# Introduction to Yasuura’s Method of Modal Expansion with Application to Grating Problems

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## Abstract

In this chapter we introduce the theory of the Yasuura’s method based on modal expansion and explain the methods of numerical computation in detail for several grating problems. After a sample problem we discuss the methods for solving two types of problems that require additional knowledge and steps, that is, scattering by a dielectric cylinder and diffraction by a grating. Some numerical results are shown to give an evidence of an experimental rule for the number of linear equations in formulating the least-squares problem that determines the modal coefficients. After confirming the rule we show a couple of examples of practical interest, i.e., scattering by a relatively deep metal grating, plasmon surface waves on a metal grating placed in conical mounting, scattering by a metal surface modulated in two directions, and scattering by periodically located dielectric spheres. To provide supplementary explanations of particular problems, four appendices are given; H-wave scattering from a cylinder, the normal equation and related topics, conical diffraction by a dielectric grating, and comparison of modal functions and the algorithm of the smoothing procedures.

## Notes

### Acknowledgements

The authors thank Mr. BenWen Chen and Mr. Rui Gong, Centre for Optical and Electromagnetic Research, South China Academy of Advanced Optoelectronics, South China Normal University for preparing the figures in Sect. 8.3.1 including numerical computations.

One of the authors (A.M.) wish to express his thanks to Japan Society for Promotion of Science (JSPS) for partial support to the work in Sects. 8.3.2 and 8.3.5 under Grant Number JP15K06023 (KAKENHI).

Another one of the authors (Y.O.) is grateful to Prof. S. He, COER-SCNU, COER-ZJU, and JORSEP-KTH for his continuous help and encouragement.

