An efficient iterative solver for the modes in a dielectric waveguide

Two Krylov subspace based methods were used to solve the sparse matrix generated by the finite difference formulation. The use of the bi-Lanczos algorithm allows this method to be computationally competitive with other approximate methods while the use of the finite difference formulation makes this method versatile enough to handle complicated waveguide structures. The BiCG based algorithm reduces the storage requirements to O(N) and thus can handle problems with several hundred thousand calculations. The speed of this algorithm can be further enhanced hy using a non-uniform grid in the solution space outside the dieletric waveguide. By using a non-uniform grid, the size of the artificial boundary can be greatly increased which limits the error in the results near cutoff frequencies.