Abstract

We present an original poling technique that uses a purely optical excitation process. The experiment consists of a seeding-type process. Writing and probing periods are alternated. Writing periods correspond to simultaneous irradiation of the sample by the coherent superposition of the 1064-nm fundamental and the 532-nm second-harmonic light of a picosecond-pulsed Nd:YAG laser. The sample is a spin-coated film of a poly(methyl methacrylate) copolymer onto which the azo-dye molecule Disperse Red 1 is grafted. We demonstrate efficient and quasi-permanent poling of the molecules with a spatial period that satisfies the phase-matching condition for second-harmonic generation. The influence of seeding parameters such as the relative phase and the relative intensities between the writing beams is studied both theoretically and experimentally. Tensorial properties and the spatial profile of the photoinduced χ(2) are analyzed from a microscopic point of view. Dark and photostimulated relaxation processes are investigated from a chemical-physics point of view. The physical origin of the photoinduced molecular orientation process is discussed. A minimal model involving the relevant experimental parameters is developed. Numerical simulations are in agreement with the experiment.

© 1997 Optical Society of America

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    [Crossref]
  2. Z. Sekkat and M. Dumont, “Photoassisted poling of azo-dye doped polymeric films at room temperature,” Appl. Phys. B 54, 486 (1992).
    [Crossref]
  3. Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo-dye chromophores,” Nonlin. Opt. 2, 359 (1992).
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    [Crossref]
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  32. Z. Sekkat and M. Dumont, “Photoinduced orientation of azo-dyes in polymeri films. Characterization of molecular angular mobility,” Synth. Metals 54, 373 (1993).
    [Crossref]
  33. M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).
  34. F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).
  35. J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
    [Crossref]
  36. C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
    [Crossref]
  37. C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
    [Crossref]

1996 (2)

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

1995 (3)

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

1994 (2)

1993 (5)

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiart, “Light-induced second harmonic generation in azo-dye polymers,” Opt. Lett. 18, 941 (1993).
[Crossref] [PubMed]

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

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

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

1992 (4)

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Z. Sekkat and M. Dumont, “Photoassisted poling of azo-dye 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,” Nonlin. Opt. 2, 359 (1992).

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

1990 (1)

V. Mizrahi, Y. Hibidino, and G. Stegeman, “Polarisation study of photoinduced second harmonic generation in glass optical fibers,” Opt. Commun. 78, 283 (1990).
[Crossref]

1989 (1)

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

1987 (5)

1986 (1)

1981 (1)

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466 (1981).
[Crossref]

1970 (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970); M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693 (1976).
[Crossref]

1969 (1)

W. Liptay, “Electrochromism and Solvatochromism,” Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[Crossref]

Aoki, H.

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

Baranova, N. B.

N. B. Baranova and B. Ya. Zeldovich, “Extension of holography to multifrequency fields,” JETP Lett. 45, 717 (1987).

Bourbin, Y.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Broussoux, D.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Charra, F.

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

C. Fiorini, F. Charra, and J. M. Nunzi, “Six-wave mixing probe of light-induced second-harmonic generation: example of dye solutions,” J. Opt. Soc. Am. B 11, 2347 (1994).
[Crossref]

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiart, “Light-induced second harmonic generation in azo-dye polymers,” Opt. Lett. 18, 941 (1993).
[Crossref] [PubMed]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

J. M. Nunzi, C. Fiorini, F. Charra, and P. Raimond, “All-optical poling of polymers,” in Polymer Thin Films for Photonics, G. Lindsay and K. D. Singer, eds. (American Chemical Society, Washington, D.C., 1995), Chap. 18, p. 240.

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

Chastaing, E.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

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, Boston, Mass., 1987), Vol. 1, part 2, p. 23.

Delaire, J. A.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Influence of photoisomerization of azobenzene 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 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

Devaux, F.

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

Driscoll, T. J.

Dumont, M.

