Abstract

The photorefractive response with moving gratings is investigated both numerically and experimentally under varied fringe-velocity and modulation-depth (m = 0.002 to m = 1) conditions. The numerical analysis employs a finite-difference technique to model photorefractive grating dynamics. The magnitude and the phase of the space-charge field are presented in detail as functions of modulation depth, fringe velocity, and crystal parameters. Energy transfer and diffraction efficiency are found to exhibit different response characteristics with modulation and fringe velocity. Numerical results for two-wave mixing are generalized through analytical expressions that approximate the numerical solutions. Illustrative experimental results are presented for Bi12SiO20.

© 1994 Optical Society of America

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  1. P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [Crossref]
  2. B. Imbert, H. Rajbenbach, S. Mallic, J. P. Herriau, and J.-P. Huignard, Opt. Lett. 13, 327 (1988).
    [Crossref] [PubMed]
  3. S. I. Stepanov and M. P. Petrov, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 1988).
  4. K. Walsh, A. K. Powell, C. Stace, and T. J. Hall, J. Opt. Soc. Am. B 7, 288 (1990).
    [Crossref]
  5. G. C. Valley, J. Opt. Soc. Am. B 1, 868 (1984).
    [Crossref]
  6. W. A. Gillespie, Z. Q. Wang, and C. M. Cartwright, presented at the Topical Meeting on Photorefractive Materials, Effects, and Devices (Kiev, Ukraine, August 11–15, 1993).
  7. G. A. Swinburne, T. J. Hall, and A. K. Powell, IEEE Proceedings of the International Conference on Holographic Systems, Components, and Applications (Institution of Electrical Engineers, London, 1989).
  8. L. B. Au and L. Solymar, J. Opt. Soc. Am. A 7, 1554 (1990).
    [Crossref]
  9. L. B. Au and L. Solymar, Opt. Lett. 13, 660 (1988).
    [Crossref]
  10. F. Vachss and L. Hessselink, J. Opt. Soc. Am. B 5, 1814 (1988).
    [Crossref]
  11. G. A. Brost, J. Opt. Soc. Am. B 9, 1454 (1992).
    [Crossref]
  12. G. A. Brost, Opt. Commun. 96, 113 (1993).
    [Crossref]
  13. A. Marrakchi, R. V. Johnson, and A. R. Tanguay, J. Opt. Soc. Am. B 3, 321 (1986).
    [Crossref]
  14. F. Vachss and T. Y. Chang, J. Opt. Soc. Am. A 6, 1683 (1989).
    [Crossref]
  15. D. J. Webb and L. Solymar, Opt. Commun. 83, 287 (1991).
    [Crossref]

1993 (1)

G. A. Brost, Opt. Commun. 96, 113 (1993).
[Crossref]

1992 (1)

1991 (1)

D. J. Webb and L. Solymar, Opt. Commun. 83, 287 (1991).
[Crossref]

1990 (2)

1989 (1)

F. Vachss and T. Y. Chang, J. Opt. Soc. Am. A 6, 1683 (1989).
[Crossref]

1988 (3)

1986 (1)

1985 (1)

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

1984 (1)

Au, L. B.

Brost, G. A.

Cartwright, C. M.

W. A. Gillespie, Z. Q. Wang, and C. M. Cartwright, presented at the Topical Meeting on Photorefractive Materials, Effects, and Devices (Kiev, Ukraine, August 11–15, 1993).

Chang, T. Y.

F. Vachss and T. Y. Chang, J. Opt. Soc. Am. A 6, 1683 (1989).
[Crossref]

Gillespie, W. A.

W. A. Gillespie, Z. Q. Wang, and C. M. Cartwright, presented at the Topical Meeting on Photorefractive Materials, Effects, and Devices (Kiev, Ukraine, August 11–15, 1993).

Hall, T. J.

K. Walsh, A. K. Powell, C. Stace, and T. J. Hall, J. Opt. Soc. Am. B 7, 288 (1990).
[Crossref]

G. A. Swinburne, T. J. Hall, and A. K. Powell, IEEE Proceedings of the International Conference on Holographic Systems, Components, and Applications (Institution of Electrical Engineers, London, 1989).

Herriau, J. P.

Hessselink, L.

Huignard, J.-P.

