Abstract

All-optical poling (AOP) of dye molecules in polymeric materials is the result of noncentrosymmetric angular hole burning (AHB: selective depletion in the angular distribution of molecules) and of angular redistribution (AR: rotation in the reversible photoisomerization). Many publications describe successfully the photoinduced anisotropy of rodlike molecules: the cumulative building of anisotropy, after many cycles, is a necessary condition of efficiency. Other publications show the important role of the tensorial properties of molecules and fields in AOP, but they consider neither the saturation of AHB, nor AR, in multiple cycles. Here, the tensorial expression of the excitation probability, for any geometry of molecules and fields, is reevaluated (density matrix formalism) and introduced in optical pumping equations, which are solved formally (at second order) and numerically. The expected symmetries are preserved, but the saturation of optical pumping introduces new tensorial couplings, which modify χ(1) and χ(2).

© 2009 Optical Society of America

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    [CrossRef]
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  5. A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].
  6. Z. Sekkat and M. Dumont, “Photoassisted poling of azodye doped polymeric films at room temperature,” Appl. Phys. B 54, 486-489 (1992).
    [CrossRef]
  7. Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerisation of azodye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 2, 359-362 (1992).
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    [CrossRef]
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    [CrossRef] [PubMed]
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  12. A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
    [CrossRef]
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  18. M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).
  19. M. Dumont and A. El Osman, “On spontaneous and photoinduced orientational mobility of dye molecules in polymer films,” Chem. Phys. 245, 437-462 (1999).
    [CrossRef]
  20. N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
    [CrossRef]
  21. S. Brasselet and J. Zyss, “Control of the polarization dependence of optically poled nonlinear polymer films,” Opt. Lett. 19, 1464-1466 (1997).
    [CrossRef]
  22. S. Brasselet and J. Zyss, “Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media,” J. Opt. Soc. Am. B 15, 257-288 (1998).
    [CrossRef]
  23. S. Brasselet, “Processus multipolaires en optique non-linéaire dans les miliuex moléculaires,” Thèse de Doctorat (Université Paris 11, Orsay, 1997).
  24. As it appears in experiments and as it will be shown in the second paper two types of saturation are to be considered: the saturation of, AHB (which exists as soon as τCPr(Ω) is not much smaller than 1, even if there is no AR) and the saturation of, AR in the photoisomerization cycle, which appears for a much smaller pumping intensity (since, AR is in competition with diffusion in T, which is often negligible), but generally after longer times (depending of the average rotation in each cycle).
  25. M. Dumont, “Modelization of the angular redistribution in optical ordering processes in dye containing polymers,” Nonlinear Opt. 25, 195-200 (2000).
  26. M. Dumont, “New developments in optical ordering of, NLO dyes in polymers,” Proc. SPIE 4461, 179-163 (2001).
  27. S. Hosotte and M. Dumont, “Orientational relaxation of photomerocyanine molecules in polymeric films,” Synth. Met. 81, 125-127 (1996).
    [CrossRef]
  28. S. Hosotte and M. Dumont, “Photoassisted poling and orientational relaxation of dye molecules in polymers. The case of spiropyran,” Proc. SPIE 2852, 53-63 (1996).
    [CrossRef]
  29. N. Bloembergen, Nonlinear Optics (Benjamin, 1965).
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  31. Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecules angular mobility,” Synth. Met. 54, 373-381 (1993).
    [CrossRef]
  32. The formal expression, ∑m′(−)m′AT,m′a,j(n+1)ST,−m′a′,j(p), looks like the scalar product ATa,J(n+1)⋅STa′,J(p), but it is not, since these two tensors are orthogonal if a≠a′.
  33. M. Dumont, is preparing a paper to be called “Dynamics of all-optical poling of photoisomerizable molecules. II. Comparison of angular redistribution models.”
  34. J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, “Transient optically induced noncentrosymmetry in a solution of octupolar molecules,” Chem. Phys. Lett. 219, 349-354 (1993).
    [CrossRef]
  35. X. Yu, X. Zhong, Q. Li, S. Luo, and Y. Chen, “Method of improving optical poling efficiency in polymer films,” Opt. Lett. 26, 220-222 (2001).
    [CrossRef]
  36. 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 12, 2347-2358 (1994).
    [CrossRef]
  37. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
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    [CrossRef]
  43. M. Dumont, “Théorie du pompage optique avec un laser, étude expérimentale dans le cas du Néon de las réponse linéaire et de quelques effets de saturation,” Thèse de Doctorat d'état ès Sciences Physiques (Faculté des Sciences de Paris, 1971), p. 43, http://tel.archives-ouvertes.fr/docs/00/06/09/12/PDF/1971DUMONT.pdf.
  44. T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597-1617 (1978).
    [CrossRef]
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  47. Z. Sekkat and W. Knoll, “Creation of second-order nonlinear optical effects by photoisomerization of polar azo dyes in polymeric fields: theoretical study of steady-state and transient properties,” J. Opt. Soc. Am. B 12, 1855-1867 (1995).
    [CrossRef]

2005 (1)

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

2001 (2)

M. Dumont, “New developments in optical ordering of, NLO dyes in polymers,” Proc. SPIE 4461, 179-163 (2001).

X. Yu, X. Zhong, Q. Li, S. Luo, and Y. Chen, “Method of improving optical poling efficiency in polymer films,” Opt. Lett. 26, 220-222 (2001).
[CrossRef]

2000 (1)

M. Dumont, “Modelization of the angular redistribution in optical ordering processes in dye containing polymers,” Nonlinear Opt. 25, 195-200 (2000).

1999 (1)

M. Dumont and A. El Osman, “On spontaneous and photoinduced orientational mobility of dye molecules in polymer films,” Chem. Phys. 245, 437-462 (1999).
[CrossRef]

1998 (1)

1997 (2)

S. Brasselet and J. Zyss, “Control of the polarization dependence of optically poled nonlinear polymer films,” Opt. Lett. 19, 1464-1466 (1997).
[CrossRef]

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

1996 (2)

S. Hosotte and M. Dumont, “Orientational relaxation of photomerocyanine molecules in polymeric films,” Synth. Met. 81, 125-127 (1996).
[CrossRef]

S. Hosotte and M. Dumont, “Photoassisted poling and orientational relaxation of dye molecules in polymers. The case of spiropyran,” Proc. SPIE 2852, 53-63 (1996).
[CrossRef]

1995 (3)

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (1995).

