Abstract
The five coupled rate equations used to describe laterally-coupled pairs of lasers with weak coupling and unequal pumping are reduced to a new system of three equations. This enables approximate closed-form steady-state solutions and explicit expressions for the boundaries between regions of stable and unstable dynamics to be found. The results of applying these approximations to specific cases of coupled laser pairs are shown to be in good agreement with results obtained from numerical solutions of the original set of five equations as well as earlier results from the literature. In addition the approximations based on the reduced set of equations allow a systematic investigation of the effects of material, device and operating conditions on trends and novel features in the dynamics of laterally-coupled laser pairs. The algebraic results give insight into trends with parameters without the need for extensive numerical computation and should therefore be of use in modelling two-element VCSEL arrays for numerous potential applications.
Highlights
T HE study of small arrays of coupled semiconductor lasers, especially those with independent control of pumping for each element [1]–[4], is an area which is receiving increasing interest and attention
For the case of VCSEL arrays this has been driven by attractive properties including phase front engineering and beam-steering [5],[6], extended bandwidth [7],[8] and enhanced digital modulation [9],[4], with one exemplar application being the prospect of tunable ultrafast photonic oscillators [10],[11]
It is shown that the approximate solutions permit novel features of the dynamic maps to be identified and their dependence on coupling coefficient and pumping asymmetry to be explored in a systematic manner
Summary
T HE study of small arrays of coupled semiconductor lasers, especially those with independent control of pumping for each element [1]–[4], is an area which is receiving increasing interest and attention. Theoretical understanding and modelling of these effects has been largely based on coupled rate equations [1],[2],[10]–[12],[14]–[16] These apply in the general case of unequal pumping and have been extended to include asymmetry in coupling coefficient and photon and carrier lifetimes [17]. It is shown that the approximate solutions permit novel features of the dynamic maps to be identified and their dependence on coupling coefficient and pumping asymmetry to be explored in a systematic manner. These features are summarised in the concluding section IV
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