Encyclopedia of RF and microwave engineering. Vol 3

A John Wiley & Sons, Inc., Publication

ENCYCLOPEDIA OF

RF AND MICROWAVE

ENGINEERING

VOLUME 3

Kai Chang, Editor

The Encyclopedia of RF and Microwave Engineering is available online at

http://www.mrw.interscience.wiley.com/erfme

Copyright r 2005 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,

electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of

the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization

through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,

MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission

should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201)

748-6011, fax (201) 748-6008.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this

book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book

and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be

created or extended by sales representatives or written sales materials. The advice and strategies contained herein may

not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor

author shall be liable for any loss of profit or any other commercial damages, including but not limited to special,

incidental, consequential, or other damages.

For general information on our other products and services please contact our Customer Care Department within the U.S.

at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be

available in electronic format.

Library of Congress Cataloging-in-Publication Data:

Wiley encyclopedia of RF and microwave engineering, 6 Volume Set / Kai

Chang, editor-in-chief.

p. cm.

Includes bibliographical references and index.

ISBN 0-471-27053-9 (cloth)

1. Wireless communication systems–Encyclopedias. 2. Radio

circuits–Encyclopedias. 3. Microwave circuits–Encyclopedias. I. Chang,

Kai, 1948-

TK5103.2.W535 2005

621.382–dc22

2004024581

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

There has been rapid development in RF and microwave

technologies during the past decade. RF and microwave

wireless applications have been and continue to be among

the hottest growth areas. Some of the expanding activities

in these fields include wireless communications (mobile,

cellular, and satellite), wireless sensors, local area net￾works, remote control and identification, global position￾ing systems (GPS), and intelligent highway and vehicle

systems (IHVS). In view of these growing applications, it

is timely to publish this Encyclopedia of RF and Micro￾wave Engineering.

The Encyclopedia is intended to serve as a compendium

of theory, design methods, applications, overview, and

future trends of technologies. Each article is a self-con￾tained unit and is written for both non-specialists and

specialists. Some overlap among articles is allowed for

easy study. It is hoped that this Encyclopedia will provide

useful information for technicians, students, professors,

engineers, managers, and researchers in today’s rapidly

changing high-frequency technologies.

The Encyclopedia consists of six volumes. The articles

include an overview of each subject, followed by in-depth

descriptions of important theory, technology develop￾ments, and applications. Each article includes a list of

references for further study. Some articles are adapted

from the Encyclopedia of Electrical and Electronics En￾gineering. Most of these articles have been revised and

updated. Among the major topics are the following:

Electromagnetics

Numerical Methods

Wave Propagation

Transmission Lines

Transmission Line Discontinuities

RF Circuits

RF Components and Devices

RF Technology

RF Integrated Circuits

RF CMOS Technology

RF Antennas

RF Transmitters

RF Receivers

RF Systems

Microwave Passive Components and Devices

Microwave Active Components and Devices

Microwave Semiconductor Devices and Processing

Microwave Hybrid and Monolithic Integrated Circuits

Microwave Antennas

Microwave Transmitters

Microwave Receivers

Microwave Superconductors

Microwave Systems

Millimeter-wave and Submillimeter-wave Technologies

RF and Microwave Measurements

RF and Microwave Propagation

RF and Microwave Photonics

CAD Techniques and Modeling

Modulation and Demodulation Techniques

Packaging and Interconnects

Frequency Synthesizers

Signal and Baseband Processing

Solid-state Devices

Semiconductor Materials

High Power Tubes

Noise

Low Power Electronics

Radar Systems

Communication Systems

Navigation Systems

Radio Technologies

Remote Sensing

Broadcasting

The Editor would like to thank Cassie Craig of Wiley

for managing this project, and George Telecki of Wiley for

his constant encouragement and support. The Editor

would also like to thank all of the members of the Editorial

Board for their suggestions and advice.

Kai Chang

College Station, Texas

PREFACE

v

CONTRIBUTORS

VALENTINE A. AALO, Florida Atlantic University, Boca Raton,

Florida, Digital Radio

MUHAMMAD TAHER ABUELMA’ATTI, King Fahd University of

Petroleum and Minerals, Dhahran, Saudi Arabia, Intermodu￾lation Measurement

SABURO ADACHI, Tohoku Institute of Technology, Sendai, Japan,

Conical Antennas

ARLON T. ADAMS, Syracuse University, Syracuse, New York,

Helical Antennas

ILESANMI ADESIDA, University of Illinois at Urbana–Champaign,

Gallium Nitride for Electronics

R. S. ADVE, Research Associates for Defense Conversion Inc.,

Stripline Components

M. AGETHEN, Innovative Processing AG (IPAG), Duisburg, Ger￾many, Noise, High-Frequency

VOLKAN AKAN, Hacettepe University, Beytepe, Ankara, Turkey,

Conformal Mapping Techniques

IGOR ALEXEFF, The University of Tennessee, Knoxville, Tennessee,

Directive Antennas

M. ALI, University of South Carolina, Columbia, Miniaturized

Packaged (Embedded) Antennas for Portable Wireless Devices

F. ALIMENTI, University of Perugia, Perugia, Italy, Waveguide

Discontinuities

JEREMY E. ALLNUTT, Virginia Tech/Northern Virginia Center,

Falls Church, Virginia, Refraction and Attenuation in Tropo￾sphere

PLINIO R. G. ALVES, University of Brasilia, Brasilia, DF, Brazil,

Antenna Accessories

W. A. ANDERSON, State University of New York at Buffalo, Thin

Film Resistors

ILTCHO ANGELOV, Chalmers University of Technology, Frequency

Converters and Mixers

JAUME ANGUERA, Fractus S.A., Barcelona, Spain, Fractal

Shaped Antennas: A Review

RODOLFO ARANEO, University of Rome ‘‘La Sapienza,’’ Rome,

Italy, Electromagnetic Shielding

ANA GARCI´A ARMADA, Universidad Carlos III de Madrid, Ma￾drid, Spain, Wireless Communications Systems

F. ARNDT, University of Bremen, Germany, Hybrid CAD Techni￾ques; Mode-Matching Methods

S. ASHOK, The Pennsylvania State University, University Park,

Pennsylvania, Schottky Barriers

KAZUHIKO ATSUKI, University of Electro-Communications, Mi￾crostrip Lines

OSAMA O. AWADELKARIM, The Pennsylvania State University,

University Park, Pennsylvania, Smart Materials

AYDIN AYKAN, Tutzing, Germany, Calibration of Circular Loop

Antenna

EZEKIEL BAHAR, University of Nebraska—Lincoln, Lincoln, Ne￾braska, Radar Remote Sensing of Irregular Stratified Media

with Multiple-Scale Features

INDER J. BAHL, M/A-COM, Inc., Roanoke, Virginia, Monolithic

Microwave Integrated Circuits (MMICs)

D. BAILLARGEAT, IRCOM, Limoges, France, Dielectric Resonator

Filters

JAMES R. BAKER–JARVIS, U.S. Department of Commerce Boulder

Laboratories, Boulder, Colorado, Electromagnetic Ferrite Tile

Absorber

CONSTANTINE A. BALANIS, Arizona State University, Tempe,

Arizona, Antenna Parameters; Circular Waveguides

PETER BALLING, ASC, Antenna Systems Consultant, ApS,

Taastrup, Denmark, Multibeam Antennas

RAJEEV BANSAL, University of Connecticut, Storrs, Connecticut,

Submarine Antennas

JAMES BAO-YEN TSUI, Wright-Patterson Air Force Base, Ohio,

Microwave Receivers

BRUNO BARELAUD, University of Limoges, Limoges Cedex,

France, Active Filters: Tools and Techniques for Active-Filter

Design

R. M. BARRIO-GARRIDO, Universidad Carlos III de Madrid,

Stripline Components

R. BARTNIKAS, IREQ/Institut de Recherche d’Hydro-Que´bec, Di￾electric Measurement

B. BAYARD, Jean Monnet University, St. Etienne, France, Ferrite

Isolators

GERARD BEAUDIN, Observatoire de Paris—Meudon, Millimeter￾Wave Measurement

BENJAMIN BEKER, University of South Carolina, Columbia,

South Carolina, Boundary Value Problems

GERALD BENDIXEN, Rockwell Collins, Inc., Radio Navigation

O. BENDOV, Dielectric Communications, Television and FM

Broadcasting Antennas

MATTHEW BENNETT, Army Research Laboratory, Radar Equip￾ment

JENNIFER T. BERNHARD, University of Illinois at Urbana—

Champaign, Urbana, Illinois, Reconfigurable Antennas

R. M. BERTENBURG, Innovative Processing AG (IPAG), Duisburg,

Germany, Noise, High-Frequency

PRAKASH BHARTIA, Natel Engineering Co., Inc., Chatsworth,

California, Hybrids and Couplers; Strip Transmission Lines

ASOKE K. BHATTACHARYYA, Lincoln University, Linear Antennas

MAREK E. BIALKOWSKI, University of Queensland (Australia)

and National University of Singapore, Power Combiners and

Dividers

JENS BIEGERT, University of New Mexico, Albuquerque, New

Mexico, Pulse Compression

MARIO BIEY, Politecnico di Torino, Frequency-Domain Circuit

Analysis

S. BILA, IRCOM, Limoges, France, Dielectric Resonator Filters

LAURENT BILLONNET, University of Limoges, Limoges, Cedex,

France, Active Filters: Overview of Active-Filter Structures;

