# American Institute of Mathematical Sciences

March  2015, 20(2): 519-586. doi: 10.3934/dcdsb.2015.20.519

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

 1 Mathematics Research Institute, CEMPS, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom 2 Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142 3 Department of Applied Mathematics, University College Cork, Western Gateway Building, Cork, Ireland

Received  April 2014 Revised  May 2014 Published  January 2015

We investigate the solution structure and stability of a semiconductor laser receiving time-delayed and frequency-filtered optical feedback from two external filters. This system is referred to as the 2FOF laser, and it has been used as pump laser in optical telecommunication and as light source in sensor applications. The underlying idea is that the two filter loops provide a means of stabilizing and controling the laser output. The mathematical model takes the form of delay differential equations for the (real-valued) population inversion of the laser active medium and for the (complex-valued) electric fields of the laser cavity and of the two filters. There are two time delays, which are the travel times of the light from the laser to each of the filters and back.
Our analysis of the 2FOF laser focuses on the basic solutions, known as continuous waves or external filtered modes (EFMs), which correspond to laser output with steady amplitude and frequency. Specifically, we consider the EFM-surface in the $(\omega_s,\,N_s,\,dC_p)$-space of steady frequency $\omega_s$, the corresponding steady population inversion $N_s$, and the feedback phase difference $dC_p$. This surface emerges as the natural object for the study of the 2FOF laser because it conveniently catalogues information about available frequency ranges of the EFMs. We identify five transitions, through four different singularities and a cubic tangency, which change the type of the EFM-surface locally and determine the EFM-surface bifurcation diagram in the $(\Delta_1,\,\Delta_2)$-plane. In this way, we classify the possible types of the EFM-surface, which consist of a combination of bands (covering the entire $dC_p$-range) and islands (covering only a finite range of $dC_p$).
We also investigate the stability of the EFMs, where we focus on saddle-node and Hopf bifurcation curves that bound regions of stable EFMs on the EFM-surface. It is shown how these stability regions evolve when parameters are changed along a chosen path in the $(\Delta_1,\,\Delta_2)$-plane. From a viewpoint of practical interests, we find various bands and islands of stability on the EFM-surface that may be accessible experimentally.
Beyond their relevance for the 2FOF laser system, the results presented here also showcase how advanced tools from bifurcation theory and singularity theory can be employed to uncover and represent the complex solution structure of a delay differential equation model that depends on a considerable number of input parameters, including two time delays.
Citation: Piotr Słowiński, Bernd Krauskopf, Sebastian Wieczorek. Mode structure of a semiconductor laser with feedback from two external filters. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 519-586. doi: 10.3934/dcdsb.2015.20.519
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