Modeling Spatial Hole Burning and Mode Competition in Index-Guided VCSELs

摘要

We have derived and implemented a simplified dynamical multi-mode model for vertical-cavity surface-emitting lasers (VCSELs) requiring minimal computing resources while remaining realistic. A set of spatially independent rate equations with a minimum of parameters is extracted from the general spatially resolved equations and reduced to dimensionless form to expose its essentials. The shape of the carrier density distribution is approximated with a linear sum of the modal shapes which are considered to be bias independent for index-guided devices. Despite this simplification the model, which can be extended to any number of laser modes, includes important effects like spatial hole burning (SHB), carrier diffusion and inhomogeneous injection and shows good agreement with spatially resolved models. We describe the reduction and nondimensionalization of the general spatially resolved equations. The implementation inside a C++ rate-equation framework is discussed and shown to provide a powerful and flexible environment to efficiently study the contribution of different factors. The effect of spatial hole burning (SHB) and mode competition on the small signal modulation response is investigated in three basic cases. The comparison between a single mode VCSEL without SHB, a single mode VCSEL including SHB and two mode VCSEL including SHB enables us to pinpoint which contributions are due to SHB and which are due to the modal competition. Finally,the possibility of using this approach to model polarization switching in VCSELs is discussed. The unavoidable small birefringence (with elasto-optic and electro-optic contributions) present in all VCSELs leads to a doubling of each mode into two orthogonal polarization states which are almost degenerate. Although small, the birefringence causes gain differences: part of which are due to frequency dependent gain and loss of the materials and part of which are due to slightly different transverse modal shapes. As these gain differences depend on the injected current, they can be one of the causes of polarization switching.