Abstract

Large-aperture class A lasers emit stationary patterns (square vortex lattices) close to the laser threshold. We show that in class B lasers the emitted patterns are in general nonstationary. The nonstationarity of the vortex lattices is related to the self-induced motion of class B laser vortices. The vortices can oscillate, can remaining spatially separated, or can annihilate and nucleate periodically, depending on the parameters of the class B laser. In the regime of chaotic motion of the vortices, class B lasers display low-dimensional deterministic chaos of transverse modes, unlike class A lasers, for which the turbulence is of the hydrodynamic type.

© 1995 Optical Society of America

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  1. K. Staliunas and C. O. Weiss, "Tilted and standing waves and vortex lattices in class A lasers," Physica D 81, 79–93 (1995).
    [CrossRef]
  2. C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
    [CrossRef] [PubMed]
  3. K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
    [CrossRef]
  4. F. Prati, L. Zucchetti, and G. Molteni, "Rotating patterns in class-B lasers with cylindrical symmetry," Phys. Rev. A (to be published).
  5. A. T. Winfree, "Spiral waves of chemical activity," Science 175, 634–636 (1972).
    [CrossRef] [PubMed]
  6. B. C. Zykov, "Cycloidal circulation of spiral waves in excitable medium," Biofizika 31, 862–865 (1986).
  7. L. A. Lugiato, C. Oldano, and L. M. Narducci, "Cooperative frequency locking and stationary spatial structures in lasers," J. Opt. Soc. Am. B 5, 879–888 (1987).
    [CrossRef]
  8. K. Staliunas, "Laser Ginzburg–Landau equation and laser hydrodynamics," Phys. Rev. A 48, 1573 (1993).
    [CrossRef] [PubMed]
  9. P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
    [CrossRef] [PubMed]
  10. R. L. Donelly, Quantized Vortices in Helium II (Cambridge U. Press, Cambridge, 1991), Chap. 3, p. 86.
  11. K. Staliunas, "Dynamics of optical vortices in a laser beam," Opt. Commun. 90, 123–127 (1992).
    [CrossRef]
  12. N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelation," Phys. Rev. Lett. 72, 1008–1011 (1984).
    [CrossRef]
  13. K. Staliunas, V. Jarutis, and A. Berzanskis, "Vortex statistics in optical speckle fields," submitted to Opt. Commun.
  14. D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
    [CrossRef] [PubMed]
  15. K. Otsuka, "Winner-takes-all dynamics and antiphase states in modulated multimode lasers," Phys. Rev. Lett. 67, 1090–1993 (1991).
    [CrossRef] [PubMed]
  16. K. Otsuka and Y. Aizawa, "Gain circulation in multimode lasers," Phys. Rev. Lett. 72, 2701–2704 (1994).
    [CrossRef] [PubMed]

1995 (1)

K. Staliunas and C. O. Weiss, "Tilted and standing waves and vortex lattices in class A lasers," Physica D 81, 79–93 (1995).
[CrossRef]

1994 (1)

K. Otsuka and Y. Aizawa, "Gain circulation in multimode lasers," Phys. Rev. Lett. 72, 2701–2704 (1994).
[CrossRef] [PubMed]

1993 (3)

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
[CrossRef]

K. Staliunas, "Laser Ginzburg–Landau equation and laser hydrodynamics," Phys. Rev. A 48, 1573 (1993).
[CrossRef] [PubMed]

1992 (3)

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

K. Staliunas, "Dynamics of optical vortices in a laser beam," Opt. Commun. 90, 123–127 (1992).
[CrossRef]

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

1991 (1)

K. Otsuka, "Winner-takes-all dynamics and antiphase states in modulated multimode lasers," Phys. Rev. Lett. 67, 1090–1993 (1991).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

B. C. Zykov, "Cycloidal circulation of spiral waves in excitable medium," Biofizika 31, 862–865 (1986).

1984 (1)

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelation," Phys. Rev. Lett. 72, 1008–1011 (1984).
[CrossRef]

1972 (1)

A. T. Winfree, "Spiral waves of chemical activity," Science 175, 634–636 (1972).
[CrossRef] [PubMed]

Aizawa, Y.

K. Otsuka and Y. Aizawa, "Gain circulation in multimode lasers," Phys. Rev. Lett. 72, 2701–2704 (1994).
[CrossRef] [PubMed]

Berzanskis, A.

K. Staliunas, V. Jarutis, and A. Berzanskis, "Vortex statistics in optical speckle fields," submitted to Opt. Commun.

Brambilla, M.

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

Dangoisse, D.

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Donelly, R. L.

R. L. Donelly, Quantized Vortices in Helium II (Cambridge U. Press, Cambridge, 1991), Chap. 3, p. 86.

Freund, I.

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelation," Phys. Rev. Lett. 72, 1008–1011 (1984).
[CrossRef]

Glorieux, P.

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Hennenquin, D.

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Indik, R.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Jakobsen, P. K.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Jarutis, V.

