Non stationary electromagnetics

Basic Electromagnetic Effects in a Medium with Time-Varying Parameters and/or Moving Boundary 1 Initial and Boundary Value Electromagnetic Problems 1.2.1 The Non-Dispersive Background 20

NON-STATIONARY ELECTROM AGNETICS

Alexander Nerukh Nataliya Sakhnenko Trevor Benson Phillip Sewell

“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.”

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 3 books

and over 250 scientific papers Prof Nerukh’s scientific interests lie in

non-stationary 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.

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

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

Dr Mariana Nikolova Georgieva-Grosse

Polikraishte, Bulgaria

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 able 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

Reason-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 organiza- tion 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

To the memory of Prof Nikolay Khizhnyak,

founder of the approach,

and

to my wife, Elena Nerukh to my daughter, Alona Sakhnenko

I Basic Electromagnetic Effects in a Medium

with Time-Varying Parameters and/or Moving Boundary

1 Initial and Boundary Value Electromagnetic Problems

1.2.1 The Non-Dispersive Background 20

1.2.3 A Rectangular Waveguide with Perfectly

1.2.4 Axial Symmetric Green’s Function for a PlanarWaveguide with Perfect Conducting Walls 251.3 Causal Time-Spatial Interpretation of ElectromagneticField Interaction with Time-Varying Objects 271.3.1 The Volterra Integral Equation for the

Electro-Magnetic Field in a Non-Dispersive

1.3.2 Influence of a Dispersive Background on the

1.3.3 Spatial-Temporal Interpretation of the Volterra

1.3.4 Three Stages of Development ofElectromagnetic Transients in a BoundedMedium with Time-Varying Parameters 351.3.5 The Field Outside the Object 381.3.6 Three Stages of Solution of a Non-Stationary

2 Transformation of an Electromagnetic Field in an

Unbounded Medium with Time-Varying

2.1.2 Transformation of Radiation of an Extrinsic

2.1.3 Evolution of a Harmonic Wave in a MediumModulated by Repetitive Identical Pulses 782.1.4 “Intermittency” in Electromagnetic Wave

Transients in a Time-Varying Linear Medium 84

Contents ix

2.2 Change of Electromagnetic Pulse Complexity in a

2.2.2 Propagation of Electromagnetic Pulses in aMedium Modulation by Repetitive Identical

2.2.3 Propagation of Electromagnetic Pulses in aMedium with Various Time Modulations 1012.2.3.1 Pulses of “soft” transformation 1032.2.3.2 Pulses of “hard” transformation 1042.2.4 Wave Chaotic Behaviour Generated by Linear

2.3 Constitutive Equations for ElectromagneticTransients in Time-Varying Plasma 1112.3.1 Phenomenological Constitutive Relations 1122.3.2 Kinetic Description of Plasma 114

2.4 Isotropic Plasma with Changing Density 1232.4.1 Step-wise Change of Plasma 1242.4.2 Continuously Changing Plasma 1292.5 Plane Wave in Gyrotropic Plasma with “Switching On”

3.2.2 Splitting of Video Pulse in a Half-Space with

3.3 Jump Changes of Plasma Density in a PlasmaHalf-Space with a Plane Boundary 1783.3.1 Plasma Density’s Jump Change in a Half-Space 1783.3.2 Two Steps Change of Plasma Density 1823.4 The Evolution of an Electromagnetic Field in the

Dielectric Layer After Its Creation 1923.4.1 The Equation for the Resolvent 1953.4.2 The Evolution of the Electromagnetic Field inthe Layer After Its Formation 1973.5 Electromagnetic Field in a Layer with Non-Linear and

3.6 Transformation of Electromagnetic Field by a Newly

3.7 The 3D Resolvent for a Problem with a PlaneBoundary of a Dielectric Half-Space 2253.7.1 The Resolvent for the Inner Problem 2283.7.2 The Resolvent for the External

