Abstract

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

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References

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  5. J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18(16), 16955–16972 (2010).
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  9. L. Zunino, O. A. Rosso, and 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, and 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, and 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, and 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, and 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, and 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 and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
    [Crossref]
  17. C. Bandt and 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, and 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, and 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 and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
    [Crossref]
  21. D. Lenstra, B. Verbeek, and 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, and 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, and 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, and 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]

2013 (2)

A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref] [PubMed]

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

2012 (2)

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

2011 (3)

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

L. Zunino, O. A. Rosso, and 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]

2010 (1)

2009 (1)

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

2007 (2)

2005 (2)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

2004 (1)

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

2002 (2)

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

J. S. Lawrence and 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]

1998 (1)

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

1994 (1)

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[Crossref]

1993 (1)

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[Crossref]

1992 (1)

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

1985 (1)

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[Crossref]

1983 (1)

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[Crossref]

An-Bang, W.

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Arroyo-Almanza, D. A.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Cao, Y. H.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

Citrin, D. S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

D. Rontani, A. Locquet, M. Sciamanna, and 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]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Collins, J. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[Crossref]

D’Huys, O.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

Dahms, T.

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Deluca, C. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[Crossref]

Den Boef, A.

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[Crossref]

Fischer, I.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Flunkert, V.

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Gao, J. B.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

García-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Grassberger, P.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[Crossref]

Heiligenthal, S.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Hively, L. M.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

Hong, Y.

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

Jüngling, T.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Kane, D. M.

J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18(16), 16955–16972 (2010).
[Crossref] [PubMed]

J. S. Lawrence and 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]

Kanter, I.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Kantz, H.

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[Crossref]

Kinzel, W.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Krauskopf, B.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Lawrence, J. S.

J. S. Lawrence and 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]

Lee, M. W.

Lehnertz, K.

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[Crossref]

Lenstra, D.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[Crossref]

Li, L.

Locquet, A.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

D. Rontani, A. Locquet, M. Sciamanna, and 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]

Mark, J.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

Ming-Jiang, Z.

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

Mirasso, C. R.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Mørk, J.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

Ortin, S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Pierce, I.

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

Pompe, B.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Priyadarshi, S.

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

Procaccia, I.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[Crossref]

Protopopescu, V. A.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

Pu, L.

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

Rontani, D.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

D. Rontani, A. Locquet, M. Sciamanna, and 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]

Rosenstein, M. T.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[Crossref]

Rosso, O. A.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

L. Zunino, O. A. Rosso, and 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]

Roy, R.

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Schöll, E.

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Sciamanna, M.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

D. Rontani, A. Locquet, M. Sciamanna, and 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]

Shore, K. A.

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18(16), 16955–16972 (2010).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Simpson, T. B.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

Soriano, M. C.

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

L. Zunino, O. A. Rosso, and 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]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

Staniek, M.

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[Crossref]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Toomey, J. P.

Tromborg, B.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

Tung, W. W.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

VanWiggeren, G. D.

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Verbeek, B.

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[Crossref]

Wang, A.

Wang, B.

Wang, Y.

Wieczorek, S.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

Yanchuk, S.

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

Yang, Y.

Yuan, W.

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

Yun-Cai, W.

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

Zhang, B.

Zunino, L.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

L. Zunino, O. A. Rosso, and 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]

IEEE J. Quantum Electron. (6)

J. S. Lawrence and 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]

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

W. Yuan, W. Yun-Cai, L. Pu, W. An-Bang, and 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]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and 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]

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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]

IEEE J. Sel. Top. Quantum Electron. (1)

L. Zunino, O. A. Rosso, and 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]

Int. J. Bifurcat. Chaos (1)

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[Crossref]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Lett. A (1)

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[Crossref]

Phys. Rep. (1)

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and 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]

S. Heiligenthal, T. Jüngling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and 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]

Phys. Rev. Lett. (2)

