Application of the Fourier-grid method to guided-wave problems

A method recently developed for solving the Schrodinger equation is applied to dielectric waveguides. The technique, which is extremely simple to implement, involves representing the differential operator in the scalar Helmholtz equation on a grid of discrete points in coordinate space, and then diagonalizing the resulting matrix to reveal the propagation constants and field patterns of the guided modes. The square of the transverse index profile is specified directly as a diagonal matrix in coordinate space, while the matrix for the transverse Laplacian is obtained through the Fourier relationship between its diagonal form in momentum space and the equivalent representation in coordinate space. The accuracy and computational performance of this procedure is assessed for one- and two-dimensional transverse profiles. Modal refractive indices and fields computed by the grid method are found to agree well with those derived by means of other techniques