Active Filters: Tools and Techniques for Active-Filter Design

TREVOR S. BIRD, CSIRO, Sydney, Australia, Antenna Feeds

GEORGE L. BLEHER, Individual Consultants, Radiation Monitor￾ing

PETER HARING BOLIVAR, Institut fuer Halbleitertechnik II,

Aachen University, Germany, Terahertz Science, Engineering

and Systems—from Space to Earth Applications

DONALD M. BOLLE, Polytechnic University, Microwave Inte￾grated Circuits

CARMEN BORJA, Fractus S.A., Barcelona, Spain, Fractal Shaped

Antennas: A Review

RENATO G. BOSISIO, E´cole Polytechnique, Montreal, Quebec,

Canada, Quadrature Phase Shift Keying (QPSK); Six-Port

Networks

vii

MATTHYS M. BOTHA, University of Illinois at Urbana—Cham￾paign, Urbana, Illinois, Finite Element Analysis

MAURIZIO BOZZI, University of Pavia, Pavia, Italy, Periodic

Structures

G. R. BRANNER, University of California, Davis, Microwave and

Radio Frequency Multipliers

W. BROCKERHOFF, Innovative Processing AG (IPAG), Duisburg,

Germany, Noise, High-Frequency

JOSEPH A. BRUDER, Air Force Research Laboratory, Radar

Tracking

D. BUDIMIR, University of Westminster, London, United King￾dom, Waveguide Components

ROBERT J. BURKHOLDER, The Ohio State University, Columbus,

Ohio, Iterative Methods

NATHAN A. BUSHYAGER, Georgia Institute of Technology, Atlanta,

Georgia, Electromagnetics, Time-Domain

QUNSHENG CAO, University of Minnesota, Minneapolis, Minne￾sota, Multiresolution Technique

C.N. CAPSALIS, National Technical University of Athens, Athens,

Greece, Antenna Reverberation Chamber

ANTOˆ NIO CARLOS, Federal University of Rio de Janeiro, Rio de

Janeiro, Brazil, Chebyshev Filters

PAULO H. P. CARVALHO, University of Brasilia, Brasilia, DF,

Brazil, Antenna Accessories

YVES CASSIVI, Ecole Polytechnique, Montreal, Quebec, Canada,

Impedance Transformers and Matching Networks

MANUEL SIERRA CASTAN˜ ER, Universidad Politecnica de Madrid,

Madrid, Spain, Monopulse Tracking Systems

ROBERT H. CAVERLY, Villanova University, Villanova, Pennsyl￾vania, Microwave Switches

HAKI CEBI, University of Colorado at Boulder, Microwave Cir￾cuits

SALVATORE CELOZZI, University of Rome ‘‘La Sapienza,’’ Rome,

Italy, Electromagnetic Shielding

THOMAS P. CENCICH, University of Colorado at Boulder, Fre￾quency-Independent Antennas; Spiral Antennas

R. CHAIR, The University of Mississippi, Oxford, Mississippi,

Microstrip Antennas, Broadband

ANDREW K. CHAN, Texas A & M University, Radar Signal

Detection

CHI HOU CHAN, City University of Hong Kong, Kowloon, Hong

Kong, Parallel Algorithms and Computing for Large-Scale

Electromagnetic Simulation; Rough-Surface Scattering: Nu￾merical Simulations and Applications in Microwave Remote

Sensing

KA FAI CHAN, City University of Hong Kong, Kowloon, Hong

Kong SAR, China, Parallel Algorithms and Computing for

Large-Scale Electromagnetic Simulation

HENG-CHIA CHANG, RF System, Blue 7 Communications, Inc.,

Fremont, California, Phase Noise and Measurements

HONG-YEH CHANG, National Taiwan University, Taipei, Taiwan,

Monolithic Millimeter-Wave Integrated Circuits

KAI CHANG, Texas A&M University, College Station, Texas,

Gratings and Grating Antennas; Microwave Power Transmis￾sion; Rectifying Antennas (Rectennas); Ring Resonators and

Circuits; Slotline Components

KOOKRIN CHAR, Conductus, Inc., HTS Film Growth

FU-CHIARNG CHEN, University of Illinois at Urbana—Cham￾paign, Radar Antennas

GUANRONG CHEN, City University of Hong Kong, Kowloon, Hong

Kong, China, Stability of Nonlinear Systems

JI CHEN, University of Houston, Houston, Texas, Transmission

Lines and Parameters

R. S. CHEN, Nanjing University of Science and Technology,

Nanjing, P.R. China, Ferrite Circulators

RICHARD L. CHEN, University of Houston, Houston, Texas,

Transmission Lines and Parameters

S. Y. CHEN, University of Illinois at Urbana–Champaign, Ur￾bana, Illinois, Electromagnetic Subsurface Remote Sensing

VICTOR C. CHEN, U.S. Naval Research Laboratory, Washington,

DC, Radar Imaging

YINCHAO CHEN, University of South Carolina, Columbia, South

Carolina, Enhancements of the Finite-Difference Time-Domain

Method; Multiresolution Technique

YUPENG CHEN, University of Central Florida, Orlando, Florida,

BiCMOS Devices and RF Integrated Circuits

DAVID B. CHESTER, Intersil Corporation, Pulse-Shaping Circuits

KAR-MING CHEUNG, California Institute of Technology, Radio￾telemetry

WENG C. CHEW, University of Illinois at Urbana—Champaign,

Urbana, Illinois, Electromagnetic Subsurface Remote Sensing;

Green’s Function Methods; High-Frequency Transmission

Lines; Radar Antennas

CHOONSIK CHO, University of Colorado at Boulder, Microwave

Circuits

CHIU H. CHOI, University of North Florida, Jacksonville, Florida,

Bandstop Filters; Lowpass Filters

JONGHOON CHOI, University of Michigan, Ann Arbor, Michigan,

Microwave Oscillators

M. CHONGCHEAWCHAMNAN, Mahanakorn University of Technol￾ogy, Bangkok, Thailand, Radiofrequency Integrated Circuits

DURGA P. CHOUDHURY, Northeastern University, Superconduct￾ing Microwave Technology

P. K. CHOUDHURY, Multimedia University, Selangor, Malaysia,

Electromagnetic Materials

CHRISTOS G. CHRISTODOULOU, University of New Mexico, Albu￾querque, Antenna Radiation Patterns; Antennas

CHRISTOS CHRISTOPOULOS, University of Nottingham, Notting￾ham, United Kingdom, The Transmission-Line Matrix (TLM)

Method

MICHAEL T. CHRYSSOMALLIS, Democritus University of Thrace,

Xanthi, Greece, Antenna Radiation Patterns

CHUNG BOON-KUAN CHUNG, Multimedia University, Cyberjaya,

Malaysia, Antenna Anechoic Chambers

SHYH-JUNG CHUNG, National Chiao-Tung University, Hsinchu,

Taiwan, Republic of China, Van Atta Array Reflector

SERGIO A. CLAVIJO, Arizona State University, Tempe, Arizona,

Artificial Magnetic Conductor

GEORGE COKKINIDES, University of South Carolina, Columbia,

South Carolina, Boundary Value Problems

PAOLO COLANTONIO, Universita` degli Studi di Roma Tor Ver￾gata, Rome, Italy, Microwave Power Amplifiers

GERALD W. COLLINS, GW Collins Consulting, Transmitters for

Analog Television

F. M. COLLINS, Ohmcraft, Inc., Thin Film Resistors

DOMINIQUE CROS, Universite´ de Limoges, Millimeter-Wave Mea￾surement

PAUL CROZAT, Universite´Paris—Sud, Millimeter-Wave Measure￾ment

TIE JUN CUI, University of Illinois at Urbana—Champaign,

Radar Antennas

ROGER DALKE, Institute for Telecommunication Sciences, Radio

Noise

GILLES DAMBRINE, Institut d’Electronique et Microe´lectronique

du Nord (IEMN), Millimeter-Wave Measurement

DAVID J. DANIELS, ERA Technology, Surrey, United Kingdom,

Ground Penetrating Radar

VIII CONTRIBUTORS

E. DANIELSSON, Royal Institute of Technology, Kista-Stockholm,

Sweden, SiC Device Technologies

AFSHIN S. DARYOUSH, Drexel University, Philadelphia, Pennsyl￾vania, Microwave Photonics: Technological Evolutions and Its

Applications

NIROD K. DAS, Polytechnic University, Microwave Integrated

Circuits

D. K. DAS-GUPTA, University of Wales, Bangor, Wales, Dielectric

Permittivity and Loss

KENNETH DAVIES, National Oceanic and Atmospheric Adminis￾tration, Sky Wave Propagation at Medium and High Frequen￾cies

W. ALAN DAVIS, The University of Texas at Arlington, Schottky

Barrier Diodes and Their Applications

WILLIAM A. DAVIS, Virginia Tech Antenna Group, Blacksburg,

Virginia, Antenna Theory

W. R. DEAL, Northrop Grumman Space Technology, Redondo

Beach, California, Dielectric Resonator Oscillators

FRANCO DE FLAVIIS, University of California at Los Angeles,

Phase Shifters

MICHAEL P. DELISIO, University of Hawaii at Manoa, Quasiop￾tical Circuits

PETER DE MAAGT, European Space Research and Technology

Centre (ESTEC), The Netherlands, Terahertz Science, Engi￾neering and Systems—from Space to Earth Applications

