Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model

The implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model of a single-mode semiconductor laser with optical feedback are numerically investigated. α is introduced on the model in the rate equation for the complex electric field by performing an expansion of the frequency of the laser mode around its threshold value ω₀. Two different expansions have led to two different sets of Lang and Kobayashi equations; in the first set, the intensity reduction of the optical gain G is taken into account in the linearization of ω, while in the second set it is neglected. Although in the literature the investigations of semiconductor lasers with optical feedback have been based on either of these sets, they are equal only when a linear form for G is assumed. In this paper, it is shown that on which set of Lang and Kobayashi equations the investigations are based is an important fact to take into account when interpreting the results, since it is shown that if the nonlinear gain is considered, the dynamics predicted by the two sets are very different. In particular, it is shown that behavior of the external cavity modes and the stability properties of the attractors differ greatly when the gain saturation coefficient ε is varied