Abstract

The spatiotemporal dynamics of two counterpropagating beams in a photorefractive crystal with nonlocal and sluggish response is investigated in the longitudinal and one transverse dimension. A static external electric field is applied to the crystal to control the coupling strength of the two-wave mixing process. A nonautonomous linear stability analysis is performed that takes a nonconstant modulation depth into account. The onset of pattern formation for arbitrary coupling constants and pump ratios and the influence of linear absorption are discussed. Above the threshold predicted by stability analysis, running transverse waves appear in the optical near field and wandering spots appear in the corresponding far field. A nonlinear eigenmode analysis reveals the running transverse waves as secondary instabilities.

[Optical Society of America ]

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  1. W. J. Firth and C. Pare , Transverse modulational instabilities for counterpropagating beams in Kerr media , Opt. Lett. OPLEDP 13 , 1096 1098 ( 1988
    [CrossRef] [PubMed]
  2. W. J. Firth , A. Fitzgerald , and C. Pare , Transverse instabilities due to counterpropagation in Kerr media , J. Opt. Soc. Am. B JOBPDE 7 , 1087 1097 ( 1990
    [CrossRef]
  3. G. G. Luther and C. J. McKinstrie , Transverse modulational instability of counterpropagating light waves , J. Opt. Soc. Am. B JOBPDE 9 , 1047 1060 ( 1992
    [CrossRef]
  4. J. B. Geddes , R. A. Indik , J. V. Moloney , and W. J. Firth , Hexagons and squares in a passive nonlinear optical system , Phys. Rev. A PLRAAN 50 , 3471 3485 ( 1994
    [CrossRef] [PubMed]
  5. G. Grynberg , Transverse-pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 695 ( 1992
    [CrossRef]
  6. J. B. Geddes , J. V. Moloney , and R. Indik , Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields , Opt. Commun. OPCOB8 90 , 117 122 ( 1992
    [CrossRef]
  7. T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 600 ( 1993
    [CrossRef]
  8. T. Honda , Flow and controlled rotation of the spontaneous optical hexagon in KNbO 3 , Opt. Lett. OPLEDP 20 , 851 853 ( 1995
    [CrossRef] [PubMed]
  9. P. P. Banerjee , H.-L. Yu , D. A. Gregory , N. Kukhtarev , and H. J. Caulfeld , Self-organization of scattering in photorefractive KNbO 3 into a reconfigurable hexagonal spot array , Opt. Lett. OPLEDP 20 , 10 12 ( 1995
    [CrossRef] [PubMed]
  10. T. Honda and P. P. Banerjee , Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback , Opt. Lett. OPLEDP 21 , 779 781 ( 1996
    [CrossRef] [PubMed]
  11. B. Sturman and A. Chernykh , Mechanism of transverse instability of counterpropagation in photorefractive media , J. Opt. Soc. Am. B JOBPDE 12 , 1384 1386 ( 1995
    [CrossRef]
  12. A. I. Chernykh and B. I. Sturman , Threshold for pattern formation in a medium with a local photorefractive response , J. Opt. Soc. Am. B JOBPDE 14 , 1754 1760 ( 1997
    [CrossRef]
  13. M. Saffman , D. Montgomery , A. A. Zozulya , K. Kuroda , and D. Z. Anderson , Transverse instability of counterpropagating waves in photorefractive media , Phys. Rev. A PLRAAN 48 , 3209 3215 ( 1993
    [CrossRef] [PubMed]
  14. M. Saffman , A. A. Zozulya , and D. Z. Anderson , Transverse instability of energy exchanging counterpropagating two-wave mixing , J. Opt. Soc. Am. B JOBPDE 11 , 1409 1417 ( 1994