GREGORY M. DEMYASHEV, Swinburne University of Technology,

Melbourne, Australia, Microwave Resonance Plasma Source

MAKARAND DEO, Carleton University, Ottawa, Canada, Neural

Networks for Microwave Circuits

M. DE QUEIROZ, Federal University of Rio de Janeiro, Rio de

Janeiro, Brazil, Chebyshev Filters

DOMINIC DESLANDES, Ecole Polytechnique, Montreal, Quebec,

Canada, Impedance Transformers and Matching Networks

DANIE¨ L DE ZUTTER, Ghent University, Ghent, Belgium, Skin

Effect

RODOLFO E. DIAZ, Arizona State University, Tempe, Arizona,

Artificial Magnetic Conductor

JEAN-CLAUDE DIELS, University of New Mexico, Albuquerque,

New Mexico, Pulse Compression

KUNG-HAU DING, Air Force Research Laboratory, Hanscom AFB,

Massachusetts, Computational Electromagnetic Scattering

Models for Microwave Remote Sensing

PAUL E. DODD, Sandia National Laboratories, Albuquerque, New

Mexico, Radiation Effects

M. DOMEIJ, Royal Institute of Technology, Kista-Stockholm,

Sweden, SiC Device Technologies

ALAN N. DUCKWORTH, Naval Research Laboratory, Electronic

Warfare

VERNON E. DUNN, Space Systems/Loral, Microwave Isolators

J. F. DUQUE-CARRILLO, University of Extremadura, Preampli￾fiers

R. B. DYBDAL, The Aerospace Corporation, Satellite Antennas

TERRY C. EDWARDS, Engalco, Bridlington, East Yorkshire, Uni￾ted Kingdom, Microstrip Circuits

HERIBERT EISELE, University of Leeds, Leeds, United Kingdom,

Gunn or Transferred-Electron Devices and Circuits

ABDELNASSER A. ELDEK, Center of Applied Electromagnetic

Systems Research (CAESR) The University of Mississippi Uni￾versity, Mississippi, Wideband Slot and Printed Antennas

KIM ANDREW ELLIOT, High-Frequency Broadcasting

ATEF Z. ELSHERBENI, Center of Applied Electromagnetic Systems

Research (CAESR) The University of Mississippi University,

Mississippi, Wideband Slot and Printed Antennas

KIE B. EOM, The George Washington University, Washington, DC,

Radar Target Recognition

HALIT EREN, Curtin University of Technology, Bentley, Australia,

Altimeters

DANILO ERRICOLO, University of Illinois, Chicago, Geometric

Optics

KARU P. ESSELLE, Macquarie University, Sydney, New South

Wales, Australia, Dielectric Resonators

JOSEPH B. EVANS, University of Kansas, Lawrence, Kansas,

Local-Area Networks (LANs)

MAREK T. FABER, Warsaw University of Technology, Warsaw,

Poland, Microwave Parametric Amplifiers

ROBERT L. FAGALY, Tristan Technologies, SQUIDs

LU FAN, Texas A&M University, Slotline Components

J. A. BRANDA˜ O FARIA, Instituto Superior Te´cnico, Lisbon, Portu￾gal, Multiconductor Transmission Lines

ALY E. FATHY, University of Tennessee, Knoxville, PIN Diodes

ORLA FEELY, University College Dublin, Dublin, Ireland, Tran￾sient Circuit Analysis

ZHIPING FENG, University of Colorado at Boulder, Microwave

Circuits; Packaging RF Devices and Modules

EVA S. FERRE-PIKAL, University of Wyoming, Laramie, Wyoming,

Frequency Standards, Characterization

ANDREA FERRERO, Politecnico di Torino, Torino, Italy, Micro￾wave Measurements

L. B. FERTIG, Georgia Institute of Technology, Frequency Mod￾ulation

DEJAN S. FILIPOVIC, University of Colorado, Boulder, Colorado,

Frequency-Independent Antennas; Spiral Antennas

MATTHEW FOLKERT, du Treil, Lundin, & Rackley, Sarasota,

Florida, Antennas for Medium-Frequency Broadcasting

KENNETH R. FOSTER, University of Pennsylvania, Philadelphia,

Pennsylvania, Biological Effects of Radiofrequency Energy as

Related to Health and Safety

DOUGLAS R. FREY, Lehigh University, Gyrators

FABRIZIO FREZZA, ‘‘La Sapienza’’ University of Rome, Italy,

Leaky-Wave Antennas

LEE-LUENG FU, California Institute of Technology, Spaceborne

Radar

DAVID W. FUGATE, Electric Research and Management, Inc.,

Magnetic Shielding

KYOHEI FUJIMOTO, University of Tsukuba, Fujisawa, Japan,

Antennas for Mobile Communications; Integrated Antenna

Systems; Small Antennas

ADRIAN K. FUNG, University of Texas at Arlington, Arlington,

Texas, Microwave Scattering Models for Earth Terrain

CYNTHIA M. FURSE, University of Utah, Salt Lake City, Utah,

Dipole Antennas and Arrays; Monopole Antennas

VINCENT F. FUSCO, Queen’s University of Belfast, Belfast, North￾ern Ireland, Dielectric Resonator Oscillators

ALESSANDRO GALLI, ‘‘La Sapienza’’ University of Rome, Italy,

Leaky-Wave Antennas

YEOW BENG GAN, National University of Singapore, Singapore,

Electromagnetic Inverse Problems

OM P. GANDHI, University of Utah, Salt Lake City, Dipole

Antennas and Arrays; Monopole Antennas

MANUEL GARCIA, Lawrence Livermore National Laboratory,

Livermore, California, Integro-Differential Equations

FRED GARDIOL, Swiss Federal Institute of Technology, Lausanne,

Switzerland, Measurement of Near Fields Using a Modulated

Scatterer

P. GARDNER, The University of Birmingham, Edgbaston, Bir￾mingham, United Kingdom, Feedforward Amplifiers

CONTRIBUTORS IX

R. GARG, Indian Institute of Technology Kharagpur, Kharagpur,

India, Coaxial Line Discontinuities

G. G. GENTILI, Milan Polytechnic, Milan, Italy, Orthomode

Transducers

RAMEZ L. GERGES, IEEE-ITSC Standards Committee, Goleta,

California, ITS Radio Service Standards & Wireless Access in

Vehicular Environments (ITS-WAVE) at 5.9 GHZ

DAVID GESBERT, Stanford University, Antenna Arrays for Mobile

Communications

RICHARD G. GEYER, U.S. Department of Commerce, Boulder

Laboratories, Boulder, Colorado, Electromagnetic Ferrite Tile

Absorber

FRANCO GIANNINI, Universita` degli Studi di Roma Tor Vergata,

Rome, Italy, Microwave Power Amplifiers

DONALD P. GIEGLER, Individual Consultants, Radiation Moni￾toring

IAN A. GLOVER, University of Bradford, Bradford, West York￾shire, United Kingdom, Radiometry

JOHN P. GODWIN, DirecTV, Direct Satellite Television Broad￾casting

BAR-GIORA GOLDBERG, Sciteq Electronics, Inc., Frequency

Synthesizers

GARY W. GOO, The Aerospace Corporation, Military Communica￾tion

JAIDEVA C. GOSWAMI, Schlumberger Technology Corporation,

Sugar Land, Texas, Application of Wavelets to Electromagnetic

Problems

G. K. GOTHARD, Auburn University, Auburn, Alabama, Integral

Equations: Application to Open-Region Problems

VICTOR L. GRANATSTEIN, University of Maryland, College Park,

Maryland, Microwave Tubes

RICHARD R. GREENE, Technology for Communications Interna￾tional, Fremont, California, Antennas for High-Frequency

Broadcasting

JERRY MICHAEL GRIMM, JMG Consulting, Irving, Texas, Elliptic

Filters

AMADEU GRIOL, Universidad Polite´cnica de Valencia, Valencia,

Spain, Photonic Band Gap (PGB)

ANDREAS GRONEFELD, Ruhr-Universita¨t Bochum, Standing

Wave Meters and Network Analyzers

ADNAN GO¨ RU¨ R, Nig˘de University, Nig˘de, Turkey, Filter Theory

MARCO GUGLIELMI, European Space Research and Technology

Centre, Noordwijk, The Netherlands, Multimode Equivalent

Network Representations

P. GUILLON, IRCOM, Limoges, France, Dielectric Resonator

Filters

K. C. GUPTA, University of Colorado at Boulder, Microwave

Circuits; Packaging RF Devices and Modules

MRIDULA GUPTA, University of Delhi South Campus, New Delhi,

India, MOSFET Modeling

R. S. GUPTA, University of Delhi South Campus, New Delhi,

India, MOSFET Modeling

GERMA´ N GUTIERREZ, Centellax, Inc., Carlsbad, California, Vari￾able-Frequency Oscillators

