Gaussian-windowed frame based method of moments formulation of surface-integral-equation for extended apertures

Abstract

Scattering or coupling of electromagnetic beam-field at a surface discontinuity separating two homogeneous or inhomogeneous media with different propagation characteristics is formulated using surface integral equation, which are solved by the Method of Moments with the aid of the Gabor-based Gaussian window frame set of basis and testing functions. The application of the Gaussian window frame provides (i) a mathematically exact and robust tool for spatial-spectral phase-space formulation and analysis of the problem; (ii) a system of linear equations in a transmission-line like form relating mode-like wave objects of one medium with mode-like wave objects of the second medium; (iii) furthermore, an appropriate setting of the frame parameters yields mode-like wave objects that blend plane wave properties (as if solving in the spectral domain) with Green's function properties (as if solving in the spatial domain); and (iv) a representation of the scattered field with Gaussian-beam propagators that may be used in many large (in terms of wavelengths) systems. A. Shlivinski et al. / Journal of Computational Physics 00 (2015) 1-19 2cavities [8][9][10][11][12][13][14][15][16][17][18][19][20]. Both of these applications involves a rich variety of physical phenomena, which should be properly described in computational modeling.An infinite fiber bundle is a periodic structure that can be analyzed within the Floquet-Bloch framework by, for example, an efficient rigorous coupled wave analysis (RCWA), see in [21][22][23][24]. The fiber bundle can be viewed as a volume grating with a periodically discontinuous dielectric constant (see, e.g., [25][26][27][28][29]). However, the perfectly periodic case does not account for the finite structure size and does not naturally lead to extensions involving nonperiodic fiber bundles.This paper concerns with the problem of an electromagnetic beam coupling into a finite fiber bundle, which is a finite array of adjacent parallel dielectric fibers (that can be termed "unit-cells") that are fused together and can be periodic or non-periodic (see in Fig. 1). Due to the finite size and possibly non-periodicity the Floquet-Bloch framework does not directly apply to this case. Instead, we treat the whole structure as an electromagnetic scattering problem with PMCHWT-type (Poggio, Miller, Chew, Harrington, Wu and Tsai) surface integral equation without direct recourse to the periodicity [30][31][32][33]. We employ a spectral-spatial (phase-space) analysis and synthesis within a method of moment (MoM) Gabor-based Gaussian window frame formulation. In the MoM Gabor-based Gaussian window frame formulation the rigorous frame theory [34,35] was used with Gaussian window functions to establish an overcomplete Gabor frame set to be applied in a computational electromagnetics wave field framework. The Gabor frames for electromagnetic wave field expansions was used in, e.g., [36][37][38][39][40][41][42][43] to provide more numerical robustness and stability as compared to the complete Gabor expansion with...