Lang and Kobayashi phase equation and its validity for low pump

Abstract

An asymptotic theory of Lang and Kobayashi (LK) equations describing a semiconductor laser subject to optical feedback is investigated in detail. We obtain a simple third order, nonlinear, delay-differential equation for the phase of the laser field which admits multiple branches of time-periodic intensity solutions. The theory is based on typical values of LK dimensionless parameters and assumes that the pump parameter is not too small. In this paper, we examine the validity of this assumption by considering the small pump limit. We find the same phase equation as the leading problem of our asymptotic analysis but now with a stronger damping coefficient. This phase equation fails as a correct asymptotic approximation only for very low pump, close to the lasing threshold. The approximation for this case is more complicated and reveals a stronger influence of the laser intensity.