Abstract

An amplitude equation is derived for a four-wave-mixing geometry with nearly counterpropagating, mutually incoherent, nondiffracting pump beams, spatially overlapping in a photorefractive material with a nonlocal response. This equation extends the earlier linear two-dimensional theory to the weakly nonlinear regime. The analysis also starts from a more complete equation for the photorefractive effect, which leads to the prediction of novel effects especially apparent in the nonlinear regime. Precise predictions for the spatio-temporal behavior of the grating amplitude in the nonlinear regime are presented. The range of validity of the amplitude equation is studied. The characteristics of the instability in the nonlinear regime are analyzed through a front-selection analysis.

© 1997 Optical Society of America

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References

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  1. S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987).
    [CrossRef] [PubMed]
  2. D. Engin, S. Orlov, M. Segev, A. Yariv, and G. Valley, Phys. Rev. Lett. 74, 1743 (1995); M. Goulkov, S. Odoulov, and R. Trott, Ukr. Phys. J. 36, 7, 1007 (1991).
    [CrossRef] [PubMed]
  3. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
    [CrossRef]
  4. A. A. Zozulya, Opt. Lett. 16, 545 (1991); V. V. Eliseev, V. T. Tikchoncuk, and A. A. Zozulya, J. Opt. Soc. Am. B 8, 12 (1991).
    [CrossRef] [PubMed]
  5. O. V. Lyubomudrov and V. V. Shkunov, Kvant. Elektron. (Moscow) 19, 1102 (1992); S. Sternklar, Opt. Lett. 20, 3249 (1995).
  6. D. Shaw, Opt. Commun. 94, 458 (1992).
    [CrossRef]
  7. A. A. Zozulya, M. Saffman, and D. Z. Anderson, Phys. Rev. Lett. 73, 6,818 (1994).
    [CrossRef] [PubMed]
  8. M. Cronin-Golomb, Opt. Commun. 89, 276 (1992); M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993); A. A. Zozulya, M. Saffman, and D. Z. Anderson, J. Opt. Soc. Am. B JOBPDE 12, 255 (1995); M. R. Belic, J. Le-onardy, D. Timotijevic, and F. Kaiser, J. Opt. Soc. Am. B JOBPDE 12, 9 (1995).
    [CrossRef] [PubMed]
  9. P. Gunter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. 1, Chaps. 2 and 3.
  10. N. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).
  11. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
    [CrossRef]
  12. N. A. Korneev and S. L. Sochava, Opt. Commun. 115, 539 (1995).
    [CrossRef]
  13. M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
    [CrossRef] [PubMed]
  14. N. V. Bobodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. V. M. Polozkow, and A. A. Zozulya, J. Opt. Soc. Am. B 9, 1493 (1992).
    [CrossRef]
  15. A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993); Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
    [CrossRef]
  16. M. Saffman, D. Montgomer, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, Phys. Rev. A 48, 4, 3209 (1993).
    [CrossRef] [PubMed]
  17. W. van Saarloos and P. C. Hohenberg, Physica D 56, 303 (1992).
    [CrossRef]
  18. M. C. Cross, Phys. Rev. Lett. 57, 2935 (1986).
    [CrossRef] [PubMed]
  19. W. van Saarloos, Phys. Rev. A 39, 6367 (1989).
    [CrossRef] [PubMed]
  20. A. Kolmogorov, I. Petrovsky, and N. Piskunov, Bull. Univ. Moscow, Ser. Int. Sec. A 1, 1 (1937); D. G. Aronson and H. F. Weinberger, Partial Differential Equations and Related Topics, J. A. Goldstein, ed. (Springer-Verlag, Heidelberg, 1975), p. 5.
  21. E. M. Lifshitz and L. P. Ptiaevskii, Physical Kinetics(Pergamon, New York, 1981), Vol. 10, Chap. 6.
  22. A. C. Newell, Lect. Appl. Math. 15, 157 (1974).

