Modeling of defect modes in photonic crystals using the fictitious source superposition method

Abstract

We present an exact theory for modeling defect modes in two-dimensional photonic crystals having an infinite cladding. The method is based on three key concepts, namely, the use of fictitious sources to modify response fields that allow defects to be introduced, the representation of the defect mode field as a superposition of solutions of quasiperiodic field problems, and the simplification of the two-dimensional superposition to a more efficient, one-dimensional average using Bloch mode methods. We demonstrate the accuracy and efficiency of the method, comparing results obtained using alternative techniques, and then concentrate on its strengths, particularly in handling difficult problems, such as where a mode is highly extended near cutoff, that cannot be dealt with in other ways.

Modes of symmetric composite defects in two-dimensional photonic crystals

“…For a given defect refractive index n d the resonance frequencies are obtained using a full numerical calculation based on the FSS method [13][14][15] (continuous curves) and by solving Eq. (19) (dashed curves).…”

“…Accordingly they can be generalized to other models such as such as the tight binding model [12] which leads to matrices with similar profiles. To validate our results, we compare the prediction of the asymptotic techniques to a full numerical calculation based on the FSS method [13][14][15].…”

Exact modelling of generalised defect modes in photonic crystal structures

“…In this step, the BZ integration eliminates the fictitious source in all but the primary cylinder (j = 0), with the remaining source at r = r 0 = 0 available to modify the response field and so formulate the defect mode. The first extension we outline to the formulation of Wilcox et al [2] enables it to handle multiple defects. To do this, we write…”

“…In order to deal with this class of problems, we developed an exact theory [2], known as the fictitious source superposition (FSS) method, for computing defect modes in an infinite 2D lattice, applying it to the study of the long-wavelength behaviour of PC fibres (PCFs) [3], and demonstrating unambiguously that the fundamental mode was never cut off. While our original implementation [2] was a useful tool, it is nevertheless limited to handling only simple (single cylinder) defects.…”

“…While our original implementation [2] was a useful tool, it is nevertheless limited to handling only simple (single cylinder) defects. Accordingly, here, we outline a generalisation of the FSS method that can accommodate complex defects comprising the removal or modification of a multiplicity of cylinders (in either a single or multiple rows) of an infinite PC.…”