-square input noise current is simply the meansquare gate noise voltage divided by the input impedance squared (without feedback.) For small input bias/feedback resistor values, that mean-square equivalent input noise current would be just (ii)FEr= (f?i)FET/R;. However, this case is of no practical interest because the Johnson noise of that smallvalue resistor would swamp the FET noise and ruin the sensitivity. In highsensitivity amplifiers, the input impedance (without feedback) is essentially the input FET gate capacitance, CFET, in parallel with the junction capacitance of the pin photodiode, Cpin,plus any stray capacitance, C,. The equivalent input noise current is then

where the total input capacitance, CT,is the sum of the FET input capacitance, CFET,plus the photodiode capacitance, Cpin,plus C,. The physical assumption behind Eq. (18) is that in the absence of feedback, the photocurrent signal will be integrated by the input capacitance CT( V = l/CT idt);that Ri is so large (for low noise) that it can be neglected over most of the bandwidth. For example, in a 45-Mb/s receiver with Ri = 1 MQ, and C, = 1 pF, C , dominates above 80 kHz. This means that the input voltage produced by the photocurrent is inversely proportional to the frequency; since the input noise voltage is fixed, the signal-to-noise ratio is also inversely proportional to the frequency. When the signal is equalized (differentiated), the low-frequency signals (and noise) are attenuated; the high frequencies are boosted. Thus, with the signal now proportional to the input photocurrent, the noise is proportional to frequency, and the meanaquare noise is proportional to the frequency squared, as in Eq. (IS), Again, using feedback to avoid signal integration does mt change the signal-to-noise ratio versus frequency. The total equivalent man-square input nom current of the FET i s obtained by integrating over the channel-filter Entgue~cyrespagse as before:

410

GARETH F. WILLIAMS

or, changing variables from f to y

=f

/ B as before,

where

is the third Personick integral. The total mean-square equivalent input-noise current due to the input devices (pin photodiode plus FET plus bias resistor) is then (Smith and Personick, 1980) (ii>T

= (i,2>R, + ( C > e

+ (ii)FET

(22a)

or

where ( i i ) R , , ( i i ) d , and (i,Z)FET are given by Eqs. (8), (15), and (20), respectively; the Personick integrals I , and l3 are given by Eqs. (9) and (21), respectively; and I d = Id,,, Idpin and CT = Cpin CFET C,. Equation (22) for the total equivalent input noise due to the input devices contains much of the information about pin-photodiode and FET technology requirements and about amplifier design constraints. In high-sensitivity designs, the input resistor Johnson noise term (i,2)R, is made almost negligible by making Ri large. The new receiver circuits, which can use large Ri values (e.g., 1 - 10 Mi2 at 45 Mb/s) without sacrificing bandwidth, are described in Section V; earlier lower-sensitivity receiver circuits used low values of Ri to achieve the required bandwidth and were dominated by the resultant Johnson noise. (Typically Ri was from a few hundred ohms to a few thousand ohms at 45 Mb/s, giving 1000- 10,000 times more Johnson noise power.) The leakage current shot noise has been made negligible by reducing the pin photodiode and FET leakage currents. This leaves the input FET noise as the fundamental noise source; the receiver noise with low-leakage devices and state-of-the-art circuitry reduces to approximately.

+

+

+

-

(This omits the noise of the rest of the receiver circuit, which will be discussed next in Section III.B.2.) Thus, the circuit-equivalent input-noise power (or mean-square noise current) is approximately proportional to B 3 , C . f / g , and the channel noise

41 1

LIGHTWAVE RECEIVERS

factor r. One can write an input circuit figure of merit that is independent of bit rate (Smith and Personick, 1980):

The receiver optical sensitivity is inversely proportional to the root-meansquare noise current and therefore is approximately proportional to ,/%B-3’2. Note that if the mean-square noise current were proportional to B 2 rather than B 3 ,the photocurrent charge per bit would be constant. In fact, the charge per bit goes up approximately as the square root of the bit rate. 2. Complete Receiver Circuit Noise Expressions with Device Figures of Merit This subsection first derives figure-of-merit expressions for pinphotodiode, FET, and FET IC technologies, then extends the Smith and Personick noise expression of Section 3.2.1 to include the noise from the input FET load device and the rest of the receiver circuit. The resultant expression is used to calculate the theoretical receiver sensitivities of Sections III.B.3 and V.H The front-end figure of merit of Eq. (24) can be rewritten as the product of an FET technology figure-of-merit times a pin-photodiode figure-of-merit. In a typical high-performance FET technology, the source-to-drain spacing or channel length is fixed at the minimum reliable resolution of the lithography. The FET transconductance and input capacitance are both proportional to the channel width (typically a few tens to a few hundred micrometers); the gm/CFET ratio is set by the FET technology. Rewriting gm as ( & , , / c ) c F E T , where CFET determines the FET size, and taking CT = CFET Cpin C, gives

+

+

Differentiating the second bracketed term with respect to CFET says that the optimum-size FET in a given technology has a gate width such that

c,,

= Cpjnf

c, = 9CT.

(254

Assuming such an optimum-sized FET, one can now write = MFET Mpin

9

where the FET-technology figure of merit is

where

fT

= gm/2dFET

is the unity gain frequency of the FET technology.