1995 (1)

N. A. Korneev and S. L. Sochava, Opt. Commun. 115, 539 (1995).
[CrossRef]

1994 (1)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, Phys. Rev. Lett. 73, 6,818 (1994).
[CrossRef] [PubMed]

1993 (3)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

M. Saffman, D. Montgomer, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, Phys. Rev. A 48, 4, 3209 (1993).
[CrossRef] [PubMed]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

1992 (3)

1989 (1)

W. van Saarloos, Phys. Rev. A 39, 6367 (1989).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

M. C. Cross, Phys. Rev. Lett. 57, 2935 (1986).
[CrossRef] [PubMed]

1976 (1)

N. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[CrossRef]

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

Opt. Commun. (2)

N. A. Korneev and S. L. Sochava, Opt. Commun. 115, 539 (1995).
[CrossRef]

D. Shaw, Opt. Commun. 94, 458 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

M. Saffman, D. Montgomer, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, Phys. Rev. A 48, 4, 3209 (1993).
[CrossRef] [PubMed]

W. van Saarloos, Phys. Rev. A 39, 6367 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, Phys. Rev. Lett. 73, 6,818 (1994).
[CrossRef] [PubMed]

M. C. Cross, Phys. Rev. Lett. 57, 2935 (1986).
[CrossRef] [PubMed]

Physica D (1)

W. van Saarloos and P. C. Hohenberg, Physica D 56, 303 (1992).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

N. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).

Other (9)

D. Engin, S. Orlov, M. Segev, A. Yariv, and G. Valley, Phys. Rev. Lett. 74, 1743 (1995); M. Goulkov, S. Odoulov, and R. Trott, Ukr. Phys. J. 36, 7, 1007 (1991).
[CrossRef] [PubMed]

A. A. Zozulya, Opt. Lett. 16, 545 (1991); V. V. Eliseev, V. T. Tikchoncuk, and A. A. Zozulya, J. Opt. Soc. Am. B 8, 12 (1991).
[CrossRef] [PubMed]

O. V. Lyubomudrov and V. V. Shkunov, Kvant. Elektron. (Moscow) 19, 1102 (1992); S. Sternklar, Opt. Lett. 20, 3249 (1995).

M. Cronin-Golomb, Opt. Commun. 89, 276 (1992); M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993); A. A. Zozulya, M. Saffman, and D. Z. Anderson, J. Opt. Soc. Am. B JOBPDE 12, 255 (1995); M. R. Belic, J. Le-onardy, D. Timotijevic, and F. Kaiser, J. Opt. Soc. Am. B JOBPDE 12, 9 (1995).
[CrossRef] [PubMed]

P. Gunter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. 1, Chaps. 2 and 3.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993); Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

A. Kolmogorov, I. Petrovsky, and N. Piskunov, Bull. Univ. Moscow, Ser. Int. Sec. A 1, 1 (1937); D. G. Aronson and H. F. Weinberger, Partial Differential Equations and Related Topics, J. A. Goldstein, ed. (Springer-Verlag, Heidelberg, 1975), p. 5.

E. M. Lifshitz and L. P. Ptiaevskii, Physical Kinetics(Pergamon, New York, 1981), Vol. 10, Chap. 6.

A. C. Newell, Lect. Appl. Math. 15, 157 (1974).

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Figures (9)

Fig. 1
Fig. 1

2D four-wave-mixing geometry in a photorefractive crystal: Two nearly counterpropagating, mutually incoherent pump beams are denoted by thick-solid and thick-dashed arrows. The waves traveling to the right (solid arrows) and the waves traveling to the left (dashed arrows) interact through a set of shared transmission gratings.

Fig. 2
Fig. 2

k-space configuration of the interacting waves and the grating wave vector: (a), 1D theory (phase-conjugate mode) and efficient grating sharing with optimal phase matching; (b), transverse modulation δ added to the phase-conjugate mode, resulting in an inefficient grating-sharing process in which phase matching is not complete. Thick-solid arrows, pump waves; thin-solid arrows, scattered waves and the grating-wave vector.

Fig. 3
Fig. 3

Spatio-temporal behavior of the grating amplitude in the linear regime (for an infinite transverse system, with localized noise source turned on for a short time): (a), absolute instability (oscillator); (b), convective instability (amplifier). The x,t coordinates of the peak of the amplitude are shown by the dashed line.

Fig. 4
Fig. 4

Growth rate (the real part of the linear-stability spectrum) versus modulation wave number around the phase-conjugate mode for β=100, α=1: below the threshold, γ0 =2.49 (short-dashed curve); at the threshold, γ0=2.50 (long-dashed curve); above the threshold, γ0=2.51 (solid curve).

Fig. 5
Fig. 5

Effects of the wave-number dependence of the photorefractive effect on the growth rate. Growth rate (the real part of the linear-stability spectrum) versus modulation wave number around the phase-conjugate mode for γ0=γ0c(α): β=10, α =1/20 (short-dashed curve); β=10, α=1/2 (long-dashed curve); β=10, α=1 (solid curve).

