Abstract

It was shown recently that the application of a dc field across a polymer film containing polar azo dye chromophores at a temperature far below that of its glass transition leads to an appreciable polar order when the azo dyes undergo cis ⇔ trans isomerization. We present a detailed theoretical study of this phenomenon based on the enhanced mobility of the azo chromophores during the isomerization process. The equations representing this phenomenological theory are solved by recurrence relations of Legendre polynomials, and both the steady state and the dynamics are investigated. Analytical expressions are derived for the photoinduced polar order and its related anisotropy for both cis and trans molecular distributions.

© 1995 Optical Society of America

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  1. B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).
  2. J. Zyss and D. S. Chemla, “Quadratic nonlinear optics and optimization of the second order nonlinear optical response of molecular crystals,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, New York, 1987), 1, p. 3.
  3. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
    [CrossRef]
  4. M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, and A. Dienes, “Second-harmonic generation and absorption studies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
    [CrossRef]
  5. S. Barry and D. Soane, “Poling of polymeric thin films at ambient temperatures for second-harmonic generation,” Appl. Phys. Lett. 58, 1134 (1991).
    [CrossRef]
  6. Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” Ph.D. dissertation (Université Paris-Sud, Paris, 1992).
  7. Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
    [CrossRef]
  8. Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo dye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol. B 2, 359 (1992).
  9. Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
    [CrossRef]
  10. P. M. Blanchard and G. R. Mitchell, “A comparison of photoinduced poling and thermal poling of azo-dye-doped polymer films for second order nonlinear optical applications,” Appl. Phys. Lett. 63, 2038 (1993).
    [CrossRef]
  11. H. Rau, “Photoisomerization of azobenzenes,” in Photochemistry and Photophysics, F. J. Rabeck, ed. (CRC, Boca Raton, Fla., 1990), Vol. 2, Chap. 4, pp. 119–141. This paper contains a large bibliography on photoisomerization.
  12. H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
    [CrossRef]
  13. E. Fisher, “The calculation of photostationary states in systems A⇔ B when only A is known,” J. Phys. Chem. 71, 3704 (1967).
    [CrossRef]
  14. R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
    [CrossRef]
  15. T. Todorov, L. Nikolova, and N. Tomova, “Polarization holography. 1. A new high-efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309 (1984).
    [CrossRef] [PubMed]
  16. A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
    [CrossRef]
  17. M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).
  18. R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
    [CrossRef]
  19. Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).
  20. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).
  21. J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142 (1991).
    [CrossRef]
  22. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742 (1989).
    [CrossRef]
  23. Z. Sekkat and W. Knoll, “Stationary state and dynamics of birefringence and nonlinear optical properties induced by electric field poling in polymeric films,” Ber. Bunsenges. Phys. Chem. 98, 1231 (1994).
    [CrossRef]
  24. Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
    [CrossRef]
  25. M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
    [CrossRef]
  26. S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20, 1179 (1979).
    [CrossRef]
  27. H. Benoit, “Contribution à l’etude de l’effet Kerr presentées par les solutions diluées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).
  28. R. E. Robertson, “Effect of free volume fluctuations on polymer relaxation in the glassy state,” J. Polym. Sci. Polym. Symp. 1, 173 (1978).
  29. G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985).
  30. K. D. Singer, “Molecular polymeric materials for nonlinear optics,” in Polymers for Light Wave and Integrated Optics: Technology and Applications, L. A. Hornak, ed. (Dekker, New York, 1992), p. 321.

1994 (4)

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
[CrossRef]

Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).

Z. Sekkat and W. Knoll, “Stationary state and dynamics of birefringence and nonlinear optical properties induced by electric field poling in polymeric films,” Ber. Bunsenges. Phys. Chem. 98, 1231 (1994).
[CrossRef]

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

1993 (5)

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
[CrossRef]

P. M. Blanchard and G. R. Mitchell, “A comparison of photoinduced poling and thermal poling of azo-dye-doped polymer films for second order nonlinear optical applications,” Appl. Phys. Lett. 63, 2038 (1993).
[CrossRef]

1992 (3)

Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
[CrossRef]

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo dye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol. B 2, 359 (1992).

