Abstract

A crystal of BaTiO3 illuminated by a spatially periodic light intensity pattern will exhibit both uniform and spatially periodic photogalvanic currents. The modulated part of the intensity produces a spatially period photo-galvanic current, which creates a spatially periodic electric field in the crystal. This field, measured by two-beam coupling, is spatially in phase with the light pattern and increases monotonically with intensity, saturating at ∼450 V/cm. The spatially uniform photogalvanic current produced by the average light intensity creates a spatially uniform electric field, which is surprisingly small (≈10 V/cm) across a nominally open-circuited BaTiO3 crystal at high optical intensity. We explain the observed intensity dependence of two-beam coupling by proposing that photogalvanic currents arise with different strengths from at least two trap levels in the crystal.

© 1992 Optical Society of America

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  1. V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).
  2. V. M. Fridkin, “Review of recent work on the bulk photovoltaic effect in ferro and piezo-electrics,” Ferroelectrics 53, 169 (1984).
    [CrossRef]
  3. B. Sturman, “Dynamic holography effects in ferroelectrics induced by spatially oscillating photovoltaic currents,” J. Opt. Soc. Am. B 8, 1333 (1991).
    [CrossRef]
  4. R. M. Pierce and R. S. Cudney, “Photorefractive coupling between orthogonally polarized light beams in barium titanate,” Opt. Lett. 17, 784 (1992).
    [CrossRef] [PubMed]
  5. A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
    [CrossRef]
  6. A. Motes and J. J. Kim, “Intensity dependent absorption coefficient in photorefractive BaTiO3 crystals,” 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. R. S. Cudney, R. M. Pierce, G. D. Bacher, and J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 8, 1326 (1991).
    [CrossRef]
  9. P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
    [CrossRef]
  10. R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, “Measurement of the photorefractive phase shift,” Opt. Lett. 17, 67 (1992).
    [CrossRef] [PubMed]
  11. L. Holtmann, “A model for the non-linear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, 89 (1989).
    [CrossRef]
  12. D. Mahgerefteh and J. Feinberg, “Explanation of the app1rent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
    [CrossRef] [PubMed]
  13. S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
    [CrossRef]
  14. F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312 (1986).
    [CrossRef]
  15. P. Günter and M. Zgonik, “Clamped–unclamped electro-optic coefficient dilemma in photorefractive phenomena,” Opt. Lett. 16, 1826 (1991).
    [CrossRef]
  16. S. Sternklar, S. Weiss, and B. Fischer, “Tunable frequency shift of photorefractive oscillators,” Opt. Lett. 11, 165 (1986).
    [CrossRef] [PubMed]
  17. K. R. MacDonald, “Self-pumped phase conjugation in photorefractive barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).
  18. I. McMichael and P. Yeh, “Phase shifts of photorefractive gratings and phase-conjugate waves,” Opt. Lett. 12, 48 (1987).
    [CrossRef] [PubMed]
  19. D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light induced surface damage in barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).
  20. M. C. Gower, “Photo-induced voltages and frequency shifts in a self-pumped phase-conjugating BaTiO3 crystal,” Opt. Lett. 11, 458 (1986).
    [CrossRef] [PubMed]
  21. P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199 (1982).
    [CrossRef]
  22. T. Y. Chang, “Nonlinear optical studies of photorefractive barium titanate: parameter measurements and phase conjugation,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).
  23. D. Mahgerefteh, D. Kirillov, and J. Feinberg, “Is the hole mobility proportional to the dc dielectric constant in photorefractive BaTiO3?” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MJ4.
  24. E. J. Herbulock, M. H. Garrett, and A. R. Tanguay, “Electric field profile effects on photorefractive grating formation in bismuth silicon oxide,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper THU4.
  25. S. Ducharme, “Pyro-electro-optic phase gratings,” Opt. Lett. 15, 1791 (1991).
    [CrossRef]
  26. A. L. Smirl, K. Bohnert, G. C. Valley, and T. H. Boggess, “Formation, decay and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
    [CrossRef]

1992 (2)

1991 (4)

1990 (1)

D. Mahgerefteh and J. Feinberg, “Explanation of the app1rent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

1989 (3)

L. Holtmann, “A model for the non-linear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, 89 (1989).
[CrossRef]

P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
[CrossRef]

A. L. Smirl, K. Bohnert, G. C. Valley, and T. H. Boggess, “Formation, decay and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
[CrossRef]

1988 (1)

1987 (3)

A. Motes and J. J. Kim, “Intensity dependent absorption coefficient in photorefractive BaTiO3 crystals,” J. Opt. Soc. Am. B 4, 1379 (1987).
[CrossRef]

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
[CrossRef]

I. McMichael and P. Yeh, “Phase shifts of photorefractive gratings and phase-conjugate waves,” Opt. Lett. 12, 48 (1987).
[CrossRef] [PubMed]

1986 (3)

1984 (1)

V. M. Fridkin, “Review of recent work on the bulk photovoltaic effect in ferro and piezo-electrics,” Ferroelectrics 53, 169 (1984).
[CrossRef]

1982 (1)

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199 (1982).
[CrossRef]

1977 (1)

V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).

