The Fourier series expansion method is a useful tool to approach the problems of discontinuities in optical waveguides, and it applies to analyze the Floquet-modes of photonic crystal waveguides. This paper shows that the Floquet-mode calculation with large truncation order is limited and explains the reason. Furthermore, two techniques of the formulation are presented to relieve this limitation. One of them is a use of the symmetric properties of the Floquet-modes, and another is a use of the Rayleigh quotients to improve accuracy of eigenvalue calculation. They are validated by numerical experiments.
2. Taflove, A., Computational Electrodynamics: The Finitedifference Time-domain Method, Artech House, Boston, London, 1995.
3. Benisty, H., "Modal analysis of optical guides with two-dimensional photonic band-gap boundaries," J. Appl. Phys., Vol. 79, No. 10, 7483-7492, 1996.
4. Tanaka, H., T. Yamasaki, and T. Hosono, "Propagation characteristics of dielectric waveguides with slanted grating structure," IEICE Trans. Electron., Vol. E77-C, No. 11, 1820-1827, 1994.
5. Jia, H. and K. Yasumoto, "Modal analysis of two-dimensional photonic-crystal waveguides formed by rectangular cylinders using an improved Fourier series method," IEEE Trans. Microwave Theory and Techniques, Vol. 54, No. 2, 564-571, 2006.
6. Hosono, T., T. Hinata, and A. Inoue, "Numerical analysis of the discontinuities in slab dielectric waveguides," Radio Sci., Vol. 17, No. 1, 75-83, 1982.
7. Yamakita, J., K. Matsumoto, and K. Rokushima, "Analysis of discontinuities in anisotropic dielectric waveguides," IEICE Technical Report, EMT-93-87, Japanese, 1993.
8. Yasumoto, K., T. Miyamoto, and M. Momoda, "Full-wave analysis of optical waveguides using periodic boundary conditions," SPIE, Vol. 3666, 170-176, 1999.
9. Miyamoto, T., M. Momoda, and K. Yasumoto, "Numerical analysis for 3-dimensional optical waveguides with periodic structures using Fourier series expansion method," IEICE Trans. Electron., Vol. J86-C, No. 6, 591-600, Japanese, 2003.
10. Yasumoto, K. and K. Watanabe, "Analysis of discontinuities in two-dimensional photonic crystal waveguides using Floquet modes concept," Int. J. Microwave and Opt. Technol., Vol. 3, No. 3, 397-403, 2008.
11. Zhang, D. and H. Jia, "Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer," IEICE Trans. Electron., Vol. E90-C, No. 3, 613-622, 2007.
12. Strang, G., Linear Algebra and Its Applications, 3rd edition, Harcourt, San Diego, 1988.