    [CrossRef]
  15. M. R. Belic , J. Leonardy , D. Timotijevic , and F. Kaiser , Transverse effects in double phase conjugation , Opt. Commun. OPCOB8 111 , 99 104 ( 1994
    [CrossRef]
  16. M. R. Belic , J. Leonardy , D. Timotijevic , and F. Kaiser , Spatiotemporal effects in double phase conjugation , J. Opt. Soc. Am. B JOBPDE 12 , 1602 1616 ( 1995
    [CrossRef]
  17. N. V. Kukhtarev , V. B. Markov , S. G. Odulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals. I. Steady state , Ferroelectrics FEROA8 22 , 949 960 ( 1979
    [CrossRef]
  18. W. Krolikowski , M. R. Belic , M. Cronin-Golomb , and A. Bledowski , Chaos in photorefractive four-wave mixing with a single grating and a single interaction region , J. Opt. Soc. Am. B JOBPDE 7 , 1204 1209 ( 1990
    [CrossRef]
  19. P. Yeh , Contra-directional two-wave mixing in photorefractive media , Opt. Commun. OPCOB8 45 , 323 326 ( 1983
    [CrossRef]
  20. M. R. Belic , Comment on using the shooting method to solve boundary-value problems involving coupled-wave equations , Opt. Quantum Electron. OQELDI 16 , 551 557 ( 1984
    [CrossRef]
  21. O. Sandfuchs , J. Leonardy , F. Kaiser , and M. R. Belic , Transverse instabilities in photorefractive counterpropagating two-wave mixing , Opt. Lett. OPLEDP 22 , 498 500 ( 1997
    [CrossRef] [PubMed]
  22. J. Leonardy , F. Kaiser , M. R. Belic , and O. Hess , Running transverse waves in optical phase conjugation , Phys. Rev. A PLRAAN 53 , 4519 4527 ( 1996
    [CrossRef] [PubMed]
  23. M. R. Belic , Exact solution to photorefractive and photochromic two-wave mixing with arbitrary dependence on fringe modulation , Opt. Lett. OPLEDP 21 , 183 185 ( 1996
    [CrossRef] [PubMed]
  24. M. Kirby , Minimal dynamical systems from PDEs using Sobolev eigenfunctions , Physica D PDNPDT 57 , 466 475 ( 1992
    [CrossRef]
  25. O. Hess and E. Scho ll , Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers , Phys. Rev. A PLRAAN 50 , 787 792 ( 1994
    [CrossRef] [PubMed]
  26. G. K. Harkness , W. J. Firth , J. B. Geddes , J. V. Moloney , and E. M. Wright , Boundary effects in large-aspect-ratio lasers , Phys. Rev. A PLRAAN 50 , 4310 4317 ( 1994
    [CrossRef] [PubMed]
  27. Ch. H. Kwak , S. Y. Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 358 ( 1994
    [CrossRef]
  28. J. E. Millerd , E. M. Garmire , M. B. Klein , M. B. Klein , B. A. Wechsler , F. P. Strohkendl , and G. A. Brost , Photorefractive response at high modulation depths in Bi 12 TiO 20 , J. Opt. Soc. Am. B JOBPDE 9 , 1449 1453 ( 1992
    [CrossRef]
  29. G. A. Brost , Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO , Opt. Commun. OPCOB8 96 , 113 116 ( 1993
    [CrossRef]
  30. L. B. Au and L. Solymar , Higher harmonic gratings in photorefractive materials at large modulation with moving fringes , J. Opt. Soc. Am. A JOAOD6 7 , 1554 1561 ( 1990
    [CrossRef]

Caulfeld, H. J

Geddes, J. B

J. B. Geddes , R. A. Indik , J. V. Moloney , and W. J. Firth , Hexagons and squares in a passive nonlinear optical system , Phys. Rev. A PLRAAN 50 , 3471 3485 ( 1994
[CrossRef] [PubMed]

Kwak, Ch. H

Ch. H. Kwak , S. Y. Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 358 ( 1994
[CrossRef]

Scholl, E

O. Hess and E. Scho ll , Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers , Phys. Rev. A PLRAAN 50 , 787 792 ( 1994
[CrossRef] [PubMed]

Other (30)

