**History**

Laboratory №36 was founded in the Institute of Radiophysics and Electronics of the USSR Academy of Sciences by a resolution №135 of the AS USSR Presidium of 26.03.1980 and it was reorganized into a department of the theory of radio waves scattering and propagation (it is the department of mathematical physics since 2000).

Heads: V. G. Sologub, Professor, Dr. of Physics and Math, (1981-1987 years); Yu. K. Sirenko, Professor, Dr. of Physics and Math, (since 1988).

Now the Department has 8 employees: one doctor of sciences and four PhD researchers. Ye. A. Skirta and V. A. Abdulcadyrov (1983), N. N. Misura and G. I. Koshevoy (1985), L. A. Pazynin and A. D. Barkov (1987), V. A. Doroshenko and V. V. Yatsyk (1989), Yu. I. Krutin (1992), L. G. Velychko (1997), A. O. Petrov (2000), V. L. Pazynin (2003), K. Yu Sirenko (2007), A. I. Amosova (2009) have become PhD of sciences since its foundation.

**The main scientific trends in 1980-2004.**

*Theory of resonant scattering of electromagnetic waves: screens with a plane and axial symmetry; periodic structures (supervised by Yu. K. Sirenko and K. Yu. Sirenko).*

The scientific trend as a whole was developed by the theorists of three departments of the Institute: Dept. №11 headed by V. P. Shestopalov, a Full Member of NASU, Dept. №12 headed by Prof. A. A. Kirilenko, Dr. of Phys. & Math. and Dept. №36. As a result, a modern theory of resonant wave scattering of electromagnetic waves (frequency domain) and a spectral theory of open electrodynamic structures have been developed including a wide range of efficient and reliable methods of analytical regularization aimed at the analysis in the most difficult resonant wavelength range, discovery and detailed investigation of a large number of anomalous and resonant effects and phenomena, clear and single-valued physical interpretation of the simulation results. The following results have been achieved at the department:

- Themethods of analyticalregularizationhave been developed to solve the problems ofdiffraction anddiffractionelectronics for the screens with a flataxial symmetry(circular waveguide with slots, the length of a circular waveguide, a hole in a metaldiaphragm, conical surfaceswith slots, different types of thin beltperiodic structure). Supervised by GSologub;
- The dispersion characteristics ofopenmicrostriptransmission linesof different types have been investigated. Supervised by I. Koshevoy andV.G.Sologub;
- The spectral theory of gratings have been developed. Supervised by K. Sirenko and V.V. Yatsyk;
- The effect of existing of slow surface waves of open periodic structures in band gap (traditional classification in literature) has been discovered and analytically described completely. It has been revealed that slow waves overcome the nontransmission zones of open periodic waveguides in the process of collisions: as the parameters vary two “real” waves collide at one point of space of complex propagation constants and turn simultaneously into “complex” waves. The collisional phenomenon has been interpreted. The effects of linear “interaction” of free oscillations and eigen waves in periodic structures have been detected and studied. The “interaction” causes global or local reconstruction of the fields which in turn leads to global or local in the type, dynamics, Q-factor and other electrodynamical characteristics. Performed under supervision of K. Sirenko.

In 1989 V.G. Sologub and Yu.K. Sirenko together with a number of other developers of the problem were awarded the USSR State Prize in the field of sciences and technology for the series of works “Theory of resonant wave scattering and its applications in Radiophysics”.

*Modeling and analyzing of transient processes in open, periodic and compact resonators (supervised by Yu. K. Sirenko and V.**L. Pazynin).*

Since the mid-1990s Yu. K. Sirenko and his group started developing the design of theoretical methods of investigating transient electromagnetic fields formed by open resonant structures of different types. The objects analyzed were heterogeneities of regular waveguides, gratings, compact dielectric and metal scatterer in a free space and near local rough boundaries between two media, the emitters of broadband signals. Methods and computational techniques have been proposed and applied. The mathematical and physical results obtained by researchers of the department from their substantial contribution to the modern theory of nonsinusoidal waves.

