## Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback

Optics Express, Vol. 17, Issue 22, pp. 20124-20133 (2009)

http://dx.doi.org/10.1364/OE.17.020124

Acrobat PDF (527 KB)

### Abstract

We experimentally and numerically demonstrate the time delay (TD) signature suppression of chaotic output in a double optical feedback semiconductor laser (DOF-SL) system. Two types of TD signature suppression are demonstrated by adjusting the lengths and the feedback power ratios of the two external cavities. One can significantly eliminate all TD signatures of the DOF-SL system and the corresponding power spectrum distribution becomes quite smooth and flat, the other suppresses one of two TD signatures and remains another one.

© 2009 OSA

## 1. Introduction

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 **437**(7066), 343–346 (2005). [CrossRef]

7. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**(12), 728–732 (2008). [CrossRef]

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 **437**(7066), 343–346 (2005). [CrossRef]

10. J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. **17**(4), 920–922 (2005). [CrossRef]

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 **437**(7066), 343–346 (2005). [CrossRef]

9. R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. **41**(4), 541–548 (2005). [CrossRef]

10. J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. **17**(4), 920–922 (2005). [CrossRef]

10. J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. **17**(4), 920–922 (2005). [CrossRef]

11. M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **54**(4), R3082–3085 (1996). [CrossRef] [PubMed]

12. R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. **81**(3), 558–561 (1998). [CrossRef]

13. M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D **203**(3-4), 209–223 (2005). [CrossRef]

*et al.*proposed a theoretical investigation of TD signature suppression (TDSS) in a SOF-SL system [14

14. 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]

15. J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. **282**(15), 3153–3156 (2009). [CrossRef]

16. F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express **14**(26), 12859–12867 (2006). [CrossRef] [PubMed]

17. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode,” Opt. Lett. **25**(11), 808–810 (2000). [CrossRef]

18. A. Többens and U. Parlitz, “Dynamics of semiconductor lasers with external multicavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(1), 016210 (2008). [CrossRef] [PubMed]

*et al.*demonstrated the possibility to complicate the TD signature for the first time [19

19. M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. **152**(2), 97–102 (2005). [CrossRef]

19. M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. **152**(2), 97–102 (2005). [CrossRef]

## 2. Experimental setup

## 3. Experimental results and discussion

*l*≈477mm,

_{cav1}*l*≈461mm, respectively. The reflected optical power is monitored at point “T” (see Fig. 1) by an optical power meter and controlled by a variable optical attenuator, and is set as 9.4μw for cavity 1 and 15.3μw for cavity 2 in order to obtain almost equal feedback power ratios (FPR, defined as the ratio of the feedback light power to the laser output power) for SOF1 and SOF2. Above operation conditions can assure that the SL under SOF and DOF cases can all be rendered into chaotic states. For the SOF case, both the chaotic time series of SOF1 and SOF2 as shown in Fig. 2(A1) and Fig. 2(B1) behave intricately and irregularly. However, as observed from corresponding power spectra (Fig. 2(A2) and Fig. 2(B2)), some uniform spacing frequency peaks relevant to the TD signatures obviously emerge upon continuous spectrum background. Following these clues, the TD signature of the SOF-SL chaotic system may be retrieved [13

_{cav2}13. M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D **203**(3-4), 209–223 (2005). [CrossRef]

14. 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]

20. V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay- differential equations,” J. Opt. Technol. **72**, 373–377 (2005). [CrossRef]

21. M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **56**(5), 5083–5089 (1997). [CrossRef]

22. S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A **351**(1), 133–141 (2005). [CrossRef]

12. R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. **81**(3), 558–561 (1998). [CrossRef]

*P*(t) represents chaotic time series, Δ

*t*is the time shift. The TD signature can be retrieved from the peak location of SF curve. The MI between

*P*(t) and

*P*(t + Δt) is described by:Where

*δ*(

*P*(t),

*P*(t + Δt)) is the joint probability,

*δ*(

*P*(t)) and

*δ*(

*P*(t + Δt)) are the marginal probability densities. The peak location of MI can also characterize the TD signature.