W. J. Firth and C. Pare , Transverse modulational instabilities for counterpropagating beams in Kerr media , Opt. Lett. OPLEDP 13 , 1096 1098 ( 1988
[CrossRef] [PubMed]

W. J. Firth , A. Fitzgerald , and C. Pare , Transverse instabilities due to counterpropagation in Kerr media , J. Opt. Soc. Am. B JOBPDE 7 , 1087 1097 ( 1990
[CrossRef]

G. G. Luther and C. J. McKinstrie , Transverse modulational instability of counterpropagating light waves , J. Opt. Soc. Am. B JOBPDE 9 , 1047 1060 ( 1992
[CrossRef]

J. B. Geddes , R. A. Indik , J. V. Moloney , and W. J. Firth , Hexagons and squares in a passive nonlinear optical system , Phys. Rev. A PLRAAN 50 , 3471 3485 ( 1994
[CrossRef] [PubMed]

G. Grynberg , Transverse-pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 695 ( 1992
[CrossRef]

J. B. Geddes , J. V. Moloney , and R. Indik , Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields , Opt. Commun. OPCOB8 90 , 117 122 ( 1992
[CrossRef]

T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 600 ( 1993
[CrossRef]

T. Honda , Flow and controlled rotation of the spontaneous optical hexagon in KNbO 3 , Opt. Lett. OPLEDP 20 , 851 853 ( 1995
[CrossRef] [PubMed]

P. P. Banerjee , H.-L. Yu , D. A. Gregory , N. Kukhtarev , and H. J. Caulfeld , Self-organization of scattering in photorefractive KNbO 3 into a reconfigurable hexagonal spot array , Opt. Lett. OPLEDP 20 , 10 12 ( 1995
[CrossRef] [PubMed]

T. Honda and P. P. Banerjee , Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback , Opt. Lett. OPLEDP 21 , 779 781 ( 1996
[CrossRef] [PubMed]

B. Sturman and A. Chernykh , Mechanism of transverse instability of counterpropagation in photorefractive media , J. Opt. Soc. Am. B JOBPDE 12 , 1384 1386 ( 1995
[CrossRef]

A. I. Chernykh and B. I. Sturman , Threshold for pattern formation in a medium with a local photorefractive response , J. Opt. Soc. Am. B JOBPDE 14 , 1754 1760 ( 1997
[CrossRef]

M. Saffman , D. Montgomery , A. A. Zozulya , K. Kuroda , and D. Z. Anderson , Transverse instability of counterpropagating waves in photorefractive media , Phys. Rev. A PLRAAN 48 , 3209 3215 ( 1993
[CrossRef] [PubMed]

M. Saffman , A. A. Zozulya , and D. Z. Anderson , Transverse instability of energy exchanging counterpropagating two-wave mixing , J. Opt. Soc. Am. B JOBPDE 11 , 1409 1417 ( 1994
[CrossRef]

M. R. Belic , J. Leonardy , D. Timotijevic , and F. Kaiser , Transverse effects in double phase conjugation , Opt. Commun. OPCOB8 111 , 99 104 ( 1994
[CrossRef]

M. R. Belic , J. Leonardy , D. Timotijevic , and F. Kaiser , Spatiotemporal effects in double phase conjugation , J. Opt. Soc. Am. B JOBPDE 12 , 1602 1616 ( 1995
[CrossRef]

N. V. Kukhtarev , V. B. Markov , S. G. Odulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals. I. Steady state , Ferroelectrics FEROA8 22 , 949 960 ( 1979
[CrossRef]

W. Krolikowski , M. R. Belic , M. Cronin-Golomb , and A. Bledowski , Chaos in photorefractive four-wave mixing with a single grating and a single interaction region , J. Opt. Soc. Am. B JOBPDE 7 , 1204 1209 ( 1990
[CrossRef]

P. Yeh , Contra-directional two-wave mixing in photorefractive media , Opt. Commun. OPCOB8 45 , 323 326 ( 1983
[CrossRef]

