Abstract

The spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain are studied numerically. Some interesting spatiotemporal phenomena observed in the experiments were obtained even though intermodal gratings were neglected in the weak-field limit. These phenomena include cooperative frequency locking, spatiotemporal periodic behavior, and spatiotemporal chaos. Moreover, our results show that the spatiotemporal chaos that appears in the photorefractive oscillator can be induced by intermittence.

© 1998 Optical Society of America

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  1. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
    [CrossRef] [PubMed]
  2. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
    [CrossRef]
  3. D. Hennequin, L. Dambly, D. Dangoiss, and P. Glorieux, “Basic transverse dynamics of a photorefractive oscillator,” J. Opt. Soc. Am. B 11, 676–684 (1994).
    [CrossRef]
  4. K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
    [CrossRef] [PubMed]
  5. J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
    [CrossRef] [PubMed]
  6. D. Z. Anderson and R. Saxena, “Theory of multimode operation of a undirectional ring oscillator having photorefractive gain: weak-field limit,” J. Opt. Soc. Am. B 4, 164–176 (1987).
    [CrossRef]
  7. G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
    [CrossRef] [PubMed]
  8. B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillators in unidirectional photorefractive ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
    [CrossRef] [PubMed]
  9. L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
    [CrossRef]
  10. 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 (1988).
    [CrossRef]
  11. Z. Jun and T. Weihan, “Instability of multimode oscillation in a photorefractive ring oscillator,” Phys. Rev. A 54, 5201–5209 (1996).
    [CrossRef] [PubMed]
  12. J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystal,” Opt. Commun. 38, 249–254 (1981).
    [CrossRef]
  13. A. Marrakchi and J. P. Huignard, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
    [CrossRef]
  14. W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
    [CrossRef] [PubMed]

1996 (2)

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

Z. Jun and T. Weihan, “Instability of multimode oscillation in a photorefractive ring oscillator,” Phys. Rev. A 54, 5201–5209 (1996).
[CrossRef] [PubMed]

1995 (2)

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillators in unidirectional photorefractive ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

1991 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

1990 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

1989 (1)

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

1988 (2)

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 (1988).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

1987 (1)

1981 (2)

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystal,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

A. Marrakchi and J. P. Huignard, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Anderson, D. Z.

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Bandy, D. K.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

D’Alessandro, G.

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

Dambly, L.

Dangoiss, D.

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Glorieux, P.

Hennequin, D.

Huignard, J. P.

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystal,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

A. Marrakchi and J. P. Huignard, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Jost, B. M.

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillators in unidirectional photorefractive ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

Jun, Z.

Z. Jun and T. Weihan, “Instability of multimode oscillation in a photorefractive ring oscillator,” Phys. Rev. A 54, 5201–5209 (1996).
[CrossRef] [PubMed]

Klische, W.

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Lugiato, L. A.

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 (1988).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

Malos, J.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

Marrakchi, A.

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystal,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

A. Marrakchi and J. P. Huignard, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

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 (1988).
[CrossRef]

Oldano, C.

Oppo, G.-L.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

Pernigo, M. A.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillators in unidirectional photorefractive ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

Saxena, R.

Slekys, G.

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Staliunas, K.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Tarroja, M. F. H.

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Tredicce, J. R.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

Vaupel, M.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

Weihan, T.

Z. Jun and T. Weihan, “Instability of multimode oscillation in a photorefractive ring oscillator,” Phys. Rev. A 54, 5201–5209 (1996).
[CrossRef] [PubMed]

Weiss, C. O.

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Wellegehausen, B.

