## Switching of synchronized chaotic oscillations in a modulated solid-state ring laser

Optics Express, Vol. 2, Issue 5, pp. 198-203 (1998)

http://dx.doi.org/10.1364/OE.2.000198

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### Abstract

We study synchronization in chaotic oscillations of counterpropagating waves in a solid-state ring laser with a periodically modulated pump. The new phenomenon of spontaneous switching of in-phase- and anti-phase chaotic synchronization has been discovered in a numerical experiment.

© Optical Society of America

## 1. Introduction

1. L.M. Pecora and T.L. Carrol, “Synchronization in chaotic systems,” Phys. Rev. Lett. **64**, 821–824 (1990). [CrossRef] [PubMed]

4. H.G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. **65**, 1575–1577 (1990). [CrossRef] [PubMed]

15. W. Klische, H.R. Telle, and C.O. Weiss, “Chaos in a solid-state laser with a periodically modulated pump,” Opt. Lett. **9**, 561–564 (1984) [CrossRef] [PubMed]

15. W. Klische, H.R. Telle, and C.O. Weiss, “Chaos in a solid-state laser with a periodically modulated pump,” Opt. Lett. **9**, 561–564 (1984) [CrossRef] [PubMed]

20. I.I. Zolotoverkh, D.N. Klimenko, and E.G. Lariontsev, “Influence of periodic loss modulation on the dynamics of self-modulation oscillations in a solid-state ring laser,” Quantum Electron. **26**, 609–613 (1996). [CrossRef]

21. I.I. Zolotoverkh, D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Parametric processes and multistability in a ring chip laser with periodic pump modulation,” Quantum Electron. **26**, 914–918 (1996) [CrossRef]

14. D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Synchronization of dynamic chaos in counterpropagating ring-laser waves,” Quantum Electron. **27**, 631–634 (1997). [CrossRef]

## 2. Model of a bidirectional ring laser

_{1,2}an the spatial Fourier components of the population inversion N

_{0}and N

_{1}=

_{1}is the relaxation time of the population inversion N, N

_{th}(1+η

_{eff})/T

_{1}is the pumping rate, N

_{th}is the threshold value of N, η

_{eff}is the pump excess over the threshold, T is the cavity round trip time, σ is the laser transition cross section, and a is the saturation parameter.

_{2,1}/2 describe coupling between counterpropagating waves due to backscattering inside the cavity. We neglect the detuning of the optical frequency ω from line center, assuming it is small with respect to the linewidth of atomic transition.

_{eff}is given by the expression

14. D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Synchronization of dynamic chaos in counterpropagating ring-laser waves,” Quantum Electron. **27**, 631–634 (1997). [CrossRef]

15. W. Klische, H.R. Telle, and C.O. Weiss, “Chaos in a solid-state laser with a periodically modulated pump,” Opt. Lett. **9**, 561–564 (1984) [CrossRef] [PubMed]

_{1}=240ms, the round-trip time of the cavity is T=1.66×10

^{-10}s, the losses in a single trip through the cavity are 1-R=3.2%, the moduli of the coupling coefficients are m/2p=170 kHz, and the excess above the threshold is η=0.21. Values of parameters h and ω

_{p}varied in the numerical experiment.

_{1,2}=a|E

_{1,2}|

^{2}undergo antiphase sinusoidal self-modulation with the frequency ω

_{m}/2π=170 kHz, and the frequency of a relaxation oscillations is ω

_{r}/2π=(ω

_{c}η/QT

_{1})

^{1/2}=65 kHz.

## 3. Chaotic regimes

_{p}close to the relaxation frequency ω

_{r}(50kHz<ω

_{p}/2π<90kHz) at the modulation depths h>0.12 the lasing dynamics may become chaotic. For the values of h in the range 0.12 < h < 0.23, there is a bistability in the laser behavior: regime of dynamic chaos coexists with the regime of periodic pulse modulation (spiking mode). For the spiking mode, the counterpropagating waves have equal intensities (I

_{1}=I

_{2}). For the chaotic regime, pulsations of intensities I

_{1,2}are not synchronized.

_{1}=|E

_{1}|

^{2}is shown in Fig.1c. Here the pump modulation signal hcos(ω

_{p}t) is also shown. This chaotic regime was studied theoretically and experimentally in Ref.[14

14. D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Synchronization of dynamic chaos in counterpropagating ring-laser waves,” Quantum Electron. **27**, 631–634 (1997). [CrossRef]

_{1},ReE

_{2}) and imaginary (ImE

_{1},ImE

_{2}) parts of complex fields E

_{1}, E

_{2}.