412

GARETH F. WILLIAMS

Mpinthe pin-photodiode (and stray capacitance) figure of merit is

1 4(Cpin+ C,) The optical receiver sensitivity is, again, proportional to the square root of the product of these figures of merit; these figures of merit will be used in Section V.H to discuss the sensitivity improvements calculated for futuretechnology receivers. The FET figure of merit indicates that high-sensitivity receivers should be made with microwave or VHSIC technologies because such FETs have the highest I;s. The optical sensitivity of such receivers is approximately proportional to the square root of fT; thus, these high-frequency FETs are preferred, even at low bit rates (e.g., 10 Mb/s), but for noise, not frequencyresponse, reasons. The two presently preferred FET technologies are 1-pm gate-length GaAs MESFETs and 1-pm gate silicon MOSFETs. The two technologies offer roughly the same sensitivity in theory; the GaAs has higher g,/C, but the silicon has a lower channel noise factor (Section III.B.3). In practice, GaAs is presently somewhat superior in performance, but silicon is much less costly. The photodiode and stray-capacitance figure of merit of Eq. (28)indicates that the pin-photodiode capacitance Cpinand the stray capacitance C, must be made as small as possible. Assuming an optimum-size amplifier input FET (gate width such that CFET= Cpin+ CJ, the optical receiver sensitivity is inversely proportional to the square root of (Cpin Cs).For small C,, the optical sensitivity is inversely proportional to the square root of the pinphotodiode capacitance. Total front-end capacitances C , are presently about a picofarad or less. Thus, the pin-photodiode technology objectives are low capacitance, low leakage current, and high quantum efficiency. Low capacitance means either a - 10'5/cc) for a wide depletion region, a small diameter low doping (area), or both. For multimode systems, which are becoming obsolescent, the photodiode diameter is typically somewhat larger than the fiber core diameter for efficient optical coupling. Present photodiodes are typically 50-100 pm in diameter. However, the output of a single-mode fiber can be focused to a diffraction limited spot; this makes possible 5- to 25-pm diameter photodiodes. How small a photodiode for a single-modesystem can be made is primarily an optical coupling and device packaging problem; changing the mask to make a smaller diode would be easy. Note that the problem of efficiently coupling a single-mode fiber to a small-area optical device has already been solved for semiconductor laser transmitters. Such packages soon will be low M .

=

+

LIGHTWAVE RECEIVERS

413

cost as laser transmitters ride down the learning curve. A four-fold reduction in capacitance appears easily practical; this would correspond to a 3-dB improvement of the optical receiver sensitivity. However, for low-cost applications, the preferred alternative may be to leave the photodiode diameter large, use the simplest packaging, and concentrate on reducing the doping instead. Thus, a high-sensitivity pin-FET receiver is a combined FET technology problem (high g,/CT), pin-photodiode problem (low Cpin,low leakage current), circuit design problem (large Ri for low Johnson noise while preserving a wide bandwidth and dynamic range), and packaging problem (low stray capacitance C,, small photodiode diameter if economic). The pin-photodiode figure of merit can also be read as a total-inputcapacitance figure of merit. For an optimized receiver, Mpin= 1 / ( 2 c T ) by Eq. (25a); thus, for a given FET figure of merit, the sensitivity of an optimized receiver is inversely proportional to the square root of CT. Actual sensitivity calculations should also include corrections for the noise due to the input FET drain load device and for the noise of following stages. The Smith and Personick noise calculations are readily extendable to include these effects. In the new IC optical receivers of Section V, the input FET's load device Q1 is typically another FET. [In hybrid IC (HIC) optical receivers, the load device is typically a resistor but HIC receivers will become absolescent.] The IC load transistor QLadds an extra mean-square noise current at the drain of QI of (29) k gmLr ~ dj. where gmLis the transconductance of QL. Equation (16) can then be rewritten as (30) d(i:)cirain = 4kT(gm1+ gmL)rdf, d(i:

)L

=4

where gmlis the transconductance of input FET Q 1 . Retracing the derivation of Eqs. (17)-(20) then gives a total mean-square equivalent input noise current due to Q1 (input FET) and QL (load FET) of

where I,, the third Personick integral, is given by Eq. (21). The optimum ratio of QLsize (gate width) to Q1 size is a tradeoff. A large QL gives a higher drain current density in Q1, and therefore a higher gml;a small Q L gives less drain current noise due to Q L . Assuming an optimum QL to Q1 ratio for the particular technology, one can revise Eq. (27) to give a

414

GARETH F. WILLIAMS

figure of merit for the FET IC technology: 2

MI, =

9ml (gml -k gmdrCFET’

(32)

MI, depends only on the IC technology, so long as the input FET Q1 is scaled so that CFET= CT/2and QLis scaled for minimum noise. The receiver figure of merit now is MI, Mpin,where Mpinis given by Eq. (28) as before; the overall receiver sensitivity is approximately proportional to the square root of the receiver figure of merit. As will be shown below, the QLnoise term typically is more important than the all following stage noise terms combined; thus, Eq. (32) is a good IC technology figure of merit and gives good approximate sensitivities. Note that fine-line GaAs depletion-mode MESFETs typically require a higher drain current, hence a larger QL, than fine-line silicon enhancementmode MOS technologies. This effect cancels part of the g,/C advantage of GaAs, as does the higher channel noise factor r in GaAs. The noise of the following stages of the receiver can be represented as an equivalent stage input mean-square noise voltage of d(ef ) per unit bandwidth df. If as-l is the total voltage gain of the s- 1 stages preceding stage s, the mean-square equivalent input noise voltage due to stage s is

-

d(ef)r = d(ef )/a:-

(33) Assuming ( e f ) is constant in frequency, one can repeat the derivation of Eqs. (17)-(20), substituting d(e,Z)I for d(e:)F,T. This gives the mean-square equivalent input noise current due to stage s: (i:)s

=9 ( 2 n C T ) 2 1 3 8 3 as- 1

(34)

The total mean-square equivalent input noise current of the receiver is then 4kT

( i : ) = -I2B Rl

+ 2ql,12B + 4kTr-(2nCT)2 gm 1

13B3

where the first two terms are the input-resistor Johnson noise and the leakagecurrent shot noise, respectively, the third term is the input FET noise, the fourth term is the noise of the input FET load (QL),and the last term is the sum over the noise contributions of the following stages. Generally, the following-stage noise current is less important than the load device contribution; even the second-stage mean-square noise is divided by the first-stage

LIGHTWAVE RECEIVERS

415

voltage gain squared. Thus, Eq. (32) is a good figure of merit for comparing FET IC technologies. 3. Sensitivities of Present-Technology pin-FET Receivers