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).

Z. Sekkat and W. Knoll, “Creation of second-order nonlinear optical effects by photoisomerization of polar azo dyes in polymeric fields: theoretical study of steady-state and transient properties,” J. Opt. Soc. Am. B 12, 1855-1867 (1995).
[CrossRef]

1994 (1)

1993 (4)

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecules angular mobility,” Synth. Met. 54, 373-381 (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, 349-354 (1993).
[CrossRef]

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiart, “Light-induced second-harmonic generation in azo-dye polymers,” Opt. Lett. 18, 941-943 (1993).
[CrossRef] [PubMed]

1992 (4)

Z. Sekkat and M. Dumont, “Photoassisted poling of azodye doped polymeric films at room temperature,” Appl. Phys. B 54, 486-489 (1992).
[CrossRef]

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerisation of azodye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 2, 359-362 (1992).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azodyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188-199 (1992).
[CrossRef]

M. Canva, G. Le Saux, P. Geoges, A. Brun, F. Chaput, and J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218-220 (1992).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

1978 (1)

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

1976 (1)

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

1971 (2)

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

1963 (2)

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23-39 (1963).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

1951 (1)

H. Benoit, “Contribution à l'étude de l"effet Kerr présenté par les solutions diluées de macromolécules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

1921 (1)

F. Weigert, “Uber einen neuen Effekt der Strahlung,” Naturwiss. 9, 30 (1921).
[CrossRef]

1919 (1)

F. Weigert, “Uber Einen Neuen Effect der Strahling in Lichtempfindlichen Schichten,” Verh. Dtsch. Phys. Ges. 21, 479-483 (1919).

Arfken, G.

G. Arfken, Mathematical Methods for Physics (Academic, 1985).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Benoit, H.

H. Benoit, “Contribution à l'étude de l"effet Kerr présenté par les solutions diluées de macromolécules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

Boilot, J. P.

Brasselet, S.

S. Brasselet and J. Zyss, “Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media,” J. Opt. Soc. Am. B 15, 257-288 (1998).
[CrossRef]

S. Brasselet and J. Zyss, “Control of the polarization dependence of optically poled nonlinear polymer films,” Opt. Lett. 19, 1464-1466 (1997).
[CrossRef]

S. Brasselet, “Processus multipolaires en optique non-linéaire dans les miliuex moléculaires,” Thèse de Doctorat (Université Paris 11, Orsay, 1997).

Brun, A.

Canva, M.

Chaput, F.

Charra, F.

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (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 12, 2347-2358 (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, 941-943 (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, 349-354 (1993).
[CrossRef]

Chen, Y.

Delaire, J. A.

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

Ducloy, M.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Dumont, M.

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

M. Dumont, “New developments in optical ordering of, NLO dyes in polymers,” Proc. SPIE 4461, 179-163 (2001).

M. Dumont, “Modelization of the angular redistribution in optical ordering processes in dye containing polymers,” Nonlinear Opt. 25, 195-200 (2000).

M. Dumont and A. El Osman, “On spontaneous and photoinduced orientational mobility of dye molecules in polymer films,” Chem. Phys. 245, 437-462 (1999).
[CrossRef]

S. Hosotte and M. Dumont, “Photoassisted poling and orientational relaxation of dye molecules in polymers. The case of spiropyran,” Proc. SPIE 2852, 53-63 (1996).
[CrossRef]

S. Hosotte and M. Dumont, “Orientational relaxation of photomerocyanine molecules in polymeric films,” Synth. Met. 81, 125-127 (1996).
[CrossRef]

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecules angular mobility,” Synth. Met. 54, 373-381 (1993).
[CrossRef]

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerisation of azodye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 2, 359-362 (1992).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azodyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188-199 (1992).
[CrossRef]

Z. Sekkat and M. Dumont, “Photoassisted poling of azodye doped polymeric films at room temperature,” Appl. Phys. B 54, 486-489 (1992).
[CrossRef]

M. Dumont, is preparing a paper to be called “Dynamics of all-optical poling of photoisomerizable molecules. II. Comparison of angular redistribution models.”

M. Dumont, “Théorie du pompage optique avec un laser, étude expérimentale dans le cas du Néon de las réponse linéaire et de quelques effets de saturation,” Thèse de Doctorat d'état ès Sciences Physiques (Faculté des Sciences de Paris, 1971), p. 43, http://tel.archives-ouvertes.fr/docs/00/06/09/12/PDF/1971DUMONT.pdf.

El Osman, A.

M. Dumont and A. El Osman, “On spontaneous and photoinduced orientational mobility of dye molecules in polymer films,” Chem. Phys. 245, 437-462 (1999).
[CrossRef]

Etilé, A. C.

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

Feld, M.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

Fiorini, C.

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (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 12, 2347-2358 (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, 349-354 (1993).
[CrossRef]

C. Fiorini, “Propriétés optiques non-linéaires du second ordre induites par voie optique dans les milieux moléculaires,” Thèse de Doctorat (Université Paris 11, Orsay, 1995).

Franken, P. A.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23-39 (1963).
[CrossRef]

Froc, G.

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

Geoges, P.

Gustafson, T. K.

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

Hosotte, S.

S. Hosotte and M. Dumont, “Photoassisted poling and orientational relaxation of dye molecules in polymers. The case of spiropyran,” Proc. SPIE 2852, 53-63 (1996).
[CrossRef]

S. Hosotte and M. Dumont, “Orientational relaxation of photomerocyanine molecules in polymeric films,” Synth. Met. 81, 125-127 (1996).
[CrossRef]

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

Idiart, E.

Kajzar, F.

Knoll, W.

Le Saux, G.

Leite, J. R. R.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

Li, Q.

Luo, S.

Makushenko, A. M.

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

Margulis, W.

Messiah, A.

A. Messiah, Mécanique Quantique (Dunod, 1960), Vol. II.

Mizrahi, V.

Nakatani, K.

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

Neporent, B. S.

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

Nguyen Thi Kim, N.

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

Nunzi, J. M.

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (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 12, 2347-2358 (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, 941-943 (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, 349-354 (1993).
[CrossRef]

Osterberg, U.

Österberg, U.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Raimond, P.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (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, 941-943 (1993).
[CrossRef] [PubMed]

Sekkat, Z.

