Abstract

Two coherent light beams can couple in any absorbing material owing to a light-induced modulation of the material’s dielectric constant. In photorefractive crystals the coupling caused by these absorption gratings appears in addition to any electro-optic coupling, complicating the interpretation of data. However, in contrast with the electro-optic gratings formed by charge diffusion, absorption gratings do not necessarily vanish as the beam-crossing angle approaches zero if there is more than one absorbing level. We show that a plot of the coupling strength of the absorption gratings versus the beam-crossing angle is characterized by only three parameters, independent of the number of absorbing levels. We use absorption gratings with a two-level model to determine experimentally some important crystal parameters, including the relative density of donors and acceptors in a barium titanate crystal. Our values agree with those obtained from measurements of the bulk light-induced absorption of the crystal.

© 1991 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  2. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
    [CrossRef]
  3. L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi (A) K89, 113 (1989).
  4. D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
    [CrossRef] [PubMed]
  5. D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
    [CrossRef] [PubMed]
  6. A. Motes and J. J. Kim, “Intensity-dependent absorption coefficient in photorefractive BaTiO3crystals,” J. Opt. Soc. Am. B 4, 1379 (1987).
    [CrossRef]
  7. 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 (1988).
    [CrossRef]
  8. A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).
  9. K. Walsh, T. J. Hall, and R. E. Burge, “Influence of polarization state and absorption gratings on photorefractive two-wave mixing in GaAs,” Opt. Lett. 12, 1026 (1987).
    [CrossRef] [PubMed]
  10. R. B. Bylsma, D. H. Olson, and A. M. Glass, “Photochromic gratings in photorefractive materials,” Opt. Lett. 13, 853 (1988).
    [CrossRef] [PubMed]
  11. R. M. Pierce, R. S. Cudney, G. D. Bacher, and J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414 (1990).
    [CrossRef] [PubMed]
  12. P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989); P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1991).
    [CrossRef]
  13. A. V. Knya’zkov and M. N. Lobanov, “Absorption modulation under hologram formation in photorefractive materials,” in Topical Meeting on Photorefractive Materials, Effects and Devices II (Optical Society of America, Washington, D.C., 1990), pp. 112–113.
  14. M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
    [CrossRef]
  15. F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312 (1986).
    [CrossRef]

1990 (2)

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

R. M. Pierce, R. S. Cudney, G. D. Bacher, and J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414 (1990).
[CrossRef] [PubMed]

1989 (2)

D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
[CrossRef] [PubMed]

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi (A) K89, 113 (1989).

1988 (2)

1987 (2)

1986 (1)

1985 (1)

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
[CrossRef]

1983 (1)

A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Alekseev-Popov, A. V.

A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).

Bacher, G. D.

Brost, G. A.

Burge, R. E.

Bylsma, R. B.

Cudney, R. S.

Feinberg, J.

R. M. Pierce, R. S. Cudney, G. D. Bacher, and J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414 (1990).
[CrossRef] [PubMed]

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

Glass, A. M.

D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
[CrossRef] [PubMed]

R. B. Bylsma, D. H. Olson, and A. M. Glass, “Photochromic gratings in photorefractive materials,” Opt. Lett. 13, 853 (1988).
[CrossRef] [PubMed]

Hall, T. J.

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

Hellwarth, R. W.

F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312 (1986).
[CrossRef]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

Holtmann, L.

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi (A) K89, 113 (1989).

Jonathan, J. M. C.

Kim, J. J.

Klein, M. B.

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
[CrossRef]

Knya’zkov, A. V.

A. V. Knya’zkov and M. N. Lobanov, “Absorption modulation under hologram formation in photorefractive materials,” in Topical Meeting on Photorefractive Materials, Effects and Devices II (Optical Society of America, Washington, D.C., 1990), pp. 112–113.

Knyaz’kov, A. V.

A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Lobanov, M. N.