3.8 Fresnel Formulae in Time Domain for a PlaneInterface Between Two Dielectrics 2403.8.1 The Time-Domain Representation of the Field

in the Case of Two Dielectric Half-Spaces 2403.8.2 Expansion of the First Part of the Field with

Respect to the Dissipation Rate 2453.8.3 Spatial-Time Representation of the Fresnel

Formula for a Transmitted Field 2463.8.4 The Polarisation Relations for the Scattered

Contents xi

3.9 Inclined Incidence of a Plane Wave on a PlaneBoundary of the Time-Varying Medium 2523.9.1 The Field Caused by the Permittivity Time

3.12.4 Integral Equations for an Object Located Nearthe Boundary of the Non-Stationary Medium 292

4 Non-Stationary Behaviour of Electromagnetic Waves

4.1 Transformation of an Electromagnetic Wave by aUniformly Moving Boundary of a Medium 3014.1.1 Discrepancy of Secondary Waves and

4.1.2 Resolution of Moving Boundary “Paradoxes” 3054.2 Evolution of an Electromagnetic Wave After

Beginning of Medium Boundary Movement 3114.3 Relativistic Uniform Accelerated Movement of a

4.4 Electromagnetic Field Energy Accumulation in a

4.4.1 Increase of the Wave Amplitudes in the

4.4.2 The Energy Accumulation in the Layer 328

4.4.3 Generation of Electromagnetic Pulses by the

4.5 Scattering of Waves by an Ellipsoid with a

II Electromagnetic Transients in Time-Varying

Waveguides and Resonators

5 An Electromagnetic Field in a Metallic Waveguide with

5.1 Expansion of an Electromagnetic Field by theNon-Stationary Eigen-Functions of a Waveguide 3485.2 Equations for a Field in the Waveguide with a

Plasma Boundary After Its Start in a Waveguide 373

6 Interaction of an Electromagnetic Wave with a Plasma

6.1 Main Relations for Electromagnetic Waves in aWaveguide with a Relativistic Moving Plasma Bunch 3886.2 Characteristic Matrix for Waves in a Waveguide with a

Contents xiii

6.6.1 Integral Operators for an Initial-BoundaryValue Problem with Axial Symmetry 4226.6.2 Excitation of the Field in a Planar Waveguide

Filled by Time-Varying Plasma 4246.6.3 Circular Cylinder with Time-Varying Medium

7 Non-Stationary Electromagnetic Processes in

7.1 Wave Equations for Longitudinal and TransverseComponents in Generalised Functions 4407.2 Volterra Integral Equations for Non-Stationary

Electromagnetic Processes in Time-Varying Dielectric

7.2.1 Integral Equations for the Fields 4417.2.2 Harmonic Waves in a Waveguide 4447.3 Solution for the Problem with a Time Jump Change in

7.4 Harmonic Wave Transformation Caused by aPermittivity Change in the Waveguide Core 4517.4.1 The Early Stage of the Transient 4527.4.2 Waves Spectra Generated by a Permittivity

7.6.3 Flat Dielectric Resonator 4857.6.4 Field Evolution in a Dielectric Waveguide 487

8 Electromagnetic Transients in Microcavities with

8.1 Mathematical Tools for Solution of theInitial-Boundary Value Problem in Dielectric

8.2 Excitation of a Dielectric Resonator by External

8.3 Whispering Gallery Mode Transformation in a

8.4 Field Transformation by the Permittivity Time-Jump

8.5 Transient Plasma in a Circular Resonator 5158.6 Stratified Cylindrical Dielectric Structure 5198.7 Whispering Gallery Modes in a Circular Dielectric

Resonator with a Transient Inclusion 5228.8 Optical Coupling of Two Transient Circular Dielectric

8.9 Frequency Change of Partial Spherical Waves Induced

by Time Change of Medium Permittivity 537

8.9.2 Analysis of the Inner Field 5438.9.3 Analysis of the Exterior Field 5458.10 Evolution of Waves After Plasma Ignition in a

