## Synchronization in small networks of time-delay coupled chaotic diode lasers |

Optics Express, Vol. 20, Issue 4, pp. 4352-4359 (2012)

http://dx.doi.org/10.1364/OE.20.004352

Acrobat PDF (1150 KB)

### Abstract

Topologies of two, three and four time-delay-coupled chaotic semiconductor lasers are experimentally and theoretically found to show new types of synchronization. Generalized zero-lag synchronization is observed for two lasers separated by long distances even when their self-feedback delays are not equal. Generalized sub-lattice synchronization is observed for quadrilateral geometries while the equilateral triangle is zero-lag synchronized. Generalized zero-lag synchronization, without the limitation of precisely matched delays, opens possibilities for advanced multi-user communication protocols.

© 2012 OSA

## 1. Introduction

6. 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]

7. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**(1), 58–61 (2010). [CrossRef]

8. A. Uchida, P. Davis, and S. Itaya, “Generation of information theoretic secure keys using a chaotic semiconductor laser,” Appl. Phys. Lett. **83**(15), 3213–3215 (2003). [CrossRef]

10. G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. **88**(9), 097903 (2002). [CrossRef] [PubMed]

11. A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. **101**(15), 154102 (2008). [CrossRef] [PubMed]

14. M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(4), 045201 (2007). [CrossRef] [PubMed]

15. 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]

16. Y. Aviad, I. Reidler, W. Kinzel, I. Kanter, and M. Rosenbluh, “Phase synchronization in mutually coupled chaotic diode lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(2), 025204 (2008). [CrossRef] [PubMed]

## 2. Generalized zero-lag synchronization

_{M}= 2τ + τ

_{a}+ τ

_{b}and self-feedback delay times are T

_{A}= 2(τ + τ

_{a}) and T

_{B}= 2(τ + τ

_{b}). Such a geometry obeys the simplest necessary sum rule for synchronization, where the sum of the two self-feedback delay times is equal to twice the mutual delay time while τ

_{a}, τ

_{b}and τ can be independently varied [14

14. M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(4), 045201 (2007). [CrossRef] [PubMed]

18. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization and multiple time delays in two coupled chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **81**(3), 036215 (2010). [CrossRef] [PubMed]

19. K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **83**(5), 056211 (2011). [CrossRef] [PubMed]

20. V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(6), 7208–7213 (1998). [CrossRef]

21. Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuation regime,” Opt. Lett. **24**(22), 1570–1572 (1999). [CrossRef] [PubMed]

_{a}= τ

_{b}, ZLS with cross correlation, ρ = 0.96, is achieved, Fig. 1(c) (light blue). The correlation repeaks at integer multiples of the round trip time, 2(2τ + τ

_{a}+ τ

_{b}), the first pair of which are shown by the light blue trace in Fig. 1(d).

*any distance*of the two lasers to the BS, predicts a generalized ZLS correlation peak at Δτ = τ

_{b}-τ

_{a}[18

18. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization and multiple time delays in two coupled chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **81**(3), 036215 (2010). [CrossRef] [PubMed]

15. 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]

16. Y. Aviad, I. Reidler, W. Kinzel, I. Kanter, and M. Rosenbluh, “Phase synchronization in mutually coupled chaotic diode lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(2), 025204 (2008). [CrossRef] [PubMed]

1. E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(6), 066214 (2006). [CrossRef] [PubMed]

17. M. Rosenbluh, Y. Aviad, E. Cohen, L. Khaykovich, W. Kinzel, E. Kopelowitz, P. Yoskovits, and I. Kanter, “Spiking optical patterns and synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(4), 046207 (2007). [CrossRef] [PubMed]

_{a}-τ

_{b}~-1.95 ns. Indeed, when the two lasers are coupled as in Fig. 1(a), the cross correlation between the two lasers revealed a generalized ZLS, ρ = 0.95, at Δτ~-1.95 ns, Fig. 1(c) (blue). The notable oscillations with period ~0.3 ns in the cross correlation are attributed to the relaxation oscillation frequency of the SC lasers. Similar oscillations were also observed in simulations of the Lang-Kobayashi equations [22

22. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**(3), 347–355 (1980). [CrossRef]

3. I. Kanter, M. Zigzag, A. Englert, F. Geissler, and W. Kinzel, “Synchronization of unidirectional time delay chaotic networks and the greatest common divisor,” Europhys. Lett. **93**(6), 60003 (2011). [CrossRef]

23. I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, D. Cohen, and W. Kinzel, “Nonlocal mechanism for synchronization of time delay networks,” J. Stat. Phys. **145**(3), 713–733 (2011). [CrossRef]

25. V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizing distant nodes: a universal classification of networks,” Phys. Rev. Lett. **105**(25), 254101 (2010). [CrossRef] [PubMed]

_{A,n}= 2(τ+τ

_{a})n and T

_{B,n}= 2(τ+τ

_{b})n, where n is an integer. Because the two lasers are synchronized, both periodicities must be present in the common synchronized fluctuations of the lasers. As a consequence the correlation has additional peaks at time shifts ±T

_{A,n}and ±T

_{B,n}, with respect to the generalized ZLS peak (Fig. 1(d))

_{.}The quasi-periodicity at these two frequencies produces sidebands at ±(T

_{A,n}-T

_{B,n}) = ±2(τ

_{a}-τ

_{b}), which are observed surrounding both the generalized ZLS peak (Fig. 1(c)) and the recurring correlations at ±T

_{A,1}and ±T

_{B,1}(Fig. 1(d)). Twice the mutual feedback repetition time is precisely equal to the sum of the two self-feedback times, as in Eq. (1), which insures that all correlation peaks are at the same time shifts.

## 3. Generalized star configuration

26. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. **105**(26), 264101 (2010). [CrossRef] [PubMed]

_{k}}. Figures 1(a)-1(b) is a special case of this geometry with n = 2. Each pair (m,k) in the general configuration thus has generalized ZLS at a time shift corresponding to the delay time difference between the pair, τ

_{m}-τ

_{k}. The solutions of the time shifts for all possible pair combinations are consistent with each other even when multiple traverses are allowed. For instance, the time shift for pair (m,k), τ

_{k}-τ

_{m}, is identical to the sum of all possible delays in getting from m to k; (m,m + 1), (m + 1,m + 2), …, (k-1,k).

22. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**(3), 347–355 (1980). [CrossRef]

## 4. Generalized sub-lattice synchronization

*without self-feedback*, as shown in Fig. 2(a) . The mirror, M, of Fig. 1(a) is replaced by two additional SC lasers, LD

_{C}and LD

_{D}, with optical delay to their coupling BS, τ

_{c}and τ

_{d}, respectively. The delay time τ between the two coupling BSs is arbitrary and can be made as long as necessary by insertion of a fiber, though in the experiment described below it was maintained at a free space value of a few ns. The resulting system has quadrilateral geometry and is shown schematically in Fig. 2(b). It is important to note that the four independent delays (τ

_{a},τ

_{b},τ

_{c},τ

_{d}) in this geometry do not form an unconstrained quadrilateral, since the difference between the delays (D,A)-(A,C) has to be equal to (D,B)-(B,C).

_{a}=τ

_{b}=τ

_{c}=τ

_{d}, the geometry reduces to a square, for which sub-lattice synchronization was theoretically predicted, with the four lasers split into diagonal pairs, (A,B) and (C,D), which are ZLS [27

27. J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(3), 035202 (2007). [CrossRef] [PubMed]

_{a}+τ), similar to a face-to-face configuration of two lasers.

_{A}and LD

_{B}at a shifted time τ

_{a}-τ

_{b}~-0.03 ns and ρ = 0.94 between LD

_{C}and LD

_{D}at a shifted time of τ

_{c}-τ

_{d}~-0.195 ns. Additional attenuated correlation peaks between the diagonal lasers pairs occur at ±37.28 ns around the central generalized ZLS peaks. The adjacent laser pairs, however, have no correlation near zero time delay but have correlation (not shown) at times corresponding to an effective face to face delay time as described below. Note that the background correlation in Fig. 1(c) is slightly above zero because for these experimental parameters LFFs occur at times > ~100 ns and in order to avoid data segments containing LFFs only the highest 50% of the correlation segments are averaged.

