Abstract

A model of the photorefractive effect is presented that includes a shallow-trap level (hole or electron trap) as well as a species of deep traps with electron and hole excitation. It is shown that, if the shallow traps can trap a significant density of charge carriers, the electron–hole competition factor ξ and the photorefractive grating can switch signs as a function of intensity. This effect may be realized in BaTiO3 crystals, in which the photorefractive effect is strongly affected by electron–hole competition as well as by shallow traps. As in the case of the single-charge-carrier model, it is shown that each of the charge gratings in the shallow and deep levels consists of in-phase gratings and a screening grating. It is shown that part of the screening gratings remains nonzero at infinitly large grating spacings.

© 1992 Optical Society of America

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  1. R. Orlowski, E. Kratzig, “Holographic method for the determination of photoinduced electron and hole transport in electro-optic crystals,” Solid State Commun. 27, 1351 (1978).
    [CrossRef]
  2. M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901 (1985).
    [CrossRef]
  3. C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
    [CrossRef]
  4. F. P. Strohkendl, J. M. C. Johnathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312 (1986).
    [CrossRef]
  5. G. C. Valley, “Simultaneous electron-hole transport in photo-refractive materials,” J. Appl. Phys. 59, 3363 (1986).
    [CrossRef]
  6. S. Ducharme, J. Feinberg, “Altering the photorefractive properties of BaTiO3by reduction and oxidization at 650°C,” J. Opt. Soc. Am. B 3, 283 (1986).
    [CrossRef]
  7. R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
    [CrossRef]
  8. F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
    [CrossRef]
  9. G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
    [CrossRef]
  10. A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
    [CrossRef]
  11. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  12. G. A. Brost, R. A. Motes, J. R. Rotge, “Intensity-dependent absorption and photorefractive effects in barium titanate,” J. Opt. Soc. Am. B 5, 1879 (1988); D. A. Temple, C. Warde, “Photoinduced absorption effects in BaTiO3” in OSA Annual Meeting, Vol. II of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper THN3.
    [CrossRef]
  13. F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
    [CrossRef]
  14. D. Mahgerefteh, J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
    [CrossRef] [PubMed]
  15. P. Tayebati, “The effect of shallow traps on the dark storage of the photorefractive grating in Bi12SiO20,” J. Appl. Phys. 70, 4082 (1991).
    [CrossRef]
  16. D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990); erratum, Phys. Rev. Lett. 65, 2613 (1990).
    [CrossRef] [PubMed]
  17. L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, K89 (1989).
    [CrossRef]
  18. G. A. Brost, R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 15, 1194, (1990).
    [CrossRef] [PubMed]
  19. P. Tayebati, “Characterization and modeling of the photorefractive in Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).
  20. P. Tayebati, D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1990).
    [CrossRef]
  21. M. B. Klein, Hughes Research Laboratory, 3011 Malibu Canyon Road, Malibu, Calif. 90265 (personal communication).
  22. D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents and light induced surface damage in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif.1990).
  23. R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 8, 1326 (1990).
    [CrossRef]

1991 (1)

P. Tayebati, “The effect of shallow traps on the dark storage of the photorefractive grating in Bi12SiO20,” J. Appl. Phys. 70, 4082 (1991).
[CrossRef]

1990 (4)

1989 (6)

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

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

R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
[CrossRef]

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

1988 (3)

1986 (3)

1985 (1)

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

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

1978 (1)

R. Orlowski, E. Kratzig, “Holographic method for the determination of photoinduced electron and hole transport in electro-optic crystals,” Solid State Commun. 27, 1351 (1978).
[CrossRef]

Amrhein, P.

C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
[CrossRef]

Bacher, G. D.

Brost, G. A.

Cudney, R. S.

Ducharme, S.

Feinberg, J.

Garmier, E.

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

Gravey, P.

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

Günter, P.

C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
[CrossRef]

Hellwarth, R. W.

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

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

Holtmann, L.

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

Johnathan, J. M. C.

Klein, M.

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

Klein, M. B.

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

M. B. Klein, Hughes Research Laboratory, 3011 Malibu Canyon Road, Malibu, Calif. 90265 (personal communication).

Kratzig, E.

