Abstract

We present the first complete theoretical model that describes the formation of the space-charge field in a photorefractive material that is exposed to a sinusoidal ac electric field with a dc offset and a running light-interference pattern. The model is complete in the sense that it is valid for all time scales and hence also covers situations in which the well-known period-averaging technique is inapplicable. For large temporal frequencies of the ac electric field the solution is shown to converge with that obtained from the period-averaging technique. It is shown that by imposing a running interference pattern and a sinusoidal ac electric field simultaneously, it becomes possible to enhance certain frequency components selectively. Experiments are performed to confirm the theoretical findings, and good agreement is obtained.

© 1998 Optical Society of America

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References

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  1. Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
    [Crossref]
  2. S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292 (1985).
    [Crossref]
  3. B. Ya. Zel’dovich and O. P. Nestiorkin, “Comparative analysis of time-dependent mechanisms for recording holograms in photorefractive crystals,” J. Mosc. Phys. Soc. 1, 231 (1991).
  4. J. Kumar, G. Albanese, and W. H. Steier, “Photorefractive two-beam coupling with applied radio-frequency fields: theory and experiment,” J. Opt. Soc. Am. B 4, 1079 (1987).
    [Crossref]
  5. K. Walsh, A. K. Powell, C. Stace, and T. J. Hall, “Techniques for the enhancement of space-charge fields in photorefractive materials,” J. Opt. Soc. Am. B 7, 288 (1990).
    [Crossref]
  6. A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric AC field,” Opt. Commun. 99, 221 (1993).
    [Crossref]
  7. A. Grunnet-Jepsen, C. H. Kwak, I. Richter, and L. Solymar, “Fundamental space-charge fields for applied alternating electric fields in photorefractive materials,” J. Opt. Soc. Am. B 11, 124 (1994).
    [Crossref]
  8. H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065 (1995).
    [Crossref]
  9. V. S. Liberman and B. Ya. Zel’dovich, “Recording of a running grating by a standing interference pattern in a photorefractive crystal,” Int. J. Nonlinear Opt. Phys. 3, 39 (1994).
    [Crossref]
  10. X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
    [Crossref]
  11. V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of a direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167 (1996).
    [Crossref]
  12. P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530 (1989).
    [Crossref]

1996 (1)

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of a direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167 (1996).
[Crossref]

1995 (1)

1994 (2)

V. S. Liberman and B. Ya. Zel’dovich, “Recording of a running grating by a standing interference pattern in a photorefractive crystal,” Int. J. Nonlinear Opt. Phys. 3, 39 (1994).
[Crossref]

A. Grunnet-Jepsen, C. H. Kwak, I. Richter, and L. Solymar, “Fundamental space-charge fields for applied alternating electric fields in photorefractive materials,” J. Opt. Soc. Am. B 11, 124 (1994).
[Crossref]

1993 (1)

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric AC field,” Opt. Commun. 99, 221 (1993).
[Crossref]

1991 (1)

B. Ya. Zel’dovich and O. P. Nestiorkin, “Comparative analysis of time-dependent mechanisms for recording holograms in photorefractive crystals,” J. Mosc. Phys. Soc. 1, 231 (1991).

1990 (1)

1989 (1)

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530 (1989).
[Crossref]

1988 (1)

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
[Crossref]

1987 (1)

1985 (2)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292 (1985).
[Crossref]

Albanese, G.

Dooghin, A. V.

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric AC field,” Opt. Commun. 99, 221 (1993).
[Crossref]

Gan, X.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
[Crossref]

Grunnet-Jepsen, A.

Hall, T. J.

Huignard, J.-P.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

Johansen, P. M.

H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065 (1995).
[Crossref]

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530 (1989).
[Crossref]

Kalinin, V. A.

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of a direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167 (1996).
[Crossref]

Kumar, J.

Kwak, C. H.

Liberman, V. S.

V. S. Liberman and B. Ya. Zel’dovich, “Recording of a running grating by a standing interference pattern in a photorefractive crystal,” Int. J. Nonlinear Opt. Phys. 3, 39 (1994).
[Crossref]

Nestiorkin, O. P.

B. Ya. Zel’dovich and O. P. Nestiorkin, “Comparative analysis of time-dependent mechanisms for recording holograms in photorefractive crystals,” J. Mosc. Phys. Soc. 1, 231 (1991).

Pedersen, H. C.

Petrov, M. P.

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292 (1985).
[Crossref]

Powell, A. K.

Rajbenbach, H.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

Richter, I.

Solymar, L.

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of a direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167 (1996).
[Crossref]

A. Grunnet-Jepsen, C. H. Kwak, I. Richter, and L. Solymar, “Fundamental space-charge fields for applied alternating electric fields in photorefractive materials,” J. Opt. Soc. Am. B 11, 124 (1994).
[Crossref]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

Stace, C.

Steier, W. H.

Stepanov, S. I.

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292 (1985).
[Crossref]

Sun, Y.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
[Crossref]

Walsh, K.

Ye, S.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
[Crossref]

Zel’dovich, B. Ya.

