Abstract

We calculate the photorefractive grating amplitude in the presence of a high-frequency applied electric field in terms of the mean-square drift length of the charge carriers. We describe how a shallow-trap level and the related long effective deep-trap recombination time are detrimental to the enhancement produced by a square-wave field. We give expressions for the shallow-trap-induced reduction in the steady-state photorefractive gain as well as for its frequency dependence. The photorefractive gain reaches the same value obtained in the one-level model at low frequencies but falls almost exponentially to a shallow-trap-limited value for higher frequencies. We compare the predictions of our model with other existing models and with experimental data.

© 1996 Optical Society of America

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References

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  1. 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]
  2. C. Besson, J. M. C. Jonathan, A. Villing, G. Pauliat, and G. Roosen, “Influence of alternating field frequency on enhanced photorefractive gain in two-beam coupling,” Opt. Lett. 14, 1359–1361 (1989).
    [CrossRef] [PubMed]
  3. F. Vachss, “Frequency-dependent photorefractive response in the presence of applied ac electric fields,” J. Opt. Soc. Am. B 11, 1045–1058 (1994).
    [CrossRef]
  4. J.-Y. Moisan, N. Wolffer, O. Moine, P. Gravey, G. Martel, A. Aoudia, R. Repka, Y. Marfaing, and R. Triboulet, “Characterization of photorefractive CdTe:V high two-wave mixing gain with an optimum low-frequency periodic external electric field,” J. Opt. Soc. Am. B 11, 1655–1667 (1994).
    [CrossRef]
  5. G. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
    [CrossRef]
  6. F. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773–3780 (1989).
    [CrossRef]
  7. G. Pauliat and G. Roosen, “Photorefractive effect generated in sillenite crystals by picosecond pulses and comparison with the quasi-continuous regime,” J. Opt. Soc. Am. B 7, 2259–2267 (1990).
    [CrossRef]
  8. P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
    [CrossRef]
  9. Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, “Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 µm,” Opt. Commun. 105, 204–208 (1994).
    [CrossRef]
  10. K. Magde and G. Brost, “Influence of the ac field frequency on the photorefractive response in Bi12SiO20,” Opt. Mater. 4, 322–325 (1995).
    [CrossRef]
  11. G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.
  12. A. Grunnet-Jepsen, L. Solymar, and C. H. Kwak, “Effect of subharmonics on two-wave gain in Bi12SiO20 under alternating electric fields,” Opt. Lett. 19, 1299–1301 (1994).
    [CrossRef] [PubMed]
  13. M. Ziari, W. H. Steier, P. M. Ranon, M. B. Klein, and S. Trivedi, “Enhancement of the photorefractive gain at 1.3–1.5 µm in CdTe using alternating electric fields,” J. Opt. Soc. Am. B 9, 1461–1466 (1992).
    [CrossRef]

1995 (1)

K. Magde and G. Brost, “Influence of the ac field frequency on the photorefractive response in Bi12SiO20,” Opt. Mater. 4, 322–325 (1995).
[CrossRef]

1994 (4)

1993 (1)

P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (2)

1985 (1)

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]

1983 (1)

G. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Aoudia, A.

Belaud, Y.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, “Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 µm,” Opt. Commun. 105, 204–208 (1994).
[CrossRef]

Besson, C.

Brost, G.

K. Magde and G. Brost, “Influence of the ac field frequency on the photorefractive response in Bi12SiO20,” Opt. Mater. 4, 322–325 (1995).
[CrossRef]

G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.

Delaye, P.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, “Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 µm,” Opt. Commun. 105, 204–208 (1994).
[CrossRef]

Gravey, P.

Grunnet-Jepsen, A.

Harris, M.

G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.

Hellwarth, R. W.

P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
[CrossRef]

Jonathan, J. M. C.

Klein, M. B.

Kwak, C. H.

Larking, J.

G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.

Launay, J.-C.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, “Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 µm,” Opt. Commun. 105, 204–208 (1994).
[CrossRef]

Magde, K.

K. Magde and G. Brost, “Influence of the ac field frequency on the photorefractive response in Bi12SiO20,” Opt. Mater. 4, 322–325 (1995).
[CrossRef]

G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.

Marfaing, Y.

Martel, G.

Moine, O.

Moisan, J.-Y.

