Lang and Kobayashi phase equation

Lang and Kobayashi equations for a semiconductor laser subject to optical feedback are investigated by using asymptotic methods. Our analysis is based on the values of two key parameters, namely, the small ratio of the photon and carrier lifetimes and the relatively large value of the linewidth enhancement factor. For low feedback levels, we derive a third-order delay-differential equation for the phase of the laser field. We then show analytically and numerically that this equation admits coexisting branches of stable periodic solutions that appear at different and almost constant amplitudes. These amplitudes are proportional to the roots of the Bessel function J(1)(x). The bifurcation diagram of the phase equation is in good agreement with the numerical bifurcation diagram of the original Lang and Kobayashi equations. We interpret the onset of the periodic solutions as the emergence of a new set of external cavity modes with a more complicated time dependence.