image001Y. Sirenko, S. Strom, and N. YashinaModeling and Analysis of Transient Processes in Open Resonant Structures.NewMethodsand Techniques. New York, Springer, 2007.


The focus of electromagnetic theory is initial boundary-value and boundary-value problems for the Maxwell equations. Those are the initial models, from which, by applying mathematical methods, we should extract physical results. The modern computer-aided research process can be divided into several stages: qualitative mathematical analysis of the initial problem; the development of algorithms and implementation of the problem in software; problem-oriented computational experiments; and physical interpretation of the results. The success of the study depends in many aspects on whether sufficiently high standards of investigation can be maintained at all these stages and whether there is an ‘intellectual core’ in these investigations which  enables  us  to  gain  new  scientific  knowledge.  As an example of a

successful implementation of such an approach that has settled a long-standing conflict between theory and experiment, we can cite the development of the theory of resonant wave scattering in the frequency domain. These results have been reported, and they have served as a basis for the development of a number of essentially new functional units and devices in millimeter and sub millimeter radio-engineering, vacuum electronics and solid-state electronics, optics and spectroscopy.

The modern theory of transient electromagnetic fields is still lacking achievements that may be compared with those existing in the frequency domain, neither by the profoundness of the study, nor by the intensity of the study of electromagnetic phenomena and, as a result by their applications. However, the process of accumulation of potentialities for a break through is a process still in action. This book is devoted to just these problems; for the most part it is focused on the development and implementation of robust and efficient mathematical models for transient electromagnetic theory.

Chapter 1 has the character of a survey. Here we have collected the information that may be useful for the problems we are going to consider. We have also, up to our possibilities, tried to analyze the state of the art in time domain electromagnetics, as it existed at the time the book was written. We focused our analysis on methods, based on frequency-spatial and time-spatial representations.

The content of the Chapters 2–4 has been inspired by the statement from paper of Georgakopoulos S.V, Birtcher C.R., Balanis C.A., Renaut R.A (IEEE Ant. and Propag. Magazine, 2002, 44, 1): ‘Therefore, it becomes clear that the boundary conditions are an integral part of a PDE (partial-differential-equation) problem, and should always accompany the FDTD (finite-difference time-domain) formulation of it. This inflicts particular concerns when the problem under examination is so-called ‘open’ space or unbounded problem, e.g., radiating, scattering, etc., meaning that the domain of interest is unbounded in one or more spatial-coordinate directions. For such problems, there are no exact boundary conditions known’.

However, exact ‘absorbing’ conditions providing efficient limitation of the computational domain of finite-difference methods do exist now, since some time back. They have already taken their proper place in electromagnetic simulation in fundamental and applied electromagnetics. In 1986, scientists from Moscow State University, А.R. Maikov, A.G. Sveshnikov, and S.A. Yakunin, published in their paper the exact non-local conditions for virtual boundaries in regular semi-infinite hollow waveguides (with constant cross section), serving as channels for signals propagating from certain resonant junctions, were formulated for the fist time. Later on the approach suggested in this paper, that is based on the utilization of radiation conditions for time-spatial amplitudes of outgoing modes, has been modified for various electromagnetic problems: antenna design, analysis and synthesis problems for quasi optical open resonators with dispersive elements, electromagnetic monitoring of human environment and others. For several particular cases the problems of non-locality and the problem of corner points, points of intersection of coordinate boundaries, have been resolved in a rigorous way. The efficiency and correctness of the approaches, based on the application of the exact ‘absorbing’ conditions is verified and proved by specific tests and numerical experiments.

The major attention of the book is focused on the general theory and details of the technique that make the basis for construction of exact conditions. These conditions are incorporated into the algorithms of the solutions of initial boundary-value problems and they enable an efficient and accurate modeling of a rather wide class of problems of transient electromagnetic scattering. The exact ‘absorbing’ conditions form a natural complement to the list of conditions that have already became classical ones, those of J.-P. Berenger, B. Engquist, A. Majda, G. Mur, and others.

The analytic results of Chapters 2–5 are orientated at the application of the finite difference methods in the final computational algorithms. TheFDTDmethod, appeared in 1966 (see the canonical paperK.S. Yee) actually has created a boom in computational electromagnetics. It may serve as a perfect example of an excellent elaborated implementation of well known principles for the discretization of the curl type Maxwell equations. This scheme allowed a huge body of computational experiments to be performed, as required by electromagnetic engineering.

