Bifurcation analysis of a semiconductor laser with filtered optical feedback

We study the dynamics and bifurcations of a semiconductor laser with delayed filtered feedback, where a part of the output of the laser re-enters after spectral filtering. This type of coherent optical feedback is more challenging than the case of conventional optical feedabck from a simple mirror, but it provides additional control over the output of the semiconductor laser by means of choosing the filter detuning and the filter width. This laser system can be modelled by a system of delay differential equations with a single fixed delay, which is due to the travel time of the light outside the laser. In this paper we present a bifurcation analysis of the filtered feedback laser. We first consider the basic continuous wave states, known as the external filtered modes (EFMs), and determine their stability regions in the parameter plane of feedback strength versus feedback phase. The EFMs are born in saddle-node bifurcations and become unstable in Hopf bifurcations. We show that for small filter detuning there is a single region of stable EFMs, which splits up into two separate regions when the filter is detuned. We then concentrate on the periodic orbits that emanate from Hopf bifurcations. Depending on the feedback strength and the feedback phase two types of oscillations can be found. First, there are undamped relaxation oscillations, which are typical for semiconductor laser systems. Secondly, there are oscillations with a period related to the delay time, which have the remarkable property that the laser frequency oscillates while the laser intensity is almost constant. These frequency oscillations are only possible due to the interaction of the laser with the filter. We determine the stability regions in the parameter plane of feedback strength versus feedback phase of the different types of oscillations. In particular, we find that stable frequency oscillations are dominant for nonzero values of the filter detuning.