## Experimental and numerical study of the symbolic dynamics of a modulated external-cavity semiconductor laser |

Optics Express, Vol. 22, Issue 4, pp. 4705-4713 (2014)

http://dx.doi.org/10.1364/OE.22.004705

Acrobat PDF (4551 KB)

### Abstract

We study the symbolic dynamics of a stochastic excitable optical system with periodic forcing. Specifically, we consider a directly modulated semiconductor laser with optical feedback in the low frequency fluctuations (LFF) regime. We use a method of symbolic time-series analysis that allows us to uncover serial correlations in the sequence of intensity dropouts. By transforming the sequence of inter-dropout intervals into a sequence of symbolic patterns and analyzing the statistics of the patterns, we unveil correlations among several consecutive dropouts and we identify clear changes in the dynamics as the modulation amplitude increases. To confirm the robustness of the observations, the experiments were performed using two lasers under different feedback conditions. Simulations of the Lang-Kobayashi (LK) model, including spontaneous emission noise, are found to be in good agreement with the observations, providing an interpretation of the correlations present in the dropout sequence as due to the interplay of the underlying attractor topology, the external forcing, and the noise that sustains the dropout events.

© 2014 Optical Society of America

## 1. Introduction

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

6. A. Aragoneses, N. Rubido, J. Tiana-alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. **3**, 1778 (2013). [CrossRef] [PubMed]

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

10. S. Donati and R-H Horng, “The diagram of feedback regimes revisited,” IEEE J. Sel. Top. Quantum Electron. **19**, 1500309 (2013). [CrossRef]

11. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E **55**, 6414 (1997). [CrossRef]

12. A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. **83**, 292 (1999). [CrossRef]

13. T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,” Phys. Rev. Lett. **87**, 243901 (2001). [CrossRef] [PubMed]

14. A. Tabaka, K. Panajotov, I. Veretennicoff, and M. Sciamanna, “Bifurcation study of regular pulse packages in laser diodes subject to optical feedback,” Phys. Rev E **70**, 036211 (2004). [CrossRef]

15. J. A. Reinoso, J. Zamora-Munt, and C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E **87**, 062913 (2013). [CrossRef]

16. S. D. Cohen, A. Aragoneses, D. Rontani, M. C. Torrent, C. Masoller, and D. J. Gauthier, “Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback,” Opt. Lett. **38**, 4331 (2013). [CrossRef] [PubMed]

17. L. Junges, T. Pöschel, and J. A. C. Gallas, “Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model,” Eur. Phys. J. D. **67**, 149 (2013). [CrossRef]

18. D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. **56**, R3370 (1997). [CrossRef]

25. K. Hicke, X. Porte, and I. Fischer, “Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback,” Phys. Rev. E **88**, 052904 (2013). [CrossRef]

26. D. Baums, W. Elsässer, and E. O. Göbel, “Farey tree and Devil’s staircase of a modulated external-cavity semiconductor laser,” Phys. Rev. Lett. **63**, 155 (1989). [CrossRef] [PubMed]

27. 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**, 1695516972 (2010). [CrossRef]

28. Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E **51**, R2697–R2700 (1995). [CrossRef]

29. D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. **36**, 175 (2000). [CrossRef]

30. W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B **17**, 4123–4138 (2003). [CrossRef]

31. J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E **63**, 066218 (2001). [CrossRef]

35. T. Schwalger, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and B. Lindner, “Interspike-interval correlations induced by two-state switching in an excitable system,” Europhys. Lett. **99**, 10004 (2012). [CrossRef]

36. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347 (1980). [CrossRef]

19. J. Mulet and C. R. Mirasso, “Numerical statistics of power dropouts based on the Lang–Kobayashi mod,” Phys. Rev. E **59**, 5400 (1999). [CrossRef]

29. D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. **36**, 175 (2000). [CrossRef]

30. W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B **17**, 4123–4138 (2003). [CrossRef]

23. A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang–Kobayashi dynamics,” Phys. Rev. A **74**, 063801 (2006). [CrossRef]

