Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction

The solution of the Helmholtz equation requires the application of an exponentiated square root operator to an initial field. This operation is greatly facilitated by the introduction of a representation in which the above-mentioned operator is diagonal. The Lanczos method permits this diagonalization to be performed in a low-dimensional space and the propagation to be carried out to arbitrary order. An iteration scheme to be carried out in conjunction with the Lanczos method is also described. This scheme permits, in principle, the bound mode of a monomode waveguide to be calculated to machine accuracy. Results for some well-known test examples of rib waveguides are presented.