M. R. Belic , Comment on using the shooting method to solve boundary-value problems involving coupled-wave equations , Opt. Quantum Electron. OQELDI 16 , 551 557 ( 1984
[CrossRef]

O. Sandfuchs , J. Leonardy , F. Kaiser , and M. R. Belic , Transverse instabilities in photorefractive counterpropagating two-wave mixing , Opt. Lett. OPLEDP 22 , 498 500 ( 1997
[CrossRef] [PubMed]

J. Leonardy , F. Kaiser , M. R. Belic , and O. Hess , Running transverse waves in optical phase conjugation , Phys. Rev. A PLRAAN 53 , 4519 4527 ( 1996
[CrossRef] [PubMed]

M. R. Belic , Exact solution to photorefractive and photochromic two-wave mixing with arbitrary dependence on fringe modulation , Opt. Lett. OPLEDP 21 , 183 185 ( 1996
[CrossRef] [PubMed]

M. Kirby , Minimal dynamical systems from PDEs using Sobolev eigenfunctions , Physica D PDNPDT 57 , 466 475 ( 1992
[CrossRef]

O. Hess and E. Scho ll , Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers , Phys. Rev. A PLRAAN 50 , 787 792 ( 1994
[CrossRef] [PubMed]

G. K. Harkness , W. J. Firth , J. B. Geddes , J. V. Moloney , and E. M. Wright , Boundary effects in large-aspect-ratio lasers , Phys. Rev. A PLRAAN 50 , 4310 4317 ( 1994
[CrossRef] [PubMed]

Ch. H. Kwak , S. Y. Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 358 ( 1994
[CrossRef]

J. E. Millerd , E. M. Garmire , M. B. Klein , M. B. Klein , B. A. Wechsler , F. P. Strohkendl , and G. A. Brost , Photorefractive response at high modulation depths in Bi 12 TiO 20 , J. Opt. Soc. Am. B JOBPDE 9 , 1449 1453 ( 1992
[CrossRef]

G. A. Brost , Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO , Opt. Commun. OPCOB8 96 , 113 116 ( 1993
[CrossRef]

L. B. Au and L. Solymar , Higher harmonic gratings in photorefractive materials at large modulation with moving fringes , J. Opt. Soc. Am. A JOAOD6 7 , 1554 1561 ( 1990
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Two-wave mixing configuration in reflection geometry with an externally applied voltage V: A1, A2, pump beams; Q, grating amplitude. z indicates the direction of propagation, and x is the transverse dimension.

Fig. 2
Fig. 2

(a) Threshold curves of the applied electric field and (b) threshold frequency of the grating amplitude, as functions of the transverse wave vector K. The coupling constant is Γ0L =2.0, and the pump ratio r0=20.09. sFP, uFP, regions of stable and unstable fixed points, respectively; Hopf, the type of bifurcation as the threshold curve is crossed in the direction of the arrow.

Fig. 3
Fig. 3

(a) Bifurcation diagram of the primary instability threshold (solid curve), displaying the critical values of E0<0 as a function of Γ0L for fixed r0=20.09. Dashed curve, constant values of the intensity coupling strength γL=1.0, 2.0, 3.0, 4.0, 5.0 (left to right); sFP, and uFP+sLC, regions of stable and unstable fixed points and stable limit cycle, respectively. (b) Oscillation frequency, (c) spatial frequency. Diamonds, points where the threshold values can be obtained analytically.

Fig. 4
Fig. 4

(a) Bifurcation diagram of the primary instability threshold, displaying the critical values of E0<0 as a function of r0 for fixed Γ0L=2.0. (b) Oscillation frequency, (c) spatial frequency. Diamonds, points where the threshold values can be obtained analytically.

Fig. 5
Fig. 5

Transverse intensity and phase profiles of beam A2 when it leaves the crystal at z=0: (a), (c) below threshold for E0=-1.8Ed; (b), (d) above threshold for E0=-2.0Ed at times t and t+T/2.