Appendix A: Transformation of an Arbitrary Signal 557

Appendix B: Taking into Account Solutions of a Homogeneous

Equation in the Intermediate Evolution Stage 561

Appendix C: Lipshitz–Hankel Functions 569

Contents xv

Appendix D: The Resolvent with Cylindrical Symmetry 573

D.1 Unbounded Medium 573

D.2 The Medium with a Cylindrical Boundary 575

Appendix E: WGM Resonator with Transient Circular Inclusion 577

This book is devoted to investigations of non-stationary

elec-tromagnetic processes It contains results concerning the

non-stationary electromagnetic processes initiated by time variations of

material objects The main idea of the book can be characterized

by the phrase “Any change makes a path for other changes” from

Niccolo dei Machiavelli (1469–1527) This book offers a good

opportunity to introduce the Volterra integral equation method

for investigations of electromagnetic phenomena more widely A

systematic presentation of this method in the time domain provides

new theoretical results, and the explicit mathematical theory is

combined with examples of its application in electromagnetic

devices in microwaves, optoelectronics, and photonics, where

time-domain methods become a powerful tool for modelling Particular

consideration is given to electromagnetic transients in time-varying

media and their potential applications The approach is formulated

and electromagnetic phenomena are investigated in detail for a

hollow metal waveguide, which contains a moving dielectric or

plasma-bounded medium, dielectric waveguides with time-varying

medium inside the core, cylindrical homogeneous resonators with

time-varying medium as well as with time-varying insertions in

them, and a system of non-stationary resonators Considering

the influences of medium changes on electromagnetic fields in

optoelectronic devices is very important for the realistic description

of such devices Many electromagnetic phenomena studied in the

book may lead to numerous innovative ideas for experimentalists

and engineers developing new classes of photonic devices

This book systematises and collects almost all results obtained bythe authors since the 1970s Some of these results were published in

Russian, and some were not published at all but may be interesting

for wider electromagnetic community It is a pleasure to express our

sincere gratitude to the people who contributed to obtaining the

results during all these years, especially Peter E Minko, Oleg N Rybin,

Irina Yu Shavorykina, and Fedor V Fedotov

Alexander Nerukh Nataliya Sakhnenko

Kharkov, Ukraine

Trevor Benson Phillip Sewell

Nottingham, UK

2012

This book owes much to collaboration with researchers in the field

It is our pleasure to express our gratitude to Prof Oleg Tretiyakov,

Dr Dmitry Nerukh, Dr Peter Minko, Dr Irina Shavorikina,

Dr Konstantin Yemelyanov, Dr Oleg Rybin, Dr Fedor Fedotov,

Dr Elena Semenova, Dr Elena Smotrova, Nataliya Ruzhitskaya, Prof

Vyacheslav Buts, Prof Marian Marciniak, Dr A Al-Jarro, and Dr Ana

Vukovic

Any change in the state of a medium, for example, a change of

its material properties or a movement of its boundaries, affects

the characteristics of an electromagnetic field existing in this

medium This influence is very strong, even in the simplest

non-dispersive electromagnetic structures As there are two temporal

processes in this case, medium change and field change, the points

of their origin acquire principal importance, and the corresponding

mathematical problems become initial boundary value ones It is

evident that a dispersive structure adds new special features to the

change of the electromagnetic field state and can greatly influence

transient electromagnetic processes In practice, waveguides and

resonators, where the electromagnetic field interacts with matter

in bound areas of space constrained by waveguide or resonator

walls, are very important dispersive structures with the presence

of the walls bringing a dispersive character to electromagnetic

wave propagation in the region considered The field interaction

with a non-stationary medium acquires new features under these

conditions In addition, because of the difference between the phase

and the group velocities of the waves conditioned by the dispersion,

the importance of taking into account some initial time of the

interaction process arises This importance increases in the case

where a medium or its borders moves, when the relationship

between all three velocities, the phase and the group velocities of

the waves and the motion velocity, begin to play a significant role

Investigations of transients in waveguides have a long history, but

they concern the degradation of pulses in stationary waveguides

and, principally, metallic waveguides

Maxwell’s equations are self-consistent only for electromagneticfields in a vacuum In a general medium the constitutive equations

