Abstract

Grating erasure in the dark has different temporal characteristics for different crystals. We present experimental results of four-wave mixing in a BaTiO3 crystal, which exhibits a large increase in the diffraction efficiency for the first few seconds after the writing beams are switched off. It is clear from these results that multiple photorefractive processes that have different time constants are involved. We study two different models, including two types of charge recombination center, one with a smaller and the other with larger time constant. The rising phenomenon is explained by use of one of these models. Experimental results for three crystals that exhibit different temporal behaviors are shown, and properties of these three crystals are discussed based on calculations for this model.

© 2000 Optical Society of America

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References

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  1. G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
    [CrossRef]
  2. M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3 at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
    [CrossRef]
  3. G. A. Brost, R. A. Motes, and J. R. Rotge, “Intensity-dependent absorption and photorefractive effects in barium titanate,” J. Opt. Soc. Am. B 5, 1879–1885 (1988).
    [CrossRef]
  4. 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]
  5. I. A. Taj and T. Mishima, “Simplified numerical solution and steady-state performance of orthogonally polarized four-wave mixing in a PR medium,” J. Opt. Soc. Am. B 6, 924–931 (1999).
    [CrossRef]
  6. F. P. Strohkendl, J. M. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef]
  7. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, Berlin, 1988–1989), Chap. 6, p. 25.
  8. 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]
  9. C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
    [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. A. Blȩdowski, 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]

1999 (1)

I. A. Taj and T. Mishima, “Simplified numerical solution and steady-state performance of orthogonally polarized four-wave mixing in a PR medium,” J. Opt. Soc. Am. B 6, 924–931 (1999).
[CrossRef]

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]

1995 (1)

C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[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 (2)

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3 at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[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]

Ble¸dowski, 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]

del Pino, 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]

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.

Klein, M. B.

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3 at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

Królikowski, W.

Kujawski, A.

Kwak, C. H.

C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[CrossRef]

Lee, E.

C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[CrossRef]

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.

I. A. Taj and T. Mishima, “Simplified numerical solution and steady-state performance of orthogonally polarized four-wave mixing in a PR medium,” J. Opt. Soc. Am. B 6, 924–931 (1999).
[CrossRef]

Motes, R. A.

Park, S. Y.

C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[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, I. A.

I. A. Taj and T. Mishima, “Simplified numerical solution and steady-state performance of orthogonally polarized four-wave mixing in a PR medium,” J. Opt. Soc. Am. B 6, 924–931 (1999).
[CrossRef]

Tayebati, P.

Valley, G. C.

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

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3 at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

J. Appl. Phys. (3)

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

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3 at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[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 (4)

Opt. Commun. (2)

C. H. Kwak, S. Y. Park, and E. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal,” Opt. Commun. 115, 315–322 (1995).
[CrossRef]

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. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, Berlin, 1988–1989), Chap. 6, p. 25.

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Figures (6)

Fig. 1
Fig. 1

Four-wave mixing configuration used in the experiments. All waves have extraordinary polarization. The direction of the optic axis is shown by arrows inside the crystal (dotted arrow, 0°-cut crystal; solid arrow, 45°-cut crystal). Diff., diffracted.

Fig. 2
Fig. 2

Experimental results showing the log of DE as a function of time for three values of pump ratio ψ. Probe ratio, σ=1; angle between writing beams inside the crystal, 10°; combined intensity of both writing waves, 1.63×104 W/m2.

Fig. 3
Fig. 3

Charge-transfer diagrams for the two models. The encircled characters show the charges of donors and traps; n is neutral or no charge.

Fig. 4
Fig. 4

-D1 and -S1 with -(D1+S1) for model 1 and with -(D1-S1) for model 2 shown as functions of time for two values of NST, i.e., 3.5×1022 m-3 (dotted curve) and 10×1022m-3 (solid curve). Other parameters are given in Table 1.(a) Calculations for model 1, (b) calculations for model 2.

Fig. 5
Fig. 5

Calculations based on model 2 showing space-charge field E1 as a function of time for three values of grating vector magnitude k. NST=6×1022 m-3; the other components are as listed in Table 1.

Fig. 6
Fig. 6

Temporal evolution of DE in four-wave mixing for three crystals. Angle between writing beams inside the crystal, 10°; pump ratio, 1/5; probe ratio, 1; μe=μh=5×10-6 m2 V-1 s-1 for crystal 1 and μe=μh=10-6 m2 V-1 s-1 for crystals 2 and 3. Other parameters are taken from Table 1 and Table 2. (a) Experimental results, (b) calculations based on model 2.

