Abstract

Transverse effects in unidirectional and bidirectional photorefractive ring resonators with high Fresnel numbers are investigated numerically. The onset of transverse-mode competition, as well as symmetry-breaking intermittent instabilities, is observed as the cavity detuning is varied. Phase relations and frequency shifts in the intracavity field, in particular, antisymmetric response with respect to the frequency shift and cavity detuning, are considered. The buildup of ordered transverse patterns whose temporal dynamics are either periodic or chaotic is followed. The temporal evolution of chaotic mode oscillation is identified as a crisis-induced intermittency at a heteroclinic tangency.

© 1998 Optical Society of America

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References

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  1. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  2. A. Yariv and S.-K. Kwong, Opt. Lett. 10, 454 (1985); S.-K. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
    [CrossRef] [PubMed]
  3. M. Petrović and M. R. Belić, Phys. Rev. A 52, 671 (1995); M. Belić, Asian J. Phys. 4, 52 (1995); M. Belić, M. Petrović, and F. Kaiser, Opt. Commun. OPCOB8 123, 657 (1996).
    [CrossRef]
  4. D. Hennequin, L. Dambly, D. Dangoisse, and P. Glorieux, J. Opt. Soc. Am. B 11, 676 (1994); L. Dambly and H. Zeghlache, Phys. Rev. A 49, 4043 (1994).
    [CrossRef] [PubMed]
  5. B. M. Jost and B. E. A. Saleh, J. Opt. Soc. Am. B 11, 1864 (1994); Phys. Rev. A 51, 1539 (1995).
    [CrossRef]
  6. Z. Chen and N. B. Abraham, Appl. Phys. B: 60, 5183 (1995); Z. Chen, D. McGee, and N. B. Abraham, J. Opt. Soc. Am. B 13, 1482 (1996).
    [CrossRef]
  7. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, Phys. Rev. Lett. 65, 2531 (1990); 67, 3749 (1991); F. T. Arecchi, Physica D 51, 450 (1991).
    [CrossRef] [PubMed]
  8. D. Z. Anderson and R. Saxena, J. Opt. Soc. Am. B 4, 164 (1987).
    [CrossRef]
  9. M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, Opt. Commun. 111, 99 (1994); J. Opt. Soc. Am. B 12, 1602 (1995).
    [CrossRef]
  10. A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, Phys. Rev. A 52, 4167 (1995).
    [CrossRef] [PubMed]
  11. F. T. Arecchi, S. Boccaletti, G. P. Puccioni, P. L. Ramazza, and S. Residori, Chaos 4, 491 (1994).
    [CrossRef] [PubMed]
  12. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
    [CrossRef]
  13. M. Belić, “Some aspects of nonlinear interaction of light and matter,” Ph.D. dissertation (City College of New York, New York, 1980; unpublished); J. Moloney, M. Belić, and H. M. Gibbs, Opt. Commun. 41, 379 (1982); M. Lax, G. P. Agrawal, M. Belić, B. J. Coffey, and W. L. Louisell, J. Opt. Soc. Am. A 2, 731 (1985).
    [CrossRef]
  14. S. R. Liu and G. Indebetouw, J. Opt. Soc. Am. B 9, 1507 (1992); Opt. Commun. 101, 442 (1993); D. Korwan and G. Indebetouw, Opt. Commun. OPCOB8 129, 205 (1996); J. Opt. Soc. Am. B JOBPDE 13, 1473 (1996).
    [CrossRef]
  15. R. Vautard and M. Ghil, Physica D 35, 395 (1989); J. Leonardy, F. Kaiser, M. Belić, and O. Hess, Phys. Rev. A 53, 4519 (1996).
    [CrossRef] [PubMed]
  16. M. Münkel, O. Hess, and F. Kaiser, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E (to be published).
  17. M. Sauer and F. Kaiser, Phys. Rev. E 54, 2468 (1996).
    [CrossRef]
  18. C. Grebogi, E. Ott, F. Romeiras, and J. A. Yorke, Phys. Rev. A 36, 5365 (1987).
    [CrossRef] [PubMed]
  19. F. T. Arecchi, S. Boccaletti, G. B. Mindlin, and C. Perez Garcia, Phys. Rev. Lett. 69, 3723 (1992).
    [CrossRef] [PubMed]
  20. J. Leonardy, F. Kaiser, M. Belić, and D. Timotijević, Opt. Commun. 132, 279 (1996).
    [CrossRef]
  21. C. Gu and P. Yeh, J. Opt. Soc. Am. B 8, 1428 (1991).
    [CrossRef]
  22. M. Petrović and M. Belić, J. Opt. Soc. Am. B 12, 1028 (1995).
    [CrossRef]
  23. M. R. MacDonald and J. Feinberg, Phys. Rev. Lett. 55, 821 (1985); M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986); B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. IEJQA7 25, 550 (1989); G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. IEJQA7 25, 201 (1989).
    [CrossRef] [PubMed]
  24. J. Leonardy, F. Kaiser, M. Belić, and O. Hess, Phys. Rev. A 53, 4519 (1996).
    [CrossRef] [PubMed]
  25. E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1994); P. Mandel and J. Wang, Opt. Lett. 19, 533 (1994).
    [CrossRef] [PubMed]