and boundary conditions significantly complicate both the

for-mulation and the solution of electromagnetic problems Such

problems become even more complex when the media are not only

inhomogeneous but are also time-varying Such a situation can be

met when considering the propagation of electromagnetic signals in

dielectric or semiconductor waveguides, in particular in the context

of modulators, pulsed lasers and frequency conversion The proper

description and investigation of the physics of these phenomena are

motivated by their significant importance to optical communication

technology; the interactions between microwave and optical pulses

and active semiconductor media in waveguides have therefore

received considerable attention in recent years The solution of such

electromagnetic problems has demanded accurate time-domain

techniques, some variants of which have received widespread

attention in the literature, mainly owing to their computational

superiority for solving wide-band problems in comparison with

frequency-domain methods Unfortunately, most of these techniques

are focussed upon numerical calculations and are not suitable for

identifying the general features of the phenomena This is especially

true for the important case of understanding the behaviour of the

guided modes supported by dielectric optical waveguides, a central

task in the simulation of integrated optical components

In 1958, F.R Morgenthaler revealed that a temporal change inthe permittivity of an unbounded medium transforms a primary

harmonic plane wave to new secondary ones having different

frequencies but the same wave number as the primary wave This

general feature is also observed when a plane wave is normally

incident onto a plane interface between two media, the permittivity

of one of which changes abruptly However, in this case the spatial

structure of waves also becomes more complex Nevertheless, the

monochromatic character of the secondary waves is not disturbed

if the medium is non-dissipative The picture of such phenomena

becomes even more complex in the case of the oblique incidence

of an electromagnetic wave onto a plane boundary with a

time-varying medium In this case, not only does the structure of the

system of monochromatic waves become more complex, but a

continuous wave spectrum also appears All the circumstances just

discussed arise in a dielectric waveguide with time-varying media A

Introduction 3

time-domain integral equation technique is presented in this paper

to take into account, in one formulation, a complex combination of

boundary and initial conditions as well as permitting the medium

parameters to change in time Investigations are made by using

the evolution approach developed in this book This approach is

also applied to the investigation of the interaction of a guided

wave with a medium moving in a rectangular waveguide with

perfectly conducting walls The relativistic movement of a

non-dispersive medium, as well as effects caused by a double-dispersion

mechanism (i.e., waveguide and plasma dispersions) are considered

The need to consider the interaction of optical beams with varying media is becoming ever more common Applications, such as

time-the production of terahertz sources are exploiting time-the phenomena

observed in such circumstances and moreover, as data rates

increase, designers of switched lasers and modulators and similar

devices must confront the consequences of these interactions There

is a significant literature considering the simple case of plane waves

interacting with time changes in the parameters of open and

semi-open regions However, to date, the practically important case of

time-variant materials in spatially limited and optically confining

waveguides has received far less attention The principal objective

of this work is to provide a formal, non-numerical, framework

within which to investigate this case and it shall be shown that

certain general conclusions regarding the nature of the optical field

in these circumstances can be demonstrated This is clearly an

important pre-cursor to the detailed numerical analysis of specific

configurations in the design of a wide variety of novel devices

The book is organised as follows The essential point forelaborating a common approach to the investigation of transient

electromagnetic phenomena is the evolutionary character of such

phenomena and the initial moment, when the non-stationary

behaviour starts, which takes an important meaning Introduction

of the initial moment for the non-stationary behaviour is dictated in

many cases by a necessity to separate the moment of “switching on”

the field and the moment of the non-stationary behaviour beginning

The non-stationary behaviour, which starts at some certain moment

of time, is accompanied by the appearance of a transient

(non-harmonic) field, so-called “transients.” These transients can form a

significant part of the total field for a long time However, they fall

from the field of vision of a stationary approach when all periodic

processes are assumed to start at the infinite past It should be noted

that a commonly used approximation of the adiabatic “switching on”

