Non stationary electromagnetics

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 3 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



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CRC Press
Taylor & Francis Group
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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)
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Visit the Taylor & Francis Web site at

and the CRC Press Web site at



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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.

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Contents

Preface
Acknowledgements
Introduction

xvii
xix
1

I. Basic Electromagnetic Effects in a Medium
with Time-Varying Parameters
and/or Moving Boundary
1 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

9

10
10

12
14
16
17
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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
2 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
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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
3 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

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x Contents

3.3

3.4

3.5

3.6
3.7

3.8

3.2.2 Splitting of Video Pulse in a Half-Space with
Time-Varying Conductivity
Jump Changes of Plasma Density in a Plasma
Half-Space with a Plane Boundary
3.3.1 Plasma Density’s Jump Change in a Half-Space
3.3.2 Two Steps Change of Plasma Density
The Evolution of an Electromagnetic Field in the
Dielectric Layer After Its Creation
3.4.1 The Equation for the Resolvent
3.4.2 The Evolution of the Electromagnetic Field in
the Layer After Its Formation
Electromagnetic Field in a Layer with Non-Linear and
Time-Varying Medium
3.5.1 Integral Equations to the Problem
3.5.2 Algorithm for Calculation of an Integral
Solution

3.5.3 Numerical Results
3.5.4 Comparison of the FDTD and Volterra Integral
Equations in Time-Domain Approaches
3.5.5 Complexity of Electromagnetic Pulse Passing a
Layer of Non-Linear Medium
Transformation of Electromagnetic Field by a Newly
Created Plasma Layer
The 3D Resolvent for a Problem with a Plane
Boundary of a Dielectric Half-Space
3.7.1 The Resolvent for the Inner Problem
3.7.2 The Resolvent for the External
Problem
Fresnel Formulae in Time Domain for a Plane
Interface Between Two Dielectrics
3.8.1 The Time-Domain Representation of the Field
in the Case of Two Dielectric Half-Spaces
3.8.2 Expansion of the First Part of the Field with
Respect to the Dissipation Rate
3.8.3 Spatial-Time Representation of the Fresnel
Formula for a Transmitted Field
3.8.4 The Polarisation Relations for the Scattered
Field

174
178
178
182
192
195
197

199
199
203
208
213
217
221
225
228
237
240
240
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Contents

3.9 Inclined Incidence of a Plane Wave on a Plane
Boundary of the Time-Varying Medium
3.9.1 The Field Caused by the Permittivity Time
Jump
3.9.2 The Field Caused by the Boundary Presence

Only
3.9.3 The Evolution of the Refracted Field
3.9.4 The Field Outside the Non-Stationary Medium
3.10 Refocusing of the Point Source Radiation by the Plane
Boundary of the Time-Varying Dielectric
3.11 Formation of Point Source Image by Time Change of
Plasma
3.12 The Electromagnetic Field in a “Double” Time-Varying
Inhomogeneity
3.12.1 The Generalised Wave Equation for a Problem
with a “Double” Inhomogeneity
3.12.2 Green’s Function for a Complex Medium
3.12.3 Green’s Function for the Problem with an
Emerging Plane Boundary
3.12.4 Integral Equations for an Object Located Near
the Boundary of the Non-Stationary Medium
4 Non-Stationary Behaviour of Electromagnetic Waves
Caused by the Movement of a Medium Boundary
4.1 Transformation of an Electromagnetic Wave by a
Uniformly Moving Boundary of a Medium
4.1.1 Discrepancy of Secondary Waves and
Boundary Condition Numbers
4.1.2 Resolution of Moving Boundary “Paradoxes”
4.2 Evolution of an Electromagnetic Wave After
Beginning of Medium Boundary Movement
4.3 Relativistic Uniform Accelerated Movement of a
Medium Boundary
4.4 Electromagnetic Field Energy Accumulation in a
Collapsing Dielectric Layer
4.4.1 Increase of the Wave Amplitudes in the

Collapsing Layer
4.4.2 The Energy Accumulation in the Layer

252
254
258
260
268
269
274
282
282
285
287
292

301
301
303
305
311
315
325
325
328

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Contents

4.4.3 Generation of Electromagnetic Pulses by the
Collapsing Layer
4.5 Scattering of Waves by an Ellipsoid with a
Time-Varying Surface

331
334

II. Electromagnetic Transients in Time-Varying
Waveguides and Resonators
5 An Electromagnetic Field in a Metallic Waveguide with
a Moving Medium
5.1 Expansion of an Electromagnetic Field by the
Non-Stationary Eigen-Functions of a Waveguide
5.2 Equations for a Field in the Waveguide with a
Non-Stationary Insertion
5.3 Vibration of a Boundary of a Plane Dielectric
Resonator
5.4 Uniform Movement of a Dielectric Layer in the
Presence of Waveguide Dispersion
5.5 Penetration of an Electromagnetic Wave Through