B. Imbert, H. Rajbenbach, S. Mallic, J. P. Herriau, and J.-P. Huignard, Opt. Lett. 13, 327 (1988).
[Crossref] [PubMed]

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Imbert, B.

Johnson, R. V.

Mallic, S.

Marrakchi, A.

Petrov, M. P.

S. I. Stepanov and M. P. Petrov, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 1988).

Powell, A. K.

K. Walsh, A. K. Powell, C. Stace, and T. J. Hall, J. Opt. Soc. Am. B 7, 288 (1990).
[Crossref]

G. A. Swinburne, T. J. Hall, and A. K. Powell, IEEE Proceedings of the International Conference on Holographic Systems, Components, and Applications (Institution of Electrical Engineers, London, 1989).

Rajbenbach, H.

B. Imbert, H. Rajbenbach, S. Mallic, J. P. Herriau, and J.-P. Huignard, Opt. Lett. 13, 327 (1988).
[Crossref] [PubMed]

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Réfrégier, P.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Solymar, L.

D. J. Webb and L. Solymar, Opt. Commun. 83, 287 (1991).
[Crossref]

L. B. Au and L. Solymar, J. Opt. Soc. Am. A 7, 1554 (1990).
[Crossref]

L. B. Au and L. Solymar, Opt. Lett. 13, 660 (1988).
[Crossref]

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Stace, C.

Stepanov, S. I.

S. I. Stepanov and M. P. Petrov, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 1988).

Swinburne, G. A.

G. A. Swinburne, T. J. Hall, and A. K. Powell, IEEE Proceedings of the International Conference on Holographic Systems, Components, and Applications (Institution of Electrical Engineers, London, 1989).

Tanguay, A. R.

Vachss, F.

F. Vachss and T. Y. Chang, J. Opt. Soc. Am. A 6, 1683 (1989).
[Crossref]

F. Vachss and L. Hessselink, J. Opt. Soc. Am. B 5, 1814 (1988).
[Crossref]

Valley, G. C.

Walsh, K.

Wang, Z. Q.

W. A. Gillespie, Z. Q. Wang, and C. M. Cartwright, presented at the Topical Meeting on Photorefractive Materials, Effects, and Devices (Kiev, Ukraine, August 11–15, 1993).

Webb, D. J.

D. J. Webb and L. Solymar, Opt. Commun. 83, 287 (1991).
[Crossref]

J. Appl. Phys. (1)

P. Réfrégier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

J. Opt. Soc. Am. A (2)

L. B. Au and L. Solymar, J. Opt. Soc. Am. A 7, 1554 (1990).
[Crossref]

F. Vachss and T. Y. Chang, J. Opt. Soc. Am. A 6, 1683 (1989).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

G. A. Brost, Opt. Commun. 96, 113 (1993).
[Crossref]

D. J. Webb and L. Solymar, Opt. Commun. 83, 287 (1991).
[Crossref]

Opt. Lett. (2)

Other (3)

S. I. Stepanov and M. P. Petrov, in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, New York, 1988).

W. A. Gillespie, Z. Q. Wang, and C. M. Cartwright, presented at the Topical Meeting on Photorefractive Materials, Effects, and Devices (Kiev, Ukraine, August 11–15, 1993).

G. A. Swinburne, T. J. Hall, and A. K. Powell, IEEE Proceedings of the International Conference on Holographic Systems, Components, and Applications (Institution of Electrical Engineers, London, 1989).

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Figures (16)

Fig. 1
Fig. 1

Amplitude, imaginary component, and phase of the fundamental component of the space-charge field for various values of the modulation index. The applied field was Ea = 5 kV/cm.

Fig. 2
Fig. 2

Normalized imaginary component of the space-charge field versus beam ratio. The value of Im(E1) corresponds to that at the modulation-optimized fringe velocity.

Fig. 3
Fig. 3

Calculated correction function versus modulation index for moving gratings. The solid curve is a fit of the numerical results to the equation given in the figure.

Fig. 4
Fig. 4

Calculated correction function versus modulation index for moving gratings. Curve a, μτ = 1.9 × 10−7 cm2/V, Λ = 3 μm, EA = 10 kV/cm; curve b, μτ = 1.9 × 10−7 cm2/V, Λ = 20 μm, EA = 5 kV/cm; curve c, μτ = 1.9 × 10−7 cm2/V, Λ = 10 μm, EA = 5 kV/cm; curve d, μτ = 6.3 × 10−7 cm2/V, Λ = 30 μm, EA = 10 kV/cm. The solid curves are fits of the numerical results to the equation f(m0 = 1/a[1 − exp(−am)]. The values of a determined from the fit were (a) 0.83, (b) 2.9, (c) 3.8, (d) 8.9.

