Abstract

We show that the relative phase of the grating components has a large effect on the temporal behavior of four-wave mixing. When the two main components of the grating are in phase, the temporal behavior, especially during grating erasure, is totally different from that when the two components are out of phase. We give experimental results of orthogonally polarized four-wave mixing for two different BaTiO3 crystals that are in good agreement with our theoretical calculations. In our calculations we use a simplified numerical method to solve the coupled-wave equations, and the temporal analysis is based on a model with multiple charge recombination.

© 2001 Optical Society of America

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References

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  1. A. Taj and T. Mishima, “Orthogonally polarized four-wave mixing involving two noncoupling waves photorefractive crystal,” J. Opt. Soc. Am. B 15, 2132–2142 (1998).
    [CrossRef]
  2. A. Taj and T. Mishima, “A simplified numerical solution and steady state performance of orthogonally polarized four-wave mixing in a PR medium,” J. Opt. Soc. Am. B 16, 924–931 (1999).
    [CrossRef]
  3. M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
    [CrossRef]
  4. A. Taj, P. Xie, and T. Mishima, “Anomalous temporal behavior of photorefractive wave mixing owing to the existence of both positive and negative charge carriers,” J. Opt. Soc. Am. B 17, 1740–1748 (2000).
    [CrossRef]
  5. G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
    [CrossRef]
  6. G. A. Brost, R. A. Motes, and J. R. Rotge, “Intensity dependent absorption and photorefractive effects in barium titi-nate,” J. Opt. Soc. Am. B 5, 1879–1885 (1988).
    [CrossRef]
  7. P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20 and BaTiO3 with shallow traps,” J. Opt. Soc. Am. B 8, 1053–1064 (1991).
    [CrossRef]
  8. F. P. Strohkendl, J. M. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef]
  9. A. Blódowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in the single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544–1547 (1989).
    [CrossRef]
  10. M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
    [CrossRef]
  11. P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  12. P. Xie, I. A. Taj, and T. Mishima, “Effect of thermal noise on the temporal behavior of the photorefractive effect,” J. Opt. Soc. Am. B (to be published).

2000 (1)

1999 (1)

1998 (1)

1996 (1)

M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
[CrossRef]

1991 (1)

1989 (1)

1988 (1)

1986 (2)

F. P. Strohkendl, J. M. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
[CrossRef]

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

1985 (1)

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1984 (1)

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

Blódowski, A.

Brost, G. A.

Carrascosa, M.

M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
[CrossRef]

Cronin-Golomb, M.

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]

Hellwarth, R. W.

Huignard, J. P.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Jonathan, J. M.

Królikowski, W.

Kujawski, A.

Limeres, J.

M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
[CrossRef]

Mahgerefteh, D.

Mishima, T.

Motes, R. A.

Pino, M. del.

M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
[CrossRef]

Rajbenbach, H.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Réfrégier, P.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Rotge, J. R.

Solymar, L.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Strohkendl, F. P.

Taj, A.

Tayebati, P.

Valley, G. C.

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

White, J. O.

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

Xie, P.

Yariv, A.

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

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. Appl. Phys. (2)

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals in moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (1)

M. del. Pino, J. Limeres, and M. Carrascosa, “Time evolution of the photorefractive phase conjugation process in BaTiO3,” Opt. Commun. 131, 211–218 (1996).
[CrossRef]

Opt. Lett. (1)

Other (1)

P. Xie, I. A. Taj, and T. Mishima, “Effect of thermal noise on the temporal behavior of the photorefractive effect,” J. Opt. Soc. Am. B (to be published).

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Figures (4)

Fig. 1
Fig. 1

Configuration of OPP FWM used in our experiments. The direction of the optic axis shown in this figure results in IPCG in our crystals. The opposite direction of the optic axis results in OPCG.

Fig. 2
Fig. 2

Temporal evolution of DE in OPP FWM in crystal 1 for different values of the pump ratio: (a) experimental results, (b) calculations (parameters are taken from Table 1). Both IPCG and OPCG are shown. The angle between writing beams inside the crystal (ϕ1-ϕp) is 12.5°, the probe ratio is 1, and the combined intensity of the two writing waves is 1.66×104 W/m2.

Fig. 3
Fig. 3

Temporal evolution of DE in OPP FWM for crystal 2 for different values of the pump ratio: (a) experimental results, (b) calculations (parameters are taken from Table 1). The experimental conditions are same as for crystal 1.

Fig. 4
Fig. 4

Calculations of grating erasing time τER, which is the time comsumed while the DE is decreased from its steady-state value to 10% of its steady-state value, shown as a function of pump ratio. The probe ratio is 1, and the combined intensity of both writing waves is 1.6×104 W/m2. Calculations for (1) crystal 1, (ϕ1-ϕp)=5°; (2) crystal 1, (ϕ1-ϕp)=15°; (3) crystal 2, (ϕ1-ϕp)=5°; and (4) crystal 2, (ϕ1-ϕp)=15°.

Equations (33)

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dA1dr=-Q1Ap+α2A1,
dA2*dr=-Q2Ad*-α2A2*,
dAp*dr=+Q1A1*+α2Ap*,
dAddr=+Q2A2-α2Ad,
Q1=πλr1peffn1 cosϕp-ϕ12iE1(t),
Q2=πλrd2effn2 cosϕd-ϕ22iE1(t),
E1(t)iqkε[D1(t)-S1(t)],
D1(t)t=-εkEDENhqt0h(ED+E>μh)I1I0[1+βD/(sDI0)]-(ED+ENh)t0h(ED+Eμh)D1(t)+ENht0h(ED+Eμh)S1(t),
S1(t)t=-εkEDENeqt0e(ED+Eμe)I1I0[1+βS(sSI0)]-(ED+ENe)t0e(ED+Eμe)S1(t)+ENet0e(ED+Eμe)D1(t),
Aqaq exp(iθq),
I1=[eˆ1·eˆpa1ap exp(iθ0)+eˆ2·eˆdada2]×expi[(θd-θ2)],
θ0(θ1-θp)-(θd-θ2)ρI-ρII,
Q(r, t)=q(r, t)exp[i(θi-θp)].
a1I1+p sin u,
a2I2+d cos v,
apI1+p cos u,
adI2+p sin v,
I1+p(r)=I1(r)+Ip(r)=[I1(r=r)+Ip(r=r)]×[exp α(r-r)],
I2+d(r)=I2(r)+Id(r)=I2(r=0)exp(-αr).
ur=-q1(r, t),
vr=q2(r, t)cos θ0.
I1=(eˆ1·eˆp)I1+p(r=r)exp[α(r-r)]sin 2u+(eˆ2·eˆd)I2(r=0)exp(-αr)×sin 2v cos θ0,
I0=[I1(r=r)+Ip(r=r)]exp[α(r-r)]+I2(r=0)exp(-αr).
DEId(r=r)I2(r=0)=sin2 v(r=r)exp(-αr),
ψI2(r=0)/I1(r=r),
σIp(r=r)I1(r=r)=cot2 u(r=r).
v(r=0)=0,u(r=r)=cot-1/σ.
EDkBTkq,
EμhγDND0μhk,EμeγSNS0μek,
ENhqND0(NDT-ND0)εkNDT,
ENeqNS0(NST-NS0)εkNST,
t0hND0NDT(sDI0+βD),
t0eNS0NST(sSI0+βS).

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