Abstract

A new cascading technique has been devised to yield solutions for the electric space-charge field that is induced in a photorefractive medium when a sinusoidal electric field and a running light interference pattern are imposed simultaneously. The cascading iteration technique offers rapid convergence to the exact solution. In addition, an advantage of the technique is that it can be used in situations when more than one frequency is present in the problem at hand. The results compare favorably with those of experiments after only a few cascading iterations, and a new low-frequency resonance is found in the space-charge field.

© 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 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (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, 292–295 (1985).
    [CrossRef]
  3. K. Walsh, A. K. Powell, C. Stace, and T. J. Hall, “Techniques for enhancement of space-charge fields in photorefractive materials,” J. Opt. Soc. Am. B 7, 288–303 (1990).
    [CrossRef]
  4. 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, 124–131 (1994).
    [CrossRef]
  5. I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
    [CrossRef]
  6. A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric ac fields,” Opt. Commun. 99, 221–224 (1993).
    [CrossRef]
  7. V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167–169 (1996).
    [CrossRef]
  8. H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065–1073 (1995).
    [CrossRef]
  9. P. M. Johansen and H. C. Pedersen, “Photorefractive space-charge field with running grating and applied sinusoidal ac electric field: solution for all time scales,” J. Opt. Soc. Am. B(to be published).
  10. D. J. Webb, K. Shcherbin, V. A. Kalinin, J. Takacs, and L. Solymar, “First experimental demonstration of a new technique for photorefractive grating enhancement,” in Photorefractive Materials, Effects and Devices, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper TA2, pp. 352–354.
  11. T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
    [CrossRef]

1997 (1)

I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
[CrossRef]

1996 (1)

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167–169 (1996).
[CrossRef]

1995 (2)

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065–1073 (1995).
[CrossRef]

1994 (1)

1993 (1)

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric ac fields,” Opt. Commun. 99, 221–224 (1993).
[CrossRef]

1990 (1)

1985 (2)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (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, 292–295 (1985).
[CrossRef]

Aubrecht, I.

I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
[CrossRef]

Dooghin, A. V.

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric ac fields,” Opt. Commun. 99, 221–224 (1993).
[CrossRef]

Grunnet-Jepsen, A.

I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
[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, 124–131 (1994).
[CrossRef]

Hall, T. J.

Huignard, J. P.

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

Johansen, P. M.

Kalinin, V. A.

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167–169 (1996).
[CrossRef]

Kwak, C. H.

Mann, M.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

McClelland, T. E.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

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, 292–295 (1985).
[CrossRef]

Powell, A. K.

Rajbenbach, H.

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

Refregier, Ph.

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

Richter, I.

Ringhofer, K. H.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Solymar, L.

I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
[CrossRef]

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167–169 (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, 124–131 (1994).
[CrossRef]

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

Stace, C.

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, 292–295 (1985).
[CrossRef]

Sturman, B. I.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Walsh, K.

Webb, D. J.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Zel’dovich, B. Ya.

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric ac fields,” Opt. Commun. 99, 221–224 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

V. A. Kalinin and L. Solymar, “Space charge fields in photorefractive crystals in the presence of direct current and alternating current fields: a new resonance,” Appl. Phys. Lett. 68, 167–169 (1996).
[CrossRef]

J. Appl. Phys. (1)

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

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

Opt. Commun. (4)

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985).
[CrossRef]

I. Aubrecht, L. Solymar, and A. Grunnet-Jepsen, “Space charge field enhancement in photorefractive materials by applied sinusoidal fields: an approximate analytical solution,” Opt. Commun. 139, 73–76 (1997).
[CrossRef]

A. V. Dooghin and B. Ya. Zel’dovich, “Two-wave mixing in photorefractive crystals at asymmetric ac fields,” Opt. Commun. 99, 221–224 (1993).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sellenites,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Other (2)

P. M. Johansen and H. C. Pedersen, “Photorefractive space-charge field with running grating and applied sinusoidal ac electric field: solution for all time scales,” J. Opt. Soc. Am. B(to be published).

D. J. Webb, K. Shcherbin, V. A. Kalinin, J. Takacs, and L. Solymar, “First experimental demonstration of a new technique for photorefractive grating enhancement,” in Photorefractive Materials, Effects and Devices, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper TA2, pp. 352–354.

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Figures (6)

Fig. 1
Fig. 1

Schematic setup. The two recording beams from an argon laser at 514 nm are slightly shifted in temporal frequency by Ω, hence inducing a running interference grating. The externally applied electric field consists of a dc part and a sinusoidal ac part. The readout is made by recording of the diffracted signal from a He–Ne laser at 633 nm on a photodetector. BSO, bismuth silicon oxide.

Fig. 2
Fig. 2

Scaled diffraction efficiency as a function of time for various cascading iterations. The curves were generated for Ω=0, a value of the applied field amplitude of E0=Eη=106 V/m, and a frequency of the ac electric field of η=1070 rad/s.

Fig. 3
Fig. 3

Oscilloscope traces (reproduced from Ref. 9) of the diffraction efficiency versus time for different values of Ω. In the applied electric field (thin curve, right-hand axis) is also shown. Traces are shown for E0=Eη=106 V/m and η=1070 rad/s.

Fig. 4
Fig. 4

Theoretical plots of the scaled diffraction efficiency versus time for four values of Ω. The curves are plotted for the same parameters as the experimental ones in Fig. 3.

Fig. 5
Fig. 5

Scaled modulus of the space-charge (sc) field versus frequency: (a), (b), (c), and (d) are plotted for ratios of Eη/E0 of 1/7, 3/7, 5/7, and 1, respectively. Dashed curves represent the fundamental component of the space-charge field, and solid curves are produced from a full cascaded approach.

