Exact Solution of Facet Reflections for Guided Modes in High-Refractive-Index-Contrast Sub-Wavelength Waveguide Via a Fourier Analysis and Perturbative Series Summation: Derivation and Applications

Facet reflections at the waveguide-air interface for strongly-guiding waveguides with sub-wavelength scale dimensions do not follow the usual Snell´s law. Significant amount of reflected power can be channeled into higher order modes as well as radiation modes. This paper shows for the first time how the exact analytical solution of the facet reflection can be obtained by using a new technique based on Fourier analysis and perturbative series summation without the need for approximation or iteration. The proposed analysis enables the distribution of power reflected into various guided and radiation modes to be readily computed. Through this technique, a spectral overlapping criterion and a coupling matrix are derived that analyze effectively the power distribution among all the strongly and weakly-coupled radiation modes in an end-facet reflection. Accurate pre-determination of the number of radiation modes for efficient computation without compromising resultant accuracy is achieved. More importantly, the anomalous wave reflection behaviors at the facet of a strongly-guiding waveguide are presented. These include anomalous high radiation modes coupling as a function of cladding refractive index not reported before. This paper further includes an exemplary illustration of the analysis based on a symmetric planar nano-waveguide with high refractive index contrast for both TE and TM polarization under fundamental incident mode.