Abstract

We investigate contradirectional two-wave mixing with partially coherent waves in photorefractive crystals. By use of a statistical theory on linear systems, a general formulation of the problem in the space and frequency domain is derived and implemented numerically. We obtain results on beam intensity and mutual coherence. The results on the enhancement of mutual coherence are compared with previous theoretical results on simpler cases and with experimental measurements. Excellent agreements are achieved. The results also indicate that the effective interaction length can be significantly longer than the coherence length of the waves.

© 1997 Optical Society of America

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References

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  1. See, for example, P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  2. See, for example, P. Gunter and J.-P. Huignard, eds., Photorefractive Materials and Devices I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
  3. B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
    [CrossRef]
  4. V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
    [CrossRef]
  5. S. C. De La Cruz, S. MacCormack, J. Feinberg, Q. B. He, H. K. Liu, and P. Yeh, “Effect of beam coherence on mutually pump phase conjugators,” J. Opt. Soc. Am. B 12, 1363 (1995).
    [CrossRef]
  6. Q. B. He and P. Yeh, “Photorefractive mutually pumped phase conjugation with partially coherent beams,” Appl. Phys. B 60, 47 (1995).
    [CrossRef]
  7. R. Hofmeister, A. Yariv, and S. Yagi, “Spectral response of fixed photorefractive grating interference filters,” J. Opt. Soc. Am. A 11, 1342 (1994).
    [CrossRef]
  8. N. V. Bogodaev, L. I. Ivleva, A. S. Korshunov, N. M. Polozkov, and V. V. Shkunov, “Increase of light-beam coherence by two-wave mixing in photorefractive crystals,” J. Opt. Soc. Am. B 10, 2287 (1993).
    [CrossRef]
  9. X. Yi, S. H. Lin, P. Yeh, and K. Y. Hsu, “Contradirectional two-wave mixing with partially coherent waves in photorefractive crystals,” Opt. Lett. 21, 1123 (1996).
    [CrossRef] [PubMed]
  10. See, for example, J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3.
  11. P. Yeh, “Fundamental limit of the speed of photorefractive effect and its impact on device applications and material research,” Appl. Opt. 26, 602 (1987).
    [CrossRef] [PubMed]
  12. R. Saxena, F. Vachss, I. McMichael, and P. Yeh, “Diffraction properties of multiple-beam photorefractive gratings,” J. Opt. Soc. Am. B 7, 1210 (1990).
    [CrossRef]
  13. R. Saxena, C. Gu, and P. Yeh, “Properties of photorefractive gratings with complex coupling constants,” J. Opt. Soc. Am. B 8, 1047 (1991).
    [CrossRef]
  14. T. Y. Chang and R. W. Hellwarth, “Optical phase conjugation by backscattering in barium titanate,” Opt. Lett. 10, 108 (1985).
    [CrossRef]
  15. B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).
  16. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
    [CrossRef]

1996 (1)

1995 (3)

S. C. De La Cruz, S. MacCormack, J. Feinberg, Q. B. He, H. K. Liu, and P. Yeh, “Effect of beam coherence on mutually pump phase conjugators,” J. Opt. Soc. Am. B 12, 1363 (1995).
[CrossRef]

Q. B. He and P. Yeh, “Photorefractive mutually pumped phase conjugation with partially coherent beams,” Appl. Phys. B 60, 47 (1995).
[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (1)

1990 (1)

1989 (1)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

1987 (1)

1985 (1)

T. Y. Chang and R. W. Hellwarth, “Optical phase conjugation by backscattering in barium titanate,” Opt. Lett. 10, 108 (1985).
[CrossRef]

1978 (1)

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

1972 (1)

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Bogodaev, N. V.

Chang, T. Y.

T. Y. Chang and R. W. Hellwarth, “Optical phase conjugation by backscattering in barium titanate,” Opt. Lett. 10, 108 (1985).
[CrossRef]

De La Cruz, S. C.

Faizullov, F. S.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Feinberg, J.

Fischer, B.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

Gu, C.

He, Q. B.