24. J. Zamora-Munt, C. Masoller, and J. Garcia-Ojalvo, “Transient low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A **81**, 033820 (2010). [CrossRef]

*i.e.*, the LFFs are a transient dynamics that dies out when the trajectory finds a stable cavity mode, in deterministic simulations of the LK model). Therefore, it is remarkable that, in spite of the fact that the inclusion of noise is required for simulating sustained LFFs, the model adequately reproduces the symbolic dynamics, and in particular, the correlations present in the sequence of dropouts, and how they vary with the modulation amplitude.

## 2. Experimental setup and LFF dynamics with current modulation

*f*= 17 MHz and the modulation amplitude varies from 0 mV to 78 mV in steps of 7.8 mV (from 0% to 4% of the dc current in steps of 0.4%). For each modulation amplitude, five measurements of 3.2 ms were recorded. The time series contains between 74,000 and 207,000 dropouts, at low and high modulation amplitude, respectively.

_{mod}*f*= 2 MHz and the modulation amplitude varies from 0 mV to 150 mV in steps of 10 mV (from 0% to 24% of the dc current in steps of 1.6%). The time series contain between 8,000 and 19,000 dropouts, at low and high modulation amplitude, respectively. While, for the 1550 nm laser, the modulation frequency is about one order of magnitude smaller than for the 650 nm laser, the relation with the characteristic time-scale of the LFF dynamics, given by the average inter-dropout interval 〈Δ

_{mod}*T*〉 is about the same: for the 650 nm laser, 〈Δ

*T*〉 = 365 ns and thus 〈Δ

*T*〉 ×

*f*= 6.2. For the 1550 nm laser, 〈Δ

_{mod}*T*〉 = 2.55

*μ*s and 〈Δ

*T*〉 ×

*f*= 5.1.

_{mod}*T*(IDIs), and the return maps, Δ

_{i}*T*vs Δ

_{i}*T*

_{i}_{+1}, for four modulation amplitudes for the 650 nm laser. As it has been reported in the literature the dropouts tend to occur at the same phase in the drive cycle with current modulation, and the IDIs are multiples of the modulation period [11

11. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E **55**, 6414 (1997). [CrossRef]

29. D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. **36**, 175 (2000). [CrossRef]

30. W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B **17**, 4123–4138 (2003). [CrossRef]

**36**, 175 (2000). [CrossRef]

11. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E **55**, 6414 (1997). [CrossRef]

**36**, 175 (2000). [CrossRef]

*T*

_{i}_{+1}vs Δ

*T*are almost symmetric, suggesting that Δ

_{i}*T*

_{i}_{+1}< Δ

*T*and Δ

_{i}*T*

_{i}_{+1}> Δ

*T*are equally probable; however, in Sec. 4 we will demonstrate that the modulation induces correlations in the Δ

_{i}*T*sequence, induced by the modulation, which can not be inferred from these plots.

_{i}## 3. Lang and Kobayashi model and method of symbolic time-series analysis

*E*and the carrier density

*N*are given by [36

36. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347 (1980). [CrossRef]

*τ*and

_{p}*τ*are the photon and carrier lifetimes respectively,

_{N}*α*is the line-width enhancement factor,

*G*is the optical gain,

*G*=

*N*/(1 +

*ε*|

*E*|

^{2}) (with

*ε*being a saturation coefficient),

*μ*is the pump current parameter,

*η*is the feedback strength,

*τ*is the feedback delay time,

*ω*

_{0}

*τ*is the feedback phase, and

*β*is the noise strength, representing spontaneous emission.

