Mode structure of a semiconductor laser with feedback from two external filters

Abstract

We investigate the solution structure and stability of a semiconductorlaser receiving time-delayed and frequency-filtered optical feedbackfrom two external filters. This system is referred to asthe 2FOF laser, and it has been used as pump laser in opticaltelecommunication and as light source in sensor applications.The underlying idea is that the two filter loops provide a means ofstabilizing and controling the laser output. The mathematicalmodel takes the form of delay differential equationsfor the (real-valued) population inversion of the laser active mediumand for the (complex-valued) electric fields of the laser cavityand of the two filters. There are two time delays, which are thetravel times of the light from the laser to each of the filters andback. Our analysis of the 2FOF laser focuses on the basic solutions,known as continuous waves or external filtered modes (EFMs), whichcorrespond to laser output with steady amplitude and frequency.Specifically, we considerthe EFM-surface in the $(\omega_s,\,N_s,\,dC_p)$-space of steady frequency$\omega_s$, the corresponding steady population inversion $N_s$, andthe feedback phase difference $dC_p$. This surface emerges as thenatural object for the study of the 2FOF laserbecause it conveniently catalogues informationabout available frequency ranges of the EFMs. We identify fivetransitions, through four different singularities and a cubic tangency, whichchange the type of the EFM-surface locally anddetermine the EFM-surface bifurcation diagram in the$(\Delta_1,\,\Delta_2)$-plane. In this way, we classify thepossible types of the EFM-surface, which consist of a combination ofbands (covering the entire $dC_p$-range) and islands (covering only afinite range of $dC_p$). We also investigate the stability of the EFMs, where we focus onsaddle-node and Hopfbifurcation curves that bound regions of stable EFMs on theEFM-surface. It is shown how these stability regions evolve whenparameters arechanged along a chosen path in the $(\Delta_1,\,\Delta_2)$-plane.From a viewpoint of practicalinterests, we find various bands and islands of stability on theEFM-surface that may be accessible experimentally. Beyond their relevance for the 2FOF laser system, the resultspresented here also showcase how advanced tools frombifurcation theory and singularitytheory can be employed to uncover and represent the complexsolution structure of a delay differential equation model thatdepends on a considerable number of input parameters, including twotime delays.