Light propagation in finite and infinite photonic crystals: The recursive Green’s function technique

Abstract

We report a computational method based on the recursive Green's function technique for calculation of light propagation in photonic crystal structures. The advantage of this method in comparison to the conventional finite-difference time domain (FDTD) technique is that it computes Green's function of the photonic structure recursively by adding slice by slice on the basis of Dyson's equation. This eliminates the need for storage of the wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances the computational speed. The second advantage of this method is that it can easily account for the infinite extension of the structure both into air and into the space occupied by the photonic crystal by making use of the so-called ``surface Green's functions.'' This eliminates the spurious solutions (often present in the conventional FDTD methods) related to, e.g., waves reflected from the boundaries defining the computational domain. The developed method has been applied to study scattering and propagation of the electromagnetic waves in the photonic band-gap structures including cavities and waveguides. Particular attention has been paid to surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the surface and very high $Q$ factor of the surface state. This effect can be employed as an operational principle for surface-mode lasers and sensors.