1974 (1)

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Bacher, G. D.

Belinicher, V. I.

V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).

Boggess, T. H.

Bohnert, K.

Brost, G. A.

Cescato, L.

P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
[CrossRef]

Chang, T. Y.

T. Y. Chang, “Nonlinear optical studies of photorefractive barium titanate: parameter measurements and phase conjugation,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

Cudney, R. S.

Ducharme, S.

S. Ducharme, “Pyro-electro-optic phase gratings,” Opt. Lett. 15, 1791 (1991).
[CrossRef]

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
[CrossRef]

Feinberg, J.

R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, “Measurement of the photorefractive phase shift,” Opt. Lett. 17, 67 (1992).
[CrossRef] [PubMed]

R. S. Cudney, R. M. Pierce, G. D. Bacher, and J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 8, 1326 (1991).
[CrossRef]

D. Mahgerefteh and J. Feinberg, “Explanation of the app1rent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
[CrossRef]

D. Mahgerefteh, D. Kirillov, and J. Feinberg, “Is the hole mobility proportional to the dc dielectric constant in photorefractive BaTiO3?” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MJ4.

Fischer, B.

Frejlich, J.

P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
[CrossRef]

Fridkin, V. M.

V. M. Fridkin, “Review of recent work on the bulk photovoltaic effect in ferro and piezo-electrics,” Ferroelectrics 53, 169 (1984).
[CrossRef]

Garrett, M. H.

E. J. Herbulock, M. H. Garrett, and A. R. Tanguay, “Electric field profile effects on photorefractive grating formation in bismuth silicon oxide,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper THU4.

Glass, A. M.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Gower, M. C.

Günter, P.

P. Günter and M. Zgonik, “Clamped–unclamped electro-optic coefficient dilemma in photorefractive phenomena,” Opt. Lett. 16, 1826 (1991).
[CrossRef]

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199 (1982).
[CrossRef]

Hellwarth, R. W.

Herbulock, E. J.

E. J. Herbulock, M. H. Garrett, and A. R. Tanguay, “Electric field profile effects on photorefractive grating formation in bismuth silicon oxide,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper THU4.

Holtmann, L.

L. Holtmann, “A model for the non-linear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, 89 (1989).
[CrossRef]

Jonathan, J. M. C.

Kim, J. J.

Kirillov, D.

D. Mahgerefteh, D. Kirillov, and J. Feinberg, “Is the hole mobility proportional to the dc dielectric constant in photorefractive BaTiO3?” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MJ4.

MacDonald, K. R.

K. R. MacDonald, “Self-pumped phase conjugation in photorefractive barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).

Magno García, P.

P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
[CrossRef]

Mahgerefteh, D.

D. Mahgerefteh and J. Feinberg, “Explanation of the app1rent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

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

D. Mahgerefteh, D. Kirillov, and J. Feinberg, “Is the hole mobility proportional to the dc dielectric constant in photorefractive BaTiO3?” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MJ4.

Malinovskii, V. K.

V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).

McMichael, I.

Motes, A.

Motes, R. A.

Negran, T. J.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Neurgaonkar, R. R.

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
[CrossRef]

Pierce, R. M.

Rotge, J. R.

Smirl, A. L.

Sternklar, S.

Strohkendl, F. P.

Sturman, B.

Sturman, B. I.

V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).

Tanguay, A. R.

E. J. Herbulock, M. H. Garrett, and A. R. Tanguay, “Electric field profile effects on photorefractive grating formation in bismuth silicon oxide,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper THU4.

Valley, G. C.

von der Linde, D.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Weiss, S.

Yeh, P.

Zgonik, M.

Appl. Phys. Lett. (1)

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Ferroelectrics (1)

V. M. Fridkin, “Review of recent work on the bulk photovoltaic effect in ferro and piezo-electrics,” Ferroelectrics 53, 169 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, “Electro-optic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. QE-23, 2116 (1987).
[CrossRef]

J. Appl. Phys. (1)

P. Magno García, L. Cescato, and J. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47 (1989).
[CrossRef]

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

Opt. Lett. (8)

Phys. Rep. (1)

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

D. Mahgerefteh and J. Feinberg, “Explanation of the app1rent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

L. Holtmann, “A model for the non-linear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, 89 (1989).
[CrossRef]

Sov. Phys. JETP (1)

V. I. Belinicher, V. K. Malinovskii, and B. I. Sturman, “Photogalvanic effect in a crystal with polar axis,” Sov. Phys. JETP 46, 362 (1977).

