It is well understood that transverse magnetic (TM) polarized optical beams could not be considered by the FFT-BPM because of the presence of the mixed derivatives of the magnetic field and refractive index in the relevant wave equation. Such mixed derivatives could be resolved by transforming the TM problem to an equivalent TE one via an ”equivalent refractive index” formalism. Unfortunately, if the refractive index of the original TM problem has step-like discontinuities along the optical confinement direction (transverse to the propagation direction), the ”equivalent index” in the transformed TE problem will exhibit a Dirac-delta distribution at the planes of these discontinuities because the ”equivalent index” involves second derivative of the inverse of the original refractive index of the TM problem. The Dirac-delta distribution could be eliminated by approximating the step-like original refractive index by a smoothed one, and hence the derivative in the equivalent index could be evaluated analytically as well as numerically; this eliminates the spike-like behavior of the equivalent index at the plane of the discontinuities. In this paper, we present a wide variety of smoothing functions that exist in the literature and assess their effects on the final numerical results of the problem under consideration. The comparative study presented here will help the interested researchers when deciding which smoothing function is appropriate for a specific problem. |