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo-dyes in polymeri 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, “Influence of photoisomerization of azobenzene derivatives in polymeric thin films on second harmonic generation: towards applications in molecular optoelectronics,” Mol. Cryst. Liq. Cryst. 235, 251 (1993).
[Crossref]

Z. Sekkat and M. Dumont, “Photoassisted poling of azo-dye 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,” Nonlin. Opt. 2, 359 (1992).

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Esselin, S.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Fersing, L.

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” in Digest of the Thirteenth International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), p. 18.

Fiorini, C.

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

C. Fiorini, F. Charra, and J. M. Nunzi, “Six-wave mixing probe of light-induced second-harmonic generation: example of dye solutions,” J. Opt. Soc. Am. B 11, 2347 (1994).
[Crossref]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

J. M. Nunzi, C. Fiorini, F. Charra, and P. Raimond, “All-optical poling of polymers,” in Polymer Thin Films for Photonics, G. Lindsay and K. D. Singer, eds. (American Chemical Society, Washington, D.C., 1995), Chap. 18, p. 240.

Froc, G.

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).

Fukuda, A.

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

Gabriagues, J. M.

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” in Digest of the Thirteenth International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), p. 18.

Gosselin, J.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Hibidino, Y.

V. Mizrahi, Y. Hibidino, and G. Stegeman, “Polarisation study of photoinduced second harmonic generation in glass optical fibers,” Opt. Commun. 78, 283 (1990).
[Crossref]

Hosotte, S.

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).

Idiart, E.

Ishikawa, K.

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 3, p. 98.

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970); M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693 (1976).
[Crossref]

Kajzar, F.

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970); M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693 (1976).
[Crossref]

Kusyk, M. G.

Le Barny, P.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Liptay, W.

W. Liptay, “Electrochromism and Solvatochromism,” Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[Crossref]

Lorin, A.

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

Loucif-Saibi, R.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

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

Margulis, W.

Mizrahi, V.

V. Mizrahi, Y. Hibidino, and G. Stegeman, “Polarisation study of photoinduced second harmonic generation in glass optical fibers,” Opt. Commun. 78, 283 (1990).
[Crossref]

Nakatani, K.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Influence of photoisomerization of azobenzene 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 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

Natansohn, A.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Nunzi, J. M.

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

C. Fiorini, F. Charra, and J. M. Nunzi, “Six-wave mixing probe of light-induced second-harmonic generation: example of dye solutions,” J. Opt. Soc. Am. B 11, 2347 (1994).
[Crossref]

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiart, “Light-induced second harmonic generation in azo-dye polymers,” Opt. Lett. 18, 941 (1993).
[Crossref] [PubMed]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

J. M. Nunzi, C. Fiorini, F. Charra, and P. Raimond, “All-optical poling of polymers,” in Polymer Thin Films for Photonics, G. Lindsay and K. D. Singer, eds. (American Chemical Society, Washington, D.C., 1995), Chap. 18, p. 240.

Nunzi, J.-M.

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

Ohmori, Y.

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466 (1981).
[Crossref]

Österberg, U.

Philipp, H. R.

H. R. Philipp, “Silicon dioxyde type α,” in Handbook of optical constants of solids, E. D. Pawlik, ed. (Academic, Orlando, Fla., 1985), p. 179.

Pocholle, J. P.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Raffy, J.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Raimond, P.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiart, “Light-induced second harmonic generation in azo-dye polymers,” Opt. Lett. 18, 941 (1993).
[Crossref] [PubMed]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

J. M. Nunzi, C. Fiorini, F. Charra, and P. Raimond, “All-optical poling of polymers,” in Polymer Thin Films for Photonics, G. Lindsay and K. D. Singer, eds. (American Chemical Society, Washington, D.C., 1995), Chap. 18, p. 240.

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

Rau, H.

H. Rau, “Photoisomerization of azobenzenes,” in Photochemistry and Photophysics, J. F. Rabek, ed. (CRC, Boca Raton, Fla., 1989), Vol. 2, Chap. 4, p. 119.

Robin, P.

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Rochon, P.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Samuel, I. D. W.

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

Sasaki, Y.

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466 (1981).
[Crossref]

Sekkat, Z.

