## Complexity in pulsed nonlinear laser systems interrogated by permutation entropy |

Optics Express, Vol. 22, Issue 15, pp. 17840-17853 (2014)

http://dx.doi.org/10.1364/OE.22.017840

Acrobat PDF (1563 KB)

### Abstract

Permutation entropy (PE) has a growing significance as a relative measure of complexity in nonlinear systems. It has been applied successfully to measuring complexity in nonlinear laser systems. Here, PE and weighted permutation entropy (WPE) are discovered to show an unexpected inversion to higher values, when characterizing the complexity at the characteristic frequencies of nonlinear drivers in laser systems, for output power sequences which are pulsed. The cause of this inversion is explained and its presence can be used to identify when irregular dynamics transform into a fairly regular pulsed signal (with amplitude and timing jitter). When WPE is calculated from experimental output power time series from various nonlinear laser systems as a function of delay time, both the minimum value of WPE, and the width of the peak in the WPE plot are shown to be indicative of the level of amplitude variation and timing jitter present in the pulsed output. Links are made with analysis using simulated time series data with systematic variation in timing jitter and/or amplitude variations.

© 2014 Optical Society of America

## 1. Introduction

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

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]

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

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

6. M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A **369**(2), 439–462 (2006). [CrossRef]

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

7. 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]

8. 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]

8. 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. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express **22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

11. J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A **82**(1), 013819 (2010). [CrossRef]

13. 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]

14. S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**(3), 033810 (2005). [CrossRef]

15. Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. **100**(1), 013907 (2008). [CrossRef] [PubMed]

16. N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. **45**(11), 1388–1395 (2009). [CrossRef]

17. D. M. Kane and J. P. Toomey, “Variable pulse repetition frequency output from an optically injected solid state laser,” Opt. Express **19**(5), 4692–4702 (2011). [CrossRef] [PubMed]

18. J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express **17**(9), 7592–7608 (2009). [CrossRef] [PubMed]

19. T. Ackemann, N. Radwell, C. McIntyre, G. L. Oppo, and W. J. Firth, “Self pulsing solitons: A base for optically controllable pulse trains in photonic networks?” in Transparent Optical Networks (ICTON), 2010 12th International Conference on, 2010), 1–4. [CrossRef]

## 2. Permutation entropy

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

9. 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*and ordinal pattern delay

*τ*. There are

*D*! possible permutations for a vector of length

*D*, so in order to obtain reliable statistics the length of the time series

*N*should be much larger than

*D*[21

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

*τ.*This is the time separation between values used to construct the vector from which the ordinal pattern is determined. Its value corresponds to a multiple of the signal sampling period. For a given time series

*D*, and ordinal pattern delay

*τ*, we consider the vectorAt each time

*s*the ordinal pattern of this vector can be converted to a unique symbol

*D*! possible permutations

*π*. The normalized permutation entropy is then defined as the normalized Shannon entropy

_{i}*S*associated with the permutation probability distribution

*P*,This normalized permutation entropy gives values

22. B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **87**(2), 022911 (2013). [CrossRef] [PubMed]

*X*, which matches a certain permutation, contributing equally to the entropy, each is weighted using the variance by,Where

_{s}*X*defined in Eq. (2). This definition retains the general properties of the original PE algorithm whilst incorporating amplitude information and more robustness to noise. In this study we employ this weighted PE as it is more appropriate for analysis of pulse-like time series [22

_{s}22. B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **87**(2), 022911 (2013). [CrossRef] [PubMed]

## 3. Experimental results

### 3.1 Results for conventional delayed optical feedback

*τ*quantifies the relative complexity of the time series on different time scales. This analysis has been previously applied to output power time series from semiconductor lasers with optical feedback (SLWOF) operating in a chaotic region of its parameter space, and has identified time scales associated with key frequencies inherent to the system being studied [9

9. 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]

10. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express **22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

*N*= 20,000 and we use ordinal pattern length

*D*= 5. This ensures

*τ*

_{ext}) and relaxation oscillation (

*τ*

_{RO}) frequencies, and integer fractions of these [10

10. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express **22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

*τ*

_{ext}), the period corresponding to the relaxation oscillation frequency (

*τ*

_{RO}) and multiples or fractions of these.

*τ*that correspond to the pulse period. A number of examples of this type of behavior have been observed and analyzed. They are describe in more detail in the following sections.

