Abstract

The spatial fidelity and the reflectivity of the externally pumped phase conjugation by four-wave mixing in photorefractive crystals are numerically studied by a three-dimensional analysis. The results are given for both the case that the input one-dimensional rectangular amplitude of the image-bearing beam has a finite extent in the plane of incidence of the four extraordinarily polarized beams and the case that the amplitude has a finite extent in the orthogonal plane. The fidelity and the reflectivity versus the input pump ratio, the moving velocity of the crystal, and the externally applied electric field are analyzed. Although one can enhance the reflectivity by moving the crystal or by applying an electric field, the output phase deviates increasingly from the conjugate phase.

© 1997 Optical Society of America

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References

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  1. P. Günter, “Holography, coherent light amplification, and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
    [CrossRef]
  2. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
    [CrossRef]
  3. P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. II, Chaps. 5 and 6.
  4. K. R. Macdonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
    [CrossRef] [PubMed]
  5. P. Xie, J. H. Dai, and H. J. Zhang, “Multigrating optical phase conjugation with considerations of phase effects,” J. Opt. Soc. Am. B 9, 2240–2247 (1992).
    [CrossRef]
  6. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102–7107 (1996).
    [CrossRef] [PubMed]
  7. G. C. Valley, “Two-wave mixing with applied field and a moving grating,” J. Opt. Soc. Am. B 1, 868–873 (1984).
    [CrossRef]
  8. J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
    [CrossRef]
  9. G. G. Valley, “Evolution of phase-conjugate waves in stimulated photorefractive backscattering,” J. Opt. Soc. Am. B 9, 1440–1448 (1992).
    [CrossRef]
  10. Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).
  11. M. R. Belic and W. Krolikowski, “Multigrating optical phase conjugation: numerical results,” J. Opt. Soc. Am. B 6, 901–909 (1989).
    [CrossRef]
  12. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Numerical studies of externally-pumped phase conjugation in photorefractive crystals with applied electric fields,” Opt. Commun. 130, 302–306 (1996).
    [CrossRef]

1996 (2)

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Numerical studies of externally-pumped phase conjugation in photorefractive crystals with applied electric fields,” Opt. Commun. 130, 302–306 (1996).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102–7107 (1996).
[CrossRef] [PubMed]

1994 (1)

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

1992 (2)

1989 (1)

1985 (1)

K. R. Macdonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

1984 (2)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

G. C. Valley, “Two-wave mixing with applied field and a moving grating,” J. Opt. Soc. Am. B 1, 868–873 (1984).
[CrossRef]

1982 (1)

P. Günter, “Holography, coherent light amplification, and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
[CrossRef]

1981 (1)

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

Belic, M. R.

Bernasconi, P.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

Dai, J. H.

Feinberg, J.

K. R. Macdonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

Günter, P.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

P. Günter, “Holography, coherent light amplification, and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
[CrossRef]

P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. II, Chaps. 5 and 6.

Huignard, J. P.

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. II, Chaps. 5 and 6.

Krolikowski, W.

Macdonald, K. R.

K. R. Macdonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Marrakchi, A.

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

Valley, G. C.

Valley, G. G.

Wang, P. Y.

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102–7107 (1996).
[CrossRef] [PubMed]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Numerical studies of externally-pumped phase conjugation in photorefractive crystals with applied electric fields,” Opt. Commun. 130, 302–306 (1996).
[CrossRef]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

Xie, P.

Yariv, A.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

Zgonik, M.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

Zhang, H. J.

Zhu, Y.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
[CrossRef]

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

J. Synthetic Cryst. (1)

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synthetic Cryst. 23, 242 (1994) (in Chinese).

Opt. Commun. (2)

J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Numerical studies of externally-pumped phase conjugation in photorefractive crystals with applied electric fields,” Opt. Commun. 130, 302–306 (1996).
[CrossRef]

Phys. Rep. (1)

P. Günter, “Holography, coherent light amplification, and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

K. R. Macdonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Other (1)

P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1989), Vol. II, Chaps. 5 and 6.

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Figures (11)

Fig. 1
Fig. 1

Schematic of externally pumped phase conjugation by four-wave mixing in 45°-cut photorefractive BaTiO3:Ce with an externally applied electric field E0 and a constant moving speed V.

Fig. 2
Fig. 2

(a) Reflectivity and (b) fidelity of the phase conjugation versus the input pump ratio r for case A. Dashed curves are for α=10° and solid curves for α=-10°.

Fig. 3
Fig. 3

(a) Reflectivity and (b) fidelity of the phase conjugation versus the input pump ratio r for case B. Dashed curves are for α=10° and solid curves for α=-10°.

Fig. 4
Fig. 4

Reflectivity as a function of the input pump ratio r for the plane-wave solution. The dashed curve is for α=10° and the solid curve for α=-10°.

Fig. 5
Fig. 5

(a) Reflectivity and (b) fidelity of the phase conjugation versus the parameter U0 for case A. α=10°.

Fig. 6
Fig. 6

(a) Reflectivity and (b) fidelity of the phase conjugation versus the parameter U0 for case B. α=10°.

Fig. 7
Fig. 7

(a) Reflectivity and (b) fidelity of the phase conjugation versus the external field E0 for case A. α=10°.

Fig. 8
Fig. 8

(a) Reflectivity and (b) fidelity of the phase conjugation versus the external field E0 for case B. α=10°.

Fig. 9
Fig. 9

Output phase changes of beam 3 in spatial frequency space for case A. Dashed curves correspond to the input phase changes of image beam 4. (a), (b) Input pump ratios r=0.1 and r=1, respectively, at α=10°; (c) r=10 at α=-10°.

