Solution of the Scalar Wave Equation for Arbitrarily Shaped Waveguides by Two Dimensional Fourier Analysis.

A new method for calculating the modes of arbitrarily shaped dielectric waveguides is presented. It consists of expanding the field in a two dimensional Fourier series. The expansion is used to convert the scalar wave equation into a matrix eigenvalue equation. To facilitate calculation of the matrix elements, the waveguide geometry is approximated by a number of rectangles of constant refractive index. The accuracy of the method is demonstrated by calculating of the mode of a circular optical fiber and comparing it with the exact solution. The method fitted well experimental data on dielectric film waveguides on silicon, including data on waveguide to fiber butt-coupling loss, waveguide far field angles and the coupling length of directional couplers.