SUSAN C. HAGNESS, University of Wisconsin-Madison, Madison,

Wisconsin, Finite-Difference Time-Domain Analysis

PETER S. HALL, The University of Birmingham, Edgbaston,

Birmingham, UK, Microstrip Antenna Arrays

RONALD E. HAM, Consultant, Microwave Detectors

PETER H. HANDEL, University of Missouri St. Louis, Noise, Low￾Frequency

KEN-yA HASHIMOTO, Chiba University, Chiba, Japan, Surface

Acoustic Wave Filters

HITOSHI HAYASHI, Nippon Telegraph and Telephone Corporation,

Yokosuka, Kanagawa, Japan, Radiofrequency Identification

(RFID) Systems

I.I. HERETAKIS, National Technical University of Athens, Athens,

Greece, Antenna Reverberation Chamber

K. HESS, University of Illinois, High-Field Effects

DAVID A. HILL, National Institute of Standards and Technology,

Radiowave Propagation Ground Effects

KURT HILTY, Swiss Federal Office of Metrology, Bern-Wabern,

Switzerland, Attenuation Measurement

JACKIE E. HIPP, Southwest Research Institute, San Antonio,

Texas, Radio Direction Finding

TAN-HSIUNG HO, National Chiao-Tung University, Hsinchu,

Taiwan, Republic of China, Van Atta Array Reflector

ALI KO¨ KSAL HOCAOGLU, Information Technologies Research

Institute, Gebze, Kocaeli, Turkey, Buried-Object Detection

WOLFGANG J.R. HOEFER, University of Victoria, Victoria, BC,

Canada, Electromagnetic Modeling; Finlines

CHRISTOPHER L. HOLLOWAY, U.S. Department of Commerce

Boulder Laboratories, Boulder, Colorado, Electromagnetic Fer￾rite Tile Absorber

ERIC L. HOLZMAN, Northrop Grumman Corporation, Baltimore,

Maryland, Microstrip Transitions; Waveguide Oscillators

WEI HONG, Southeast University, Nanjing, China, Feedback

Oscillators

WILLIAM E. HORD, Microwave Applications Group, Santa Maria,

CA, Ferrite Phase Shifters

HOTON HOW, Hotech, Inc., Belmont, Massachusetts, Magnetic

Microwave Devices; Microstrip Antennas

DAVID A. HOWE, National Institute of Standards and Technology

(NIST), Boulder, Colorado, Frequency Stability

LUNG-HWA HSIEH, Texas A&M University, College Station,

Texas, Ring Resonators and Circuits

JOHN HUANG, Jet Propulsion Laboratory, Pasadena, California,

Reflectarray Antenna

ROBERT D. HUNSUCKER, RP Consultants, Klamath Falls, Ore￾gon, Electromagnetic Waves in Ionosphere

BERNARD HUYART, Ecole Nationale Supe´rieure des Te´le´commu￾nications (ENST), Millimeter-Wave Measurement

MOHAMED IBNKAHLA, Queen’s University, Kingston, Ontario,

Canada, Analytical and Adaptive Modeling of Nonlinear High￾Power Amplifiers; MIMO Systems for Wireless Communications

T. KORYU ISHII, Marquette University, Milwaukee, Wisconsin,

Distributed Amplifiers

YUKIO ITO, The Pennsylvania State University, Piezoelectricity

TATSUO ITOH, UCLA, Slotlines

R. LAWRENCE IVES, Calabazas Creek Research, Inc., Saratoga,

California, Gyrotrons

DAVID R. JACKSON, University of Houston, Houston, Texas,

Leaky Modes and High-Frequency Effects in Microwave Inte￾grated Circuits

RAMAKRISHNA JANASWAMY, University of Massachusetts, Am￾herst, Massachusetts, Signal Fading in Radiocommunications

BERNARD JARRY, University of Limoges, Limoges, Cedex, France,

Active Filters: Overview of Active-Filter Structures; Active

Filters: Tools and Techniques for Active-Filter Design

DAVID C. JENN, Naval Postgraduate School, Radar Cross Section

A. R. JHA, JHA Technical Consulting Services, Cerritos, CA,

Cryogenic Electronics

JIAN-MING JIN, University of Illinois at Urbana—Champaign,

Urbana, Illinois, Finite Element Analysis; Green’s Function

Methods

J. RALPH JOHLER, Johler Associates, Sky Wave Propagation at

Low Frequencies

X CONTRIBUTORS

ROBERT T. JOHNK, U.S. Department of Commerce, Boulder

Laboratories, Boulder, Colorado, Electromagnetic Ferrite Tile

Absorber

RICHARD L. JOHNSON, Southwest Research Institute, San Anto￾nio, Texas, Radio Direction Finding

WILLIAM T. JOINES, Duke University, Guided Electromagnetic

Waves

EDWARD V. JULL, University of British Columbia, Vancouver,

British Columbia, Canada, Horn Antennas

DARKO KAJFEZ, University of Mississippi, University, Missis￾sippi, Q Factor

MOTOHISA KANDA, National Institute of Standards and Technol￾ogy, Electromagnetic Field Measurement

NORIAKI KANEDA, Lucent Technologies, Holmdel, New Jersey,

The Yagi–Uda Antenna

WENG LOCK KANG, The University of Tennessee, Knoxville,

Tennessee, Directive Antennas

U. KARACAOGLU, INTEL Corp., ICG Wireless Networking

Group—WPD, San Diego, California, Radio-frequency Inte￾grated Circuits

NEMAI CHANDRA KARMAKAR, Monash University, Clayton, Vic￾toria Australia, Adaptive Array Antennas; Coaxial Lines and

Waveguides; Electromagnetic-Bandgap-Assisted Bandpass Fil￾ters

ROGER KAUL, U.S. Army Research Laboratory, Microwave Limit￾ers

W. NEILL KEFAUVER, Lockheed Martin, Antenna Compact

Range; Antenna Testing and Measurements

YUNJIN KIM, California Institute of Technology, Pasadena, Cali￾fornia, Radar Remote Sensing

KATSUJI KIMURA, NEC Corporation, Mixer Circuits

CLIFFORD A. KING, Lucent Technologies, Negative Resistance

AHMED A. KISHK, University of Mississippi, University, Missis￾sippi, Method of Moments

FRANCIS J. KLEMM, Naval Research Laboratory, Electronic War￾fare

R. D. KOILPILLAI, Ericsson, Inc., Radiowave Propagation in

Multipath Channels

ERIK L. KOLLBERG, Chalmers University of Technology, Fre￾quency Converters and Mixers

GE´ ZA KOLUMBA´ N, Budapest University of Technology and

Economics, Hungary, Phase-Locked Loops

SHOZO KOMAKI, Osaka University, Osaka, Japan, Radio-on-Fiber

(RoF) Systems

MYUNG JIN KONG, University of South Carolina, Columbia,

South Carolina, Boundary Value Problems

S.-M. KOO, Royal Institute of Technology, Kista-Stockholm,

Sweden, SiC Device Technologies

V. KOMAROV, Washington State University, Permittivity and

Measurement

N.K. KOUVELIOTIS, National Technical University of Athens,

Athens, Greece, Antenna Reverberation Chamber

PIOTR KOWALCZYK, Gdansk University of Technology, Gdansk,

Poland, Helmholtz Equations

DENNIS KOZAKOFF, Devry University, Alpharetta, Georgia, Aper￾ture Antennas

CLIFFORD M. KROWNE, Microwave Technology Branch, Wa￾shington, DC, Left-Handed Materials for Microwave Devices

and Circuits

A. KUMAR, High Speed Networking Laboratory, Detroit, Michi￾gan, Personal Area Networking with Bluetooth

B.P. KUMAR, Sacramento State University, Microwave and Radio

Frequency Multipliers

RONALD KWOK, California Institute of Technology, Spaceborne

Radar

AKHLESH LAKHTAKIA, Pennsylvania State University, University

Park, Pennsylvania, Anisotropy and Bianisotropy; Chirality

KING WAI LAM, City University of Hong Kong, Kowloon, Hong

Kong SAR, China, Parallel Algorithms and Computing for

Large-Scale Electromagnetic Simulation

PAOLO LAMPARIELLO, ‘‘La Sapienza’’ University of Rome, Italy,

Leaky-Wave Antennas

JOY LASKAR, Georgia Institute of Technology, Atlanta, RF/Wire￾less Packaging

GIANLUCA LAZZI, North Carolina State University, Dipole An￾tennas and Arrays; Monopole Antennas

W. MARSHALL LEACH JR., Georgia Institute of Technology,

Atlanta, Georgia, Radio-frequency Circuit Noise

CHUNG-YI LEE, Qualcomm, Inc., Slotlines

JAY K. LEE, Syracuse University, Syracuse, New York, Helical

Antennas

JIN-FA LEE, The Ohio State University, Columbus, Ohio, Iterative

Methods

JONG-WOOK LEE, Kyung Hee University, Korea, Gallium Nitride

for Electronics

K. F. LEE, University of Mississippi, Oxford, Dual- and Multi￾frequency Microstrip Antennas; Microstrip Antennas, Broad￾band

RICHARD Q. LEE, NASA Glenn Research Center, Cleveland, Ohio,

Notch Antennas

SUNGJAE LEE, Purdue University, West Lafayette, Indiana, Low￾Noise Amplifiers: Device Noise Characterization and Design

WILLIAM C.Y. LEE, AirTouch Communication, Cellular Radio

Y. C. LEE, University of Colorado, Chong-IL Park, Kyocera

America, Inc., Packaging RF Devices and Modules

LAWRENCE M. LEEMIS, The College of William & Mary, Williams￾burg, Virginia, Monte Carlo Simulation in Reliability

PATT LEGGATT, History of Wireless Communication

K. W. LEUNG, City University of Hong Kong, Kowloon, Hong Kong

SAR, Dielectric Resonator Antennas

CURT A. LEVIS, Ohio State University, Columbus, Ohio, Friis

Free-Space Transmission Formula; Radiowave Propagation

Concepts

RALPH LEVY, R. Levy Associates, La Jolla, California, Waveguide

Directional Couplers; Waveguide Junctions

LE-WEI LI, National University of Singapore, Kent Ridge, Singa￾pore, Complex Media

QIN LI, University of Washington, Seattle, Washington, Computa￾tional Electromagnetic Scattering Models for Microwave

Remote Sensing; Rough-Surface Scattering: Numerical Simu￾lations and Applications in Microwave Remote Sensing