R. Orlowski, E. Kratzig, “Holographic method for the determination of photoinduced electron and hole transport in electro-optic crystals,” Solid State Commun. 27, 1351 (1978).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Mahgerefteh, D.

P. Tayebati, D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1990).
[CrossRef]

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990); erratum, Phys. Rev. Lett. 65, 2613 (1990).
[CrossRef] [PubMed]

D. Mahgerefteh, J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[CrossRef] [PubMed]

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents and light induced surface damage in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif.1990).

Maillard, A.

R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Medrano, C.

C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
[CrossRef]

Motes, R. A.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Orlowski, R.

R. Orlowski, E. Kratzig, “Holographic method for the determination of photoinduced electron and hole transport in electro-optic crystals,” Solid State Commun. 27, 1351 (1978).
[CrossRef]

Ozkul, C.

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

Partovi, A.

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

Picoli, G.

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

Pierce, R. M.

Rotge, J. R.

Rupp, R. A.

R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Strohkendl, F. P.

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

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

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

Tayebati, P.

P. Tayebati, “The effect of shallow traps on the dark storage of the photorefractive grating in Bi12SiO20,” J. Appl. Phys. 70, 4082 (1991).
[CrossRef]

P. Tayebati, D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1990).
[CrossRef]

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

P. Tayebati, “Characterization and modeling of the photorefractive in Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).

Valley, G. C.

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

G. C. Valley, “Simultaneous electron-hole transport in photo-refractive materials,” J. Appl. Phys. 59, 3363 (1986).
[CrossRef]

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

Vieux, V.

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Voit, E.

C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
[CrossRef]

Walter, J.

R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
[CrossRef]

Appl. Phys. A (1)

R. A. Rupp, A. Maillard, J. Walter, “Impact of the sub-linear photoconductivity law on the interpretation of holographic results in BaTiO3,” Appl. Phys. A 49, 259–268 (1989).
[CrossRef]

Appl. Phys. Lett. (1)

A. Partovi, E. Garmier, G. C. Valley, M. Klein, “Photorefractive characterization of the deep traps in semi-insulating GaAs,” Appl. Phys. Lett. 55, 2701 (1989).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

J. Appl. Phys. (7)

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

G. Picoli, P. Gravey, C. Ozkul, V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: a new mechanism of resonance,” J. Appl. Phys. 66, 3798 (1989).
[CrossRef]

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

C. Medrano, E. Voit, P. Amrhein, P. Günter, “Optimization of the photorefractive properties of KNbO3crystals,” J. Appl. Phys. 64, 4668 (1988).
[CrossRef]

G. C. Valley, “Simultaneous electron-hole transport in photo-refractive materials,” J. Appl. Phys. 59, 3363 (1986).
[CrossRef]

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

P. Tayebati, “The effect of shallow traps on the dark storage of the photorefractive grating in Bi12SiO20,” J. Appl. Phys. 70, 4082 (1991).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. Lett. (1)

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990); erratum, Phys. Rev. Lett. 65, 2613 (1990).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

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

Solid State Commun. (1)

R. Orlowski, E. Kratzig, “Holographic method for the determination of photoinduced electron and hole transport in electro-optic crystals,” Solid State Commun. 27, 1351 (1978).
[CrossRef]

Other (3)

P. Tayebati, “Characterization and modeling of the photorefractive in Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).

M. B. Klein, Hughes Research Laboratory, 3011 Malibu Canyon Road, Malibu, Calif. 90265 (personal communication).

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents and light induced surface damage in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif.1990).

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Figures (7)

Fig. 1
Fig. 1

Energy-band diagram of crystals with (a) hole and (b) electron shallow traps.

Fig. 2
Fig. 2

Intensity dependence of the absorption of holes relative to that of electrons, R, and square of the inverse hole diffusion length, Kh2 (represented by the same curve).

Fig. 3
Fig. 3

Intensity dependence of the electron–hole competition term ξ(I0) at k2 = 100 μm2.

Fig. 4
Fig. 4

Electron–hole competition as a function of k2 at various intensities: (a) 1 mW/cm2, (b) 400 mW/cm2, (c) 700 mW/cm2, (d) 2 W/cm2.

Fig. 5
Fig. 5

iE1/m plotted as a function of k2 at intensities (a) 1 mW/cm2, (b) 400 mW/cm2, (c) 700 mW/cm2, (d) 2 W/cm2.