Fig. 6
Fig. 6

Strong dependence of the nonlinear coefficient on the grating wave number; the curve shows the nonlinear coefficient, g0, of the amplitude equation versus the ratio of the grating wave number (angle between the pump beams) and the Debye wave number, α.

Fig. 7
Fig. 7

Convective spatio-temporal behavior of the grating amplitude in the nonlinear regime: (a), Infinite transverse system with localized noise source, turned on for a short time; the x,t coordinates of the two fronts are shown by the dashed lines. The two fronts are traveling in the positive-x direction with velocities different from the group velocity. (b), Semi-finite transverse system with nonlocalized, time-independent noise source. Only the right front develops; the grating amplitude is the amplified noise from the left boundary. (c), Transition from convective to absolute instability in the semi-infinite system depicted in (b). At time t* the coupling constant is increased above the transition value. A left front develops, creating a much sharper left edge for the disturbance. The grating amplitude becomes independent of the noise.

Fig. 8
Fig. 8

Family of front solutions predicted by the marginal stability approach to the front selection for ε=0.05 from the amplitude equation, Eq. (11) (short-dashed curves), and from the starting equations Eq. (1) (solid curves). The horizontal long-dashed line illustrates the group velocity. The maxima of the curves correspond to the selected front solutions for the left fronts, βκ>0, and the right fronts, βκ<0. There are two branches of solutions predicted by the starting equations for the left fronts.

Fig. 9
Fig. 9

Results of the linear front-selection analysis: spatial-decay rate and velocities of the selected front solutions versus normalized coupling constant ε, calculated from starting equations (solid curves) and from the amplitude equation (dashed curves).

Equations (73)

