Abstract

We analyze the phase-conjugate reflectivity and the amplification property of cross-polarized four-wave mixing with a photorefractive BaTiO3 crystal in consideration of the pump-beam depletion. The coupled-wave equations of cross-polarized four-wave mixing with ordinary writing beams and an extraordinary reading beam are derived and solved with the grating-integral method. The results of calculations of the reflectivity and the amplification factors will be compared with the result under the undepleted-pumps approximation. In addition, to achieve high conversion efficiency, the optimum conditions for the intensity ratio and the angle configurations of the probe and pump beams are also discussed.

© 1998 Optical Society of America

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References

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  1. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and application of FWM in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12 (1984).
    [CrossRef]
  2. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  3. M. Esselbach, A. Kiessling, H. Rehn, B. Fleck, and R. Kowarscik, “Transient phase measurement using a self-pumped phase-conjugate mirror as an optical-novelty filter,” J. Opt. Soc. Am. B 14, 846 (1997).
    [CrossRef]
  4. O. Ikeda, “Low-pass, high-pass, and bandpass spatial-filtering characteristics of a BaTiO3 phase conjugator,” J. Opt. Soc. Am. B 4, 1387 (1987).
    [CrossRef]
  5. I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
    [CrossRef]
  6. T. M. Das and G. C. Bhar, “Phase-conjugate bistability andmultistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019 (1994).
    [CrossRef]
  7. H. Kong, C. Lin, A. M. Biernacki, and M. Cronin-Golomb, “Photorefractive phase conjugation with orthogonally polarized pumping beams,” Opt. Lett. 13, 324 (1988).
    [CrossRef] [PubMed]
  8. W. Wu, S. Campbell, S. Zhou, and P. Yeh, “Polarization-encoded optical logic operations in photorefractive media,” Opt. Lett. 18, 1742 (1993).
    [CrossRef] [PubMed]
  9. A. Roy and K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate FWM in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987 (1994).
    [CrossRef]
  10. M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
    [CrossRef]
  11. M. Saito, A. Okamoto, Y. Takayama, and M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” Photorefractive Materials, Effects and Devices (IEEE, New York, 1997), pp. 204–207.
  12. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486 (1982).
    [CrossRef] [PubMed]
  13. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313 (1982).
    [CrossRef] [PubMed]

1997

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

M. Esselbach, A. Kiessling, H. Rehn, B. Fleck, and R. Kowarscik, “Transient phase measurement using a self-pumped phase-conjugate mirror as an optical-novelty filter,” J. Opt. Soc. Am. B 14, 846 (1997).
[CrossRef]

1994

T. M. Das and G. C. Bhar, “Phase-conjugate bistability andmultistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019 (1994).
[CrossRef]

A. Roy and K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate FWM in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987 (1994).
[CrossRef]

1993

1992

I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
[CrossRef]

1988

1987

1984

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

1982

Bhar, G. C.

T. M. Das and G. C. Bhar, “Phase-conjugate bistability andmultistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019 (1994).
[CrossRef]

Biernacki, A. M.

Campbell, S.

Cronin-Golomb, M.

Das, T. M.

T. M. Das and G. C. Bhar, “Phase-conjugate bistability andmultistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019 (1994).
[CrossRef]

Esselbach, M.

Feinberg, J.

Fischer, B.

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

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313 (1982).
[CrossRef] [PubMed]

Fleck, B.

Ikeda, O.

Khoo, I. C.

I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
[CrossRef]

Kiessling, A.

Kong, H.

Kowarscik, R.

Li, H.

I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
[CrossRef]

Liang, Y.

I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
[CrossRef]

Lin, C.

Okamoto, A.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

Rehn, H.

Roy, A.

A. Roy and K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate FWM in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987 (1994).
[CrossRef]

Saito, M.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

Sato, K.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

Singh, K.

A. Roy and K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate FWM in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987 (1994).
[CrossRef]

Takayama, Y.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

White, J. O.

