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.

Figure 1

Grid of real-space sampling points generated for an array of cylindrical air holes of radius $0.45\Lambda$ in silica as described in Sec. 4, calculated here with FFT grid size $128\times 128$. To show the grid clearly, every fourth point is plotted to obtain a grid of dimension $32\times 32$. The unit cell is marked.

Figure 2

Smoothed dielectric function of the test system described in Sec. 4 plotted in a direction normal to the dielectric interface; the radius $r$ is measured from the center of the hole. Exact Gaussian smoothing of comparable width (FWHM $2.8\times {10}^{-3}\Lambda$) is shown.

Figure 3

Convergence of the eight highest-$\beta$ modes of the test system of Sec. 4 as FFT grid size is increased. In each plot the solid horizontal line is the result from a KKR-based method correct to 4 d.p. (Refs. 33, 34), the dashed lines show convergence of the conventional method, and the mixed dot/dash lines show the GCC results. The Gaussian smoothing widths for the conventional results are (in order of increasing dash length) ${10}^{-3}\Lambda$, $6\times {10}^{-3}\Lambda$, and ${10}^{-2}\Lambda$. The approximate equivalent widths for the GCC curves are (again in order of increasing dash length) ${10}^{-3}\Lambda$ and $2.5\times {10}^{-3}\Lambda$. Inset figures show Poynting vector magnitude (large in black regions). The modes at $\beta \Lambda =12.2654$, 11.0734, 9.0329, 8.0086, and 7.7931 are degenerate pairs; only one of each pair is shown.