A. V. Knya’zkov and M. N. Lobanov, “Absorption modulation under hologram formation in photorefractive materials,” in Topical Meeting on Photorefractive Materials, Effects and Devices II (Optical Society of America, Washington, D.C., 1990), pp. 112–113.

Mahgerefteh, D.

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Motes, A.

Motes, R. A.

Nolte, D. D.

D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
[CrossRef] [PubMed]

Odoulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Olson, D. H.

D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
[CrossRef] [PubMed]

R. B. Bylsma, D. H. Olson, and A. M. Glass, “Photochromic gratings in photorefractive materials,” Opt. Lett. 13, 853 (1988).
[CrossRef] [PubMed]

Pierce, R. M.

Rotge, J. R.

Saikin, A. S.

A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Strohkendl, F. P.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

Tayebati, P.

P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989); P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1991).
[CrossRef]

Valley, G. C.

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Walsh, K.

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

J. Appl. Phys. (2)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980); J. Appl. Phys. 52, 537(E) (1981).
[CrossRef]

M. B. Klein and G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
[CrossRef]

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

Opt. Lett. (4)

Phys. Rev. Lett. (2)

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

D. D. Nolte, D. H. Olson, and A. M. Glass, “Nonequilibrium screening of the photorefractive effect,” Phys. Rev. Lett. 63, 891 (1989).
[CrossRef] [PubMed]

Phys. Status Solidi (A) (1)

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi (A) K89, 113 (1989).

Sov. Tech. Phys. Lett. (1)

A. V. Alekseev-Popov, A. V. Knyaz’kov, and A. S. Saikin, “Recording volume amplitude-phase holograms in a lead–lanthanum zirconate–titanate ceramic,” Sov. Tech. Phys. Lett. 9, 475 (1983).

Other (2)

P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989); P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1991).
[CrossRef]

A. V. Knya’zkov and M. N. Lobanov, “Absorption modulation under hologram formation in photorefractive materials,” in Topical Meeting on Photorefractive Materials, Effects and Devices II (Optical Society of America, Washington, D.C., 1990), pp. 112–113.

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Figures (7)

Fig. 1
Fig. 1

(a) Diffusive charge transport; (b) no diffusion.

Fig. 2
Fig. 2

Two-beam coupling absorption grating gain per unit length, γabs, versus internal half-angle θint in BaTiO3 (the Free crystal) at six intensities for λ = 488 nm. The solid curves are fits to Eq. (8). The gain does not vanish as the crossing angle goes to zero, but it reaches an offset value Φ. In the inset the offset can clearly be seen to be negative. The magnitude of the offset and the absorption-grating gain both vary with intensity.

Fig. 3
Fig. 3

Two-beam coupling gain per unit length at λ = 458 nm and λ = 488 nm in BaTiO3 (the Free crystal) at I = 0.6 W/cm2 with (a) the pure absorption-grating configuration and (b) the absorption-plus-electro-optic grating configuration. In (b) the sign of the crossing angle is the same as the sign of kg · ĉ, where the positive direction of 8 is chosen as in Ref. 2. The solid curves in (a) and (b) are fits to Eqs. (8) and (21), respectively. The insets show the beam-coupling geometry used for these two experiments. For a given wavelength the gain for both electro-optic and absorption measurements at kg = 0 is the same, corresponding to the absorption-grating offset discussed in the text.

Fig. 4
Fig. 4

Absorption-grating gain per unit length as a function of incident intensity at λ = 488 nm at an external crossing half-angle of 0.4° in the Free crystal of BaTiO3. [Note that at this angle the gain is due only to the offset Φ(I), to within a maximum error <0.004 cm−1.] The solid curve is a four-parameter fit to a two-level model using Eq. (17). The temperature was (17.7 ± 0.2)°C.

Fig. 5
Fig. 5

Inverse screening length k0 in units of 2/c versus incident intensity at λ = 488 nm in the Free crystals of BaTiO3. The data points were obtained from curve fits to the data in Fig. 2. In the two-level model the magnitude of k0 is expected to saturate at high intensities and to reach a constant value at low intensities.

