Highly accurate modelling of generalised defect modes in photonic crystals using the fictitious source superposition method

We present an exact theory for modelling the modes of two-dimensional photonic crystals with a truly infinite cladding. It builds on our previous implementation of the fictitious source superposition method for simple point defects that is based on three key ideas: the use of fictitious sources to modify response fields so defects can be introduced; the representation of the defect mode as a superposition of solutions of quasiperiodic field problems; and the reduction of the two-dimentioanl superposition to a highly efficient, one-dimentional average. here we describe an elegant extension to model a compound, multiple scatterer, defect in a given layer, and then build on this to construct an arbitrary defect from multiple layers of compound defects. We demonstrate the method's accuracy and efficiency, and present two and three-dimentional results. We also demonstrate the strengths of the method for difficult problems, such as where a mode is hgihly extended near cutoff