### 3.2 Vertical-cavity surface-emitting laser with frequency-selective feedback

16. N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. **45**(11), 1388–1395 (2009). [CrossRef]

16. N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. **45**(11), 1388–1395 (2009). [CrossRef]

19. T. Ackemann, N. Radwell, C. McIntyre, G. L. Oppo, and W. J. Firth, “Self pulsing solitons: A base for optically controllable pulse trains in photonic networks?” in Transparent Optical Networks (ICTON), 2010 12th International Conference on, 2010), 1–4. [CrossRef]

*τ*at multiples of ~0.3 ns, seen in Fig. 2(b). The peaks indicate a higher irregularity at the round-trip frequency and harmonics/sub-harmonics. Interestingly, the peaks are not sitting on a flat background but there are indications for an undershoot before and after the main peak, i.e. the peaks might be interpreted as a central narrow spike on a broader minimum. This observation will be supported by the discussion of the data in Sec. 3.3.

### 3.3 Solid state laser with optical injection

14. S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**(3), 033810 (2005). [CrossRef]

17. D. M. Kane and J. P. Toomey, “Variable pulse repetition frequency output from an optically injected solid state laser,” Opt. Express **19**(5), 4692–4702 (2011). [CrossRef] [PubMed]

*ω*= −1.849 and normalized injection strength

*K*= 0.226. Both these experimentally measured quantities were normalized to the angular relaxation oscillation frequency (see [14

14. S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**(3), 033810 (2005). [CrossRef]

*τ*equal to multiples of the pulse period. This matches the previous observation for the VCSEL of an increase in complexity at ordinal pattern delays corresponding to the pulse period, however there is a much more noticeable drop either side of the peak than was observed in the VCSEL plot in Fig. 2(b).

### 3.4 Discussion of experimental results

*τ*close to, but not equal to the pulse period. These are the dips seen either side of the peaks in Figs. 2(b) and 3(b).

## 4. Simulating random amplitude and temporal variations to pulses

*π*or 158 sampling points. For this clean signal, where each pulse has exactly the same amplitude, the permutation entropy drops to zero when the ordinal pattern delay

*τ*exactly matches the pulse period (

*π*). In this situation all values of the sub vector are equal and therefore the probability distribution has only one element with a value of 1, with the remaining

*D*! possible ordinal patterns having a probability of zero. To investigate the effect of unequal pulse amplitudes, each pulse was multiplied by a very small random number (between 1 and 10

^{−5}and 1 + 10

^{−5}). The plot comparing the WPE calculated from the clean and perturbed signals is shown in Fig. 4(b). The sharp peak at

*τ*= 158 (pulse period) observed for the perturbed signal is similar to that observed in the pulsed laser systems in Fig. 2 and Fig. 3. This provides a first indication that the WPE peak can be caused by inequalities in the pulse amplitudes.

### 4.1 Amplitude noise

*τ*= 158, increases linearly as seen in Fig. 6(a). This is observed as a result of the time series points adjacent to the pulse peak becoming larger than the peak of neighboring pulses.

*τ*= 158. This invariance of the maximum WPE value to amplitude noise is expected since it is only based on the relative order of the points. The value of the WPE at the peak is mostly indicative of the complexity of the set of random numbers used to generate the pulse amplitudes for that particular noise level. This is illustrated by the plot in Fig. 6(b) which shows the WPE calculated on the random numbers used to generate the simulated pulse amplitudes with delay

*τ*= 1 (

*D*= 4) and the WPE of the time series with delay

*τ*= 158 (

*D*= 4). Since the number of points being analyzed here is

*N*= 128 (i.e. the number of pulses), an ordinal pattern length of

*D*= 4 was used in this case in order to satisfy the requirement that

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

*τ*= 158 (red dots) is the same as the WPE calculated on the set of random numbers used for pulse amplitudes (black squares).

### 4.2 Temporal noise

## 5. Discussion

**22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

*τ*= 4.5 ns) and Fig. 9(a) the time series from which it was calculated shows fully developed chaotic operation at 60 mA. At an injection current of 51 mA the PE plot in Fig. 9(e) show the inverted peak resulting from the pulsed laser operation shown in Fig. 9(e).

## 6. Comparing stochastic simulations with nonlinear laser dynamics

11. J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A **82**(1), 013819 (2010). [CrossRef]

13. 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]

**72**(3), 033810 (2005). [CrossRef]

19. T. Ackemann, N. Radwell, C. McIntyre, G. L. Oppo, and W. J. Firth, “Self pulsing solitons: A base for optically controllable pulse trains in photonic networks?” in Transparent Optical Networks (ICTON), 2010 12th International Conference on, 2010), 1–4. [CrossRef]

*D*= 4 and ordinal pattern delay

*τ*= 1. Since we are only using the pulse amplitudes and pulse intervals, rather than the full time series, we have much lower

*N*(see second column in Table 1). As such we use

*D*= 4 so that

**22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

18. J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express **17**(9), 7592–7608 (2009). [CrossRef] [PubMed]

23. 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]