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

1991 (2)

J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142 (1991).
[CrossRef]

S. Barry and D. Soane, “Poling of polymeric thin films at ambient temperatures for second-harmonic generation,” Appl. Phys. Lett. 58, 1134 (1991).
[CrossRef]

1990 (1)

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

1989 (2)

1984 (1)

1979 (1)

S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20, 1179 (1979).
[CrossRef]

1978 (1)

R. E. Robertson, “Effect of free volume fluctuations on polymer relaxation in the glassy state,” J. Polym. Sci. Polym. Symp. 1, 173 (1978).

1971 (1)

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

1967 (1)

E. Fisher, “The calculation of photostationary states in systems A⇔ B when only A is known,” J. Phys. Chem. 71, 3704 (1967).
[CrossRef]

1951 (1)

H. Benoit, “Contribution à l’etude de l’effet Kerr presentées par les solutions diluées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985).

Aust, E. F.

Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).

Barry, S.

S. Barry and D. Soane, “Poling of polymeric thin films at ambient temperatures for second-harmonic generation,” Appl. Phys. Lett. 58, 1134 (1991).
[CrossRef]

Benoit, H.

H. Benoit, “Contribution à l’etude de l’effet Kerr presentées par les solutions diluées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Blanchard, P. M.

P. M. Blanchard and G. R. Mitchell, “A comparison of photoinduced poling and thermal poling of azo-dye-doped polymer films for second order nonlinear optical applications,” Appl. Phys. Lett. 63, 2038 (1993).
[CrossRef]

Büchel, M.

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Burland, D. M.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
[CrossRef]

Chemla, D. S.

J. Zyss and D. S. Chemla, “Quadratic nonlinear optics and optimization of the second order nonlinear optical response of molecular crystals,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, New York, 1987), 1, p. 3.

Davydov, B. D.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Delaire, J. A.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

Derkacheva, L. D.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Dhenaut, C.

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

Dienes, A.

Doi, M.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).

Dumina, V. V.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Dumont, M.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
[CrossRef]

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo dye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol. B 2, 359 (1992).

Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
[CrossRef]

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
[CrossRef]

Edwards, S. F.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).

Fisher, E.

E. Fisher, “The calculation of photostationary states in systems A⇔ B when only A is known,” J. Phys. Chem. 71, 3704 (1967).
[CrossRef]

Froc, G.

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
[CrossRef]

Garito, A. F.

S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20, 1179 (1979).
[CrossRef]

Gauglitz, G.

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

Gosselin, J.

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

Greiner, G.

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

Higgins, B. G.

Hosotte, S.

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
[CrossRef]

King, L. A.

Knoesen, A.

Knoll, W.

Z. Sekkat and W. Knoll, “Stationary state and dynamics of birefringence and nonlinear optical properties induced by electric field poling in polymeric films,” Ber. Bunsenges. Phys. Chem. 98, 1231 (1994).
[CrossRef]

Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Koreneva, L. G.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Kowel, S. T.

Kuzyk, M. G.

Lalama, S. J.

S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20, 1179 (1979).
[CrossRef]

Loucif-Saibi, R.

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

Meier, H.

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

Menzel, H.

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Miller, R. D.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
[CrossRef]

Mitchell, G. R.

P. M. Blanchard and G. R. Mitchell, “A comparison of photoinduced poling and thermal poling of azo-dye-doped polymer films for second order nonlinear optical applications,” Appl. Phys. Lett. 63, 2038 (1993).
[CrossRef]

Mortazavi, M. A.

Nakatani, K.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

Nathansohn, A.

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

Nikolova, L.

Orendi, H.

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Rau, H.

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

H. Rau, “Photoisomerization of azobenzenes,” in Photochemistry and Photophysics, F. J. Rabeck, ed. (CRC, Boca Raton, Fla., 1990), Vol. 2, Chap. 4, pp. 119–141. This paper contains a large bibliography on photoisomerization.

Robertson, R. E.

R. E. Robertson, “Effect of free volume fluctuations on polymer relaxation in the glassy state,” J. Polym. Sci. Polym. Symp. 1, 173 (1978).

Rochon, P.

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

Sanokhina, M. A.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Sekkat, Z.

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Z. Sekkat and W. Knoll, “Stationary state and dynamics of birefringence and nonlinear optical properties induced by electric field poling in polymeric films,” Ber. Bunsenges. Phys. Chem. 98, 1231 (1994).
[CrossRef]

Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
[CrossRef]

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo dye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol. B 2, 359 (1992).

Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
[CrossRef]

Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” Ph.D. dissertation (Université Paris-Sud, Paris, 1992).

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
[CrossRef]

Singer, K. D.

M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742 (1989).
[CrossRef]

K. D. Singer, “Molecular polymeric materials for nonlinear optics,” in Polymers for Light Wave and Integrated Optics: Technology and Applications, L. A. Hornak, ed. (Dekker, New York, 1992), p. 321.

Soane, D.

S. Barry and D. Soane, “Poling of polymeric thin films at ambient temperatures for second-harmonic generation,” Appl. Phys. Lett. 58, 1134 (1991).
[CrossRef]

Todorov, T.

Tomova, N.

Walsh, C. A.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
[CrossRef]

Wu, J. W.

Xie, S.

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

Zahn, H. E.

Zhabostinskii, M. E.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Zolin, V. F.

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Zyss, J.

J. Zyss and D. S. Chemla, “Quadratic nonlinear optics and optimization of the second order nonlinear optical response of molecular crystals,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, New York, 1987), 1, p. 3.

Ann. Phys. (Paris) (1)

H. Benoit, “Contribution à l’etude de l’effet Kerr presentées par les solutions diluées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Sekkat and M. Dumont, “Photoassisted poling of azo dyes doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
[CrossRef]

Appl. Phys. Lett. (2)

S. Barry and D. Soane, “Poling of polymeric thin films at ambient temperatures for second-harmonic generation,” Appl. Phys. Lett. 58, 1134 (1991).
[CrossRef]

P. M. Blanchard and G. R. Mitchell, “A comparison of photoinduced poling and thermal poling of azo-dye-doped polymer films for second order nonlinear optical applications,” Appl. Phys. Lett. 63, 2038 (1993).
[CrossRef]

Ber. Bunsenges. Phys. Chem. (1)

Z. Sekkat and W. Knoll, “Stationary state and dynamics of birefringence and nonlinear optical properties induced by electric field poling in polymeric films,” Ber. Bunsenges. Phys. Chem. 98, 1231 (1994).
[CrossRef]

Chem. Mater. (1)

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized poly(methyl methacrylate) films,” Chem. Mater. 5, 229 (1993).
[CrossRef]

Chem. Phys. Lett. (1)

Z. Sekkat, M. Büchel, H. Orendi, H. Menzel, and W. Knoll, “Photoinduced alignment of azobenzene moieties in the side chains of polyglutamate films,” Chem. Phys. Lett. 220, 497 (1994).
[CrossRef]

Chem. Rev. (1)

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Chem. (2)

H. Rau, G. Greiner, G. Gauglitz, and H. Meier, “Photochemical quantum yields in the A(+hν) ⇔ B(+hν, Δ) system when only the spectrum of A is known,” J. Phys. Chem. 94, 6523 (1990).
[CrossRef]

E. Fisher, “The calculation of photostationary states in systems A⇔ B when only A is known,” J. Phys. Chem. 71, 3704 (1967).
[CrossRef]

J. Polym. Sci. Polym. Symp. (1)

R. E. Robertson, “Effect of free volume fluctuations on polymer relaxation in the glassy state,” J. Polym. Sci. Polym. Symp. 1, 173 (1978).

Macromolecules (1)

A. Nathansohn, P. Rochon, J. Gosselin, and S. Xie, “Azo polymers for reversible optical storage. 1. Poly(4′-((2-(acryloyloxy)ethyl)ethylamino)-4-nitroazobenzene),” Macromolecules 25, 2268 (1992).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

R. Loucif-Saibi, K. Nakatani, C. Dhenaut, J. A. Delaire, Z. Sekkat, and M. Dumont, “Influence of photoisomerization of azobenzenes derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[CrossRef]

Mol. Cryst. Liq. Cryst. Sci. Technol. B (1)

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo dye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol. B 2, 359 (1992).

Nonlin. Opt. (1)

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photo-induced orientation and orientational relaxation of azo dyes in polymeric films,” Nonlin. Opt. 5, 395 (1993).

Opt. Spectrosc. (USSR) (1)

B. D. Davydov, L. D. Derkacheva, V. V. Dumina, M. E. Zhabostinskii, V. F. Zolin, L. G. Koreneva, and M. A. Sanokhina, “Charge transfer and harmonic generation in molecular crystals,” Opt. Spectrosc. (USSR) 30, 274 (1971).