20. V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(6), 7208–7213 (1998). [CrossRef]

22. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**(3), 347–355 (1980). [CrossRef]

_{th}=1.2, similar to the experimental operating parameters. The time delays for the simulated quadrilateral geometry were arbitrarily selected to be τ

_{a}=2 ns, τ

_{b}=4 ns, τ

_{c}=3 ns, τ

_{d}=7 ns and τ=11 ns, thus yielding a ZLS delay between laser A and B, τ

_{a}-τ

_{b}=2 ns002C and between lasers C and D, τ

_{c}-τ

_{d}=4 ns, Fig. 3(a) . To achieve a better understanding of this type of generalized synchronization, the quadrilateral geometry is transformed into a square, Fig. 3(b), in the following way: Since we know that laser D has a generalized ZLS with laser C of 4 ns we move along the laser emission line of laser D by 4 ns towards A and B, to create a virtual laser, that would be in ZLS with C. Similarly we follow the laser B emission to a virtual position by moving it 2 ns towards C and D so that virtual laser B is now in ZLS with laser A. We have thus created a square configuration, indicated by the dashed line in Fig. 3(b). From sub-lattice synchronization in a square geometry [27

27. J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(3), 035202 (2007). [CrossRef] [PubMed]

## 5. Equilateral triangle – smallest network with ZLS

*absence of self-feedback*; an equilateral triangle, which was previously examined in the presence of self-feedback [28

28. C. González, C. Masoller, M. Torrent, and J. García-Ojalvo, “Synchronization via clustering in a small delay-coupled laser network,” Europhys. Lett. **79**(6), 64003 (2007). [CrossRef]

_{a}=τ

_{b}=τ

_{c}. Experimental results, with p = 1.1, show ZLS between all three pairs of lasers, ρ

_{AB}= 0.93, ρ

_{BC}= 0.92, ρ

_{AC}= 0.91 at zero time shift, as shown in Fig. 4(c), consistent with simulations.

## 6. Conclusion

## Acknowledgments

## References and links

1. | E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

2. | O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos |

3. | I. Kanter, M. Zigzag, A. Englert, F. Geissler, and W. Kinzel, “Synchronization of unidirectional time delay chaotic networks and the greatest common divisor,” Europhys. Lett. |

4. | R. Vardi, A. Wallach, E. Kopelowitz, M. Abeles, S. Marom, and I. Kanter, “Synthetic reverberating activity patterns embedded in networks of cortical neurons,” Arxiv preprint arXiv:1201.0339 (2012). |

5. | M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. |

6. | 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 |

7. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics |

8. | A. Uchida, P. Davis, and S. Itaya, “Generation of information theoretic secure keys using a chaotic semiconductor laser,” Appl. Phys. Lett. |

9. | I. Kanter, M. Butkovski, Y. Peleg, M. Zigzag, Y. Aviad, I. Reidler, M. Rosenbluh, and W. Kinzel, “Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography,” Opt. Express |

10. | G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. |

11. | A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. |

12. | B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. |

13. | B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, and R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

14. | M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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

16. | Y. Aviad, I. Reidler, W. Kinzel, I. Kanter, and M. Rosenbluh, “Phase synchronization in mutually coupled chaotic diode lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

17. | M. Rosenbluh, Y. Aviad, E. Cohen, L. Khaykovich, W. Kinzel, E. Kopelowitz, P. Yoskovits, and I. Kanter, “Spiking optical patterns and synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

18. | M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization and multiple time delays in two coupled chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

20. | V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

21. | Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuation regime,” Opt. Lett. |

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

23. | I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, D. Cohen, and W. Kinzel, “Nonlocal mechanism for synchronization of time delay networks,” J. Stat. Phys. |

