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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1713–1725

Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy

J. P. Toomey and D. M. Kane  »View Author Affiliations

Optics Express, Vol. 22, Issue 2, pp. 1713-1725 (2014)

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A semiconductor laser with delayed optical feedback is an experimental implementation of a nominally infinite dimensional dynamical system. As such, time series analysis of the output power from this laser system is an excellent test of complexity analysis tools, as applied to experimental data. Additionally, the systematic characterization of the range and variation in complexity that can be obtained in the output power from the system, which is available to be used in applications like secure communication, is of interest. Output power time series from a semiconductor laser system, as a function of the optical feedback level and the laser injection current, have been analyzed for complexity using permutation entropy. High resolution maps of permutation entropy as a function of optical feedback level and injection current have been achieved for the first time. This confirms prior research that identifies a coherence collapse region which is found to be uninterrupted with respect to any embedded islands with different dynamics. The results also show new observations of low optical feedback dynamics which occur in a region below that for coherence collapse. The map of the complexity shows a strong dependence on the delay time used in the permutation entropy calculation. Short delay times, which sample information at the complete measurement bandwidth, produce maps with drastically different systematic variation in complexity throughout the coherence collapse region, compared to maps generated with a delay time that matches the optical feedback delay. Evaluating the complexity with a permutation entropy delay equal to the external cavity delay produces results consistent with the notion of weak/strong chaos, as well as categorizing the dynamics as being of high complexity where the external cavity delay time is harder to identify. These are both desirable features for secure communication applications. The results also show permutation entropy as a function of delay time can be used to detect key frequencies driving the dynamics, including any that may exist due to, or arise from, technicalities of device fabrication and/or noise. A more complete insight into complexity as measured by permutation entropy is gained by considering multiple delay times.

© 2014 Optical Society of America

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.1540) Lasers and laser optics : Chaos
(140.5960) Lasers and laser optics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

Original Manuscript: September 26, 2013
Revised Manuscript: November 8, 2013
Manuscript Accepted: January 2, 2014
Published: January 17, 2014

J. P. Toomey and D. M. Kane, "Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy," Opt. Express 22, 1713-1725 (2014)

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  1. D. M. Kane and K. A. Shore, eds., Unlocking Dynamical Diversity: Feedback Effects on Semiconductor Lasers (John Wiley & Sons, 2005).
  2. S. Wieczorek, B. Krauskopf, T. B. Simpson, D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005). [CrossRef]
  3. J. S. Lawrence, D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. 38(2), 185–192 (2002). [CrossRef]
  4. J. Mørk, B. Tromborg, J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992). [CrossRef]
  5. J. P. Toomey, D. M. Kane, M. W. Lee, K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18(16), 16955–16972 (2010). [CrossRef] [PubMed]
  6. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer Series in Optical Sciences (Springer-Verlag, Berlin, 2006), Vol. 111.
  7. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef] [PubMed]
  8. G. D. VanWiggeren, R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]
  9. L. Zunino, O. A. Rosso, M. C. Soriano, “Characterizing the Hyperchaotic Dynamics of a Semiconductor Laser Subject to Optical Feedback Via Permutation Entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011). [CrossRef]
  10. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007). [CrossRef] [PubMed]
  11. W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, Z. Ming-Jiang, “Can Fixed Time Delay Signature be Concealed in Chaotic Semiconductor Laser With Optical Feedback?” IEEE J. Quantum Electron. 48(11), 1371–1379 (2012). [CrossRef]
  12. A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013). [CrossRef] [PubMed]
  13. S. Priyadarshi, I. Pierce, Y. Hong, K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012). [CrossRef]
  14. M. T. Rosenstein, J. J. Collins, C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993). [CrossRef]
  15. H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994). [CrossRef]
  16. P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983). [CrossRef]
  17. C. Bandt, B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002). [CrossRef] [PubMed]
  18. Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004). [CrossRef] [PubMed]
  19. M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, C. R. Mirasso, “Time Scales of a Chaotic Semiconductor Laser With Optical Feedback Under the Lens of a Permutation Information Analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011). [CrossRef]
  20. M. Staniek, K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007). [CrossRef]
  21. D. Lenstra, B. Verbeek, A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985). [CrossRef]
  22. J. P. Toomey and D. M. Kane, “Low level optical feedback in semiconductor lasers as a tool to identify nonlinear enhancement of device noise,” Proceedings 2010 Conference on Optoelectronic and Microelectronic Materials & Devices (COMMAD 2010), 55–5656 (2010). [CrossRef]
  23. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (Cambridge University Press, Cambridge, 2004).
  24. S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011). [CrossRef] [PubMed]
  25. M. C. Soriano, X. Porte, D. A. Arroyo-Almanza, C. R. Mirasso, and I. Fischer, “Experimental distinction of weak and strong chaos in delay-coupled semiconductor lasers,” in Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO EUROPE/IQEC), 2013)
  26. S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, W. Kinzel, “Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(1), 012902 (2013). [CrossRef] [PubMed]
  27. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009). [CrossRef]

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