Preparatory work, based sometimes on sophisticated analytical derivations can transform the algorithm based on FDTD method into not only a universal and powerful but also an accurate and efficient tool, providing researchers with new knowledge and profound insight into transient phenomena. We hope that results, presented in Chapters 6 and 7 may be considered as a support for this conclusion.

Chapter 5 deals with new algorithmization schemes for simple (canonical) and complicated (chain of the junctions of simple problems) initial boundary-value problems of the electromagnetic theory of waveguides. The approaches suggested in this Chapter are based on the description of the scattering properties of inhomogeneities of regular waveguides in terms of the transform operators of the signals’ evolutionary basis that is qualitatively equal for all guiding structures. The corresponding approaches in the frequency domain are widely known. The modification of these prototypes and the solution to the technical and methodological problems arising due to the specific character of the time domain are the major issues highlighted in this Chapter.

In Chapters 6 and 7 we put forward our ideas about the most efficient approaches to the study of physical peculiarities of resonant scattering both in time and frequency domain. We also discuss methods for solving complicated applied problems of model synthesis of resonant quasi-optics. We describe these ideas for real models that have as a principal element a periodic diffraction grating.

Electromagnetic analysis of open periodic resonators and waveguides (gratings) is always associated with solution to important applied problems; the efficiency and further development of some promising topics substantially depend on how deep the characteristics of such scattering and wave directing objects are studied. Special attention should be given to theoretical studies that significantly simplify or even make unnecessary the experimental analysis and can be used for model synthesis and hardware optimization. This timely problem is partially solved in terms of the wave diffraction theory for the frequency domain, which enabled one to efficiently analyze many cases that are interesting from practical point of view, and in the process gather a lot of experience concerning abnormal and resonance scattering modes.

However, many problems in electromagnetic theory of gratings still remain unsolved, and the traditional approaches and methods are not capable of providing the solution to them. One such problem is concerned with studying the physical nature and analytical description of various resonance and anomalous space-frequency and space-time field transformations. Its efficient solution is provided by a mathematical analysis of the peculiar features of the analytic continuation of the solutions to the elliptic boundary-value problems into the domain of complex, usually physically unrealizable parameter values. Here, unlike the traditional problems of the frequency domain, the closest attention should be paid, not to the regular points from the intervals of parameter variation, where the corresponding operators are boundedly invertible, but rather mostly to the complementary sets, i.e. spectra, and the analysis of such phenomena and their behavior in complex space.

Maybe the very first attempt to use complex frequencies in electromagnetics should be dated to 1884, when J.J. Thomson analyzed free field oscillations in the exterior of a perfectly conducting sphere. The oscillations that satisfied the condition of ‘outgoing radiation’ increased exponentially in space, which was a reason for criticism: H. Lamb alleged the problem to be ill-posed. The effect of the ‘exponential catastrophe’ is still keeping many researches busy solving non-self-adjoint spectral boundary-value problems, although the question is completely settled by turning to space-time representations in the analysis: every divergent oscillation is associated with an exponential time-dependent factor that covers the coordinate-dependence in any space point.

Studies of dispersion relations and spectrum analysis is now a dominating trend in many areas of physics. It is concentrated on analytical properties of various functions describing physical phenomena, as they are extended into the domain of complex, ‘non-physical’ argument values. This means that although the physical meaning is inherent only in such notions as frequency and energy, nevertheless, the complex values of these parameters, that are never actually realized, are used in the analysis of the behavior of the system.

Dispersion relations result from the ‘self-adjoint’ or ‘non-self-adjoint’ theory of ‘open’ (‘exterior’) elliptic boundary-value problems. In the first case, the solution is sought in the space L2 (the — L2 theory), and in the second case, the solution is subject to the conditions of ‘outgoing radiation’ (along the real axis). C.L. Dolph has emphasized that there are no strict rules for using one theory or another, and, at the same time, he notes that the analysis of non-self-adjoint problems contributes substantially to the understanding of the case and provides a solid ground for many existing scattering theories. The — L2 theory seems to be more convenient and elegant in use from the mathematical point of view, but the more generalized approach based on the analysis of non-self-adjoint problems, proves to be a more powerful and capable tool for yielding useful, physically relevant representations.

Diffraction gratings are amongst the most popular objects in classical electromagnetic theory. What is the reason for that? There are several, but as the principal one we may consider the dispersive property of the grating: the diffraction grating is the most universal dispersive (frequency selective) element within the total frequency range that is exploited nowadays. As a result, the design and modeling of gratings of various configurations are always in demand. And as a personal reason we can add that the electromagnetic theory of diffraction gratings is just beautiful, describing a large variety of physical phenomena and their promising applications.

