Natalia Sidorchuk (Ryazantseva)
PhD.
Cooperative work:
Vladimir Yachin – problem formulation, technique development, analysis of results.
Natalia Sidorchuk (Ryazantseva) – derivation of analytic expressions, development of algorithms and computer programs, numerical investigations, analysis of results.
The technique based on rigorous integral equations of electromagnetics has been developed for calculation of scattering characteristics of doublyperiodic magnetodielectric structures. These structures uderly the artificial electromagnetic crystals (also known as photonic crystals) having commercial and military application in the realm of handheld wireless communications, global positioning systems (GPSantenas), etc.
The Galerkin technique is applied to reduce the initial equations to a set of secondorder differential ones with constant coefficients. The algorithm enables us to obtain the scattered field characteristics in a wide frequency range, including a resonance region. Computer program is written in FORTRAN language.


The figures represent the periodic structure example and calculation results for polarization characteristics of the reflected wave. The grating bounded by two uniform media consists of chiral elements. To perform the calculation these elements are composed of identical rectangular parallelepipeds, the periodic cell is partitioned into 11x11 segments of this kind. Dielectric permittivity of the grating elements is e =  9.5 + j1.19 (gold at wavelength l = 620 nm), dielectric permittivity of the substrate is e_{s} = 15.3 + j0.172 (silicon at the same wavelength). The periodic cell is a square and L_{x} = L_{y} = 2.46 mm, the grating elements thickness is h = 100 nm = 0.04L_{x}. At this rate, the frequency parameter, i.e. ratio L_{x} / l = 3.968. The polarization states of the firstorder diffracted waves are presented in the figures.


The figures show the reflection coefficient of the conducting screen with crossshaped openings versus the periodtowavelength ratio (at normal incidence of the wave). Two curves, namely, solid and dashed ones are obtained for the conductor with e = 10 + 424000000j (at wavelength l » 1.8cm) and the screen thickness h = 0.0003L_{x} when we take into account different numbers of spatial harmonics (the number is defined by expression (2N+1)^{2}); the curve marked with circles is calculated for the screen of gold with e = 157.92 + 21.414j (at wavelength l » 2mm) when the screen thickness is h = 0.15L_{x}. For reference, the dashed curve with long strokes calculated for the case of perfectly conducting screen, is presented (B.J. Rubin and H.L. Bertony. IEEE Trans. Antennas Propag. 1983, AP31, No. 6, pp. 829836).


With the technique presented above the problem solution can be obtained in the case when material parameters of the doublyperiodic structure vary smoothly over a periodic cell. For that we have to tend the number of segments of periodic cell partitioning to infinity and the segment sizes to zero at the same time. The procedure is carried out analytically and does not result in grid size reduction for the algorithm. The figures demonstrate results of reflection coefficient calculations in the case of the conducting screen with circular openings (at normal incidence of the wave). The openings are placed in points of oblique (60°) coordinate grids. The curves are calculated for the values of material parameters identical to that for the conducting screen with crossshaped openings and have the same marking. The curve for reference to perfectly conducting screen is taken from paper “C.C. Chen. IEEE Trans. Microwave Theory Tech. 1971, MTT19, No. 5, pp. 475481”.