of a process at the infinite past can easily lead to indefiniteness in the

problem formulation because of the irreversibility of non-stationary

phenomenon Therefore, investigation of the non-stationary

elec-tromagnetic phenomena should be based on the equations, which

include general representation of the medium parameters, where

an inhomogeneity has a shape and medium properties inside it

that are time-dependent The mathematical technique relating the

theory of transient electromagnetic phenomena should contain a

description of both continuous and abrupt changes of both the

field functions and the medium parameters This technique has also

to take into account the correlation between spatial and temporal

changes in the media Such a correlation occurs, for example, when

a medium boundary moves in space In this case a sharp time-jump

of the medium parameters occurs at every fixed point passed by the

medium boundary

The theory of generalised functions is an adequate mathematicaltechnique for treating such problems The generalised functions

describe uniformly continuous and discontinuous functions of the

field and media parameters Applying this theory to the classical

electromagnetic equations means a substitution of the generalised

derivatives instead of the conventional (classical) derivatives with

corresponding modification of Maxwell’s equations The

mathe-matical formulation of a non-stationary electromagnetic problem

into a differential equation in the space of generalised functions

and then conversion of a differential equation into an integral

one is given in Chapter 1 This allows all conditions for the

fields on the discontinuity surfaces (boundaries) to be included

directly into the equations, as well as the moments at which the

time-varying parameters change The causal time-spatial evolution

of an electromagnetic field and a technique developed for the

consideration of such problems are presented

The main phenomena caused by a time-change of an unboundedmedium are considered in Chapter 2 It is shown that modulation

of the medium by a finite chain of medium permittivity time

Introduction 5

disturbances can lead to the appearance of chaotic behaviour

in some field characteristics This is estimated by calculation of

statistical complexity, the Hurst’s index and the Lyapunov exponent

A dispersive medium is represented by a plasma with an abrupt

change of density, and by a magnetised plasma whose magnetisation

is switched on at some moment of time A wide variety influences

that a medium boundary can have on electromagnetic transients

is considered in Chapter 3 The normal incidence of a plane

electromagnetic wave onto a plane boundary of a dielectric or

plasma, created at some moment of time, is considered The

resolvent operators are derived for the 3D case of a medium plane

boundary and this allows Fresnel’s formulas to be obtained in the

time domain By virtue of these operators a new effect of secondary

wave focusing by a non-stationary medium plane boundary is

investigated

The interaction of electromagnetic waves with a medium withmoving boundary is investigated in Chapter 4 Moving boundary

“paradoxes” that occur when the number of supposed waves does

not correspond to the number of boundary conditions are resolved

Peculiarities of the wave interaction with a moving boundary whose

movement begins at zero moment of time are also investigated in

this chapter A sharp origin of uniform movement is considered, as

well as continuous relativistic uniform accelerated movement when

a velocity changes from zero to a relativistic value Investigation of a

collapsing dielectric layer reveals energy accumulation in the layer

and the generation of electromagnetic pulses

The influence of a moving medium on a guiding wave in arectangular waveguide with perfectly conducting walls, as well as

the wave evolution, is considered in Chapter 5 The transformation

of the guiding wave in the waveguide filled by a uniform dielectric

relativistic moving along the waveguide is investigated Using this

problem the possibility to model various phenomena concerned

with the interaction of electromagnetic waves with a boundary

of a relativistic moving medium in the presence of waveguide

dispersion is shown The interaction of the electromagnetic field

with a plasma “flashed” at zero moment of time in the waveguide

and then uniformly expanding is considered It is shown in Chapter

6 that interplay between phase and group velocities can enhance the

interaction of an electromagnetic wave with a plasma bunch moving

in a metallic waveguide and make this interaction very effective for

small values of the bunch

All the effects considered in the previous chapters arise indielectric or plasma waveguides and resonators with time-varying