This subsection calculates theoretical sensitivities for present-technology silicon and GaAs FET IC receivers and compares the two technologies. It also includes sensitivity results from the literature. Section V.H will extend the calculations of this section to include sensitivity calculations for probable future-technology IC receivers. For the numerical sensitivity calculations, g,/C is taken as 70 mS/pF for 1-pm silicon MOSFETs and 90 mS/pF for 1-pm GaAs MESFETs; r is taken as 1.2 for the silicon and 1.5 for the GaAs. Since the FET technology figure of merit is g,/(CT) by Eq. (27), the silicon and GaAs sensitivities are essentially equal in theory; GaAs FETs have higher transconductances, but silicon FETs have lower channel noise. In practice, GaAs designs are presently a few decibels more sensitive. The Personick integrals I , and I , can be roughly estimated by remembering that the channel-filter bandwidth is typically 0.56 times the bit rate B for synchronous detection NRZ receivers. Thus, taking IF(y)(= 1 for y c 0.56 and F ( y ) = 0 for y > 0.56 in Eq. (9) for I , gives I , = 0.56; using this approximation in Eq. (21) for I , gives I , = 0.059. In fact, I , is taken as 0.564 and 1, as 0.0868 (Paski, 1980b). Values for I , and I , for different input and output (filtered) pulse shapes are found in Personick (1973) and in Smith and Personick (1980). Figure 7 shows theoretical digital optical-receiver sensitivities versus bit rate for 1-pm gate length GaAs and silicon IC receivers using InGaAs pin photodiodes. The optical wavelength is 1.3 pm. The calculations assume the new high-sensitivity, micro-FET feedback IC receiver designs, as described in Sections V.A, V.B, and V.F, in which the input/feedback resistor noise is almost negligible; the sensitivities were calculated on paper designs using Eq. (35) for the total receiver noise (see Section V.H). Figure 7 assumes a l-pF total front-end capacitance, e.g., Cpin= 0.40 pF, C, = 0.10 pF, C,,, = 0.5 pF; the silicon input FET then has a transconductance of 35 mS; the GaAs input FET has a transconductance of 45 mS. The photodiode leakage current is taken as 1 nA at 20°C and 15 nA at a maximum operating temperature of 85°C (the 15x increase assumes that the leakage is a G-R current via midgap states.) The bit-error rate is Since the theoretical sensitivities of the silicon and GaAs FET IC receivers are essentially the same, only one curve is plotted for both. Note also that the calculated sensitivities go approximately as B - 3 / 2 ; the first two terms in Eq. (35) for the input noise squared are negligible except at low bit rates; the

416

GARETH F. WILLIAMS

m 0

A

/

a -40 J

-0

& -50 0 W I-

s -60 !Y W

0

--

10 Mb/s

100

I Gb/s

BIT RATE

FIG. 7. Theoretical sensitivities of present-technology pin-FET lightwave receivers. Dots show GaAs FET receiver measurements from the literature; triangles show silicon FET receiver measurements.

others go as B 3 except for the (small) following-stage noise summation, which increases slightly faster than B 3 because the gain per stage is less for wider bandwidth stages. Finally, note that the calculated sensitivities are competitive with those calculated for present APD receivers to 100-200 Mb/s. Section V.H presents further detail on these IC optical receiver sensitivity calculations and extends them to include sensitivity calculations for probable future-technology IC receivers. It also presents scaling laws for the basic IC design of Section V.F and uses the device figures of merit from the last section for an intuitive discussion for the calculated sensitivity improvements to be expected from the future-technology receiver designs. Figure 7 also shows the best experimentally achieved pin-FET receiver sensitivities as of this writing. The GaAs FET receiver measurements at 34 Mb/s, 140 Mb/s, 280 Mb/s, and 565 Mb/s were by D. R. Smith et al. (1982); the 45-Mb/s measurement was by Williams and LeBlanc (1986); the 1Gb/s measurement was by Linke et al. (1983).The 45-Mb/s silicon MOSFET receiver measurement was by Ogawa et al. (1982); the 800-Mb/s silicon result was by Abidi et al. (1984). All were measured at room temperature. Unfortunately, the silicon MOSFET optical receivers are presently a few decibels less sensitive than either theory or RMS noise measurements would indicate. This may be because fine-line silicon FETs were developed as a digital IC technology in which noise was not important. The fine-line GaAs MESFET technology was developed as an analog technology for microwave receivers. Therefore, the GaAs noise problems were important and were solved.

LIGHTWAVE RECEIVERS

417

At present, silicon FETs appear preferable for high-volume, low-cost designs in which a modest sensitivity penalty is acceptable; GaAs takes over for higher-sensitivity, higher-cost designs (e.g., long-haul transmission systems). Silicon may well take over the high-sensitivity applications when its noise problems are solved. On the other hand, the GaAs technology may become cheaper and its scale of integration may be increased. The two technologies are in a race with each other; the silicon technologists must improved their sensitivity; the GaAs technologists must reduce their costs. GaAs FET IC technologies are also presently preferred over silicon for optical fiber receivers above 1 Gb/s, both because of the higher saturation velocity in GaAs and because GaAs FET ICs have lower parasitic capacitances. Short-channel GaAs MESFETs are fabricated on a semiinsulating substrate; the parasitic capacitances to the substrate are very small (Sze, 1981). Short-channel silicon MOSFETs are presently junction isolated and have the full junction capacitance to the substrate (Sze, 1981).In addition, the gates of silicon MOS transistors overlap both the source and the drain; GaAs MESFETs do not have this overlap capacitance; the Schottky gate metallization defines the channel and therefore cannot overlap the source and drain.

C. Avalanche Photodiode Receiver Noise and Sensitivity Calculations In an avalanche photodiode (APD), the primary photocurrent is multiplied by impact ionization in the p-n junction, which is operated reversebiased near breakdown. The multiplied photocurrent signal goes to the receiver circuit. Consider an APD in which the primary photocurrent is multiplied (on average) by a factor (M). If the multiplication process were noiseless and the APD leakage current negligible, the optical signal power required by the receiver would be decreased by a factor (M) and the optical receiver sensitivity increased by a factor (M). In fact, the multiplication process is noisy because it is the result of random impact ionizations. The equivalent mean-square primary photocurrent noise d(i,2), per bandwidth df due to the APD is the shot noise of the photocurrent I,, plus leakage current 1, times the McIntyre excess noise factor F ( ( M ) ) (McIntyre, 1966): (36) d(i,t>D = 2 q U p i l + W ( < W ) d J For noiseless multiplication, F ( < M >) = 1, and the noise is just the photocurrent plus leakage-current shot noise; F((M)) is the factor by which the real avalanche multiplication increases the noise over that of a noiseless multiplication.

