DOI: https://doi.org/10.15407/rpra26.03.239

M. E. Kaliberda, L. M. Lytvynenko, S. A. Pogarsky


Purpose: Problem of the H-polarized plane wave diffraction by the structure, which consists of two semi-infinite strip gratings, is considered. The gratings are placed in the same plane. The gap between the gratings is arbitrary. The purpose of the paper is to develop the operator method to the structures, which scattered fields have both discrete and continuous spatial spectra.

Design/methodology/approach: In the spectral domain, in the domain of the Fourier transform, the scattered field is expressed in terms of the unknown Fourier amplitude. The field reflected by the considered structure is represented as a sum of two fields of currents on the strips of semi-infinite gratings. The operator equations are obtained for the Fourier amplitudes. These equations use the operators of reflection of semi-infinite gratings, which are supposed to be known. The field scattered by a semi-infinite grating can be represented as a sum of plane and cylindrical waves. The reflection operator of a semi-infinite grating has singularities at the points, which correspond to the propagation constants of plane waves. Consequently, the unknown Fourier amplitudes of the fi eld scattered by the considered structure also have singularities. To eliminate these latter, the regularization procedure has been carried out. As a result of this procedure, the operator equations are reduced to the system of integral equations containing the integrals, which should be understood as the Cauchy principal value and Hadamar finite part integrals. The discretization has been carried out. As a result, the system of linear equations is obtained, which is solved with the use of the iterative procedure.

Findings: The operator equations with respect to the Fourier amplitudes of the field scattered by the structure, which consists of two semi-infinite gratings, are obtained. The computational investigation of convergence has been made. The near and far scattered fields are investigated for different values of the grating parameters.

Conclusions: The effective algorithm to study the fields scattered by the strip grating, which has both discrete and continuous spatial spectra, is proposed. The developed approach can be an effective instrument in solving a series of problems of antennas and microwave electronics.

Key words: semi-infinite grating, operator method, singular integral, hypersingular integral, regularization procedure

Manuscript submitted 23.06.2021

Radio phys. radio astron. 2021, 26(3): 239-249


1. SHESTOPALOV, V. P., LYTVYNENKO, L. M., MASALOV, S. A. and SOLOGUB, V. G., 1973. Wave diffraction by gratings. Kharkiv, Ukraine: Kharkiv State University Press. (in Russian).

2. MATSUSHIMA, A., NAKAMURA, Y. and TOMINO, S., 2005. Application of Integral Equation Method to Metal-plate Lens Structures. Prog. Electromagn. Res. vol. 54, pp. 245‒262. DOI: https://doi.org/10.2528/PIER05011401

3. MUNK, B. A., 2000. Frequency Selective Surfaces: Theory and Design. New York: John Wiley & Sons, Inc. DOI: https://doi.org/10.1002/0471723770

4. FEL’D, Y. N., 1958. Electromagnetic Wave Diffraction by Semi-infinite Grating. Radiotekhnika i Elektronoka. vol. 13, no. 7, pp. 882‒889. (in Russian).

5. FEL’D, Y. N., 1955. On infinite systems of linear algebraic equations connected with problems on semi-infinite periodic structures. Doklady AN USSR. vol. 102, no. 2, pp. 257–260. (in Russian).

6. HILLS, N. L. and KARP, S. N., 1965. Semi-infinite Diffraction Gratings–I. Commun. Pure Appl. Math. vol. 18, is. 1-2, pp. 203‒233. DOI: https://doi.org/10.1002/cpa.3160180119

7. HILLS, N. L., 1965. Semi-Infinite Diffraction Gratings. II. Inward Resonance. Commun. Pure Appl. Math. vol. 18, is. 3, pp. 389‒395. DOI: https://doi.org/10.1002/cpa.3160180302

8. WASYLKIWSKYJ, W., 1973. Mutual coupling effects in semi-infinite arrays. IEEE Trans. Antennas Propag. vol. 21, is. 3, pp. 277‒285. DOI: https://doi.org/10.1109/TAP.1973.1140507