ZHONG XIN LI, City University of Hong Kong, Rough-Surface

Scattering: Numerical Simulations and Applications in Micro￾wave Remote Sensing

ERNESTO LIMITI, Universita` degli Studi di Roma Tor Vergata,

Rome, Italy, Microwave Power Amplifiers

KUN-YOU LIN, National Taiwan University, Taipei, Taiwan,

Monolithic Millimeter-Wave Integrated Circuits

BERNABE´ LINARES-BARRANCO, National Microelectronics Center

(CNM), Harmonic Oscillators, Circuits

CLINTON L. LINGREN, Individual Consultants, Radiation Mon￾itoring

LIANGHONG LIU, Virginia Commonwealth University, Richmond,

Virginia, Modulation-Doped FETs

QING HUO LIU, Duke University, Durham, North Carolina, Fast

Fourier Transforms and Nufft

CONTRIBUTORS XI

REN-CHIEH LIU, National Taiwan University, Taipei, Taiwan,

Monolithic Millimeter-Wave Integrated Circuits

JOHN LODGE, Communications Research Centre, Ottawa, On￾tario, Mobile Satellite Communications

ALBERTO ASENSIO LOPEZ, Universidad Politecnica de Madrid,

Madrid, Spain, Monopulse Tracking Systems

JAVIER RAMOS LO´ PEZ, Universidad Carlos III de, Madrid,

Madrid, Spain, Wireless Communications Systems

WALT LOWERY, Symetrix, Inc., Radio Broadcast Studio Equip￾ment

L. LUCCI, University of Florence, Florence, Italy, Corrugated Horn

Antennas

ERNST LUEDER, University of Stuttgart, Filter Synthesis

PAOLO LUGLI, University Rome, Monte Carlo Analysis

ANDRE LUITEN, University of Western Australia, Nedlands, Aus￾tralia, Q-Factor Measurement

K. M. LUK, City University of Hong Kong, Kowloon, Hong Kong,

Dual- and Multifrequency Microstrip Antennas; Microstrip

Antennas, Broadband

JOHANN-FRIEDRICH LUY, Munich Technical University and

DaimlerChrysler Research, Germany, Digital Microwave Recei￾vers; GeSi Alloys and Devices

E. M. MA, Cypress Semiconductor Inc. (Minnesota), Thin Film

Resistors

ZHEWANG MA, University of Electro-Communications, Microstrip

Lines

TOM G. MACKAY, University of Edinburgh, Edinburgh, United

Kingdom, Anisotropy and Bianisotropy

KOICHI MAEZAWA, Nagoya University, Nagoya, Japan, Resonant

Tunneling Diodes

ROBERT J. MAILLOUX, AFRL/SNH, Hanscom Air Force Base,

Massachusetts, Antenna Scanning Arrays

IVAN A. MAIO, Politecnico di Torino, Frequency-Domain Circuit

Analysis

C. L. MAK, City University of Hong Kong, Kowloon, Hong Kong,

Microstrip Antennas, Broadband

J. A. G. MALHERBE, University of Pretoria, Pretoria, South

Africa, Lens Antennas

DONALD C. MALOCHA, University of Central Florida, Surface

Acoustic Wave Applications

GIULIANO MANARA, University of Pisa, Pisa, Italy, Generalized

Scattering Matrix Technique

G. F. MANES, University of Florence, Italy, Gallium Arsenide

Technology and Applications

CHRIS MANN, Rutherford Appleton Laboratory, UK, Terahertz

Science, Engineering and Systems—from Space to Earth Ap￾plications

RAAFAT R. MANSOUR, University of Waterloo, Waterloo, Ontario,

Canada, High-Temperature Superconductors; Multiplexers

RICARDO MARQUE´ S, University of Seville, Seville, Spain, Ferrite

Loaded Waveguides

JAVIER MARTI´, Universidad Polite´cnica de Valencia, Valencia,

Spain, Photonic Band Gap (PGB)

ALEJANDRO MARTI´NEZ, Universidad Polite´cnica de Valencia,

Valencia, Spain, Photonic Band Gap (PGB)

K. T. MATHEW, Cochin University of Science and Technology,

Cochin, India, Perturbation Theory

WOLFGANG MATHIS, University of Hannover, Germany, Oscilla￾tor Design

VLADIMIR MATIJASEVIC, Conductus, Inc., HTS Film Growth

ARVYDAS MATULIONIS, Semiconductor Physics Institute, Noise,

Hot Carrier Effects

J. H. MCCLELLAN, Georgia Institute of Technology, Frequency

Modulation

JAMES W. MCCLYMONDS, Raytheon Systems Company, Impatt

Diodes and Circuits

DAVID J. MCLAUGHLIN, University of Massachusetts, Amherst,

Massachussets, Radar Polarimetry

KENNETH K. MEI, City University of Hong Kong and Zhejiang

University, Hangzhou, China, Maxwellian Circuits; MEI

Method

LEONARDO R. A. X. MENEZES, University of Brasilia, Brasilia,

DF, Brazil, Antenna Accessories

JAVIER GISMERO MENOYO, Universidad Politecnica de Madrid,

Madrid, Spain, Monopulse Tracking Systems

WOLFGANG S. MENZEL, University of Ulm, Ulm, Germany, Fin￾line Components

FRANCISCO MESA, University of Seville, Seville, Spain, Leaky

Modes and High-Frequency Effects in Microwave Integrated

Circuits

KRZYSZTOF ARKADIUSZ MICHALSKI, Texas A&M University,

College Station, Texas, Coulomb Gauge in Electromagnetics;

Electromagnetic Field Computation in Planar Multilayers

ALAN MICKELSON, University of Colorado, Boulder, Colorado,

Active Antennas

KONSTANTINOS MISIAKOS, NCSR ‘‘Demokritos,’’ Athens, Greece,

Diodes

RAJ MITTRA, Pennsylvania State University, University Park,

Pennsylvania, Enhancements of the Finite-Difference Time￾Domain Method; Generalized Scattering Matrix Technique;

Multiresolution Technique

RYAN Y. MIYAMOTO, University of Hawaii at Manoa, Honolulu,

Hawaii, Retrodirective Systems

ANANDA SANAGAVARAPU MOHAN, University of Technology, Syd￾ney, New South Wales, Australia, Dielectric Resonators

ANURAAG MOHAN, University of Delaware, Newark, Delaware,

Galerkin Method (Rayleigh–Ritz Method)

MOHAMMAD NURUNNABI MOLLAH, Khulna University of Engi￾neering and Technology, Khulna, Bangaladesh, Electromag￾netic-Bandgap-Assisted Bandpass Filters

MAURO MONGIARDO, University of Perugia, Perugia, Italy, Trans￾verse Resonance Technique

AGOSTINO MONORCHIO, University of Pisa, Pisa, Italy, General￾ized Scattering Matrix Technique

FERNANDO J. S. MOREIRA, Federal University of Minas Gerais,

Belo Horizonte, Brazil, Electromagnetic Wave Propagation

CHIP MORGAN, CMBE, Inc., Low-Power Broadcasting

D. P. MORGAN, Impulse Consulting, Northampton, United King￾dom, Surface Acoustic Wave Devices

ANTONIO MORINI, Universita` di Ancona, Ancona, Italy, Direc￾tional Couplers

HADIS MORKOC, , Virginia Commonwealth University, Richmond,

Virginia, Modulation-Doped FETs

STANLEY A. MOROZ, Naval Research Laboratory, Electronic

Warfare

AMIR MORTAZAWI, University of Michigan, Ann Arbor, Michigan,

Microwave Oscillators

MICHAL MROZOWSKI, Gdansk University of Technology, Gdansk,

Poland, Helmholtz Equations

TH. MUELLER, DaimlerChrysler Research, Ulm, Germany, Digi￾tal Microwave Receivers

BRIAN MULKEEN, University College Dublin, Dublin, Ireland,

Transient Circuit Analysis

BLAINE T. MURAKAMI, University of Hawaii at Manoa, Honolulu,

Hawaii, Retrodirective Systems

RANDALL L. MUSSELMAN, USAF Academy, Colorado Springs,

Colorado, Electromagnetic Wave Scattering

BRENT A. MYERS, Intersil Corporation, Pulse-Shaping Circuits

XII CONTRIBUTORS

VIJAY NAIR, Intel Corporation, Hillsboro, Oregon, Low-Noise

Amplifiers

KRISHNA NAISHADHAM, Massachusetts Institute of Technology,

Lexington, Massachusetts, Parameter Estimation from Electro￾magnetic Simulations Using Signal Models

HISAMATSU NAKANO, Hosei University, Koganei, Tokyo, Japan,

Slot Antennas

MYSORE NARAYANAN, Miami University, Oxford, Ohio, Wideband

Amplifiers

RAFAEL NAVAS-GONZA´ LEZ, Institute of Microelectronics of Se￾ville, Seville, Spain, Voltage-to-Frequency Converters

JEAN-MICHEL NEBUS, Universite de Limoges, Millimeter-Wave

Measurement

R. NESTI, National Institute for Astrophysics, Florence, Italy,

Corrugated Horn Antennas; Orthomode Transducers

ROBERT D. NEVELS, Texas A&M University, College Station,

Texas, Coulomb Gauge in Electromagnetics, The

H. K. NG, City University of Hong Kong, Kowloon, Hong Kong

SAR, Dielectric Resonator Antennas

KWOK K. NG, Lucent Technologies, Negative Resistance

LAM NGUYEN, Army Research Laboratory, Radar Equipment

TIEN M. NGUYEN, The Aerospace Corporation, Military Commu￾nication

KENNETH V. NOREN, University of Idaho, Moscow, Idaho, Elliptic

Filters

ROBERT NOWLIN, Arizona State University East, Mesa, Arizona,

Attenuators

MICHAEL W. NURNBERGER, Navy Research Laboratory, Washing￾ton, University of Colorado, Boulder, Colorado, Frequency-in￾Dependent Antennas

