Abstract

We investigate the dynamical behavior of a class-B, bidirectional, solid-state ring laser with a square-wave modulated pump. Our treatment includes the coupling of oppositely directed traveling wave modes via backscattering in addition to their coupling via the gain medium. We find that depending on the pump ratio and the depth and frequency of modulation, the intensity waveforms of the two oppositely directed modes may exhibit periodic, quasi-periodic, and chaotic behavior. We also find that although the periodic waveforms of mode intensities are antisynchronized, chaotic waveforms may be synchronized or unsynchronized. A detailed map of different operating regimes as functions of frequency and depth of modulation is presented. Curves are presented to illustrate the behavior.

© 2006 Optical Society of America

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  1. G. V. Perevdentseva, P. A. Khandokin, and Ya. I. Khanin, "Theory of a single-frequency solid-state ring laser," Sov. J. Quantum Electron. 10, 71-74 (1980).
    [CrossRef]
  2. D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
    [CrossRef]
  3. P. A. Khandokin and Ya. I. Khanin, "Instabilities in a solid-state ring laser," J. Opt. Soc. Am. B 2, 226-231 (1985).
    [CrossRef]
  4. P. A. Khandokin and Ya. I. Khanin, "Chaotic Dynamics of a YAG:Nd laser with a ring resonator," Sov. J. Quantum Electron. 18, 1248-1251 (1988).
    [CrossRef]
  5. P. A. Khandokin and Ya. I. Khanin, "Interaction between relaxation oscillations and occurrence of instabilities in a class-B bidirectional laser with a nonreciprocal ring cavity," Quantum Electron. 26, 34-36 (1996).
    [CrossRef]
  6. W. Klische, H. R. Telle, and C. O. Weiss, "Chaos in a solid-state laser with a periodically modulated pump," Opt. Lett. 9, 561-563 (1984).
    [CrossRef] [PubMed]
  7. W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985).
    [CrossRef] [PubMed]
  8. N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).
  9. D. N. Klimenko, N. V. Kravtsov, E. G. Lariontsev, and V. V. Firsov, "Synchronisation of dynamic chaos in counterpropagating ring-laser waves," Quantum Electron. 27, 631-634 (1997).
    [CrossRef]
  10. E. G. Lariontsev, "Switching of synchronized chaotic oscillations in a modulated solid-state ring laser," Opt. Express 2, 198-203 (1998).
    [CrossRef] [PubMed]
  11. P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).
  12. H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
    [CrossRef] [PubMed]
  13. F. Hollinger and C. Jung, "Single-longitudinal-mode laser as a discrete dynamical system," J. Opt. Soc. Am. B 2, 218-225 (1985).
    [CrossRef]
  14. N. B. Abraham and C. O. Weiss, "Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser," Opt. Commun. 68, 437-441 (1988).
    [CrossRef]
  15. T. M. Shen and G. P. Agrawal, "Pulse-shape effects on frequency chirping in single-frequency semiconductor lasers under current modulation," J. Lightwave Technol. LT-4, 497-503 (1986).
    [CrossRef]
  16. M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).
  17. S. Singh and L. Mandel, "Mode competition in a homogeneously broadened ring laser," Phys. Rev. A 20, 2459-2463 (1979).
    [CrossRef]
  18. L. Mandel, R. Roy, and S. Singh, "Optical bistability effects in a dye ring laser," in Optical Bistability, C.M.Bowden, M.Ciftan, and H.Robl, eds. (Plenum, 1981), pp. 127-150.
  19. P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
    [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 and E. G. Lariontsev, "Influence of the amplitude nonreciprocity of the cavity on the characteristics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 604-608 (1996).
    [CrossRef]
  22. 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]
  23. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
    [CrossRef]
  24. A. A. Tsonis, Chaos: From Theory to Applications (Plenum, 1992).

1998 (1)

1997 (1)

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

1996 (4)

P. A. Khandokin and Ya. I. Khanin, "Interaction between relaxation oscillations and occurrence of instabilities in a class-B bidirectional laser with a nonreciprocal ring cavity," Quantum Electron. 26, 34-36 (1996).
[CrossRef]

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]

I. I. Zolotoverkh and E. G. Lariontsev, "Influence of the amplitude nonreciprocity of the cavity on the characteristics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 604-608 (1996).
[CrossRef]

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]

1993 (1)

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).