S. C. De La Cruz, S. MacCormack, J. Feinberg, Q. B. He, H. K. Liu, and P. Yeh, “Effect of beam coherence on mutually pump phase conjugators,” J. Opt. Soc. Am. B 12, 1363 (1995).
[CrossRef]

Q. B. He and P. Yeh, “Photorefractive mutually pumped phase conjugation with partially coherent beams,” Appl. Phys. B 60, 47 (1995).
[CrossRef]

Hellwarth, R. W.

T. Y. Chang and R. W. Hellwarth, “Optical phase conjugation by backscattering in barium titanate,” Opt. Lett. 10, 108 (1985).
[CrossRef]

Hofmeister, R.

Hsu, K. Y.

Ivleva, L. I.

Kivshar, Y. S.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Korshunov, A. S.

Lin, S. H.

Liu, H. K.

MacCormack, S.

McMichael, I.

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Polozkov, N. M.

Popovichev, V. I.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Ragul’skii, V. V.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Saxena, R.

Shkunov, V. V.

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Sternklar, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

Vachss, F.

Wang, V.

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

Weiss, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

Yagi, S.

Yariv, A.

Yeh, P.

Yi, X.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Appl. Opt. (1)

Appl. Phys. B (1)

Q. B. He and P. Yeh, “Photorefractive mutually pumped phase conjugation with partially coherent beams,” Appl. Phys. B 60, 47 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

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

J. Opt. Soc. Am. B (4)

Mod. Phys. Lett. B (1)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Opt. Eng. (1)

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

Opt. Lett. (2)

Zh. Éksp. Teor. Fiz. (1)

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in SBS,” Zh. Éksp. Teor. Fiz. 15, 160 (1972).

Other (3)

See, for example, J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3.

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

See, for example, P. Gunter and J.-P. Huignard, eds., Photorefractive Materials and Devices I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).

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Figures (13)

Fig. 1
Fig. 1

Two-wave mixing in photorefractive crystals modeled as a linear system with the signal-wave entrance plane and the pump-wave entrance plane as two input planes and any arbitrary plane inbetween as the output plane.

Fig. 2
Fig. 2

Signal-wave intensity gain as a function of the optical-path difference at the signal-wave entrance plane in the nondepleted-pump regime. The dashed curve is the numerical solution. The solid curve is the approximate analytical solution.

Fig. 3
Fig. 3

Mutual coherence of the two waves at the signal-wave exit plane as a function of the optical path difference at the signal-wave entrance plane in the nondepleted-pump regime. The dashed curve is the numerical solution. The solid curve is the approximate analytical solution.

Fig. 4
Fig. 4

Signal intensity, pump intensity, and the normalized mutual coherence as a function of position z in the photorefractive medium for partially coherent waves (solid curves) and monochromatic waves (dashed curves).

Fig. 5
Fig. 5

(a) Signal-intensity gain and (b) the normalized mutual coherence of the two waves at the pump-wave entrance plane (z=L) as functions of the optical path difference at the signal-wave entrance plane (z=0) for a coupling constant γ =3 cm-1 and various intensity ratios between the signal wave and the pump wave. Note that the curve for a coupling constant γ=3 cm-1 and β1 is the same as that for a coupling constant γ=0 cm-1 and an arbitrary β.

Fig. 6
Fig. 6

Signal intensity I1(z) (solid curve) and the grating profile Q(z)/I0(z) (dashed curve) as functions of the position z inside the photorefractive medium for an incident intensity ratio β=1 and a coupling constant γ=20 cm-1.

Fig. 7
Fig. 7

Signal intensity I1(z) (solid curve) and the grating profile Q(z)/I0(z) (dashed curve) as functions of the position z inside the photorefractive medium for an incident intensity ratio β =10-4 and coupling constants (a) γ=10 cm-1 and (b) γ =20 cm-1.

Fig. 8
Fig. 8

Normalized spectra of the amplified signal wave (solid curve) and the depleted pump wave (dashed curve) at their respective exit planes for an incident intensity ratio β=10-4 and coupling constants (a) γ=10 cm-1 and (b) γ=20 cm-1.

