The Lanczos recursion algorithm is used to determine forward-propagating solutions for both the paraxial and Helmholtz wave equations for longitudinally invariant refractive indices. By eigenvalue analysis it is demonstrated that the method gives extremely accurate solutions to both equations.

We show that the coupling efficiency from a laser diode (LD) to an optical fiber through a ball lens can be calculated accurately using the exact solution to Maxwell's equations for the scattering of a beam from a dielectric sphere. Our calculated results agree closely with coupling measurements from an asymmetric LD for two different ball lenses.

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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 abovementioned operator is diagonal. The Lanczos method permits this diagonalization to be performed in a lowdimensional 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.