1996 (3)

M. Sauer and F. Kaiser, Phys. Rev. E 54, 2468 (1996).
[CrossRef]

J. Leonardy, F. Kaiser, M. Belić, and D. Timotijević, Opt. Commun. 132, 279 (1996).
[CrossRef]

J. Leonardy, F. Kaiser, M. Belić, and O. Hess, Phys. Rev. A 53, 4519 (1996).
[CrossRef] [PubMed]

1995 (2)

M. Petrović and M. Belić, J. Opt. Soc. Am. B 12, 1028 (1995).
[CrossRef]

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

1994 (1)

F. T. Arecchi, S. Boccaletti, G. P. Puccioni, P. L. Ramazza, and S. Residori, Chaos 4, 491 (1994).
[CrossRef] [PubMed]

1992 (1)

F. T. Arecchi, S. Boccaletti, G. B. Mindlin, and C. Perez Garcia, Phys. Rev. Lett. 69, 3723 (1992).
[CrossRef] [PubMed]

1991 (1)

1987 (2)

D. Z. Anderson and R. Saxena, J. Opt. Soc. Am. B 4, 164 (1987).
[CrossRef]

C. Grebogi, E. Ott, F. Romeiras, and J. A. Yorke, Phys. Rev. A 36, 5365 (1987).
[CrossRef] [PubMed]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Chaos (1)

F. T. Arecchi, S. Boccaletti, G. P. Puccioni, P. L. Ramazza, and S. Residori, Chaos 4, 491 (1994).
[CrossRef] [PubMed]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

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

Opt. Commun. (1)

J. Leonardy, F. Kaiser, M. Belić, and D. Timotijević, Opt. Commun. 132, 279 (1996).
[CrossRef]

Phys. Rev. A (3)

J. Leonardy, F. Kaiser, M. Belić, and O. Hess, Phys. Rev. A 53, 4519 (1996).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, F. Romeiras, and J. A. Yorke, Phys. Rev. A 36, 5365 (1987).
[CrossRef] [PubMed]

A. A. Zozulya, G. Montemezzani, and D. Z. Anderson, Phys. Rev. A 52, 4167 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Sauer and F. Kaiser, Phys. Rev. E 54, 2468 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

F. T. Arecchi, S. Boccaletti, G. B. Mindlin, and C. Perez Garcia, Phys. Rev. Lett. 69, 3723 (1992).
[CrossRef] [PubMed]

Other (14)

M. R. MacDonald and J. Feinberg, Phys. Rev. Lett. 55, 821 (1985); M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986); B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. IEJQA7 25, 550 (1989); G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. IEJQA7 25, 201 (1989).
[CrossRef] [PubMed]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, Opt. Commun. 111, 99 (1994); J. Opt. Soc. Am. B 12, 1602 (1995).
[CrossRef]