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Influence of photoisomerization of azobenzene 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 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

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

Z. Sekkat and M. Dumont, “Photoassisted poling of azo-dye 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,” Nonlin. Opt. 2, 359 (1992).

Singer, K. D.

Sohn, J. E.

Stegeman, G.

V. Mizrahi, Y. Hibidino, and G. Stegeman, “Polarisation study of photoinduced second harmonic generation in glass optical fibers,” Opt. Commun. 78, 283 (1990).
[Crossref]

Stolen, R. H.

Takezoe, H.

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

Tom, H. W. K.

Torkelson, J. M.

J. G. Victor and J. M. Torkelson, “On measuring the distribution of local free volume in glassy polymers by photochromic and fluorescence techniques,” Macromolecules 20, 2242 (1987).
[Crossref]

Victor, J. G.

J. G. Victor and J. M. Torkelson, “On measuring the distribution of local free volume in glassy polymers by photochromic and fluorescence techniques,” Macromolecules 20, 2242 (1987).
[Crossref]

Xie, S.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 12, p. 377.

Zeldovich, B. Ya.

N. B. Baranova and B. Ya. Zeldovich, “Extension of holography to multifrequency fields,” JETP Lett. 45, 717 (1987).

Zyss, J.

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, “Light-induced second-harmonic generation in an octupolar dye,” Opt. Lett. 20, 24 (1995).
[Crossref]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

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, Boston, Mass., 1987), Vol. 1, part 2, p. 23.

Angew. Chem. Int. Ed. Engl. (1)

W. Liptay, “Electrochromism and Solvatochromism,” Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[Crossref]

Appl. Phys. B (1)

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

Appl. Phys. Lett. (2)

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azo-aromatic polymers,” Appl. Phys. Lett. 60, 4 (1992).
[Crossref]

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466 (1981).
[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 1 doped and functionalized poly(methylmethacrylate) films,” Chem. Mater. 5, 229 (1993).
[Crossref]

Chem. Phys. Lett. (1)

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349 (1993).
[Crossref]

J. Appl. Phys. (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970); M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693 (1976).
[Crossref]

J. Nonlin. Opt. Phys. Mater. (1)

C. Fiorini, J. M. Nunzi, F. Charra, I. D. W. Samuel, and J. Zyss, “Photoinduced nonlinear octupolar polarization: transient and permanent regimes,” J. Nonlin. Opt. Phys. Mater. 5, 653 (1996).
[Crossref]

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

JETP Lett. (1)

N. B. Baranova and B. Ya. Zeldovich, “Extension of holography to multifrequency fields,” JETP Lett. 45, 717 (1987).

Jpn. J. Appl. Phys. 1 (1)

H. Aoki, K. Ishikawa, H. Takezoe, and A. Fukuda, “Photoinduced destruction of polar structure in dye-pendant polymers studied by second harmonic generation,” Jpn. J. Appl. Phys. 1 35, 168 (1996).
[Crossref]

Macromolecules (1)

J. G. Victor and J. M. Torkelson, “On measuring the distribution of local free volume in glassy polymers by photochromic and fluorescence techniques,” Macromolecules 20, 2242 (1987).
[Crossref]

Mol. Cryst. Liq. Cryst. (1)

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

Nonlin. Opt. (3)

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerization of azo-dye chromophores,” Nonlin. Opt. 2, 359 (1992).

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry,” Nonlin. Opt. 9, 339 (1995).

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlin. Opt. 9, 327 (1995).

Opt. Commun. (1)

V. Mizrahi, Y. Hibidino, and G. Stegeman, “Polarisation study of photoinduced second harmonic generation in glass optical fibers,” Opt. Commun. 78, 283 (1990).
[Crossref]

Opt. Lett. (5)

Phys. Rev. Lett. (1)

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, “Picosecond light-induced noncentrosymmetry in a dye solution,” Phys. Rev. Lett. 68, 2440 (1992).
[Crossref] [PubMed]

Rev. Tech. Thomson-CSF (1)

D. Broussoux, E. Chastaing, S. Esselin, P. Le Barny, P. Robin, Y. Bourbin, J. P. Pocholle, and J. Raffy, “Organic materials for nonlinear optics,” Rev. Tech. Thomson-CSF 20, 151 (1989).

