Computation of propagation in adiabatically tapered dielectric structures based on eigenfunction expansions: application to (active) optical devices

An eigenfunction expansion method is presented which uses the complete set of Hermite-Gauss (HG) functions to obtain the required solution of the propagation problems and has certain advantages, as discussed. This method may also be considered as a perturbation method of analysis since the HG functions are the solutions of a longitudinally uniform waveguide with a parabolically varying transverse refractive index distribution. Note that the HG functions form a complete and discrete set for the function space of interest namely that corresponding to square integrable functions. As a proof of its effectiveness the HG function expansion method is applied to analyse the fields in a variety of longitudinally non-uniform passive devices. The extension of this approach to the to the analysis of active optical devices requires a self-consistent solution to be determined to take into account both the non-uniform device geometry and the non-linear interaction of the optical field with the inversion population distribution in the device. Further, compactness of the analysis scheme for the overall model is achieved by demonstrating that the HG method is also very effective in solving the carrier diffusion equation. In addition, the merits of the collocation numerical procedure have been utilised to reduce the complexity of the formalism.