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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
By
Progress In Electromagnetics Research, Vol. 64, 293-316, 2006
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, ···, UQ 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 Uq (q = 1, 2, ···, 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, ···, LQ, 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.
Citation
Michitoshi Ohtsu Yoichi Okuno Akira Matsushima 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
http://www.jpier.org/PIER/pier.php?paper=06071401
References

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

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

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.

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

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.

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

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.

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.

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

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

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

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.

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

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.

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.

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.

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.

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.

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