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

On the basis of two optical fibers with an optional lateral and longitudinal displacement in a homogeneous background medium, we describe a general full-vectorial Wilson-basis discretized mode-matching method that evaluates the converged electromagnetic fields after all resonances in the possible gap cavity have settled. Wilson basis functions feature strong localization in both the spatial and the spectral domain, which allows for efficient modal electromagnetic field expansions, adequate truncation of field propagation operators, and sparse translation operators, which in turn allow to make ad hoc electromagnetic field-matching operators. For physical contact connections between single-mode fibers with a mode-field diameter mismatch, we obtain attenuation curves that are right between those obtained from approximation methods that either effectively match the electric field or the magnetic field under the assumption of a vanishing reflection. For fibers separated by a growing gap, constructive and destructive interference patterns in the cavity are computed by the successive application of Love's equivalence principle and the propagation operator. By leveraging the physical width of the Wilson basis functions and the stepsize of the propagation operator, the initial operators may be reused in solving the interface problem for other wavelengths. For multi-mode fiber connections, we provide attenuation curves on a modal electromagnetic field level, as well as for overfilled and core-confined target encircled-flux compliant launches. A comparison to geometrical-optics based approaches shows attenuation differences in the order of several hundredth of a dB, and although that is small, it is significant for modern connection attenuation specifications. In the final example of connections between regular and trench-assisted multi-mode fibers, we notice that the relative change in the cumulative near-field power distribution can be significant, despite a marginally small attenuation. The influence of a core diameter and/or numerical aperture mismatch can be examined with a deliberate lateral misalignment.