Tables (2)

Tables Icon

Table 1 Photorefractive Parameters of Crystalsa

Tables Icon

Table 2 Properties of Three Crystals at λ=514.5nm

Equations (54)

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E1(t)t+1τE1(t)=1τE1ss,
1τ=1τe+1τh.
E1ss=i ENEDm(τh-τe)2(ED+EN)(τh+τe),
D0(t)t=-(sD I0+βD)D0(t)+γD×[NE-D0(t)-S0(t)][NDT-D0(t)],
S0(t)t=-(sS I0+βS)S0(t)+γS×[NE-D0(t)-S0(t)][NST-S0(t)],
NENDT+NST-NA.
nh0(t)NE-D0(t)-S0(t).
EDkBTkq,
EμDγD[NDT-D0(t)]μhk,EμSγS[NST-S0(t)]μhk.
D1(t)t=AD m2-BDD1(t)+CDM1(t),
AD-[sD I0D0(t)(ED+EμS)-EμDsS I0S0(t)](ED+EμD+EμS),
BD(ED+EμS)[(sD I0+βD)+γDnh0(t)]+EμDnh0(t)μhq/ε(ED+EμD+EμS),
CDEμD[(sS I0+βS)+γSnh0(t)]-EμDnh0(t)μhq/ε(ED+EμD+EμS),
m=2I1/I0.
S1(t)t=AS m2-BSS1(t)+CSD1(t),
AS-[sS I0S0(t)(ED+EμD)-EμSsD I0D0(t)](ED+EμD+EμS),
BS(ED+EμD)[(sS I0+βS)+γSnh0(t)]+EμSnh0(t)μhq/ε(ED+EμD+EμS),
CSEμS[(sD I0+βD)+γDnh0(t)]-EμSnh0(t)μhq/ε(ED+EμD+EμS).
D1=BS AD+CD ASBSBD-CDCS m2,
S1=CS AD+BD ASBSBD-CDCS m2.
E1(t)iqkε[D1(t)+S1(t)].
ΔNND0-NS0.
EμhγDND0μhk,EμeγSNS0μek,
ENhqND0(NDT-ND0)εkNDT,
ENeqNS0(NST-NS0)εkNST,
t0hND0NDT(sD I0+βD),
t0eNS0NST(sS I0+βS),
D1(t)t=-εkEDENhqt0h(ED+Eμh)×m2[1+βD/(sD I0)]-(ED+ENh)t0h(ED+Eμh)D1(t)+ENht0h(ED+Eμh)S1(t),
S1(t)t=-εkEDENeqt0e(ED+Eμe)×m2[1+βS/(sS I0)]-(ED+ENe)t0e(ED+Eμe)S1(t)+ENet0e(ED+Eμe)D1(t).
 D1=-εkENhq(ED+ENe+ENh)×ED+ENe1+βD/(sD I0)+ENe1+βS/(sS I0) m2,
S1=-εkENeq(ED+ENe+ENh)×ED+ENh1+βS/(sS I0)+ENh1+βD/(sD I0) m2.
E1(t)iqkε[D1(t)-S1(t)].
f(m)=(1/a)[1-exp(-am)],
ND(x, t)t=-[sD I(x, t)+βD]ND(x, t)+γDnh(x, t)[NDT-ND(x, t)],
NS(x, t)t=-[sS I(x, t)+βS]NS(x, t)+γSnh(x, t)×[NST-NS(x, t)],
Jx(x, t)=μhqnh(x, t)Ex(x, t)-kBTμh nh(x, t)x,
Jx(x, t)x=-q [ND(x, t)+NS(x, t)+nh(x, t)]t.
Ex(x, t)x=q[ND(x, t)+NS(x, t)+nh(x, t)-(NDT+NST-NA)]ε.
ND(x, t)=D0(t)+[D1(t)exp(-ikx)+D1*(t)exp(ikx)],
NS(x, t)=S0(t)+[S1(t)exp(-ikx)+S1*(t)exp(ikx)],
nh(x, t)=nh0(t)+[nh1(t)exp(-ikx)+nh1*(t)exp(ikx)],
ExSC(x, t)=E0+[E1(t)exp(-ikx)+E1*(t)exp(ikx)],
I(x, t)=I0+I1 exp(-ikx)+I1* exp(ikx),
ND(x, t)t=-[sD I(x, t)+βD]ND(x, t)+γDnh(x, t)×[NDT-ND(x, t)],
NS(x, t)t=-[sS I(x, t)+βS]NSi(x, t)+γSne(x, t)×[NST-NS(x, t)],
Jh(x, t)=μhqnh(x, t)Ex(x, t)-kBTμh nh(x, t)x,
Je(x, t)=μeqne(x, t)Ex(x, t)+kBTμe ne(x, t)x,
Jx(x, t)=Jh(x, t)+Je(x, t),
Jh(x, t)x=-q [ND(x, t)+nh(x, t)]t,
Je(x, t)x=q [NS(x, t)+ne(x, t)]t.
Ex(x, t)x=q[ND(x, t)-NS(x, t)+nh(x, t)-ne(x, t)-(NDT-NST-ΔN)]ε.
ne(x, t)=ne0(t)+[ne1(t)exp(-ikx)+ne1*(t)exp(ikx)].
D0(t)=-nh0(t)+NDT-ND0,
S0(t)=-ne0(t)+NST-NS0.

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