Synth. Metals (1)

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

Other (10)

F. Charra, C. Fiorini, A. Lorin, J.-M. Nunzi, and P. Raimond, “Procédé de fabrication de structures en polyméres organiques, transparentes, auto-organisées pour la conversion de fréquence optique,” French patent9503616 (March28, 1995).

H. R. Philipp, “Silicon dioxyde type α,” in Handbook of optical constants of solids, E. D. Pawlik, ed. (Academic, Orlando, Fla., 1985), p. 179.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 3, p. 98.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

H. Rau, “Photoisomerization of azobenzenes,” in Photochemistry and Photophysics, J. F. Rabek, ed. (CRC, Boca Raton, Fla., 1989), Vol. 2, Chap. 4, p. 119.

Schott glass data reference handbook.

J. M. Nunzi, C. Fiorini, F. Charra, and P. Raimond, “All-optical poling of polymers,” in Polymer Thin Films for Photonics, G. Lindsay and K. D. Singer, eds. (American Chemical Society, Washington, D.C., 1995), Chap. 18, p. 240.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 12, p. 377.

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, Boston, Mass., 1987), Vol. 1, part 2, p. 23.

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” in Digest of the Thirteenth International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), p. 18.

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Figures (18)

Fig. 1
Fig. 1

Block diagram of the experimental setup. (a) A photodiode triggered by part of the ω-reading beam synchronizes the fast sampler system used for measuring the intensity of the generated SH beam, which is detected by the PM tube. The BK7 glass plate is held upon a rotating stage for adjustment of the relative phase between the two writing beams. P, polarizers; F, interference filter at 532 nm; R, dielectric mirror for the fundamental beam; S, shutter; synchronized with the insertion of the green-blocking RG 630 Schott filter; Phd, photodiode measuring the evolution of the optical density of the sample as a function of the seeding time (in the direction parallel to the excitation direction); KG3, Schott filter cutting the fundamental beam; HP, Hewlett-Packard. (b) Modification of the previous setup permitting in situ testing of the induced birefringence. The laser is a cw He–Ne laser (λ=632.8 nm, 1 mW); Fs, filters at 632.8 nm; HP, Hewlett-Packard; Ps, polarizers; PhD, photodiode.

Fig. 2
Fig. 2

Real-time growth and lifetime of the SHG signal (in arbitrary units). Negative times correspond to the seeding-time preparation process. At time zero the seeding process is stopped. Positive times are associated with the study of the temporal stability of the induced χ(2) susceptibility. The sample was a spin-coated thin film of DR1–MMA 35/65 with a thickness of 0.5 µm. Its optical density at 532 nm was 1.5. The peak intensity of the fundamental beam was a few gigawatts per square centimeter. The SH seeding beam was much less intense, 10100 MW/cm2.

Fig. 3
Fig. 3

Intensity of the generated SH signal (in arbitrary units) after 20 min of preparation time as a function of the phase difference ΔΦ between the writing beams at frequencies ω and 2ω. The reference (ΔΦ=0) is arbitrarily taken at normal incidence upon the BK7 plate. The solid curve is a fit to Eq. (4). The curve in the inset corresponds to modulation amplitude C as a function of the sample thickness. The modulation amplitude is defined as C=(ImaxSHG-IminSHG)/(ImaxSHG+IminSHG), where ImaxSHG and IminSHG are the extremal values, as a function of ΔΦ, of the SH intensity generated at saturation.

Fig. 4
Fig. 4

Influence of the phase difference ΔΦ on the sign of the polarity of the photoinduced order. The intensity of the SHG signal is given in arbitrary units. As for Fig. 3, the sample was a 0.1-µm-thick film of DR1–MMA 35/65. ΔΦ=ΔΦMax corresponds to the optimum phase difference. As measured preliminarily, in two distinct prepared areas, both parameters ΔΦ=ΔΦMax and ΔΦ=ΔΦMax+π led to the same SH signal intensity. The inset shows variations of the photoinduced birefringence during the seeding and reconstruction periods.

