Abstract

We study four-wave mixing with orthogonally polarized pumps (OPP-FWM) in photorefractive crystal for both transmission and reflection gratings, taking pump depletion into consideration. We simplify the coupled wave equations in an analytical way, which makes qualitative understanding of the performance of OPP-FWM easier. Then we solve the simplified equations numerically by a shooting method. Properties such as the coupling efficiency of the writing beams, the diffraction efficiency, and the diffracted beam reflectivity are calculated as functions of the beam angles, the probe ratio, and the pump ratio. Some of the calculated results are compared with the experimental results. Two principal concepts, i.e., how the diffraction efficiency and the reflectivity are affected by changes in the relative phase of the two grating components and by the feedback effect that is due to the coupling of writing waves, are clearly explained with experimental proof. We also show that, because it requires a different form of the coupled wave equations, OPP-FWM shows a different angular dependence and has a much higher efficiency at larger angles between probe and pumps, as compared with the usual geometry of four-wave mixing in which all four waves are either ordinary or extraordinary.

© 1999 Optical Society of America

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References

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  1. S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
    [CrossRef]
  2. A. Roy and K. Singh, J. Mod. Opt. 41, 987 (1994).
    [CrossRef]
  3. H. Kong, C. Lin, A. M. Biernacki, and M. Cronin-Golomb, Opt. Lett. 13, 324 (1988).
    [CrossRef] [PubMed]
  4. M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
    [CrossRef]
  5. I. Jacques and C. Judd, Numerical Analysis (Chapman & Hall, London, 1987).
  6. I. A. Taj and T. Mishima, Trans. Inst. Electron. Inf. Commun. Eng. C-I J81-C-I (6), 376–378 (1998).
  7. I. A. Taj and T. Mishima, J. Opt. Soc. Am. B 15, 2132 (1998).
    [CrossRef]
  8. H. Matsuoka, A. Okamoto, M. Takamura, and K. Sato, J. Opt. Soc. Am. B 15, 1545 (1998).
    [CrossRef]
  9. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  10. M. Zgonik, K. Nakagawa, and P. Günter, J. Opt. Soc. Am. B 12, 1416 (1995).
    [CrossRef]

1998 (3)

1995 (1)

1994 (1)

A. Roy and K. Singh, J. Mod. Opt. 41, 987 (1994).
[CrossRef]

1988 (1)

1985 (1)

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

1984 (1)

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Biernacki, A. M.

Cronin-Golomb, M.

H. Kong, C. Lin, A. M. Biernacki, and M. Cronin-Golomb, Opt. Lett. 13, 324 (1988).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Fisher, B.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Günter, P.

Kong, H.

Lin, C.

Matsuoka, H.

Mishima, T.

I. A. Taj and T. Mishima, Trans. Inst. Electron. Inf. Commun. Eng. C-I J81-C-I (6), 376–378 (1998).

I. A. Taj and T. Mishima, J. Opt. Soc. Am. B 15, 2132 (1998).
[CrossRef]

Nakagawa, K.

Okamoto, A.

Petrov, M. P.

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Roy, A.

A. Roy and K. Singh, J. Mod. Opt. 41, 987 (1994).
[CrossRef]

Sato, K.

Singh, K.

A. Roy and K. Singh, J. Mod. Opt. 41, 987 (1994).
[CrossRef]

Stepanov, S. I.

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Taj, I. A.

I. A. Taj and T. Mishima, Trans. Inst. Electron. Inf. Commun. Eng. C-I J81-C-I (6), 376–378 (1998).

I. A. Taj and T. Mishima, J. Opt. Soc. Am. B 15, 2132 (1998).
[CrossRef]

Takamura, M.

White, J. O.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Yariv, A.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Zgonik, M.

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

J. Mod. Opt. (1)

A. Roy and K. Singh, J. Mod. Opt. 41, 987 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Opt. Lett. (1)

Trans. Inst. Electron. Inf. Commun. Eng. C-I (1)

I. A. Taj and T. Mishima, Trans. Inst. Electron. Inf. Commun. Eng. C-I J81-C-I (6), 376–378 (1998).

Other (2)

I. Jacques and C. Judd, Numerical Analysis (Chapman & Hall, London, 1987).

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

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Figures (7)

Fig. 1
Fig. 1

Geometry for OPP-FWM in photorefractive crystal for (a) a transmission grating and (b) a reflection grating.

Fig. 2
Fig. 2

(a) Configuration 1: DE and CE as functions of ϕd-ϕ2 (solid curves) for IPCG, γ1=2 cm-1, γ2=6 cm-1, and (dotted curves) for OPCG, γ1=-2 cm-1, γ2=-6 cm-1. (b) Configuration 2: DE and CE as functions of ϕp-ϕ1, (solid curves) for IPCG, γ1=6 cm-1, γ2=2 cm-1, and (dotted curves) for OPCG, γ1=-6 cm-1, γ2=-2 cm-1.