Phys. Rev. A (1)

S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20, 1179 (1979).
[CrossRef]

Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. (1)

Z. Sekkat, E. F. Aust, and W. Knoll, “Room temperature photo-induced polar order. A new method for poling polymeric films containing polar azo dyes for second order applications,” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 35, 176 (1994).

Synth. Metals (1)

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
[CrossRef]

Other (7)

Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” Ph.D. dissertation (Université Paris-Sud, Paris, 1992).

J. Zyss and D. S. Chemla, “Quadratic nonlinear optics and optimization of the second order nonlinear optical response of molecular crystals,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, New York, 1987), 1, p. 3.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).

H. Rau, “Photoisomerization of azobenzenes,” in Photochemistry and Photophysics, F. J. Rabeck, ed. (CRC, Boca Raton, Fla., 1990), Vol. 2, Chap. 4, pp. 119–141. This paper contains a large bibliography on photoisomerization.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985).

K. D. Singer, “Molecular polymeric materials for nonlinear optics,” in Polymers for Light Wave and Integrated Optics: Technology and Applications, L. A. Hornak, ed. (Dekker, New York, 1992), p. 321.

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” in Photopolymers and Applications in Holography, Optical Data Storage, Optical Sensors and Interconnects, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2042, 2 (1993).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Trans ⇔ cis isomerization of azobenzenes. (b) Simplified model of the molecular states. σt and σc are the cross sections for absorption of one photon by a molecule in the trans and cis states, respectively; γ is the thermal relaxation rate; and Φct and Φtc are the quantum yields of photoisomerization.

Fig. 2
Fig. 2

Heavy black lines indicate the laboratory coordinates axes (1, 2, 3); gray lines indicate the molecular principal axes (x, y, z) (solid lines) and (x′, y′, z′) (dashed lines) for two different orientations of the molecular symmetry long axis that is along the z (z′) axis. The poling dc field is applied along the 3 axis. The pumping light beam may be linearly polarized along the 3 axis and also circularly polarized in the {1, 2} plane, or it may be unpolarized, propagating in the direction of the 3 axis.

Fig. 3
Fig. 3

(a) Onset of the usual thermal poling that occurs without requiring photoisomerization (I = 0 or τp = ∞), if the mobility of the trans molecules is appreciable (τDt = 10). Onset of photoinduced poling when the pumping light beam is turned on (τp = 1) with a greatly reduced (τDt = 104) trans mobility and for different cis molecular mobility [(b) τDc = 10, (c) τDc = 0.1]. The inset shows the numerical values of the physical parameters that appear in the following order: (τDc, τDt, τp, e r t, Q) (see text for definitions). The time constants are normalized by the cis lifetime, and μtE/kT = 1. T1 and C1, respectively, characterize the polar order of the trans and the cis molecular distributions. χ 333 ( 2 ) is the normalized component along the 3 axis of the second-order susceptibility of the whole molecular angular distribution. For all figures, the time is normalized by the cis lifetime.

Fig. 4
Fig. 4

Onset of photoinduced poling with and without retention of molecular orientation memory during the cis ⇒ trans back isomerization. (a) Q = 0, (b) Q = 0.3. See Fig. 3 for an explanation of the insets.

Fig. 5
Fig. 5

Effect of the pumping light intensity (or τp) on the onset of a photoinduced molecular polar order. (a) τp = 5, (b) τp = 0.5. See Fig. 3 for an explanation of the insets.

Fig. 6
Fig. 6

Onset of photoinduced poling for e r t = 1/4. See Fig. 3 for an explanation of the insets.

Fig. 7
Fig. 7

Evolution of the polar order when the pumping light is switched off with the dc field applied (a) when the steady state of PIP is pursued to saturation and Q = 0.8, (b) when the steady state of PIP is not pursued to saturation and Q = 0.8, (c) when the steady state of PIP is not pursued to saturation and Q = 0. T1, C1, and χ 333 ( 2 ) are the same as in Fig. 3.

Fig. 8
Fig. 8

Experimental curve from Ref. 8, showing the evolution of the Pockels coefficient of a DR1 functionalized PMMA film from attenuated total reflection electro-optic modulation. The sudden jumps up and down of the signal when the dc field is switching on and off originate from the third-order nonlinearity χ(3). In this figure E0 refers to a poling voltage and is different from the strength of the poling field E defined in Eq. (4). The inset shows a similar experiment7 performed with DR1-doped PMMA. For more details see Refs. 7 and 8. Redrawn by permission of Gordon & Breach, from Ref. 9.