418

GARETH F. WILLIAMS

If the avalanche is initiated by injection of photocarriers at one side of the avalanche region, F ( ( M ) ) is given by (McIntyre, 1966)

[

F ( ( M ) ) = ( M ) 1 - (1 - k)

(y; ‘)’I.

(37)

where k is the ratio of the electron and hole ionization coefficients. In silicon, electrons have the higher ionization coefficient; the 0.8-pm wavelength silicon APDs use photoelectron initiated multiplication, and k is the ratio of the hole ionization coefficient to the electron ionization coefficient.In InGaAs/InP 1.3to 1.6-pm wavelength APDs, the avalanche is photo-hole initiated and k is the ratio of the electron ionization coefficient to that of holes. In silicon APDs, k is typically 0.02 (Melchior et al., 1978);in InGaAs/InP APDs, k is typically 0.4 at present. The APD is not the “solid state equivalent of a photomultiplier” because both electrons and holes can impact ionize in an APD; in a photomultiplier, only electrons impact ionize (on the dynodes); there are no holes in a vacuum. In an electron-initiated APD, the electrons of the primary avalanche travel downstream, creating electron-hole pairs by impact ionization; the resultant holes travel upstream and can create more electrons, thus initiating secondary electron avalanches, etc. This hole feedback makes avalanche carrier multiplication much noisier than photomultiplier electron multiplication. When the electron-hole feedback loop-gain becomes unity, the avalanche is selfsustaining and the diode breaks down. The higher the multiplication, the closer to breakdown and the noisier the multiplication. For k = 0 (best case), only one carrier ionizes and the APD multiplication process is similar to that of a photomultiplier. In this limit, F ( ( M ) ) becomes 1 F((M))=2 -(M). For k = 1 (worst case), F ( ( M ) ) becomes F((M)) = (M);

(39)

the avalanche multiplication is then more noisy due to the carrier feedback. In general, the lower the k, the less noisy the multiplication. Thus, the silicon APDs (k 0.02) are much less noisy for a given multiplication than the InGaAs APDs ( k 0.5); unfortunately, silicon is transparent below 1.1 pm and cannot be used for 1.3- to 1.6-pm APDs. Following Smith and Personick (1980), the total equivalent mean-square primary photocurrent noise (if)ph is the APD noise integrated over the receiver bandwidth plus the equivalent mean-square input noise current ( i : )T

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419

LIGHTWAVE RECEIVERS

of the receiver amplifier, divided by (M)2: = 2q(iph

+ z8)F((M))zlB

( i 3 T +7 ,

(40) (M) where the integral over frequency is I I B ; I, is the first Personick integral. The receiver amplifier circuit noise (i;)* is given by the expressions of Section 1II.B. When a logic-zero bit is transmitted, the optical signal, hence the photocurrent iph, is ideally zero; the zero-level mean-square noise then is due only to the APD leakage current and to the amplifier noise. When a logic one bit is transmitted, the mean-square noise is increased by 2 q l p h l F((M))Z, B, which is the noise due to the avalanche multiplication of the one-level photocurrent. Since an APD receiver's one-level noise is greater than its zerolevel noise, the decision threshold D is typically set closer to the zero level than to the one level. In practice, the optical transmitter is not completely turned off during zero bits, for reasons of transmitter response speed and (for laser transmitters) for reasons of optical frequency stability. Take the zero-bit optical signal level P(0) as a fraction r of the one-bit optical signal level P(1): (i:>ph

where (iph(0)) is the expected photocurrent at the sampling instant for a zero bit; (iph(l)) is the photocurrent for one bit. r is called the transmitter optical extinction ratio. Ideally, r should be zero; in practice, r is may be as high as 0.2.This means a smaller photocurrent signal component for a given average optical power. In addition, the zero-level photocurrent is the functional equivalent of a leakage current and adds a corresponding noise term to both the zero and one signals. Taking the avalanche-noise amplitude distribution as approximately Gaussian, the error probability or bit-error rate (BER) is given by Eq. (3): BER = f erfc(Q/fi) where, for a zero bit, and ith is the decision threshold and (i:o)'iz is the root-mean-square noise for a zero bit. Similarly, for a one bit,

[Eq. (3), Fig. 4b]. Qo = Q1 = 6 gives a bit-error rate of Following Smith and Personick (1980), taking Qo = Q1 equal to the Q required for the desired, BER, and using the APD receiver noise equation (40)

420

GARETH F. WILLIAMS

to give ( i i 0 ) l / * and ( i i l ) l / ’ in Eqs. (42) and (43), yields the photocurrent signal required for the desired BER. Assuming that zero bits and one bits are equally probable, and converting the photocurrent to an input optical power, gives an equation for the average optical power required for a given BER:

(44) where Q is the photocurrent signal-to-average-noise ratio for the given BER, multiplied by a prefactor that is unity for zero (ideal) extinction ratio r:

(i,2)T is the equivalent mean-square amplifier noise, Idm is the leakage current of the APD that undergoes multiplication, and F ( ( M ) ) is the McIntyre excess noise factor of Eq. (37). Note that a high multiplication ( N ) reduces the effect of the receiver amplifier noise but gives more avalanche multiplication noise and a higher F(( M)) in Eq. (40).A low (M) gives lower avalanche multiplication noise but increases the effect of amplifier noise. The optimum (M) is determined by this trade-off. In theory, Eq. (44)can be differentiated to find the optimum gain; the result is an unmanageable expression of no particular physical interest. In practice, one is much better off using a minimum finder program to find the optimum gain (M ) and the minimum qfj numerically. InGaAs/InP heterostructure APDs are presently preferred for 1.3- to 1.6-pm applications because they offer low leakage currents and a reasonable k-ratio. The light is absorbed in a low-field InGaAs (E, = 0.73 eV) layer, and the avalanche multiplication takes place in a high-field InP layer (E, = 1.35 eV). The use of a wide-gap avalanche region avoids the tunneling current problem of all-InGaAs APDs (Nishida et al., 1979; Kanbe et al., 1980; Susa et at., 1980). Until recently, these APDs were slow due to hole trapping by the InGaAs-to-InP valence-band step; this problem was solved by adding several intermediate band-gap layers between the InGaAs and the InP or by continuously grading the composition between InGaAs and InP (Matsushima et at., 1982; Campbell et at., 1983). Figure 8 shows theoretical InGaAs/InP APD receiver sensitivities versus bit rate. The figure assumes that the APD is used with an optimized 1-pm gate

