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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 5 — Mar. 2, 1998
  • pp: 198–203
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Switching of synchronized chaotic oscillations in a modulated solid-state ring laser

Evguenii G. Lariontsev  »View Author Affiliations


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

In most cases, identical (in-phase) synchronization has been dealt with. In this regime the corresponding variables in coupled systems coincide. In the case of anti-phase synchronization, the corresponding variables have equal moduli and opposite signs. In both these cases one can speak about “full synchronization” of chaotic oscillations. In Refs. [11–13

11. A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization in a population of globally coupled chaotic oscillators,” Europhys. Lett. 34, 165–170 (1996). [CrossRef]

] the phenomenon of phase synchronization of chaotic oscillations was found. This effect was defined as the appearance of entrainment between the phases of oscillators, while their amplitudes remain chaotic and non-correlated.

This paper concerns chaotic dynamics of modulated single-mode class-B lasers. A response of such lasers to low frequency modulation of parameters was studied in many publications. In particular, the formation and evolution of strange chaotic attractors was considered in Refs. [15–21

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]

]. It was shown that parametric excitation of relaxation oscillations reduces the threshold for onset of dynamic chaos in Fabry-Perot [15–17

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]

] and ring lasers [20

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

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]

]. Synchronization of dynamic chaos in counterpropagating ring-laser waves was observed 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]

]. In this Ref. regions of existence of strange attractors of two types, synchronized and nonsynchronized, were found in the plane of the system parameters (depth and frequency of pump modulation). The present paper is devoted to a more detailed consideration of properties of the synchronized attractor. The main purpose of this paper is to demonstrate in a numerical experiment with a model of a solid-state ring laser (SSRL) the new phenomenon of spontaneous switching of in-phase and anti-phase chaotic synchronization.

2. Model of a bidirectional ring laser

The counterpropagating waves traveling in opposite directions inside a ring cavity interact with one another and may have different amplitudes and frequencies. We restrict our consideration to the case of single mode generation in each direction. The counterpropagating waves are coupled due to the backscattering by optical inhomogeneities in the cavity. Interference between the counterpropagating waves causes periodic spatial variation (along the cavity axis) of the energy density of the optical field. If the population inversion is saturated by the radiation field in a ring cavity, periodic structures (gratings) are induced in active medium. The population inversion N can be represented in the form of a sum of the spatial harmonics. Self-diffraction of the counterpropagating waves by the induced gratings causes the nonlinear coupling of these waves.

For a description of bidirectional lasing we use a set of equations derived in semiclassical theory of a SSRL for the complex field amplitudes E1,2 an the spatial Fourier components of the population inversion N0 and N1=N*1:

dE1,2dt=12ωQE1,2+im2E2,1+σl2T(N0E1,2+NE2,1),
T1dN0dt=Nth(1+ηeff)N0(1+à(E12+E22))àN+E1E2*àNE1*E2,
T1dN+dt=N+(1+à(E12+E22))àN0E1E2*,
N0=1L0LNdz,N±=1L0LNe±i2kzdz,N=N+*,
(1)

where T1 is the relaxation time of the population inversion N, Nth(1+ηeff)/T1 is the pumping rate, Nth is the threshold value of N, ηeff 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.

The losses for counterpropagating waves are taken to be equal, Q is the quality factor of the cavity. The terms imE2,1/2 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.

We report here theoretical investigations of dynamical chaos in an SSRL with periodic pump modulation. In the presence of pump modulation, the effective pump intensity ηeff is given by the expression

ηeff=η+hcos(ωpt),
(2)

where η is the excess of the pump intensity above the threshold in the absence of pump modulation, h is the modulation depth.

The system of equations was solved numerically by the Runge-Cutta method of the eighth order with double precision. The laser parameters were the same as for the experimentally investigated Nd: YAG SSRL with a monolithic cavity [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]

,15

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]

]: the relaxation time of the population difference is T1=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 ωp varied in the numerical experiment.

For these parameters, in the absence of pump modulation (h=0) intensities of the counterpropagating waves I1,2=a|E1,2|2 undergo antiphase sinusoidal self-modulation with the frequency ωm/2π=170 kHz, and the frequency of a relaxation oscillations is ωr/2π=(ωcη/QT1)1/2=65 kHz.

3. Chaotic regimes

In the region of the pump modulation frequencies ωp close to the relaxation frequency ωr (50kHz<ωp/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 (I1=I2). For the chaotic regime, pulsations of intensities I1,2 are not synchronized.

At the modulation depths h>0.23, synchronization in chaotic intensities of counterpropagating waves could be observed. For synchronized chaotic oscillations, the laser radiation in both directions consists of a nearly periodic sequence of pulses with chaotic amplitudes. The typical time domain behavior of the normalized intensity I1=|E1|2 is shown in Fig.1c. Here the pump modulation signal hcos(ωpt) 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]

]. In this paper, we study the behavior of a real (ReE1,ReE2) and imaginary (ImE1,ImE2) parts of complex fields E1, E2.