Equations (49)

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d n t ( Ω ) d t = - I ϕ t c [ σ t + ( σ t - σ t ) cos 2 θ ] n t ( Ω ) + I ϕ c t n c ( Ω ) [ σ c + ( σ c - σ c ) cos 2 θ ] × P c t ( Ω Ω ) d Ω + 1 τ c Q ( Ω Ω ) n c ( Ω ) × d Ω + D t R · [ R n t ( Ω , t ) + n t ( Ω , t ) R U / k T ] , d n c ( Ω ) d t = - I ϕ c t [ σ c + ( σ c - σ c ) cos 2 θ ] n c ( Ω ) - 1 τ c n c ( Ω ) + I ϕ t c n t ( Ω ) × [ σ t + ( σ t - σ t ) cos 2 θ ] P t c ( Ω Ω ) d Ω + D c R · [ R n c ( Ω , t ) + n c ( Ω , t ) R U / k T ] ,
D = ( k T ) / ξ ,
R v × ,
U = - μ E cos θ ,
n t ( Ω ) d Ω = N t ,         n c ( Ω ) d Ω = N c , N t + N c = N , P c t , t c ( Ω Ω ) d Ω = 1 ,         Q ( Ω Ω ) d Ω = 1 ,
n t ( Ω ) = 1 2 π n = 0 2 n + 1 2 T n P n ( cos θ ) , n c ( Ω ) = 1 2 π k = 0 2 k + 1 2 C k P k ( cos θ ) .
T n = 0 π n t ( θ ) P n ( cos θ ) sin θ d θ , C n = 0 π n c ( θ ) P n ( cos θ ) sin θ d θ .
P c t ( χ ) = 1 2 π q = 0 2 q + 1 2 P q c t P q ( cos χ ) , P t c ( χ ) = 1 2 π m = 0 2 q + 1 2 P q t c P q ( cos χ ) , Q ( χ ) = 1 2 π q = 0 2 m + 1 2 Q m P m ( cos χ ) ,
T 0 = N t ,             C 0 = N c ,             P 0 t c = P 0 c t = Q 0 = 1.
d d x [ ( x 2 - 1 ) d P n ( x ) d x ] = n ( n + 1 ) P n ( x ) , d d x [ ( x 2 - 1 ) P n ( x ) ] = ( n + 1 ) ( n + 2 ) 2 n + 1 P n + 1 ( x ) - n ( n - 1 ) 2 n + 1 P n - 1 ( x ) , x 2 ( 2 n + 1 ) P n ( x ) = ( n + 1 ) ( n + 2 ) 2 n + 3 P n + 2 ( x ) + [ ( n + 1 ) 2 2 n + 3 + n 2 2 n - 1 ] P n ( x ) + n ( n - 1 ) 2 n - 1 P n - 2 ( x ) ,
0 2 π P n ( cos χ ) d Φ = 2 π P n ( cos θ ) P n ( cos θ ) ,
d T n d t = - I t { T } + I c P n c t { C } + γ Q n C n + n ( n + 1 ) D t [ - T n + u t 2 n + 1 ( T n - 1 - T n + 1 ) ] , d C n d t = I t P n t c { T } - I c { C } - γ C n + n ( n + 1 ) D c [ - C n + u t 2 n + 1 ( C n - 1 - C n + 1 ) ] ,
{ T } = { 3 e r t κ n + · T n + 2 + [ 1 + e r t ( 3 κ n - 1 ) ] · T n + 3 e r t κ n - · T n - 2 } , { C } = { 3 e r c κ n + · C n + 2 + [ 1 + e r c ( 3 κ n - 1 ) ] · C n + 3 e r c κ n - · C n - 2 } , κ n + = ( n + 1 ) ( n + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) ,             κ n = 2 n 2 + 2 n - 1 ( 2 n - 1 ) ( 2 n + 3 ) , κ n - = n ( n - 1 ) ( 2 n - 1 ) ( 2 n + 1 ) , u t , c = μ t , c E / k T ,             I t , c = σ ¯ t , c I .