Z. Sekkat and W. Knoll, “Creation of second-order nonlinear optical effects by photoisomerization of polar azo dyes in polymeric fields: theoretical study of steady-state and transient properties,” J. Opt. Soc. Am. B 12, 1855-1867 (1995).
[CrossRef]

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecules angular mobility,” Synth. Met. 54, 373-381 (1993).
[CrossRef]

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azodyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188-199 (1992).
[CrossRef]

Z. Sekkat and M. Dumont, “Photoassisted poling of azodye doped polymeric films at room temperature,” Appl. Phys. B 54, 486-489 (1992).
[CrossRef]

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerisation of azodye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 2, 359-362 (1992).

Sharma, R. M.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

Sheffield, R. L.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2003).

Sipe, J. E.

Stegeman, G. I.

Stolbova, O. V.

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

Stolen, R. H.

Tom, H. W. K.

Ward, J. F.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23-39 (1963).
[CrossRef]

Weigert, F.

F. Weigert, “Uber einen neuen Effekt der Strahlung,” Naturwiss. 9, 30 (1921).
[CrossRef]

F. Weigert, “Uber Einen Neuen Effect der Strahling in Lichtempfindlichen Schichten,” Verh. Dtsch. Phys. Ges. 21, 479-483 (1919).

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yee, T. K.

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yu, X.

Zhong, X.

Zyss, J.

S. Brasselet and J. Zyss, “Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media,” J. Opt. Soc. Am. B 15, 257-288 (1998).
[CrossRef]

S. Brasselet and J. Zyss, “Control of the polarization dependence of optically poled nonlinear polymer films,” Opt. Lett. 19, 1464-1466 (1997).
[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, 349-354 (1993).
[CrossRef]

Ann. Phys. (Paris) (1)

H. Benoit, “Contribution à l'étude de l"effet Kerr présenté par les solutions diluées de macromolécules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Appl. Phys. B (1)

Z. Sekkat and M. Dumont, “Photoassisted poling of azodye doped polymeric films at room temperature,” Appl. Phys. B 54, 486-489 (1992).
[CrossRef]

Chem. Phys. (1)

M. Dumont and A. El Osman, “On spontaneous and photoinduced orientational mobility of dye molecules in polymer films,” Chem. Phys. 245, 437-462 (1999).
[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, 349-354 (1993).
[CrossRef]

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

Mol. Cryst. Liq. Cryst. (1)

N. Nguyen Thi Kim, M. Dumont, J. A. Delaire, and K. Nakatani, “Orientation of azo-dye molecules in polymer films, via photoisomerization: dichroism measurements and second harmonic generation,” Mol. Cryst. Liq. Cryst. 430, 249-256 (2005).
[CrossRef]

Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. (1)

Z. Sekkat and M. Dumont, “Poling of polymer films by photoisomerisation of azodye chromophores,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 2, 359-362 (1992).

Naturwiss. (1)

F. Weigert, “Uber einen neuen Effekt der Strahlung,” Naturwiss. 9, 30 (1921).
[CrossRef]

Nonlinear Opt. (3)

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced noncentrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339-347 (1995).

M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327-338 (1995).

M. Dumont, “Modelization of the angular redistribution in optical ordering processes in dye containing polymers,” Nonlinear Opt. 25, 195-200 (2000).

Opt. Lett. (7)

Opt. Spectrosc. (3)

B. S. Neporent and O. V. Stolbova, “Reversible orientation photochromism in viscous solutions of complex organic substances,” Opt. Spectrosc. 14, 331-336 (1963) B. S. Neporent and O. V. Stolbova[Opt. Spektrosk. 14, 624-633 (1963) (in Russian)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. I. Model of the system,” Opt. Spectrosc. 31, 295-299 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 557-564 (1971)].

A. M. Makushenko, B. S. Neporent, and O. V. Stolbova, “Reversible orientation photochromism and photoisomerization of aromatic azo compound. II. Azobenzene and substituted azobenzene derivatives,” Opt. Spectrosc. 31, 397-401 (1971) A. M. Makushenko, B. S. Neporent, and O. V. Stolbova[Opt. Spektrosk. 31, 741-748 (1971)].

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction of light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Phys. Rev. A (3)

A. C. Etilé, C. Fiorini, F. Charra, and J. M. Nunzi, “Phase-coherent control of the molecular polar order using dual-frequency interferences between circularly polarized beams,” Phys. Rev. A 56, 3888-3896 (1997).
[CrossRef]

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. M. Sharma, and M. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151-1168 (1976).
[CrossRef]

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

Proc. SPIE (4)

M. Dumont, S. Hosotte, G. Froc, and Z. Sekkat, “Orientational manipulation of chromophores through photoisomerization,” Proc. SPIE 2042, 2-13 (1993).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azodyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188-199 (1992).
[CrossRef]

M. Dumont, “New developments in optical ordering of, NLO dyes in polymers,” Proc. SPIE 4461, 179-163 (2001).

S. Hosotte and M. Dumont, “Photoassisted poling and orientational relaxation of dye molecules in polymers. The case of spiropyran,” Proc. SPIE 2852, 53-63 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23-39 (1963).
[CrossRef]

Synth. Met. (2)

Z. Sekkat and M. Dumont, “Photoinduced orientation of azo dyes in polymeric films. Characterization of molecules angular mobility,” Synth. Met. 54, 373-381 (1993).
[CrossRef]

S. Hosotte and M. Dumont, “Orientational relaxation of photomerocyanine molecules in polymeric films,” Synth. Met. 81, 125-127 (1996).
[CrossRef]

Verh. Dtsch. Phys. Ges. (1)

F. Weigert, “Uber Einen Neuen Effect der Strahling in Lichtempfindlichen Schichten,” Verh. Dtsch. Phys. Ges. 21, 479-483 (1919).

Other (12)

C. Fiorini, “Propriétés optiques non-linéaires du second ordre induites par voie optique dans les milieux moléculaires,” Thèse de Doctorat (Université Paris 11, Orsay, 1995).

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

S. Brasselet, “Processus multipolaires en optique non-linéaire dans les miliuex moléculaires,” Thèse de Doctorat (Université Paris 11, Orsay, 1997).

As it appears in experiments and as it will be shown in the second paper two types of saturation are to be considered: the saturation of, AHB (which exists as soon as τCPr(Ω) is not much smaller than 1, even if there is no AR) and the saturation of, AR in the photoisomerization cycle, which appears for a much smaller pumping intensity (since, AR is in competition with diffusion in T, which is often negligible), but generally after longer times (depending of the average rotation in each cycle).