_{cav1}≈3.25ns and τ

_{cav2}≈3.15ns) can be clearly extracted from the peaks location of SF and MI curves in Fig. 3(A)-(B). However, for the DOF case as shown in Fig. 3(C1) and Fig. 3(C2), both SF and MI curves show no significant peaks at τ

_{cav1}and τ

_{cav2}, and all the TD signatures have almost been depressed into background. Therefore, such a DOF configuration can afford a possible TD signature concealment scheme under certain conditions. In addition, there are some small troughs with a period of 0.2ns in these SF curves, which characterizes the relaxation oscillation period (τ

_{RO}) of the SL as τ

_{RO}≈0.2ns [14

14. 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]

23. C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. **128**(4-6), 363–376 (1996). [CrossRef]

## 4. Theoretical simulation and analysis

*E*and the average carrier number

*N*in the active region, can be expressed as:

*κ*and

_{cav1}*κ*denote the feedback strengths of external cavity 1 and external cavity 2,

_{cav1}*τ*and

_{cav1}*τ*are the delay times corresponding to the round-trip time of the external cavity 1 and the external cavity 2,

_{cav2}*β*is the linewidth-enhancement factor.

*G*(t) =

*g*(

*N*(t)−

*N*)/(1 + ε

_{0}*E*(t)

*) is the gain coefficient, where*

^{2}*g*is differential gain coefficient, ε is the gain saturation coefficient and

*N*is the transparency carrier number.

_{0}*ω*is the angular frequency of SL,

_{0}*γ*is the photon loss rate.

_{p}*γ*= 1/

_{p}*τ*, where

_{p}*τ*is the photon lifetime.

_{p}*τ*is the round-trip time in SL intracavity,

_{L}*F*(t) is the spontaneous-emission noise,

*J*is the injection carrier rate,

*τ*is the carrier lifetime. The relaxation oscillation period

_{N}*τ*is an intrinsic damping time of the free-running laser and could be estimated by

_{RO}*τ*≈2π(

_{RO}*gE*

^{2}/

*τ*)

_{p}^{-1/2}. Equation (3) can be solved by using the fourth-order Runge-Kutta algorithm, and the parameters are set as:

*β*= 4,

*ω*= 1.216 × e

_{0}^{15}rad/s,

*τ*= 4.2ps,

_{p}*τ*= 8.5ps,

_{L}*τ*= 1.6ns,

_{N}*g*= 2 × 10

^{4}s

^{−1},

*F*(t) = 0,

*N*= 1.25 × 10

_{0}^{8}, and

*ε*= 1 × 10

^{−7}. For comparison with experimental results,

*J*is set as 1.6

*J*with

_{th}*τ*≈0.2ns,

_{RO}*τ*is fixed as 3.2ns and

_{cav1}*κ*= 0.04.

_{cav1}*τ*and

_{cav2}*κ*are variable for different considerations.

_{cav2}*τ*= 3.2ns,

_{cav1}*τ*= 3.12ns,

_{cav2}*κ*= 0.04,

_{cav1}*κ*= 0.05, respectively. For the SOF situation ((A) and (B)), clear TD peaks can be seen. But for the DOF configuration ((C)), the entire TD signature is attenuated significantly. Comparing this diagram with Figs. 2-3, the numerical calculations relatively successfully demonstrate the experimental results. Further calculations show that by reducing

_{cav2}*τ*and increasing

_{cav2}*κ*to suitable value, the second type TDSS as shown in Fig. 5 can also be simulated.

_{cav2}*τ*= 3.2ns and

_{cav1}*κ*=

_{cav1}*κ*= 0.04 and

_{cav2}*τ*varies from 0.4ns to 4ns. As shown in this diagram, the diagonal line Δt≈

_{cav2}*τ*and the vertical line Δt≈3.2ns reveal the TD signatures of cavity 2 and of cavity 1. Meantime, there exist numerous relatively weak peaks and valleys with a period of about 0.1ns≈1/2

_{cav2}*τ*. Two obvious twisted regions can be observed around

_{RO}*τ*≈1.6ns and

_{cav2}*τ*≈3.2ns. For the region around

_{cav2}*τ*≈3.2ns, the TD signatures become very weak at several separated sub-regions such as

_{cav2}*τ*≈2.92ns,

_{cav2}*τ*≈3.12ns,

_{cav2}*τ*≈3.28ns and

_{cav2}*τ*≈3.48ns, which indicates possible TDSS sub-regions. Interestingly, these TDSS sub-regions does not locate at the point where

_{cav2}*τ*equals to

_{cav2}*τ*strictly, but periodically distribute around

_{cav1}*τ*and roughly follow relations:

_{cav1}*τ*-

_{cav2}*τ*≈-3/2

_{cav1}*τ*, −1/2

_{RO}*τ*, 1/2

_{RO}*τ*, 3/2

_{RO}*τ*, respectively. Here, we name them as type I TDSS. As for the region around

