Non-symmetrical vortex beam shaping in VECSEL laser arrays

We propose and numerically test, to our knowledge, a novel concept for asymmetric vortex beam generation in a degenerate vertical external cavity surface emitting laser (DVECSEL). The method is based on a phase-locking ring array of lasers created inside a degenerate cavity with a binary amplitude mask containing circular holes. The diffraction engineering of the mask profile allows for controlling the complex coupling between the lasers. The asymmetry between different lasers is introduced by varying the hole diameters corresponding to different lasers. Several examples of masks with non-uniform or uniform circular holes are investigated numerically and analytically to assess the impact of non-uniform complex coupling coefficients on the degeneracy between the vortex and anti-vortex steady states of the ring laser arrays. It is found that the in-phase solution always dominates irrespective of non-uniform masks. The only solution to make one particular vortex solution dominant over other possible steady-state solutions consists of imprinting the necessary phase shift among neighboring lasers in the argument of their coupling coefficients. We also investigate the role of the Henry factor inherent to the use of a semiconductor active medium in the probabilities to generate vortex solutions. Analytical calculations are performed to generalize a formula previously reported [Opt. Express 30, 15648 (2022)], for the limiting Henry factor to cover the case of complex couplings.