Fig. 9
Fig. 9

Normalized mutual coherence of the two waves as functions of position z for an incident intensity ratio β=10-4 and for coupling constants (a) γ=10 cm-1 (solid curve) and γ =0 cm-1 (dashed curve) and (b) γ=20 cm-1 (solid curve) and γ=0 cm-1 (dashed curve).

Fig. 10
Fig. 10

Phase-conjugation reflectivity as a function of the coherence length of the incident beam for various value of the coupling constant γ.

Fig. 11
Fig. 11

Experimental setup. The distances L1 and L2 are the optical path length of the signal wave and the pump wave from laser source to the signal-wave incident plane z=0, respectively. L2ref is the optical path length of reference wave E2ref from the laser source to the signal output plane z=L.

Fig. 12
Fig. 12

Interference pattern of the signal wave and the reference wave at the output plane P1 (a) without photorefractive coupling and (b) with photorefractive coupling. Note the increase of fringe visibility that is due to the coupling.

Fig. 13
Fig. 13

Signal-intensity gain as a function of the optical path difference of the two waves at the signal-wave entrance plane (z=0). The dots are experimental data, and the solid curve is the theoretical data.

Equations (53)

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E(z, t)=E1(z, t)exp(-iω0t+ik0z)+E2(z, t)exp(-iω0t-ik0z),
Γ11(z, τ)=E11(z, Δω)exp(-iΔωτ)dΔω,
Γ22(z, τ)=E22(z, Δω)exp(-iΔωτ)dΔω,
Γ12(z, τ)=E12(z, z, Δω)exp(-iΔωτ)dΔω,
I1(z)Γ11(z, 0)=E11(z, Δω)dΔω,
I2(z)Γ22(z, 0)=E22(z, Δω)dΔω,
Γ12(z, 0)=E12(z, z, Δω)dΔω.
δn=-iγ2cω0Q(z, t)I0(z)exp(2ik0z)+c.c.,
τphQ(z, t)t+Q(z, t)=E1(z, t)E2*(z, t),
Q(z, t)Q(z, t)=Γ12(z, 0).
E˜=E˜1(z, ω)exp(-iωt+ikz)+E˜2(z, ω)exp(-iωt-ikz),
E˜1(z, ω)z=γ2I0(z)Q(z)E˜2(z, ω)exp(-2iΔkz)-α2E˜1(z, ω),
E˜2(z, ω)z=γ2I0(z)Q*(z)E˜1(z, ω)exp(2iΔkz)+α2E˜2(z, ω),
H11(z, ω)=H11(z, ω)exp(iΔkz),
H12(z, ω)=H12(z, ω)exp(-iΔkz),
H21(z, ω)=H21(z, ω)exp(iΔkL)exp(iΔkz),
H22(z, ω)=H22(z, ω)exp(iΔkL)exp(-iΔkz),
E11(z, Δω)=H11(z, ω)H11*(z, ω)E11(z=0, Δω)+H21(z, ω)H21*(z, ω)E22(z=L, Δω)+H11(z, ω)H21*(z, ω)×E12(z=0, z=L, Δω)exp(-iΔkL)+H21(z, ω)H11*(z, ω)×[E12(z=0, z=L, Δω)×exp(-iΔkL)]*,