_{RO}*τ*≈1.6ns, there also appear some slight TDSS sub-regions around

_{cav2}*τ*≈1.44ns,

_{cav2}*τ*≈1.54ns,

_{cav2}*τ*≈1.64ns and

_{cav2}*τ*≈1.74ns, respectively, and these sub-regions roughly satisfy following relations: 2

_{cav2}*τ*-

_{cav2}*τ*≈-3/2

_{cav1}*τ*, −1/2

_{RO}*τ*, 1/2

_{RO}*τ*, 3/2

_{RO}*τ*. Here, we name them as type II TDSS. Further calculations show that by selecting suitable feedback strength, the TDSS effect, which suppresses the TD signature of one external cavity and remains TD signature of another external cavity, can be strengthened. Above theoretical results reveal that there exist some special regions where two types of TDSS happens just as observed in our experiments.

_{RO}*τ*= 3.2ns and

_{cav1}*τ*= 3.12ns) and type II TDSS (

_{cav2}*τ*= 3.2ns and

_{cav1}*τ*= 1.64ns). From these two maps (Figs. 10 (A1)-(B1)), one can see that the theoretical results verifies previous experimental observations to certain degree. Furthermore, Figs. 10 (A2)-(B2) give the variation of the amplitude

_{cav2}*ρ*, which is the maximum of the SF peak in a time shift window around 3.2ns, with the feedback strength. These two curves also confirm experimental tendency as shown in Fig. 4 and Fig. 6.

## 5. Conclusion

*τ*plays an important role during TDSS. We hope that this work will be helpful for enhancing security in chaotic cryptosystems. Also, this work may offer some useful insights for the nonlinear dynamical properties of such DOF-SL system.

_{RO}## Acknowledgments

## References and links

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 |

2. | V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. |

3. | G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express |

4. | J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express |

5. | F. Y. Lin and J. M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. |

6. | F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. |

7. | A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics |

8. | X. F. Wang, G. Q. Xia, and Z. M. Wu, “Theoretical investigations on the polarization performances of current-modulated VCSELs subject to weak optical feedback,” J. Opt. Soc. Am. B |

9. | R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. |

10. | J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. |

11. | M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

12. | R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. |

13. | M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D |

14. | 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. |

15. | J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. |

16. | F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express |

17. | F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode,” Opt. Lett. |

18. | A. Többens and U. Parlitz, “Dynamics of semiconductor lasers with external multicavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. |

20. | V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay- differential equations,” J. Opt. Technol. |

21. | M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

22. | S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A |

23. | C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 9, 2009

Revised Manuscript: October 15, 2009

Manuscript Accepted: October 16, 2009

Published: October 20, 2009

**Citation**

Jia-Gui Wu, Guang-Qiong Xia, and Zheng-Mao Wu, "Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback," Opt. Express **17**, 20124-20133 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20124

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### References

- 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 437(7066), 343–346 (2005). [CrossRef]
- V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. 30(7), 1537–1541 (1994). [CrossRef]
- G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005). [CrossRef] [PubMed]
- J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17(15), 12619–12626 (2009). [CrossRef] [PubMed]
- F. Y. Lin and J. M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. 40(6), 682–689 (2004). [CrossRef]
- F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004). [CrossRef]
- A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
- X. F. Wang, G. Q. Xia, and Z. M. Wu, “Theoretical investigations on the polarization performances of current-modulated VCSELs subject to weak optical feedback,” J. Opt. Soc. Am. B 26(1), 160–168 (2009). [CrossRef]
- R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005). [CrossRef]
- J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (2005). [CrossRef]
- M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996). [CrossRef] [PubMed]
- R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998). [CrossRef]
- M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005). [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]
- J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282(15), 3153–3156 (2009). [CrossRef]
- F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express 14(26), 12859–12867 (2006). [CrossRef] [PubMed]
- F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode,” Opt. Lett. 25(11), 808–810 (2000). [CrossRef]
- A. Többens and U. Parlitz, “Dynamics of semiconductor lasers with external multicavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(1), 016210 (2008). [CrossRef] [PubMed]
- M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005). [CrossRef]
- V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay- differential equations,” J. Opt. Technol. 72, 373–377 (2005). [CrossRef]
- M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997). [CrossRef]
- S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (2005). [CrossRef]
- C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. 128(4-6), 363–376 (1996). [CrossRef]

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