The history of the study of diffraction gratings is a great story and it deserves that a special book is written about it. There are many prominent scientists and scientific schools that made their contributions that can not be over estimated nowadays. We should like to emphasize here that the results presented in this book, especially in Chapters 6 and 7, are based on the results and inspired by Kharkov (Ukraine) scientific school that made considerable contribution to the mathematical background and profound study of electromagnetic phenomena of diffraction theory. This book should in particular be seen as an effort to make that work better known outside the Former Soviet Union.

Yuriy Sirenko, Staffan Strom, Nataliya Yashina

image007Y. Sirenko and S. Strom (eds.). Modern Theory of Gratings.Resonant Scattering: Analysis Techniques and Phenomena. New York, Springer, 2010.


The first publication about the discovery of diffraction grating by the American astronomer D. Rittenhouse dates back to 1786. It was not noticed by the scientific community of the day, and in the history of science the optician J. Fraunhofer was considered to be the creator of the diffraction grating (1821). Theoretical studies of this device, characterized by amazing dispersion properties were started by F.M. Schwerd in 1835. In those days spectral analysis was coming into being. The needs from this new area stimulated making gratings with progressive enhancement of the resolution, and they encouraged relevant theoretical and experimental studies. The outstanding achievements of H.A. Rowland must be mentioned here. He developed  a  machine  capable  of  making  quite  fine  diffraction   gratings

(1882). Also, he suggested making ruling lines on a concave spherical surface and as a result spectrum dispersion and sharpness were elevated to a level that had not been seen before.

The progress in several scientific and technological fields is to a large extent guided by the performance of the presently available gratings which are so sophisticated that sometimes they seem to have little to do with their predecessors from the 19th century. Polarization converters and phase changers, filters and multiplexers, quantum and solid state oscillators, open quasi-optical dispersion resonators and power compressors – these are only a few applications of periodic structures which astonish us (up to now!) by their capabilities for controlled polarization, spatial and frequency selection of signals.

Different operating frequency ranges call for gratings differing in characteristic size (length of a period), and in their way of achieving the operating mode. The range is so wide that, say, if one end is a standard echelette optical reflection grating (3600 lines per millimeter on a 40[cm]x40[cm] aluminium sheet) the other could be the antenna array of the unique decameter radio telescope UTR-2 developed and fabricated by the academician S.Ya. Braude’s team at the Institute of Radio Physics and Electronics of the Ukrainian Academy of Sciences in 1966. This antenna field is developed by two multi-component arrays. The first one, 1800[m] long and 53[m] wide, consists of 1440 wideband components making up six meridian aligned rows. The other, 900[m] long and 40[m] wide, is normal to the first one and carries six rows of 600 dipoles. All the dipoles (8[m] long and 1.8[m] across wire cylinders) are horizontally arranged at a height of 3.5[m] and east-west oriented.

Few countries could afford equipment for ruling optical gratings with thousands lines per millimeter. This process, expensive and time-consuming, failed to satisfy growing practical requirements. Rather good results have been achieved in making replicas of mechanically produced originals. An idea that diffraction gratings can be manufactured with the aid of holography was suggested by Yu.N. Denisyuk in 1962. The idea has been developed into holographic gratings intensively used in the making of spectral instruments. The advantages of holographic gratings consist in the fact that such gratings are free from grating spirits (i.e. high orders caused by periodicity deviation), they are characterized by little occasional light diffusion and are easy to produce. Naturally, to get desired diffraction characteristics from holographic gratings is more difficult than, e.g., getting them from ruled echelette gratings whose geometry uniquely depends on the so-called blaze angle. Holographic gratings rank below ruled gratings in diffraction efficiency but, according to many authors, their wave front quality in a working order (harmonic) is better. In addition, several observation were made in the 1980’s that the employment of certain schemes of hologram recording and subsequent photoresist processing opened the way for design of blazed gratings, including echelettes.

Evidently effective employment of diffraction gratings cannot be achieved without thorough theoretical and experimental research into their diffraction properties. The investigations began early in the 20thcentury. R.W. Wood improved the diffraction grating by shaping the grooves to specific geometries. On this basis, he launched systematic studies of the energy distribution among different harmonics and experimentally found the property of anomalous scattering. Lord Rayleigh was the first to expand the field scattered from the grating into a series of plane waves. Studying the echelette wave diffraction in theoretical terms, he developed an approximate technique (known as the Rayleigh method) which has been one of the most widely used until rigorous techniques became available.

