Abstract

We investigate the effect of beam temporal coherence on four-wave mixing via transmission gratings in photorefractive media. We take into account the propagation of mutual coherence and self-coherence in space and the evolution of mutual coherence and self-coherence with time. Analytical solutions for nondepleted-pump cases and numerical results for general cases are obtained. We find that the mutual coherence of the signal beam and the pump beam can be enhanced or reduced, depending on the coupling constant and the beam ratio of the signal beam and the pump beam. A decrease of the initial mutual coherence can lead to a lower phase-conjugate reflectivity in the nondepleted-pump case. Even with low initial mutual coherence, high phase-conjugate reflectivity can still be obtained in the case with pump depletion. As a result of the shift of the grating profile, partial coherence can lead to an enhancement of the phase-conjugate reflectivity when the signal–pump-beam ratio is small. We also found that when the initial value of the phase-conjugate beam is zero, the phase-conjugate beam and the pump beam are in phase during propagation. Results for beam intensity and mutual coherence are presented and discussed to provide an insight into four-wave mixing with partially temporally coherent waves.

© 1999 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. M. Cronin-Golomb, H. Kong, and W. Krolikowski, “Photorefractive two-beam coupling with light of partial spatiotemporal coherence,” J. Opt. Soc. Am. B 9, 1698 (1992).
    [CrossRef]
  3. H. Kong, C. Wu, and M. Cronin-Golomb, “Photorefractive two-beam coupling with reduced spatiotemporal coherence,” Opt. Lett. 16, 1183 (1991).
    [CrossRef] [PubMed]
  4. N. V. Bogodaev, L. I. Ivleva, A. S. Korshunov, N. M. Plolzkov, and V. V. Shkunov, “Increase of light-beam coherence by two-wave mixing in photorefractive crystal,” J. Opt. Soc. Am. B 10, 2287 (1993).
    [CrossRef]
  5. 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]
  6. X. Yi, C. Yang, P. Yeh, S.-H. Lin, and K. Y. Hsu, “General solution of contradirectional two-wave mixing with partially coherent waves in photorefractive crystals,” J. Opt. Soc. Am. B 14, 1396 (1997).
    [CrossRef]
  7. X. Yi and P. Yeh, “Two-wave mixing with partially coherent waves in high-speed photorefractive media,” J. Opt. Soc. Am. B 14, 2885 (1997).
    [CrossRef]
  8. Q. B. He and P. Yeh, “Photorefractive mutually pumped phase conjugation with partially coherent beams,” Appl. Phys. B: Photophys. Laser Chem. 60, 47 (1995).
    [CrossRef]
  9. S. D. L. Cruz, S. MacCormack, J. Feinberg, Q. B. He, H. Liu, and P. Yeh, “Effect of beam coherence on mutually pump phase conjugators,” J. Opt. Soc. Am. B 12, 1363 (1995).
    [CrossRef]
  10. W. Krolikowski, “Photorefractive four-wave mixing with partially coherent light beams,” Appl. Phys. B: Photophys. Laser Chem. 65, 541 (1997).
    [CrossRef]
  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. C. Gu and P. Yeh, “Reciprocity in photorefractive wave mixing,” Opt. Lett. 16, 455 (1991).
    [CrossRef] [PubMed]

1997 (3)

1996 (1)

1995 (2)

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

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

1993 (1)

1992 (1)

1991 (2)

1987 (1)

Bogodaev, N. V.

Cronin-Golomb, M.

Cruz, S. D. L.

Feinberg, J.

Gu, C.

He, Q. B.

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

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

Hsu, K. Y.

Ivleva, L. I.

Kong, H.

Korshunov, A. S.

Krolikowski, W.

W. Krolikowski, “Photorefractive four-wave mixing with partially coherent light beams,” Appl. Phys. B: Photophys. Laser Chem. 65, 541 (1997).
[CrossRef]

M. Cronin-Golomb, H. Kong, and W. Krolikowski, “Photorefractive two-beam coupling with light of partial spatiotemporal coherence,” J. Opt. Soc. Am. B 9, 1698 (1992).
[CrossRef]

Lin, S.-H.

Liu, H.

MacCormack, S.

Plolzkov, N. M.

Shkunov, V. V.