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

18. J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express **17**(9), 7592–7608 (2009). [CrossRef] [PubMed]

23. 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]

## 7. Conclusion

9. 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]

**22**(2), 1713–1725 (2014). [CrossRef] [PubMed]

*τ*just below and just above the pulse period, sample the rising and falling edges of the pulses and are represented by the same ordinal patterns. This increased occurrence of certain patterns results in a drop in the permutation entropy at these delays. When the values of

*τ*matches the pulse period then the WPE will either drop to zero, if the pulse amplitudes are equal, or will increase to a value which quantifies the entropy of the probability distribution of the pulse amplitudes. In the case where the pulses have randomly varying amplitude the WPE will approach 1. This feature was observed in WPE plots for output power time series recorded from a vertical cavity surface emitting laser with frequency selective feedback [16

**45**(11), 1388–1395 (2009). [CrossRef]

**72**(3), 033810 (2005). [CrossRef]

*τ*, emerges as a signal to identify pulsed dynamics (albeit with some level of amplitude and/or temporal noise) in nonlinear laser systems where otherwise this might have been missed if researchers are not studying the detail of the time series data.

## Acknowledgments

## References and links

1. | P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” 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 |

3. | H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A |

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

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

6. | M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A |

7. | 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. |

8. | 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. |

9. | 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. |

10. | J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express |

11. | J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A |

12. | 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 Stat. Nonlin. Soft Matter Phys. |

13. | 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. |

14. | S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A |

15. | Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. |

16. | N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. |

17. | D. M. Kane and J. P. Toomey, “Variable pulse repetition frequency output from an optically injected solid state laser,” Opt. Express |

18. | J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express |

19. | T. Ackemann, N. Radwell, C. McIntyre, G. L. Oppo, and W. J. Firth, “Self pulsing solitons: A base for optically controllable pulse trains in photonic networks?” in Transparent Optical Networks (ICTON), 2010 12th International Conference on, 2010), 1–4. [CrossRef] |

20. | N. Radwell, “Characteristics of a cavity soliton laser based on a VCSEL with frequency selective feedback,” PhD Thesis (University of Strathclyde, 2010). |

21. | M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos |

22. | B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.3580) Lasers and laser optics : Lasers, solid-state

(140.5960) Lasers and laser optics : Semiconductor lasers

(140.3538) Lasers and laser optics : Lasers, pulsed

(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 24, 2014

Revised Manuscript: May 5, 2014

Manuscript Accepted: July 4, 2014

Published: July 16, 2014

**Citation**

Joshua P. Toomey, Deborah M. Kane, and Thorsten Ackemann, "Complexity in pulsed nonlinear laser systems interrogated by permutation entropy," Opt. Express **22**, 17840-17853 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-17840

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### References

- P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D9(1-2), 189–208 (1983). [CrossRef]
- M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D65(1-2), 117–134 (1993). [CrossRef]
- H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A185(1), 77–87 (1994). [CrossRef]
- H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (Cambridge University Press, Cambridge, 2004).
- C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett.88(17), 174102 (2002). [CrossRef] [PubMed]
- M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A369(2), 439–462 (2006). [CrossRef]
- 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]
- 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]
- J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express22(2), 1713–1725 (2014). [CrossRef] [PubMed]
- J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A82(1), 013819 (2010). [CrossRef]
- 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 Stat. Nonlin. Soft Matter Phys.84(2), 026202 (2011). [CrossRef] [PubMed]
- 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]
- S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A72(3), 033810 (2005). [CrossRef]
- Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett.100(1), 013907 (2008). [CrossRef] [PubMed]
- N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron.45(11), 1388–1395 (2009). [CrossRef]
- D. M. Kane and J. P. Toomey, “Variable pulse repetition frequency output from an optically injected solid state laser,” Opt. Express19(5), 4692–4702 (2011). [CrossRef] [PubMed]
- J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express17(9), 7592–7608 (2009). [CrossRef] [PubMed]
- T. Ackemann, N. Radwell, C. McIntyre, G. L. Oppo, and W. J. Firth, “Self pulsing solitons: A base for optically controllable pulse trains in photonic networks?” in Transparent Optical Networks (ICTON), 2010 12th International Conference on, 2010), 1–4. [CrossRef]
- N. Radwell, “Characteristics of a cavity soliton laser based on a VCSEL with frequency selective feedback,” PhD Thesis (University of Strathclyde, 2010).
- M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos17(10), 3729–3733 (2007). [CrossRef]
- B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.87(2), 022911 (2013). [CrossRef] [PubMed]
- J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express18(16), 16955–16972 (2010). [CrossRef] [PubMed]

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