The work along this trend has resulted in developing a new universal method of calculating nonstationary fields in nonclassical (unbounded) domains. The exact absorbing conditions are correctly included in the standard computational patterns of the finite-difference method for reducing equivalently the original open problems to closed ones. These conditions efficiently limit the computational domain and do not distort the physics of the process modeled by mathematical tools.

An approach has been developed, which leans upon description of the scattering properties of the regular inhomogeneities of waveguide tracts and free space in terms of transform operators qualitatively identical for all structures with a discrete spatial spectrum of evolutionary basis of the signals. In the frequency domain this approach is known as the method of generalized scattering matrices. It is widely used at present in constructing integrating shells of advanced automated simulation systems for profound researcher and important applied problems. The new modes of algorithmization of initial boundary-value problems are “replicas” of the patterns used in the method of generalized scattering matrices modified to allow for the specific of time domain.

The research in this trend have been supported by individual grants (Yu. K. Sirenko) of the Royal Swedish Academy of Sciences (1996 – 1999) and the International Science Foundation (1998 – 1999). Yurii Sirenko was awarded in “The Best Researcher and Teacher” nomination.

*Inverse problems of electrodynamic theory of gratings (**supervised by**Yu. K. Sirenko and L. G. Velychko)*

In the beginning of the 1990s inverse problems of electrodynamic theory of gratings were among the main trends in the department. The problems of visualization of perfectly reflective periodic structures with an arbitrary ruling profile have been considered within the limits of the single-site and single-frequency pattern. Good results have been obtained in long- and medium- wavelength ranges of variations the main geometrical parameters. In medium-wavelength range the period and the profiling depth is up to 5 and 2 wavelengths, respectively.

The elaborated quasilinearization ideas used to solve the visualization problems have allowed several approaches to solving the problems of synthesis of reflective gratings with the pre-assigned diffraction characteristics in the frequency range or/and in the intervals of angles of the coming plane wave.

The similar approach has been used to obtain an exact solution of the problem of synthesis of periodic dielectric layers generating a pre-assigned field in the radiation zones at a pre-assigned frequency and a fixed direction of the incoming excitation wave. Periodic nearly-free (lamellated) surfaces with the radiation characteristics suitable for relativistic diffraction electronics have been synthesized.

*Wave scattering from statistically rough surfaces and inhomogeneous media. Strict models of waves scattering theory (**supervised by**L. A.**Pazynin**and V.**L.**Pazynin)*

From 1989 to 2000 the assessment problem of the mean electromagnetic field of the electric dipole near statically rough, on the average spherical surface with small surface impedance were investigating in the department in corporation with the other departments of the Institute. The solution of the problem has been obtained as series in spherical harmonics (the illuminated zone) and in envelopes or spiral waves (the shade zone). The last representation is important in an applied aspect, where the sphericity is taken into account in the problems of beyond-the-horizon propagation above the ground. In order to construct the solution, the known in quantum scattering theory Clebsch-Gordan coefficients were generalized to the complex values of their indexes, and the corresponding infinite series were summed efficiently.

In 1997 the results of the study of the mean field of a plane wave incident on the limited randomly fluctuated domain with the fuzzy boundary was published. This domain is the statistically heterogeneous plane-laminated medium with a constant mid-value of the refractive index and dispersion varying in the pre-assigned direction by analogy with an asymmetrical Epstein layer. An analytical solution of the corresponding Dyson equation in bifocal approximation points to a partial reflection of the coherent signal. An analysis of the transient radiation of the longitudinal magnetic dipole flying through the fuzzy boundary between two homogeneous media at a constant speed adjoins to this trend methodologically. The obtained solution has allowed significantly refine the known “sharpness” criterion of the fuzzy boundary when charged particle flies into the dense medium.