The formal expression, ∑m′(−)m′AT,m′a,j(n+1)ST,−m′a′,j(p), looks like the scalar product ATa,J(n+1)⋅STa′,J(p), but it is not, since these two tensors are orthogonal if a≠a′.

M. Dumont, is preparing a paper to be called “Dynamics of all-optical poling of photoisomerizable molecules. II. Comparison of angular redistribution models.”

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2003).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

A. Messiah, Mécanique Quantique (Dunod, 1960), Vol. II.

G. Arfken, Mathematical Methods for Physics (Academic, 1985).

M. Dumont, “Théorie du pompage optique avec un laser, étude expérimentale dans le cas du Néon de las réponse linéaire et de quelques effets de saturation,” Thèse de Doctorat d'état ès Sciences Physiques (Faculté des Sciences de Paris, 1971), p. 43, http://tel.archives-ouvertes.fr/docs/00/06/09/12/PDF/1971DUMONT.pdf.

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Figures (8)

Fig. 1
Fig. 1

Diagram of levels and transitions. The lifetime of excited bands is very short and their population is negligible. In the present paper, the optical excitation of C is considered in the formal equations, but ignored, as nonresonant, in their resolution.

Fig. 2
Fig. 2

Time evolution of the Cartesian (curves with markers at calculated points) and of the spherical (without markers) components of χ ( 1 ) (dashed lines, left axis) and χ ( 2 ) (solid lines, right axis), for rodlike molecules, with different writing fields polarizations (according to Table 2). Parameters are: A = S ( 2 ) = 0.05 , B = S ( 3 ) = 10 5 , G = S ( 4 ) = 2.10 8 , τ C = 10 , I 2 ω = 1 , I ω = 1000 , 1 τ D , T = 0 . The reading susceptibility tensors are proportional to the writing ones and normalized ( A ( n ) = 1 ) . χ 0 ( 1 ) 0 is not drawn as it is proportional to the total population N T . For Z Z and R–L writing fields, horizontal dashed lines indicate the stationary solution obtained from Eq. (27): they clearly appear as the asymptotes of the transient solutions for the Cartesian components. The X X case has not been represented as it is simply obtained from the Z Z case, by rotation of axes. Let us notice the change of sign of χ 0 ( 2 ) 3 in the Z Z case and of χ 1 ( 2 ) 3 in the X Y and R–R cases.

Fig. 3
Fig. 3

Time evolution of the Cartesian and of the spherical components of χ ( 1 ) and χ ( 2 ) , for plane octupolar molecules and isotropic octupolar molecules, with Z Z and X Y writing fields polarizations. Drawing conventions and parameters are the same as in Fig. 2. With plane molecules and Z Z fields, the growing of anisotropy, the increase of N T for t > 3 τ C , and the decrease of χ 0 ( 2 ) 3 for t > 7 τ C , are due to the gathering of molecules in a centrosymmetric arrangement in the { x , y } plane. With plane molecules and X Y fields, the difference between χ x x ( 1 ) and χ y y ( 1 ) is due to the anisotropy of the two-photon excitation.

Fig. 4
Fig. 4

Polar representation of the second-harmonic intensity as a function of the polarization angle, ϕ, of the reading beam at ω. Upper row: I x 2 ω and I y 2 ω , measured through X and Y analyzers, respectively. Lower row: I tot 2 ω measured without an analyzer and I R A 2 ω measured through an analyzer rotating with the reading polarization. In each diagram the largest signal is normalized to one. The first column is calculated with χ 1 ( 2 ) 1 (or χ 1 ( 2 ) 3 ): these diagrams are obtained at any time in the case of R–R pumping. ( χ 3 ( 2 ) 3 = 0 ) . The diagrams of the second column correspond to χ 3 ( 2 ) 3 alone and are observed at any time for R–L excitation (if β J = 3 0 ). In this case, I tot 2 ω does not depend on ϕ (polarization independent SHG, as emphasized in [22]). The third column corresponds to octupolar molecules ( β J = 1 = 0 ) , with X X fields: it is valid at any time, since χ 3 ( 2 ) 3 χ 1 ( 2 ) 3 = 5 3 is constant. The fourth column represents the X Y pumping of octupolar molecules, at the first perturbation order: except for isotropic octupolar molecules, these diagrams are modified by saturation, which changes the ratio χ 3 ( 2 ) 3 χ 1 ( 2 ) 3 . All diagrams have been drawn with Δ Φ = φ 2 ω 2 φ ω = 0 . For circular polarizations, this means that both fields are parallel to X, at t = 0 . If Δ Φ 0 , χ ( 2 ) is rotated by Δ Φ for R–R pumping and by Δ Φ 3 for R–L pumping: all diagrams are rotated except I x 2 ω and I y 2 ω , in the R–R case, which are distorted. For X X and X Y , the shape of diagrams does not change, but the intensity of SHG is multiplied by cos 2 Δ Φ .

Fig. 5
Fig. 5

Polar diagrams representing I x 2 ω and I y 2 ω (upper graphs) and I tot 2 ω and I R A 2 ω (lower graphs), with X X writing fields, for rodlike molecules. The left-hand diagrams correspond to the first order, Eq. (22), while the right-hand ones are calculated from the stationary solution, Eq. (27).

Fig. 6
Fig. 6

Same as Fig. 5 for rodlike molecules with X Y writing fields.

Fig. 7
Fig. 7

B 3 B 1 (or B 1 B 3 ), as a function of time. At the first order [Eq. (22)], B 1 and B 3 are equal to β J = 1 2 and β J = 3 2 , respectively. Here they are used as fitting parameters, when Eq. (35) is reported in the expressions (32) of I x 2 ω ( ϕ ) and I y 2 ω ( ϕ ) . For each t value these expressions are fitted to the curves, I x 2 ω ( ϕ ) and I y 2 ω ( ϕ ) , obtained by using the computed values of χ m ( 2 ) J ( t ) shown in Figs. 2, 3. The procedure is the same as that used in [22, 23] for fitting the theoretical expressions to experimental measurements. As expected, the first point ( t = Δ t : first order) gives B 3 B 1 = β J = 3 2 β J = 1 2 = 2 3 , for rodlike molecules, and B 1 = β J = 1 2 = 0 , for octupolar molecules. With time, these quantities vary: B 3 B 1 changes sign for rodlike molecules with X X fields, while B 1 no longer vanishes for plane octupolar molecules, with X Y fields.