Fig. 5
Fig. 5

Amplitude of the SHG signal for two low-concentration polymer rods of DR1–MMA. The chromophores concentration (2.8×10-3 % in monomers) is the same in both samples.

Fig. 6
Fig. 6

(a) Dependence of the initial growth of the induced χ(2) susceptibility on the intensities of the writing beams (in arbitrary units). The sample was a 0.1-µm-thick film of DR1–MMA 35/65. The intensities of the writing beams were varied by slight changes in the power supply of the laser amplifier, which, because the SH seeding beam results from frequency doubling of the fundamental beam inside the KDP crystal, can vary the quantity |Eω2E2ω| without changing the ratio |E2ω/Eω2| that is typical of the polarization efficiency. (b) Influence of the relative intensities of the writing beams at frequencies ω and 2ω on the efficiency of the poling process. The sample was a 0.3-µm-thick film of DR1–MMA 35/65. The fundamental beam intensity was kept constant, and the SH beam intensity was varied by slight changes in the KDP crystal’s adjustment. Each curve corresponds to an optimized phase difference between the writing beams. SH seeding intensities are indexed in laboratory units.

Fig. 7
Fig. 7

Intensity of the SHG after saturation as a function of the relative intensities of the writing beams. Samples were spin-coated films of DR1–MMA 35/65 with different thicknesses. Each experimental point corresponds to an optimized adjustment of the phase difference ΔΦ of the writing beams.

Fig. 8
Fig. 8

Spatial profile (in the plane of the film) of the induced χ(2) susceptibility. The sample was a spin-coated thin film of thickness 0.3 µm of DR1–MMA 35/65. The polarization direction of the probe beam is the same as that of the writing beams. The probe beam diameter was 100 µm. This profile yields a full width at half-height of 1 mm, in good agreement with the writing-beam diameters.

Fig. 9
Fig. 9

Polar map of the induced χ(2) susceptibility, corresponding to the square root of the SHG intensity. Analysis was performed a few minutes after the poling process was stopped. The solid curve corresponds to the cos(δ) theoretical fit.

Fig. 10
Fig. 10

Dependence of the SH signal amplitude on the polarization angle of the fundamental reading beam. Case 1 corresponds to the signal generated in the direction parallel to the polarization direction of the writing beams. Case 2 corresponds to a signal generated in the perpendicular direction. Solid curves were fitted with the following theoretical dependences: IxSHG|χxxx(2) cos2 θ+χxyy(2) sin2 θ| and IySHG|χyxy(2) sin(2θ)|, where IxSHG and IySHG are the SH intensities generated in directions x (polarizer P1) and y (polarizer P2).

Fig. 11
Fig. 11

Temporal growth and decay of the second-order susceptibility χ(2) and birefringence Δn displayed on a semilogarithmic scale. (The same data are shown in the inset on a linear scale). The sample was DR1–MMA 35/65 with a thickness of 0.1 µm, seeded under optimal preparation conditions. After the seeding process was stopped, SHG was probed periodically by the reading beam at frequency ω. The laser was switched off between measurements, the period between successive measurements was 30 s. τ3τΔn where τ and τΔn are, respectively, the lifetimes of the photoinduced susceptibility χ(2) and of the photoinduced anisotropy.

Fig. 12
Fig. 12

Erasure of the photoinduced polar orientation with monochromatic irradiation at frequency ω. Negative times correspond to seeding of the sample. A comparison is shown between dark (circles) and photostimulated (squares) decays. The curve in the inset shows the spatial profile of the photoinduced χ(2). The spatial profile was obtained by SHG inside the sample with a beam at fundamental frequency strongly attenuated before focusing. The spatial profile was measured before (circles) and after (squares) irradiation of the central polarized zone with an intense and focused beam at fundamental frequency (103 more intense than the beam used for the χ(2) reading).