Fig. 10
Fig. 10

Output phase changes (solid curves) of beam 3 in spatial frequency space for case A. Dashed curves correspond to the conjugate phase change of input beam 4. (a), (b), (c) Parameter U0=0.2, 0.6, and 1, respectively. r=1 and α=10°.

Fig. 11
Fig. 11

Output phase changes (solid curves) of beam 3 in spatial frequency space for case B. Dashed curves correspond to the conjugate phase change of input beam 4. (a), (b), (c) External field E0=20, 40, and 60, respectively, in units of volts per millimeter. r=1 and α=10°.

Equations (26)

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Ej(x, y, z, t)=Aj(x, y, z)exp[i(±kz-ωt)]+c.c.,
Ej(x, z, t)Aj(z)exp[±i(kxx+kzz)-iωt]+c.c.,
Aj(x, y, z)=qxqyfj(qx, qy, z)×exp{i[qxx+qyy+(βq-k)z]},
cos αdA1(z)dz=-θ1θ2 γ(-α, 0, θ1, θ2)I0(z)×[A1(z)f4*(θ1, θ2, z)+A2*(z)f3(θ1, θ2, z)]f4(θ1; θ2, z),
cos αdA2*(z)dz=-θ1θ2 γ(-α, 0, θ1, θ2)I0(z)×[A1(z)f4*(θ1, θ2, z)+A2*(z)f3(θ1, θ2, z)]f3*(θ1, θ2, z),
cos θ1 cos θ2df3(θ1, θ2, z)dz=γ(-α, 0, θ1, θ2)I0(z)[A1(z)f4*(θ1, θ2, z)+A2*(z)f3(θ1, θ2, z)]A2(z)-θ1θ2γ(θ1, θ2, θ1,θ2)I0(z)×[f4(θ1, θ2, z)f4*(θ1, θ2, z)+f3*(θ1, θ2, z)f3(θ1, θ2, z)]f3*(θ1, θ2, z),
cos θ1 cos θ2df4*(θ1, θ2, z)dz=γ(-α, 0, θ1, θ2)I0(z)[A1(z)f4*(θ1, θ2, z)+A2*(z)f3(θ1, θ2, z)]A1*(z)-θ1θ2 γ(θ1, θ2, θ1, θ2)I0(z)×[f4(θ1, θ2, z)f4*(θ1, θ2, z)+f3*(θ1, θ2, z)f3(θ1, θ2, z)]f4(θ1, θ2, z),
I0(z)=|A1(z)|2+|A2(z)|2+θ1θ2|f3(θ1, θ2, z)|2+θ1θ2|f4(θ1, θ2, z)|2,
γ(θ1, θ2, θ1, θ2)=-iωno32creff(θ1, θ2, θ1, θ2)×Esc(θ1, θ2, θ1, θ2)cos(θ1-θ1),
reff(θ1, θ2, θ1, θ2)={no4r13[sin β(cos θ2 sin θ1-cos θ2 sin θ1)+cos β(cos θ2 cos θ1-cos θ2 cosθ1)]×cos(β-θ1)cos(β-θ1)+no2ne2r42[cos β(cos θ2 sin θ1-cos θ2 sin θ1)-sin β(cos θ2 cos θ1-cos θ2 cos θ1)]sin(2β-θ1-θ1)+ne4r33[sin β(cos θ2 sin θ1-cos θ2 sin θ1)+cos β(cos θ2 cos θ1-cos θ2 cos θ1)]×sin(β-θ1)sin(β-θ1)}k/(no3neK).
K=k[(cos θ2 sin θ1-cos θ2 sin θ1)xˆ+(sin θ2-sin θ2)yˆ+(cos θ2 cos θ1-cos θ2 cos θ1)zˆ],
Esc(θ1, θ2, θ1, θ2)=Eq(Ed-iE0)E0+i(Eq+Ed)+[iE0-(Eq+Ed)]KVτ,
E0=E0 cos(E0·K),
V=V cos(V·K),
cos(E0·K)=cos(V·K)=[(cos θ2 sin θ1-cos θ2 sin θ1)cos(α/2)+(cos θ2 cos θ1-cos θ2 cos θ1)×sin(α/2)]k/K,
ε=εa sin2 χc+εc cos2 χc,
cos χc=[sin β(cos θ2 sin θ1-cos θ2 sin θ1)+cos β(cos θ2 cos θ1-cos θ2 cos θ1)]k/K.
FI=θ1θ2f4(θ1, θ2, 0)f3(θ1, θ2, 0)+c.c.2θ1θ2|f3(θ1, θ2, 0)|2 θ1θ2|f4(θ1, θ2, 0)|21/2.
FI=A4(x, y, 0)A3(x, y, 0)dxdy+c.c.2|A3(x, y, 0)|2dxdy|A4(x, y, 0)|2dxdy1/2.
R=θ1θ2|f3(θ1, θ2, 0)|2θ1θ2|f4(θ1, θ2, 0)|2.
Ai(x, y, 0)=const.0-d/2<xd/2x-d/2orx>d/2,y(-, +).
Ai(x, y, 0)=const.0-d/2<yd/2y-d/2ory>d/2,x(-, +).
cos(E0·K)|caseA=cos(θ1/2)
cos(E0·K)|caseB=sin(α/2)(1+cos θ2)[2(1-cos θ2 cos α)]1/2,
cos(E0·K)|caseB<1+cos θ2<cosθ2=cos(E0·K)|caseA,
cos(E0·K0)|caseB-cos(E0·K)|caseB>cos(E0·K0)|caseA-cos(E0·K)|caseA.

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