Fig. 8
Fig. 8

Double-Feynmann diagrams representing the resonant terms in 2 ω ( 2 ) (a.1 and a.2, with m and n 0 ) and in ω ( 2 ) (b.1 and b.2, with m 0 ), as in Eqs. (A17, A19), respectively. Each curve represents one interaction with light, which adds a new frequency and a new denominator. If, after the interaction p, the matrix element is ρ m n ( p ) ( ω p ) , the new denominator is L m n i ω p and ω p is the sum of all the frequencies from the p interactions. With some conventions, it is possible to write exactly each term [39, 43, 44]. All the above diagrams contain the resonant denominator L m 0 2 i ω (from the second order, in a.1 and a.2 and from the first order in b.1 and b.2).

Tables (2)

Tables Icon

Table 1 Tensorial Components of Molecular Susceptibilities a

Tables Icon

Table 2 Field Tensors and Nonvanishing Tensorial Components of the Susceptibilities a

Equations (108)

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

n ( Ω ) = N 8 π 2 Δ t Pr ( Ω ) ,
d n T ( Ω ) d t = Φ T C Pr T ( Ω ) n T ( Ω ) + Φ C T R C T P ( Ω Ω ) Pr C ( Ω ) n C ( Ω ) d Ω + 1 τ C R C T R ( Ω Ω ) n C ( Ω ) d Ω + ( d n T ( Ω ) d t ) Diff ,
d n C ( Ω ) d t = Φ T C R T C P ( Ω Ω ) Pr T ( Ω ) n T ( Ω ) d Ω Φ C T Pr C ( Ω ) n C ( Ω ) 1 τ C n C ( Ω ) + ( d n C ( Ω ) d t ) Diff ,
[ n T ( Ω ) + n C ( Ω ) ] d Ω = N T + N C = N ,
R T C P ( Ω Ω ) d Ω = 1 ,
R T C P ( Ω Ω ) d Ω = 1 ,
Pr ( Ω ) = 1 2 Im { α Ω ( 2 ω ) } Re { E 2 ω * E 2 ω } + 1 Im { β Ω ( 2 ω = ω + ω ) } Re { E 2 ω * E ω E ω } + 1 4 Im { γ Ω ( ω = ω + ω ω ) } Re { E ω * E ω * E ω E ω } .
A ( 2 ) = α , A ( 3 ) = β , A ( 4 ) = γ ,
S ( 2 ) = 1 2 Im { α ( 2 ω ) } ,
S ( 3 ) = 1 Im { β ( 2 ω ) } ,
S ( 4 ) = 1 4 Im { γ ( ω ) } ,
E ( 2 ) = Re { E 2 ω * E 2 ω } , E ( 3 ) = Re { E 2 ω * E ω E ω } , E ( 4 ) = Re { E ω * E ω * E ω E ω } ,
S Ω , i 1 i n ( n ) = I 1 I n R i 1 I 1 ( Ω ) R i n I n ( Ω ) S I 1 I n ( n ) ,
S ( n ) = a , J n m = J + J S m a , J ( n ) B Ω , m a , J ( n ) ,
E ( n ) = a , J n m = J + J E m a , J ( n ) B m a , J ( n ) .
R Ω ( B m a , J ) = B Ω , m a , J = m = J J B m a , J D m m J ( Ω ) ,
S Ω , m a , J ( n ) = m = J J D m m J ( Ω ) S m a , J ( n ) .
Pr ( Ω ) = n S Ω ( n ) E ( n )
= n , i 1 i n I 1 I n R i 1 I 1 ( Ω ) R i n I n ( Ω ) S I 1 I n ( n ) E i 1 i n ( n ) ,
Pr ( Ω ) = n , a , J , m ( ) m S Ω , m a , J ( n ) E m a , J ( n ) = n , a , J , m , m ( ) m D m m J ( Ω ) S m a , J ( n ) E m a , J ( n ) .
n T ( Ω ) = J , m , m 2 J + 1 8 π 2 T J m m D m m J ( Ω ) , n C ( Ω ) = J , m , m 2 J + 1 8 π 2 C J m m D m m J ( Ω ) ,
T J m m = n T ( Ω ) D m m J * ( Ω ) d Ω , C J m m = n C ( Ω ) D m m J * ( Ω ) d Ω .
χ T , i 1 i n ( n 1 ) = I 1 I n A I 1 I n ( n ) R i 1 I 1 ( Ω ) R i n I n ( Ω ) n T ( Ω ) d Ω ,
χ T , m ( n ) a , J = m A T , m a , J ( n + 1 ) D m m J ( Ω ) n T ( Ω ) d Ω = m ( ) m m T J , m , m A T , m a , J ( n + 1 ) ,
d T J m m d t = L n n T L n n P J m m L n n T + L n n m C L n n P J m m L n n C R C T , m m P , J + m ( 1 τ C C J m m R C T , m m R , J T J m m d T , m m J ) ,
d C J m m d t = L n n m T L n n P J m m L n n T R T C , m m P , J L n n C L n n P J m m L n n C 1 τ C C J m m m C J m m d C , m m J ,