0 = I c ( P n c t P n t c - 1 ) { C } + [ γ ( Q n P n t c - 1 ) - n ( n + 1 ) D c ] C n + u c n ( n + 1 ) 2 n + 1 ( C n - 1 - C n + 1 ) .
0 = ( P n c t P n t c - 1 ) { C } ,             { T } = P n c t { C } .
δ n C n = u c ( C n - 1 - C n + 1 ) ,
δ n = ( 2 n + 1 ) [ 1 + ( 1 - Q n P n t c ) τ D c / τ c ] , τ D c = 1 / n ( n + 1 ) D c .
A 1 c = C 1 C 0 = u c δ 1 + u c 2 / δ 2 ,             A 2 c = C 2 C 0 = u c 2 / δ 2 δ 1 + u c 2 / δ 2 .
( Δ n + C n + 2 + Δ n - C n - 2 ) + Δ n C n = u c ( C n - 1 - C n + 1 ) ,
Δ n ± = 3 e r c I c ( 1 - P n t c P n c t ) ( 2 n + 1 ) τ D c κ n ± , Δ n = I c ( 1 - P n t c P n c t ) [ 1 + e r c ( 3 κ n - 1 ) ] ( 2 n + 1 ) τ D c + δ n ,
A 1 c = u c ( Δ 2 + Δ 2 - ) / Δ 1 Δ 2 + u c 2 / Δ 1 ,             A 2 c = - Δ 2 - + u c 2 / Δ 1 Δ 2 + u c 2 / Δ 1 .
A 1 t = Q 1 + 2 e r t A 2 t 1 + 4 e r t / 5 A 1 c , A 2 t = Q A 2 c - 2 e r t / 5 1 + 2 e r t ( 2 / 7 - Q A 2 c ) .
C 1 ( t ) = d [ λ 0 λ 2 ( λ 2 - λ 0 ) exp ( - λ 2 t ) + 1 ( λ 0 - λ 2 ) exp ( - λ 0 t ) + 1 λ 2 ] , T 1 ( t ) = ( a λ 2 - λ 1 + b λ 0 - λ 1 - c λ 1 ) exp ( - λ 1 t ) - a λ 2 - λ 1 exp ( - λ 2 t ) - b λ 0 - λ 1 exp ( - λ 1 t ) + c λ 1 ,
λ 0 = 1 / τ c + 1 / τ p ,             λ 1 = 1 / τ p + 1 / τ D t , λ 2 = 1 / τ c + 1 / τ D c ,
1 / τ p = σ ¯ t Φ t c I ,             τ p = τ p ( 1 + 4 e r t / 5 ) ,
a = d Q τ c λ 0 λ 2 ( λ 2 - λ 0 ) , b = d Q τ c 1 λ 0 - λ 2 + N c τ D t u t 3 , c = d λ 2 Q τ c + N t τ D t u t 3 , d = N c τ D c u c 3 .
C 1 = d λ 2 = N c 1 + τ D c / τ c u c 3 , T 1 = c λ 1 = N t 1 + τ D t / τ p × ( u t 3 + Q N c N t 1 τ D c + τ D t + τ c / τ D t u c 3 ) ,
N t = N 1 + τ c / τ p ,             N c = N - N t .
χ 333 ( 2 ) ( t ) = 3 χ 113 ( 2 ) ( t ) = 3 5 [ β z z z t , * T 1 ( t ) + β z z z c , * C 1 ( t ) ] ,
C 1 ( t ) = a 10 exp ( - λ 0 t ) + a 11 exp ( - λ 1 t ) + a 12 exp ( - λ 2 t ) , C 2 ( t ) = a 20 exp ( - λ 0 t ) + a 21 exp ( - λ 1 t ) + a 22 exp ( - λ 2 t ) ,
λ 0 = 1 / τ c ,             λ 1 , 2 = [ ( 2 + τ D c / τ c ) ( 1 - u c 2 / 5 ) 1 / 2 ] / τ D c , a 10 = g 1 0 + g 2 0 ,             a 11 = g 1 - g 1 0 ,             a 12 = g 2 - g 2 0 , a 20 = h 1 c g 1 0 + h 2 c g 2 0 ,             a 21 = h 1 c a 11 ,             a 22 = h 2 c a 12 ,
