Adaptive curvilinear coordinates in a plane-wave solution of Maxwell's equations in photonic crystals

A method is described to compute the modes propagating at a given frequency in dielectric systems that are periodic in two dimensions and uniform in the third dimension, using a plane-wave basis expressed in a system of generalized curvilinear coordinates. The coordinates are adapted to the structure under consideration by increasing the effective plane-wave cutoff in the vicinity of the interfaces between dielectrics, where the electromagnetic fields vary most rapidly. The favorable efficiency and convergence properties of the method are shown by comparison with the conventional plane-wave formulation of Maxwell’s equations. Although the method is developed to study propagation in photonic crystal fibers, it is also applicable more generally to plane-wave modal solutions of structured dielectrics.