P J m m L n n T = Φ T C n , a , K , q , q ( ) q + m m ( 2 L + 1 ) ( L K J n q m ) ( L K J n q m ) S T , q a , K ( p ) E q a , K ( p ) ,
P J m m L n n C = Φ C T n , a , K , q , q ( ) q + m m ( 2 L + 1 ) ( L K J n q m ) ( L K J n q m ) S C , q a , K ( p ) E q a , K ( p ) ,
( d T J m m d t ) Exc. ( 1 ) = P J m m 000 T N = Φ T C N 2 J + 1 p , a ( ) m S T , m a , J ( p ) E m a , J ( p ) .
C J m m ( 1 ) = P J m m 000 T R T C P , J N τ C 1 + τ C d C J ,
T J m m ( 1 ) = P J m m 000 T N d T J [ 1 R T C P , J R C T R , J 1 + τ C d C J ] .
Δ ( 1 ) χ T , m ( n ) a , J = Φ T C N 2 J + 1 m ( ) m A T , m a , J ( n + 1 ) p , a S T , m a , J ( p ) E m a , J ( p ) Δ t .
Δ ( 1 ) χ T , m ( 2 ) J ( ω r ) = 1 Φ T C N 2 J + 1 n , a ( β T J ( ω r ) ( Im { β T ( 2 ω ) } ) J ) E m s s , J ( 3 ) Δ t , J = 1 , 3 .
( d T J m m d t ) Exc. ( 2 ) = K 1 , q 1 , q 1 P J m m K 1 q 1 q 1 T T K 1 q 1 q 1 ( 1 ) p 1 , a 1 , K 1 , q 1 , q 1 p 2 , a 2 , K 2 , q 2 , q 2 ( ) m ( K 1 K 2 J q 1 q 2 m ) ( K 1 K 2 J q 1 q 2 m ) S T , q 2 a 2 , K 2 ( p 2 ) E q 2 a 2 , K 2 ( p 2 ) S T , q 1 a 1 , K 1 ( p 1 ) E q 1 a 1 , K 1 ( p 1 ) 1 2 J + 1 p 1 , a 1 , K 1 p 2 , a 2 , K 2 ( ) m ( S a 1 K 1 ( p 1 ) S a 2 K 2 ( p 2 ) ) m J ( E a 2 K 2 ( p 2 ) E a 1 K 1 ( p 1 ) ) m J .
Δ ( 2 ) χ T , m ( n ) a , J 1 2 J + 1 p 1 , a 1 , K 1 , m p 2 , a 2 , K 2 ( ) m A m a , J ( n + 1 ) ( S a 1 K 1 ( p 1 ) S a 2 K 2 ( p 2 ) ) m J ( E a 2 K 2 ( p 2 ) E a 1 K 1 ( p 1 ) ) m J .
Δ ( 3 ) χ T , m ( n ) a , J 1 2 J + 1 p 1 , a 1 K 1 , p 2 , a 2 , m K 2 , p 3 , a 3 , K 3 , L ( ) m A m a , J ( n + 1 ) ( ( S a 1 K 1 ( p 1 ) S a 2 K 2 ( p 2 ) ) L S a 3 K 3 ( p 3 ) ) m J [ E a 3 K 3 ( p 3 ) ( E a 2 K 2 ( p 2 ) E a 1 K 1 ( p 1 ) ) L ] m J .
0 = Pr T ( Ω ) n T ( Ω ) + τ C 1 R C T R ( Ω Ω ) n C ( Ω ) d Ω ,
0 = R T C P ( Ω Ω ) Pr T ( Ω ) n T ( Ω ) d Ω 1 τ C n C ( Ω ) + ( d n C ( Ω ) d t ) Diff .
n C ( Ω ) = N C ( 8 π 2 ) 1 , n T ( Ω ) = N C ( 8 π 2 τ C Pr T ( Ω ) ) 1 ,
N C = N [ 1 + ( 8 π 2 τ C Pr T ( Ω ) ) 1 d Ω ] 1 .
n T ( Ω ) = K Pr T ( Ω ) 1 , n C ( Ω ) = K Pr C ( Ω ) 1 , K = N [ ( Pr T ( Ω ) 1 + Pr C ( Ω ) 1 ) d Ω ] 1 .
n T ( Ω , t ) t = Pr T ( Ω ) n T ( Ω , t ) + 1 8 π 2 τ C N C + 1 τ D , T ( n T ( Ω , t ) + 1 8 π 2 N T ( t ) ) ,
N C ( t ) t = Pr T ( Ω ) n T ( Ω , t ) d Ω 1 τ C N C ( t ) .
p ( n ) ( Ω ) = i 1 i n , I 1 I n R i 1 I 1 ( Ω ) R i n I n ( Ω ) s I 1 I n ( n ) e i 1 i n ( n ) ,
Pr ( Ω ) = A I 2 ω p ( 2 ) ( Ω ) + B I ω I 2 ω p ( 3 ) ( Ω ) + G I ω 2 p ( 4 ) ( Ω ) .
I x 2 ω P x 2 = [ χ x x x ( 2 ) E x 2 + χ x y y ( 2 ) E y 2 ] 2 = [ 1 30 ( χ 1 ( 2 ) 3 2 χ 1 ( 2 ) 1 ) ( 3 cos 2 ϕ + sin 2 ϕ ) 1 2 χ 3 ( 2 ) 3 cos 2 ϕ ] 2 ,
I y 2 ω P y 2 = [ 2 χ y x y ( 2 ) E x E y ] 2 = [ 1 30 ( χ 1 ( 2 ) 3 2 χ 1 ( 2 ) 1 ) + 1 2 χ 3 ( 2 ) 3 ] 2 sin 2 2 ϕ .
I tot 2 ω = I x 2 ω + I y 2 ω ,
I R A 2 ω = [ P x cos ϕ + P y sin ϕ ] 2 .
χ m ( 2 ) 1 = 1 3 B 1 E m 1 ( 3 ) , χ m ( 2 ) 3 = 1 7 B 3 E m 3 ( 3 ) .
χ ( 2 ) J = 3 χ ( 2 ) J = 1 = 3 7 B 3 B 1 ( e J = 3 ( 3 ) e J = 1 ( 3 ) ) X X = 6 7 B 3 B 1 .
E ω = Re ( E ω e i ω t ) , E 2 ω = Re ( E 2 ω e 2 i ω t ) .
ψ ( t ) = 0 + a ( 1 ) ( t ) 0 + b ( 1 ) ( t ) 1 + a ( 2 ) ( t ) 0 + b ( 2 ) ( t ) 1 ,
