SIMPLE FINITE-DIFFERENCE ALGORITHM FOR CALCULATING WAVE-GUIDE MODES

A simple algorithm for solving the finite-difference (FD) equations fo r the mode eigenvalues and field distributions of a linear waveguide i s presented. By applying the discretized Helmholtz operator column-by- column to an index distribution defined on an N x N grid, a matrix who se size is only a few times N x N is obtained. This yields a reduction in computation time and space compared with the other classical FD ap proaches which involve an N2 x N2 matrix. Our method is tested against problems for which the exact solutions are known, and we find a high degree of accuracy. Despite the existence of fast algorithms for the t reatment of the classical N2 x N2 matrix, our new algorithm presents s ome advantages over existing FD methods of comparable speed, including : the ability to find all the modes and associated field profiles, ver y high numerical stability, and no numerical approximations in the pro cedure. In addition, some general optimum expressions for the domain s ize and density of grid points which are consistent with the desired p recision are provided, and apply to any FD method including ours.