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ie-iα2x2e+αxi
=-(1+αxe)2te-eiαxe-αxixe,
exp[j(kL-β)z](zA-jβA-jβx2A+j2γ0eA)+c.c.
=0,
exp[-j(kL-β)z](zB+jβB+jβx2B-j2γ0eB)+c.c.
=0.
e=exp[j(2+δ)x]exp(-jΩt)e¯δ(z)+c.c.
A=exp(-jx)+exp[j(1+δ)x]exp(-jΩt)a¯δ(z)+c.c.
B=exp(-jx)+exp[j(1+δ)x]exp(-jΩt)b¯δ(z)+c.c.
1jγ˜/2jγ˜/2j2γ0z+Δ0-j2γ00z-Δe¯δa¯δb¯δ=0,
a¯δ=Γ/2C1 exp(ηz)+Γ/2C2 exp(-ηz),
b¯δ=(η-Δ-Γ/2)C1 exp(ηz)-(η+Δ+Γ/2)C2×exp(-ηz),
e¯δ=-jγ˜[(η-Δ)C1 exp(ηz)-(η+Δ)C2 exp(-ηz)].
-2/Γ=sinh(η)/η
-jΩ=-1-α2(2+δ)2-γ0α(2+δ)(-2/Γ),
-2/Γ=1+Δ/3+4/45 Δ2+19/945 Δ3+8/2025 Δ4+O(Δ5).
Re(-jΩ)=-(1+α24+2αγ0)-δ(α24+αγ0)+δ2(32/45 αγ0β2-α2)+O(δ3),
Im(-jΩ)=-δ(4/3 αγ0β)-δ2(4/3 αγ0β)+O(δ3).
Re(-jΩ)=-2α(γ0-γ0c)+32/45 β2δ2αγ0+O(β3δ3),
γ1c=γ0c+(1-4α2)2/(1+4α2)45/128 1/β2+O(1/β4),
δ1c=(1-4α2)/(1+4α2)45/64 1/β2+O(1/β4),
ν1c=(1-4α2)/(1+4α2)15/16 α/β+O(1/β3).
τ0tA+τ0s0xA=εA+2s+ξ02(1+jc1)x2A-g0|A|2A.
Ap¯Bp¯=1-|A(x, t)|2(1+4α2)2/α2z2(1-z)2+o(ε3/2),
ab=jA(x, t)(1+4α2)/αz(1-z)exp(jx)+o(ε3/2),
e=A(x, t)exp(j2x)-jA(x, t)2(1+8α2)2α(1+16α2)×exp(j4x)+o(ε3/2),
A=Ap¯(z, t)exp(-jx)+a(z, x, t),
B=Bp¯(z, t)exp(-jx)-b(z, x, t).
τ0-1=γ0c[γ0(-jΩ)]0,γ0c=(1+4α2),
s0=[δ Im(-jΩ)]0,γ0c=(2/3)β(1+4α2),
τ0-1ξ02(1+jc1)
=-1/2[δ2(-jΩ)]0,γ0c=α2+16/45 β2(1+4α2)+(1/3)jβ(1+4α2).
g0=1/3-(1+28α2+208α4+512α6)/[1+16α2)4(1+4α2)3].
Ωiqκ=0,
κ=-/+ε1/2/[ξ0(c12+1)1/2],
q=-c1κ,
v=s0+/-ε1/22ξ0(c12+1)1/2/τ0.
εa1/2=s0τ0/[2ξ0(c12+1)1/2].
dΩdδ=dΩdΔdΔdδ=0.
A=Ap¯(z, t)exp(-jx)+a(z, x, t),
B=Bp¯(z, t)exp(-jx)+b(z, x, t).
X=ε1/2x,T=εt.
xx+ε1/2X,xt+εT.
a=ε1/2a0+εa1+
b=ε1/2b0+εb1+
e=ε1/2e0+εe1+
i=1+ε1/2i0+εi1+
Ap¯=1+εAp1¯+
Bp¯=1+εBp1¯+
Le0a0b0+αxi0j2γ0ce0 exp(-jx)-j2γ0ce0 exp(-jx)=0,
L=1-α2x2000z-jβ-jβx2000z+jβ+jβx2,
a0=a0¯ exp(jx)=j(1+4α2)/αA0(T)z exp(jx),
b0=b0¯ exp(jx)=j(1+4α2)/αA0(T)(1-z)exp(jx),
e0=e0¯ exp(2jx)+c.c.=A0(T)exp(2jx)+c.c.,
L0e0¯a0¯b0¯=1jγ˜/2jγ˜/2j2γ0cz0-2γ0c0ze0¯a0¯b0¯.
z1001Ap1¯Bp1¯=j2γ0ca0¯ e0*¯-b0¯ e0*¯,
Ap1¯Bp1¯=-|A0(T)|2(1+4α2)2/α2z2(1-z)2.
Le1a1b1+αxi1j2γ0ce1 exp(-jx)-j2γ0ce1 exp(-jx)=-αe0xe0-e0i0-α2xe0xi0+α2i0x2e0-j2γ0ce0a0j2γ0ce0a0.
a1=a1¯ exp(jx),b1=b1¯ exp(jx),
e1=e10¯ exp(2jx)+e11¯ exp(4jx)+c.c.
e11¯=-jA0(T)2/(2α)(1+8α2)/(1+16α2).
i2=i20+i21+is=1/2(a2s¯+b2s¯)exp(2jx)+1/2(Ap1*¯a0¯+Bp1*¯b0¯)+s2 exp(2jx)+c.c.
Le2a2sb2s+αxi20j2γ0ce20¯ exp(-jx)-j2γ0ce20¯ exp(-jx)={-te0¯-e11¯i0*¯(1+24α2)-2jα(e11¯ e0*¯+e02¯i0*¯+i21¯+s2)}exp(2jx)+c.c.(-j2γ0ce0¯Ap1¯-j2γ0ce0¯)exp(jx)(j2γ0ce0¯Bp1¯+j2γ0ce0¯)exp(jx).
(1+4α2)L0e20¯a2s¯b2s¯=-TA0+A0|A0|2(1+28α2+208α4+512α6)/[(1+16α2)4α2]-A0|A0|2(1+4α2)3[z3+(1-z)3]/(α2)-2jαs2-jA0|A0|2(1+4α2)4/α3z2+jA0(1+4α2)/α+jA0|A0|2(1+4α2)4/α3(1-z2)-jA0(1+4α2)/α.
L0+e0+¯a0+¯b0+¯=1-j2γ0cj2γ0c-jγ˜/2-z0-jγ˜/20-ze0+¯a0+¯b0+¯=0,
e0+¯a0+¯b0+¯=1-j{α/(1+4α2)}(z-1)-j{α/(1+4α2)}z.
(x1, x2, x3)|(y1, y2, y3)=01dzi3xi*yi.
-TA0+A0(1+4α2)-j2αs2+A0|A0|2
×(1+28α2+208α4+512α6)[(1+16α2)4α2]-(1+4α2)3(3α2).
τ0tA=εA+2s-g0|A|2A.
te+ne+xn=0,
xe=N+-1,
nN+=i.

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