M. Belić, “Some aspects of nonlinear interaction of light and matter,” Ph.D. dissertation (City College of New York, New York, 1980; unpublished); J. Moloney, M. Belić, and H. M. Gibbs, Opt. Commun. 41, 379 (1982); M. Lax, G. P. Agrawal, M. Belić, B. J. Coffey, and W. L. Louisell, J. Opt. Soc. Am. A 2, 731 (1985).
[CrossRef]

S. R. Liu and G. Indebetouw, J. Opt. Soc. Am. B 9, 1507 (1992); Opt. Commun. 101, 442 (1993); D. Korwan and G. Indebetouw, Opt. Commun. OPCOB8 129, 205 (1996); J. Opt. Soc. Am. B JOBPDE 13, 1473 (1996).
[CrossRef]

R. Vautard and M. Ghil, Physica D 35, 395 (1989); J. Leonardy, F. Kaiser, M. Belić, and O. Hess, Phys. Rev. A 53, 4519 (1996).
[CrossRef] [PubMed]

M. Münkel, O. Hess, and F. Kaiser, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E (to be published).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

A. Yariv and S.-K. Kwong, Opt. Lett. 10, 454 (1985); S.-K. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[CrossRef] [PubMed]

M. Petrović and M. R. Belić, Phys. Rev. A 52, 671 (1995); M. Belić, Asian J. Phys. 4, 52 (1995); M. Belić, M. Petrović, and F. Kaiser, Opt. Commun. OPCOB8 123, 657 (1996).
[CrossRef]

D. Hennequin, L. Dambly, D. Dangoisse, and P. Glorieux, J. Opt. Soc. Am. B 11, 676 (1994); L. Dambly and H. Zeghlache, Phys. Rev. A 49, 4043 (1994).
[CrossRef] [PubMed]

B. M. Jost and B. E. A. Saleh, J. Opt. Soc. Am. B 11, 1864 (1994); Phys. Rev. A 51, 1539 (1995).
[CrossRef]

Z. Chen and N. B. Abraham, Appl. Phys. B: 60, 5183 (1995); Z. Chen, D. McGee, and N. B. Abraham, J. Opt. Soc. Am. B 13, 1482 (1996).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, Phys. Rev. Lett. 65, 2531 (1990); 67, 3749 (1991); F. T. Arecchi, Physica D 51, 450 (1991).
[CrossRef] [PubMed]

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1994); P. Mandel and J. Wang, Opt. Lett. 19, 533 (1994).
[CrossRef] [PubMed]

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Figures (18)

Fig. 1
Fig. 1

Photorefractive ring oscillators: (a) unidirectional, (b) bidirectional. Aj are the slowly varying envelopes, and Q is the amplitude of the transmission grating.

Fig. 2
Fig. 2

(a) Intracavity intensity I40 as a function of cavity detuning Ψext, for δ=0, Γ0=2, and R=0.9 and in the plane-wave case. The unit of intensity is the incident pump intensity. (b) Frequency offset Ω between the pump and the intracavity field as a function of cavity detuning Ψext.

Fig. 3
Fig. 3

Same as Fig. 2(a) but for different Γ0 and across one resonance. The values of Γ0 going from the inside to the outside profiles are 1, 2, 4, 6, and 10. (b) Same as Fig. 2(b).

Fig. 4
Fig. 4

(a) Same as Fig. 3(a) but for different τδ and for Γ0=10. The values of τδ are as follows: solid curve, τδ=0; dotted curve, τδ=-0.03; dashed curve, τδ=-0.47; dashed–dotted curve, τδ=-0.8. (b) Corresponding frequency offset Ω. The other parameters are as in Fig. 2.

Fig. 5
Fig. 5

Schematic representation of different ST states obtained by variation of cavity detuning Ψext. The bars denote spatially antisymmetric states.

Fig. 6
Fig. 6

Beam profiles of signal beam I40=I2(x, z=0): (a) state 1 (Ψext=0.091π), (b) state 2̅ (Ψext=0.094π), (c) state 3 (Ψext=0.097π), (d) state 4̅ (Ψext=0.1π).

Fig. 7
Fig. 7

ST dynamics of RA40: (a)–(d) correspond to stationary states (a)–(d) in the same order in Fig. 6.

