Abstract
We develop a numerical algorithm that computes the Green's function of Maxwell equation for a 2D finite-size photonic crystal, composed of rods of arbitrary shape. The method is based on the boundary integral equation, and a Nyström discretization is used for the numerical solution. To provide an exact solution that validates our code we derive multipole expansions for circular cylinders using our integral equation approach. The numerical method performs very well on the test case. We then apply it to crystals of arbitrary shape and discuss the convergence.
Highlights
A photonic crystal is a periodic dielectric structure that has the feature that there are prohibited frequencies for the propagation of electromagnetic waves inside
In [3] and [5] the method was used for transmission calculations. This was extended in [1] to construct the two-dimensional Greens function and local density of states (LDOS) for finite-sized two-dimensional photonic crystals composed of circular cylinders of infinite length
The method is used for deriving the multipole expansions for the case of circular cylinders
Summary
A photonic crystal is a periodic dielectric structure that has the feature that there are prohibited frequencies for the propagation of electromagnetic waves inside. For infinite structures, the LDOS vanishes inside a complete band gap Since their introduction [2], the study of photonic crystals has increased significantly in the past decade and many techniques have been used for the computation of the spectra ([3], [4]) and the analysis of the LDOS [1]. In [3] and [5] the method was used for transmission calculations This was extended in [1] to construct the two-dimensional Greens function and LDOS for finite-sized two-dimensional photonic crystals composed of circular cylinders of infinite length. Integral equation techniques were used for the study of photonic crystals [8] where a system of two integral equations with two unknown functions on the boundary of each rod was obtained by applying Green’s theorems to a periodic problem. In the conclusion, we summarize the results and discuss future applications
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