AARON T. OHTA, University of Hawaii at Manoa, Honolulu,

Hawaii, Retrodirective Systems

ERIK O¨ JEFORS, Uppsala University, Uppsala, Sweden, Mono￾lithic Antennas

FRANK OLYSLAGER, Ghent University, Ghent, Belgium, Skin

Effect

BAN-LEONG OOI, National University of Singapore, Singapore,

Printed Inductors

GEOFFREY C. ORSAK, Southern Methodist University, Noise and

Interference Modeling

JOHN M. OSEPCHUK, Concord, Massachusetts, Magnetrons; Mi￾crowave Heating

M. O¨ STLING, Royal Institute of Technology, Kista-Stockholm,

Sweden, SiC Device Technologies

CANER OZDEMIR, Mersin University, Mersin, Turkey, Synthetic

Aperture Radar

HASAN PADAMSEE, Cornell University, Superconducting Cavity

Resonators

SHANTANU KUMAR PADHI, Monash University, Australia, Coax￾ial Lines and Waveguides

R. K. PANDEY, The University of Alabama, Tuscaloosa, Alabama,

Ferroelectric Materials

CONSTANTINOS PAPADIAS, Lucent Technologies, Antenna Arrays

for Mobile Communications

ROBERT K. PARKER, Naval Research Laboratory, Washington,

D.C., Microwave Tubes

D. P. PATEL, Naval Research Laboratory, Microwave Ferroelec￾tric Devices

CLAYTON R. PAUL, Mercer University, Macon, Georgia, Coupled

Transmission Lines; Electromagnetic Compatibility

AROGYASWAMI PAULRAJ, Stanford University, Antenna Arrays

for Mobile Communications

S. J. PEARTON, University of Florida, Gainesville, Florida, Pas￾sivation

JOSE´ CARLOS PEDRO, University of Aveiro, Portugal, Intermo￾dulation

G. PELOSI, University of Florence, Italy, Corrugated Horn Anten￾nas; Orthomode Transducers

ODILON M. C. PEREIRA-FILHO, Federal University of Minas

Gerais, Belo Horizonte, Brazil, Electromagnetic Wave Propaga￾tion

MANUEL SIERRA PEREZ, Universidad Politecnica de Madrid,

Madrid, Spain, Monopulse Tracking Systems

EDWARD PETERSON, Arizona State University East, Mesa,

Arizona, Attenuators

STEPHANE PINEL, Georgia Institute of Technology, Atlanta, RF/

Wireless Packaging

UMBERTO PISANI, Politecnico di Torino, Torino, Italy, Microwave

Measurements

GREESHMA PISHARODY, University of Delaware, Newark, Dela￾ware, Galerkin Method (Rayleigh–Ritz Method)

J. M. POND, Naval Research Laboratory, Microwave Ferroelectric

Devices

PROTAP PRAMANICK, K & L Microwave, Inc., Salisbury, Mary￾land, Hybrids and Couplers; Strip Transmission Lines

JAYASIMHA PRASAD, Maxim, San Jose, California, Heterojunc￾tion Bipolar Transistor

CARLES PUENTE, Fractus S.A., Barcelona, Spain, Fractal Shaped

Antennas: A Review

ANYONG QING, National University of Singapore, Singapore,

Electromagnetic Inverse Problems

RONALD RACKLEY, du Treil, Lundin, & Rackley, Sarasota, Flor￾ida, Antennas for Medium-Frequency Broadcasting

MARK RADER, The University of Tennessee, Knoxville, Tennessee,

Directive Antennas

KALADHAR RADHAKRISHNAN, Intel Corporation, Chandler, Ari￾zona, High-Frequency Transmission Lines

M. RAHMAN, Hankel Transforms

JAIME RAMIREZ-ANGULO, New Mexico State University, Las

Cruces, New Mexico, Bandpass Filters

R. KEITH RANEY, Johns Hopkins University, Laurel, Maryland,

Radar Altimetry

N. RANGANATHAN, University of South Florida, Integrated Cir￾cuits

J. B. L. RAO, Naval Research Laboratory, Microwave Ferro￾electric Devices

S. M. RAO, Auburn University, Auburn, Alabama, Integral

Equations: Application to Open-Region Problems

JAMES C. RAUTIO, Sonnet Software, Inc., Phoenix, New York,

Applied Numerical Electromagnetic Analysis for Planar High￾Frequency Circuits

LEONHARD M. REINDL, Albert Ludwigs Universitat, Freiburg,

Germany, Surface Acoustic Wave Delay Lines

MARC RESSLER, Army Research Laboratory, Radar Equipment

R. REUTER, Innovative Processing AG (IPAG), Duisburg, Ger￾many, Noise, High-Frequency

PATRICK REYNAERT, KULeuven ESAT-MICAS, Leuven, Belgium,

Mixed-Signal CMOS RF Integrated Circuits

DAVID F. RIVERA, Naval Undersea Warfare Center, Newport,

Rhode Island, Submarine Antennas

I.D. ROBERTSON, University of Leeds, Leeds, United Kingdom,

Radiofrequency Integrated Circuits

CHRISTOPHER T. RODENBECK, Texas A&M University, College

Station, Texas, Gratings, Grating Antennas

ARMANDO A. RODRIGUEZ, Arizona State University, Tempe,

Arizona, Missile Guidance

A´ NGEL RODRI´GUEZ-VA´ ZQUEZ, University of Seville, Spain, Har￾monic Oscillators, Circuits; Voltage-to-Frequency Converters

CONTRIBUTORS XIII

ULRICH L. ROHDE, Synergy Microwave Corporation, Paterson,

New Jersey, Phase-Locked Oscillators and Frequency Synthe￾sizers

ARYE ROSEN, Drexel University, Philadelphia, Pennsylvania,

PIN Diodes

TULLIO ROZZI, Universita` di Ancona, Ancona, Italy, Directional

Couplers

THOMAS E. RUDEN, Newton Highlands, Massachusetts, Magne￾trons

CLEMENS C.W. RUPPEL, Albert Ludwigs Universitat, Freiburg,

Germany, Surface Acoustic Wave Delay Lines

ANDREW RUSEK, Oakland University, Rochester, Michigan, Re￾flectometers, Time-Domain

PETER RUSSER, Munich Technical University, Germany, GeSi

Alloys and Devices; Oscillator Design

ANDERS RYDBERG, Uppsala University, Uppsala, Sweden, Mono￾lithic Antennas

VICTOR RYZHII, University of Aizu, Heterostructures Devices

MAGDALENA SALAZAR-PALMA, Madrid Polytechnic University,

Spain, Direction of Arrival Estimation and Adaptive Processing

Using a Conformal Phased Array; Stripline Components

PABLO SANCHIS, Universidad Polite´cnica de Valencia, Valencia,

Spain, Photonic Band Gap (PGB)

CARLES SANS, Agrotech Biotecnologia Aplicada SA, Spain, Mi￾crowave Superconductor Devices

TAPAN K. SARKAR, Syracuse University, Syracuse, New York,

Direction of Arrival Estimation and Adaptive Processing Using

a Conformal Phased Array; Stripline Components

RONAN SAULEAU, University of Rennes, Rennes, France, Fabry–

Perot Resonators

B. SAUVIAC, Jean Monnet University, St. Etienne, France, Ferrite

Isolators

MANOJ SAXENA, University of Delhi, New Delhi, India, MOSFET

Modeling

ROLF SCHAUMANN, Portland State University, Portland, Oregon,

Circuit Tuning

BURKHARD SCHIEK, Ruhr-Universita¨t Bochum, Standing Wave

Meters and Network Analyzers

DOMINIQUE M. M.-P. SCHREURS, K.U.Leuven, Leuven-Heverlee,

Belgium, Microwave Solid-State Devices

THOMAS F. SCHUBERT Jr., University of San Diego, San Diego,

California, Feedback Amplifiers

HERMANN SCHUMACHER, University of Ulm, Ulm, Germany,

Silicon-Germanium (Si-Ge)

LEONARD M. SCHWAB, Germaine Development Corp., Smith

Chart

JAMES R. SCHWANK, Sandia National Laboratories, Albuquer￾que, New Mexico, Radiation Effects

S. SELLERI, University of Florence, Italy, Corrugated Horn An￾tennas; Orthomode Transducers

DIPAK L. SENGUPTA, University of Detroit Mercy, Detroit, Michi￾gan, Conformal Antennas

FRANCISCO J. GONZA´ LEZ SERRANO, Universidad Carlos III de

Madrid, Madrid, Spain, Wireless Communications Systems

FRED W. SEXTON, Sandia National Laboratories, Albuquerque,

New Mexico, Radiation Effects

L. SHAFAI, University of Manitoba, Winnipeg, Manitoba, Canada,

Dielectric Loaded Antennas

MARTY R. SHANEYFELT, Sandia National Laboratories, Albu￾querque, New Mexico, Radiation Effects

ARVIND K. SHARMA, TRW, Redondo Beach, California, Cavity

Resonators

WILLIAM M. SHERRILL, Southwest Research Institute, San Anto￾nio, Texas, Radio Direction Finding