_{sp}*μ*=

*μ*

_{0}+

*a*sin(2

*πf*), where

_{mod}t*a*is the modulation amplitude,

*f*is the modulation frequency, and

_{mod}*μ*

_{0}is the dc current. Simulations of 2 ms were performed. The intensity time-series were averaged over a moving window of 1 ns to simulate the bandwidth of the experimental detection system. The averaged time series contained between 12,000 and 30,000 dropouts for low and high modulation amplitude, respectively. The best agreement with the dynamics found in the experimental data was for

*μ*

_{0}= 1.01,

*f*= 21 MHz,

_{mod}*ε*= 0.01,

*k*= 300 ns

^{−1},

*τ*= 5 ns,

*γ*= 1 ns

^{−1},

*β*= 10

_{sp}^{−4}ns

^{−1},

*η*= 10 ns

^{−1}, and

*α*= 4. For these parameters 〈Δ

*T*〉 = 127 ns and 〈Δ

*T*〉 ×

*f*= 2.7.

_{mod}1. C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. **88**, 174102 (2002). [CrossRef] [PubMed]

*D*are defined by considering the relative length of

*D*consecutive IDIs [see Fig. 2(a)]. For

*D*= 2 there are two OPs: Δ

*T*< Δ

_{i}*T*

_{i}_{+1}gives word ‘01’ and Δ

*T*> Δ

_{i}*T*

_{i}_{+1}gives word ‘10’; for

*D*= 3 there are six OPs: Δ

*T*< Δ

_{i}*T*

_{i}_{+1}< Δ

*T*

_{i}_{+2}gives ‘012’, Δ

*T*

_{i}_{+2}< Δ

*T*

_{i}_{+1}< Δ

*T*gives ‘210’, etc. This symbolic transformation keeps the information about correlations present in the dropout sequence, but neglects the information contained in the duration of the IDIs. The words are formed by consecutive non-superposing IDIs (

_{i}*i.e.*, for

*D*= 2, Δ

*T*, Δ

_{i}*T*

_{i}_{+1}define one word and Δ

*T*

_{i}_{+2}, Δ

*T*

_{i}_{+3}define the next one). Then, the probabilities of the different words are computed in each time series.

*D*is much longer than the correlation length, most words will appear in the sequence with similar probabilities. In addition, we need to consider the length of the time-series, because the number of possible words increases with

*D*as

*D*!, and for large

*D*values, long time series will be needed for computing the word probabilities with robust statistics. Here, we recorded long time series of dropouts and the main limitation for the value of

*D*comes from the large level of stochasticity of the LFF dynamics, which results in correlations among only few consecutive dropouts. Thus, we limit the ordinal analysis to

*D*= 2 and

*D*= 3 words. We will show that the LFF symbolic dynamics is such that the analysis with words of

*D*= 3 allows us to uncover correlations which are not seen with

*D*= 2 words.

3. N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison,” Phys. Rev. E **84**, 026202 (2011). [CrossRef]

*D*= 2 words, the TP analysis can uncover correlations among five consecutive dropouts, and thus allows us to extract information about the memory of the system in a longer time scale. The TPs can be normalized in two different ways: normalized for all transitions, such that the sum of all possible TPs is one, ∑

_{i,j}*TP*

_{i}_{→}

*= 1, or normalized for each word*

_{j}*i*, such that ∑

_{j}*TP*

_{i}_{→}

*= 1 ∀*

_{j}*i*. We compute these two sets of TPs corresponding to

*D*= 2 words. For the first set, the TPs are normalized such that TP

_{(01→01)}+TP

_{(01→10)}= 1 and TP

_{(10→01)}+TP

_{(10→10)}= 1. For the second set:

## 4. Results

*D*= 2 (a, b, c) and

*D*= 3 (d, e, f), vs. the modulation amplitude, for the 650 nm laser (a, d), for the 1550 nm laser (b, e), and for the simulated time series (c, f). The gray region indicates probability values consistent with the null hypothesis (NH) that the words are equally probable, and thus, that there are no correlations among the dropouts. In other words, probability values outside the gray regions are not consistent with a uniform distribution of word probabilities and reveal serial correlations in the IDI sequence. It can be noticed that the gray region is narrower in (a, d) than in (b, e) and (c, f). This is due to the fact that the number of dropouts recorded for the 650 nm laser is much larger than for the 1550 nm laser (the corresponding delay times being 4.7 ns and 25 ns respectively), and is also larger than the number of dropouts in the simulated data.