Fig. 6
Fig. 6

Measured bulk absorption versus incident intensity in the Free crystal at λ = 488 nm, ordinary polarization, and T = (17.8 ± 0.2)°C. The solid curve is a fit to Eq. (20).

Fig. 7
Fig. 7

Total (absorption-plus-electro-optic) two-beam coupling gain per unit length versus kg in the Free crystal of BaTiO3, obtained with the geometry of Fig. 3(b). The data are curve fits to Eq. (21).

Equations (22)

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N j E t = ( β j + s j I ) N j F - γ j n N j E ,
j = e μ n E k B T μ n ,
j = 1 l ( ξ j F N j F t + ξ j E N j E t ) ± n t + · j e = 0 ,
· E = e ɛ 0 ɛ [ j = 1 l ( ξ j F N j F + ξ j E N j E ) ± n ] ,
N j 1 F ( I , k g ) = m N j eff ( I ) ( s j I β j + s j I ) k g 2 / k 0 2 ( I ) + ϕ j ( I ) 1 + k g 2 / k 0 2 ( I ) ,
ϕ j = 1 - ( β j + s j I s j I ) i = 1 l k 0 i 2 k 0 2 ( s i I β i + s i I ) .
ɛ traps ( x ) = j = 1 l p j F N j F ( x ) + p j E N j E ( x ) ,
γ abs ( I , k g ) = Γ abs ( I ) [ k g / k 0 ( I ) ] 2 + Φ ( I ) 1 + [ k g / k 0 ( I ) ] 2 e ^ 1 . e ^ 2 * 2 ,
Γ abs ( I ) ω n c j = 1 l N j eff ( I ) ( s j I β j + s j I ) Im ( Δ p j )
γ abs ( k g ) = Γ abs ( k g / k 0 ) 2 1 + ( k g / k 0 ) 2 e ^ 1 · e ^ 2 * 2 .
Φ ( I ) - ω n c j = 1 l N j eff ( I ) ϕ j ( I ) ( s j I β j + s j I ) Im ( Δ p j ) .
Φ ( I ) = - ω n c ɛ ɛ 0 k B T e 2 i = 1 l j = 1 l k 0 i 2 ( I ) k 0 j 2 ( I ) k 0 2 ( I ) ( s j I β j + s j I ) × Im ( Δ p j - Δ p i ) .
Δ α ( I ) = ω n c j = 1 l [ N j 0 F ( I ) - N j 0 F ( 0 ) ] Im ( Δ p j ) .
γ eo ( I , k g ) = Γ eo ( I ) 2 k g / k 0 ( I ) 1 + k g 2 / k 0 2 ( I ) e ^ 1 · e ^ 2 * 2 ,
Γ eo ( I ) = ω 2 n c k B T e r eff j = 1 l k 0 j 2 ( I ) k 0 ( I ) ( s j I β j + s j I ) ,
γ = 1 L ln [ I probe ( with a grating present ) I probe ( without a grating present ) ] ,
Φ ( I ) = ω n c β A β A + s A I N D 0 F Im ( Δ p D - Δ p A ) N A N A - N D 0 F + N D N D - N D 0 F .
N D 0 F ( I ) = N D + N A ± [ ( N D + N A ) 2 - 4 N D N A G ] 1 / 2 2 G ,
G 1 - γ A s D γ D s A s A I β A + s A I .
N A / N D = 0.97 ± 0.02 , β A / s A = 0.95 ± 0.06 W / cm 2 , ( γ A / γ D ) ( s D / s A ) = 0.031 ± 0.006 ,             and ( ω / n c ) N D Im ( Δ p D - Δ p A ) = - 6.2 ± 0.5 cm - 1 .
α ( I ) = α 0 + ( ω / n c ) ( N D 0 F - N A ) ( Δ p D - Δ p A )
γ total ( I ) = γ eo ( I ) + γ abs ( I ) 2 Γ eo ( I ) [ k g / k 0 ( I ) ] + Φ ( I ) .

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