Abstract

The band transport model describes charge statics and dynamics within photorefractive materials. Here a steady-state solution is found for the setting in which an arbitrary optical beam illuminates a photorefractive material. The specific case of a Gaussian beam is investigated. The effects of intensity and beam waist are discussed. A comparison is made with previous studies of this problem. It is found that, when the rate of photoexcitation is less than that of thermal excitation, the band transport model can be linearized. In this case the charge density and the space-charge field are proportional to the intensity of the light. On the other hand, when the rate of photoexcitation is greater than that of thermal excitation, the linear solution to the band transport model no longer applies. In this case the charge density and the space-charge field are independent of the intensity and dependent only on the size of the beam waist.

© 1992 Optical Society of America

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References

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  1. A. M. Glass, Opt. Eng. 17, 470 (1978).
    [Crossref]
  2. G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
    [Crossref]
  3. G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
    [Crossref]
  4. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
    [Crossref]
  5. D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
    [Crossref]
  6. M. G. Moharan and L. Young. J. Appl. Phys. 48, 3230 (1977).
    [Crossref]
  7. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
    [Crossref]
  8. J. Feinberg and G. D. Bacher, Appl. Phys. Lett. 48, 570 (1986).
    [Crossref]
  9. D. A. Rockwell and C. G. Giuliano, Opt. Lett. 11, 147 (1986).
    [Crossref]
  10. M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
    [Crossref]
  11. S. Sternklar, S. Weiss, M. Segev, and B. Fischer, Opt. Lett. 11, 528 (1986).
    [Crossref] [PubMed]
  12. M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
    [Crossref]
  13. J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).
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    [Crossref] [PubMed]
  15. C. J. Gaeta, R. C. Lind, W. P. Brown, and C. R. Giuliano, Opt. Lett. 13, 1093 (1988).
    [Crossref] [PubMed]
  16. D. Statman and B. Liby, J. Opt. Soc. Am. B 6, 1884 (1989).
    [Crossref]
  17. T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
    [Crossref]
  18. J. A. Khoury and R. W. Eason, J. Mod. Opt. 36, 369 (1989).
    [Crossref]
  19. J. Feinberg, J. Opt. Soc. Am. 72, 46 (1982).
    [Crossref]
  20. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
    [Crossref]
  21. R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 56, 40 (1985).
    [Crossref]
  22. M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
    [Crossref]
  23. M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
    [Crossref]
  24. M. Cronin-Golomb, H. Kong, and B. S. Chen, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 24.
  25. D. von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).
    [Crossref]
  26. C. Xu, “Theoretical investigation of the effects of a Gaussian laser beam on photorefractive materials,” Ph.D. dissertation (University of New Mexico, Albuquerque, N.M., 1991).

1990 (2)

M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
[Crossref]

M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
[Crossref]

1989 (2)

D. Statman and B. Liby, J. Opt. Soc. Am. B 6, 1884 (1989).
[Crossref]

J. A. Khoury and R. W. Eason, J. Mod. Opt. 36, 369 (1989).
[Crossref]

1988 (3)

1987 (1)

J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).

1986 (6)

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

D. A. Rockwell and C. G. Giuliano, Opt. Lett. 11, 147 (1986).
[Crossref]

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, Opt. Lett. 11, 528 (1986).
[Crossref] [PubMed]

J. Feinberg and G. D. Bacher, Appl. Phys. Lett. 48, 570 (1986).
[Crossref]

M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
[Crossref]

M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
[Crossref]

1985 (1)

R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 56, 40 (1985).
[Crossref]

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

1983 (1)

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

1982 (1)

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

1978 (1)

A. M. Glass, Opt. Eng. 17, 470 (1978).
[Crossref]

1977 (1)

M. G. Moharan and L. Young. J. Appl. Phys. 48, 3230 (1977).
[Crossref]

1975 (1)

D. von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).
[Crossref]

1972 (1)

D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

Albers, J.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Amodei, J. J.

D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

Bacher, G. D.

J. Feinberg and G. D. Bacher, Appl. Phys. Lett. 48, 570 (1986).
[Crossref]

Brown, W. P.

Chen, B. S.

M. Cronin-Golomb, H. Kong, and B. S. Chen, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 24.

Clayton, C. M.

Cronin-Golomb, M.

M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

M. Cronin-Golomb, H. Kong, and B. S. Chen, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 24.

Eason, R. W.

J. A. Khoury and R. W. Eason, J. Mod. Opt. 36, 369 (1989).
[Crossref]

Feinberg, J.

J. Feinberg and G. D. Bacher, Appl. Phys. Lett. 48, 570 (1986).
[Crossref]

J. Feinberg, J. Opt. Soc. Am. 72, 46 (1982).
[Crossref]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Fischer, B.

M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
[Crossref]

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, Opt. Lett. 11, 528 (1986).
[Crossref] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

Fisher, B.

