Prof Reinhold Pregla University of Hagen, Germany “This rigorous and, at the same time, easy-to-understand explanation of non-stationary electromagnetic phenomena will be of great interest to researchers from the physical science community.” Prof Elena Romanova Saratov State University, Russia “This magnificent work guides readers through the mysterious world of non-stationary electromagnetics Its very first sentence catches them and sets free their imagination to expect and see the newly discovered sides of our nature.” Dr Mariana Nikolova Georgieva-Grosse Polikraishte, Bulgaria Prof Georgi Nikolov Georgiev St Cyril and St Methodius University of Veliko Tarnovo, Bulgaria This book is devoted to investigations of non-stationary electromagnetic processes It offers a good opportunity to introduce the Volterra integral equation method more widely to the electromagnetic community The explicit mathematical theory is combined with examples of its application in electromagnetic devices, optoelectronics, and photonics, where time-domain methods become a powerful tool for modelling Many of the electromagnetic phenomena that are studied in the book may lead to numerous new ideas for experimentalists and engineers developing new classes of photonic devices Alexander Nerukh is head of the Department of Higher Mathematics, Kharkov National University of Radioelectronics, Ukraine He has published books and over 250 scientific papers Prof Nerukh’s scientific interests lie in nonstationary and nonlinear electrodynamics, and he has collaborated with the University of Nottingham and Aston University in these fields Nataliya Sakhnenko is associate professor at the Department of Higher Mathematics, Kharkov National University of Radioelectronics She has held joint research with the University of Nottingham and the University of Jena Her current research interests are in time-domain problems of photonics, plasmonics, and metamaterials Phillip Sewell is professor of electromagnetics in the Faculty of Engineering, University of Nottingham His research interests involve analytical and numerical modelling of electromagnetic problems, with application to optoelectronics, electromagnetic compatibility, and electrical machines He has published  approximately 500 papers V250 ISBN-13 978-981-4316-44-6 Nerukh | Sakhnenko Benson | Sewell Trevor Benson is director of the George Green Institute for Electromagnetics Research, University of Nottingham His research interests include experimental and numerical studies of electromagnetic fields and waves, lasers and amplifiers, nanoscale photonic circuits, and electromagnetic compatibility He is author or co-author of more than 600 journal and conference papers NON-STAT IONA RY ELECTROM AGNETICS “This is the first comprehensive book on this topic Scientists working on the electromagnetic field theory in general, too, will find a lot of interesting material here.” NON-STAT IONA RY ELECTROM AGNETICS Alexander Nerukh Nataliya Sakhnenko Trevor Benson Phillip Sewell This page intentionally left blank CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20120829 International Standard Book Number-13: 978-9-81436-424-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com July 30, 2012 11:10 PSP Book - 9in x 6in To the memory of Prof Nikolay Khizhnyak, founder of the approach, and to my wife, Elena Nerukh —A.N to my daughter, Alona Sakhnenko —N.S 00-Alexander–prelims This page intentionally left blank July 30, 2012 11:10 PSP Book - 9in x 6in 00-Alexander–prelims Contents Preface Acknowledgements Introduction xvii xix I Basic Electromagnetic Effects in a Medium with Time-Varying Parameters and/or Moving Boundary Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium 1.1 Generalised Wave Equation for an Electromagnetic Field in a Time-Varying Medium with a Transparent Object 1.1.1 Generalised Derivatives 1.1.2 Initial and Boundary Conditions for Electromagnetic Fields in a Time-Varying Medium 1.1.3 Maxwell’s Equations in Generalised Derivative Representation 1.1.4 Generalised Wave Equation for the Case of a Non-Dispersive Background 1.1.5 Generalised Wave Equation for the Case of a Dispersive Background 1.2 Fundamental Solutions (Green’s Functions) to Maxwell’s Equations 1.2.1 The Non-Dispersive Background 1.2.2 The Dispersive Background 1.2.3 A Rectangular Waveguide with Perfectly Conducting Walls 10 10 12 14 16 17 20 20 22 23 July 30, 2012 11:10 PSP Book - 9in x 6in 00-Alexander–prelims viii Contents 1.2.4 Axial Symmetric Green’s Function for a