Abstract
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.
Highlights
SPIN-VCSELs allow the control of output polarisation by injection of spin-polarised carriers
A more accurate model to account for these non-local anisotropies is based on a matrix approach which yields the spatial distribution of the electromagnetic modes of the spin-VCSEL [6,20]
A comprehensive spin-flip model (SFM)-based numerical analysis of high-frequency polarisation oscillations and modulation in optically-pumped spin-VCSELs has given a deeper understanding of the influence of birefringence and spin relaxation rates [33]. It should be clear from the discussion above that, more complex models are becoming available to aid the design and analysis of spin-VCSELs for specific applications, the more basic and intuitive SFM approach is still extremely useful and versatile for exploring their dynamics
Summary
Proposed advantages of these devices over conventional lasers, the promise of very high speed operation [2] has recently excited much attention. A more accurate model to account for these non-local anisotropies is based on a matrix approach which yields the spatial distribution of the electromagnetic modes of the spin-VCSEL [6,20] This model is well suited to model the effects of an integrated surface grating in order to tailor the high-speed performance of the device [6]. A comprehensive SFM-based numerical analysis of high-frequency polarisation oscillations and modulation in optically-pumped spin-VCSELs has given a deeper understanding of the influence of birefringence and spin relaxation rates [33] It should be clear from the discussion above that, more complex models are becoming available to aid the design and analysis of spin-VCSELs for specific applications, the more basic and intuitive SFM approach is still extremely useful and versatile for exploring their dynamics. In the present contribution a three-variable reduction of the SFM equations for spin-VCSELs is presented and used to derive steady-state solutions and stability boundaries for dynamic states
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