media that are considered in Chapters 7 and 8 The dielectric

waveguide with a non-linear material is considered, along with

the excitation of whispering gallery modes (WGMs) by an external

transient source Temporal changes of the permittivity inside the

resonator have been investigated as well For an initial wave

without an external source (e.g., a WGM) the change of the material

permittivity leads to a resonant frequency shift that does not depend

on the initial light intensity and is proportional to the fractional

change of the refractive index for material switching in the whole

cavity For the case of refractive index switching in a circular coaxial

region, or in a ring region near the rim, we observe a dependence of

the frequency shift on the degree of overlap between the initial field

and the transient region Breaking the symmetry of the structure via

a transient circular inclusion leads to a rotation of the field pattern

Features that distinguish the time-spatial behaviour of the field in

coupled resonators from those found in a single one are investigated

Part I

Basic Electromagnetic Effects in

a Medium with Time-Varying Parameters and/or Moving

Boundary

7

Chapter 1

Initial and Boundary Value

Electromagnetic Problems in a

Time-Varying Medium

An essential point for elaborating a common approach to the

inves-tigation of transient electromagnetic phenomena is the evolutionary

character of such phenomena, and an initial moment, when the

non-stationary condition starts, takes an important meaning The

introduction of the non-stationary initial moment is dictated in

many cases by a necessity to separate the moment of “switching

on” the field and the 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 so-called transients can exist for a

long time, being a significant part of the total field However, they

fall out of the field of vision of a stationary approach when all

periodic processes are assumed to start at the infinite past It

should be noted that the commonly used approximation of an

adiabatic “switching on” of a process at the infinite past can easily

lead to 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, and Phillip Sewell

Copyright c 2013 Pan Stanford Publishing Pte Ltd.

ISBN 978-981-4316-44-6 (Hardcover), 978-981-4364-24-9 (eBook)

www.panstanford.com

investigation of non-stationary electromagnetic phenomena should

be based on equations which include a general representation

of the medium parameters, where an inhomogeneity has a

time-dependent shape and time-time-dependent medium properties inside it

A mathematical approach to the theory of transient electromagnetic

phenomena should contain a description of both continuous and

abrupt changes of both the field functions and the medium

parameters This technique also has to take into account the

correlation between spatial and temporal changes in the media

Such a correlation occurs, for example, when a medium boundary

moves in space In this case a sharp time jump of the medium

parameters occurs at every fixed point passed by the medium

boundary

The theory of generalised functions [1–6] is an adequate matical technique for treating such problems The generalised func-

mathe-tions describe uniformly continuous and discontinuous funcmathe-tions of

the field and media parameters Applying this theory to the classical

electromagnetic equations means a substitution of the generalised

derivatives instead of the conventional (classical) derivatives with a

corresponding modification of Maxwell’s equation

In this chapter a non-stationary electromagnetic problem ismathematically formulated as a differential equation in a gener-

alised function space This allows all conditions for the fields on the

discontinuity surfaces (boundaries) as well as parameter time jumps

to be included directly into the equations

1.1 Generalised Wave Equation for an Electromagnetic

Field in a Time-Varying Medium with a Transparent Object

1.1.1 Generalised Derivatives

To use the space of generalised functions one must consider the

generalised derivatives [1–6], instead of the classic one Assume

a vector-function a(t , r) has a discontinuity on an arbitrary

time-varying surface, S(t), and that its jump value is equal to [a] S =

a − a−, where a+ is the magnitude of the vector function on the

positive side of this surface This side is determined by a normal

vector n, as shown in Fig 1.1:

Generalised Wave Equation for an Electromagnetic Field 11

Figure 1.1. The orientation of the normal vector on the discontinuity

a surface delta-function, and u nis a velocity component normal to a

certain surface domain Here and later, bold characters are used for

where [a]t=0= a(t = +0) − a(t = −0).