Nouchi, P.

P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
[CrossRef]

Partanen, J. P.

P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
[CrossRef]

Pauliat, G.

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]

Ranon, P. M.

Repka, R.

Roosen, G.

Solymar, L.

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, 292–295 (1985).
[CrossRef]

Strohkendl, F.

F. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773–3780 (1989).
[CrossRef]

Triboulet, R.

Trivedi, S.

Vachss, F.

Valley, G.

G. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Villing, A.

Wolffer, N.

Ziari, M.

J. Appl. Phys. (1)

F. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773–3780 (1989).
[CrossRef]

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

Opt. Commun. (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]

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, “Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 µm,” Opt. Commun. 105, 204–208 (1994).
[CrossRef]

Opt. Eng. (1)

G. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Opt. Lett. (2)

Opt. Mater. (1)

K. Magde and G. Brost, “Influence of the ac field frequency on the photorefractive response in Bi12SiO20,” Opt. Mater. 4, 322–325 (1995).
[CrossRef]

Phys. Rev. B (1)

P. Nouchi, J. P. Partanen, and R. W. Hellwarth, “Simple transient solutions for photoconduction and the space-charge field in a photorefractive material with shallow traps,” Phys. Rev. B 47, 15581–15587 (1993).
[CrossRef]

Other (1)

G. Brost, K. Magde, J. Larking, and M. Harris, “Investigation of the frequency-dependent photorefractive response with alternating electric fields in BSO,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, DC., 1995), paper MPB2, pp. 144–157.

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Figures (2)

Fig. 1
Fig. 1

Definition of the relevant time constants. τ1 is the time constant that describes trapping into the deep level, τ2 is the time constant that describes trapping into the shallow level, and τth is the thermal excitation time from shallow traps.

Fig. 2
Fig. 2

Calculated space-charge field amplitude Esc/m as a function of the applied-field frequency in the presence of shallow traps. Monte Carlo simulation with the following parameters: τ1 = 1.25 ns, τth = 0.5 ms, effective trap density N = 4 × 1016 cm3, effective dielectric constant = 10.3, grating spacing Λg = 10 µm, applied-field amplitude E = 10 kV/cm. The horizontal dashed line corresponds to the one-level case, whereas the other curves indicate the behavior in the presence of a shallow-trap level with a characteristic trapping time τ2 from 2.22 to 0.022 ns.

Equations (24)

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E sc = m E q E D E q + E D ,
E q = e N 0 K g ,
E D = K g k B T e + μ τ E 2 1 + K g 2 L D 2 ,
d n d t = D d 2 n d x 2
D = L 2 2 τ .
D d n d x = μ n E 0 .
E D = K g 2 μ τ L D 2 + L E 2 .
P ( t ) = 1 τ exp t / τ ,
L E 2 = 2 θ 0 μ τ E t 2 d t ,
E D = K g k B T e + μ τ E 2 .
E D = K g k B T e + L E 2 2 τ μ 1 + K g 2 L D 2 .
P f t = n = 0 1 τ exp n θ + t τ = 1 τ exp t / τ 1 1 exp θ / τ ,
L E 2 = 0 θ L 2 t P f t d t ,
L + / 2 t = μ E 2 t 0 t t t s t s 2 d t s = μ E t 2 3 .
L 2 t | t < θ / 2 = μ E t 2 1 4 3 t θ .
L E 2 = μ 2 E 2 0 θ / 2 t 2 1 4 t 3 θ P f t + P f θ t d t ,
L E 2 = 2 μ E τ 2 1 4 τ θ 1 exp θ / 2 τ 1 + exp θ / 2 π .
τ 0 1 = τ 1 1 + τ 2 1 .
p 2 = τ 1 / τ 1 + τ 2
τ deep = τ 1 1 + τ th / τ 2 .
μ t = μ 1 + τ th / τ 2 1 .
L E 2 = 2 μ τ 0 E 2 + p 2 2 μ τ 0 E 2 + p 2 2 μ τ 0 E 2 + = 2 μ τ 0 E 2 1 1 p 2 ,
L E 2 = 2 μ τ 1 E 2 1 + τ 1 / τ 2 = L E 2 | one level 1+ τ 1 / τ 2 .
τ 2 I = τ 2 I = 0 + τ th N st n I ,

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