Application of finite element methods to photonic crystal modelling

Photonic crystals are materials with periodically modulated dielectric constant, through which certain frequencies of electromagnetic radiation cannot propagate; the luminary analogues of semiconductors. The modes admitted by photonic crystals can be investigated effectively using the finite element method with the assistance of the Bloch-Floquet theorem, by considering a unit cell of the material and imposing periodic boundary conditions. Along with the Dirichlet and metric matrices, a third type of elemental matrix emerges. The types of results that are of interest to photonic crystal manufacturers are introduced and presented; in this context, the benefits of using subspace iteration techniques to solve the eigensystems are discussed.

[1]  Sergio Pissanetzky,et al.  Sparse Matrix Technology , 1984 .

[2]  Jian-Ming Jin,et al.  The Finite Element Method in Electromagnetics , 1993 .

[3]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[4]  M. Petyt.  Large order structural eigenanalysis techniques: 1989, by N. S. Sehmi, Chichester: John Wiley & Sons Limited. Price £39·95; pp. 223. ISBN 0-745-804-802 , 1992 .

[5]  Experimental investigation of photonic crystal waveguide devices and line-defect waveguide bends , 2000 .

[6]  S. Pissanetzky.  Sparse Matrix Algebra , 1984 .

[7]  D. Larkman,et al.  Photonic crystals , 1999, International Conference on Transparent Optical Networks (Cat. No. 99EX350).

[8]  E. Yablonovitch,et al.  Inhibited spontaneous emission in solid-state physics and electronics. , 1987, Physical review letters.

[9]  Michel Piché,et al.  Photonic bandgaps in periodic dielectric structures , 1993 .

[10]  C Gough,et al.  Introduction to Solid State Physics (6th edn) , 1986 .

[11]  Villeneuve,et al.  Photonic band gaps in two-dimensional square and hexagonal lattices. , 1992, Physical review. B, Condensed matter.

[12]  C. Kittel.  Introduction to solid state physics , 1954 .

[13]  Alan Jennings,et al.  Matrix Computation for Engineers and Scientists , 1977 .

[14]  P. Kuchment,et al.  Asymptotic methods for thin high-contrast two-dimensional PBG materials , 1999 .

[15]  P. Kuchment.  Floquet Theory for Partial Differential Equations , 1993 .

[16]  Edward L. Wilson,et al.  Numerical methods in finite element analysis , 1976 .