1988 (3)

P. A. Khandokin and Ya. I. Khanin, "Chaotic Dynamics of a YAG:Nd laser with a ring resonator," Sov. J. Quantum Electron. 18, 1248-1251 (1988).
[CrossRef]

N. B. Abraham and C. O. Weiss, "Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser," Opt. Commun. 68, 437-441 (1988).
[CrossRef]

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

1986 (1)

T. M. Shen and G. P. Agrawal, "Pulse-shape effects on frequency chirping in single-frequency semiconductor lasers under current modulation," J. Lightwave Technol. LT-4, 497-503 (1986).
[CrossRef]

1985 (5)

F. Hollinger and C. Jung, "Single-longitudinal-mode laser as a discrete dynamical system," J. Opt. Soc. Am. B 2, 218-225 (1985).
[CrossRef]

P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).

P. A. Khandokin and Ya. I. Khanin, "Instabilities in a solid-state ring laser," J. Opt. Soc. Am. B 2, 226-231 (1985).
[CrossRef]

W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985).
[CrossRef] [PubMed]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

1984 (1)

1982 (1)

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

1981 (1)

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

1980 (1)

G. V. Perevdentseva, P. A. Khandokin, and Ya. I. Khanin, "Theory of a single-frequency solid-state ring laser," Sov. J. Quantum Electron. 10, 71-74 (1980).
[CrossRef]

1979 (1)

S. Singh and L. Mandel, "Mode competition in a homogeneously broadened ring laser," Phys. Rev. A 20, 2459-2463 (1979).
[CrossRef]

Abraham, N. B.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

N. B. Abraham and C. O. Weiss, "Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser," Opt. Commun. 68, 437-441 (1988).
[CrossRef]

Agrawal, G. P.

T. M. Shen and G. P. Agrawal, "Pulse-shape effects on frequency chirping in single-frequency semiconductor lasers under current modulation," J. Lightwave Technol. LT-4, 497-503 (1986).
[CrossRef]

Andreyev, P. A.

P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).

Christian, W.

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

Firsov, V. V.

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

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]

Hoffer, L. M.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

Hollinger, F.

Ivanov, D. V.

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

Jung, C.

Khandokin, P. A.

P. A. Khandokin and Ya. I. Khanin, "Interaction between relaxation oscillations and occurrence of instabilities in a class-B bidirectional laser with a nonreciprocal ring cavity," Quantum Electron. 26, 34-36 (1996).
[CrossRef]

P. A. Khandokin and Ya. I. Khanin, "Chaotic Dynamics of a YAG:Nd laser with a ring resonator," Sov. J. Quantum Electron. 18, 1248-1251 (1988).
[CrossRef]

P. A. Khandokin and Ya. I. Khanin, "Instabilities in a solid-state ring laser," J. Opt. Soc. Am. B 2, 226-231 (1985).
[CrossRef]

G. V. Perevdentseva, P. A. Khandokin, and Ya. I. Khanin, "Theory of a single-frequency solid-state ring laser," Sov. J. Quantum Electron. 10, 71-74 (1980).
[CrossRef]

Khanin, Ya. I.

P. A. Khandokin and Ya. I. Khanin, "Interaction between relaxation oscillations and occurrence of instabilities in a class-B bidirectional laser with a nonreciprocal ring cavity," Quantum Electron. 26, 34-36 (1996).
[CrossRef]

P. A. Khandokin and Ya. I. Khanin, "Chaotic Dynamics of a YAG:Nd laser with a ring resonator," Sov. J. Quantum Electron. 18, 1248-1251 (1988).
[CrossRef]

P. A. Khandokin and Ya. I. Khanin, "Instabilities in a solid-state ring laser," J. Opt. Soc. Am. B 2, 226-231 (1985).
[CrossRef]

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

G. V. Perevdentseva, P. A. Khandokin, and Ya. I. Khanin, "Theory of a single-frequency solid-state ring laser," Sov. J. Quantum Electron. 10, 71-74 (1980).
[CrossRef]

Klimenko, D. N.

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

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]

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]

Klische, W.

W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985).
[CrossRef] [PubMed]

W. Klische, H. R. Telle, and C. O. Weiss, "Chaos in a solid-state laser with a periodically modulated pump," Opt. Lett. 9, 561-563 (1984).
[CrossRef] [PubMed]

Kravtsov, N. V.

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

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]

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).

Kruzhalov, S. V.

P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).

Lamb, W. E.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Lariontsev, E. G.

E. G. Lariontsev, "Switching of synchronized chaotic oscillations in a modulated solid-state ring laser," Opt. Express 2, 198-203 (1998).
[CrossRef] [PubMed]

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

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]

I. I. Zolotoverkh and E. G. Lariontsev, "Influence of the amplitude nonreciprocity of the cavity on the characteristics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 604-608 (1996).
[CrossRef]

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]

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).

Lett, P.

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

Lippi, G. L.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

Mandel, L.