In 1999 the mathematical model of the irregular planar waveguide, the surface complex impedance of the waveguide’s walls varies continuously simulating a smooth transition between the regular sections. The analytical expressions for the transformation coefficients of eigenmodes of the regular waveguide section in its transition region have been obtained. The consistent assessment of an error is known in the theory of smooth irregular waveguides of adiabatic approximation has been first presented. It has been discovered the regime of anomalously high transformation of the basic zero mode to the first one in the adiabatically smooth transition caused by mutual transformation of zero- and first modes near their degeneracy regime.

The generalization of the model of the isotropic transition Epstein layer to biisotropic, plane-layered media has been formulated. The existence of reflectionless modes of a plane wave through such medium is evident from the expressions for the reflection and transmission coefficients. The analysis of the known J.Radlow solution of the diffraction problem of a plane wave on a plane’s quarter has been performed. The incorrectness of the solution has been detected and corrected.

A model of plane-layered, smoothly inhomogeneous medium involving the domains with ordinary and double-negative media has been proposed. The analysis of the obtained solution has showed that the known effect of the negative refraction in the isotropic double-negative medium is the direct consequence of Maxwell’s equations and the energy conservation law.

*Experimental study**of the slow surface waves propagation in open transmission lines in the millimeter wavelength range and investigation their diffraction phenomenon by periodic and quasiperiodic structures (supervised by A. P. Evdokimow, V. V. Kryzhanovsky )*

The experimental group of the department investigated the excitation, canalization and transformation of slow surface waves of open transmission lines in the millimeter wavelength range. The ridged dielectric waveguide has been studied in detail. It can be used in the diffraction radiation antennas to obtain the level of the side-lobe level below that –30 dB. The exciters of slow surface waves in the waveguides with the losses below that 0.1 dB have been synthesized.

The study of the transformation laws of the surface waves into the volume waves for new combinations of different diffraction gratings and open metal-dielectric waveguides allowed to expand significantly the set of advanced radiating structures. These findings are used to develop the electrodynamic patterns of the scanning and non-scanning diffraction radiation antennas in the centimeter and millimeter wavelength ranges for different radiometric and radar systems.

**The main scientific trends in 2005-2014**

- 1.
*T**he method of the exact**absorbing conditions for solving initial boundary-value problems in the theory of open, compact, periodic and waveguide resonators: elaboration and implementation (supervised by K. Yu. Sirenko and Yu. K. Sirenko)*

The method solves one of the most important and complex theoretical problems of modern computational electrodynamics. It reduces the open model initial-boundary value problems (the domain of analysis of this problems is infinite in one or more spatial directions) to equivalent closed one. It allows us to formulate correctly and to solve numerically the fundamental and applied problems of analysis arising in a study of the transient and steady-state processes in open resonant electrodynamic structures.

The theory of the method and the results of its implementation are presented in [1-8]. The original exact local and nonlocal absorbing conditions for pulse waves coming through virtual boundaries in the cross sections of regular waveguides and through coordinate virtual boundaries in free space are stated there. The conditions are formulated for the scalar (plane and axially-symmetric) and vector electrodynamic problems. The concept of the virtual feed waveguides intended for solving a lot of practical problems of pulse and monochromatic waves radiation has been developed. The range of the physical effects of the detected and investigated in detail physical effects involved in anomalous and regular spatial-temporal and spatial-frequency transformations of the electromagnetic fields has been significantly extended.

In particular, the following properties have been completely studied: (a) the features of symmetric pulse and monochromatic waves radiation by axially-symmetric structures with feeding circular and coaxial waveguides; (b) the slot resonances in *TEM*-waves in the narrow segments of coaxial and radial waveguides, significantly expanding the facilities of structurally simple nodes of the waveguides and emitters; (c) the effects of the mode and frequency-mode layering of the broadband signals at the stepped connections of the waveguides with different cross-section and on conical plugs in circular and coaxial waveguides; (d) the effects of diffraction radiation in finite plane and axially-symmetric periodic structures; (d) the abilities to control the band changes of open compact resonators.

