Electromagnetic reflection–transmission problems in a Wilson basis: fiber-optic mode-matching to homogeneous media

The Wilson basis features strong localization in both the spatial and the spectral domain. This enables us to efficiently describe high-frequency wavefields through a parsimonious set of coefficients. By choosing a single universal basis to expand fields, one effectively detaches scattering problems from the specific design of optical waveguides and components that form an optical interface. Equipped with a sparse, diagonally-dominant translation operator, the Wilson basis functions are convenient building blocks to address scattering problems in a more general setting. The physical interface may be reconfigured, while preserving the computational effort of the initial expansion. In this paper, we demonstrate optical reflection–transmission problems for interfaces between optical fibers and homogeneous media. In particular, we treat the construction of one-way propagating electromagnetic fields in the Wilson basis that are generated by Wilson-basis discretized equivalent dipole-source distributions. The Green’s function spectral integrals benefit from the strong localization to achieve good convergence. The decomposition of a wavefield in one-way forward and backward propagating wavefields is the result of careful construction of equivalent sources, and is effectively a Wilson-basis discretized Poincaré–Steklov operator. The decomposition of each and every guided fiber mode to one-way forward and backward propagating fields in homogeneous space can be accomplished in such a way that the boundary conditions are satisfied. These one-way propagating fields subsequently serve as building blocks for the decomposition of arbitrary incident fields, so that the scattering problems are properly solved. The reflection due to a guided mode as excitation is the same as the reflection due to the specific excitation of the same but backward propagating mode up to the accuracy of the numerical quadratures. Upon illuminating the fiber through a complex-source beam wavefield for a number of lateral steps, the Wilson basis formulation immediately produces the corresponding change in the modal power distribution.