Abstract

The route to chaos through type I intermittency is observed in a gain-modulated cw CO2 laser at a low frequency, 120 Hz. As the current is decreased, the output follows the pattern quasi-periodicity → type I intermittency → chaos → laser off. We characterize the chaos by obtaining the Kolmogorov entropy and prove that the pattern is caused by the unstable discharges.

© 1993 Optical Society of America

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References

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  1. H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett. 53A, 77 (1975).
  2. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 256 (1979).
    [Crossref]
  3. See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
    [Crossref] [PubMed]
  4. D. L. MacFarlane and L. W. Casperson, “Pulse-train instabilities in a mode-locked argon laser: experimental studies,” J. Opt. Soc. Am. B 4, 1777 (1987).
    [Crossref]
  5. L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62 (1985).
    [Crossref]
  6. K. A. Shore, “Amplification properties of dynamic instabilities: Hopf bifurcation in semiconductor lasers,” J. Opt. Soc. Am. B 5, 1211 (1988), and references therein.
    [Crossref]
  7. For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
    [Crossref] [PubMed]
  8. C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
    [Crossref]
  9. D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
    [Crossref] [PubMed]
  10. P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983); see S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990) for further details.
    [Crossref]

1989 (1)

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

1988 (1)

1987 (2)

D. L. MacFarlane and L. W. Casperson, “Pulse-train instabilities in a mode-locked argon laser: experimental studies,” J. Opt. Soc. Am. B 4, 1777 (1987).
[Crossref]

D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
[Crossref] [PubMed]

1985 (2)

L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62 (1985).
[Crossref]

See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
[Crossref] [PubMed]

1983 (2)

C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
[Crossref]

P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983); see S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990) for further details.
[Crossref]

1979 (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 256 (1979).
[Crossref]

1975 (1)

H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett. 53A, 77 (1975).

Biswas, D. J.

D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
[Crossref] [PubMed]

Casperson, L. W.

Chatterjee, U. K.

D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
[Crossref] [PubMed]

Dangoisse, D.

See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
[Crossref] [PubMed]

Dev, V.

D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
[Crossref] [PubMed]

Ghazzawi, A. M.

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

Glorieux, P.

See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
[Crossref] [PubMed]

Godone, A.

C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
[Crossref]

Grassberger, P.

P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983); see S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990) for further details.
[Crossref]

Green, C.

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

Haken, H.

H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett. 53A, 77 (1975).

Ikeda, K.

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 256 (1979).
[Crossref]

MacFarlane, D. L.

Midavaine, T.

See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
[Crossref] [PubMed]

Olafsson, A.

C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
[Crossref]

Pernigo, M. A.

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

Procaccia, I.

P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983); see S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990) for further details.
[Crossref]

Quel, E. J.

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

Shore, K. A.

Tredicce, J. R.

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

Weiss, C. O.

C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 256 (1979).
[Crossref]

Phys. Lett. (1)

H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett. 53A, 77 (1975).

Phys. Rev. A (3)

C. O. Weiss, A. Godone, and A. Olafsson, “Route to chaotic emission in a cw He–Ne laser,” Phys. Rev. A 28, 892 (1983).
[Crossref]

D. J. Biswas, V. Dev, and U. K. Chatterjee, “Experimental observation of oscillatory instabilities and chaos in a gain-modulated single-mode cw CO2laser,” Phys. Rev. A 35, 456 (1987).
[Crossref] [PubMed]

P. Grassberger and I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983); see S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990) for further details.
[Crossref]

Phys. Rev. Lett. (2)

See T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989 (1985) for active Q switching; M. Tachikawa, K. Tanni, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077 (1988) for passive Q switching.
[Crossref] [PubMed]

For an example, see J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, and M. A. Pernigo, “Spatial and temporal instabilities in a CO2laser,” Phys. Rev. Lett. 62, 1274 (1989).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Quasi-periodic output and its spectrum at 20 mA (arbitrary vertical scale): a, output signal (0.1 s/division), b, spectrum (20 Hz/division).

Fig. 2
Fig. 2

Route to chaos through type I intermittency (arbitrary vertical scale): a and b, quasi-periodicity; c, chaos (0.1 s/division); d, spectrum of c (20 Hz/division).

Fig. 3
Fig. 3

Discharge currents and their laser output (vertical scale is arbitrary, and horizontal scale is 20 ms/division): a, laser output of intermittency; b, its discharge current; c, laser output before the laser is off; d, its discharge current.

Fig. 4
Fig. 4

Graph of ln[Cp(r)] versus ln(r): a, intermittency; b, chaos.

Fig. 5
Fig. 5

Kolmogorov (KOMO.) entropies of laser outputs: a, intermittency; b, chaos.

Equations (3)

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C p ( r ) = lim n 1 n 2 i j n H [ r | x p ( i ) x p ( j ) | ] ,
K p ( r ) = 1 τ ln [ C p ( r ) C p + 1 ( r ) ] ,
a = ln [ C p ( r ) ] / ln ( r ) .

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