Analytical Models of Stochastic Scattering Loss for TM and TE Modes in Dielectric Waveguides Exhibiting Exponential Surface Roughness, and a Validation Methodology in 3D FDTD

The dielectric slab waveguide (DSW) has been the subject of much analysis in the past decades, with many important applications in high-speed high-bandwidth electromagnetic propagation; however, nearly all analysis of the DSW's stochastic propagation loss $\alpha$ (dB/cm), associated with random surface roughness of its sidewalls, has revolved around the transverse-electric (TE) mode of operation.

This work derives analytical expressions for $\alpha$ in the transverse-magnetic (TM) mode, and correlates it against numerical experiments in the method of finite-difference time-domain (FDTD) in both two-dimensional (2D) and three-dimensional (3D) space. In this work, we introduce the novel \textit{singular effective impedance} (SEI) formulation for normalization of the TM mode $\alpha$ and verify its accuracy through FDTD which shows near 0\% mean-error in both TM and TE modes. We perform several stochastic numerical experiments in 3D FDTD, where the DSW's width is set to be much larger than its height, and show that $\alpha$ correlates quite well against the proposed 2D analytical model and 2D FDTD. Finally, analysis of FDTD-computed data reveals that existing analytical models of $\alpha$ may be missing higher-order interaction terms involving correlation length $L_c$ (m) of the surface roughness.