Three-Variable Reduction of the Spin-Flip Model for Spin-VCSELs

Recent increased interest in spin-polarised vertical-cavity surface-emitting lasers (spin-VCSELs) as potential high-speed sources has spurred research into the analysis of their dynamics. This has often been explored by combining the spin-flip model (SFM) with numerical methods. However, numerical simulation does not readily expose key dependencies and to date there is a lack of accessible closed-form analytical results for the steady-state solutions and dynamic stability boundaries. Thus in the present contribution we address this and show that, for zero dichroism, the five rate equations of the SFM can be reduced to a set of three. These can be solved in the steady-state in terms of the intensity and polarisation of the pump (optical or electrical), and the ellipticity of the output. Additionally, a small-signal analysis leads to analytic results for the boundaries between stable and unstable operation in the plane of pump ellipticity versus pump intensity. Comparison of the results from these expressions with those from numerical bifurcation and continuation methods shows very good agreement. The accuracy of the reduced set of equations is confirmed by comparing the results with those from the full set of SFM equations. In the limiting case of very high birefringence, as would be required for a potential THz source, a simple algebraic relation is derived for the spin relaxation rate in terms of other parameters. Hence we find that the range of spin relaxation rates to achieve THz oscillations is very limited. The relative simplicity of the present approach thus offers a rapid, intuitive and convenient route to study the dynamics of spin-VCSELs.