LIGHTWAVE RECEIVERS

42 1

-40

n w

L I-"

w n

-50 nGoAs/TnP APD

-60 10 Mb/s

100 BIT RATE

1 Gb/s

FIG.8. Theoretical InGaAs/InP APD receiver sensitivities. 1 = 1.3 pm, I , = 3 nA at 20"C, 45 nA at 85"C, CT = 1 pF, high-sensitivity receiver ICs. Dots show APD receiver measurements from the literature.

technology GaAs FET receiver amplifier, which is the most sensitive amplifier type at present. It also assumes a total receiver input capacitance of 1 p F (this includes the APD capacitance), a primary leakage current at 20°C of 3 nA, and an APD k-ratio of 0.4. Figure 8 shows the pin-FET receiver sensitivities from Fig. 7 for comparison, plus the 20°C and 85°C InGaAs/InP APD sensitivities. The 85°C leakage current is taken as 45 nA. This assumes that the APD leakage current is a generation-recombination current via midgap states in the InGaAs, which gives a 15x increase in leakage current from 20°C to 85°C. Figure 8 also shows InGaAs/InP APD-FET receiver measurements from the literature. The measurements at 420 Mb/s and 1 Gb/s were by Campbell et al. (1983); those at 2 Gb/s and 4 Gb/s were by Kasper et af. (1985). All are the best values at those bit rates, as of this writing; all used the APD of Campbell et al.. Note, however, that these measurements were made at room temperature; however, in field use, the maximum operating temperature is typically 85°C. Unless the receiver and APD temperature is controlled, the 85°C sensitivity is what matters in practice. Figure 8 shows that present InGaAs/InP 1.3- to 1.6-pm APDs in theory offer little sensitivity advantage below 100 Mb/s at a maximum operating temperature of 85°C. This is due to the noise caused by the avalanche multiplication of the primary leakage current; an APD without leakage

422

E

m

GARETH F. WILLIAMS

-20

85°C

s

a

W

in5 2

I

-30

-40

0

t 0

-50

W

10 Mb/s

100

1 Gb/s

B I T RATE

FIG.9. Theoretical Ge APD receiver sensitivities. I = 1.3 pm,I, = 100 nA at 20”C, 1 pA at 8 5 T ,CT = 1 pF, high-sensitivity receiver ICs.

current would offer a sensitivity improvement at all bit rates. In addition, reducing the leakage current is presently more important than improving the k-ratio for bit rates less than 500 Mb/s at 85°C. Germanium APDs have also been used in 1.3- to 1.6-pm optical receivers (Mikawa et al., 1981). Present Ge APDs have very high leakage currents but until recently were faster than InGaAs/InP APDs and therefore once were preferred for Gb/s applications. Figure 9 shows theoretical germanium APD sensitivities versus bit rate assuming 16 = 100 nA at 20°C 16 = 1 pA at 85°C k = 0.5, and a GaAs IC front end. The InGaAs/InP APD is presently superior to the germanium APD at every bit rate. In fact, at 85”C, InGaAs pin photodiodes are superior to germanium APDs for bit rates below 300 Mb/s.

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IV. FIRST-AND SECOND-GENERATION LIGHTWAVE RECEIVERS

This section traces the evolution of lightwave receivers from the early voltage-amplifier and transimpedance designs, in which the sensitivity was limited by circuit noise, up through the present versions of the integrating or “high impedance” receiver, in which the sensitivity is limited primarily by the fundamental device noise. Unfortunately, the high-sensitivity, integrating receivers require equalization, with its attendant problems, and have very limited dynamic ranges. Therefore, almost all commercial systems still use the transimpedance receiver, despite its lower sensitivity. In time, both transim-

LIGHTWAVE RECEIVERS

423

pedance receivers and integrating receivers will be displaced by the highsensitivity, nonintegrating active-feedback IC receivers of Section V. Subsection 1V.A describes the early voltage-amplifier optical receivers. These receivers had very low sensitivities but were the first lightwave receivers. Subsection 1V.B describes the simple integrating optical receivers (Personick, 1973; Goell, 1974), which can achieve very high sensitivities but are vulnerable to saturation on long strings of ones or zeroes in the data. This problem was solved by encoding the data stream (Brooks, 1980); upgraded versions of these receivers have been introduced in a few 1.3-pm transmission systems (Hooper et af.,1980; D. R. Smith et a]., 1980). Subsection 1V.C describes conventional transimpedance receivers. These receivers do not integrate the signal but are less sensitive than integrating receivers. These were first used in 0.8-pm transmission systems; a silicon APD detector was used to make up for the lower sensitivity (R. G. Smith et al., 1978). A bipolar IC version for 1.3-pm systems has achieved good sensitivities at bit rates of several hundred Mb/s while using a nonmultiplying pin photodiode (Paski, 1980; Snodgrass and Klinman, 1984). Subsection 1V.D describes integrating transimpedance receivers (Gloge et al., 1980; Ogawa et al., 1983). These receivers are as sensitive as the simple integrating receivers of Subsection 1V.B and d o not require encoding of the data stream because they integrate the signal over less of the bandwidth. However, the integration pole frequency is temperature sensitive; these receivers are difficult to equalize reliably, e.g., if the temperature can vary over a commercial temperature range of - 25°C to + 85°C. A. Voltage-Amplijier Optical Receivers

Figure 10a shows a generalized schematic of a voltage-amplifier optical receiver. The photocurrent develops a voltage across a series load resistor RL; this voltage is then amplified by a voltage amplifier of gain A. Ideally, the output voltage is vOut = ARLiph.Note, however, that RL must be small enough to keep the photocurrent signal from being integrated by the photodiode capacitance Cpinplus the capacitance Cx due to the input transistor of the amplifier, plus any stray input capacitance C,. Explicitly, R , must be less than the capacitive impedance l/(oc,)over the desired bandwidth, where C , = Cpin+ C , + C, is the total capacitance seen by the photocurrent (Fig. lob). For a bandwidth, or pole frequency, greater than the bit rate B,