In the evolution of grating theory one can identify several key periods. One falls within the last decades of the 20th century, characterized by the fact that relevant theoretical problems were approached using classical mathematical disciplines: mathematical physics, computing mathematics, theory of differential and integral equations, etc. That the grating became a subject of adequate mathematical simulation has opened up new opportunities for reliable physical analysis and also new avenues of attack, on a rigorous theoretical base, on numerous applied problems. At this stage the modern electromagnetic theory of gratings was greatly contributed by the radio physical schools of Marseille, France (R. Petit, D. Maystre, M. Neviere, P. Vincent, A. Roger,J. Chandezon et al.) and Kharkov, Ukraine (V.P. Shestopalov, L.N. Litvinenko, S.А. Masalov, V.G. Sologub, А.А. Kirilenko et al.). The key chapters of the book largely proceed from their achievements from the early 1960’s and onwards. Obviously, the growth of the research in the above-mentioned schools was heavily influenced by the results from other scientific centers the world over. We will address the most significant of them.

The methodology of modern radio physics is based on mathematical simulation and numerical experiment and it is realized by solving boundary value (frequency domain) and initial boundary value (time domain) problems for Maxwell’s equations. Time domain approaches (see, e.g., Chapter 4) offer more versatility and are more suited for the analysis of sophisticated electromagnetic structures of interest for applications. As a rule, the calculations here are reduced to realization of explicit schemes (the schemes with the sequential passage of the time levels). There is a good agreement between the calculated results (results from analysis of electromagnetic field space-time transformations) and general human perception – the time domain is free of some idealizations which are peculiar to the frequency domain. Moreover, time domain results are easy to change into the amplitude-frequency characteristics in the prefixed range of the frequency parameter k=2π/λ, where  is the free space wavelength. However, time-domain methods are not used as extensively as one would expect for getting physical results proper. Thus, for example, all power of the most popular at the moment FDTD-method is mainly applied to solution of particular engineering problems.

Far more examples of systematic and fruitful theoretical treatment can be met in time harmonic electromagnetics whose problems have been addressed much earlier in rigorous formulation. The last decades of the 20th century have brought some special powerful techniques for analysis and synthesis of various electromagnetic objects. Numerous physical phenomena accompanying processes of monochromatic wave’s radiation, propagation and scattering have been identified, interpreted and implemented into design of novel devices. Such advances have been assured by the fact that new theoretical methods have been developed being oriented to the solution to the specific applied problems. They have accounted the peculiarities of problems of interest and hence have provided with not only qualitative information, but they created also the base for qualitative analysis with further generalization. As an example one may consider the authentic analytic regularization procedures (see Chapter 2) outperforming other frequency domain techniques in resonance situations. Actually, nearly all profound physical results gained from electromagnetic theory of gratings are due to usage of analytic regularization procedures.

The long-wave specific case (k=l/λ<<1, l is the grating period length), ending up with solutions of simple analytic representations and convenient approximations, has been understood most comprehensively by frequency domain methods. Here the approach grounded on equivalent boundary conditions possessing in the general case anisotropic properties is widely applied (B.Ya. Моyzhes, L.А. Vainshtein, V.М. Аstapenko and G.D. Мalyuzhinetz; Ye.I. Nefedov and А.N. Sivov; et al.). The theory of dense gratings based on this approach takes into account the influence of shape and relative size of grating’s elements, the presence of sharp boundaries in the dielectric filling and allows one to make a correct limit transition as the conductors come infinitely close. A key point in the solution of the diffraction problem based on this theory is a search for the reflection and the propagation coefficients in terms of powers of a small parameter k via considering a relevant static problem. The long-wavelength diffraction is implemented in many modern superhigh frequency devices and units thus the relevant theoretical studies are of current importance. Simple and convenient analytic representations are very useful for the designers and, at the same time, they are an aid to general nature interpretations contributing to the electromagnetic theory of gratings. An example is the effect observed by G.D. Мalyuzhinetz in the 1940’s: given a certain angle of incidence on a dense grating arranged by metal bars of nonzero thickness, a plane H-polarized wave propagates through it with no reflection.

