Advertisement

Symmetry-Breaking Bifurcations in Laser Systems with All-to-All Coupling

Conference paper

Abstract

We consider a system of n semiconductor lasers with all-to-all coupling that is described using the Lang-Kobayashi rate equations. The lasers are coupled through their optical fields with delay arising from the finite propagation time of the light from one laser to another. As a consequence of the coupling structure, the resulting system of delay differential equations is equivariant under the symmetry group S n ×S1. Since symmetry gives rise to eigenvalues of higher multiplicity, implementing a numerical bifurcation analysis to our laser system is not straightforward. Our results include the use of the equivariance property of the laser system to find symmetric solutions, and to correctly locate steady-state and Hopf bifurcations. Additionally, this method identifies symmetry-breaking bifurcations where new branches of solutions emerge.

Notes

Acknowledgements

This work was funded by the UP System Emerging Interdisciplinary Research Program (OVPAA-EIDR-C03-011). The author also acknowledged the support of the UP Baguio through RLCs during the A.Y. 2014–2015.

References

  1. 1.
    Agrawal, G.P., Dutta, N.K.: Long-Wavelength Semiconductor Lasers. Van Nostrand Reinhold, New York (1986)CrossRefGoogle Scholar
  2. 2.
    Alsing, P.M., Kovanis, V., Gavrielides, A., Erneux, T.: Lang and Kobayashi phase equation. Phys. Rev. A 53, 4429–4434 (1996)CrossRefGoogle Scholar
  3. 3.
    Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: A MATLAB package for bifurcation analysis of delay differential equations. Technical report TW-330. Department of Computer Science, K.U. Leuven, Leuven (2001)Google Scholar
  4. 4.
    Erzgraber, H., Krauskopf, B., Lenstra, D.: Compound laser modes of mutually delay-coupled lasers. SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Birkhauser Verlag, Basel (2002)CrossRefMATHGoogle Scholar
  6. 6.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1988)CrossRefMATHGoogle Scholar
  7. 7.
    Gray, G.R., Ryan, A.T., Agrawal, G.P., Gage, E.C.: Control of optical-feedback-induced laser intensity noise in optical data recording. Opt. Eng. 32, 739–745 (1993)CrossRefGoogle Scholar
  8. 8.
    Gray, G.R., Ryan, A.T., Agrawal, G.P., Gage, E.C.: Optical-feedback-induced chaos and its control in semiconductor lasers. In: SPIE’s 1993 International Symposium on Optics, Imaging, and Instrumentation, San Diego, pp. 45–57. International Society for Optics and Photonics (1993)Google Scholar
  9. 9.
    Lang, R., Kobayashi, K.: External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron 16, 347–355 (1980)CrossRefGoogle Scholar
  10. 10.
    Mork, J., Tromborg, B., Mark, J.: Chaos in semiconductor lasers with optical feedback: theory and experiment. IEEE J. Quantum Electron 28, 93–108 (1992)CrossRefGoogle Scholar
  11. 11.
    Sano, T.: Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback. Phys. Rev. A 50, 2719–2726 (1994)CrossRefGoogle Scholar
  12. 12.
    van Tartwijk, G.H.M., Levine, A.M., Lenstra, D.: Sisyphus effect in semiconductor lasers with optical feedback. IEEE J. Select. Top. Quantum Electron. 1, 466–472 (1995)CrossRefGoogle Scholar
  13. 13.
    Verduyn Lunel, S.M., Krauskopf, B.: The mathematics of delay equations with an application to the Lang-Kobayashi equation. In: Krauskopf, B., Lenstra, D. (eds.) Fundamental Issues of Nonlinear Laser Dynamics. AIP Conference Proceedings, vol. 548. American Institute of Physics, New York (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Philippines BaguioBaguio CityPhilippines

Personalised recommendations