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. |

25. | V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizing distant nodes: a universal classification of networks,” Phys. Rev. Lett. |

26. | J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. |

27. | J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

28. | C. González, C. Masoller, M. Torrent, and J. García-Ojalvo, “Synchronization via clustering in a small delay-coupled laser network,” Europhys. Lett. |

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.3325) Lasers and laser optics : Laser coupling

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 13, 2011

Revised Manuscript: January 17, 2012

Manuscript Accepted: January 20, 2012

Published: February 7, 2012

**Citation**

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, "Synchronization in small networks of time-delay coupled chaotic diode lasers," Opt. Express **20**, 4352-4359 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4352

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### References

- E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(6), 066214 (2006). [CrossRef] [PubMed]
- O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos18(3), 037116 (2008). [CrossRef] [PubMed]
- I. Kanter, M. Zigzag, A. Englert, F. Geissler, and W. Kinzel, “Synchronization of unidirectional time delay chaotic networks and the greatest common divisor,” Europhys. Lett.93(6), 60003 (2011). [CrossRef]
- R. Vardi, A. Wallach, E. Kopelowitz, M. Abeles, S. Marom, and I. Kanter, “Synthetic reverberating activity patterns embedded in networks of cortical neurons,” Arxiv preprint arXiv:1201.0339 (2012).
- M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett.106(22), 223901 (2011). [CrossRef] [PubMed]
- 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. Photonics2(12), 728–732 (2008). [CrossRef]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010). [CrossRef]
- A. Uchida, P. Davis, and S. Itaya, “Generation of information theoretic secure keys using a chaotic semiconductor laser,” Appl. Phys. Lett.83(15), 3213–3215 (2003). [CrossRef]
- I. Kanter, M. Butkovski, Y. Peleg, M. Zigzag, Y. Aviad, I. Reidler, M. Rosenbluh, and W. Kinzel, “Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography,” Opt. Express18(17), 18292–18302 (2010). [CrossRef] [PubMed]
- G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett.88(9), 097903 (2002). [CrossRef] [PubMed]
- A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett.101(15), 154102 (2008). [CrossRef] [PubMed]
- B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett.107(3), 034102 (2011). [CrossRef] [PubMed]
- B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, and R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.80(5), 056205 (2009). [CrossRef] [PubMed]
- M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(4), 045201 (2007). [CrossRef] [PubMed]
- 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]
- Y. Aviad, I. Reidler, W. Kinzel, I. Kanter, and M. Rosenbluh, “Phase synchronization in mutually coupled chaotic diode lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(2), 025204 (2008). [CrossRef] [PubMed]
- M. Rosenbluh, Y. Aviad, E. Cohen, L. Khaykovich, W. Kinzel, E. Kopelowitz, P. Yoskovits, and I. Kanter, “Spiking optical patterns and synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(4), 046207 (2007). [CrossRef] [PubMed]
- M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization and multiple time delays in two coupled chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.81(3), 036215 (2010). [CrossRef] [PubMed]
- K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(5), 056211 (2011). [CrossRef] [PubMed]
- V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics58(6), 7208–7213 (1998). [CrossRef]
- Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuation regime,” Opt. Lett.24(22), 1570–1572 (1999). [CrossRef] [PubMed]
- R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16(3), 347–355 (1980). [CrossRef]
- I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, D. Cohen, and W. Kinzel, “Nonlocal mechanism for synchronization of time delay networks,” J. Stat. Phys.145(3), 713–733 (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]
- V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizing distant nodes: a universal classification of networks,” Phys. Rev. Lett.105(25), 254101 (2010). [CrossRef] [PubMed]
- J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010). [CrossRef] [PubMed]
- J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(3), 035202 (2007). [CrossRef] [PubMed]
- C. González, C. Masoller, M. Torrent, and J. García-Ojalvo, “Synchronization via clustering in a small delay-coupled laser network,” Europhys. Lett.79(6), 64003 (2007). [CrossRef]

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