Transition to the generalised derivatives allows the conditionsfor the fields on the surfaces and at the time points to be included

directly into Maxwell’s equations, where the medium parameters

are discontinuous These conditions are given by the terms in

the square brackets in Eqs 1.1.1 and 1.1.2 Compared with the

continuous medium case, the equation form remains unchanged

almost everywhere To show this, let us consider the classical

Maxwell’s equations in a continuous medium These equations have

the following form in an SI system:

where E is the electric field strength, B is the magnetic flux density,

P and M are vectors of medium electric and magnetic polarisations,

j is a conductivity current, ε0 = 10−9

36π [F·m−1] and μ0 =

4π · 10−7 [H·m−1] are the permittivity and permeability of free

space, respectively, and √ε10μ0 = c = 3 · 108[m/s] is the velocity

of light in vacuum The polarisations, P and M, and the electric field

flux density D and the magnetic field strength H are connected in a

Here, 1+κ  ε= ε,  κ εis an operator of an electrical susceptibility, ε is

an operator of a relative permittivity of the medium, and analogously

1+κ  μ =μ,   κ μis an operator of a magnetic susceptibility andμ is 

an operator of a relative permeability of the medium Asκ  εandκ  μ

are assumed to be operators, the relations (Eq 1.1.4) are general

ones and they describe all possible media, including dispersive and

anisotropic ones

1.1.2 Initial and Boundary Conditions for Electromagnetic

Fields in a Time-Varying Medium

It is convenient to determine the conditions for the field on a

discontinuity surface on the basis of Maxwell’s equations in integral

form [7, 8, 9] They have an invariant form independent of the way

the medium parameters change:

Generalised Wave Equation for an Electromagnetic Field 13

where j and q are bulk densities of current and charge in a laboratory

frame of reference It follows from Eq 1.1.5 that the boundary

conditions have the following form for an arbitrarily moving surface

[9, 10]:

n × [H]s + u n[Dtan]s = γ−1(n × (i× n))

n × [E]s − u n[Btan]s = 0 [Dn]s = σ[B

n]s = 0, (1.1.6)

where Dtan, Btanare the components of the fields tangential to the

surface,γ−1 = 1− β2is the relativistic factor,β n = u n /c, and i

andσare the surface current and charge densities in the intrinsic

frame of reference of the moving surface domain, respectively

The electric field flux density, as well as the magnetic induction,remains continuous at time jumps of the medium features This

follows from the classical Maxwell’s equations (Eq 1.1.3) together

with the limiting value of the spatial derivatives of the field Indeed,

integrating the relation curl

Analogously, the equation curlE= −∂B

∂B

∂t t=0= − [curlE] t=0= 0.

The electric flux density derivative remains continuous only when

medium’s magnetic features are continuous Analogously, the

magnetic induction is continuous in the case when the medium

electric features are continuous If this is not the case, then the initial

condition for the magnetic induction derivative can be derived as

follows The equation curlE= −∂B

∂t in the case when the permittivity

changes in time, E(t)= D(t)

Returning to the differential form of Maxwell’s equations (Eq 1.1.3),

we see that the second equation in Eq 1.1.3 is not changed when the

classical derivatives are replaced by the generalised ones, while the

first equation gains an additional term determined by the surface

current i = v n σ+ γ−1(n × (i× n)) in the laboratory frame of

Generalised Wave Equation for an Electromagnetic Field 15

In these equations, all the derivatives are generalised; therefore,

they readily contain the boundary conditions for the fields on

the discontinuity surfaces [11] that distinguishes them from

Eq 1.1.3 The discontinuity surface S(t) restricts the region V (t)

and is moving in the general case Equation 1.1.11 forms the

basis for further description and investigation of electromagnetic

phenomena in time-varying inhomogeneous media

Merging the two equations results in a wave equation in ageneralised derivative representation This equation describes an

electromagnetic field in a medium whose parameters can vary

arbitrarily in time as well as in space (this variation includes the

arbitrarily moving surface as well) [12]:

inside V (t) and equal to zero outside this region, we can define

the generalised functions P, M and j that describe the medium

electromagnetic representation in the whole space:

