Springer Series in Electrophysics Volume 18 Edited by Walter Eng!

Springer Series in Electrophysics

Editors: Gunter Ecker Walter Engl Leopold B. Felsen

Volume 1 Structural Pattern Recognition By T. Pavlidis

Volume 2 Noise in Physical Systems Editor: D. Wolf

Volume 3 The Boundary-Layer Method in Diffraction Problems By V. M. Babic, N. Y. Kirpicnikova

Volume 4 Cavitation and Inhomogeneities in Underwater Acoustics Editor: W. Lauterborn

Volume 5 Very Large Scale Integration (VLSI) Fundamentals and Applications Editor: D. F. Barbe 2nd Edition

Volume 6 Parametric Electronics An Introduction By K-H. L6cherer, C. D. Brandt

Volume 7 Insulating Films on Semiconductors Editors: M. Schulz, G. Pensl

Volume 8 Theoretical Fundamentals of the Ocean Acoustics By L. Brekhovskikh, Y. P. Lysanov

Volume 9 Principles of Plasma Electrodynamics By A. F. Alexandrov, L. S. Bogdankevich, A. A. Rukhadze

Volume 10 Ion Implantation Techniques Editors: H. Ryssel, H. Glawischnig

Volume 11 Ion Implantation: Equipment and Techniques Editors: H. Ryssel, H. Glawischnig

Volume 12 Fundamentals of VLSI Technology Editor: Y. Tarui

Volume 13 Physics of Highly Charged Ions By R. K Janev, L. P. Presnyakov, V. P. Shevelko

Volume 14 Plasma Physics for Physicists By I. A. Artsimovjch, R. Z. Sagdeev

Volume 15 Relativity and Engineering By 1. Van Bladel

Volume 16 Elements of Random Process Theory By S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky

Volume 17 Concert Hall Acoustics By Y. Ando

Volume 18 Planar Circuits for Microwaves and Lightwaves by T. Okoshi

Volume 19 Physics of Shock Waves in Gases and Plasmas By M. Liberman, A. Velikovich

Volume 20 Kinetic Theory of Particles and Photons By 1. Oxenius

Takanori Okoshi

Planar Circuits for Microwaves and Lightwaves

With 138 Figures

Springer-Verlag Berlin Heidelberg New York Tokyo

Professor Dr. Takanori Okoshi Department of Electronic Engineering, University of Tokyo Bunkyo-ku, Tokyo 113, Japan

Series Editors: Professor Dr. Gunter Ecker Ruhr-Universitat Bochum, Theoretische Physik, Lehrstuhl I, Universitatsstrasse 150, D4630 Bochum-Querenburg, Fed. Rep. of Germany

Professor Dr. Walter Engl Institut fUr Theoretische Elektrotechnik, Rhein.-Westf. Technische Hochschule, Templergraben 55, D-5100 Aachen, Fed. Rep. of Germany

Professor Leopold B. Felsen Ph.D. Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201, USA

ISBN-13: 978-3-642-70085-9 e-ISBN-13: 978-3-642-70083-5 DOl: 10.1007/978-3-642-70083-5

Library of Congress Cataloging in Publication Data. Okoshi, Takanori, 1932-. Planar circuit for microwaves and lightwaves. (Springer series in electrophysics ; v. 18). Bibliography: p. Includes index. 1. Microwave integrated circuits. 2. Integrated optics. I. Title. II. Series. TK7876.0415 1985 621.381'32 84-22147

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungs­gesellschaft Wort", Munich.

© Springer-Verlag Berlin Heidelberg 1985

Softcover reprint of the hardcover 1 st edition 1985

The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: K+V Fotosatz, 6124 Beerfelden. Offset printing: Beltz Offsetdruck, 6944 Hemsbach/Bergstr. Bookbinding: J. Schaffer OHG, 6718 Griinstadt. 2153/3130-543210

to my late mother

Preface

Until recently, three principal classes had been known in the electrical cir­cuitry. They were as follows:

1) The lumped-constant circuit, which should be called a zero-dimensional circuit, in the sense that the circuit elements are much smaller in size as compared with the wavelength in all three spatial directions.

2) The distributed-constant circuit, which should be called a one-dimensional circuit, in the sense that the circuit elements are much smaller than the wavelength in two directions but comparable to the wavelength in one di­rection.

3) The waveguide circuit, which should be called a three-dimensional circuit, in the sense that the circuit elements are comparable to the wavelength in all three directions.