Of tremendous interest for physics and applications and a great problem for analysis is the resonance case k=O(1), i.e. the case when the wavelength is comparable with the grating period. When computer resources were limited, the research into the resonance domain had been restricted to some specific or limiting situations. They were studied by V.S. Ignatovskiy, E.A.N. Whitehead, F. Berz, J.F. Carlson, A.E. Heins, G.L. Baldwin, L.А. Vainshtein, V. Twersky, Yu.P. Lysanov and others. These researchers laid a solid ground for the modern theory of resonant wave scattering by periodic structures. Indeed, the ideas and achievements gained in the 1940’s to 1960’s are traced in almost every present-day method of mathematical modeling oriented to the numerical experiment. First of all, it is the method of partial domains (or mode matching method) whose first fruitful implementation can be seen in L.N. Deryugin’s works. Next are the potential-theory based methods (integral equation techniques) whose today’s technique (N. Amitay, V. Galindo and C.P. Wu; А.S. Il’inskiy and T.N. Galishnikova; А.I. Sukhov; Z.Т. Nazarchuk, and others) is based on quasi-periodic Green’s function derived by V. Twersky. At a point of equivalent reformulation of the original boundary value problem, the authors of some analytic numerical methods (Ye.V. Avdeev and G.V. Voskresenskiy; R. Mittra and T. Itoh; S.М. Zhurav, and others) address, either implicitly or not, to the technique and the results of the analytic solution to canonic diffraction problems similar to those considered by E.A.N. Whitehead, F. Berz, and others. Only few such problems have been solved rigorously. The most popular ones (see, e.g., works by E. Luneburg and K. Westpfahl; V.D. Luk’yanov; L.A. Vainshtein and V.I. Vol’man) have always been those about half-plane gratings and planar strip gratings. The enduring interest in elementary structures whose diffraction characteristics have long been thoroughly studied for arbitrary geometrical parameters and frequencies, from the long to the short wave regions, is indeed reasonable. The main significance of these considerations and the most valueable aspects of the outcomes consist in the search for new ideas and approaches and proving their potentials to be used in more sophisticated situations which are far from standard.

In closing the theme of continuity, it should be mentioned that the numerical solution of the problems concerning the plane wave diffraction by periodic corrugated surfaces has been the most frequently attempted by invoking the Rayleigh method (Rayleigh hypothesis). There are methods that take the Rayleigh representations for the scattered field and extend them in a straightforward manner from their region of validity directly to the grating surface. Furthermore, there are methods, prompted by the Rayleigh hypothesis, but resting on the fundamental results of I.N. Vekua about completeness of some systems of functions on curved contours. In the first case, difficulties in the proof of the principal step (it is necessary to study singularities of the analytic continuation of the Rayleigh representation as a function of space coordinates) can be overcome only for shallow gratings, with groove profile described by a sufficiently smooth one-valued function (see works of A.G. Kyurkchan). And even then a correct truncation of the resulting infinite system of algebraic equations for unknown amplitudes of the field space harmonics is not possible. In the other case, a principal feasibility exists to construct special linear combinations of functions that are asymptotically close to the solutions of the corresponding diffraction problems throughout the whole scattering domain. The central problem – development of stable computation schemes – is solved then with the adaptive (assignable by the groove shape and  k value) collocation technique.

In many works of electromagnetic theory of gratings the modern scientific methodology chain “object  -> mathematical model  -> algorithm  -> numerical experiment  -> physical interpretation of the results   ->formulation of general conclusions and recommendations” breaks somewhere in the middle, at a level of standard illustrations of the efficiency of the algorithm. But nevertheless, after L.N. Deryugin’s work who analyzed (in terms of some particular cases) the surface and double surface resonances on the comb gratings, issues do appear, which inform of experimental, analytic, and numerical results, concerning:

  • threshold phenomena (A. Hessel and A.A. Oliner; B.М. Bolotovskiy and А.N. Lebedev; E.А. Yakovlev and М.V. Robachevskiy);
  • semi-transparent grating effects of total resonant transition and reflection of plane waves (Ye.V. Avdeev and G.V. Voskresenskiy; А.F. Chaplin and А.D. Khzmalyan; R.S. Zaridze, and G.М. Тalakvadze; Yu.P. Vinichenko, А.А. Lemanskiy and М.B. М
  • effects of total nonspecular wave reflection by catoptric structures (E.V. Jull and G.R. Ebbeson; J.R. Andrewarsha, J.R. Fox and I.J. Wilson; S.N. Vlasov and Ye.V. Koposova, and others).

Some authors (see, for example, works of E.V. Jull, D.C.W. Hi, N.C. Beaulieu, and P. Facq) have raised a very important question about the differences between the ideal (infinitely extending structure in the plane wave field) and actual (finite excitation field spot on the infinite periodic structure or finite structure in the plane wave field) operating modes of the grating.