The most important recent developments in this trend include: (a) a mathematical justification of the method of exact absorbing conditions, in particular, the proof of theorems on the unique solubility of the modified problems [11,12]; (b) the development of the so-called Block FFT-Based Acceleration Scheme for the implementation of standard convolution operation reducing the computation time by two or three degrees [9,10]; (c) the elaboration of effective discrete finite element method for solving initial boundary value problems of electrodynamics [13,14] on the basis of the absorbing conditions.

*New efficient**approaches to solving the problems of analysis and synthesis of the energy compressors in the centimete and millimeter wavelength ranges: development and implementation (supervised by Yu. K. Sirenko and V. L. Pazynin)*

The practical implementation schemes of model synthesis of the direct-flow active compressors in the rectangular, circular and coaxial waveguides segments [15] have been developed. Resonant and distributed switches (locks) have been synthesized to provide the efficient energy accumulator by the compressor and its quick discharge into single mode excurrent waveguides and in a free space [6, 15–17]. The characteristics of high-power short pulses by the simplest monopolistic structures with a coaxial feed waveguide have been studied. It has been suggested and calculated a scheme of the synthesis of new phased array antennas when each their radiating element is both an active compressor at the same time [18]. The physical characteristics of transient electromagnetic energy accumulator, switching the compressor mode from the accumulation mode to energy discharge mode, a high-power short pulses radiation in a free space [16, 18] have been first studied in detail. The effective methods of the analysis and the synthesis of open resonators, i.e., resonant microwave power accumulators [4, 19] have been designed and tested. It has been discovered a number of new effects when resonators-accumulators are excited by long quasi-monochromatic pulses through the beyond-cutoff diaphragm. For example, there is always a point of time when the amplitude of the reflected wave in the feed waveguide is reduced to zero during the fine adjustment of the off-site generator to a resonant frequency of the accumulator. The accounting of these effects allows one to control the characteristics of the synthesized compressors such as the coefficient of efficiency, the degree of compression, and the power capacity of the formed pulse [16].

The method of rigorous definition of the laws of amplitude and frequency modulation of electromagnetic pulse subjected to passive compression inside the linear dispersion waveguide tract [10]. The method consists in inversion the history of process and solving the model initial boundary-value problem with the excitation pulse that must be obtained at the compressor discharge. The time profile of the input pulse is obtained in this way (accurate within the inverse replacement of time variable). The method was tested for the sections of regular waveguides with arbitrary cross section, the possibility of pulse compression in these structures is proved. This compression is more than a hundredfold larger with an efficiency coefficient approximate to unit.

*The construction of the**analytical basis for the correct formulation and an efficient numerical solution of the initial boundary-value and boundary problems of electromagnetic theory of two-dimensionally periodic structures (supervised by Yu. K. Sirenko and L. G. Velichko)*

Serious model is one-dimensional periodic gratings appeared in the second half of the XX century, when the corresponding theoretical problem was started to consider from the point view of classical disciplines like mathematical physics, computational mathematics, the theory of differential and integral equations. The improving of the models quality and the possibilities of computer technology is greatly accelerated new physical results obtaining, its implementation in the engineering practice, the development of new devices in the millimeter and submillimeter wavelength ranges, and new materials manufacturing with the size of the inclusions from micro- to nanometers. Nowadays, not all capabilities including the polarization, the frequency and spatial selections of electromagnetic signals are used in full. The potential of the classic two-dimensional models is almost completed now – three-dimensional vector models including the specifics of terahertz and optical wavelength ranges are central in the theory and in the practice.