V. S. Liberman and B. Ya. Zel’dovich, “Recording of a running grating by a standing interference pattern in a photorefractive crystal,” Int. J. Nonlinear Opt. Phys. 3, 39 (1994).
[Crossref]

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric AC field,” Opt. Commun. 99, 221 (1993).
[Crossref]

B. Ya. Zel’dovich and O. P. Nestiorkin, “Comparative analysis of time-dependent mechanisms for recording holograms in photorefractive crystals,” J. Mosc. Phys. Soc. 1, 231 (1991).

Appl. Phys. Lett. (1)

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of a direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167 (1996).
[Crossref]

IEEE J. Quantum Electron. (1)

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530 (1989).
[Crossref]

Int. J. Nonlinear Opt. Phys. (1)

V. S. Liberman and B. Ya. Zel’dovich, “Recording of a running grating by a standing interference pattern in a photorefractive crystal,” Int. J. Nonlinear Opt. Phys. 3, 39 (1994).
[Crossref]

J. Appl. Phys. (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45 (1985).
[Crossref]

J. Mosc. Phys. Soc. (1)

B. Ya. Zel’dovich and O. P. Nestiorkin, “Comparative analysis of time-dependent mechanisms for recording holograms in photorefractive crystals,” J. Mosc. Phys. Soc. 1, 231 (1991).

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

Opt. Commun. (3)

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292 (1985).
[Crossref]

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric AC field,” Opt. Commun. 99, 221 (1993).
[Crossref]

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive media,” Opt. Commun. 66, 155 (1988).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Magnitude of the coefficients |ap| from Eq. (12), plotted in the range p=[-5, 5]. In both curves the fringe spacing is Λ=18 μm, the optical modulation is m=0.98, the temporal frequency of the ac field is η=10 s-1, the amplitude of the dc field is E0=106 v/m, and the ac field is fully modulated, i.e., Eη=E0. (a) Positive value of E0, (b) negative value of E0.

Fig. 2
Fig. 2

Normalized diffraction efficiency, ρ, versus time for different values of the ac field frequency, η. The straight line represents ρ obtained from the average solution, Eq. (9). E0 and Ω both equal zero. A 100 by 100 matrix was used to generate the plot.

Fig. 3
Fig. 3

Solid curve represents normalized diffraction efficiency, ρ, (left axis) versus time (bottom axis). Dots represent scaled amplitude components, |ap|/EM,K, (right axis) versus p (top axis). E0=Eη=10 kV/cm and η=1070 s-1. A 20 by 20 matrix was used to generate the plot.

Fig. 4
Fig. 4

Schematic presentation of the setup used for verification of the theoretical results.

Fig. 5
Fig. 5

Measured oscilloscope traces of the diffraction efficiency for the same six values of η as in Fig. 3. E0 and Ω both equal zero.

Fig. 6
Fig. 6

Measured oscilloscope traces of (i) the diffraction efficiency (thick curve, left axis) and (ii) the applied electric field (thin curve, right axis) for different values of Ω. E0=-10 kV/cm, Eη=10 kV/cm, and η=1070 s-1.    

Fig. 7
Fig. 7

Diffraction efficiency versus the applied dc electric field. The thick curve was obtained theoretically from Eq. (7), whereas the dots and the thin curve were obtained experimentally. The unit of the diffraction efficiency was scaled conveniently to obtain the best fit between the theoretical curve and the points obtained from experiment. The parameters are Ω=0, η=12 600 s-1, Eη=4 kV/cm, and E0, which is varied from -10 to 10 kV/cm.

Tables (1)

Tables Icon

Table 1 Relevant Material Parameters12 for Bi12SiO20 a

Equations (20)

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E˙(t)+Γ(t)E(t)=F(t),
Γ(t)=ω0 Ea(t)+i(ED+Eq)Ea(t)+i(ED+EM)-iΩ,
F(t)=12 mω0Eq ED-iEa(t)Ea(t)+i(ED+EM).
ED=kBTKe,EM=1τμK,Eq=eNAε0εsK.
Ea(t)=E0+Eη sin(ηt),
E(t)=FΓ [1-exp(Γt)],
Γ=1EΣ [(ω0-iΩ)EΣ+iω0(Eq-EM)],
F=-i mω02 Eq1-i EMEΣ,
EΣ=(E0+iED+iEM)1-EηE0+iED+iEM2.
Es-s=FΓ=-i mω02 Eq EΣ-iEM(ω0-iΩ)EΣ-iω0(Eq-EM).
E(t)=p=-ap exp(ipηt),
p=-[ω0(E0+iED+iEq)+(Ω-pη)(ED+EM-iE0)]ap-Eη2 [(p+1)η-(Ω+iω0)]ap+1+Eη2 [(p-1)η-(Ω+iω0)]ap-1exp(ipηt)
=-i2 mω0EqE0+iED-i Eη2×[exp(iηt)-exp(-iηt)],
ρ=10 |ap|2EM2,
rτ=2πητg,
τg=[Re(Γ)]-1,
Esc=K[a0 exp(-iηt)+a1]exp(iKr),
a0=-ω0Eσ 12 mEq(E0+iED)-i Eη2 a1,
a1=14 mEqEη iEσ-(η+iω0)(E0+iED)Eσ(E0+iED+iEq)+i Eη24 (η+iω0),
Eσ=ω0(E0+iED+iEq]+η(ED+EM-iE0).

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