M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
[Crossref]

M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
[Crossref]

Gaeta, C. J.

Giuliano, C. G.

Giuliano, C. R.

Glass, A. M.

A. M. Glass, Opt. Eng. 17, 470 (1978).
[Crossref]

D. von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).
[Crossref]

Goltz, J.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Hellwarth, R. W.

R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 56, 40 (1985).
[Crossref]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Herden, A.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Khoury, J. A.

J. A. Khoury and R. W. Eason, J. Mod. Opt. 36, 369 (1989).
[Crossref]

Klein, M. B.

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Klumb, H.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Kong, H.

M. Cronin-Golomb, H. Kong, and B. S. Chen, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 24.

Kramer, M. A.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

Laeri, F.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Liby, B.

Lind, R. C.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

McFarlane, R. A.

J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).

Moharan, M. G.

M. G. Moharan and L. Young. J. Appl. Phys. 48, 3230 (1977).
[Crossref]

Mullen, R. A.

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 56, 40 (1985).
[Crossref]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

Ophir, Y.

M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
[Crossref]

M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
[Crossref]

Rockwell, D. A.

Rytz, D.

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

Segev, M.

M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
[Crossref]

M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
[Crossref]

M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
[Crossref]

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, Opt. Lett. 11, 528 (1986).
[Crossref] [PubMed]

Sifuentes, S.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

Staebler, D. L.

D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

Statman, D.

Sternklar, S.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Tschudi, T.

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

Ury, I.

M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
[Crossref]

Valley, G. C.

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

von der Linde, D.

D. von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).
[Crossref]

Wechsler, B.

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

Weiss, S.

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, Opt. Lett. 11, 528 (1986).
[Crossref] [PubMed]

M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
[Crossref]

White, J. O.

J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

Xu, C.

C. Xu, “Theoretical investigation of the effects of a Gaussian laser beam on photorefractive materials,” Ph.D. dissertation (University of New Mexico, Albuquerque, N.M., 1991).

Yariv, A.

M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

Young, L.

M. G. Moharan and L. Young. J. Appl. Phys. 48, 3230 (1977).
[Crossref]

Annu. Rev. Mater. Sci. (1)

G. C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler, Annu. Rev. Mater. Sci. 18, 165 (1988).
[Crossref]

Appl. Opt. (1)

Appl. Phys. (1)

D. von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).
[Crossref]

Appl. Phys. Lett. (5)

M. Segev, Y. Ophir, and B. Fisher, Appl. Phys. Lett. 56, 1086 (1990).
[Crossref]

J. Feinberg and G. D. Bacher, Appl. Phys. Lett. 48, 570 (1986).
[Crossref]

M. Segev, S. Weiss, and B. Fischer, Appl. Phys. Lett. 50, 1397 (1986).
[Crossref]

M. Cronin-Golomb, A. Yariv, and I. Ury, Appl. Phys. Lett. 48, 1240 (1986).
[Crossref]

J. O. White, G. C. Valley, and R. A. McFarlane, Appl. Phys. Lett. 50, 890 (1987).

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979);Ferroelectrics 22, 961 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

T. Tschudi, A. Herden, J. Goltz, H. Klumb, F. Laeri, and J. Albers, IEEE J. Quantum Electron. QE-22, 1493 (1986).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[Crossref]

J. Appl. Phys. (4)

D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

M. G. Moharan and L. Young. J. Appl. Phys. 48, 3230 (1977).
[Crossref]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 56, 40 (1985).
[Crossref]

J. Mod. Opt. (1)

J. A. Khoury and R. W. Eason, J. Mod. Opt. 36, 369 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

M. Segev, Y. Ophir, and B. Fisher, Opt. Commun. 77, 265 (1990).
[Crossref]

Opt. Eng. (2)

A. M. Glass, Opt. Eng. 17, 470 (1978).
[Crossref]

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Opt. Lett. (3)

Other (2)

M. Cronin-Golomb, H. Kong, and B. S. Chen, in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C.), p. 24.

C. Xu, “Theoretical investigation of the effects of a Gaussian laser beam on photorefractive materials,” Ph.D. dissertation (University of New Mexico, Albuquerque, N.M., 1991).

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Figures (4)

Fig. 1
Fig. 1

(A) Carrier density in BaTiO3 for I0 of a, 0 W/cm2; b, 0.025 W/cm2; c, 0.05 W/cm2; d, 0.075 W/cm2; and e, 0.1 W/cm2. The beam waist is 1.0 × 10−3 cm. (B) Total charge density in BaTiO3 for I0 of a, 0 W/cm2; b, 0.025 W/cm2; c, 0.05 W/cm2; d, 0.075 W/cm2; and e, 0.1 W/cm2. The beam waist is 1.0 × 10−3 cm. (C) Space-charge field in BaTiO3 for I0 of a, 0 W/cm2; b, 0.025 W/cm2; c, 0.05 W/cm2; d, 0.075 W/cm2; and e, 0.1 W/cm2. The beam waist is 1.0 ×10−3 cm.

