Computational methods for determining the complex propagation constants of leaky waveguide modes have become so powerful and so readily available that it is possible to use these methods with little understanding of what they are calculating. We compare different computational methods for calculating the propagation constants of the leaky modes, focusing on the relatively simple context of a W-type slab waveguide. In a lossless medium with infinite transverse extent, a direct determination of the leaky mode by using mode matching is compared with complete mode decomposition. The mode matching method is analogous to the multipole method in two dimensions. We then compare these results with a simple finite-difference scheme in a transverse region with absorbing boundaries that is analogous to finite-difference or finite-element methods in two dimensions. While the physical meaning of the leaky modes in these different solution methods is different, they all predict a nearly identical evolution for an initial, nearly confined mode profile over a limited spatial region and a limited distance. Finally, we demonstrate that a waveguide that uses bandgap confinement with a central defect produces analogous results.