Other (5)

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

K. R. MacDonald, “Self-pumped phase conjugation in photorefractive barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).

T. Y. Chang, “Nonlinear optical studies of photorefractive barium titanate: parameter measurements and phase conjugation,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

D. Mahgerefteh, D. Kirillov, and J. Feinberg, “Is the hole mobility proportional to the dc dielectric constant in photorefractive BaTiO3?” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MJ4.

E. J. Herbulock, M. H. Garrett, and A. R. Tanguay, “Electric field profile effects on photorefractive grating formation in bismuth silicon oxide,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper THU4.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup for measuring the two-beam coupling.

Fig. 2
Fig. 2

Imaginary gIeo and real gReo parts of the purely electro-optic coupling in the crystal FREE versus kg at a fixed, high intensity (13 W/cm2). λ = 488 nm, T = 17.7 ± 0.2°C. The inset shows the polarization of the beams and the direction of the grating wave vector.

Fig. 3
Fig. 3

gIe0 in crystal FREE versus light intensity I at two internal beam crossing half-angles θint. The solid curve is the best fit to the two-level model discussed in the text. The other curves are best fits assuming that only one of the levels contributes to the photogalvanic effect. λ = 488 nm; T = 17.7 ± 0.2°C.

Fig. 4
Fig. 4

Photogalvanic current measured externally with an electrometer in crystal CHIP versus light intensity I. The solid line is the best fit to the two-level model; the dotted line is a best fit to a simple linear dependence of the photogalvanic current on intensity. λ = 488 nm; T = 21.4 ± 0.2°C.

Equations (63)