I 1/(27CC,B). (46) Such voltage-amplifier receivers have low sensitivities because of the RL

Johnson noise current of the small-value RL needed to avoid signal integration

424

GARETH F. WILLIAMS

FIG. 10. Voltage amplifier receiver: (a) receiver circuit, (b) equivalent circuit.

by CT(Personick, 1973b);by Eq. (8), 2

4kT

= -I2& RL

(47)

Consider a 44.7-Mb/s design example; take CT = 4 pF (typical for early receivers). By equation (46), RI I 8 9 0 R. Assuming a noiseless voltage amplifier,so that the Johnson noise associated with R, is the only noise source, the best possible sensitivity for the voltage-amplifier receiver at 1.3 pm [by Eqs. (1) and (811 is only -35 dBm optical. The state-of-the-art receiver sensitivity at that bit rate is -51.7 dBm, an improvement of 16.7 dB optical or 33 dB electrical over the voltage-amplifier receiver circuit. If C, for the voltage-amplifier receiver were reduced to the 1 pF of present state-of-the-art receivers, the sensitivity penalty still would be 13.7 dB optical. The simple voltage amplifier optical receiver now is not preferred for any application and has disappeared from the literature. B. Integrating Optical Receivers

The integrating receiver or “high-impedance” receiver (Fig. 1la) can achieve excellent sensitivities because R , is made large enough so that its Johnson noise is negligible, (Personick, 1973a, 1973b; Goell, 1974; Hooper

LIGHTWAVE RECEIVERS

425

et al., 1980). However, the photocurrent signal is then integrated by C,; the signal must be differentiated (equalized) later, as shown. Waveforms are shown in Fig. 11b. For 45-Mb/s receivers using state-of-the-art photodiodes and 1-pm gate-length GaAs FETs or silicon MOSFETs, RL should be 2 1 MR for best sensitivity. For R , = 1 MR and CT = 1 pF, the resultant input current-to-voltage pole is 160 kHz (Eq. 46); above 160 kHz the photocurrent signal is integrated by C,. Thus the output signal, uo, is integrated over most of the signal bandwidth, even though the voltage gain, A, is flat. For the 45 Mb/s NRZ case, the Nyquist signal bandwidth I,B is -25 MHz; thus, the signal is integrated over seven octaves, from 160 kHz to 25 MHz. The equalizer must differentiate the signal over these same seven octaves; to do this, the equalizer zero frequency is set equal to the input pole frequency. The equalization technique of Fig. 1 l a introduces a noise penalty. The reason is that the passive equalizer attenuates the signal; the peak attenuation is theequalization ratio I , B / f , = 25 MHz/l60 KHz = 156X for the45-Mb/s example. This signal attenuation enhances the effect of noise in stages following the equalizer, resulting in an equalizer noise penalty; here, if A is less than 156, the low-frequency signal voltage after the equalizer is smaller than the signal at the amplifier input! Active equalization, which would reduce this

EQUALIZER

n

.

: *TIME

FIG. 11. Integrating receiver with equalizer:(a) circuit, (b) waveforms.

426

GARETH F. WILLIAMS

noise penalty, is difficult for these high frequencies and large equalization ratios, due to stability problems. In theory, this equalizer noise penalty may be reduced by increasing the gain, A, before the equalizer. However, for a random bit stream (maximum information content), increasing A reduces the photocurrent for which saturation of the integrating amplifier on long strings of ones (or zeroes if ac coupled) causes an unacceptable bit-error rate. This reduces the dynamic range, resulting in a sensitivity versus dynamic-range tradeoff. Even with minimal dynamic range, A is still limited and the equalizer noise penalty is still appreciable at high bit rates. A second sensitivity versus dynamic-range tradeoff is involved in choosing RL. A high R, improves sensitivity by reducing the input Johnson noise current but, for the maximum-information-content random-bit-stream, increases the probability of saturation on long strings of ones or zeroes because the integration pole frequency, fp, is lower. A low R, reduces sensitivity but improves dynamic range; both the Johnson noise and fp are increased. Both sensitivity versus dynamic-range tradeoffs may be reduced by encoding the data stream to limit the number of consecutive ones or zeroes; a 7B/8B encoding technique (Brooks, 1980) is used by the British Post Office (BPO) and others. In this technique, eight bits are transmitted for every seven bits of data; the redundancy allows the total disparity (difference between the number of ones and zeroes transmitted) to be kept close to zero. This removes the very-low-frequency signal components; large equalization ratios are not needed; the equalizer attenuation is reduced, and the allowable gain before the equalizer is increased; the equalizer noise penalty almost vanishes, and RL can be made large. These encoding techniques make the simple integrating receiver of Fig. 11 practical for transmission applications. However, there is a small sensitivity penalty because more bits must be transmitted to pass the same amount of information. If the receiver is FET noise limited, the mean square noise current goes as B 3 (Eq. 35) and the optical sensitivity goes as B-3'2. A 7B/8B code requires an 8/7 higher bit rate, reducing the optical sensitivity by a factor (7/8)312or 0.9 dB optical. In addition, for encoding schemes in which any error in an eight-bit transmitted block causes all seven recovered data bits to be in error, the bit-error rate (BER) is eight times larger for a given signal-to-noise ratio. Using Eq. (3) or Fig. 5 for the BER versus signal-to-noise ratio says that this is equivalent to an additional encoding sensitivity penalty of 0.2 dB optical for a total encoding sensitivity penalty of 1.1 dB. If a seven data bit plus one sign bit-encoding scheme is used, only a sign bit error will cause seven data bit errors; the BER is only doubled; the additional sensitivity penalty is then only 0.08 dB, for a total encoding sensitivity penalty of 1 dB.

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427

LIGHTWAVE RECEIVERS

C. Transimpedance Optical Receivers

Transimpedance receivers were developed to increase the sensitivity achievable without signal integration; the ideal transimpedance circuit of Fig. 12a theoretically allows R, to be increased to the same value as in an integrating receiver, but without integrating the signal. The reason is that R , (now R F )is connected around the voltage gain element; this negative feedback raises the input pole frequency. For the ideal circuit of Fig. 12a,

ir

=

= uln(1

+A)

Rf RF This produces a virtual input resistance (Fig. 12b) of a

re should be made small enough to keep the photocurrent signal from being integrated by the receiver input capacitance CT;explicitly, re should be less than the capacitive impedance l/(oC,) over the signal bandwidth. Thus, the receiver bandwidth is simply the input pole frequency due to re in parallel with C, :

re=

RF/(A+I 1

FIG.12. Ideal transimpedancereceiver: (a) circuit, (b) equivalent circuit.