Fig. 2
Fig. 2

(A) Carrier density in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−3 cm. (B) Total charge density in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−3 cm. (C) Space-charge field in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−3 cm.

Fig. 3
Fig. 3

(A) Carrier density in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−1 cm. (B) Total charge density in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−1 cm. (C) Space-charge field in BaTiO3 for I0 of a, 0.1 W/cm2; b, 1 W/cm2; c, 10 W/cm2; d, 100 W/cm2; and e, 1000 W/cm2. The beam waist is 1.0 × 10−1 cm.

Fig. 4
Fig. 4

Charge density and space-charge field amplitudes versus σI0/β in BaTiO3. l D of a, 5.8 × 10−2; b, 1.8 × 10−2; c, 5.8 × 10−3; d, 1.8 × 10−3; and e, 5.8 × 10−4.

Tables (2)

Tables Icon

Table 1 Parameters Used for Calculations of the Photorefractive Effecta

Tables Icon

Table 2 Screening Lengths and Rate Constants

Equations (30)

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

t ( N D + n ) + 1 e · j = 0 ( continuity ) ,
N D + t = ( σ h ν I + β ) ( N D N D + ) γ R n N D + ( excitation and recombination ) ,
j = e μ n E + e D n
· D = e ( N D + N A n ) ,
D = 0 E .
I = I 0 exp ( x 2 / 2 w 2 ) .
t c = 0 / e μ N A ,
l D 2 = D t c .
τ ( Ñ ñ ) + j ξ = 0 ( continuity ) ,
Ñ τ = ( σ Ĩ + β ) ( Ñ D Ñ ) γ ñ Ñ ( excitation and recombination ) ,
j = ñ + l D 2 ñ ξ
ξ = Ñ ñ 1 .
τ = t t c , ξ = x w , l D = l D w , = E μ t c w , j = j t c e N A ,
σ = σ / w 2 , β = β t c , γ = γ R N A t c .
Ĩ = ( I / h ν ) t c w 2 ;
Ĩ = Ĩ 0 exp ( ξ 2 / 2 ) .
ñ + 1 Ñ D ( β + σ Ĩ ) γ ñ + β + σ Ĩ = l D 2 2 ( ln ñ ) ξ 2 + j ss 1 ñ 2 ñ ξ ,
Ñ = Ñ D 1 + γ ñ / ( β + σ Ĩ ) ,
ñ ( ξ ) = ñ T + ñ I .
ñ T = 1 2 { ( 1 + β γ ) + [ ( 1 β γ ) 2 + 4 Ñ D β γ ] 1 / 2 } Ñ D β γ .
ñ I = ñ I 0 ( ξ ) [ Ĩ ( ξ ) / Ĩ 0 ] .
ñ I = ñ I 0 ( ξ ) exp ( ξ 2 / 2 ) .
ñ I 0 ñ ph = 1 2 { ( σ Ĩ 0 γ ) + [ ( 1 σ Ĩ 0 γ ) 2 + 4 Ñ D σ Ĩ 0 γ ] 1 / 2 } Ñ D σ Ĩ 0 γ .
ñ 2 + b ñ + c = 0 ,
b = 1 + σ Ĩ + β γ l D 2 { ñ I ( ξ 2 1 ) ñ T exp ( ξ 2 / 2 ) + ñ I [ ξ n I ñ T exp ( ξ 2 / 2 ) + ñ I ] 2 } , c = σ Ĩ + β γ ( 1 Ñ D l D 2 { ñ I ( ξ 2 1 ) ñ T exp ( ξ 2 / 2 ) + ñ I [ ξ n I ñ T exp ( ξ 2 / 2 ) + ñ I ] 2 } ) ,
ñ I 0 = ( ñ ñ T ) exp ( ξ 2 / 2 )
Q = l D 2 2 ( ln ñ ) ξ 2 = l D 2 [ ñ ph ñ T ( 1 ξ 2 ) exp ( ξ 2 2 ) + ( ñ ph ñ T ) 2 ξ 2 exp ( ξ 2 ) ]
= l D 2 ( ln ñ ) ξ = l D 2 ñ ph ñ T ξ exp ( ξ 2 2 ) .
Q = l D 2 2 ( ln ñ ñ 2 = l D 2 [ ( 1 ξ 2 ) + ξ 2 exp ( ξ 2 2 ) ] l D 2
= l D 2 ( ln ñ ) ñ = l D 2 ξ .

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