E22(z, Δω)=H12(z, ω)H12*(z, ω)E11(z=0, Δω)+H22(z, ω)H22*(z, ω)E22(z=L, Δω)+H12(z, ω)H22*(z, ω)×E12(z=0, z=L, Δω)exp(-iΔkL)+H22(z, ω)H12*(z, ω)×[E12(z=0, z=L, Δω)×exp(-iΔkL)]*,
E12(z, z, Δω)=exp(i2Δkz){H11(z, ω)H12*(z, ω)×E11(z=0, Δω)+H21(z, ω)H22*(z, ω)×E22(z=L, Δω)+H11(z, ω)H22*(z, ω)×E12(z=0, z=L, Δω)×exp(-iΔkL)+H21(z, ω)H12*(z, ω)×[E12(z=0, z=L, Δω)×exp(-iΔkL)]*}.
Ess(Δω)=4(π ln 2)1/2δωexp-2(ln 2)1/2Δωδω2.
E11(z=0, Δω)=βEss(Δω),
E22(z=0, Δω)=Ess(Δω),
E12(z1=0, z2=L, Δω)=βEss(Δω)exp(-ik0L)×exp(-iωtd),
E12(z, z, Δω)=Ess(Δω)exp(-ikL)×exp(-iωtd)exp(2iΔkz).
Γ12(z, τ)=Γsτ+td+nL-2nzcexp(-iω0td),
ddzI1=γI1I2I1+I2-αI1,
ddzI1=γI1I2I1+I2+αI1.
E11(z, Δω)=E˜1(z, ω)E˜1*(z, ω),
E22(z, Δω)=E˜2(z, ω)E˜2*(z, ω),
E12(z, z, Δω)=E˜1(z, ω)E˜2*(z, ω)exp(2iΔkz),
E11(z, Δω)z=γ2I0(z)[Q(z)E12*(z, z, Δω)+Q*(z)E12(z, z, Δω)]-αE11(z, Δω),
E22(z, Δω)z=γ2I0(z)[Q(z)E12*(z, z, Δω)+Q*(z)E12(z, z, Δω)]+αE22(z, Δω),
E12(z, z, Δω)z=2iΔkE12(z, z, Δω)+γ2I0(z)Q(z)×[E11(z, Δω)+E22(z, Δω)]-αE12(z, z, Δω).
Γ12(z, τ)z=-2ncΓ12(z, τ)τ+γ2Γ12(z, 0)I1+I2×[Γ11(z, τ)+Γ22(z, τ)],
Γ11(z, τ)z=γ2Γ12(z, 0)I1+I2Γ12*(z, -τ)+γ2Γ12*(z, 0)I1+I2Γ12(z, τ)-αΓ11(z, τ),
Γ22(z, τ)z=γ2Γ12(z, 0)I1+I2Γ12*(z, -τ)+γ2Γ12*(z, 0)I1+I2Γ12(z, τ)+αΓ22(z, τ).
z[Γ11(z, τ)-Γ22(z, τ)]=0.
z[I1(z)-I2(z)]=0.
Γ12(z=0, τ)βΓss(τ+δt)exp(-iω0δt),
Γ11(z=0, τ)βΓss(τ),
Γ22(z=0, τ)Γss(τ),
Γss(τ)=exp-δωτ4(ln 2)1/22
I1(z)I1(0)Γ11(z, 0)Γ11(0, 0)=Γ12(0, 0)Γ12*(0, 0)Γ11(0, 0)Γ22(0, 0)×[exp(γz)-1]exp(-αz)+exp(-αz),
γ12(z)Γ12(z, 0)[Γ11(z, 0)Γ22(z, 0)]1/2=1Γ12*(0, 0)Γ12(0, 0)[1-exp(-γz)]+Γ11(0, 0)Γ22(0, 0)Γ122(0, 0)exp(-γz)1/2.
Q(z, t)=1τph-tE1(z, t)E2*(z, t)expt-tτphdt.
Q(z, t)=E1(z, t)E2*(z, t)Γ12(z, 0).
|Q(z, t)-Q(z, t)|2=|Q(z, t)|2-|Q(z, t)|2.
|Q(z, t)|2=1τph2-t-tF(z, t)F*(z, t)×expt-tτphexpt-tτphdtdt,
|Q(z, t)|2=12τph2-0dt1 expt1τpht1-t1dt2R(z, t2).
|Q(z, t)-Q(z, t)|2=12τph2-0dt1 expt1τpht1-t1dt2[R(z, t2)-|Γ12(z, 0)|2].
|Q(z, t)-Q(z, t)|2<mI1(z)I2(z)Δtτph.
Q(z, t)Q(z, t)=Γ12(z, 0).

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