Computational methods for finite thickness photonic crystals

Abstract

We develop a computational algorithm for finite thickness photonic crystals and apply it to photonic crystal slabs and artificial opals. The algorithm is not limited to solving only photonic crystals but can be applied to any electromagnetic problem, which is periodic in a plane and has a finite thickness in the perpendicular direction. An application in cylindrical coordinates is also presented. The method is based on the diagonalized form of Maxwell's equations, in which one spatial direction is distinguished and isolated. We show that this formulation is especially suitable for problems, which are periodic along two and non-periodic along the third axis. The fields are expanded in a combination of several bases, planewaves in the transversal plane, finite differences and eigenvectors in the perpendicular direction. The method is applied to both eigenmode and excitation problems. Furthermore, we develop an efficient scheme for computing the reflection of arbitrarily shaped and polarised beams from the surface of a periodic media. A great emphasis is laid on accuracy and efficiency. The resulting equations are solved using iterative techniques together with problem adapted, operator level preconditioners. Finally, we compare the computations to measurement results obtained from artificial opals.