Fig. 8
Fig. 8

Temporal change of the transverse profiles of intracavity field A40: (a) intermittent chaotic state C1 for Ψext=0.095π, intensity distribution; (b) distribution of the real part; (c) periodic mode oscillation P1 for Ψext=0.096π, intensity distribution; (d) real-part distribution.

Fig. 9
Fig. 9

(a) Power spectrum (arbitrary units) of chaotic state C1. The spectrum is calculated for the complex electric field A40(x=-0.74, t), and the frequency Ω is given in units of 1/τ. Dashed lines represent the frequencies of the stationary oscillation of states 1 (Ω1=-0.0053) and 3 (Ω3=-0.018). (b) Power spectrum of periodic oscillation P1.

Fig. 10
Fig. 10

Complex eigenmodes of chaotic oscillation C¯2 in Fig. 6: left, real and imaginary parts of eigenmodes p(i)(x); right, corresponding moduli |p(i)|2.

Fig. 11
Fig. 11

Time expansion coefficients |a(i)|2 of the complex eigenmodes in Fig. 10.

Fig. 12
Fig. 12

Temporal development of symmetry-breaking index S: (a) state C¯2, (b) state P¯2.

Fig. 13
Fig. 13

(a) Integrated intracavity intensities I40 (solid curves) and I3d (dashed curves) of the bidirectional ring as functions of frequency-shift parameter δ for three values of external mirror reflectivities R: 0.5, 0.7, 0.9 (from the inside to the outside). (b) Corresponding frequency offset Ω. The solid curve corresponds to the highest value of the reflectivity; the dotted curve, to the lowest.

Fig. 14
Fig. 14

(a) Integrated intracavity intensities I40 (solid curve) and I3d (dashed curve) of the bidirectional ring as functions of frequency-shift parameter δ in the plane-wave case. (b) Same as (a) but for the transverse case.

Fig. 15
Fig. 15

(a) Transverse intensity profiles of intracavity intensities I4d (solid curve) and I3d (dashed curve) in the PC case. (b) Corresponding phases. Note the different scales for the intensities.

Fig. 16
Fig. 16

(a) Same as Fig. 15, for τδ=-0.5. (b) Same as (a), for τδ=0.5. Note the different scales for the intensities.

Fig. 17
Fig. 17

Time signal at the center of beams I30 and I4d when an external electric field is applied to the crystal: (a) E0=1.8 kV/cm, periodic dynamics; (b) E0=3 kV/cm, irregular dynamics. Arrows indicate the time interval during which the snapshots in Figs. 18(b) and 18(c) were taken.

Fig. 18
Fig. 18

Transverse patterns of intracavity intensities I30 and I4d. (a) E0=1.8 kV/cm, corresponding to Fig. 17(a). The snapshots were taken across one period, from t=11τ to t=12.4τ. (b) E0=3 kV/cm, corresponding to Fig. 17(b). (c) Continuation of (b). The time interval between two consecutive snapshots in (b) and (c) equals 2τ, starting from t=17τ.

Equations (15)

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zA1+βKˆ  A1+iϕ2A1=QA4-αA1,
zA2+βKˆ  A2-iϕ2A2=Q¯A3+αA2,
zA3-βKˆ  A3-iϕ2A3=-QA2+αA3,
zA4-βKˆ  A4+iϕ2A4=-Q¯A1-αA4,
τtQ+Q=Γ0I (A1A¯4+A¯2A3),
A1(x, y, z=0)=C1G(-ξ; x-β/2, y),
A2(x, y, z=d)=C2G(ξ; x+β/2, y),
A4(x, y, z=0)=R exp(ik0L)FSP[A4(x, y, z=d)],
A3(x, y, z=d)=R exp(-ik0L)FSP[A3(x, y, z=0)],
FSP(A3/4)=FT-1[exp(iϕk2L/d)FT(A3/4)],
k0L=2Mπ+Ψext,
Γ=Γ01+iτδ,
Γ0=2.0,R=0.9,ϕ=3×10-4,
α=0.1,β=0.
Γ0=4.0,ϕ=0.01,α=0,β=10-4.

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