Also nowadays the diffraction grating is still one of the central objects of electromagnetic analysis. Independent of how comprehensive the progress in our understanding of the grating is, continued research in this direction remains very important, indeed practical needs and the intrinsic logic of development of the modern grating theory present us with new problems, sending us to seek and hopefully find ways to their solution. Just so was formed during the recent years a new line of investigation, which is partially considered in this book (see Chapter 5) and associated with the analysis, synthesis and determination of equivalent parameters of artificial materials – layers and coatings which have a periodic structure and properties exhibited by natural materials in exceptional cases only.

Generally speaking, the book reflects those results which, in our opinion, are able to further pursue electromagnetic theory of gratings in pace with today’s requirements of fundamental and applied science. The book gives the reader quite a comprehensive idea of:

  • spectral theory of gratings (Chapter 1) giving reliable grounds for physical analysis of space-frequency and space-time transformations of the electromagnetic field in open periodic resonators and waveguides;
  • authentic analytic regularization procedures (Chapter 2) that, in contradistinction to the traditional frequency-domain approaches, fit perfectly for the analysis of resonant wave scattering processes;
  • parametric Fourier method and C-method (Chapter 3) oriented on the effective numerical analysis of transformation properties of periodic interfaces and multilayer conformal arrays;
  • new rigorous methods for analysis of special-temporal transformations of electromagnetic field that are grounded on the construction and incorporation into the standard finite-difference computational schemes the so-called exact absorbing boundary conditions (Chapter 4);
  • new solution variants to the homogenization problem (Chapter 5) – the central problem arising in the synthesis of metamaterials and metasurfaces;
  • new physical and applied results (Chapters 2 to 5) about pulsed and monochromatic wave resonant scattering by periodic structures, including structures loaded on dielectric layers or chiral and left-hand medium layers, etc.

The authors hope that the reader will find that the discussed physical and applied results are presented in an illuminating way. Thus, for example, some figures in Chapter 4 are accompanied by .exe-files which enable to watch in dynamics the space-time transformations of the electromagnetic field close to finite and infinite periodic structures. The archive with these files will be placed at web-page The book is intended for researchers and postgraduate students in computational electrodynamics and optics, theoretical and applied radio physics. The material is also suitable for undergraduate courses in physics, computational physics and applied mathematics.

The authors are representatives of a series of large European scientific and educational centers: Royal Institute of Technology, Stockholm, Sweden (Staffan Ström, the editor and the co-author of Chapter 4);Blaise Pascal University, Clermont-Ferrand, France (Jean Chandezon and Gerard Granet – Chapter 3); Usikov Institute of Radio Physics and Electronics of the National Academy of Sciences of Ukraine, Kharkov, Ukraine (Petr Melezhik – Sections 2.2, 2.4, 2.5; Anatoliy Poyedinchuk – Sections 2.1, 2.2, 2.4, 2.5, 3.6; Yuriy Sirenko – the editor, the author and co-author of Chapters 1, 4 and Section 2.3; Yuriy Tuchkin – Sections 2.1, 2.6; and Nataliya Yashina – Chapter 4 and Sections 2.4, 2.5, 3.6); Lund University, Lund, Sweden (Daniel Sjöberg – Chapter 5).

In this book, they are united by their profound interest in periodic structures, an area whose study has always been associated with burning scientific and engineering problems for the last one and a half hundred years.

Yuriy Sirenko and Staffan Ström



image042В.Кравченко, К.Сиренко, Ю.СиренкоПреобразование и излучение электромагнитных волн открытыми резонансными структурами. Моделирование ианализ переходных и установившихся процессов. Москва, Физматлит, 2011.


Современная методология получения новых знаний, основными составляющими которой являются математическое моделирование и вычислительный эксперимент, реализуется в теории электромагнитного поля через решение краевых (частотная область) и начально-краевых (временная область) задач для уравнений Максвелла. Направление, связанное с исследованием процессов излучения, распространения и рассеяния импульсных волн, развивается в последние годы более интенсивно. Прежде всего, потому, что для проектирования ряда перспективных устройств техники  связи,  электроники и радиолокации потребовались надежные

сведения о пространственно-временных и пространственно-частотных трансформациях поля в достаточно сложных электродинамических структурах, а возможности традиционных подходов частотной области в этом отношении либо существенно ограничены, либо уже исчерпаны. В подходах временной области привлекает также и то, что они:

Вместе с тем, в теории неустановившихся электромагнитных полей существует ряд проблем, не получивших к настоящему моменту времени универсальных, обоснованных и практически реализуемых решений, и это сказывается на качестве модельных построений, ограничивает возможности методов временной области по изучению физики переходных процессов и закономерностей пространственно-временных трансформаций импульсных волн. Это, прежде всего, проблема корректного и эффективного ограничения пространства счета в так называемых открытых задачах, т.е. в задачах, область анализа которых уходит на бесконечность вдоль одного или нескольких пространственных направлений. Перечень можно продолжить проблемой дальней зоны, проблемой больших и отдаленных источников поля и др.

