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

Abstract

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.

Electromagnetic Mode Matching in a Wilson Basis --- Optical Fiber Connections with a Gap

“…The spatial translation is denoted by index n and the modulation by index ℓ. We adopted the algorithm of Daubechies et al [5] for the construction of θ(x) and gave a streamlined summary in [11]. With the aid of the forward Fourier transformationw…”

“…This can be circumvented by expressing every modal expansion in terms of the same complete set of orthogonal basis functions, i.e., a universal interface. We have discussed such a universal interface in terms of Wilson basis functions in [10,11]. The Wilson basis functions are duo-localized, i.e., they decay exponentially both in the spatial and in the spectral domain [5,37].…”

“…The Wilson basis functions are duo-localized, i.e., they decay exponentially both in the spatial and in the spectral domain [5,37]. In the Wilson basis, high-frequency electromagnetic wavefields can be represented through a parsimonious multidimensional set of coefficients [1,11]. In [10,11], we introduced the Wilson basis universal interface approach, applied it to the electromagnetic scattering of incident modes at the end-face of a multi-mode fiber, and reciprocally, the coupling of light into a fiber.…”

“…In the Wilson basis, high-frequency electromagnetic wavefields can be represented through a parsimonious multidimensional set of coefficients [1,11]. In [10,11], we introduced the Wilson basis universal interface approach, applied it to the electromagnetic scattering of incident modes at the end-face of a multi-mode fiber, and reciprocally, the coupling of light into a fiber. The major advantage of our Wilson-basis approach from a computational mode-matching perspective, is that it becomes irrelevant whether the coefficients represent fields that are associated with a round fiber, a rectangular dielectric waveguide or any other physical medium for which a complete set of (modal) transverse electromagnetic field solutions can be computed.…”

“…The successive application of Love's equivalence principle and the propagation operator allows for the evaluation of the electromagnetic fields at integer multiples of the selected distance almost effortlessly [18]. It is convenient that the coupling matrix between Wilson basis functions and laterally displaced Wilson basis functions is sparse [11], so that any adjustment of the lateral alignment of any of the the physical media (for instance a fiber) with respect to the interface is easily evaluated in the Wilson basis prior to executing the mode-matching procedure.…”