428

GARETH F. WILLIAMS

Thus, the ideal transimpedance feedback configuration would theoretically increase the input current-to-voltage pole frequency or bandwidth by a factor of the gain, A, plus 1. Ideally this pole could be placed above the signal bandwidth by increasing A, eliminating integration of the signal. This would eliminate the need for equalization or coding and thus eliminate the associated noise penalty. Unfortunately, real feedback resistors include a parasitic feedback capacitance C, shunting the physical RF (Fig. 13a). For frequencies greater than fpR = 1/(2nR~C,), the feedback resistor acts like a feedback capacitor and the signal is integrated even for large A. (The exact pole frequency is fp = 1/[2dF(CR + C,/(A + l))] by inspection of Fig. 13b). In addition, even if an ideal feedback resistor had been available, the hybrid IC gain elements used in most conventional transimpedance receivers do not have enough voltage gain, A, to place the input pole frequency, f p , above the passband. Adding stages would add extra phase shift inside the feedback loop and typically would cause instability problems. Thus, the usual approach was to use a low-value R, to avoid signal integration due to C, and to reduce the gain required; the resultant Johnson noise penalty was accepted. The 0.8-pm wavelength, 45-Mb/s receivers of R. G. Smith et al. (1978) used R, = 4 kR, and used a silicon APD to make up the sensitivity loss. The later 1.3-pm wavelength, 274-Mb/s design of Ogawa

"0

0

re

= RF/(Atl)

FIG.13. Real transimpedance receiver: (a) circuit with parasitic capacitances, (b) equivalent circuit.

429

LIGHTWAVE RECEIVERS

and Chinnock (1979) used RF = 5 kR and accepted a modest signal integration in return for a somewhat smaller sensitivity penalty. Paski (1980) demonstrated both approaches in his experimental silicon bipolar receiver ICs; the commercial silicon bipolar receiver IC by Snodgrass and Klinman (1984) was designed to avoid signal integration. D. Integrating Transimpedance Optical Receivers

The transimpedance amplifier circuit now is sometimes used as an integrating receiver (Gloge et al., 1980; Ogawa et al., 1983); this reduces the large equalization ratio often required with the simple integrating amplifier of Fig. 11. The best conventional high-value resistors have C, 0.05 pF. Thus, for the 45 Mb/s case discussed earlier, taking R , = 1 MR, the amplifier must integrate the signal abovej, = 3.2 MHz. Equalization is still required, though the equalization ratio is typically 10-20 times less than for the simple integrating receiver example of Fig. 11. The equalizer noise penalty and the dynamic range/sensitivity tradeoffs for a nonencoded bit stream are correspondingly improved; at low bit rates (B < 100 Mb/s), further improvement might be obtained by active equalization. However, equalization noise is entirely eliminated for all bit rates by the nonintegrating receivers of Section V; the IC implementations are also much cheaper to manufacture than equalized receivers. Both the simple integrating amplifiers of the BPO (Hooper et al., 1980; D. R.Smith ef al., 1980) and the transimpedance integrating amplifiers of Ogawa et al. (1983) use a forward voltage amplifier with a FET input transistor followed by a bipolar junction transistor (BJT) cascode. The folded cascode of Ogawa et al. is illustrated in Fig. 14.The input transistor Q1is a l-micron gatelength GaAs FET for low noise; this is followed by a level shifting P N P cascode stage Q2 and an emitter follower output buffer Q 3 .The input voltage, F,,, creates a signal current Fngmlat the drain of Ql.The emitter of Q2 absorbs most of this current because it is a forward-biased diode and acts like a short. The Q1drain signal current passes through Qz to Rez; the amplifier voltage gain is then A = -gmlre~, (5 1)

-

where gmlis the input FET transconductance. Typically these amplifiers are implemented in hybrid IC (HIC) form (- 1 mil conductor and resistor patterns on a ceramic substrate with discrete semiconductor device chips). For the BPO -type receiver, the input resistor would be connected to a bias source; for a transimpedance design, the input resistor is connected to the output. One problem with GaAs FET transimpedance integrating amplifiers using the voltage gain element of Fig. 14 is that their response pole frequency, f,, is

430

GARETH F. WILLIAMS

. I

\I

I

I I

I I

I

I

"in

I I

I

I

' '-h v

s

FIG.14. FET-BJT cascode voltage amplifier.

approximately inversely proportional to the absolute temperature. For a -25°C to + 85°C temperature range in the field, f, would vary by 1.45:l. The equalizer zero frequency must track this variation in the receiver pole frequency for reliable operation; such tracking is difficult to achieve in practice. This problem arises because f, is approximately proportional to the voltage gain A (Eq. 50), A is proportional to the input FET transconductance gml (Eq. 51), and gml is proportional to the electron velocity in the channel. Now the electron velocity in GaAs near 300°K is approximately inversely proportional to the absolute temperature, due to phonon scattering (Ruch and Fawcett, 1970). This applies from low fields through to velocity saturation. Thus, f , goes approximately as 1/Ttoo. The simple integrating receiver with restricted disparity encoding and limited equalization used by the BPO (Hooper et al. 1980; D.R. Smith et al., 1980) neatly side-steps the equalization tracking problem. The encoding removes the low frequenciesfrom the signal; since no feedback is used, the pole frequency (above which the signal is integrated) is below the lowest frequency signal component. Exactly where the pole is does not matter; the equalizer can simply differentiate over the entire spectrum of the signal. The BPO design also allows ac coupling throughout the amplifier. Of course, the nonintegrating, high-sensitivity active-feedback receivers of Section V avoid equalization problems entirely. Finally, for both types of integrating receiver, the maximum photocurrent is still restricted by the dc drop across RF,even if encoding is used. For the 45-Mb/s case cited, assuming 5 V supplies, this corresponds to a maximum