The principal subject of this book is the analysis and design (synthesis) theories for another circuit class which appeared in the late 1960s and became common in the 1970s. This new circuit class is

4) the planar circuit, which should be called a two-dimensional circuit, in the sense that the circuit elements are much smaller in size as compared with the wavelength in one direction, but comparable to the wavelength in the other two directions.

As described in the Introduction, the planar circuit concept has become not only technically significant, but also necessary in microwave engineering because of the advent of millimeter-wave integrated circuits and various low­impedance semiconductor microwave devices. The technical significance of the planar circuit is still increasing at present because it is now an inevitable tool in the exact analysis and design of microwave/millimeter-wave integrated circuits. Chapters 2 - 8 are devoted to the analysis and design of such "planar circuits" to be used in the microwave and millimeter-wave regions.

In the final two chapters (Chaps. 9, 10) a different circuit category is dealt with, that is,

5) the optical planar circuit, whose dimension is comparable to the wave­length in one derection, but is much larger than the wavelength in the other two directions.

VIII Preface

Such a circuit category is also becoming technically significant because it plays an important role in some optical integrated circuits.

We should note that despite an apparent resemblance of their names, these two types of electromagnetic circuits, i.e., the planar circuit and the optical planar circuit, require entirely different analysis and design approaches.

The research of the planar circuit theory was started in Italy (S. Ridella et al.) and Japan (T. Okoshi) independently in 1968. In the author's case, the concept of the planar circuit first came to mind when he was asked to give a tutorial talk about the" general principle in the design of Gunn and IMP ATT oscillator circuits" at a symposium held in a hilltop hotel in the Za6 Moun­tains, Japan, on August 28 - 29, 1968. At that time it was not yet easy to give a general guiding principle of the circuit design of this sort because the im­pedance characteristics of Gunn and IMP A TT diodes had not been well known. The only undoubted fact about the impedance characteristics was that these devices had very low negative impedance as compared with electron­beam devices (tubes) which had been predominant in the microwave engineer­ing until that time. Therefore, the author was obliged to construct the talk mainly on how to design strip line or waveguide circuits having very low im­pedance levels, for use in Gunn and IMP ATT oscillators as their resonant cir­cuits. Preparation for the talk led the author to the concepts of the open­boundary and short-boundary planar circuits having arbitrary shapes, which are described in Chaps. 3 and 4, respectively.

Throughout the 1970s and 1980s, many scientists in the world (Japan, Italy, the United States, Canada, India, Brazil, and so forth) joined the re­search of planar circuits. In addition to its analysis, even the computer-aided synthesis of planar circuits having prescribed characteristics has become pos­sible, as will be found in Chaps. 6, 7. The system of theories of electrical cir­cuitry, ranging from the zero-dimensional through three-dimensional, was completed finally by the addition of the planar circuit concept.

On the other hand, the significance of the concept of optical planar cir­cuits, as defined above, came to be recognized by optoelectronics specialists in the 1970s. Frankly speaking, the author himself has contributed very little to the progress in this area. Descriptions in Chaps. 9, 10, therefore, are based principally upon the work performed by several excellent scientists in various countries of the world.

Tokyo, December 1984 Takanori Okoshi

Acknowledgments

The author is indebted to a large number of people for the work on which this book is based. In the earliest stage of the planar circuit research in my laboratory at the University of Tokyo, an im­portant contribution was made by Professor Tanroku Miyoshi, presently with Kobe University, Japan. When I decided to start planar circuit research in 1968, he was a graduate student in my laboratory working for the M. S. degree. He got his M. S. degree in March 1969, and from April 1969 through March 1972 he concentrated entirely upon planar circuit research as his Ph. D. thesis work. His brilliance and effort brought forth various analysis techniques of open-boundary planar circuits described in Chaps. 2, 3.

Later, Professor Miyoshi and I wrote a book entitled Planar Circuits (in Japanese), which was published by Ohm Publishing Company, Tokyo, in 1975. In 1978, this book received the honor of being awarded the Excellent Book Prize by the Institute of Electronics and Communication Engineers of Japan (IECE Japan). Some parts of this English book are similar in content to the Japanese edition; these are the first halves of Chaps. 2, 3, and most of Chap. 4. (Other parts, i.e., about three-quarters of the entire volume, have been newly written on the basis of developments of the theory in the past nine years.) The present author would like to thank the coauthor of the previous Japanese edition, Professor Tanroku Miyoshi, who has generously permitted me to use in this book some figures and descriptions in the Japanese edition.