## References

- 1.M. Bass (ed.),
*Handbook of Optics; Volume II — Devices, Measurements, and Properties*, 2nd edn. (McGraw-Hill, 1995)Google Scholar - 2.R.H.T. Bates, Analytic constraints on electromagnetic field computations. IEEE Trans. Microw. Theory Tech.
**MTT-23**(8), 605–623 (1975)Google Scholar - 3.R.H.T. Bates, J.R. James, I.N.L. Gallett, R.F. Millar, An overview of point matching. Radio Electron. Eng.
**43**(3), 193–200 (1973)CrossRefGoogle Scholar - 4.A. Boag, Y. Leviatan, A. Boag, Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current method. Radio Sci.
**23**(4), 612–624 (1988)ADSCrossRefGoogle Scholar - 5.G.P. Bryan-Brown, J.R. Sambles, M.C. Hutley, Polarization conversion through the excitation of surface plasmons on a metallic grating. J. Modern Opt.
**37**(7), 1227–1232 (1990)ADSCrossRefGoogle Scholar - 6.A.P. Calderón, The multipole expansion of radiation fields. J. Ration. Mech. Anal. (J. Math. Mech.)
**3**, 523–537 (1954)Google Scholar - 7.M. Cadilhac, R. Petit, On the diffraction problem in electromagnetic theory: a discussion based on concepts of functional analysis including an example of practical application, in
*Huygens’ Principle 1690–1990: Theory and Applications, Studies in Mathematical Physics*, ed. by H. Blok, et al. (Elsevier, Amsterdam, 1992)Google Scholar - 8.G. Hass, L. Hardley, Optical properties of metal, in
*American Institute of Physics Handbook*, ed. by D.E. Gray, 2nd ed. (McGraw-Hill, 1963), pp. 6–107Google Scholar - 9.J.P. Hugonin, R. Petit, M. Cadilhac, Plane-wave expansions used to describe the field diffracted by a grating. J. Opt. Soc. Am.
**71**(5), 593–598 (1981)ADSCrossRefGoogle Scholar - 10.H. Ikuno, K. Yasuura, Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method. Radio Sci.
**13**(6), 937–946 (1978)ADSCrossRefGoogle Scholar - 11.H. Ikuno, M. Gondoh, M. Nishimoto, Numerical analysis of electromagnetic wave scattering from an indented body of revolution. Trans. IEICE Electron.
**E74-C**(9), 2855–2863 (1991)Google Scholar - 12.T. Inagaki, J.P. Goudonnet, J.W. Little, E.T. Arakawa, Photoacoustic study of plasmon-resonance absorption in a bigrating. J. Opt. Soc. Am. B
**2**(3), 433–439 (1985)ADSCrossRefGoogle Scholar - 13.M. Kawano, H. Ikuno, M. Nishimoto, Numerical analysis of 3-D scattering problems using the Yasuura method. Trans. IEICE Electron.
**E79-C**(10), 1358–1363 (1996)Google Scholar - 14.A.N. Kolmogorov, S.V. Fomin,
*Elements of the Theory of Functions and Functional Analysis*(Dover, New York, 1999)Google Scholar - 15.C.L. Lawson, R.J. Hanson,
*Solving Least Squares Problems*(Prentice-Hall, New Jersey, 1974)Google Scholar - 16.T. Matsuda, D. Zhou, Y. Okuno, Numerical analysis of plasmon-resonance absorption in a bisinusoidal metal grating. J. Opt. Soc. Am. A
**19**(4), 695–701 (2002)Google Scholar - 17.A. Matsushima, Y. Momoka, M. Ohtsu, Y. Okuno, Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation. Progr. Electromagn. Res.
**69**, 305–322 (2007)CrossRefGoogle Scholar - 18.R.F. Millar, Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers. Radio Sci.
**8**(8–9), 785–796 (1973)ADSMathSciNetCrossRefGoogle Scholar - 19.H. Nakano, Frequency-independent antennas: spirals and log-periodics, in
*Modern Antenna Handbook*, ed. by C.A. Balanis (Wiley, New Jersey, 2008), pp. 263–323Google Scholar - 20.Y. Nakata, M. Koshiba, M. Suzuki, Finite-element analysis of plane wave diffraction from dielectric gratings. Trans. IEICE Jpn.
**J69-C**(12), 1503–1511 (1986)Google Scholar - 21.M. Neviér, The homogeneous problems, in
*Electromagnetic Theory of Gratings*, ed. by R. Petit (Springer, Berlin, 1980), pp. 123–157Google Scholar - 22.M. Ohtsu, Y. Okuno, A. Matsushima, T. Suyama, A Combination of up- and down-going Floquet modal functions used to describe the field inside grooves of a deep grating. Progr. Electromagn. Res.
**64**, 293–316 (2006)CrossRefGoogle Scholar - 23.Y. Okuno, A numerical method for solving edge-type scattering problems. Radio Sci.
**22**(6), 941–946 (1987)ADSCrossRefGoogle Scholar - 24.Y. Okuno, The mode-matching method, in
*Analysis Methods in Electromagnetic Wave Problems*, ed. by E. Yamashita (Artech House, 1990), pp. 107–138Google Scholar - 25.Y. Okuno, An introduction to the Yasuura method, in
