Description:
In this paper, we present a detailed and rigorous derivation of the delay differential equations of the spin-flip model for vertical external cavity lasers with a semiconductor saturable absorption mirror. This model describe mode-locked semiconductor lasers in the ring-resonator geometry with unidirectional lasing. This contribution completes a previous communication [Vladimirov et al. Opt. Lett., 45, 252 (2020)], and we further complete the analytical derivation by taking into account phase and amplitude anisotropies and the resulting different delay times for orthogonal linear polarizations. We show evidence of the coexistence of two linearly polarized frequency combs generation with slightly different repetition rates due to the birefringence-induced time-offlight difference.