LIGHTWAVE RECEIVERS

-

43 1

photocurrent of is 2 to 4 PA, or a dynamic range of 22 dB. Thus, the best integrating receiver dynamic range still implies the need for field-installed attenuators in many applications. N

V. ACTIVEFEEDBACK LIGHTWAVE RECEIVER CIRCUITS A. Introduction The new active-feedback receivers of this section are the first lightwave receiver circuits to achieve high sensitivities without integrating the signal and the first to achieve wide dynamic ranges without compromising the sensitivity. In addition, the IC designs can be realized very inexpensively for commercial applications. The high sensitivity, plus the response, dynamic range, and cost advantages mean that these new lightwave receivers are suitable for localarea-network, datalink, and loop-plant applications, as well as for the longhaul transmission systems addressed by most previous receiver designs. The previous lightwave receiver designs of Section IV offered a choice between high-sensitivity circuits that integrate the signal and lower-sensitivity transimpedance circuits that do not. The high-sensitivity circuits were eventually adopted for a few leading-edge transmission systems; however, because they integrate the signal, they have limited dynamic range and require subsequent equalization, with its attendant problems and sensitivity penalty. In addition, the data stream often must be encoded, introducing another modest sensitivity penalty. Finally, it would be almost impossible to extend the dynamic range of these integrating receivers; most possible techniques would vary the input pole frequency above which the signal is integrated, thus requiring a tracking equalizer. The active-feedback receivers described in this section comprise a basic nonintegrating high-sensitivity receiver, plus a dynamic-range-extender/ automatic-gain-control (AGC) circuit. The basic receiver can be realized using either the capacitive feedback circuit or the micro-FET feedback circuit; the former is preferred for hybrid IC designs, the latter is preferred for monolithic IC designs. Both basic receivers offer excellent sensitivity, but their dynamic range is little better than that of previous designs. Accordingly, the dynamic range extender circuits are necessary for most loop, localarea-network, and datalink applications and are useful for many transmission applications. These active-feedback receivers are somewhat more sensitive than the integrating receivers of the prior art; the equalizer and its noise penalty are eliminated, encoding and its noise penalty are unnecessary, and the dynamicrange-extender circuits permit the basic receivers to be optimized solely for

432

GARETH F. WILLIAMS

sensitivity, eliminating all sensitivity versus dynamic-range tradeoffs. The dynamic range is essentially unlimited (54 dB optical, 108 dB electrical demonstrated) and extends to higher optical powers than available from present transmitters. A hybrid IC active-feedback receiver was demonstrated and monolithic IC active feedback receivers were proposed by Williams (1982, 1985); the first such IC was demonstrated by Fraser et al. (1983). IC versions are now in production by AT&T Technologies(Morrison, 1984;Steininger and Swanson, 1986), and a l.ir-Gb/s implementation has been announced. (Dorman et al., 1987). However, these commercial designs are beyond the scope of this section. In theory, the ideal transimpedance amplifier of Fig. 12a (Section 1V.C) could have been used for the basic nonintegrating, high-sensitivity receiver. However, as was discussed in Section IV.C, previous attempts to realize this basic receiver encountered two problems, neither of which was solved in those designs. Both problems were caused by the need to use a large-value feedback resistor R , for low Johnson noise. The first problem was that the gain elements used did not have enough voltage gain to avoid signal integration by the receiver input capacitance if a large-value feedback resistor was used. Adding stages to these designs would have added phase shift within the feedback loop, which typically would cause ringing or instability. Solutions to this problem will be covered in the sections following. The second, and more significant, problem was that conventional largevalue feedback resistors act like feedback capacitors over most of the bandwidth, owing to their parasitic shunt capacitance CR (Fig. 13a). This causes signal integration over most of the bandwidth; thus, using a large-value conventional feedback resistor for high sensitivity gives the integrating transimpedance receiver of Section 1V.D. To avoid this problem in a typical 45-Mb/s design with R, = 2 MQ would require CRto be less than CkW1 pF, which is impossible with conventional feedback resistors. There are two ways to solve the problem of the parasitic feedback capacitance, C,, and make a nonintegrating high-sensitivity receiver; C, can either be eliminated by use of a nonconventional feedback resistor or CRc a be used as part of the feedback element. Section V,B discusses the micro-FETfeedback basic receiver, in which CRis eliminated by using a special-design micro-FET as the feedback resistor; these designs are preferred for IC implmentations. Section V.C discusses the capacitive feedback basic reccmr, in whit24 C, is used as part of the feedback element; these designs ale pse, fened for hybrid IC (HIC) implementations. Section V.D then describes the dynamic-range-extender/AGC circuits used with thaw two b a ~ creccivet

LIGH'I'WAVI: REC'EIVEHS

433

circuits to form the complete high-sensitivity, wide-dynamic-range, nonintegrating receivers. The remaining four scctions discuss complete active-feedback receiver designs. Section V.E describes a hybrid IC design using the capacitive-feedback basic receiver circuit. Section V.F describes FET IC designs using the microFET-feedback basic receiver. The IC designs are preferred for commercial applications. Two principal IC circuit examples are presented, with a discussion of NMOS, CMOS, and GaAs implementations. Scction V.G discusses how these 1C receiver designs are scaled to different bit rates; Section V.H then presents calculations of IC receiver sensitivities for bit rates between 10 Mb/s and 4 Gb/s. These calculations include scnsitivities of receivers using present pin-photodiodc and FET IC technologies, plus sensitivities to be expected from probable future receivers, given expected advances in pin-photodiode and FET IC technologies. The scnsitivity improvements expected in these future-technology receivers are then explained on the basis of the device figures-of-merit of Section III.B.2. Finally, note that although the new active-feedback receiver designs behave much like the ideal transimpedance receivers of theory, thecircuits are quite different. Both active-feedback designs use unconventional feedback elements never contemplated in the simple ideal-transimpedancc receiver; for csitmpk. both include irn itctivc coniponcnt (c.g.,;I tritnsistor)in thc feedback circuit. A second dilli-rcncc is that h ~ t hiictivc-f~cdhiickreccivers tisuitlly include one of the new dynainic-raiigc-e~tetider~ AGC circuits, which givcs these receivers thc tlytiitnjic rangc needed for most loop, Icxal-area-network. iiiid ilatalink applicittions. The new A