Analytical framework for the steady state analysis of wavelength-dependent and intensity-dependent interaction of multiple monochromatic beams in semiconductor photonic structures with multiple active and passive sections

The design, optimization, and parametric studies of photonic integrated circuits typically require steady state optical field profiles. Previously, a planewave-based eigenmode expansion and mode-matching analysis was shown to be an efficient method in calculating the steady-state optical field profile of a monochromatic beam in 2-D/3-D geometries with passive media. In this paper, we extend this approach for steady state analyses of photonic integrated circuits with both passive as well as active media, by enabling the interaction of multiple monochromatic beams with the medium via a self-consistent solution of the Maxwell's equations and a modified Bloch equations-based analytical framework. Through the Bloch equations-based framework, we treat the active medium as an ensemble of two-level Bloch atoms with varying density of states to characterize the bandstructure of semiconductors. This enables us to track carrier transitions between conduction and valence bands induced by the light-matter interaction over the entire bandwidth of excited carriers, and yield the correct quasi equilibrium Fermi-Dirac distributions. The numerical solution of obtaining the carrier distributions at each spatial location in active media is simplified by reducing the modified Bloch equations with multiple unknowns to a single equation with just one unknown, to enhance the computational efficiency of the approach. The calculated carrier distributions are then used to evaluate the spatial profile of the complex refractive index seen by each monochromatic beam along with its optical field profile. The applicability of the method to 2-D structures of complex geometry is demonstrated by obtaining the spatial profiles of refractive indices seen by two simultaneously interacting monochromatic beams along with their optical field profiles in a waveguide with multiple passive and active sections.