Fig. 3
Fig. 3

Contour plot of percentage DE as a function of the grating vector angle ϕg and the difference of the beam angles (ϕ1-ϕp) in configuration 1. Calculations of coupling coefficients are included. NA=2×1016 cm-3, r=0.25 cm.

Fig. 4
Fig. 4

3D plot of percentage DE as a function of the grating vector angle ϕg and the difference of the beam angles (ϕ1-ϕp) in configuration 1. NA=2×1016 cm-3, r=0.25 cm.

Fig. 5
Fig. 5

DE and DBR as functions of |ϕ1-ϕp|, for both IPCG and OPCG cases, configuration 1, and a transmission grating: (a) theoretical calculations, (b) experimental results.

Fig. 6
Fig. 6

DE and DBR for IPCG as functions of the pump ratio ψ for different values of the probe ratio σ at ϕ1=255.43°, ϕ2=75.35°, and ϕp=275.61° (configuration 1 and transmission grating): (a) theoretical calculations, (b) experimental results.

Fig. 7
Fig. 7

DE and DBR for OPCG as functions of the pump ratio ψ for different values of the probe ratio σ at ϕ1=75.05°, ϕ2=254.97°, and ϕp=95.75° (configuration 1 and transmission grating): (a) theoretical calculations, (b) experimental results.

Equations (43)

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dA1dr=-γ1eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0Ap,
dA2*dr=-γ2eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0Ad*,
dAp*dr=γ1eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0A1*,
dAddr=γ2eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0A2.
Aiai exp(jθi),
ddr(a12+ap2)=0orI1+IpI1+p=const.,
ddr(a22+ad2)=0orI2+IdI2+d=const.,
a1I1+p sin q,a2I2+d cos t,
apI1+p cos q,adI2+d sin t.
θ0(θ1-θp)-(θd-θ2)ρI-ρII,
tanθ02=expC1-γ2eˆ1·eˆpI1+pI0+0r sin(2q)cot(2t)dr+γ1eˆ2·eˆdI2+dI0+0r sin(2t)cot(2q)dr.
θ0=0whenγ2ispositive,
θ0=πwhenγ2isnegative.
dqdr=-γ1[X sin(2q)+Y sin(2t)cos θ0],
dtdr=γ2[X sin(2q)cos θ0+Y sin(2t)],
Xeˆ1·eˆpI1+p2I0,Yeˆ2·eˆdI2+d2I0.
ψI2(r=0)I1(r=r)=a22(r=0)a12(r=r),
σIp(r=r)I1(r=r)=ap2(r=r)a12(r=r)=tan2 q(r=r).
t(r=0)=0,q(r=r)=cot-1 σ.
CEI1(r=0)-I1(r=r)Ip(r=r)=sin2 q(r=0)-sin2 q(r=r)cos2 q(r=r),
DEId(r=r)I2(r=0)=sin2 t(r=r),
DBRId(r=r)Ip(r=r)=ad2(r=r)ap2(r=r)=ψDEσ.
dA1dr=γ1eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0Ap,
dA2*dr=γ2eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0Ad*,
dAp*dr=γ1eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0A1*,
dAddr=γ2eˆ1·eˆpA1Ap*+eˆ2·eˆdAdA2*I0A2.
ddr(a12-ap2)=0orI1-IpI1-p=const.,
ddr(a22-ad2)=0orI2-IdI2-d=const.
a1I1-p cosh q,
a2I2-d cosh t,
apI1-p sinh q,
adI2-d sinh t.
I0=I1-p cosh(2q)+I2-d cosh(2t).
tanθ02=expC1+γ2eˆ1·eˆpI1-p×+0rsinh(2q)[tanh(t)-coth(t)]2I0dr+γ1eˆ2·eˆdI2-d×+0rsinh(2t)[tanh(q)-coth(q)]2I0dr.
θ0=0whenγ2ispositive(IPCG),
θ0=πwhenγ2isnegative(OPCG).
dqdr=γ1[eˆ1·eˆpI1-p sinh(2q)+eˆ2·eˆdI2-d sinh(2t)cos θ0]2[I1-p cosh(2q)+I2-d cosh(2t)],
dtdr=γ2[eˆ1·eˆpI1-p sinh(2q)cos θ0+eˆ2·eˆdI2-d sinh(2t)]2[I1-p cosh(2q)+I2-d cosh(2t)].
ψI2(r=r)I1(r=0)=a22(r=r)a12(r=0)
σIp(r=r)I1(r=0)=ap2(r=r)a12(r=0).
CE=I1(r=r)-I1(r=0)Ip(r=r)=cosh2 q(r=r)-cosh2 q(r=0)sinh2 q(r=r),
DEId(r=r)I2(r=r)=tanh2 t(r=r),
DBRId(r=r)Ip(r=r)=ad2(r=r)ap2(r=r)=ψ DEσ.

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