*D*= 2, for small and for high modulation amplitude. However, the analysis with

*D*= 3, reveals that, for high modulation, the probabilities are outside the gray region, revealing correlations among four consecutive IDIs. We note that there are two groups of words, one less probable (‘012’, ‘210’) and one more probable (‘021’,‘102’, ‘120’, ‘201’), resulting, for

*D*= 2, in the same probabilities for ’01’ as for ’10’. With

*D*= 3, the less probable words are those which imply three consecutively increasing or decreasing IDIs and this can be understood in the following terms: strong enough modulation forces a rhythm in the LFF dynamics, and three consecutively increasing or decreasing intervals imply a loss of synchrony with the external rhythm, and thus, are less likely to occur.

*D*= 2 words, depicted in Fig. 4, we obtain information about correlations among five consecutive dropouts. This analysis is statistically more robust than computing the probabilities of the 24 words of length

*D*= 4.

## 5. Conclusions

18. D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. **56**, R3370 (1997). [CrossRef]

## Acknowledgments

## References and links

1. | C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. |

2. | O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, “Distinguishing noise from chaos,” Phys. Rev. Lett. |

3. | N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison,” Phys. Rev. E |

4. | L. Zunino, M. C. Soriano, and O. A. Rosso, “Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach,” Phys. Rev. E |

5. | M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. |

6. | A. Aragoneses, N. Rubido, J. Tiana-alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. |

7. | D. Lenstra, B. H. Verbeek, and A. J. Den Boef, “Coherence collapse in single-mode semiconductor-lasers due to optical feedback,” IEEE J. Quantum. Electron. |

8. | K. Lüdge, editor, |

9. | D. M. Kane and K. A. Shore, eds. |

10. | S. Donati and R-H Horng, “The diagram of feedback regimes revisited,” IEEE J. Sel. Top. Quantum Electron. |

11. | M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E |

12. | A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. |

13. | T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,” Phys. Rev. Lett. |

14. | A. Tabaka, K. Panajotov, I. Veretennicoff, and M. Sciamanna, “Bifurcation study of regular pulse packages in laser diodes subject to optical feedback,” Phys. Rev E |

15. | J. A. Reinoso, J. Zamora-Munt, and C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E |

16. | S. D. Cohen, A. Aragoneses, D. Rontani, M. C. Torrent, C. Masoller, and D. J. Gauthier, “Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback,” Opt. Lett. |

17. | L. Junges, T. Pöschel, and J. A. C. Gallas, “Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model,” Eur. Phys. J. D. |

18. | D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. |

19. | J. Mulet and C. R. Mirasso, “Numerical statistics of power dropouts based on the Lang–Kobayashi mod,” Phys. Rev. E |

20. | M. Sciamanna, C. Masoller, N. B. Abraham, F. Rogister, P. Mégret, and M. Blondel, “Different regimes of low-frequency fluctuations in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B |

21. | J. F. Martínez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Experimental deterministic coherent resonance,” Phys. Rev. Lett. |

22. | Y. Hong and K. A. Shore, “Statistical measures of the power dropout ratio in semiconductor lasers subject to optical feedback,” Opt. Lett. |

23. | A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang–Kobayashi dynamics,” Phys. Rev. A |

24. | J. Zamora-Munt, C. Masoller, and J. Garcia-Ojalvo, “Transient low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A |

25. | K. Hicke, X. Porte, and I. Fischer, “Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback,” Phys. Rev. E |

26. | D. Baums, W. Elsässer, and E. O. Göbel, “Farey tree and Devil’s staircase of a modulated external-cavity semiconductor laser,” Phys. Rev. Lett. |