Fig. 13
Fig. 13

Erasure of the photoinduced polar orientation with monochromatic irradiation at frequency 2ω. The sample was a 0.1-µm-thick film of DR1–MMA 35/65. After seeding, this sample was irradiated continuously with a monochromatic beam at frequency 2ω (the fluence used here was a few megawatts per cubic centimeter, which is the fluence of the 2ω seeding beam). The inset shows the evolution of the photoinduced birefringence under the same conditions.

Fig. 14
Fig. 14

Experimental setup for the study of space-charge effects. The sample was a 0.3-µm-thick film of DR1–MMA 35/65. (b) Inset corresponds to the in-plane polarized zone.

Fig. 15
Fig. 15

Theoretical dependence of the induced χ(2) susceptibility on the propagation length inside the sample. The induced χ(2) was calculated from the order parameters A1 after decomposition of the molecular distribution N(Ω) as a function of the Legendre polynomials: χ(2)α3/5 A1+2/5 A3. The simulated curve corresponds to a sample with the same index dispersion as the DR1–MMA 35/65. The absorption coefficient used for the calculations was taken 10 times lower than in the case of the DR1–MMA 35/65.

Fig. 16
Fig. 16

Influence of the phase difference between the writing beams at frequencies ω and 2ω on the SHG intensity I2ωSHG. We used a 0.1 µm-thick sample with the same characteristics (index dispersion, absorption coefficient) as for the DR1–MMA 35/65 sample in Fig. 4. Both optimized and unoptimized relative intensities between the writing beams at ω and 2ω were considered: γ=γmax (circles) and γ=5γmax (triangles). Numerical simulation data were obtained after integration of all the contributions induced inside the material. Both curves are normalized to unity. The solid curve is a sinusoidal fit to numerical simulation data.

Fig. 17
Fig. 17

Representation of the SHG intensity as a function of the relative intensities between the writing beams at frequencies ω and 2ω. The value γ=1 corresponds to equal excitation probabilities by one- or two-photon absorption processes. As for the previous simulated curves, we considered a material with the same characteristics as those of DR1–MMA 35/65. Three sample thicknesses were considered that correspond to the situations studied experimentally (see Fig. 7).

Fig. 18
Fig. 18

Dependence of the SHG signal on the sample thickness l in the case of optimized seeding conditions (filled circles). As for the curves shown in Figs. 16 and 17, we considered a material with the same characteristics as the DR1–MMA 35/65. The optimal intensity ratio γ, measured on the front face of the sample (z=0) and corresponding to every investigated sample thickness is given on the right-hand scale. It fits an exp(-α/2z) law (dashed curve), as expected from the sample absorption at frequency 2ω.

Equations (39)