Since the early 1970s, a number of facuity members, assistants, graduate students, and under­graduate students joined the planar circuit research at the University of Tokyo. Only the names of the most important of these people are, in chronological order, Miss S. Kitazawa, Mr. T. Takeuchi, Mr. Y. Uehara, Prof. J.-P. Hsu, Prof. M. Saito, Dr. F. Kato, Mr. T. Imai, and Mr. K. Ito. In addition to the above people, Professors E. Yamashita of Electro-Communication Univer­sity and Y. Kobayashi of Saitama University contributed much to the refinement of the theory through discussions at the "Planar Circuit Colloquium" held bimonthly for about ten years in Tokyo.

Finally, the author heartily thanks his secretaries, Miss M. Onozuka and Miss N. Matsunaga, who typed the manuscript and its versions many times. Their beautiful work encouraged the author greatly. Without their devoted assistance this book would never have been completed.

December, 1984 Takanori Okoshi

Contents

1. Introduction ............................................... 1 1.1 Seven Ranks in Electrical and Optical Circuitry .............. 1

1.1.1 Conventional Four Ranks. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Comparison of Circuit Dimensions ................... 2 1.1.3 Nonconventional Three Ranks. . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Planar Circuits .................................... 3 1.1.5 Optical Planar Circuits ............................. 5 1.1.6 Long-Waveguide Circuit... ..... .... ... .... .... .... . 5

1.2 Classification and Technical Significance of Planar Circuits ... 6 1.2.1 Three Basic Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 The Neumann and Dirichlet Problems ................ 7 1.2.3 Technical Significance of the Planar Circuit Concept .... 8

1.3 History of Planar Circuit Research. . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 History of the Earliest Stage. . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 History After 1972 ................................. 9

1.4 History of Optical Planar Circuit Research . . . . . . . . . . . . . . . . . . 9 1.5 Purpose and Organization of This Book .................... 9

2. Analysis of Planar Circuits Having Simple Shapes ............... 10 2.1 Background............................................ 10 2.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Boundary Conditions for Cases when External Ports Are

Absent. .... .... . ... ..... . .... . ...... . ... .... ... .. 12 2.2.3 Eigenfunction Expansion ........................... 13 2.2.4 A Simple Example of the Solution .................... 14 2.2.5 Boundary Condition at Ports ........................ 16

2.3 Derivation of Circuit Characteristics ....................... 16 2.3.1 Definition of Terminal Voltage and Current ........... 16 2.3.2 Circuit Characteristics Expressed in Terms

of Green's Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 Expansion of Green's Function by Eigenfunctions ...... 19

2.4 Examples of Analysis Based on Green's Function ............ 20 2.4.1 Rectangular Circuit ................................ 20 2.4.2 Circular Circuit ................................... 23

XII Preface

2.4.3 Triangular Circuit ................................. 24 2.4.4 Annular Circuit ................................... 27 2.4.5 Circular and Annular Sectors ........................ 28 2.4.6 Open-Ended Stripline (Comparison with Distributed-

Constant Line Theory) ............................. 28 2.5 Determination of Equivalent Circuit Parameters Based on

Energy Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Equivalent Circuit of a Multiport Planar Circuit ............. 31 2.7 Validity of the Open-Boundary Assumption ... . . . . . . . . . . . . . . 32 2.8 Examples of Planar Circuits Having Simple Shapes. . . . . . . . . . . 33

2.8.1 Circular Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8.2 Coupled-Mode Filter Using a Single Circular Resonator. 35 2.8.3 Planar 3-dB Hybrid Using a Circular Resonator ........ 39

2.9 Summary .............................................. 42

3. Analysis of Planar Circuits Having Arbitrary Shapes ............. 43 3.1 Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Basic Formulation ofthe Contour-Integral Method. . . . . . . . . . . 44 3.3 Circuit Parameters of an Equivalent NPort ................. 45 3.4 Transfer Parameters of a Two-Port Circuit. . . . . . . . . . . . . . . . . . 47 3.5 Numerical Computation Procedure. . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Description of Circuit Pattern ....................... 49 3.5.2 Computation of rijand eij ........................... 50 3.5.3 Computation of Matrix Elements uij and hij ............ 51 3.5.4 Computation of Transfer Parameters ................. 51 3.5.5 Computation ofInput Admittance and 812 . . . . . • . . . • • . . 52 3.5.6 RF Voltage Along the Circuit Periphery. . . . . . . . . . . . . . . 53

3.6 Examples of Computer Analysis by the Contour-Integral Method................................................ 53 3.6.1 One-Port Circular Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6.2 Two-Port Circular Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.3 Two-Port Square Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6.4 Irregularly Shaped Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6.5 Computer Time ................................... 59