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, “Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings,” Phys. Rev. Lett. 107(23), 234102 (2011).
[Crossref] [PubMed]

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Physica D (2)

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[Crossref]

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[Crossref]

Science (1)

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Semicond. Sci. Technol. (1)

S. Priyadarshi, I. Pierce, Y. Hong, and K. A. Shore, “Optimal operating conditions for external cavity semiconductor laser optical chaos communication system,” Semicond. Sci. Technol. 27(9), 094002 (2012).
[Crossref]

Other (5)

D. M. Kane and K. A. Shore, eds., Unlocking Dynamical Diversity: Feedback Effects on Semiconductor Lasers (John Wiley & Sons, 2005).

J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer Series in Optical Sciences (Springer-Verlag, Berlin, 2006), Vol. 111.

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)

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]

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (Cambridge University Press, Cambridge, 2004).

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Figures (10)
Fig. 1
Fig. 1 Semiconductor laser with optical feedback setup.

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Fig. 2
Fig. 2 Dynamics maps of (a) RMS amplitude and (b) permutation entropy (D = 5, τ = 2) as a function of optical feedback level and laser injection current. The white curve shows the laser threshold injection decreasing with increasing optical feedback.

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Fig. 3
Fig. 3 Time series and corresponding FFT spectra for different regions of the parameter space as marked in Fig. 2. (i) Iinj = 68.7 mA, TAOM = 0.644, (ii) Iinj = 58.5 mA, TAOM = 0.194, (iii) Iinj = 68.2 mA, TAOM = 0.09.

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Fig. 4
Fig. 4 Probability distribution of all possible m! ordinal patterns for (a) low permutation entropy region: Iinj = 58.5 mA, TAOM = 0.194, (b) High permutation entropy region: Iinj = 68.7 mA, TAOM = 0.644, and (c) Noise (laser operating below threshold): Iinj = 45.0 mA, TAOM = 0.065.

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Fig. 5
Fig. 5 Experimental laser output power time series in different regions of the parameter space where the dynamics are of a relatively (a) low frequency (region ii) and (b) high frequency (region i), respectively. Vectors for generating the ordinal permutation probability distribution for short (blue circles) and long (red squares) values of delay τ are shown.

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Fig. 6
Fig. 6 Permutation entropy as a function of delay τ and ordinal pattern length D, as used in the calculation, for output power time series from the parameter space with strong injection and feedback (Iinj = 68.7 mA, TAOM = 0.644).

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Fig. 7
Fig. 7 Permutation entropy as a function of injection current and optical feedback level using (a) τ = τext = 90 and D = 5, (b) τ = τext/2 = 45 and D = 5.

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Fig. 8
Fig. 8 Map of the autocorrelation function (ACF) (a) peak amplitude, and (b) peak delay. The black arrow indicates the external cavity delay of 90 sampling periods = 4.5 ns.

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Fig. 9
Fig. 9 Autocorrelation function (ACF) for Iinj = 70 mA and (a) TAOM = 0.650, (b) TAOM = 0.451, (c) TAOM = 0.250, and (d) TAOM = 0.150.

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Fig. 10
Fig. 10 Permutation entropy (D = 5) as a function of delay and injection current for several fixed feedback levels (a) TAOM = 0.548, (b) TAOM = 0.350, (c) TAOM = 0.245, (d) TAOM = 0.147. White arrows indicate external cavity features, blue arrows indicate relaxation oscillation features, and green arrows indicate the splitting external cavity peaks.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

S[ P ]=− ∑ i=1 M p i ln p i .
s→( x s−( D−1 )τ , x s−( D−2 )τ ,..., x s−τ , x s ).
x s− r 0 τ ≥ x s− r 1 τ ≥...≥ x s− r D−2 τ ≥ x s− r D−1 τ .
ℋ S [ P ]= S[ P ] S max = − ∑ i=1 D! p( π i ) lnp( π i ) lnD! .

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