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j pg ( x ) = β : E opt ( x ) E opt * ( x ) ,
j pg ( I ) = p I = ĉ κ α I .
j pg ( I ) = p ( I ) I = ĉ κ ( I ) α ( I ) I ,
j pg ( I ) = e ĉ ( d opt s I N F + d therm β N F + d rec γ n N E ) ,
n = ( s I + β ) N F / ( γ N E ) .
j pg ( I ) = e ĉ s I N F ( d opt d rec ) = e ĉ s I N F d ,
j pg ( I ) = e ĉ I j d j s j N j F ( I ) p ( I ) I ,
E p ( z ) = E p ( 0 ) exp ( g eo z ) ,
g eo ( k g , I ) = ω 2 c n ord 3 r 13 k B T e × η ( I ) [ k g + i k drift ( I ) ] + i k p g ( I ) + i k Δ pg ( I ) 1 + ( { k g [ k g + i k drift ( I ) i k uniform ( I ) ] } / [ k 0 2 ( I ) ] ) .
I ( z ) = I 0 ( 1 + m cos k g z ) ,
j pg [ I ( z ) ] = p [ I ( z ) ] I ( z ) = p [ I 0 ( 1 + m cos k g z ) ] I 0 ( 1 + m cos k g z ) = p ( I 0 ) I 0 + p ( I 0 ) I 0 m cos k g z + ( d p d I I = I 0 ) I 0 2 m cos k g z + ( terms of higher order in m ) .
g eo ( k g , I ) = ω 2 c n ord 3 r 13 k B T e { η ( I ) [ k g + i k drift ( I ) ] + i k pg ( I ) + i k Δ pg ( I ) } .
I p ( Ω ) = 2 μ I p 0 exp ( g R l ) sin ( g l l ) ,
I p ( Ω ) = ( μ 2 / 2 ) I p 0 [ exp ( g R l ) cos ( g l l ) 1 ] ,
g R eo ( k g ) = ω 2 c n ord 3 r 13 k B T e η k g ,
g I eo = ω 2 c n ord 3 r 13 k B T e ( η k drift + k pg + k Δ pg ) .
E in phase = k B T e ( η k drift + k pg + k Δ pg ) = g I eo 2 c ω n ord 3 r 13 .
k pg ( I ) = e I k B T μ n 0 ( I ) [ s A d A N A ; 0 F ( I ) + s D d D N D ; 0 F ( I ) ] ,
E 0 ( I ) = E pv ( I ) = j pg ( I ) σ ( I ) = I μ n 0 ( I ) [ s A d A N A ; 0 F ( I ) + s D d D N D ; 0 F ( I ) ] .
k drift = k pg ,
( ω 2 c n ord 3 r 13 μ γ D N D ) d D = 1.4 cm 1 , ( ω 2 c n ord 3 r 13 μ γ A N D ) d A = 0.5 cm 1 .
μ γ D N D μ γ A N D μ τ 2 ,
Λ A 2 = ( k B T / e ) μ τ A ,
Λ D 2 = ( k B T / e ) μ τ D ,
k pg ( I ) = k B T e [ d A Λ A 2 ( I ) s A I s A I + β A + d D Λ D 2 ( I ) ] ;
N j E t = ( β j + s j : E opt E opt * ) N j F γ j N j E n ,
j = e n μ E SC k B T μ n + j pg ,
j ( ξ j F N j F t + ξ j E N j E t ) ± n t + j e = 0 ,
( 0 E ) = e j ( ξ j F N j F + ξ j E N j E ) ,
j pg ( x ) = e j N j F χ j : E opt ( x ) E opt * ( x ) ,
E opt ( x ) = E r ( x ) ê r exp ( i k r x ) + E p ( x ) ê p exp ( i k p x ) ,
N j F , E = N j ; 0 F , E + Re [ N j ; 1 F , E exp ( i k g x ) ] ,
n = n 0 + Re [ n 1 exp ( i k g x ) ] ,
j = j 0 + Re [ j 1 exp ( i k g x ) ] ,
E = E 0 + Re [ E 1 exp ( i k g x ) ] ,
k g k p k r = ( ω / c ) ( n p ŝ p n r ŝ r ) .
E 1 = ± i k ̂ g k B T e ( k g ± i k drift ) ( j m j k 0 j 2 / k 0 2 ) ± i 2 E p E r * I k pg ± i ( j m j k s j j k s j j m j k 0 j 2 k 0 2 ) 1 + { [ k g ( k g ± i k drift i k uniform ) ] / k 0 2 } ,
k drift = e k B T ( k ̂ g μ E 0 ) ( k ̂ g μ k g ) ,
k o j 2 = e 2 k B T 0 ( k ̂ g k ̂ g ) N j eff ,
N j eff = N j ; 0 F N j ; 0 E N j ; 0 F + N j ; 0 E ,
k 0 2 = j k o j 2 ,
k pg = e I k B T n 0 ( k ̂ g μ k ̂ g ) j N j ; 0 F k ̂ g χ j : ê p ê r * ,
k uniform = j k s j ,
k s j = e k B T n 0 k ̂ g μ k ̂ g n j eff ( | E p | 2 k ̂ g χ j : ê p ê p * + | E r | 2 k ̂ g χ j : ê r ê r * ) ,
m j = 2 E p E r * s j : ê p ê r * s j : ( | E p | 2 ê p ê p * + | E r | 2 ê r ê r * ) + β j .
E p ( z ) = E p ( 0 ) exp ( g eo z ) ,
g eo = i ω 2 n p c ê p * [ ( ω ) R E 1 ( ω ) ] ê r E r .
j m j k 0 j 2 / k 0 2 = m η ( I ) ,
j m j k s j j k s j j m j k 0 j 2 k 0 2 = m ( j s j I s j I + β j k s j η ( I ) j k s j ) m k Δ pg ( I ) ,
η ( I ) j s j I s j I + β j k 0 j 2 k 0 2 .
g eo ( k g , I ) = ω 2 c n ord 3 r 13 k B T e × η ( I ) [ k g + i k drift ( I ) ] + i k pg ( I ) + i k Δ pg ( I ) 1 + { k g [ k g + i k drift ( I ) i k uniform ( I ) ] / k 0 2 ( I ) } ,
N A ; 0 F + N D ; 0 F = N A ,
n 0 = ( s A I + β A ) N A ; 0 F γ A N A ; 0 E = S D I N D ; 0 F γ D N D ; 0 E .
N D ; 0 F = N D + N A ± [ ( N D + N A ) 2 4 N D N A G ] 1 / 2 2 G ,
G ( I ) 1 γ A s D γ D s A s A I s A I + β A .
k pg ( I ) = e I k B T μ n 0 ( I ) [ s A d A N A ; 0 E ( I ) + s D d D N D ; 0 F ( I ) ] ,
j total ( z ) = j dirft ( z ) + j pg ( z ) = σ ( z ) E ( z ) + p ( z ) I ( z ) ,
E ( z ) = E 0 + E 1 m cos k g z + ,
σ ( z ) = σ dark + a ( I 0 ) I 0 + [ a ( I 0 ) + ( d a d I ) I 0 I 0 ] I 0 m cos k g z + ,
p ( z ) = p ( I 0 ) + ( d p d I ) I 0 I 0 m cos k g z + .
E 0 = p ( I 0 ) I 0 σ dark + a ( I 0 ) I 0 ,
E 1 = p ( I 0 ) I 0 + I 0 2 ( d p / d I ) I 0 + E 0 [ a ( I 0 ) I 0 + I 0 2 ( d a / d I ) I 0 ] σ dark + a ( I 0 ) I 0 .
E 1 ( high intensity ) p ( I 0 ) I 0 p ( I 0 ) I 0 a ( I 0 ) I 0 = 0 .

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