Plasma Boundary After Its Start in a Waveguide
6 Interaction of an Electromagnetic Wave with a Plasma
Bunch Moving in a Metallic Waveguide
6.1 Main Relations for Electromagnetic Waves in a
Waveguide with a Relativistic Moving Plasma Bunch
6.2 Characteristic Matrix for Waves in a Waveguide with a
Plasma Layer
6.3 Frequency Multiplication and Amplitude
Amplification
6.4 Enhanced Reflectivity from the Moving Plasma Bunch
6.5 Resonance Effects in a Stratified Plasma Cluster
Moving in a Waveguide
6.5.1 The Characteristic Matrix for Stratified Plasma
Cluster
6.5.2 Resonance Effects
6.6 Axial Symmetric Electromagnetic Fields in a Planar
Metallic Waveguide

347
348
354
359
364
373

387
388
397
403
407

412
413
415
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Contents

6.6.1 Integral Operators for an Initial-Boundary
Value Problem with Axial Symmetry
6.6.2 Excitation of the Field in a Planar Waveguide
Filled by Time-Varying Plasma
6.6.3 Circular Cylinder with Time-Varying Medium
in Plate-Parallel Waveguide
7 Non-Stationary Electromagnetic Processes in
Time-Varying Dielectric Waveguides
7.1 Wave Equations for Longitudinal and Transverse
Components in Generalised Functions
7.2 Volterra Integral Equations for Non-Stationary
Electromagnetic Processes in Time-Varying Dielectric
Waveguides
7.2.1 Integral Equations for the Fields
7.2.2 Harmonic Waves in a Waveguide
7.3 Solution for the Problem with a Time Jump Change in

the Waveguide Core Permittivity
7.4 Harmonic Wave Transformation Caused by a
Permittivity Change in the Waveguide Core
7.4.1 The Early Stage of the Transient
7.4.2 Waves Spectra Generated by a Permittivity
Time Jump
7.5 Transformation of a Wave in a Nonlinear Dielectric
Waveguide
7.5.1 Step-Like Description of Field Evolution
7.5.2 The Step-Resolvent Method for the Waveguide
7.5.3 Calculation Scheme for Time-Step
Approximation
7.5.4 Evolution of the Electromagnetic Wave
After Switching off Non-Linearity in the
Waveguide
7.6 Two Ways for Calculation of Field Evolution in
Dielectric Waveguide: Via Brillouin- or Eigen-Waves
7.6.1 Elastic Oscillations
7.6.2 Differential Formulation of Initial and
Boundary Value Electromagnetic Problem in a
Dielectric Waveguide

422
424
426

435
440

441

441
444
445
451
452
453
460
460
462
470

473
477
478

481

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Contents

7.6.3 Flat Dielectric Resonator
7.6.4 Field Evolution in a Dielectric Waveguide
8 Electromagnetic Transients in Microcavities with
Time-Varying Material Properties
8.1 Mathematical Tools for Solution of the
Initial-Boundary Value Problem in Dielectric
Cylindrical Resonators
8.1.1 The Integral Approach
8.1.2 The Differential Approach
8.2 Excitation of a Dielectric Resonator by External
Transient Source
8.3 Whispering Gallery Mode Transformation in a
Transient Dielectric Resonator
8.4 Field Transformation by the Permittivity Time-Jump
in a Dielectric Resonator
8.5 Transient Plasma in a Circular Resonator
8.6 Stratified Cylindrical Dielectric Structure
8.7 Whispering Gallery Modes in a Circular Dielectric
Resonator with a Transient Inclusion
8.8 Optical Coupling of Two Transient Circular Dielectric
Resonators
8.9 Frequency Change of Partial Spherical Waves Induced
by Time Change of Medium Permittivity
8.9.1 Field Representation
8.9.2 Analysis of the Inner Field
8.9.3 Analysis of the Exterior Field
8.10 Evolution of Waves After Plasma Ignition in a
Sphere
8.10.1 Solution to the Problem
8.10.2 The Evolutionary Process

485
487

495

497
498
501
504
507
513
515
519
522
530
537
539
543
545
548
548
550

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


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Contents

Appendix D: The Resolvent with Cylindrical Symmetry
D.1 Unbounded Medium
D.2 The Medium with a Cylindrical Boundary

573
573
575

Appendix E: WGM Resonator with Transient Circular Inclusion

577

Index

585

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Preface

This book is devoted to investigations of non-stationary electromagnetic processes. It contains results concerning the nonstationary 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 timedomain 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 by
the authors since the 1970s. Some of these results were published in
Russian, and some were not published at all but may be interesting

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xviii Preface

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


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Acknowledgements

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.

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Alexander-Introduction

Introduction

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 nondispersive 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 electromagnetic

fields in a vacuum. In a general medium the constitutive equations


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Alexander-Introduction

2 Introduction

and boundary conditions significantly complicate both the formulation 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 in
the 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 timevarying 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


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Alexander-Introduction

Introduction

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 nondispersive 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 timevarying media is becoming ever more common. Applications, such as
the production of terahertz sources are exploiting 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 semiopen 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 for
elaborating 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 (nonharmonic) field, so-called “transients.” These transients can form a

3


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Alexander-Introduction

4 Introduction

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 electromagnetic 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 mathematical
technique 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 mathematical 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 unbounded
medium are considered in Chapter 2. It is shown that modulation
of the medium by a finite chain of medium permittivity time