h 1 , 2 c , t = 3 [ 1 ( 1 - u c , t 2 / 5 ) 1 / 2 ] / u c , t , g 1 , 2 0 = N c / τ D c λ 1 , 2 - λ 0 u c [ 1 ± ( 1 - u c 2 / 5 ) 1 / 2 ] 6 ( 1 - u c 2 / 5 ) 1 / 2 , g 1 = ( h 2 c C 1 - C 2 ) / ( h 2 c - h 1 c ) , g 2 = ( - h 1 c C 1 + C 2 ) / ( h 2 c - h 1 c ) ,
T 1 ( t ) = t 1 ( t ) + t 2 ( t ) , T 2 ( t ) = h 1 t t 1 ( t ) + h 2 t t 2 ( t ) ,
t 1 , 2 ( t ) = t 1 , 2 0 + t 1 , 2 1 exp ( - λ 2 t ) + t 1 , 2 2 exp ( - λ 1 t ) + t 1 , 2 3 exp ( - λ 0 t ) + ( t 1 , 2 - i = 0 3 t 1 , 2 i ) × exp ( - λ 1 , 2 t ) , λ 1 , 2 = [ 2 ( 1 - u t 2 / 5 ) 1 / 2 ] / τ D t .
t 1 , 2 0 = E 1 , 2 0 / λ 1 , 2 ,             t 1 , 2 1 = E 1 , 2 1 / ( λ 1 , 2 - λ 2 ) , t 1 , 2 2 = E 1 , 2 2 / ( λ 1 , 2 - λ 1 ) ,             t 1 , 2 3 = E 1 , 2 3 / ( λ 1 , 2 - λ 0 ) , E 1 0 = N u t h 2 t / 3 τ D t ( h 2 t - h 1 t ) ,             E 2 0 = - ( h 1 t / h 2 t ) E 1 0 , E 1 , 2 i = ( ± h 2 , 1 t X 1 i X 2 i ) / τ D i ( h 2 t - h 1 t )             ( i = 1 , 2 , 3 ) ,
X 1 , 2 1 = Q a 1 , 22 τ D t / τ c ,             X 1 , 2 2 = Q a 1 , 21 τ D t / τ c , X 1 3 = Q a 10 τ D t / τ c - N c u t / 3 , X 2 3 = Q a 20 τ D t / τ c .
1 D t d A n t d t = - n ( n + 1 ) A n t .
A n t ( t ) = A n t ( ) exp ( - t / τ n ) ,             τ n = 1 / n ( n + 1 ) D t .
cos χ = cos θ cos θ + sin θ sin θ cos Φ ,
P n ( cos χ ) = P n ( cos θ ) P n ( cos θ ) + 2 m = 1 n ( n - m ) ! ( n + m ) ! × P n m ( cos θ ) P n m ( cos θ ) cos m Φ .
0 2 π P n ( cos χ ) d Φ = 2 π P n ( cos θ ) P n ( cos θ ) .
P i ( ω ) = P i 0 + χ i j ( 1 ) ( - ω , ω ) E j ω + χ i j k ( 2 ) ( - ω , ω 1 , ω 2 ) E j ω 1 E k ω 2 + χ i j k l ( 3 ) ( - ω , ω 1 , ω 2 , ω 3 ) E j ω 1 E k ω 2 E l ω 3 + .
p I = μ I + α I J F J + β I J K F J F K + γ I J K L F J F K F L + ,
G ( θ ) = n = 0 2 n + 1 2 A n P n ( cos θ ) ,
A n = P n ( cos θ ) = - 1 + 1 d ( cos θ ) G ( θ ) P n ( cos θ ) .
χ 33 ( 1 ) = N α ¯ ω ( 1 + 2 r ω A 2 ) , χ 11 ( 1 ) = N α ¯ ω ( 1 - r ω A 2 ) ,
χ 333 ( 2 ) = N β z z z * ( 3 5 A 1 + 2 5 A 3 ) , χ 113 ( 2 ) = N β z z z * ( 1 5 A 1 - 1 5 A 3 ) ,
χ 3333 ( 3 ) = N γ z z z z * ( 1 5 + 4 7 A 2 + 8 35 A 4 ) , χ 2233 ( 3 ) = χ 1133 ( 3 ) = N γ z z z z * ( 1 15 + 4 21 A 2 - 4 35 A 4 ) , χ 1111 ( 3 ) = 3 χ 1122 ( 2 ) = 3 N γ z z z z * ( 1 15 - 2 21 A 2 + 1 35 A 4 ) ,
N ( Ω ) = ( N / 2 π ) G ( θ ) ,

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