Fig. 6
Fig. 6

Modulation-normalized deep-level concentration gratings N1/m plotted as a function of k2 at intensities (a) 1 mW/cm2 (b) 400 mW/cm2, (c) 700 mW/cm2, (d) 2 W/cm2.

Fig. 7
Fig. 7

Modulation-normalized shallow-level concentration gratings M1/m plotted as a function of k2 at intensities (a) 1 mW/cm2, (b) 400 mW/cm2, (c) 700 mW/cm2, (d) 2 W/cm2.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

N D - t = s h I ( N D - N D - ) - γ h n h N D - - s e I N D - + γ e n e ( N D - N D - ) ,
M t = - ( s T I + β ) M + γ T n h ( M T - M ) ,
t n h + 1 e z j h = s h I ( N D - N D - ) - γ h n h N D - + ( s T I + β ) M - γ T n e ( M T - M ) ,
t n e - 1 e z j h = s e I N D - - γ e n e ( N D - N D - ) ,
· E = - ( e / ) × ( n e - n h + N D - - N F - M ) ,
j h = e μ h n h E - μ h k B T z n h ,
j e = e μ e n e E + μ e k B T z n e ,
I = I 0 Re [ 1 + m exp ( i k z ) ] .
N D - = N F + N 0 + Re [ N 1 exp ( i k z ) ] ,
M = M 0 + Re [ M 1 exp ( i k z ) ] ,
n 0 e M 0 ,             n 0 h M 0 .
N 0 M 0 .
n e 1 - n h 1 N 1 - M 1 .
N 0 = M 0 = 1 2 [ ρ ( I 0 ) - 1 ] ( [ ρ ( I 0 ) ( N DF + M T ) + N F ] - { [ ρ ( I 0 ) ( N DF + M T ) + N F ] 2 - 4 ρ ( I 0 ) [ ρ ( I 0 ) - 1 ] N DF M T } 1 / 2 ) ,
ρ ( I 0 ) = s h γ T s T γ h 1 [ 1 + β / s T I 0 ] ,             N D F N D - N F .
( - e ) N 1 = - m e N E k 2 ξ k 2 + k 0 2 + S ,
( e ) M 1 = m e M E k 2 k 2 + k 0 2 [ ( 1 - ξ ) 1 + k 2 / K h 2 - 1 1 + β / s T I 0 ] - S ,
S = - m e M E k 0 D 2 k 2 + k 0 2 [ 1 1 + s T I 0 / β + k 2 k 2 + K h 2 ( ξ - 1 ) ] .
N E = ( N D F - N 0 ) ( N F + N 0 ) N D ,             M E = M 0 ( M T - M 0 ) M T .
k 0 D 2 = e 2 N E / ( k B T ) ,             k 0 T 2 = e 2 M E / ( k B T ) .
k 0 2 = k 0 D 2 + k 0 T 2 ,
K h 2 = e / ( k B T μ h ) γ h ( N F + N 0 ) , K e 2 = e / ( k B T μ e ) γ e ( N D F - N 0 ) .
ξ = ( 1 - C ) / ( 1 + C ) , C = R ( k 2 + K h 2 ) / ( k 2 + K e 2 ) ,
R = s e ( N F + N 0 ) / [ s h ( N DF - N 0 ) ] .
E 1 = i e k ( N 1 - M 1 ) = i m k B T e k k 2 + k 0 2 [ ξ k 0 D 2 + k 0 T 2 × ( ξ - 1 1 + k 2 / K h 2 + 1 1 + β / s T I 0 ) ] .
Γ abs = h ν ( s D N 1 - s T M 1 ) ,
N F = 0.18 + 10 16 cm - 3 , M T = 18 × 10 16 cm - 3 , N D = 10 19 cm - 3 , β = 10 s - 1 , β / s T = 10 W / cm 2 , α = [ s h ( N D - N F ) + s e N F ] h ν = 1.7 cm - 1 , R ( I 0 = 0 ) = s e N F / s h ( N D - N F ) = 0.05 , K h 2 ( I 0 = 0 ) = 2 μ m - 2 , K e 2 = 400 μ m - 2 , γ T / γ D = 10 , = 135 0 ,

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