Planar Waveguide with Perfect Conducting Walls 1.3 Causal Time-Spatial Interpretation of Electromagnetic Field Interaction with Time-Varying Objects 1.3.1 The Volterra Integral Equation for the Electro-Magnetic Field in a Non-Dispersive Background 1.3.2 Influence of a Dispersive Background on the Integral Equation Form 1.3.3 Spatial-Temporal Interpretation of the Volterra Integral Equation 1.3.4 Three Stages of Development of Electromagnetic Transients in a Bounded Medium with Time-Varying Parameters 1.3.5 The Field Outside the Object 1.3.6 Three Stages of Solution of a Non-Stationary Problem 1.4 The Resolvent Method for Solving the Integral Equation 1.4.1 Impulse Representation of Operators 1.4.2 Kernels of the Integral Equations for Typical Media 1.4.3 The Resolvent Method Transformation of an Electromagnetic Field in an Unbounded Medium with Time-Varying Parameters 2.1 Transformation of a Plane Electromagnetic Wave in a Non-Dispersive Medium 2.1.1 Splitting of a Plane Harmonic Wave into Two New Ones with a Shifted Frequency by a Time Jump in Medium Parameters 2.1.2 Transformation of Radiation of an Extrinsic Source 2.1.3 Evolution of a Harmonic Wave in a Medium Modulated by Repetitive Identical Pulses 2.1.4 “Intermittency” in Electromagnetic Wave Transients in a Time-Varying Linear Medium 25 27 27 32 34 35 38 39 40 40 43 49 61 62 64 69 78 84 July 30, 2012 11:10 PSP Book - 9in x 6in 00-Alexander–prelims Contents 2.2 Change of Electromagnetic Pulse Complexity in a Time-Varying Medium 2.2.1 Complexity of the Signals 2.2.2 Propagation of Electromagnetic Pulses in a Medium Modulation by Repetitive Identical Pulses 2.2.3 Propagation of Electromagnetic Pulses in a Medium with Various Time Modulations 2.2.3.1 Pulses of “soft” transformation 2.2.3.2 Pulses of “hard” transformation 2.2.4 Wave Chaotic Behaviour Generated by Linear Systems 2.3 Constitutive Equations for Electromagnetic Transients in Time-Varying Plasma 2.3.1 Phenomenological Constitutive Relations 2.3.2 Kinetic Description of Plasma 2.3.3 Gyrotropic Plasma 2.3.4 Moving Plasma 2.4 Isotropic Plasma with Changing Density 2.4.1 Step-wise Change of Plasma 2.4.2 Continuously Changing Plasma 2.5 Plane Wave in Gyrotropic Plasma with “Switching On” Magnetising Field 2.5.1 Basic Equations 2.5.2 The Resolvent for the Integral Equation 2.5.3 The Case of the Arbitrary Time-Varying Magnetic Field Approximation 2.5.4 The Transformation of a Plane Wave 2.5.5 The Transformation of the Plasma Oscillations Influence of Medium Plane Boundaries on Electromagnetic Transients 3.1 A Resolvent for an Initial Boundary Value 1D Problem in a Dielectric 3.2 Electromagnetic Field in a Half-Restricted Time-Varying Medium 3.2.1 Transformation of a Plane Wave 92 92 94 101 103 104 106 111 112 114 116 119 123 124 129 134 135 137 138 140 144 151 153 157 157 ix July 6, 2012 10:33 PSP Book - 9in x 6in 09-Alexander-App References References Vladimirov, V S (1967) Equations of Mathematical Physics (in Russian), Nauka, Moscow, 437 p Bateman, H., and Erdely, A (1954) Tables of Integral Transforms, McGraw-Hill Book Comp., Inc., New York, Toronto, London, 344 p Prudnikov, A P., Brychkov, Y A., and Marichev, O I (1986) Integrals and Series, vol 2: Special Functions, Gordon and Breach Science Publishers, New York, 752 p Morgenthaler, F R (1958) Velocity modulation of electromagnetic waves, IRE Trans Microw Theory Tech., MTT-6, 167–172 Fante, R L (1971) Transmission of electromagnetic waves into timevarying media, IEEE Trans Antenn Propag., AP-19(3), 417–424 Krasilnikov, V N., and Lutchenko, L N (1972) Principle of apparent position of interface and generalization of V.A Fock’s reflected formulae onto the case of moving boundaries (in Russian), Probl Diffraction Wave Propag., 12, 150–157, Leningrad Pao, H.-Y., Dvorak, S L., and Dudley, D G (1996) IEEE Trans Antenn Propag., 44(7), 925–932 Dvorak, S L., and Dudley, D G (1995) IEEE Trans Electromagn Compatibility, 37(2), 192–200 Dvorak, S L., and Kuester, E F (1990) J Comput Phys., 87(2), 301–327 583 This page intentionally left blank July 11, 2012 11:57 PSP Book - 9in x 6in Colour Insert Figure 2.20 Alexander-colour-Insert July 11, 2012 11:57 C2 PSP Book - 9in x 6in Colour Insert Figure 3.47 Figure 4.14 Alexander-colour-Insert July 11, 2012 11:57 PSP Book - 9in x 6in Alexander-colour-Insert Colour Insert Figure 4.15 Figure 4.16 C3 July 11, 2012 11:57 C4 PSP Book - 9in x 6in Colour Insert Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Alexander-colour-Insert July 11, 2012 11:57 PSP Book - 