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

S. Singh and L. Mandel, "Mode competition in a homogeneously broadened ring laser," Phys. Rev. A 20, 2459-2463 (1979).
[CrossRef]

L. Mandel, R. Roy, and S. Singh, "Optical bistability effects in a dye ring laser," in Optical Bistability, C.M.Bowden, M.Ciftan, and H.Robl, eds. (Plenum, 1981), pp. 127-150.

Mandel, P.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

Martorin, I. I.

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

Mello, T.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

Pakhomov, L. N.

P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).

Perevdentseva, G. V.

G. V. Perevdentseva, P. A. Khandokin, and Ya. I. Khanin, "Theory of a single-frequency solid-state ring laser," Sov. J. Quantum Electron. 10, 71-74 (1980).
[CrossRef]

Pikovsky, A. S.

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

Roy, R.

L. Mandel, R. Roy, and S. Singh, "Optical bistability effects in a dye ring laser," in Optical Bistability, C.M.Bowden, M.Ciftan, and H.Robl, eds. (Plenum, 1981), pp. 127-150.

Sargent, M.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Scully, M. O.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Shelaev, A. N.

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).

Shen, T. M.

T. M. Shen and G. P. Agrawal, "Pulse-shape effects on frequency chirping in single-frequency semiconductor lasers under current modulation," J. Lightwave Technol. LT-4, 497-503 (1986).
[CrossRef]

Singh, S.

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

S. Singh and L. Mandel, "Mode competition in a homogeneously broadened ring laser," Phys. Rev. A 20, 2459-2463 (1979).
[CrossRef]

L. Mandel, R. Roy, and S. Singh, "Optical bistability effects in a dye ring laser," in Optical Bistability, C.M.Bowden, M.Ciftan, and H.Robl, eds. (Plenum, 1981), pp. 127-150.

Swift, J. B.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

Swinney, H. L.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

Telle, H. R.

Tsonis, A. A.

A. A. Tsonis, Chaos: From Theory to Applications (Plenum, 1992).

Vastano, J. A.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

Weiss, C. O.

N. B. Abraham and C. O. Weiss, "Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser," Opt. Commun. 68, 437-441 (1988).
[CrossRef]

W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985).
[CrossRef] [PubMed]

W. Klische, H. R. Telle, and C. O. Weiss, "Chaos in a solid-state laser with a periodically modulated pump," Opt. Lett. 9, 561-563 (1984).
[CrossRef] [PubMed]

Wolf, A.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

Yu Petrun'kin, V.

P. A. Andreyev, S. V. Kruzhalov, L. N. Pakhomov, and V. Yu Petrun'kin, "Stability conditions for one-frequency oscillation in ring lasers," Sov. J. Commun. Technol. Electron. 30, 131-133 (1985).

Zeghlache, H.

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

Zolotoverkh, I. I.

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]

I. I. Zolotoverkh and E. G. Lariontsev, "Influence of the amplitude nonreciprocity of the cavity on the characteristics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 604-608 (1996).
[CrossRef]

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]

J. Lightwave Technol. (1)

T. M. Shen and G. P. Agrawal, "Pulse-shape effects on frequency chirping in single-frequency semiconductor lasers under current modulation," J. Lightwave Technol. LT-4, 497-503 (1986).
[CrossRef]

J. Opt. Soc. Am. B (2)

Laser Phys. (1)

N. V. Kravtsov, E. G. Lariontsev, and A. N. Shelaev, "Oscillation regimes of ring solid-state lasers and possibilities for their stabilization, Laser Phys. 3, 21-62 (1993).

Opt. Commun. (1)

N. B. Abraham and C. O. Weiss, "Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser," Opt. Commun. 68, 437-441 (1988).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. (1)

P. Lett, W. Christian, S. Singh, and L. Mandel, "Macroscopic quantum fluctuations and first-order phase transition in a laser," Phys. Lett. 47, 1892-1895 (1981).
[CrossRef]

Phys. Lett. A (1)

D. V. Ivanov, Ya. I. Khanin, I. I. Martorin, and A. S. Pikovsky, "Chaos in a solid-state laser with periodically modulated losses," Phys. Lett. A 89, 229-230 (1982).
[CrossRef]

Phys. Rev. A (3)

W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985).
[CrossRef] [PubMed]

H. Zeghlache and P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, "Bidirectional ring laser: stability analysis and time-dependent solutions," Phys. Rev. A 37, 470-497 (1988).
[CrossRef] [PubMed]

S. Singh and L. Mandel, "Mode competition in a homogeneously broadened ring laser," Phys. Rev. A 20, 2459-2463 (1979).
[CrossRef]

Physica D (1)

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
[CrossRef]

Quantum Electron. (5)

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]

