Wilson basis expansions of electromagnetic wavefields: a suitable framework for fiber optics

To detach large vectorial full-wave electromagnetic scattering problems from the specific, dissimilar bases that are typically associated with the optical waveguiding structures comprising an optical interface, we propose and demonstrate the expansion of wavefields in a single universal Wilson basis. We construct a Wilson basis by following the derivation in the paper by Daubechies, Jaffard and Journé. This basis features exponentially decaying basis functions in both the spatial and spectral domain. The strong localization in phase-space, with basis functions that strongly resemble wavefields, allow for efficient expansions of high-frequency electromagnetic fields. In a Wilson basis, the interface scattering problem is effectively separated from the physical configuration. For the evaluation of multiple, laterally displaced interface configurations, one may reuse electromagnetic fields in the Wilson basis, because the translation operator is sparse and diagonally dominant. We consider actual reflection-transmission problems comprising optical fibers and homogeneous media in the Wilson basis framework in a companion paper. There, the localization in the spectral domain aids the conversion of the numerical scheme to generate electromagnetic fields in homogeneous media due to Wilson-basis discretized electromagnetic sources. In this paper, we review the Wilson basis construction, demonstrate the expansion of modal electromagnetic fields in an optical fiber, and complex-source beams that are tilted with respect to the optical axis. Even for largely tilted beams (up to 60°), despite being highly oscillatory in the cross-sectional plane, the fields are well represented by a finite number of higher-order Wilson basis functions.