Книга посвящена математическому моделированию и физическому анализу переходных и установившихся процессов в открытых резонансных электродинамических структурах, формирующих, направляющих, рассеивающих и излучающих импульсные и монохроматические электромагнитные волны. Ее основные темы:

Известные эвристические и приближенные решения проблемы, связанной с переходом к конечным областям анализа в открытых задачах временной области, базируются, в основном, на использовании так называемых AbsorbingBoundaryConditions (ABCs) и PerfectlyMatchedLayers (PMLs). Главный недостаток этих решений – непрогнозируемое поведение вычислительных ошибок при больших значениях времени наблюдения  и, как следствие, отсутствие гарантий правильности получаемых результатов в ситуациях, связанных с резонансным рассеянием волн.

В книге развит подход, позволяющий объективно оценить и минимизировать погрешности, возникающие при замене открытых начально-краевых задач задачами закрытыми. Его основу составляют построение и включение в вычислительную схему метода конечных разностей точных поглощающих условий, т.е. условий, добавление которых к оригинальной начально-краевой задаче никак не сказывается на ее решении. Коротко историю этого подхода можно изложить следующим образом. В 1986 г. А.Р. Майков, А.Г. Свешников и С.А. Якунин публикуют работу, в которой первыми формулируют точные нелокальные условия для виртуальных границ в поперечном сечении регулярных полубесконечных полых волноводов – каналов, по которым распространяются сигналы, формируемые каким-либо волноводным узлом. Позднее их результат, в основе которого лежит использование условий излучения для пространственно-временных амплитуд парциальных составляющих (мод) несинусоидальных волн, уходящих от области локализации эффективных источников и рассеивателей, был модифицирован и развит применительно к самым разным задачам теоретической и прикладной радиофизики. Рассмотрены волноводные и антенные задачи; задачи анализа и синтеза квазиоптических открытых дисперсионных резонаторов, задачи распространения волн в среде, окружающей человека в его повседневной деятельности, и т.д. Для ряда частных случаев строго решены проблемы нелокальности и угловых точек – точек пересечения виртуальных координатных границ. Эффективность и корректность подхода подтверждена результатами вычислительных экспериментов и решением тестовых задач.

Аналитические результаты этой книги (см. главы II–V: нелокальные и локальные точные поглощающие условия для различных структур и в различных системах координат; алгоритмы решения проблемы дальней зоны и проблемы больших и отдаленных источников поля; пространственно-временной аналог метода обобщенных матриц рассеяния и др.) ориентированы на использование в вычислительных схемах метода конечных разностей. Действительная история и теория этого метода, конечно же, гораздо богаче тех кратких и упрощенных версий, которые обычно излагаются в изданиях «electromagnetic community». Finite-DifferenceTime-DomainMethod (FDTD-метод), с появлением которого в 1966 г. (см. каноническую работу K.S. Yee) здесь вполне справедливо связывают начало вычислительного бума, – пример хорошо продуманной реализации известных принципов при дискретизации роторных уравнений Максвелла, реализации, порожденной требованиями практики, появлением реальных перспектив осуществления огромных объемов вычислений за приемлемый промежуток времени. Но работа K.S. Yee не вызвала лавину теоретических результатов, как это часто утверждают. Правильнее было бы сказать, что она способствовала существенному расширению круга исследователей, владеющих основами метода, способных применить его без грубых ошибок и грамотно распорядиться ресурсами современных компьютеров при получении конкретных данных, характеризующих конкретные электродинамические элементы и узлы. Серьезно теорией конечно-разностных методов и разработкой принципов их использования в прикладной математике и вычислительной физике занимались те, чьи имена, к сожалению, мы не найдем ни в одном из обзоров, ни в одной из обобщающих работ, посвященных FDTD-методу.

J.L. Volakis и D.B. Davidson – редакторы раздела EMProgrammer’sNotebookв журнале Antennas & PropagationMagazine – в предисловии к одной из статей журнала (Ant. andPropag. Magazine. – 2002. – Vol.44, no.1) характеризуютFDTD-методкак «one of the workhorses of computational electromagnetics».Конечно же, в этих словах – только признание надежности и полезности соответствующего подхода, но на практике ему действительно чаще всего отводится лишь рутинная, объемная счетная работа. Вместе с тем, хорошо подготовленный вычислительный эксперимент может превращать алгоритмы метода конечных разностей в универсальный, эффективный и достаточно тонкий инструмент получения новых знаний о физике переходных и установившихся процессов. Обоснованию этого утверждения посвящены пять заключительных глав книги. Наиболее интересные результаты этих глав перечислены ниже.