27. | J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express |

28. | Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E |

29. | D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. |

30. | W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B |

31. | J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E |

32. | F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. |

33. | J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E |

34. | J. M. Buldú, D. R. Chialvo, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Ghost resonance in a semiconductor laser with optical feedback,” Europhys. Lett. |

35. | T. Schwalger, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and B. Lindner, “Interspike-interval correlations induced by two-state switching in an excitable system,” Europhys. Lett. |

36. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.2020) Lasers and laser optics : Diode lasers

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 21, 2014

Revised Manuscript: February 6, 2014

Manuscript Accepted: February 7, 2014

Published: February 21, 2014

**Virtual Issues**

Physics and Applications of Laser Dynamics (2014) *Optics Express*

**Citation**

Andrés Aragoneses, Taciano Sorrentino, Sandro Perrone, Daniel J. Gauthier, M. C. Torrent, and Cristina Masoller, "Experimental and numerical study of the symbolic dynamics of a modulated external-cavity semiconductor laser," Opt. Express **22**, 4705-4713 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4705

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### References

- C. Bandt, B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102 (2002). [CrossRef] [PubMed]
- O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, M. A. Fuentes, “Distinguishing noise from chaos,” Phys. Rev. Lett. 99, 154102 (2007). [CrossRef] [PubMed]
- N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison,” Phys. Rev. E 84, 026202 (2011). [CrossRef]
- L. Zunino, M. C. Soriano, O. A. Rosso, “Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach,” Phys. Rev. E 86, 046210 (2012). [CrossRef]
- M. C. Soriano, L. Zunino, L. Larger, I. Fischer, C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. 36, 2212 (2011). [CrossRef] [PubMed]
- A. Aragoneses, N. Rubido, J. Tiana-alsina, M. C. Torrent, C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013). [CrossRef] [PubMed]
- D. Lenstra, B. H. Verbeek, A. J. Den Boef, “Coherence collapse in single-mode semiconductor-lasers due to optical feedback,” IEEE J. Quantum. Electron. 21, 674–679 (1985). [CrossRef]
- K. Lüdge, editor, Nonlineal Laser Dynamics. From Quantum Dots to Cryptography (Wiley-VCH, 2011). [CrossRef]
- D. M. Kane, K. A. Shore, eds. Unlocking Dynamical Diversity (John Wiley & Sons, 2005). [CrossRef]
- S. Donati, R-H Horng, “The diagram of feedback regimes revisited,” IEEE J. Sel. Top. Quantum Electron. 19, 1500309 (2013). [CrossRef]
- M. Giudici, C. Green, G. Giacomelli, U. Nespolo, J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414 (1997). [CrossRef]
- A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. 83, 292 (1999). [CrossRef]
- T. Heil, I. Fischer, W. Elsäßer, A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,” Phys. Rev. Lett. 87, 243901 (2001). [CrossRef] [PubMed]
- A. Tabaka, K. Panajotov, I. Veretennicoff, M. Sciamanna, “Bifurcation study of regular pulse packages in laser diodes subject to optical feedback,” Phys. Rev E 70, 036211 (2004). [CrossRef]
- J. A. Reinoso, J. Zamora-Munt, C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E 87, 062913 (2013). [CrossRef]
- S. D. Cohen, A. Aragoneses, D. Rontani, M. C. Torrent, C. Masoller, D. J. Gauthier, “Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback,” Opt. Lett. 38, 4331 (2013). [CrossRef] [PubMed]
- L. Junges, T. Pöschel, J. A. C. Gallas, “Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model,” Eur. Phys. J. D. 67, 149 (2013). [CrossRef]
- D. W. Sukow, J. R. Gardner, D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. 56, R3370 (1997). [CrossRef]
- J. Mulet, C. R. Mirasso, “Numerical statistics of power dropouts based on the Lang–Kobayashi mod,” Phys. Rev. E 59, 5400 (1999). [CrossRef]
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