3.7 Analyses Based on Eigenfunction Expansion ................ 60 3.7.1 Advantages and Disadvantages of Eigenfunction-

Expansion Approaches ............................. 60 3.7.2 Methods for Solving Eigenvalue Problems ............. 61 3.7.3 Silvester's Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.7.4 Solution of an Eigenvalue Problem by Variational Method 63 3.7.5 Rayleigh-RitzMethod .............................. 64 3.7.6 Finite-ElementMethod ............................. 65

3.8 Summary .............................................. 67

Contents XIII

4. Short-Boundary Planar Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1 Background ........ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 68 4.2 Principle of Analysis .................................... 69 4.3 Short-Boundary Planar Circuit Having Two Coaxial

Coupling Ports ......................................... 70 4.3.1 Basic Equation .................................... 70 4.3.2 Simplification of the Basic Equation. . . . . . . . . . . . . . . . . . 72 4.3.3 Derivation of Admittance Parameters. . . . . . . . . . . . . . . . . 73 4.3.4 Derivation of Transfer Parameters ................... 74

4.4 Short-Boundary Planar Circuit Having Two Waveguide Coupling Ports ......................................... 75 4.4.1 Basic Equation .................................... 75 4.4.2 Simplification of the Basic Equation .................. 76

4.5 Examples of Numerical Analysis .......................... 78 4.5.1 Short-Circuited Radial Line ......................... 78 4.5.2 Uniform Waveguide Section. . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5.3 Waveguide Section Including a Thick Inductive Window. 81 4.5.4 Waveguide Sections Including Corners. . . . . . . . . . . . . . . . 83 4.5.5 Waveguide Section Including Post. . . . . . . . . . . . . . . . . . . . 84

4.6 Higher-Order Mode Consideration at Reference Planes ....... 84 4.7 Summary .............................................. 86

5. Segmentation Method ....................................... 87 5.1 Background ..................................... '. . . . . . . 87 5.2 Theory of Segmentation Method Using S Matrices ........... 88

5.2.1 Basic Concepts .................................... 88 5.2.2 Interface Network ................................. 89 5.2.3 Computation of the S Matrix ........................ 90 5.2.4 Reduction of Computer Time in S Matrix Computation . . 91

5.3 Theory of Segmentation Method Using Z Matrices ........... 93 5.3.1 Basic Equations ................................... 93 5.3.2 Computation of the Z Matrix ........................ 94 5.3.3 Reduction of Computer Time in ZMatrix Computation. 95

5.4 Summary. . ... .... .... ... ..... ..... .... ... ... . ... .... .. 96

6. Trial-and-Error Synthesis of Optimum Planar Circuit Pattern ..... 97 6.1 Background............................................ 97 6.2 Method of Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.1 Principle of the Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2.2 Computer Analysis ................................ 99 6.2.3 Starting Circuit Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.4 Figure of Merit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

XIV Contents

6.2.5 Variation of Characteristics by Modification of Circuit Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2.6 Optimum Circuit Pattern ........................... 103 6.3 Comparison with Experiment ............................. 104 6.4 Computer Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Summary .............................................. 105

7. Fully Computer-Oriented Synthesis of Optimum Planar Circuit Pattern ............................................. 106 7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Method of Synthesis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2.1 Outline of Synthesis Process. . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2.2 Analysis of Frequency Characteristics . . . . . . . . . . . . . . . . . lOS 7.2.3 Pattern Variables.... . .... . ..... . ... . ... . ... .. . . . .. lOS 7.2.4 Evaluation Function ............................... 109 7.2.5 Algorithm for Optimization of Circuit Pattern ......... 110

7.3 Parameters and Computational Techniques in an Actual Example of Synthesis ......................... 111 7.3.1 Number of Pattern Variables ........................ 111 7.3.2 Reduction of Computer Time Taking Advantage of

Double Symmetry ................................. 111 7.3.3 Number of Sampling Points Along Periphery .......... 112 7.3.4 Parameters in Evaluation Function ................... 112 7.3.5 Relative Widths of External Striplines . . . . . . . . . . . . . . . . . 112 7.3.6 Assumption of Uniform Current at Ports .............. 113

7.4 Results of Synthesis ..................................... 113 7.4.1 Process of Optimization ............................ 113 7.4.2 Optimized Circuit Patterns .......................... 114

7.5 Experimental Verification.... . ... . . . ... . ... . ... . ... .... . . 114 7.5.1 Circuit Design and Structure. . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5.2 Result of Measurement ............................. 117

7.6 Further Improvement of Frequency Characteristics by Addition of External Circuits ..................................... 117 7.6.1 Types of External Circuits. .... . . .... .... ....... . . ... 11S 7.6.2 Optimization of Parameters ......................... 119 7.6.3 Result of Optimization ............................. 120 7.6.4 Obtained Frequency Characteristics .................. 120