9. NISHIMOTO, M. and IKUNO, H., 2001. Numerical analysis of plane wave diffraction by a semi-infinite grating. IEEJ Trans. Fundam. Mater. vol. 121, is. 10, pp. 905‒910. DOI: https://doi.org/10.1541/ieejfms1990.121.10_905

10. CAPOLINO, F. and ALBANI, M., 2009. Truncation Effects in a Semi-infinite Periodic Array of Thin Strips: A Discrete Wiener-Hopf Formulation. Radio Sci. vol. 44, is. 2, id RS2S91. DOI: https://doi.org/10.1029/2007RS003821

11. NISHIMOTO, M. and IKUNO, H., 1999. Analysis of Electromagnetic Wave Diffraction by a Semi-infinite Strip Grating and Evaluation of End-Effects. Prog. Electromagn. Res. vol. 23, pp. 39‒58. DOI: https://doi.org/10.2528/PIER98101602

12. KALIBERDA, M., LYTVYNENKO, L. and POGARSKY, S., 2017. Method of singular integral equations in diffraction by semi-infinite grating: H-polarization case. Turk. J. Elec. Eng. Comp. Sci. vol. 25, no. 6, pp. 4496‒4509. DOI: https://doi.org/10.3906/elk-1703-170

13. KALIBERDA, M. E., LYTVYNENKO, L. N and, POGARSKY, S. A., 2018. Singular integral equations in diffraction problem by an infinite periodic strip grating with one strip removed: E–polarization case. J. Electromagn. Waves Appl. vol. 32, is. 3, pp. 332‒346. DOI: https://doi.org/10.1080/09205071.2017.1383943

14. KALIBERDA, M. E., LYTVYNENKO, L. N. and POGARSKY, S. A., 2019. Electromagnetic interaction of two semi-infinite coplanar gratings of flat PEC strips with arbitrary gap between them. J. Electromagn. Waves Appl. vol. 33, is. 12, pp. 1557‒1573. DOI: https://doi.org/10.1080/09205071.2019.1615996

15. LYTVYNENKO, L. M., REZNIK, I.I.and LYTVYNENKO, D. L., 1991. Wave scattering by semi-infinite periodic structure. Doklady AN Ukr. SSR. no. 6, pp. 62–67. (in Russian).

16. KALIBERDA, M. E., LITVINENKO, L. N. and POGARSKII, S. A., 2009. Operator Method in the Analysis of Electromagnetic Wave Diffraction by Planar Screens. J. Commun. Technol. Electron. vol. 54, no. 9, pp. 975‒981. DOI: https://doi.org/10.1134/S1064226909090010

17. VOROBIOV, S. N., LITVINENKO, L. N. and PROSVIRNIN, S. L., 1996. Electromagnetic Wave Diffraction by Finite Extent Structure Consisting of Nonequidistant Strips Having Different Width. Comparison of Full-wave Spectral and Operator Method. Radio Phys. Radio Astron. vol. 1, no.1, pp. 110‒118. (in Russian).

18. VOROBYOV, S. N. and LYTVYNENKO, L. M., 2011. Electromagnetic Wave Diffraction by Semi-Infinite Strip Grating. IEEE Trans. Antennas Propag. vol. 59, is. 6, pp. 2169‒2177. DOI: https://doi.org/10.1109/TAP.2011.2143655

19. LYTVYNENKO, L. M., KALIBERDA, M. E. and POGARSKY, S. A., 2013. Wave Diffraction by Semi-Infinite Venetian Blind Type Grating. IEEE Trans. Antennas Propag. vol. 61, is. 12, pp. 6120‒6127. DOI: https://doi.org/10.1109/TAP.2013.2281510

20. FELSEN, L. B. and MARCUVITZ, N., 1973. Radiation and Scattering of Waves. Upper Saddle River, NJ, USA: Prentice-Hall.


semi-infinite grating; operator method; singular integral; hypersingular integral; regularization procedure

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