Wu, C.

Yang, C.

Yeh, P.

Yi, X.

Appl. Opt. (1)

Appl. Phys. B: Photophys. Laser Chem. (2)

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

W. Krolikowski, “Photorefractive four-wave mixing with partially coherent light beams,” Appl. Phys. B: Photophys. Laser Chem. 65, 541 (1997).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Lett. (3)

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1

FWM in photorefractive media by the transmission gratings approximation. The gratings are formed by beam pair (A1, A2), (A3, A4), or both.

Fig. 2
Fig. 2

Self-coherent functions Γ11(z, 0) and Γ44(z, 0) versus position z in nondepleted FWM. The coupling constant is γL=4.0, with beam ratios β21=10, β31=1.0, and β41=0. (a) Analytical results, (b) numerical results.

Fig. 3
Fig. 3

Phase-conjugate reflectivity as a function of coupling constant γL.

Fig. 4
Fig. 4

Spatial variation of the four self-coherence functions inside the photorefractive medium for various coupling conditions: I1(0)=1, I2(0)=1, I3(L)=1, I4(L)=0. (a) Positive coupling constant γL=10, (b) negative coupling constant γL=-10.

Fig. 5
Fig. 5

Spatial variation of the four self-coherence functions and the mutual-coherence functions inside the photorefractive medium in the case with pump depletion: I1(0)=1, I2(0)=10, I3(L)=1, I4(L)=0, γL=-15. Solid curves Γ12n(0, 0)=1.0; dashed curves, Γ12n(0, 0)=0.5.

Fig. 6
Fig. 6

Amplitudes of the index gratings in the photorefractive medium with pump depletion with the same parameters as in Fig. 5. The dotted curves are obtained with the conventional approximation in which coupling and propagation of the coherence function are neglected. (a) Initial normalized mutual coherence Γ12n(0, 0)=0.8, (b) initial normalized mutual coherence Γ12n(0, 0)=0.2.

Fig. 7
Fig. 7

Phase-conjugation reflectivity as a function of the coupling constants with various parameters: a, I1(0)=1, I2(0)=1, I3(L)=10; b, I1(0)=1, I2(0)=10, I3(L)=1; c, I1(0)=I2(0)=I3(L)=1.0.

Fig. 8
Fig. 8

Dependence of phase conjugation on initial normalized mutual coherence Γ12n(0, 0) (solid curves, left-hand scale; dashed curves, right-hand scale). (a) Small coupling constants, I1(0)=1, I2(0)=1, I3(L)=1. (b) Large coupling constants [solid curves I1(0)=1, I2(0)=1, I3(L)=10; dashed curves, I1(0)=1, I2(0)=10, I3(L)=1].

Equations (51)