9in x 6in Alexander-colour-Insert Colour Insert Figure 6.14 Figure 6.15 C5 July 11, 2012 11:57 C6 PSP Book - 9in x 6in Colour Insert Figure 8.2 Figure 8.4 Figure 8.5 Alexander-colour-Insert July 11, 2012 11:57 PSP Book - 9in x 6in Alexander-colour-Insert Colour Insert Figure 8.6 Figure 8.8 C7 July 11, 2012 11:57 C8 PSP Book - 9in x 6in Colour Insert Figure 8.9 Figure 8.12 Alexander-colour-Insert July 11, 2012 11:57 PSP Book - 9in x 6in Alexander-colour-Insert Colour Insert Figure 8.15 Figure 8.17 C9 July 11, 2012 11:57 C10 PSP Book - 9in x 6in Colour Insert Figure 8.18 Figure 8.20 Alexander-colour-Insert July 11, 2012 11:57 PSP Book - 9in x 6in Alexander-colour-Insert Colour Insert Normalized Frequency Shift Figure 8.23 Normalized cavity-to-cavity separation Figure 8.24 C11 July 11, 2012 11:57 C12 PSP Book - 9in x 6in Colour Insert Figure 8.28 Alexander-colour-Insert Prof Reinhold Pregla University of Hagen, Germany “This rigorous and, at the same time, easy-to-understand explanation of non-stationary electromagnetic phenomena will be of great interest to researchers from the physical science community.” Prof Elena Romanova Saratov State University, Russia “This magnificent work guides readers through the mysterious world of non-stationary electromagnetics Its very first sentence catches them and sets free their imagination to expect and see the newly discovered sides of our nature.” Dr Mariana Nikolova Georgieva-Grosse Polikraishte, Bulgaria Prof Georgi Nikolov Georgiev St Cyril and St Methodius University of Veliko Tarnovo, Bulgaria This book is devoted to investigations of non-stationary electromagnetic processes It offers a good opportunity to introduce the Volterra integral equation method more widely to the electromagnetic community The explicit mathematical theory is combined with examples of its application in electromagnetic devices, optoelectronics, and photonics, where time-domain methods become a powerful tool for modelling Many of the electromagnetic phenomena that are studied in the book may lead to numerous new ideas for experimentalists and engineers developing new classes of photonic devices Alexander Nerukh is head of the Department of Higher Mathematics, Kharkov National University of Radioelectronics, Ukraine He has published books and over 250 scientific papers Prof Nerukh’s scientific interests lie in nonstationary and nonlinear electrodynamics, and he has collaborated with the University of Nottingham and Aston University in these fields Nataliya Sakhnenko is associate professor at the Department of Higher Mathematics, Kharkov National University of Radioelectronics She has held joint research with the University of Nottingham and the University of Jena Her current research interests are in time-domain problems of photonics, plasmonics, and metamaterials Phillip Sewell is professor of electromagnetics in the Faculty of Engineering, University of Nottingham His research interests involve analytical and numerical modelling of electromagnetic problems, with application to optoelectronics, electromagnetic compatibility, and electrical machines He has published  approximately 500 papers V250 ISBN-13 978-981-4316-44-6 Nerukh | Sakhnenko Benson | Sewell Trevor Benson is director of the George Green Institute for Electromagnetics Research, University of Nottingham His research interests include experimental and numerical studies of electromagnetic fields and waves, lasers and amplifiers, nanoscale photonic circuits, and electromagnetic compatibility He is author or co-author of more than 600 journal and conference papers NON-STAT IONA RY ELECTROM AGNETICS “This is the first comprehensive book on this topic Scientists working on the electromagnetic field theory in general, too, will find a lot of interesting material here.” NON-STAT IONA RY ELECTROM AGNETICS Alexander Nerukh Nataliya Sakhnenko Trevor Benson Phillip Sewell ... indefiniteness in the problem formulation because of the irreversibility of the non- stationary phenomenon Therefore, an Non- Stationary Electromagnetics Alexander Nerukh, Nataliya Sakhnenko, Trevor Benson,... indefiniteness in the problem formulation because of the irreversibility of non- stationary phenomenon Therefore, investigation of the non- stationary electromagnetic phenomena should be based on the equations,... moment of the beginning of non- stationary behaviour The non- stationary state, which starts at some definite moment of time, is accompanied by the appearance of a transient (non- harmonic) field These