Fig. 6
Fig. 6

Spatiotemporal dynamics above the instability threshold at E0=-2.0Ed. Running transverse waves in the near field for (a) I1 and (b) I2, and (c), (d) wandering spots in the far field; the pump beams have been subtracted. Γ0L=2.0, f=0.025, r0=20.09, αL=0.

Fig. 7
Fig. 7

(a) Bifurcation diagram of the primary instability threshold (solid curve) displaying the critical values of E0<0 as a function of αL for fixed Γ0L=2.0 and r0=20.09. (b) Oscillation frequency, (c) spatial frequency. Asterisks, results obtained from numerical simulations. For the spatial frequency one obtains different values for I1 (⋄) and I2 (□). The dashed curve in (b) indicates the correction for the frequency given by stability analysis.

Fig. 8
Fig. 8

Frequency behavior above threshold for different values of αL: (*) 0.0, (△) 0.3, (□) 0.5, obtained from numerical simulations. Solid curves, cubic splines through the data points; dashed curve, onset of pattern formation given by stability analysis.

Fig. 9
Fig. 9

Transverse intensity profiles for E0/Ed= (a) -2.2, (b) -2.3, (c) -2.6 at times t and t+T/2.

Fig. 10
Fig. 10

Spatial distribution of the slowly varying envelope of |Q| within the crystal at one instant of time for E0/Ed= (a) -2.0, (b) -2.6.

Fig. 11
Fig. 11

Spectra of normalized eigenvalues λ(i) (in percent) for the intensity patterns of (a) I1 and (b) I2 with (●) E0/Ed=-2.0 and (□) E0/Ed=-2.6.

Fig. 12
Fig. 12

(a), (d) Spatiotemporal substructures: the two dominating eigenmodes (b) with λ(1)80.22% and (e) with λ(2)19.39% and (c), (f) their time-dependent expansion coefficients of the running transverse wave of Fig. 6(b).

Fig. 13
Fig. 13

Spatial Fourier spectra of the two largest eigenmodes from Fig. 12(b) (solid curve) and from Fig. 12(e) (dashed curve).

Fig. 14
Fig. 14

Behavior of (a) the basic spatial frequency K0 and (b) the frequency gap ΔK as functions of the applied field strength for (⋄) beam I1 and (□) beam I2. Solid curves, cubic splines through the data points.

Equations (20)

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zA1+ifx2A1+αA1=-QA2,
-zA2+ifx2A2+αA2=Q*A1.
τtQ+ηQ=Γ A1A2*|A1|2+|A2|2,
η=Ed+Eq+iE0EM+Ed+iE0,
Γ=Γ01+EqEd Ed+iE0EM+Ed+iE0,
A1(x, z=0)=C1 exp(-x2/w02),
A2(x, z=L)=C2 exp(-x2/w02).
A1(z, x, t)=A10(z)[1+a(z, x, t)],
A2(z, x, t)=A20(z)[1+b(z, x, t)],
Q(z, x, t)=Q0(z)[1+q(z, x, t)].
za=A(z, K, λ)a(z, K, λ).
A=U-1m02γ+(1-m02)g(λ)-fK20-s1-m02h(λ)m02β+(1-m02)h(λ)+fK200s1-m02g(λ)s1-m02g(λ)00-h(λ)-fK2s1-m02h(λ)0fK2g(λ)U.
g(λ)=λ2 Γeτeλτe+1+Γe*τe*λτe*+1,
h(λ)=λ2i Γeτeλτe+1-Γe*τe*λτe*+1.
m0(z)=2[I10(z)I20(z)]1/2I10(z)+I20(z)
F(L)=exp0L Tr A(s)ds×z=0L D(z).
a(zout)=S(K, λ)a(zin),
coshγ-g2+cos(χ1)cos(χ2)+p sinc(χ1)sinc(χ2)=0,
Γ0L=2.0,r0=20.09,f=0.025.
δI(x, t)=ia(i)(t)p(i)(x).

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