volume | PIER Journals

Abstract

For dielectric periodic gratings, we propose the combination of a spectral-domain volume integral equation and Fourier factorization rules to address the Gibbs phenomenon caused by jumps in both the fields and the permittivity. From a theoretical point of view we discuss two ways to overcome the computational complexity caused by the inverse rule by changing the fundamental unknowns of the underlying electromagnetic problem. The resulting numerical system is solved iteratively and the corresponding matrix-vector product has an O(NMlogM) complexity, where M is the number of Fourier modes and N is the number of sample points in the longitudinal direction.

1. Lalanne, P. and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am A, Vol. 13, No. 4, 779-784, 1996.

2. Li, , L., "Use of Fourier series in the analysis of discontinuous periodic structures ," J. Opt. Soc. Am. A, Vol. 13, No. 9, 1870-1876, 1996.

3. Li, , L., "New formulation of the Fourier modal method for crossed surface-relief gratings ," J. Opt. Soc. Am. A, Vol. 14, No. 10, 2758-2767, 1997.

4. Popov, , E. and M. Neviere, "Maxwell equations in Fourier space: Fast-converging formulation for di®raction by arbitrary shaped, periodic, anisotropic media ," J. Opt. Soc. Am. A, Vol. 18, No. 11, 2886-28894, 2001.

5. Schuster, T., J. Ruoff, N. Kerwien, S. Ra°er, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings ," J. Opt. Soc. Am. A, Vol. 24, No. 9, 2880-2890, 2007.

6. Yang, , H.-Y., D., R. Diaz, and N. G. Alexopoulos, "Reflection and transmission of waves from multilayer structures with planar implanted periodic material blocks ," J. Opt. Soc. Am. B, Vol. 14, No. 10, 2513-2521, Oct. 1997.

7. Shi, , Y. and C. H. Chan, "Multilevel Green's function interpolation method for analysis of 3-D frequency selective structures using volume/surface integral equation," J. Opt. Soc. Am. A, Vol. 27, No. 2, 308-318, 2010.

8. Van Beurden, , M. C., B. P. de Hon, and , "Electromagnetic modelling of antennas mounted on a band-gap slab | discretisation issues and domain and boundary integral equations ," Proc. of the Int. Conf. on Electromagnetics in Advanced Applications ICEAA'03, 637-640, 2003.

9. Chang, , Y.-C., G. Li, H. Chu, and J. Opsal, "Eff cient ¯nite-element, Green's function approach for critical-dimension metrology of three-dimensional gratings on multilayer ¯lms," J. Opt. Soc. Am. A, Vol. 23, No. 3, 638-645, 2006.

10. Magath, , T., "Coupled integral equations for diffraction by pro¯led, anisotropic, periodic structures ," IEEE Trans. Antennas Propag., Vol. 54, No. 2, 681-686, 2006.

11. Van Beurden, , M. C., "Fast convergence with spectral volume integral equation for crossed block-shaped gratings with improved material interface conditions ," J. Opt. Soc. Am. A, Vol. 28, No. 11, 2269-2278, 2011.

12. Kailath, T., I. Koltracht, and , "Matrices with block Toeplitz inverses," Linear Algebra Appl., Vol. 75, 145-153, 1986.

13. GÄotz, , P., T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express, Vol. 16, No. 22, 17295-17301, 2008.

14. Gohberg, , I., I. Koltracht, and , "Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation," Numer. Math., Vol. 47, 237-288, 1985.

15. Magath, T. and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A, Vol. 22, No. 11, 2405-2418, 2005.

16. Sleijpen, , G. L. G. and aD. R. Fokkema, "BiCGstab for linear equations involving unsymmetric matrices with complex spectrum," Electron. Trans. Numer. Anal., Vol. 1, 11-32, 1993.

17. Johnson, S. G. and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett., Vol. 77, No. 22, 3490-3492, 2000.

Prof. Martijn Constant van Beurden