The three-dimensional vector models [5, 20–23] are constructed on the basis of the space-time concepts and are intended for grid implementation of the finite-difference method or finite elements method when they performed numerically. The advantages of time-domain approaches: (a) they are free from frequency domain idealizations, (b) they are universal and do not limit the geometrical and material parameters of the study objects, (c) they lead to explicit computational algorithms that do not require to inverse of some operators, (d) the obtained results can be easy converted into the standard amplitude-frequency characteristics of the gratings.

Obviously, that the computational algorithms solving the gratings problems should be stable and convergent, the computational errors should be predictable, and the numerical results are aimed at unique physical interpretation. It is important that the theoretical analysis at all modeling stages (the statement of boundary and initial boundary-value problems, the determination of their correctness classes, the study of quality characteristics features of the analytic continuation of solutions of model problems into the domain of complex values of real parameters, etc.) would be performed at a high level.

In [5,20-23] a number of analytical results are presented. These results are the theoretical basis for effective numerical study of two-dimensional periodic structures. The model problems of electrodynamic theory of gratings are correctly stated. It has been solved the question of correctly and efficiently reducing of computational domain in the problems characterizing the space-time transformations of electromagnetic field by two-dimensional periodic structures. The important characteristics and properties of transient and steady state fields in the regular part of the rectangular cross-section Floquet channel have been specified and analyzed. The strict consequences of the Poynting complex power theorem (the energy balance relations) and the Lorentz lemma (the reciprocity relations) for two-dimensional periodic gratings of finite thickness excited by *TE*– or *TM*-waves have been originally presented. The method for transport operators (the space-time analogue of the generalized scattering matrix method) intended for significantly reducing the computational resources for calculation of multilayer periodic structures or the structures on thick substrates has been elaborated. A number of important problems of the spectral theory of two-dimensional periodic gratings have been solved. The obtained result is important for correct physical analysis of the processes of resonance scattering of pulsed and monochromatic waves.

*Development of**n**ew**analytical methods for solving actual problems of radiophysics (supervised by L. A. Pazynin)*

It has been suggested a strict model of a ring waveguide with constant cross-section and the alternating surface impedance distribution of one of its walls. For the class of the circular hodographs of the surface impedance an analytical solution of the excitation problem of the waveguide of this kind has been proposed. The obtained solution is used to simulation the effect of “the circle slipping”and finding out the causes of it. This effect is known from the observations on the long traces of VLF range in waveguide Earth-ionosphere [24]. Numerical experiments have been shown that this effect is the result of a linear “interaction” between two primary modes of the waveguide near their degeneracy regime. There is not connection with the diffraction transformation of the primary mode into higher order on spatial inhomogeneity of the waveguide wall, as suggested before. It has been specified the threshold behavior of this phenomenon and the necessary conditions for its appearance. A new method for the analysis of the transient electromagnetic field generated by a pulse linear current whose currier is located near the plane boundary between two dielectric media has been suggested. In contrast to the Cagniard-de Hoop method, which is widely used to study transient electromagnetic, acoustic and seismic waves, the new approach is performed on the basis of the transformation of the integral representation for the field in the space of two complex variables. This transformation allowed one to replace the construction of auxiliary the Cagniard contour by using the standard procedure of solving an algebraic equation. A new type of the field interpretation as an integral over the finite contour has been obtained. The method can be generalized to the case of multi-layer media and arbitrary sources [25].

The possibility of the radical distortion of the radar image of a perfectly conducting sphere by deposition of a special coating on the surface fabricated from an artificial material has been strictly proved. A radial distribution of the permittivity and the permeability of the coating has been found, wherein the scattered field everywhere outside of the object location is exactly the same with the scattered field of a perfectly conducting sphere of the previously specified arbitrary smaller radius [26]. The requirements for the material parameters of the distorting covering are much simpler than in the case of masking coating.