P= χ(P1− Pex)+ Pex

M= χ(M1− Mex)+ Mex (1.1.13)

j= χj1+ jextr

where the values with the index “1” are defined inside the region

V (t) and those with the index “ex” are defined outside of it This

external region is further referred to as a “background medium”

jextr is a current describing extrinsic sources of the field (see

According to the main idea of the approach originated by Khizhnyak

[13], the left-hand side of this equation has the form as in the

background, and the right-hand side is distinct from zero inside

Figure 1.2. The medium description and the arrangement of an

inhomo-geneity and field sources in the general problem formulation

the region V (t) only This equation allows consideration of the

electromagnetic problem for an arbitrary inhomogeneity placed

in various backgrounds We will consider two such cases: a

non-dispersive background, and a plasma as an example of non-dispersive

one

1.1.4 Generalised Wave Equation for the Case of a

Non-Dispersive Background

First we consider the generalised wave equation in the

non-dispersive background that is described by the relative permittivity

and permeabilityε and μ, respectively In this case

Pex= ε0(ε − 1)E Mex= 1

μ0

(1− μ−1)B. (1.1.15)

If these operators commute with the operators ∂t ∂ and curl, then

we have for the background polarisations in Eq 1.1.14

Generalised Wave Equation for an Electromagnetic Field 17

Now the generalised wave equation takes the form

where all parameters of the region V (t) are collected in the

right-hand side of this equation Introducing the shortright-hand notation

wave-phase velocity in the background

1.1.5 Generalised Wave Equation for the Case of a

Dispersive Background

Equation 1.1.19 describes the electromagnetic field in the case of

the object placed in the uniform non-dispersive medium Let us now

consider the case of a dispersive background, the simplest and most

applicable of which is a cold isotropic plasma It is known that such

a plasma is described by the constitutive relations

where ω e = N e2/mε0 is a plasma frequency, N is a density

of electrons, and e and m are the electron charge and mass,

Inside the object, this equation has the right-hand side

defined in the whole considered space

completely, even though they contain only the electric field The

magnetic field B can be determined from the second equation in

Eq 1.1.3 according to which B= − t

is defined uniquely if the condition of zero-value limit is fulfilled for

the electromagnetic field when t→ −∞

Generalised Wave Equation for an Electromagnetic Field 19

The right-hand side of Eqs 1.1.19 and 1.1.25 depend on the

extrinsic current, jextr, and on the inhomogeneity presence, which

is described by the functionχ in Q1 If both these components are

absent, then the right-hand side is equal to zero, and the equation

describes a free-space electromagnetic field E0 in the background

without sources This field is a general solution to a homogeneous

equation in the non-dispersive medium

inho-restricted in the past by the moment−t∞, where t∞can be assumed

as some great value It means that field sources start operating after

the moment −t∞ This assumption allows considering the

right-hand side of Eqs 1.1.19 and 1.1.25 as finite functions, and to apply

the theory of generalised functions to this equation According to

this theory the formal solutions to the equations for time-varying

and moving medium (Eq 1.1.19 or 1.1.25) can be written as a

convolution of the fundamental solution to this equation with the

right-hand side of the latter

Initial conditions for electromagnetic fields in plasma have

another form Continuity of D+ − D− = 0 follows directly from

Maxwell’s equations as well as B+ − B− = 0 or H+ − H− = 0

if magnetic properties do not change The equation∂D ∂t = ε0∂E

where ¯j is a jump of current at zero moment of time It gives the

continuity for the electric field E+ − E− = 0 as well as for the

... cases: a

non- dispersive background, and a plasma as an example of non- dispersive

one

1.1.4 Generalised Wave Equation for the Case of a

Non- Dispersive...

Non- Dispersive Background

First we consider the generalised wave equation in the

non- dispersive background that is described by the relative permittivity

and permeabilityε... 1.1.19 describes the electromagnetic field in the case of

the object placed in the uniform non- dispersive medium Let us now

consider the case of a dispersive background, the simplest