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2006-08-29

A Combination of Up- and Down-Going Floquet Modal Functions Used to Describe the Field Inside Grooves of a Deep Grating

Michitoshi Ohtsu,

Dr. Michitoshi Ohtsu

Kumamoto University

Japan

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Yoichi Okuno,

Dr. Yoichi Okuno

Department of Electrical and Computer Engineering

Kumamoto University

Japan

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Akira Matsushima,

Prof. Akira Matsushima

Fukuoka Institute of Technology

Japan

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Taikei Suyama

Professor Taikei Suyama

Department of Electrical and Computer Engineering

Kumamoto University

Japan

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Progress In Electromagnetics Research, Vol. 64, 293-316, 2006

doi:10.2528/PIER06071401 | Google Scholar

Abstract

An effective computational method based on a conventional modal-expansion approach is presented for solving the problem of diffraction by a deep grating. The groove depth can be the same as or a little more than the grating period. The material can be a perfect conductor, a dielectric, or a metal. The method is based on Yasuura's modal expansion, which is known as a least-squares boundary residual method or a modified Rayleigh method. The feature of the present method is that: (1) The semi-infinite region U over the grating surface is divided into an upper half plane U₀ and a groove region UG by a fictitious boundary (a horizontal line); (2) The latter is further divided into shallow horizontal layers U₁, U₂, Â·Â·Â·, U_Q again by fictitious boundaries; (3) An approximate solution in U₀ is defined in a usual manner, i.e., a finite summation of up-going Floquet modal functions with unknown coefficients, while the solutions in U_q (q = 1, 2, Â·Â·Â·, Q) include not only the up-going but also the down-going modal functions; (4) If the grating is made of a dielectric or a metal, the semi-infinite region L below the surface is partitioned similarly into L₀, L₁, Â·Â·Â·, L_Q, and approximate solutions are defined in each region; (5) A huge-sized least squares problem that appears in finding the modal coefficients is solved by the QR decomposition accompanied by sequential accumulation. The method of solution for a grating made of a perfect conductor is described in the text. The method for dielectric gratings can be found in an appendix. Numerical examples include the results for perfectly conducting and dielectric gratings.

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Michitoshi Ohtsu, Yoichi Okuno, Akira Matsushima, and Taikei Suyama, "A Combination of Up- and Down-Going Floquet Modal Functions Used to Describe the Field Inside Grooves of a Deep Grating," Progress In Electromagnetics Research, Vol. 64, 293-316, 2006.
doi:10.2528/PIER06071401

References

1. Yasuura, K. and T. Itakura, "Approximation method for wave functions," Kyushu Univ. Tech. Rep., Vol. 38, No. 1, 72-77. Google Scholar

2. Yasuura, K. and T. Itakura, "Complete set of wave functions," Kyushu Univ. Tech. Rep., Vol. 38, No. 4, 378-385, 1966. Google Scholar

3. Yasuura, K. and T. Itakura, "Approximate algorism by complete set of wave functions," Kyushu Univ. Tech. Rep., Vol. 39, No. 1, 51-56, 1966. Google Scholar

4. Yasuura, K., "A view of numerical methods in diffraction problems," Progress in Radio Science, 257-270, 1971. Google Scholar

5. Davies, J. B., "A least-squares boundary residual method for the numerical solution of scattering problems," IEEE Trans., Vol. MTT-21, No. 2, 99-104, 1973. Google Scholar

6. Van den Berg, P. M., "Reflection by a grating: Rayleigh methods," J. Opt. Soc. Am., Vol. 71, No. 10, 1224-1229, 1981. Google Scholar

7. Hugonin, J. P., R. Petit, and M. Cadilhac, "Plane-wave expansions used to describe the field diffracted by a grating," J. Opt. Soc. Am., Vol. 71, No. 5, 593-598, 1981. Google Scholar

8. Ikuno, H. and K. Yasuura, "Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method," Radio Sci., Vol. 13, 937-946, 1978. Google Scholar

9. Okuno, Y. and K. Yasuura, "Numerical algorithm based on the mode-matching method with a singular-smoothing procedure for analysing edge-type scattering problems," IEEE Trans. Antennas Propagat., Vol. AP-30, 580-587, 1982.
doi:10.1109/TAP.1982.1142858 Google Scholar

10. Zaki, K. A. and A. R. Neureuther, "Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization," IEEE Trans. Antennas Propagat., Vol. AP-19, No. 2, 208-214, 1971.
doi:10.1109/TAP.1971.1139908 Google Scholar

11. Zaki, K. A. and A. R. Neureuther, "Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization," IEEE Trans. Antennas Propagat., Vol. AP-19, No. 6, 747-751, 1971.
doi:10.1109/TAP.1971.1140031 Google Scholar

12. Moharam, M. G. and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am., Vol. 72, No. 10, 1385-1392, 1982. Google Scholar

13. Lippmann, B. A., "Note on the theory of gratings," J. Opt. Soc. Am., Vol. 43, No. 5, 1953. Google Scholar

14. Millar, R. F., "The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers," Radio Sci., Vol. 8, No. 9, 785-796, 1973. Google Scholar

15. Chandezon, J., M. T. Dupuis, G. Cornet, and D. Maystre, "Multicoated gratings: a diffraction formalism applicable in the entire optical region," J. Opt. Soc. Am., Vol. 72, No. 7, 839-846, 1982. Google Scholar

16. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of planar grating diffraction," J. Opt. Soc. Am., Vol. 71, No. 7, 811-818, 1981. Google Scholar

17. Matsuda, T., D. Zhou, and Y. Okuno, "Numerical analysis of plasmon-resonance absorption in bisinusoidal metal gratings," J. Opt. Soc. Am. A, Vol. 19, No. 4, 695-699, 2002. Google Scholar

18. Matsuda, T. and Y. Okuno, "A numerical analysis of plane-wave diffraction from a multilayer-overcoated grating," IEICE, Vol. J76- C-I, No. 6, 206-214, 1993. Google Scholar

19. Lawson, C. L. and R. J. Hanson, Solving Least-Squares Problems, Prentice-Hall, 1974.