A wavelet formulation of the finite-difference method: Full-vector analysis of optical waveguide junctions

We have developed an efficient, large-stencil finite-difference scheme of the time-dependent Maxwell's curl equations based on the wavelet-collocation formulation in the time-domain. The proposed scheme enables, for the first time within a limited computational resource, full-vector analysis of three-dimensional rib waveguides that are typically used in integrated planar optical devices. The formulation takes advantage of compactly-supported interpolating bases to expand and represent the electric and magnetic fields. Moreover, unlike the well-known beam propagation methods, the numerical scheme is based on the first-principle algorithm with no explicit approximation, and thus rigorous and versatile for various types of boundary conditions. We demonstrate the efficiency of the method by first analyzing a straight rib-waveguide and examining the convergence of the results. Then we investigate a Y-shaped junction structure that is electrically too large to analyze with the conventional finite-difference time-domain scheme.