Pr ( 0 1 ) = 1 ψ ( t ) 2 ω 10 = b ( 1 ) 2 + 2 Re ( b ( 1 ) * b ( 2 ) ) + b ( 2 ) 2 ω 10 ,
d 4 W d t d 3 r = E d P d t t = 1 4 E * d P d t t + c.c. ,
P = n T ( Ω ) ( Ω ) d Ω ,
d 4 W d t d 3 r = ( d 4 W d t d 3 r ) ω + ( d 4 W d t d 3 r ) 2 ω = 1 2 Re ( i ω E ω * P ω ) + 1 2 Re ( 2 i ω E 2 ω * P 2 ω ) ,
2 ω ( 1 ) = α Ω ( 2 ω ) E 2 ω e 2 i ω t ,
ω ( 3 ) = γ Ω ( ω = ω ω + ω ) ( E ω * E ω E ω ) e i ω t ,
2 ω ( 2 ) = β Ω ( 2 ω = ω + ω ) : ( E ω E ω ) e 2 i ω t ,
ω ( 2 ) = 2 β Ω ( ω = 2 ω ω ) : ( E ω * E 2 ω ) e i ω t .
Pr ( 1 ) ( Ω ) = ( 4 ω ) 1 Re ( 2 i ω E 2 ω * 2 ω ( 1 ) ) = ( 4 ) 1 Re [ i α Ω i j ( 2 ω ) E 2 ω i * E 2 ω j + i α Ω j i * ( 2 ω ) E 2 ω j E 2 ω i * ] = ( 2 ) 1 Im [ α Ω i j ( 2 ω ) ] Re [ E 2 ω i * E 2 ω j ] .
Pr ( 2 ) ( Ω ) = ( 4 ) 1 Im [ ( 2 β Ω i j k ( 2 ω = ω + ω ) 2 β Ω k i j * ( ω = 2 ω ω ) ) E 2 ω i * E ω j E ω k ] .
β Ω i j k ( 2 ω = ω + ω ) = β Ω k i j * ( ω = 2 ω ω ) .
ρ ( p + 1 ) t = 1 i [ H 0 , ρ ( p + 1 ) ] 1 i [ μ E , ρ ( p ) ] ( ρ ( p + 1 ) t ) Relax ,
ρ m n ( p + 1 ) t = L m n ρ m n ( p + 1 ) + q , r i q ( μ m r ρ r n ( p ) ρ m r ( p ) μ r n ) e i ω q t ( m n ) ,
ρ m m ( p + 1 ) t = γ m ρ m m ( p + 1 ) + q , r i q ( μ m r ρ r m ( p ) ρ m r ( p ) μ r m ) e i ω q t ( m 0 ) ,
ρ 00 ( p + 1 ) t = q , r i q ( μ 0 r ρ r 0 ( p ) ρ 0 r ( p ) μ r 0 ) e i ω q t .
ω q = ω p ,
ω p ( p ) e i ω p t = 2 ( m , r μ m r ρ r m ( p ) ) ω p e i ω p t = A ω p ( p ) q = 1 p E ω q e i ω q t .
ρ m 0 ( 1 ) = ρ 0 m ( 1 ) * = ω 1 i ( q 1 μ m 0 ) L m 0 i ω 1 ( e i ω 1 t e L m 0 t ) t Γ m 0 1 ω 1 i ( q 1 μ m 0 ) L m 0 i ω 1 e i ω 1 t .
1 2 ω 1 ( 1 ) = m ( μ 0 m ρ m 0 ( 1 ) + μ m 0 ρ 0 m ( 1 ) ) ω 1 = m ( i μ 0 m ( q 1 μ m 0 ) Γ m 0 + i ( ω m 0 ω 1 ) i μ m 0 ( q 1 μ 0 m ) Γ m 0 i ( ω m 0 + ω 1 ) ) .
α i j ( 2 ω ) = 1 m μ 0 m i μ m 0 j ( ω m 0 2 ω ) i Γ m 0 ,
Im ( α i j ( 2 ω ) ) = 1 m Γ m 0 μ 0 m i μ m 0 j ( ω m 0 2 ω ) 2 + Γ m 0 2 .
( ρ 00 ( 2 ) t ) ω = 0 = m 0 ( ρ m m ( 2 ) t ) ω = 0 = 1 4 2 m 0 ( ( E 2 ω μ 0 m ) ( E 2 ω μ m 0 ) L m 0 2 i ω + ( E 2 ω μ 0 m ) ( E 2 ω μ m 0 ) L m 0 * + 2 i ω ) = 1 2 2 m 0 Γ m 0 μ 0 m i μ m 0 j ( ω m 0 2 ω ) 2 + Γ m 0 2 Re [ E 2 ω i * E 2 ω j ] = 1 2 Im [ α i j ( 2 ω ) ] Re [ E 2 ω i * E 2 ω j ] .
ρ m n ( 2 ) = ω 1 , ω 2 e i ( ω 1 + ω 2 ) t e L m n t L m n i ( ω 1 + ω 2 ) ( ( q 2 μ m 0 ) ( q 1 μ 0 n ) L n 0 * i ω 1 + ( q 1 μ m 0 ) ( q 2 μ 0 n ) L m 0 i ω 1 ) ,
ρ m 0 ( 2 ) = ρ 0 m ( 2 ) * = ω 1 , ω 2 e i ( ω 1 + ω 2 ) t L m 0 i ( ω 1 + ω 2 ) ( ( q 1 μ m 0 ) ( q 2 μ 00 ) L m 0 i ω 1 n 0 ( q 2 μ m n ) ( q 1 μ n 0 ) L n 0 i ω 1 ) ,
ρ 00 ( 2 ) = ω 1 , ω 2 m 0 1 e i ( ω 1 + ω 2 ) t i ( ω 1 + ω 2 ) ( ( q 2 μ 0 m ) ( q 1 μ m 0 ) L m 0 i ω 1 + ( q 1 μ 0 m ) ( q 2 μ m 0 ) L m 0 * i ω 1 ) ,
1 2 ω 3 = ω 1 + ω 2 ( 2 ) = n , m ω 1 + ω 2 = ω 3 ( μ n m ρ m n ( 2 ) ) = n 0 , m ω 1 + ω 2 = ω 3 { μ n m ( q 2 μ m 0 ) ( q 1 μ 0 n ) ( L m n i ω 3 ) ( L n 0 * i ω 1 ) + μ m n ( q 1 μ n 0 ) ( q 2 μ 0 m ) ( L n m i ω 3 ) ( L n 0 i ω 1 ) μ 0 m ( q 2 μ m n ) ( q 1 μ n 0 ) ( L m 0 i ω 3 ) ( L n 0 i ω 1 ) μ m 0 ( q 1 μ 0 n ) ( q 2 μ n m ) ( L m 0 * i ω 3 ) ( L n 0 * i ω 1 ) } .
2 ω ( 2 ) = 1 2 2 m 0 μ 0 m L m 0 2 i ω ( ( E ω μ m 0 ) ( E ω μ 00 ) L m 0 i ω n 0 ( E ω μ m n ) ( E ω μ n 0 ) L n 0 i ω ) ,