K. Staliunas, V. Jarutis, and A. Berzanskis, "Vortex statistics in optical speckle fields," submitted to Opt. Commun.

Lepers, C.

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Louvergneux, E.

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Lugiato, L. A.

Moloney, J. V.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Molteni, G.

F. Prati, L. Zucchetti, and G. Molteni, "Rotating patterns in class-B lasers with cylindrical symmetry," Phys. Rev. A (to be published).

Narducci, L. M.

Newell, A. C.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Oldano, C.

Otsuka, K.

K. Otsuka and Y. Aizawa, "Gain circulation in multimode lasers," Phys. Rev. Lett. 72, 2701–2704 (1994).
[CrossRef] [PubMed]

K. Otsuka, "Winner-takes-all dynamics and antiphase states in modulated multimode lasers," Phys. Rev. Lett. 67, 1090–1993 (1991).
[CrossRef] [PubMed]

Prati, F.

F. Prati, L. Zucchetti, and G. Molteni, "Rotating patterns in class-B lasers with cylindrical symmetry," Phys. Rev. A (to be published).

Shvartsman, N.

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelation," Phys. Rev. Lett. 72, 1008–1011 (1984).
[CrossRef]

Staliunas, K.

K. Staliunas and C. O. Weiss, "Tilted and standing waves and vortex lattices in class A lasers," Physica D 81, 79–93 (1995).
[CrossRef]

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
[CrossRef]

K. Staliunas, "Laser Ginzburg–Landau equation and laser hydrodynamics," Phys. Rev. A 48, 1573 (1993).
[CrossRef] [PubMed]

K. Staliunas, "Dynamics of optical vortices in a laser beam," Opt. Commun. 90, 123–127 (1992).
[CrossRef]

K. Staliunas, V. Jarutis, and A. Berzanskis, "Vortex statistics in optical speckle fields," submitted to Opt. Commun.

Tarroja, F. M. H.

K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
[CrossRef]

Telle, H. R.

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

Weiss, C. O.

K. Staliunas and C. O. Weiss, "Tilted and standing waves and vortex lattices in class A lasers," Physica D 81, 79–93 (1995).
[CrossRef]

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
[CrossRef]

Winfree, A. T.

A. T. Winfree, "Spiral waves of chemical activity," Science 175, 634–636 (1972).
[CrossRef] [PubMed]

Zucchetti, L.

F. Prati, L. Zucchetti, and G. Molteni, "Rotating patterns in class-B lasers with cylindrical symmetry," Phys. Rev. A (to be published).

Zykov, B. C.

B. C. Zykov, "Cycloidal circulation of spiral waves in excitable medium," Biofizika 31, 862–865 (1986).

Biofizika (1)

B. C. Zykov, "Cycloidal circulation of spiral waves in excitable medium," Biofizika 31, 862–865 (1986).

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

K. Staliunas, F. M. H. Tarroja, and C. O. Weiss, "Transverse mode locking, antilocking and self-induced dynamics of class-B lasers," Opt. Commun. 102, 69–75 (1993).
[CrossRef]

K. Staliunas, "Dynamics of optical vortices in a laser beam," Opt. Commun. 90, 123–127 (1992).
[CrossRef]

Phys. Rev. A (4)

D. Dangoisse, D. Hennenquin, C. Lepers, E. Louvergneux, and P. Glorieux, "Two-dimensional optical lattices in a CO2 laser," Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, "Restless optical vortex," Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

K. Staliunas, "Laser Ginzburg–Landau equation and laser hydrodynamics," Phys. Rev. A 48, 1573 (1993).
[CrossRef] [PubMed]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, "Space–time dynamics of wide-gain-section lasers," Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

K. Otsuka, "Winner-takes-all dynamics and antiphase states in modulated multimode lasers," Phys. Rev. Lett. 67, 1090–1993 (1991).
[CrossRef] [PubMed]

K. Otsuka and Y. Aizawa, "Gain circulation in multimode lasers," Phys. Rev. Lett. 72, 2701–2704 (1994).
[CrossRef] [PubMed]

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelation," Phys. Rev. Lett. 72, 1008–1011 (1984).
[CrossRef]

Physica D (1)

K. Staliunas and C. O. Weiss, "Tilted and standing waves and vortex lattices in class A lasers," Physica D 81, 79–93 (1995).
[CrossRef]

Science (1)

A. T. Winfree, "Spiral waves of chemical activity," Science 175, 634–636 (1972).
[CrossRef] [PubMed]

Other (3)

K. Staliunas, V. Jarutis, and A. Berzanskis, "Vortex statistics in optical speckle fields," submitted to Opt. Commun.

F. Prati, L. Zucchetti, and G. Molteni, "Rotating patterns in class-B lasers with cylindrical symmetry," Phys. Rev. A (to be published).