*Development and implementation of**new efficient experimental and theoretical approaches to solving the problems of analysis and synthesis of the elements and the components of the unique radiation diffraction antennas (supervised by A. P. Evdokymov)*

The results of the work in this trend are presented in the paper [27] and, in part, in two chapters of the monograph [28]. This paper works are devoted to the construction of antennas for aerospace systems basing. A plane array antenna with a combined technique of beam scanning and its focusing aimed at running in the multi-channel radiometer [29] is among the results in [27]. The bandwidth of the receiving signal is 10%, the carrier frequency is 100 GHz, the scan sector is and the number of formed beams is 32. This characteristics obtaining has been made due to the preliminary detailed study of the properties of the reflective gratings of different types.

In 2006, the broadband antenna with a large (up to 15 m) aperture for the radar of 3.1 cm of side-looking artificial Earth satellite with the operational switching swaths [30] was developed. The electrodynamic pattern of the antenna with wide abilities of swaths choice by the locator has been elaborated. There is no need to use and to expand the long length of waveguide paths within the limits of the pattern. The validity of made decisions and the possibility of their implementation are proved by the construction and the testing of the experimental model of the antenna in the millimeter wavelength range. The range of works on the study of the possibility the parabolic mirror with the aperture of 15 m and the width of 1.1 m on-orbit delivery have been done. The laying variant of the mirror under the blister of the rocket carrier “Cyclone” has been tested in the construction department “Yuzhnoye”. The realizability of the project has been clearly demonstrated, but the further works in this trend were stopped for lack of funding.

The series of experimental studies of new diffraction radiation antennas with electromechanical scanning for modern security systems and to avoid the vehicles incident have been performed. The removable removable reflective gratings placed into the field of the main wave of the electric type of ridged dielectric waveguide have been optimized; the medium-priced variants of antennas with good characteristics of the directional diagram and the sufficiently large sector of continuous or discrete scanning beam [31] have been suggested and tested.

The development of the diffraction radiation antenna with a large aperture for the airdrome locator is required the elaboration and implementation of new methods of synthesis of radiators with the amplitude and phase distribution of the current density of the field specified at all their length. This problem has been solved successfully [32]. The implementation of cosine-amplitude distribution at the low phase distortions via the length of linear feeding antenna (see Fig. 4) allowed constructing the antenna of aerodrome locator with law losses and low side-lobes level. The antenna has the following characteristics: the frequency is equal to 36 GHz, the aperture is equal to 2000×2000 мм^{2}, the width of the directional diagram in the azimuth and elevation planes is 0.27° and 0.24°, respectively, the gain equal to 45 dB.

The antennas of modern radar systems, especially with a large aperture, need high-quality waveguide rotative joints. The theoretical studies of the department are intended for elaboration the original broadband rotative joint with the wavelength equals to 8 mm [33]. The inclusion of additional choke grooves whose parameters were determined in numerical experiments made possible to conserve the allowable level of losses, even with significant increase in a radial gap. The long-term exploitation of the joint is guaranteed, all its major electromagnetic characteristics conserve at the same time.

The influence the main characteristics of the eigen modes of ridged and planar dielectric waveguides of closely spaced periodic scatterers, i.e., the reflective and printed-circuit gratings has been studied in detail in performing of fundamental research works on developing the grounds for the elaboration of modern high-tech antennas in millimeter and centimeter wavelength ranges. In particular, the experimental model of the planar antenna with the aperture of 1000×200 мм^{2} has been manufactured and studied on this basis [34]. The aerial tuning was performed to provide the cosine-square pedestal distribution of the field lengthwise of the aperture with the coincidence of –14 dB on its edges. The width of the directional diagram in diffraction radiation antennas is 2.95° and in horizontal plane is equal to 0.64°. The gain is equal to 41.4 dB at the level of the side-lobes, in the horizontal plane it is equal –31.4 dB. The possibility of the directional diagrams forming in the diffraction radiation antennas with the level of side-lobes below then -30 dB is proved experimentally. The functionability of the antenna remains in the frequency band (34 ±0.5) GHz when a minor angle-frequency depending of the beam position: times 1% of frequency changing.