I. I. Zolotoverkh and E. G. Lariontsev, "Influence of the amplitude nonreciprocity of the cavity on the characteristics of self-modulation oscillations in a solid-state ring laser," Quantum Electron. 26, 604-608 (1996).
[CrossRef]

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]

P. A. Khandokin and Ya. I. Khanin, "Interaction between relaxation oscillations and occurrence of instabilities in a class-B bidirectional laser with a nonreciprocal ring cavity," Quantum Electron. 26, 34-36 (1996).
[CrossRef]

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

Sov. J. Commun. Technol. Electron. (1)

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[CrossRef]

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[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Time evolution of the counterpropagating mode intensities in the self-modulation regime for pump ratio r = 1.5 and backscattering coefficient b 2 π = 100 kHz . The two signals are sinusoidal and out of phase by π.

Fig. 2
Fig. 2

Regions of dynamical behavior in the parameter space spanned by modulation depth h m and frequency ν m . Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in this paper are for pump ratio r = 1.5 and b 2 π = 115 kHz .

Fig. 3
Fig. 3

Regions of dynamical behavior according to the synchronization of the intensity waveforms in the h m ν m plane.

Fig. 4
Fig. 4

(a) Pump modulation and intensity waveforms (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane in the quasi-periodic regime (Point B in Figs. 2, 3). Notice the strong presence of self-modulation and modulation frequencies in the power spectrum.

Fig. 5
Fig. 5

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane at the onset of chaos (Point C in Figs. 2, 3).

Fig. 6
Fig. 6

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane in the chaotic regime (Point D in Figs. 2, 3). Note the disappearance of the self-modulation frequency at 115 kHz from the power spectrum.

Fig. 7
Fig. 7

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane for synchronized chaos (Point S in Figs. 2, 3). Note the absence of the self-modulation frequency at 115 kHz from the power spectrum.

Fig. 8
Fig. 8

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane for quasi-periodic signals (Point G in Figs. 2, 3).

Fig. 9
Fig. 9

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane for unsynchronized chaos (Point H in Figs. 2, 3). Note the presence of dominant frequencies against a continuous background in the power spectrum.

Fig. 10
Fig. 10

(a) Pump modulation and intensity waveforms, (b) the power spectrum, and (c) the trajectory in the I 1 I 2 plane for well-developed chaos (Point W in Figs. 2, 3). Note the absence of any dominant frequencies in the power spectrum.

Fig. 11
Fig. 11

Variations of the largest Lyapunov exponent λ max κ with modulation depth h m for different modulation frequencies.

Fig. 12
Fig. 12

Variations of the information dimension D I with modulation depth h m for different modulation frequencies.

Equations (20)

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E ( z , t ) = ε ̂ [ E 1 ( t ) e i ( k z ω t ) + E 2 ( t ) e i ( k z + ω t ) ] ,
D ( z , t ) = D o ( t ) + D + 2 ( t ) e i 2 k z + D 2 ( t ) e i 2 k z + D + 4 ( t ) e i 4 k z .
d E 1 d t = κ E 1 + i b E 2 + ω 2 2 ε o γ ( D o E 1 + D + E 2 ) ,
d E 2 d t = κ E 2 + i b E 1 + ω 2 2 ε o γ ( D o E 2 + D E 1 ) .
1 γ d D o d t = W D o [ 1 + s ( E 1 2 + E 2 2 ) ] s D + E 1 * E 2 s D E 1 E 2 * ,
1 γ d D + d t = s D o E 1 E 2 * D + [ 1 + s ( E 1 2 + E 2 2 ) ] .
s = 4 2 2 γ γ .
ω 2 D th 2 ε o γ = κ .
f j = s E j , d o = D o D th , d + = D + D th , τ = κ t , a = γ κ ,
f ̇ 1 = ( d o 1 ) f 1 + ( i b κ + d + ) f 2 ,
f ̇ 2 = ( d o 1 ) f 2 + ( i b κ + d ) f 1 ,
d ̇ o = a [ r d o ( 1 + f 1 2 + f 2 2 ) d + f 1 * f 2 d f 1 f 2 * ] ,
d ̇ + = a [ d o f 1 f 2 * d + ( 1 + f 1 2 + f 2 2 ) ] .
r = W W t h .
I 1 f 1 2 ( r 1 ) 2 ( 1 + cos b t ) ,
I 2 f 2 2 ( r 1 ) 2 ( 1 cos b t )
d ε d t = J ε ,
D I = j + i = 1 j Λ i Λ j + 1 ,
i = 1 j Λ i > 0 , i = 1 j + 1 Λ i < 0 .
r = r o + h m sign [ cos ( 2 π ν m τ κ ) ] ,

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