_{1},ImE

_{1}) (a), (ReE

_{1},ReE

_{2}) (b), and (ImE

_{1},ImE

_{2}) (c). One can see from Fig.2b, 2c that the points of the phase space portrait are on two straight lines: E

_{1}=E

_{2}(in-phase chaotic synchronization) and E

_{1}=-E

_{2}(anti-phase chaotic synchronization). The typical time domain behavior of ReE

_{1}and ReE

_{2}is shown in Fig.1a and Fig.1b, respectively. One can see from Fig.1 that during each period of modulation the switch is observed from in-phase (antiphase) chaotic synchronization to anti-phase (in-phase) chaotic synchronization.

_{1}-ϕ

_{2}for counterpropagating waves. The phase difference Φ is constant during the pulses of radiation and jumps on the value π between the neighboring pulses. The electric field inside the cavity represents a standing wave in each pulse. Nodes of a standing wave in a previous pulse shift to positions of antinodes in the next one.

_{p}/2π=50 kHz, h=0.24, the Lyapunov exponents λ

_{i}multiplied by T

_{1}are equal:{λ

_{i}T

_{1}}|={2.8;0.92;0;0;-1.2;-4.1;-8.7;-3.8}. There are five Lyapunov exponents for which their sum s=λ

_{1}+λ

_{2}+…+λ

_{5}>0. Assuming that the Kaplan-Yorke conjecture holds for this attractor, the information dimension D

_{I}is equal to D

_{I}=5+s/|λ

_{6}|=5.61.

## 4. Conclusion

**27**, 631–634 (1997). [CrossRef]

_{1,2}and ImE

_{1,2}, respectively.

## Acknowledgments

## References and links

1. | L.M. Pecora and T.L. Carrol, “Synchronization in chaotic systems,” Phys. Rev. Lett. |

2. | L.M. Pecora and T.L. Carrol, “Driving systems with chaotic signals,” Phys. Rev. A |

3. | I.I. Blekhman, P.S. Landa, and M.G. Rosenblum, “Synchronization and chaotization in interacting dynamical systems,” Appl. Mech. Rev. |

4. | H.G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. |

5. | R. Roy and K.S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. |

6. | T. Sugawara, M. Tachikawa, T. Tsukamoto, and V. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. |

7. | V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. , |

8. | C.R. Mirasso, P. Colet, and P. Garcia-Fernandez “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photonics Technol. Lett. |

9. | L. Rahman, G. Li, and F. Tian, “Remoute synchronization of high-frequency chaotic signals in semiconductor lasers for secure communications,” Opt. Commun. |

10. | V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic lasers by optical feedback for cryptographic applications,” IEEE J. Quantum Electron. |

11. | A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization in a population of globally coupled chaotic oscillators,” Europhys. Lett. |

12. | G.V. Osipov, A. Pikovsky, M. Rosenblum, and J. Kurths, “Phase synchronization effects in a lattice of nonidentical Rossler oscillators,” Phys. Rev. E |

13. | U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, “Experimental observation of phase synchronization,” Phys. Rev. E |

14. | D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Synchronization of dynamic chaos in counterpropagating ring-laser waves,” Quantum Electron. |

15. | W. Klische, H.R. Telle, and C.O. Weiss, “Chaos in a solid-state laser with a periodically modulated pump,” Opt. Lett. |

16. | G.P. Puccioni, A. Poggi, W. Gadomski, J.R. Tredicce, and F.T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. |

17. | J.R. Tredicce, F.T. Arecchi, G.P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low-dimensional chaos in a modulated homogeneously broadened single-mode laser,” Phys. Rev. A |

18. | F. Papoff, D. Dangoisse, E. Poite-Hanoteau, and P. Glorieux, “Chaotic transients in a CO2 laser with modulated parameters: critical slowing-down and crisis-unduced intermittency,” Opt. Commun. |

19. | N.G. Solari, E. Eschnazi, R. Gilmore, and J.R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. |

20. | I.I. Zolotoverkh, D.N. Klimenko, and E.G. Lariontsev, “Influence of periodic loss modulation on the dynamics of self-modulation oscillations in a solid-state ring laser,” Quantum Electron. |

21. | I.I. Zolotoverkh, D.N. Klimenko, N.V. Kravtsov, E.G. Lariontsev, and V.V. Firsov, “Parametric processes and multistability in a ring chip laser with periodic pump modulation,” Quantum Electron. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.1540) Lasers and laser optics : Chaos

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 28, 1998

Revised Manuscript: December 23, 1997

Published: March 2, 1998

**Citation**

Evguenii Lariontsev, "Switching of synchronized chaotic oscillations in
a modulated solid-state ring laser," Opt. Express **2**, 198-203 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-5-198