GRANT S. SHIROMA, University of Hawaii at Manoa, Honolulu,

Hawaii, Retrodirective Systems

WAYNE A. SHIROMA, University of Hawaii at Manoa, Honolulu,

Quasioptical Circuits; Retrodirective Systems

Y.H. SHUM, City University of Hong Kong, Kowloon, Hong Kong,

Dual- and Multifrequency Microstrip Antennas

MOSHE SIDI, Technion—Israel Institute of Technology, Multiple￾Access Schemes

FRANKLIN C. SILVA, University of Brasilia, Brasilia, DF, Brazil,

Antenna Accessories

RAINEE N. SIMONS, NASA Glenn Research Center, Cleveland,

Ohio, Coplanar Stripline Transitions; Coplanar Waveguide

Components

GORDON G. SINCLAIR, Technology for Communications Interna￾tional, Fremont, California, Antennas for High-Frequency

Broadcasting

O. N. SINGH, Banaras Hindu University, Varanasi, India, Elec￾tromagnetic Materials

KAZIMIERZ SIWIAK, TimeDerivative, Inc., Coral Springs, Florida,

Loop Antennas; Ultra-Wideband Radio

MARK A. SLETTEN, Naval Research Laboratory, Washington, DC,

Radar Polarimetry

PETER M. SMITH, McMaster University, Hamilton, Canada,

Acoustic Microwave Devices

CHARLES E. SMITH, Center of Applied Electromagnetic Systems

Research (CAESR) The University of Mississippi University,

Mississippi, Wideband Slot and Printed Antennas

CHRISTOPHER M. SNOWDEN, Filtronic Compound Semiconduc￾tors and University of Leeds, Shipley, West Yorkshire, United

Kingdom, III–V Semiconductors

RICHARD V. SNYDER, RS Microwave, Butler, New Jersey, Fre￾quency-Division Multiplexers

JORDI SOLER, Fractus S.A., Barcelona, Spain, Fractal Shaped

Antennas: A Review

SAMEER SONKUSALE, Tufts University, Medford, Massachusetts,

A/D and D/A Conversion Architectures and Techniques

ROBERTO SORRENTINO, University of Perugia, Perugia, Italy,

Transverse Resonance Technique; Waveguide Discontinuities

ANTHONY E. SPEZIO, Naval Research Laboratory, Electronic

Warfare

S. SRIDHAR, Northeastern University, Superconducting Micro￾wave Technology

DANIELA STAICNLESCU, Georgia Institute of Technology, Atlanta,

RF/Wireless Packaging

MANFRED STECHER, Rhode & Schwarz GmbH & Co.KG, Mag￾netic Field Measurement

MICHIEL STEYAERT, KULeuven ESAT-MICAS, Leuven, Belgium,

Mixed-Signal CMOS RF Integrated Circuits

BERND H. STRASSNER II, Sandia National Laboratories, Albu￾querque, New Mexico, Microwave Power Transmission; Rectify￾ing Antennas (Rectennas)

WARREN L. STUTZMAN, Virginia Tech Antenna Group, Blacks￾burg, Virginia, Antenna Theory

BINGZHI SU, University of Colorado, Chong-IL Park, Kyocera

America, Inc., Packaging RF Devices and Modules

ALMUDENA SUA´ REZ, University of Cantabria, Santander, Canta￾bria, Spain, Nonlinear Circuit Analysis; Nonlinear Circuit

Design

YOUNG-HO SUH, Mimix Broadband Inc., Houston, Texas, Copla￾nar Stripline (CPS) Components

SAMIA A. SULIMAN, The Pennsylvania State University, Univer￾sity Park, Pennsylvania, Smart Materials

AHMED IYANDA SULYMAN, Queen’s University, Kingston, Ontario,

Canada, MIMO Systems for Wireless Communications

XIV CONTRIBUTORS

CHEN SUN, Nanyang Technological University, Singapore, Adap￾tive Array Antennas

RAJI SUNDARARAJN, Arizona State University East, Mesa, Ar￾izona, Attenuators

JAMES A. SVOBODA, Clarkson University, Potsdam, New York,

Circuit Stability

ALLEN TAFLOVE, Northwestern University, Evanston, Illinois,

Finite-Difference Time-Domain Analysis

R. S. TAHIM, RST Scientific Research, Inc., New York, Microwave

Mixers

PAUL J. TALLERICO, Los Alamos National Laboratory, Klystron

JAMES M. TALLEY, Naval Research Laboratory, Electronic War￾fare

MICHAEL A. TAMAMOTO, University of Hawaii at Manoa, Hono￾lulu, Hawaii, Retrodirective Systems

J. TANG, Washington State University, Permittivity and Mea￾surement

SERIOJA O. TATU, E´cole Polytechnique, Montreal, Quebec, Ca￾nada, Quadrature Phase Shift Keying (QPSK)

ALEXANDER L. TAUBE, Swinburne University of Technology,

Melbourne, Australia, Microwave Resonance Plasma Source

F. J. TEGUDE, Innovative Processing AG (IPAG), Duisburg,

Germany, Noise, High-Frequency

FERNANDO L. TEIXEIRA, The Ohio State University, Columbus,

Ohio, Electromagnetic Wave Propagation; High-Frequency

Transmission Lines

MANOS M. TENTZERIS, Georgia Institute of Technology, Atlanta,

Georgia, Application of Wavelets to Electromagnetic Problems;

Electromagnetics, Time-Domain; RF/Wireless Packaging

VALERIA TEPPATI, Politecnico di Torino, Torino, Italy, Microwave

Measurements

MARCO A. B. TERADA, New Mexico State University, Las Cruces,

New Mexico, Reflector Antennas

D. G. THOMAS JR., University of California, Davis, Microwave

and Radio Frequency Multipliers

TUAN TON, Army Research Laboratory, Radar Equipment

GUIDO TORELLI, University of Pavia, Preamplifiers

ALESSANDRO TOSCANO, ‘‘Roma Tre’’ University, Rome, Italy,

Method of Lines

P. T. TRAKADAS, National Technical University of Athens, Athens,

Greece, Antenna Reverberation Chamber

WALTER J. TRYBULA, SEMATECH, Austin, Texas, Surface Mount

Technology

FENG-CHI E. TSAI, University of Queensland, Queensland, Aus￾tralia, Power Combiners and Dividers

LEUNG TSANG, University of Washington, Seattle, Computational

Electromagnetic Scattering Models for Microwave Remote Sen￾sing; Rough-Surface Scattering: Numerical Simulations and

Applications in Microwave Remote Sensing

GEORGE A. TSIHRINTZIS, University of Piraeus, Piraeus, Greece,

Radar Applications

HAIPING TSOU, California Institute of Technology, Radiotele￾metry

KATSUTOSHI TSUKAMOTO, Osaka University, Osaka, Japan,

Radio-on-Fiber (RoF) Systems

AHMET SERDAR TU¨ RK, Information Technologies Research In￾stitute, Gebze, Kocaeli, Turkey, Buried-Object Detection

KENJI UCHINO, The Pennsylvania State University, Piezoelectri￾city

PYOTR YA. UFIMSTEV, University of California at Irvine, Califor￾nia, Backscatter

SHAMBHU J. UPADHYAYA, State University of New York at Buffalo,

Noise Generators

RUEDIGER VAHLDIECK, Institut fu¨r Feldtheorie und Ho¨chstfre￾quenztechnik and Swiss Federal Institute of Technology, Zurich,

Switzerland, Electromagnetic Modeling; Finlines

ANDRE´ VANDER VORST, Universite´ Catholique de Louvain, Lou￾vain-la-Neuve, Belgium, Waveguides

ROBERT B. VAN DOVER, Bell Labs, Lucent Technologies, Magnetic

Materials

JAKOB J. VAN ZYL, California Institute of Technology, Pasadena,

California, Radar Remote Sensing

RODNEY G. VAUGHAN, Simon Fraser University, Burnaby, British

Columbia, Canada, Mobile Radio Channels

LUCIO VEGNI, ‘‘Roma Tre’’ University, Rome, Italy, Method of

Lines

RAJU D. VENKATARAMANA, University of South Florida, Inte￾grated Circuits

S. VERDEYME, IRCOM, Limoges, France, Dielectric Resonator

Filters

GUY VERNET, Universite´ Paris—Sud, Millimeter-Wave Measure￾ment

FERNANDO VIDAL-VERDU´ , University of Ma´laga, Ma´laga, Spain,

Voltage-to-Frequency Converters

D. VINCENT, Jean Monnet University, St. Etienne, France, Ferrite

Isolators

OSAMU WADA, Kobe University, Japan, Indium Phosphide (InP)

P. F. WAHID, University of Central Florida, Orlando, Florida,

Antennas

H. R. WALKER, Pegasus Data Systems, Inc., Edison, New Jersey,

Intermediate-Frequency Amplifiers

FRED L. WALLS, Total Frequency, Boulder, Colorado, Frequency

Standards, Characterization

CHARLES C. WANG, The Aerospace Corporation, Military Com￾munication

CHI WANG, Orbital Sciences Corporation, Dulles, Virginia, Comb￾line Filters

D. X. WANG, City University of Hong Kong, Kowloon, Hong Kong,

P.R. China, Ferrite Circulators

HUEI WANG, National Taiwan University, Taipei, Taiwan, Mono￾lithic Millimeter-Wave Integrated Circuits

LIHONG V. WANG, Texas A&M University, College Station, Texas,

Medical Imaging with Microwave: Thermoacoustic Tomography

MINSHENG WANG, Texas Instruments Incorporated, Radar Signal

Detection

S. WANG, Washington State University, Permittivity and Mea￾surement

ZEYI WANG, Tsinghua University, Beijing, China, Capacitance

Extraction

ANDREW WEBB, University of Illinois at Urbana–Champaign,

Urbana, Magnetic Resonance Imaging

KEVIN J. WEBB, Purdue University, West Lafayette, Indiana,

Low-Noise Amplifiers: Device Noise Characterization and De￾sign

RALF WEBER, Ericsson Eurolab Deutschland GmbH, Mobile

Communication

DANIEL S. WEILE, University of Delaware, Newark, Delaware,

Galerkin Method (Rayleigh–Ritz Method)