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

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313 (1982).
[CrossRef] [PubMed]

Wu, W.

Yariv, A.

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

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase conjugation,” Opt. Lett. 7, 313 (1982).
[CrossRef] [PubMed]

Yeh, P.

Zhou, S.

IEEE J. Quantum Electron.

I. C. Khoo, Y. Liang, and H. Li, “Coherent-beam amplification and polarization switching in a birefringent medium—photorefractive crystal,” IEEE J. Quantum Electron. 28, 1816 (1992).
[CrossRef]

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

J. Mod. Opt.

A. Roy and K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate FWM in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

T. M. Das and G. C. Bhar, “Phase-conjugate bistability andmultistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019 (1994).
[CrossRef]

Opt. Rev.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686 (1997).
[CrossRef]

Other

M. Saito, A. Okamoto, Y. Takayama, and M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” Photorefractive Materials, Effects and Devices (IEEE, New York, 1997), pp. 204–207.

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

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Figures (8)

Fig. 1
Fig. 1

(a) Optical geometry of CPFWM. Aj (j=14) denotes the complex field amplitude. Θ is the relative angle between the c axis and the z axis. The z axis is defined as the interaction direction and the bisector of the angle between A1 and A4. (b) Wave vector and polarization state of each wave. kj, θj, and e^j denote the wave vector, the angle between kj and the z axis, and the polarization unit vector of Aj, respectively. Each θj is a phase-matching angle that satisfies the Bragg condition.

Fig. 2
Fig. 2

(a) Dependence of the phase-conjugate reflectivity R under the UPA case on pump ratio and pu for various values of orientation angle Θ. At Θ=225°, the largest reflectivity is obtained. (b) Dependence of the amplification factor of the probe beam m on pu for various values of Θ. Signal amplification occurs in the range 0°<Θ<180°.

Fig. 3
Fig. 3

Phase-conjugate reflectivity R versus Θ. (a) R for various values of pr, where pu=0.1 is assumed. Calculation result under the UPA case is also plotted. (b) R for various values of pu, where pr=1 is assumed. The peak of R in the region 180°<Θ<360° divided into two peaks.

Fig. 4
Fig. 4

Three-dimensional plot showing the reflectivity R with pu and pr for (a) Θ=45° and (b) Θ=90°. At Θ=45° the reflectivity is larger than that of the arrangement Θ=90°.

Fig. 5
Fig. 5

Amplification factor of the probe beam m versus Θ. (a) m for various values of pr, where pu=0.1 is assumed. Calculation result under the UPA case is also plotted. (b) m for various values of pu, where pr=1 is assumed. The amplification becomes large as pu decreases.

Fig. 6
Fig. 6

Distribution of the normalized intensity of each beam along the z axis for (a) Θ=45°, 225° and (b) Θ=60°, 240°. pu=1 and pr=1 are assumed. At Θ=240° the largest reflectivity is obtained.

Fig. 7
Fig. 7

Reflectivity R versus θ2, which is the relative angle between the backward pump beam and the z axis. pu=1 and pr=1 are assumed. When Θ=225°, the highest conversion is obtained at θ2=1° and θ2=5.5°.

Fig. 8
Fig. 8

Amplification factor m versus θ2. pu=1 and pr=1 are assumed. When Θ=225°, the most effective wave mixing is realized at θ2=5.5° because the depletion of the probe beam is small and highest conversion is obtained.