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### References

- L.M. Pecora, T.L. Carrol, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).<BR> [CrossRef] [PubMed]
- L.M. Pecora, T.L. Carrol, "Driving systems with chaotic signals," Phys. Rev. A 44, 2374-2383 (1991).<BR> [CrossRef] [PubMed]
- I.I. Blekhman, P.S. Landa, M.G. Rosenblum, "Synchronization and chaotization in interacting dynamical [CrossRef]
- H.G. Winful, L. Rahman, "Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers," Phys. Rev. Lett. 65, 1575-1577 (1990).<BR> [CrossRef] [PubMed]
- R. Roy, K.S. Thornburg, "Experimental synchronization of chaotic lasers," Phys. Rev. Lett. 72, 2009-2011 (1994).<BR> [CrossRef] [PubMed]
- T. Sugawara, M. Tachikawa, T. Tsukamoto, V. Shimizu, "Observation of synchronization in laser chaos," Phys. Rev. Lett. 72 , 3502-3504 (1994).<BR> [CrossRef] [PubMed]
- V. Annovazzi-Lodi, S. Donati, A. Scire, "Synchronization of chaotic injected-laser systems and its application to optical cryptography," IEEE J. Quantum Electron., 32, 953-959 (1996).<BR> [CrossRef]
- C.R. Mirasso, P. Colet, P. Garcia-Fernandez "Synchronization of chaotic semiconductor lasers: application to encoded communications," IEEE Photonics Technol. Lett. 8, 299-301 (1996).<BR> [CrossRef]
- L. Rahman, G. Li, F. Tian, "Remoute synchronization of high-frequency chaotic signals in semiconductor lasers for secure communications," Opt. Commun. 138, 91-94 (1997) .<BR> [CrossRef]
- V. Annovazzi-Lodi, S. Donati, A. Scire, "Synchronization of chaotic lasers by optical feedback for cryptographic applications," IEEE J. Quantum Electron. 33, 1449-1454 (1997).<BR> [CrossRef]
- A. Pikovsky, M. Rosenblum, J.Kurths, "Synchronization in a population of globally coupled chaotic oscillators, Europhys. Lett. 34, 165-170 (1996).<BR> [CrossRef]
- G.V.Osipov, A. Pikovsky, M. Rosenblum, J.Kurths, "Phase synchronization effects in a lattice of nonidentical Rossler oscillators," Phys. Rev. E 55, 2353-2361 (1997).<BR> [CrossRef]
- U.Parlitz, L.Junge, W.Lauterborn, L.Kocarev, "Experimental observation of phase synchronization," Phys. Rev. E 54, 2115-2117 (1996).<BR> [CrossRef]
- D.N.Klimenko, N.V.Kravtsov, E.G.Lariontsev, V.V. Firsov, "Synchronization of dynamic chaos in counterpropagating ring-laser waves," Quantum Electron. 27, 631-634 (1997).<BR> [CrossRef]
- W. Klische, H.R. Telle, C.O. Weiss, "Chaos in a solid-state laser with a periodically modulated pump," Opt. Lett. 9, 561-564 (1984)<BR> [CrossRef] [PubMed]
- G.P. Puccioni, A. Poggi, W. Gadomski, J.R. Tredicce, F.T. Arecchi, "Measurement of the formation and evolution of a strange attractor in a laser," Phys. Rev. Lett. 55, 339-342 (1985).<BR> [CrossRef] [PubMed]
- J.R. Tredicce, F.T. Arecchi, G.P. Puccioni, A. Poggi, W. Gadomski, "Dynamic behavior and onset of low-dimensional chaos in a modulated homogeneously broadened single-mode laser," Phys. Rev. A 34, 2073-2081 (1986).<BR> [CrossRef] [PubMed]
- F. Papoff, D. Dangoisse, E. Poite-Hanoteau, P. Glorieux, "Chaotic transients in a CO2 laser with modulated parameters: critical slowing-down and crisis-unduced intermittency," Opt. Commun. 67, 358-362 (1988) .<BR> [CrossRef]
- N.G. Solari, E. Eschnazi, R. Gilmore, J.R. Tredicce, "Influence of coexisting attractors on the dynamics of a laser system," Opt. Commun. 64, 49-53 (1987) .<BR> [CrossRef]
- I.I..Zolotoverkh, D.N.Klimenko, E.G.Lariontsev, "Influence of periodic loss modulation on the dynamics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 609-613 (1996).<BR> [CrossRef]
- I.I..Zolotoverkh, D.N.Klimenko, N.V.Kravtsov, E.G.Lariontsev, V.V. Firsov, "Parametric processes and multistability in a ring chip laser with periodic pump modulation," Quantum Electron. 26, 914-918 (1996)<BR> [CrossRef]

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