Fig. 5
Fig. 5

Correction parameter versus enhancement parameter R.

Fig. 6
Fig. 6

Calculated effective gain coefficient versus grating period for various input beam ratios and crystal thicknesses. Curve a, Γ = ΓS; curve b, β = 106, d = 1 mm; curve c, β = 106, d = 10 mm; curve d, β = 102, d = 1 mm; curve e, β = 102, d = 10 mm.

Fig. 7
Fig. 7

Calculated effective gain coefficient versus crystal thickness for various values of modulation. The velocity at the crystal entrance is νopt. Each solid curve includes absorption. Each dashed curve includes both absorption and optical activity.

Fig. 8
Fig. 8

Variation of the effective gain coefficient versus normalized fringe velocity at the crystal entrance for m = 0.2.

Fig. 9
Fig. 9

Calculated temporal evolution of the fundamental component of the space-charge field at the optimum velocity for two-wave mixing for various values of modulation.

Fig. 10
Fig. 10

Schematic diagram of the experimental arrangement for characterizing the photorefractive response with moving gratings in BSO. AO, acousto-optic cell; DET, detector.

Fig. 11
Fig. 11

Measured two-beam coupling gain coefficient versus beam ratio in BSO. The solid curve is a fit of the function f(m) = 1/a[1 − exp(−am)] to the experimental data.

Fig. 12
Fig. 12

Measured two-wave mixing gain coefficient versus frequency detuning for various values of the modulation index.

Fig. 13
Fig. 13

Measured normalized diffraction efficiency versus frequency detuning for various values of the modulation index.

Fig. 14
Fig. 14

Comparison of the measured two-wave mixing gain and diffraction efficiency versus frequency detuning for (a) m = 0.6. and (b) m = 0.9.

Fig. 15
Fig. 15

Measured two-wave mixing gain coefficient versus grating period for m = 0.006 and m = 0.2.

Fig. 16
Fig. 16

Experimental temporal evolution of the diffraction signal for various values of modulation.

Equations (19)

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d N 1 / d t = - ( S 1 I + β 1 ) N 1 + γ 1 ( N 1 T - N 1 ) N h ,
d N h / d t = - d N 1 / d t ,
N h / d t = d N h / d t - · j h ,
j h = μ h N h E - D N h ,
· E = ρ / 0 ,
I ( x , t ) = I 0 { 1 + m sin [ K ( x - v t ) ] } ,
E 1 = i m ( - E A + i E D ) / D 1 ,
D 1 = - E A / E q + b [ 1 + ( E D / E M ) ] + i [ 1 + ( E D / E Q ) + ( b E A / E M ) ] , E D = K k B T / q ,             E Q = q N e / ( 0 K ) , E M = γ R N e / μ K , b = K v τ di e ,
v opt = A [ 1 + ( 1 - G / A 2 H ) 1 / 2 ] / K τ di e ,
A = ( E A / E Q ) + E D / E A [ 1 + ( E D / E Q ) ] ,             B = E M / E D , c = E M / E D ( 1 + E D / E Q ) ,             f = ( E D + E M ) / E A , H = 1 + f 2 ,             G = 2 A f B - 2 A c - c 2 - B 2 .
R = E 0 / ( E A + E D ) ,
Im [ E 1 ( m ) ] = f ( m ) E 0 .
f ( m ) = 1 / a [ 1 - exp ( - a m ) ] .
a = 1.43 R - 0.85.
d I R / d z = - Γ S [ f ( m , v ) / m ] [ I R I S / ( I R + I S ) ] - α I R ,
d I S / d z = Γ S [ f ( m , v ) / m ] [ I R I S / ( I R + I S ) ] - α I S ,
Γ S = 2 π r eff n 3 E 0 / λ cos ( θ )
Γ eff = ( 1 / d ) ln [ γ 0 β / ( β + 1 - γ 0 ) ] ,
τ e = [ f ( m ) / m ] τ g .

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