Fig.1 Time-dependence of ReE1 (a), ReE2 (b), and |E|2 - (1), hcos(ωpt) - (2) (c) for the chaotic oscillations in a modulated SSRL. The results were obtained by numerically integrating the system of equations (1) with ωp/2π=50 kHz, h=0.24.
Fig.2. Projections of the chaotic pulsations on the planes of variables (ReE1, ImE1) (a), (ReE1 , ReE2) (b), and (ImE1, ImE2) (c). The results were obtained by numerically integrating the system of equations (1) with ωp/2π=50 kHz, h=0.24.

Fig. 2 shows projections of the phase space portrait for this regime onto the planes of variables (ReE1,ImE1) (a), (ReE1,ReE2) (b), and (ImE1,ImE2) (c). One can see from Fig.2b, 2c that the points of the phase space portrait are on two straight lines: E1=E2 (in-phase chaotic synchronization) and E1=-E2 (anti-phase chaotic synchronization). The typical time domain behavior of ReE1 and ReE2 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.

A spectrum of the Lyapunov exponents and the information dimension of the strange attractor have been found. For the regime of identical synchronization at ωp/2π=50 kHz, h=0.24, the Lyapunov exponents λi multiplied by T1 are equal:{λiT1}|={2.8;0.92;0;0;-1.2;-4.1;-8.7;-3.8}. There are five Lyapunov exponents for which their sum s=λ12+…+λ5>0. Assuming that the Kaplan-Yorke conjecture holds for this attractor, the information dimension DI is equal to DI=5+s/|λ6|=5.61.

4. Conclusion

In conclusion, we would like to discuss possible ways of experimental observation of the considered effect. As it was shown above, the behavior of intensities of the counterpropagating waves in the regime of synchronized chaos was studied 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]

]. To observe switchings between in-phase and anti-phase states of chaotic synchronization, one can compare the interference fringes formed by the counterpropageting waves in different chaotic pulses. Another possibility for observing this effect may be realized by heterodyne detection. One must mix the output waves from the ring laser with a local oscillator wave. In such an experiment, one can detect “in-phase” and “quadrature-phase” field components of the counterpropagating waves which correspond to ReE1,2 and ImE1,2, respectively.

Acknowledgments

This research was supported financially by the Russian Foundation for Basic Research.

References and links

1.

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

2.

L.M. Pecora and T.L. Carrol, “Driving systems with chaotic signals,” Phys. Rev. A 44, 2374–2383 (1991). [CrossRef] [PubMed]

3.

I.I. Blekhman, P.S. Landa, and M.G. Rosenblum, “Synchronization and chaotization in interacting dynamical systems,” Appl. Mech. Rev. 48, 733–752 (1995). [CrossRef]

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]

5.

R. Roy and K.S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2011 (1994). [CrossRef] [PubMed]

6.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and V. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3504 (1994). [CrossRef] [PubMed]

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. , 32, 953–959 (1996). [CrossRef]

8.

C.R. Mirasso, P. Colet, and P. Garcia-Fernandez “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photonics Technol. Lett. 8, 299–301 (1996). [CrossRef]

9.

L. Rahman, G. Li, and F. Tian, “Remoute synchronization of high-frequency chaotic signals in semiconductor lasers for secure communications,” Opt. Commun. 138, 91–94 (1997). [CrossRef]

10.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic lasers by optical feedback for cryptographic applications,” IEEE J. Quantum Electron. 33, 1449–1454 (1997). [CrossRef]

11.

A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization in a population of globally coupled chaotic oscillators,” Europhys. Lett. 34, 165–170 (1996). [CrossRef]

12.

G.V. Osipov, A. Pikovsky, M. Rosenblum, and J. Kurths, “Phase synchronization effects in a lattice of nonidentical Rossler oscillators,” Phys. Rev. E 55, 2353–2361 (1997). [CrossRef]

13.

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, “Experimental observation of phase synchronization,” Phys. Rev. E 54, 2115–2117 (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]

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]

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. 55, 339–342 (1985). [CrossRef] [PubMed]

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 34, 2073–2081 (1986). [CrossRef] [PubMed]

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. 67, 358–362 (1988). [CrossRef]

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. 64, 49–53 (1987). [CrossRef]

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]

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

  1. L.M. Pecora, T.L. Carrol, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).<BR> [CrossRef] [PubMed]
  2. L.M. Pecora, T.L. Carrol, "Driving systems with chaotic signals," Phys. Rev. A 44, 2374-2383 (1991).<BR> [CrossRef] [PubMed]
  3. I.I. Blekhman, P.S. Landa, M.G. Rosenblum, "Synchronization and chaotization in interacting dynamical [CrossRef]
  4. 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]
  5. R. Roy, K.S. Thornburg, "Experimental synchronization of chaotic lasers," Phys. Rev. Lett. 72, 2009-2011 (1994).<BR> [CrossRef] [PubMed]
  6. T. Sugawara, M. Tachikawa, T. Tsukamoto, V. Shimizu, "Observation of synchronization in laser chaos," Phys. Rev. Lett. 72 , 3502-3504 (1994).<BR> [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. A. Pikovsky, M. Rosenblum, J.Kurths, "Synchronization in a population of globally coupled chaotic oscillators, Europhys. Lett. 34, 165-170 (1996).<BR> [CrossRef]
  12. 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]
  13. U.Parlitz, L.Junge, W.Lauterborn, L.Kocarev, "Experimental observation of phase synchronization," Phys. Rev. E 54, 2115-2117 (1996).<BR> [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. 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]
  21. 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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