  • Разработан и реализован новый подход к анализу спектральных характеристик открытых компактных, волноводных и периодических резонаторов методами временной области (глава VI).
  • Впервые строгими методами временной области подробно исследованы щелевые резонансы – полуволновые и четвертьволновые резонансы на TEM-волнах в узких радиальных и коаксиальных щелях идеальных проводников. Возбуждение таких резонансов позволяет существенно изменять основные характеристики простых аксиально-симметричных волноводных узлов и всенаправленных антенн стандартной конфигурации (главы VII и VIII).
  • Впервые широко представлены сведения об основных электродинамических характеристиках (амплитудно-частотных и импульсных) канонических аксиально-симметричных излучателей TE0n— и TM0n-волн (монополей, зеркальных и резонансных антенн) – положено начало формированию «библиотеки элементарных излучателей», обращение к которой может упростить и ускорить решение многих прикладных задач (глава VIII).
  • В рамках плоских моделей проанализированы открытые электродинамические структуры (излучатели, работающие на эффекте преобразования поверхностных волн в объемные; сверхширокополосные и резонансные антенны; резонаторы с существенно разреженным спектром; и т.д.), каждую из которых можно рассматривать в качестве прототипа при модельном синтезе новых узлов и устройств резонансной квазиоптики, антенной техники, вакуумной и твердотельной электроники (глава IX).
  • Решены специальные и прикладные задачи, связанные с анализом и синтезом резонансных излучателей мощных коротких радиоимпульсов, компрессоров мощности, фазированных антенных решеток, и т.д. (глава X).

Книга снабжена файловым приложением, включающим:

Мы благодарим за помощь наших коллег, сотрудников отдела математической физики и отдела теории дифракции и дифракционной электроники Института радиофизики и электроники им. А.Я. Усикова Национальной академии наук Украины Анну Игоревну Амосову, Людмилу Георгиевну Величко, Вадима Леонидовича Пазынина, Елену Сергеевну Шафалюк, Наталью Петровну Яшину. Большая часть материала, включенного в книгу, – это результат нашей совместной работы в последние несколько лет.

В. Кравченко, Ю. Сиренко, К. Сиренко

book springerY. Sirenko and L. Velychko (eds.). Electromagnetic Waves in Complex Systems: Selected Theoretical and Applied Problems. New York, Springer, 2016.


Our aim in writing this manuscript was to provide young researches and graduate students with a book that combines examples of solving serious research problems in electromagnetics and original results that encourage further investigations. The book contains seven papers on various aspects of resonant wave propagation and scattering written by different authors. Each paper solves one original problem. However, all of the papers are unified by authors’ desire to show the advantages of rigorously justified approaches to all stages of the study: from problem formulation and selection of the method of attack to interpretation of the results.

A glance at the Contents will reveal a range of physical problems raised in the book. Mostly, those are the problems associated with wave propagation and scattering in natural and artificial environments or with designing the elements and units for antenna feeders. The authors invoke both theoretical (analytical and numerical) and experimental techniques for handling the problems. Considerable attention is given to the mathematical simulation issues, problems of computational efficiency and physical interpretation of the results of numerical or full-scale experiments. Most of the presented results are original and have not been published earlier.

The need for rigorous theoretical justification of mathematical modeling and computational experiments – the widely-used methodologies of obtaining new knowledge – is evident. Underformulated problems, neglect of the estimation of stability and convergence of numerical schemes cannot guarantee reliability of the results. Furthermore, the rigorous theoretical basis of the laboratory and full-scale experiments allows to conduct research saving time and material resources, to safely test simulated devices in a variety of operating conditions. To demonstrate the advantages of rigorous approaches and their realizability is the heart of the ideology of this book. And we address it to those young researchers who are going to work actively and fruitfully in the field of theoretical and applied physics, electronics and optics.

The authors of this book are mostly current or former employees of the Department of Mathematical Physics at the O.Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences (Kharkiv, Ukraine). Professor Yuriy Sirenko, who has been at the head of the department over the last 25 years, initiated the writing of this rather unusual in its conception book. He has had a major influence on it, both scientific and organizational, and managed to inspire other colleagues with his idea.

The assumed background of the reader is mostly limited to standard undergraduate topics in physics and mathematics.

Lyudmyla Velychko

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