Fig. 6
Fig. 6

Comparison of the scaled diffraction efficiency versus time calculated from the Fourier series solution from Ref. 9 (open circles) and the cascading iteration outlined in the present paper (solid curve). Both curves are plotted for the values η=1070 rad/s, Ω=1.5η, and E0=Eη=106 V/m.

Equations (40)

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I=I0[1+m cos(Kx-Ωt)],
Ea(t)=E0+Eη sin(ηt),
Esc=E(t)exp(iKx-iΩt)+c.c.,
E˙(t)+Γ(K, t)E(t)=F(K, t),
Γ(K, t)=ω0 Ea(t)+i(ED,K+Eq,K)Ea(t)+i(ED,K+EM,K)-iΩ,
F(K, t)=12 ω0mEq,K ED,K-iEa(t)Ea(t)+i(ED,K+EM,K),
ED,K=kBTKe,EM,K=1μτK,
Eq,K=eNAε0sK,ω0=NDNA sI0,
E(t)=exp-Γ(K, t)dtE(0)-F(K, t)×expΓ(K, t)dtdt,
G(ω)=12π -g(t)exp(iωt)dt,
E(ω)=A1(0)(ω)+A2(0)(ω)E(ω+η)+A3(0)(ω)E(ω-η),
A1(0)(ω)=2π2 ω0m Eq,KD(ω) (ED,K-iE0)δ(ω)-1/2Eη[δ(ω+η)-δ(ω-η)],
A2(0)(ω)=i2 EηD(ω) [ω0-i(Ω+ω+η)],
A3(0)(ω)=-i2 EηD(ω) [ω0-i(Ω+ω-η)],
D(ω)=ω0[E0+i(ED,K+Eq,K)]-i(Ω+ω)×[E0+i(ED,K+EM,K)],
E(ω+η)=A1(0)(ω+η)+A2(0)(ω+η)E(ω+2η)+A3(0)(ω+η)E(ω),
E(ω-η)=A1(0)(ω-η)+A2(0)(ω-η)E(ω)+A3(0)(ω-η)E(ω-2η).
E(ω)=A1(1)(ω)+A2(1)(ω)E(ω+2η)+A3(1)(ω)E(ω-2η),
A1(1)(ω)=A1(0)(ω)+A2(0)(ω)A1(0)(ω+η)+A3(0)(ω)A1(0)(ω-η)1-A2(0)(ω)A3(0)(ω+η)-A3(0)(ω)A2(0)(ω-η),
A2(1)(ω)=A2(0)(ω)A2(0)(ω+η)1-A2(0)(ω)A3(0)(ω+η)-A3(0)(ω)A2(0)(ω-η),
A3(1)(ω)=A3(0)(ω)A3(0)(ω-η)1-A2(0)(ω)A3(0)(ω+η)-A3(0)(ω)A2(0)(ω-η).
A1(n)(ω)=A1(n-1)(ω)+A2(n-1)(ω)A1(n-1)(ω+2n-1η)+A3(n-1)(ω)A1(n-1)(ω-2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),
A2(n)(ω)=A2(n-1)(ω)A2(n-1)(ω+2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),
A3(n)(ω)=A3(n-1)(ω)A3(n-1)(ω-2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),n=1, 2,  .
E(t)12 ω0mEq,KED,K-iE0D(0)-12 Eη exp(iηt)D(-η)+12 Eη exp(-iηt)D(η).
E(ω)=A1(n)(ω)+A2(n)(ω)E(ω+2nη)+A3(n)(ω)E(ω-2nη),
A1(n)(ω)=A1(n-1)(ω)+A2(n-1)(ω)A1(n-1)(ω+2n-1η)+A3(n-1)(ω)A1(n-1)(ω-2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),
A2(n)(ω)=A2(n-1)(ω)A2(n-1)(ω+2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),
A3(n)(ω)=A3(n-1)(ω)A3(n-1)(ω-2n-1η)1-A2(n-1)(ω)A3(n-1)(ω+2n-1η)-A3(n-1)(ω)A2(n-1)(ω-2n-1η),n=1,2,3, .
E(ω+2kη)=A1(k)(ω+2kη)+A2(k)(ω+2kη)×E(ω+2k+1η)+A3(k)×(ω+2kη)E(ω),
E(ω-2kη)=A1(k)(ω-2kη)+A2(k)(ω-2kη)E(ω)+A3(k)(ω-2kη)E(ω-2k+1η).
E(ω)=A1(k)(ω)+A2(k)(ω)A1(k)(ω+2kη)+A3(k)(ω)A1(k)(ω-2kη)1-A2(k)(ω)A3(k)(ω+2kη)-A3(k)(ω)A2(k)(ω-2kη)+A2(k)(ω)A2(k)(ω+2kη)1-A2(k)(ω)A3(k)(ω+2kη)-A3(k)(ω)A2(k)(ω-2kη) E(ω+2k+1η)+A3(k)(ω)A3(k)(ω-2kη)1-A2(k)(ω)A3(k)(ω+2kη)-A3(k)(ω)A2(k)(ω-2kη)E(ω+2k+1η).
E(ω)=A1(k+1)(ω)+A2(k+1)(ω)E(ω+2k+1η)+A3(k+1)(ω)E(ω-2k+1η).
A1(0)(ω), A1(1)(ω), A1(2)(ω), , A1(n)(ω), 
E(ω)A1(n)(ω),n=0, 1, 2,  .
|A1(0)(ω)|1,|Ai(0)(ω)|,i=2, 3,
|A1(n)(ω)|1,
|Ai(n)(ω)|2n,i=2, 3,
|A1(n)(ω)-E(ω)|2n
limnA1(n)(ω)=E(ω).

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