R. L. Donelly, Quantized Vortices in Helium II (Cambridge U. Press, Cambridge, 1991), Chap. 3, p. 86.

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Figures (11)

Fig. 1
Fig. 1

Optical field amplitude |E| and the population inversion D in (a) the 1D traveling roll pattern and (b) on the line connecting the optical field zero with the maximum population inversion of the 2D restless vortex. Plot (a) is obtained from the analytical equations (4) and (5), and plot (b) is obtained from the numerical integration of Eqs. (1) with periodic boundary conditions (β = 0). The other parameters are p = 1, γ = 0.045 (ωrel = 0.3), and γ= 5.

Fig. 2
Fig. 2

Temporal evolution of the 1D optical field pattern (a) for reflecting lateral boundaries and (b) for spherical mirrors. The transverse-mode separation Δω = 0.35; the other parameters are p = 0.5, γ = 0.16 (ωrel = 0.4), and γ = 5. The time changes by Δτ = 100.

Fig. 3
Fig. 3

(a) Acoustic and (b) optical vibrational modes of the vortex lattice obtained by numerical integration of Eqs. (1). The SVL and the fundamental mode frequency separation (in the empty resonator) are (a) Δω = 1.75 and (b) Δω = 0.4. Other parameters are γ = 0.5, p = 1, γ = 0.035 (ωrel = 0.264).

Fig. 4
Fig. 4

Snapshot of the optical field amplitude corresponding to the mixed vibrational mode of the vortex lattice obtained by numerical integration of Eqs. (1). Δω = 1; other parameters are as in Fig. 3.

Fig. 5
Fig. 5

Transverse pattern oscillation frequency v as a function of frequency separation between the fundamental mode and the SVL (in the empty resonator) Δω obtained from the numerical integration of Eqs. (1) with periodic boundary conditions. The frequency separations between all the other transverse mode families are proportional to Δω, as indicated by dashed lines. Other parameters are γ = 0.5, p = 1, and γ = 0.035 (ωrel = 0.264). The numerically calculated value of the relaxation oscillation frequency (dotted line) is smaller by a factor of ≅1.4 than that calculated analytically as a result of the linear frequency pulling.

Fig. 6
Fig. 6

(a) Snapshot of the optical field amplitude and (b) its time-averaged picture obtained from the numerical integration of Eqs. (1) with reflecting boundaries. γ = 1.25, p = 1.5, γ = 0.15 (ωrel = 0.67), Δω = 0.35, β = −22.9.

Fig. 7
Fig. 7

Mean radius of the restless vortex spatial trajectory 〈2R〉 (triangles), mean separation of the neighboring vortices 〈r〉 (crosses), and the average percentage of the vortices of the same topological charge among the nearest neighbors (circles) versus γ. 〈2R〉 is normalized to the spatial period of the SVL, 〈r〉 is normalized to the corresponding value calculated for random distribution of vortices on a plane. The plots are obtained from the numerical integration of Eqs. (1) with the reflecting boundaries, with the parameters p = 1.5, γ = 0.15 (ωrel = 0.67), Δω = 0.35, β = −2.9.

Fig. 8
Fig. 8

Stationary-field distribution in class A lasers with spherical mirrors, obtained from the numerical integration of Eqs. (1). p = 1.5, γ = γ = 2, Δω = 1, β = −10.

Fig. 9
Fig. 9

(a) Snapshot of the optical field amplitude and (b) its time-averaged picture of a class B laser with spherical mirrors, as obtained from the numerical integration of Eqs. (1). γ = 0.1; the other parameters are as in Fig. 8.

Fig. 10
Fig. 10

(a) Time dependence of the field amplitude at a particular spatial point of the transverse pattern and (b) the power spectrum of intensity modulation of a class B laser: p = 6.5, γ = 3.5, γ = 0.05, Δω = 2.0, β = −16.

Fig. 11
Fig. 11

Slopes of the log-log plots of the correlation integral for embedding dimensions 1 (bottom curve) to 40 (top curve) versus In(r) (a) for class B lasers, γ = 0.05, and (b) for class A lasers, γ = 3.5. From these plots the correlation dimensions can be estimated. Other parameters are as in Fig. 10.

Equations (13)

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E t = ( i ω C + κ ) E + κ P + i d κ 2 E ,
P t = γ P + γ E D ,
D t = γ [ ( D D 0 ) + ½ ( E * P + P * E ) ] .
E τ = ( D 1 ) E i β E + i d 2 E ,
D τ = γ [ ( D D 0 ) + | E | 2 D ] .
E τ = ( D 1 ) E i ( β d 2 ) E g ( β d 2 ) 2 E ,
D τ = γ [ ( D D 0 ) + | E | 2 D ] .
E ( x , t ) = E + exp ( ikx i ω t ) + E exp ( ikx + i ω t ) ,
D ( x , t ) = d 0 + d [ exp ( 2 ikx 2 i ω t ) + exp ( 2 ikx + 2 i ω t ) ] ,
ω 2 = γ p 4 + γ , E 0 2 = 2 p 4 + γ .
| υ | 2 = γ p d 4 | β | .
E ( x , t ) = E 0 tanh ( x / x 0 ) exp [ Φ ( x ) i ω t ] .
υ = μ ( γ d ) 1 / 2 ,

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