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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show all references

References:
[1]

V. I. Arnold, The Theory of Singularities and Its Applications,, Accademia Nazionale dei Lincei, (1991). Google Scholar

[2]

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[3]

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[4]

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[5]

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[6]

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[7]

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]

T. Erneux, A. Gavrielides, K. Green and B. Krauskopf, External cavity modes of semiconductor lasers with phase-conjugate feedback,, Phys. Rev. E, 68 (2003). doi: 10.1103/PhysRevE.68.066205. Google Scholar

[15]

T. Erneux, M. Yousefi and D. Lenstra, The Injection Laser Limit of Lasers Subject to Filtered Optical Feedback,, Proc. European Quantum Electronics Conf., (2003). Google Scholar

[16]

H. Erzgräber and B. Krauskopf, Dynamics of a filtered-feedback laser: Influence of the filter width,, Optics Letters, 32 (2007), 2441. doi: 10.1364/OL.32.002441. Google Scholar

[17]

H. Erzgräber, B. Krauskopf and D. Lenstra, Bifurcation analysis of a semiconductor laser with filtered optical feedback,, SIAM J. Appl. Dyn. Sys., 6 (2007), 1. doi: 10.1137/060656656. Google Scholar

[18]

H. Erzgräber, B. Krauskopf, D. Lenstra, A. P. A. Fischer and G. Vemuri, Frequency versus relaxation oscillations in semiconductor laser with coherent filtered optical feedback,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.055201. Google Scholar

[19]

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. A. Fischer and G. Vemuri, Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.026212. Google Scholar

[20]

B. Farias, T. P. de Silans, M. Chevrollier and M. Oriá, Frequency bistability of a semiconductor laser under a frequency-dependent feedback,, Phys. Rev. Lett., 94 (2005). doi: 10.1103/PhysRevLett.94.173902. Google Scholar

[21]

S. G. Fischer, M. Ahmed, T. Okamoto, W. Ishimori and M. Yamada, An improved analysis of semiconductor laser dynamics under strong optical feedback,, IEEE J. Quantum Electron., 9 (2003), 1265. Google Scholar

[22]

A. P. A. Fischer, O. Andersen, M. Yousefi, S. Stolte and D. Lenstra, Experimental and theoretical study of semiconductor laser dynamics due to filtered optical feedback,, IEEE J. Quantum Electron., 36 (2000), 375. Google Scholar

[23]

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