ANDREW R. WEILY, Macquarie University, Sydney, New South

Wales, Australia, Dielectric Resonators

ROBERT WEIRATHER, Harris Corporation, Quincy, Illinois,

Transmitters for Digital Television

DANIEL WEIS, Individual Consultants, Radiation Monitoring

JOSEPH F. WHITE, JFW Technology, Inc., Microwave Phase

Shifters

RAYMOND A. WILDMAN, University of Delaware, Newark, Dela￾ware, Galerkin Method (Rayleigh–Ritz Method)

CONTRIBUTORS XV

RONALD WILENSKY, Technology for Communications Interna￾tional, Fremont, California, Antennas for High-Frequency

Broadcasting

JOHN E.C. WILLIAMS, Massachusetts Institute of Technology,

Cambridge, Massachusetts, Superconducting Electromagnets

H. WONG, City University of Hong Kong, Kowloon, Hong Kong,

Microstrip Antennas, Broadband

KIN-LU WONG, National Sun Yat-Sen University, Kaohsiung,

Taiwan, Microstrip Antennas, Compact

KE WU, Ecole Polytechnique, Montreal, Quebec, Canada, Impe￾dance Transformers and Matching Networks; Six-Port Net￾works; Slow Wave Structures

KE-LI WU, The Chinese University of Hong Kong, Shatin, NT,

Hong Kong, Low Temperature Co-Fired Ceramic (LTCC) Tech￾nology in RF and Microwave Engineering

TE-KAO WU, Northrop-Grumman, Redondo Beach, California,

Frequency Selective Surfaces

THOMAS X. WU, University of Central Florida, Orlando, Florida,

BiCMOS Devices and RF Integrated Circuits

ZHIPENG WU, UMIST, Manchester, United Kingdom, Electromag￾netic Surface Waves

TADEUSZ WYSOCKI, Curtin University of Technology, Mobile

Communication

MING YAO XIA, Peking University, Beijing, China, Rough-Surface

Scattering: Numerical Simulations and Applications in Micro￾wave Remote Sensing

FUQIN XIONG, Cleveland State University, Cleveland, Ohio, Am￾plitude Shift Keying

XINYU XU, Canada, Six-Port Networks

DAO-XIAN XU, National University of Singapore, Singapore,

Printed Inductors

JIANJUN XU, Carleton University, Ottawa, Canada, Neural Net￾works for Microwave Circuits

YUAN XU, Texas A&M University, College Station, Texas, Medical

Imaging with Microwave: Thermoacoustic Tomography

MASATSUNE YAMAGUCHI, Chiba University, Chiba, Japan, Sur￾face Acoustic Wave Filters

EIKICHI YAMASHITA, University of Electro-Communications, Mi￾crostrip Lines

XIAOMIN YANG, University of Central Florida, Orlando, Florida,

BiCMOS Devices and RF Integrated Circuits

ERDEM YAZGAN, Hacettepe University, Beytepe, Ankara, Turkey,

Conformal Mapping Techniques

WEI YE, National University of Singapore, Radar Signal Proces￾sing

TAT SOON YEO, National University of Singapore, Radar Signal

Processing

WEN-YAN YIN, National University of Singapore, Kent Ridge,

Singapore, Complex Media

FRANK S. YOUNG, Electric Power Research Institute, Magnetic

Shielding

SIMON H. YUEH, California Institute of Technology, Spaceborne

Radar

WENHUA YU, Pennsylvania State University, University Park,

Pennsylvania, Enhancements of the Finite-Difference Time￾Domain Method

WENJIAN YU, Tsinghua University, Beijing, China, Capacitance

Extraction

TAE-YEOUL YUN, Hanyang University, Seoul, Korea, Piezoelectric

Transducer Controlled Circuits

E. K. N. YUNG, City University of Hong Kong, Kowloon, Hong

Kong, P.R. China, Ferrite Circulators

S. ZEADALLY, High Speed Networking Laboratory, Detroit, Michi￾gan, Personal Area Networking with Bluetooth

HANS-JU¨ RGEN ZEPERNICK, Cooperative Research Center for

Broadband Telecommunications and Networking, Mobile Com￾munication

C.-M. ZETTERLING, Royal Institute of Technology, Kista-Stock￾holm, Sweden, SiC Device Technologies

QI-JUN ZHANG, Carleton University, Ottawa, Canada, Neural

Networks for Microwave Circuits

WENGE ZHANG, University of Colorado, Chong-IL Park, Kyocera

America, Inc., Packaging RF Devices and Modules

JIANYI ZHOU, Southeast University, Nanjing, China, Feedback

Oscillators

LIN ZHOU, Duke University, North Carolina, Rough-Surface

Scattering: Numerical Simulations and Applications in Micro￾wave Remote Sensing

LEI ZHU, Nanyang Technological University, Singapore, Copla￾nar Waveguide (CPW) Transmission Lines; Microwave Filters

HERBERT ZIRATH, Chalmers University of Technology, Frequency

Converters and Mixers

TAKIS ZOURNTOS, Texas A&M University, College Station, Texas,

A/D and D/A Conversion Architectures and Techniques

ZORAN ZVONAR, Analog Devices, Wilmington, Massachusetts,

Radio System Performance

XVI CONTRIBUTORS

HANKEL TRANSFORMS

M. RAHMAN

The object of this article is to introduce integral transform

of a particular type, called the Hankel transform, and to

illustrate the use of this method by means of examples.

The treatment is that of a review article and as such is not

meant to be exhaustive; its aim is to give a concatenated

account of known results rather than present new ones.

The emphasis throughout is on those results that are of

frequent occurrence in boundary value problems of math￾ematical physics, but some indication is also given for

possible theoretical investigations.

Proofs are either omitted entirely or only the key steps

are outlined. Readers interested in rigorous proofs of some

of the statements in this article are referred to the books

by Sneddon [1,2], Davies [3], Andrews and Shivamoggi [4],

and Zayed [5].

The organization of the article is as follows. In the first

section, we illustrate the motivation behind introducing

the Hankel transform and then give a precise definition of

the Hankel transform and its inversion. The next two

sections are devoted to the derivation of some basic

properties of Hankel transforms. In the following section,

we explore the connection between Fourier and Hankel

transforms. Parseval’s relation for Hankel transforms is

then deduced. We next introduce the modified operator of

Hankel transforms. An overview of Erdelyi–Kober opera￾tors and their generalization by Sneddon and Cooke is

given. We then derive Beltrami-type relations and give a

brief account of their generalization by Sneddon. An

extensive account is given of the applications of Erdelyi–

Kober and Cooke operators to dual, triple, and quadruple

integral equations involving Hankel transforms. A num￾ber of issues that arise in connection with applications of

Hankel transforms to many physical problems is then

addressed. For the convenience of the readers, a compen￾dium is given in the last section of the basic theorems and

formulas of Hankel transforms that are of frequent occur￾rence in applications.

1. THE HANKEL TRANSFORM

The Hankel transform arises naturally as a result of using

the method of separation of variables to boundary value

problems of mathematical physics in cylindrical coordi￾nates, for example, boundary value problems for the La￾place and Helmholtz equations involving half-spaces and

regions bounded by parallel planes. In general, applica￾tion of this technique is relevant to problems leading to

the integration of equations of the type

@2f

@r2 þ

1

r

@f

@r
v2

r2 f þ Lf ¼ fðr; ...Þ

where L is a linear operator that does not contain r, and

f(r, y) is a prescribed function.

To illustrate this, let us consider the axisymmetric

solution j(r, z) of Laplace’s equation

@2f

@r2 þ

1

r

@f

@r þ

@2f

@z2 ¼ 0 ð1Þ

in the half-space r40, z40, which satisfies the boundary

condition

fðr; 0Þ ¼ fðrÞ ð2Þ

where f(r) is a prescribed function of r.

In addition, the solution of the problem must satisfy the

regularity conditions so that the field decays as R-N,

where R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 þ z2 p .

Assuming that the solution can be represented in the

separated-variable form

fðr; zÞ ¼ f1ðrÞf2ðzÞ

we find that Eq. (1) reduces to

1

f1

d2f1

dr2 þ

1

f1r

df1

dr ¼
1

f2

d2f2

dz2 ð3Þ

Since the left-hand side of Eq. (3) depends only on r while

the right-hand side only on z, we conclude that they must be

equal to a constant, say, l¼
s

2

, where s is a real quantity.

Thus, we obtain two ordinary differential equations

d2f1

dr2 þ

1

r

df1

dr þ s2f1 ¼ 0

df2

dz2
s2f2 ¼ 0

ð4Þ

The first of these equations is that of Bessel [6], whose

solution bounded at the origin is

f1ðrÞ ¼ A1ðsÞJ0ðsrÞ

where A1(s) is an arbitrary function of s and J0(sr) is the

zeroth-order Bessel function of the first kind.

On the other hand, the solution of the second relation of

Eq. (4) ensuring a decaying field is given by

f2ðzÞ ¼ A2ðsÞ e
sz

Therefore, the solution of Eq. (1) is

fðr; zÞ ¼ AðsÞJ0ðsrÞ e
sz ð5Þ

where A(s) is an arbitrary function of s. Readers can easily

verify that the other cases, namely, l ¼ 0 and l ¼ s2 (s is a

real quantity), must be ignored, since they do not ensure a

decaying field as R ! 1.

H

1893