Invariance of optimal composite waveguide geometries with respect to permittivity of the metal cladding

We optimize the threshold gain for cylindrical composite (semiconductor-dielectric-metal) waveguides (WGs) with various metal claddings. We show that the optimal dielectric width is invariant with respect to the imaginary part of the permittivity of the metal, εM'', and weakly dependent on the real part, εM'. To explain this behavior, we compare optimal geometries of WGs with different semiconductor permittivities, εG'. Results from these comparisons indicate that the optimal effective index parallels the optimal threshold gain in its relation to εM. We use our results to heuristically propose an analytical expression for the optimal threshold gain that approximates the numerical solution to within a factor of two over the range of explored εG'. Finally, we use data from our optimizations to obtain approximate analytical expressions for the optimal dielectric width and threshold gain as functions of the total WG radius.