Equations (60)

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dA1dz=-γ0I0 (A1A4*e^1e^4+A2*A3e^2e^3)A4-a1A1,
dA2dz=-γeI0 (A2A3*e^2e^3+A1A4*e^1e^4)A3+a2A2,
dA3dz=γeI0 (A2*A3e^2e^3+A1A4*e^1e^4)A2+a3A3,
dA4dz=γ0I0 (A1*A4e^1e^4+A2A3*e^2e^3)A1-a4A4.
Δk=(k4-k1)-(k2-k3)
e^1=e^4=(010)T,
e^2=(-cos θ20-sin θ2)T,
e^3=(-cos θ30sin θ3)T.
I0=j=14|Aj|2.
reffo=r13 sin[(φ1+φ4)/2],
reffe={r13no4 cos φ2 cos φ3+2r42no2ne2×cos2[(φ2+φ3)/2]+r33ne4 sin φ2 sin φ3}×sin[(φ2+φ3)/2]/(neno3).
dA1/dz=dA2/dz=0,
dA3/dz=γe(A2*A3e^2e^3+A1A4*)A2/I0,
dA4/dz=γo(A1*A4+A2A3*e^2e^3)A1/I0,
I0I1+I2,
e^1e^4=1.
R=|A3*(0)|2|A4(0)|2=γe2pu(1+pu)2 sinh(βL)α sinh(βL)+β cosh(βL)2.
m=|A4(L)|2|A4(0)|2=β exp(δL)α sinh(βL)+β cosh(βL)2,
α=(γepue^2e^3-γo)/2(1+pu),
β=α2+(puγoγee^2e^3)/(1+pu)2,
δ=(γepue^2e^3+γo)/2(1+pu),
pu=I2/I1.
dA1/dz=-γo(A1A4*+A2*A3e^2e^3)A4/I0,
dA2/dz=-γe(A2A3*e^2e^3+A1A4*)A3/I0,
dA3/dz=γe(A2*A3e^2e^3+A1A4*)A2/I0,
dA4/dz=γo(A1*A4+A2A3*e^2e^3)A1/I0.
G=(A1A4*+A2*A3e^2e^3)/I0.
G= |G|exp(-iϕ),
u=0zγo|G|dz,
v=0zγe|G|dz,
dA1/du=-exp(-iϕ)A4,
dA2/dv=-exp(iϕ)A3,
dA3/dv=exp(-iϕ)A2,
dA4/du=exp(iϕ)A1.
R=[I2(L)sin2 V]/I4(0),
V=0Lγe|G|dz.
m=[cos U+I1(0)/I4(0) sin U]2.
U=0Lγo|G|dz=(γo/γe)V.
0V (1+pu+pr)dvpr cos2 γoγe v+(1-pr)2 sin2 γoγe v+pue^2e^32 sin[2(v-V)]=γeL,
pu=I2(L)/I1(0),
pr=I4(0)/I1(0).
I1(z)=I1(0)(cos2 u+pr sin2 u-pr sin 2u),
I2(z)=I1(0)pu cos2(v-V),
I3(z)=I1(0)pu sin2(v-V),
I4(z)=I1(0)(pr cos2 u+sin2 u+pr sin 2u).
A1=A1(0)cos u-A4(0)exp(-iϕ)sin u,
A2=A2(L)cos(v-V),
A3=A2(L)exp(-iϕ)sin(v-V),
A4=A4(0)cos u+A1(0)exp(iϕ)sin u,
V=0Lγe|G|dz.
R=[I2(L)sin2 V]/I4(0),
m=[cos U+I1(0)/I4(0) sin U]2,
dv/dz=γeexp(iϕ)(A1A4*+A2*A3e^2e^3)/I0.
0v I0 exp(-iϕ)A1A4*+A2*A3e^2e^3 dv=γez.
0V I0 exp(-iϕ)A1A4*+A2*A3e^2e^3 dv=γeL.
A1A4*=I1(0)I4(0) exp(-iϕ)cos(2u)+I1(0)-I4(0)2 exp(-iϕ)sin(2u),
A2*A3e^2e^3=I2(L)e^2e^32 exp(-iϕ)sin[2(v-V)].
0V (1+pu+pr)dvpr cos2 γoγe v+(1-pr)2 sin2 γoγe v+pue^2e^32 sin[2(v-V)]=γeL,
pu=I2(L)/I1(0),
pr=I4(0)/I1(0),

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