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χind(2)E3(M, t)t=[Eω2E2ω* exp(-iΔk·M)+Eω*2E2ω exp(iΔk·M)]exp-α2z,
χind(2)(z)=χeff(2) cos(ΔΦ+Δk·z)exp-α2z,
dESHGdz=-α2ESHG+i ωχind(2)(z)Eω22nc×exp[-i(2kω-kSHG)z],
I2ωSHG(z=l)=ESHG(z=l)2=ω2deff2l24n2c210ODIω21+sinc22π llc+2 sinc2π llccos2ΔΦ+2π llc,
I2ωSHG=ω2deff24n2c210ODIω2l2,
P011/4(μ01·E2ω)(μ01·E2ω)*+(μ01·Eω)(Δμ·Eω)(μ01·Eω)*(Δμ·Eω)*16(ω)2+(μ01·E2ω)*(μ01·Eω)(Δμ·Eω)+(μ01·E2ω)(μ01·Eω)*(Δμ·Eω)*8ω,
P011-photon=P012-photon,
E2ωEω2=Δμ2ω(cos2 θ)1/2=Δμ23ω,
(ISHGQuartz)Max=4ω2n2c2ΔkQ2d112Iω2,
χΔΔΔ(2)=Ω N(Ω)βΔΔΔ(Ω)dΩ,
N(Ω)=N(θ)=l=0 2l+12AlPl(θ),
χΔΔΔ(2)=Nβiii0A3 cos3 δ+35(A1-A3)cos δ.
χxxx(2)3/5A1(t)+2/5 A3(t)3/5A1 exp(-2Dt)+2/5A3 exp(-12Dt),
Δn=nx-nyχxx(1)-χyy(1)A2(t)A2 exp(-6Dt).
dN(Ω0)dt=-A cos2 θ0N(Ω0)-B cos4 θ0N(Ω0)-C cos3 θ0N(Ω0)+A  cos2 θ1N(Ω1)P(Ω1Ω0)dΩ1+B  cos4 θ1N(Ω1)P(Ω1Ω0)dΩ1+C  cos3 θ1N(Ω1)P(Ω1Ω0)dΩ1-D2N(Ω0).
Aμ012E2ω2=Bγ2,
Bμ012Δμ2(2ω)2Eω4,
Cμ012Δμ(ω)Eω2E2ω*cos(ΔΦ+Δkz)=2Bγ cos(ΔΦ+Δkz),
N(Ω0)=N(θ0)=j 2j+12AjPj[cos(θ0)],
I1= cos2(θ1)N[cos(θ1)]P(Ω1Ω0)dΩ1=120π cos2(θ1)N[cos(θ1)]d[cos(θ1)].
I1=j 2j+14Aj 0π cos2(θ1)Pj[cos(θ1)]d[cos(θ1)].
cos2 θPj(cos θ)=αj(-2)Pj-2(cos θ)+αj(0)Pj(cos θ)+αj(+2)Pj+2(cos θ),
αj(-2)=j(j-1)(2j+1)(2j-1)
αj(0)=j2(2j+1)(2j-1)+(j+1)2(2j+3)(2j+1),
αj(+2)=(j+1)(j+2)(2j+3)(2j+1);
cos3 θPj(cos θ)=γj(-3)Pj-3(cos θ)+γj(-1)Pj-1(cos θ)+γj(+1)Pj+1(cos θ)+γj(+3)Pj+3(cos θ),
γj(-3)=j-22j-3αj(-2),
γj(-1)=j-12j-3αj(-2)+j2j+1αj(0),
γj(+1)=j+12j+1αj(0)+j+22j+5αj(+2),
γj(+3)=j+32j+5αj(+2);
cos4 θPj(cos θ)=βj(-4)Pj-4(cos θ)+βj(-2)Pj-2(cos θ)+βj(0)Pj(cos θ)+βj(+2)Pj+2(cos θ)+βj(+4)Pj+4(cos θ),
βj(-4)=j-32j-5γj(-3),
βj(-2)=j-22j-5γj(-3)+j-12j-1γj(-1),
βj(0)=j2j-1γj(-1)+j+12j+3γj(+1),
βj(+2)=j+22j+3γj(+1)+j+32j+7γj(+3),
βj(+4)=j+42j+7γj(+3).
J=0π Pj[cos(θ1)]d[cos(θ1)]=2δj,0,
dA0dt=0,
dAjdt=Aj+4-B (j+4)(j+3)(j+2)(j+1)(2j+7)(2j+5)(2j+3)(2j+1)+Aj+3-C (j+3)(j+2)(j+1)(2j+5)(2j+3)(2j+1)+Aj+2-A (j+2)(j+1)(2j+3)(2j+1)-B 2(j+1)(j+2)(2j2+6j-3)(2j-1)(2j+1)(2j+3)(2j+7)+Aj+1-C 3(j+1)(j2+2j-1)(2j-1)(2j+1)(2j+5)+Aj-j(j+1)D-A 2j2+2j-1(2j+3)(2j-1)+B 3(2j4+4j3-6j2-8j+3)(2j+5)(2j+3)(2j-1)(2j-3)+Aj-1-C 3j(j2-2)(2j+3)(2j+1)(2j-3)+Aj-2-A j(j-1)(2j+1)(2j-1)+B 2j(j-1)(2j2-2j-7)(2j+3)(2j+1)(2j-1)(2j-5)+Aj-3-C j(j-1)(j-2)(2j+1)(2j-1)(2j-3)+Aj-4-B j(j-1)(j-2)(j-3)(2j+1)(2j-1)(2j-3)(2j-5).

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