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Appl. Phys. (1)

A. Marrakchi and J. P. Huignard, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
[CrossRef]

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

Opt. Commun. (2)

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystal,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, “Spontaneous spatial pattern formation in lasers and cooperative frequency locking,” Opt. Commun. 68, 63–68 (1988).
[CrossRef]

Phys. Rev. A (6)

Z. Jun and T. Weihan, “Instability of multimode oscillation in a photorefractive ring oscillator,” Phys. Rev. A 54, 5201–5209 (1996).
[CrossRef] [PubMed]

K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillator and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

J. Malos, M. Vaupel, K. Staliunas, and C. O. Weiss, “Dynamical structures of a photorefractive oscillator,” Phys. Rev. A 53, 3559–3564 (1996).
[CrossRef] [PubMed]

G. D’Alessandro, “Spatiotemporal dynamics of a unidirectional ring oscillator with photorefractive gain,” Phys. Rev. A 46, 2791–2802 (1992).
[CrossRef] [PubMed]

B. M. Jost and B. E. A. Saleh, “Spatiotemporal dynamics of coupled-transverse-mode oscillators in unidirectional photorefractive ring resonators,” Phys. Rev. A 51, 1539–1548 (1995).
[CrossRef] [PubMed]

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2 laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3794–3797 (1991).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Pattern stability as detuning Δ11 is varied.

Fig. 2
Fig. 2

For detuning Δ11=1.0, (a) stable intensity and (b) phase distribution in a cooperative frequency-locking state. Four topological defects are contained.

Fig. 3
Fig. 3

Periodic evolution of the intensity pattern in the first half-period when modes A¯112 and A¯032 are unlocked at Δ11=0.85. Time increases from (a) to (d), and the time interval between successive images is 80s.

Fig. 4
Fig. 4

Power spectra calculated from the field at point (x, y)=(0.3, 0.3) for (a) spatiotemporal periodicity and (b) spatiotemporal chaos.

Fig. 5
Fig. 5

Intensity of mode A¯00 as a function of time with frequency spacing (a) Δd=1.0×10-5, (b) 0.9×10-5, (c) 0.8×10-5, (d) 0.05×10-5. With decreasing frequency spacing the chaos is induced by intermittence.

Fig. 6
Fig. 6

Characteristics of the intensity correlation. The correlative region C(r0; r)0.5 is indicated by the numbers.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

Pmt=-γ[(1+iΔ2)Pm-(Am+iBm)Pm+m1m2Γm1m2m2mCm1m(Am1+iBm1)×Dm2Pm2Pm2*Pm1],
Am=2Crgm+2CiΔmgm2+Δm2,
Bm=2Cigm-2CrΔmgm2+Δm2,
Dm=Am2+Bm2,
Γm1m2m2m=0ρdρ02πdϕA¯m1A¯m2A¯m2A¯m(ρ, ϕ),
Cm1m=-1/21/2dη exp[i(L/c)(ωm1-ωm)(η+1/2)],
A¯112(ρ, ϕ)=4π ρ(1-ρ2)exp(-ρ2)sin ϕ,
A¯032(ρ, ϕ)=4π 131/2ρ3 exp(-ρ2)sin 3ϕ,
Γ1111Γ1113Γ3111Γ3113=0.1492080.0172290.0172290.059683,
Γ1331Γ1333Γ3331Γ3333=0.0596830.00.00.149208.
Δ2=tanπ4-ϕ02,
Δ00=tanπ4-ϕ02g00.
iδ(Am-iBm)I˜m=-γ[(1+iΔ2)(Am-iBm)I˜m-Im+m1m2Γm1m2m2mIm2Em1Em*],
δ=mBmI˜mmAmI˜m,
δ=mΔmImmIm2Ci=0-mImmΔmIm2Cr=0.
A¯pli(ρ, ϕ)=2[2ρ2]l/2p!(p+l)!1/2×Lpl(2ρ2)exp(-ρ2)Bl(i)(ϕ),
Bl(i)(ϕ)=12πl=01π cos lϕl>0,i=11π sin lϕl>0,i=2
(p=0;l=0,17;i=1).
C(r0;r)=I(r0, t)I(r, t)-I(r0, t)I(r, t)[I2(r0, t)-I(r0, t)2]1/2[I2(r, t)-I(r, t)2]1/2,

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