*Analytical and Numerical Methods in Electromagnetic Wave Theory*, ed. by M. Hashimoto, M. Idemen, O.A. Tretyakov (Science House, 1993), pp. 515–565Google Scholar - 26.Y. Okuno, H. Ikuno, Completeness of the boundary values of equivalent sources. Mem. Fac. Eng. Kumamoto Univ.
**38**(1), 1–8 (1993)ADSGoogle Scholar - 27.Y. Okuno, H. Ikuno, Yasuura’s method, its relation to the fictitious source methods, and its advancements in the solution of 2D problems, in
*Generalized Multipole Techniques for Electromagnetic and Light Scattering*, ed. T. Wriedt (Elsevier, Amsterdam, 1999)Google Scholar - 28.Y. Okuno, T. Matsuda, T. Kuroki, Diffraction efficiency of a grating with deep grooves, in
*Proceedings of the 1995 Sino-Japanese Joint Meeting on Optical Fiber Science and Electromagnetic Theory (OFSET’95)*, vol. 1 (Tianjin, China, 1995), pp. 106–111Google Scholar - 29.Y. Okuno, T. Suyama, R. Hu, S. He, T. Matsuda, Excitation of surface plasmons on a metal grating and its application to an index sensor. Trans. IEICE Electron.
**E90-C**(7), 1507–1514 (2007)Google Scholar - 30.Y. Okuno, H. Yamaguchi, The idea of equivalent sources in the Yasuura method, in
*Proceedings 1992 International Symposium on Antennas Propagat (ISAP’92)*, vol. 1E3-2 (Sapporo, Japan, 1992)Google Scholar - 31.Y. Okuno, 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.
**30**(4), 580–587 (1982)ADSCrossRefzbMATHGoogle Scholar - 32.R. Petit (ed.),
*Electromagnetic Theory of Gratings*(Springer, Berlin, 1980)Google Scholar - 33.R. Petit, M. Cadilhac, Electromagnetic theory of gratings: some advances and some comments on the use of the operator formalism. J. Opt. Soc. Am. A
**7**(9), 1666–1674 (1990)ADSMathSciNetCrossRefGoogle Scholar - 34.H. Raether, Surafce plasmon and roughness, in
*Surface Polaritons — Electromagnetic Waves at Surfaces and Interfaces*, ed. by V.M. Agranovich, D.L. Mills (North Holland, 1982), pp. 331–403Google Scholar - 35.M. Tomita, K. Yasuura, The Rayleigh expansion theorem for the boundary value problem in two media. Kyushu Univ. Tech. Rep.
**52**(2), 142–154 (1979)Google Scholar - 36.J.R. Wait, Reflection from a wire grid parallel to a conducting plane. Can. J. Phys.
**32**, 571–579 (1954)ADSCrossRefzbMATHGoogle Scholar - 37.X. Xu, B.W. Chen, R. Gong, M. Zheng, Use of auxiliary source fields in Yasuura’s method, in
*Proceedings of the 2017 IEEE International Conference on Computational Electromagnetics (ICCEM2017)*, vol. 2C1.2 (Kumamoto, Japan, 2017)Google Scholar - 38.K. Yasuura, T. Itakura, Approximation method for wave functions (I). Kyushu Univ. Tech. Rep.
**38**(1), 72–77 (1965)Google Scholar - 39.K. Yasuura, T. Itakura, Complete set of wave functions – approximation method for wave functions (II). Kyushu Univ. Tech. Rep.
**38**(4), 378–385 (1966)Google Scholar - 40.K. Yasuura, T. Itakura, Approximation algorithm by complete set of wave functions – approximation method for wave functions (III). Kyushu Univ. Tech. Rep.
**39**(1), 51–56 (1966)Google Scholar - 41.K. Yasuura, H. Ikuno, Smoothing process on the mode-matching method for solving two-dimensional scattering problems. Mem. Fac. Eng. Kyushu Univ.
**37**(4), 175–192 (1977)Google Scholar - 42.K. Yasuura, Y. Okuno, Singular-smoothing procedure on Fourier analysis. Mem. Fac. Eng. Kyushu Univ.
**41**(2), 123–141 (1981)Google Scholar - 43.K. Yasuura, M. Tomita, Convergency of approximate wave functions on the boundary – the case of inner domain. Kyushu Univ. Tech. Rep.
**52**(1), 79–86 (1979)Google Scholar - 44.K. Yasuura, M. Tomita, Convergency of approximate wave functions on the boundary – the case of outer domain. Kyushu Univ. Tech. Rep.
**52**(1), 87–93 (1979)Google Scholar - 45.K. Yasuura, M. Tomita, Numerical analysis of plane wave scattering from dielectric cylinders. Trans. IECE Jpn.
**62-B**(2), 132–139 (1979)Google Scholar - 46.K.A. Zaki, A.R. Neureuther, Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization. IEEE Trans. Antennas Propagat.
**AP-19**(2), 208–214 (1971)Google Scholar