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Ej(z, ω)=Aj(z, ω-ω0)exp(-ikjz),
Aj(z, ω-ω0)=-+Aj(z, t)exp[i(ω-ω0)t]dt,
A1(z, t)z+1νA1(z, t)t=-γ2Q(z, t)A2(z, t)I0,
A2(z, t)z+1νA2(z, t)t=γ2Q*(z, t)A1(z, t)I0,
A3(z, t)z-1νA3(z, t)t=γ2Q(z, t)A4(z, t)I0,
A4(z, t)z-1νA4(z, t)t=-γ2Q*(z, t)A3(z, t)I0,
τpQ(z, t)t+Q(z, t)=A1(z, t)A2*(z, t)+A3(z, t)A4*(z, t),
1τp=1τ1+1τ2,
1τ1|A1|2+|A2|2,
1τ2|A3|2+|A4|2.
Γ12(z, τ)z=γ2I0Q(z)[Γ11(z, τ)-Γ22(z, τ)],
Γ11(z, τ)z=-γ2I0Q(z)Γ12*(z, -τ)-γ2I0Q*(z)Γ12(z, τ),
Γ22(z, τ)z=γ2I0Q(z)Γ12*(z, -τ)+γ2I0Q*(z)Γ12(z, τ),
Γ34(z, τ)z=γ2I0Q(z)[Γ44(z, τ)-Γ33(z, τ)],
Γ33(z, τ)z=γ2I0Q(z)Γ34*(z, -τ)+γ2I0Q*(z)Γ34(z, τ),
Γ44(z, τ)z=-γ2I0Q(z)Γ34*(z, -τ)-γ2I0Q*(z)Γ34(z, τ),
Γ13(z, τ)z=-2νΓ13(z, τ)τ-γ2I0Q(z)Γ23(z, τ)+γ2I0Q*(z)Γ14(z, τ),
Γ14(z, τ)z=-2νΓ14(z, τ)τ-γ2I0QΓ24(z, τ)-γ2I0QΓ13(z, τ),
Γ23(z, τ)z=-2νΓ23(z, τ)τ+γ2I0Q*(z)Γ13(z, τ)+γ2I0Q*(z)Γ24(z, τ),
Γ24(z, τ)z=-2νΓ24(z, τ)τ+γ2I0Q*(z)Γ14(z, τ)-γ2I0Q(z)Γ23(z, τ).
Γ11(z, t)+Γ22(z, t)=const.,
Γ33(z, t)+Γ44(z, t)=const.
A1A1*, A4A4*A2A2*, A3A3*,
A3A4*A1A2*.
Γ22(z, τ)z=Γ33(z, τ)z0,
Γ12(z, τ)z=-γ2I0Γ12(z, 0)Γ22(z, τ),
Γ11(z, τ)z=-γ2I0Γ12(z, 0)Γ12*(z, -τ)-γ2I0Γ12*(z, 0)Γ12(z, τ),
Γ34(z, τ)z=-γ2I0Γ12(z, 0)Γ33(z, τ),
Γ44(z, τ)z=-γ2I0Γ12*(z, 0)Γ34(z, τ)-γ2I0Γ12(z, 0)Γ34*(z, -τ).
Γs(τ)=exp-πΔντ2ln 22,
Γ11(0, τ)=Γs(τ),
Γ22(0, τ)=β21Γs(τ),
Γ12(0, τ)=β21Γs(τ+Δt),
Γ33(L, τ)=β31Γs(τ),
Γ44(L, τ)=β41Γs(τ),
Γ34(L, τ)=β31β41Γs(τ+Δt),
Γ11(z, τ)=12I02Γ12(0, 0)Γ12*(0, 0)[Γ22(0, τ)+Γ22*(0, -τ)]exp-γ2z-12+1I0[Γ12(0, 0)Γ12*(0, -τ)+Γ12*(0, 0)Γ12(0, τ)]exp-γ2z-1+Γ11(0, τ),
Γ22(z, τ)=Γ22(0, τ),
Γ33(z, τ)=Γ33(L, τ),
Γ44(z, τ)=12I02Γ12(0, 0)Γ12*(0, 0)[Γ33(L, τ)+Γ33*(L, -τ)]exp-γ2z-exp-γ2L2,
Γ12(z, τ)=1I0Γ12(0, 0)Γ22(0, τ)exp-γ2z-1+Γ12(0, τ),
Γ34(z, τ)=1I0Γ12(0, 0)Γ33(L, τ)×exp-γ2z-exp-γ2L,
Γ11(z, 0)=1I0Γ12(0, 0)Γ12*(0, 0)×[exp(-γz)-1]+Γ11(0, 0),
Γ22(z, 0)=Γ22(0, 0),
Γ33(z, 0)=Γ33(L, 0),
Γ33(z, 0)=1I02Γ12(0, 0)Γ12*(0, 0)Γ33(L, 0)×exp-γ2z-exp-γ2L2,
Γ12(z, 0)=Γ12(0, 0)exp-γ2z,
Γ34(z, 0)=1I0Γ12(0, 0)Γ33(L, 0)×exp-γ2z-exp-γ2L.
Γ12n(z, 0)=|Γ12(z, 0)|[Γ11(z, 0)Γ22(z, 0)]1/2=11+1Γ12n2(0, 0)-1exp(γz)1/2,
Γ34n(z, 0)=|Γ34(z, 0)|[Γ33(z, 0)Γ44(z, 0)]1/2=1.
|ρ|2=Γ44(0, 0)Γ11(0, 0)=1I02Γ33(L, 0)Γ11(0, 0)Γ12(0, 0)Γ12*(0, 0)×1-exp-γ2L2Γ12n2(0, 0)1-exp-γ2L2.

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