7.7 Evaluation of the Synthesized Circuit Patterns . . . . . . . . . . . . . . . 121 7.7.1 Comparison of Theory and Experiment ............... 121 7.7.2 Comparison with Other Characteristics ............... 121 7.7.3 Comparison with Other Trials for Synthesizing Planar

Circuits .......................................... 122 7.S Summary .............................................. 123

Contents XV

8. Planar Circuits with Anisotropic Spacing Media ................. 124 8.1 Background............................................ 124 8.2 Theories of Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2.1 Basic Equations ................................... 125 8.2.2 Analysis Based on Eigenfunction Expansion ........... 127 8.2.3 Analysis Based on a Contour-Integral Equation ........ 128

8.3 Formulations for Numerical Computation and Examples of Calculation .......................................... 129 8.3.1 Formulation for the Eigenfunction-Expansion Method .. 129 8.3.2 Examples of Calculation ............................ 131 8.3.3 Formulation for the Contour-Integral Method ......... 133 8.3.4 Examples of Calculation ............................ 135

8.4 Comparison of the Eigenfunction-Expansion and Contour-Integral Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.5 Optimum Design of Ferrite Planar Circuits. . . . . . . . . . . . . . . . . . 138 8.5.1 Technical and Historical Backgrounds ................ 138 8.5.2 Method of Numerical Analysis. . . . . . . . . . . . . . . . . . . . . . . 139 8.5.3 Optimum Design of a Disk-Shaped Circulator. . . . . . . . .. 139 8.5.4 Optimum Design of a Triangular Circulator. . . . . . . . . . . . 140 8.5.5 Modified Triangular Circulators Having Curved Sides. . . 142

8.6 Summary .............................................. 144

9. Optical Planar Circuits ...................................... 145 9.1 Background............................................ 145 9.2 Wave-Optics Approach to Optical Planar Circuits. . . . . . . . . . .. 147

9.2.1 Basic Equations ................................... 147 9.2.2 TE Modes ........................................ 148 9.2.3 TM Modes ......... '" . .... . . ... . ...... . ... .. . . ... 149 9.2.4 Vertical Field Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.3 Geometrical Optics Approach to Optical Planar Circuits ...... 151 9.3.1 Concepts of Geometrical Optics and Ray .............. 151 9.3.2 Eikonal and Eikonal Equation. . . . . . . . . . . . . . . . . . . . . . . 151 9.3.3 Ray Equation ..................................... 152

9.4 Optical Planar Circuits Having Uniform Slab Structure ....... 154 9.4.1 Model to be Considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4.2 TE-Mode Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4.3 TM-Mode Waves .................................. 156

9.5 Optical Planar Circuits Having Periodic Structures ........... 157 9.5.1 Mathematical Expression of Optical Bloch Waves. . . . . . . 157 9.5.2 Dispersion Relation and Group Velocity of Optical

Bloch Waves ...................................... 158 9.5.3 Excitation of Optical Bloch Waves ................... 160 9.5.4 Example of Measured pz-pxRelations ................ 160

XVI Contents

9.6 Planar Lenses 161 9.7 Summary............................................ 163

10. Optical Planar Circuits Having Stripelike Waveguiding Structures. 164 10.1 Background.......................................... 164 10.2 Model to be Considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.3 Geometrical Optics Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.3.1 Limitation of Geometrical Optics ................. 166 10.3.2 Propagating Condition .......................... 166

10.4 Wave-Optics Approaches .............................. 167 10.5 Beam-Propagation Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

10.5.1 Principle ...................................... 168 10.5.2 Numerical Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.5.3 Examples of Calculation Results .................. 171

10.6 Summary.. . ... . . ... ... . .... . ..... . ... .... .... .. . . . .. 174

Appendix .................................................... 175 A2.1 Derivationof(2.5) .................................... 175 A2.2 Some Characteristics of Eigenvalues and Eigenfunctions . . . . 177 A3.1 Weber's Solution Using Cylindrical Waves ............... 178 A3.2 Derivation of (3.1) .................................... 180 A3.3 Choice of the Green's Function Used in Contour-Integral

Analysis ............................................ 181 A4.1 Proof of (4.1) for a Multiply Connected Circuit Pattern .... 185 A5.1 Computational Technique for Combining n Subports to a

Single External Port .................................. 186 A8.1 Derivation of (8.5,8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A8.2 Derivation of (8.18, 19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 A9.1 Derivation of (9.27) ................................... 188 A9.2 Derivation of (9.32,33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

References ................................................... 191

Subject Index ................................................. 197