Application of a previously proposed matrix method (which can only be used for TE mode solutions) to studying the wave characteristics of TM modes is described. To derive the matrix equation for TM-mode solutions of slab waveguides, the gradient of a continuous field is defined as the sum of a continuous function and a stepwise function so that boundary conditions are satisfied. By expanding both the index profile and mode field distribution into a truncated Fourier series, a particular matrix equation for determining mode indices and mode field distributions is obtained. Such a matrix equation can be generalized asBX = 2X, where B is a constant matrix, is the propagation constant, and the vector accounts for the mode field distribution. Wave characteristics of TM modes of slab waveguides with arbitrary index profiles can thus be obtained following the solution of a linear algebra problem in a way similar to that for studying TE modes. The matrixBhere, however, as indicated by this work, is different from that for TE mode solutions. Also, it is pointed out in this paper that the matrix B for TM-mode solutions has a different form as a different kind of waveguide is considered. Numerical results presented herein show that a sufficiently high accuracy can be obtained by using the proposed method.
Relation:
IEEE Journal of Quantum Electronics 37(12): 1654-1660