β i j k ( 2 ω = ω + ω ) = 1 2 2 m 0 μ 0 m i ω m 0 2 ω i Γ m 0 ( μ m 0 k μ 00 j + μ m 0 j μ 00 k 2 ( ω m 0 ω ) n 0 μ m n k μ n 0 j + μ m n j μ n 0 k 2 ( ω n 0 ω ) ) .
ω = 2 ω ω ( 2 ) = 1 2 2 m 0 { μ 0 m ( E 2 ω μ m 0 ) ( E ω * μ 00 ) + μ 00 ( E ω * μ 0 m ) ( E 2 ω μ m 0 ) ( ω m 0 2 ω i Γ n 0 ) ( ω m 0 ω ) n 0 μ n m ( E 2 ω μ m 0 ) ( E ω * μ 0 n ) + μ 0 n ( E ω * μ n m ) ( E 2 ω μ m 0 ) ( ω m 0 2 ω i Γ m 0 ) ( ω n 0 ω ) } = 2 β k i j ( ω = 2 ω ω ) E 2 ω i E ω j * ,
β k i j ( ω = 2 ω ω ) = 1 4 2 m 0 μ m 0 i ω m 0 2 ω i Γ m 0 ( μ 00 j μ 0 m k + μ 00 k μ 0 m j ω m 0 ω n 0 ( μ 0 n j μ n m k + μ 0 n k μ n m j ω n 0 ω ) ) .
Pr ( 2 ) ( Ω ) = 1 2 3 m 0 Γ m 0 ( ω m 0 2 ω ) 2 + Γ m 0 2 Re { E 2 ω i * E ω j E ω k ( n 0 ( μ 0 m i μ m n j μ n 0 k ( ω n 0 ω ) ) μ 0 m i μ m 0 j μ 00 k ( ω m 0 ω ) ) } .
Pr ( 2 ) ( Ω ) = Γ 2 3 μ 01 i μ 10 j ( μ 11 k μ 00 k ) ( ω 10 ω ) ( ( ω 10 2 ω ) 2 + Γ 2 ) Re { E 2 ω i * E ω j E ω k } .
Pr ( 2 ) ( Ω ) = Γ 2 3 m 0 ( n 0 ( μ 0 m i μ m n j μ n 0 k ) μ 0 m i μ m 0 j μ 00 k ) ( ω 10 ω ) ( ( ω 10 2 ω ) 2 + Γ 2 ) Re { E 2 ω i * E ω j E ω k } = 1 Im { β Ω i j k ( 2 ω ) } Re { E 2 ω i * E ω j E ω k } .
( ρ 00 ( 3 ) t ) ω = 0 = m 0 ( ρ m m ( 3 ) t ) ω = 0 = Pr ( 2 ) ( Ω ) .
γ l k j i ( ω ; ω , ω , ω ) = 1 4 3 m 0 1 ( ω m 0 2 ω i Γ m ) ( μ m 0 i μ 00 j ( ω m 0 ω ) r 0 μ m r j μ r 0 i ( ω r 0 ω ) ) ( μ 0 m l μ 00 k ( ω m 0 ω ) n μ 0 n l μ n m k ( ω n 0 ω ) + n 0 μ 0 n k μ n m l ( ω m n ω ) ) 1 8 3 ω 2 m 0 1 ( ω m 0 2 ω i Γ m ) ( μ m 0 i μ 00 j + μ m 0 j μ 00 i n 0 ( μ m n j μ n 0 i + μ m n i μ n 0 j ) ) ( μ 00 k μ 0 m l + μ 00 l μ 0 m k n 0 ( μ 0 n k μ n m l + μ 0 n l μ n m k ) ) .
Pr ( 3 ) ( Ω ) = 1 4 ω Re ( i ω E ω * ω ( 3 ) ) = 1 4 Im [ γ Ω l k j i ( ω ; ω + ω + ω ) ] Re [ E ω l * E ω k * E ω j E ω i ] .
D m m J ( φ , θ , ψ ) = e i m φ r m m J ( θ ) e i m ψ ,
D m m J ( Ω ) = ( ) m m D m , m J * ( Ω ) = D m m J * ( Ω ) ,
m D m m J ( Ω ) D m m J ( Ω ) = m D m m J ( Ω ) D m m J * ( Ω ) = δ m m ,
D m m J ( Ω ) D n n K * ( Ω ) d Ω = 8 π 2 2 J + 1 δ J K δ m n δ m n ,
D n n L ( Ω ) D q q K ( Ω ) D m m J * ( Ω ) d Ω = 8 π 2 ( ) m m ( L K J n q m ) ( L K J n q m ) ,
D m 0 J ( φ , θ , ψ ) = 4 π 2 J + 1 Y J m * ( θ , φ ) ,
D 0 m J ( φ , θ , ψ ) = 4 π 2 J + 1 Y J m ( θ , ψ ) ,
D 0 0 J ( φ , θ , ψ ) = P J ( cos θ ) ,
V Q K = ( ) k 2 k 1 Q 2 K + 1 q 1 , q 2 ( k 1 k 2 K q 1 q 2 Q ) T q 1 k 1 U q 2 k 2 .
n T ( θ , φ ) = J , m 2 J + 1 4 π T J m 0 Y J m * ( θ , φ ) ,
T J m 0 = 4 π 2 J + 1 n T ( θ , φ ) Y J m ( θ , φ ) d ( cos θ ) d φ .
R C T R ( θ , φ θ , φ ) = F C T R ( ζ ) = J 2 J + 1 4 π R C T R , J P J ( cos ζ ) = J R C T R , J m = J J Y J m * ( θ , φ ) Y J m ( θ , φ ) .
R C T R ( Ω Ω ) n C ( Ω ) d Ω = J , m , J , m 2 J + 1 4 π R C T R , J C J m 0 ( Y J m * ( θ , φ ) Y J m ( θ , φ ) ) Y J m * ( θ , φ ) d ( cos θ ) d φ = J , m 2 J + 1 4 π R C T R , J C J m 0 Y J m * ( θ , φ ) ,
( d T J m 0 d t ) C T R = 1 τ C R C T R , J C J m 0 .
n T ( θ , φ ) = J ( 2 J + 1 ) ( 4 π ) 1 T J P J ( cos θ ) ,
Δ n T ( Ω ) = τ C 1 Δ t n C ( Ω δ ) R C T R ( δ ) d δ .
Δ χ T , M L ( n ) J = M M D M L M M J ( Ω ) Δ n T ( Ω ) A M M J ( n + 1 ) d Ω = τ C 1 Δ t M R D M L M J ( Ω ) n C ( Ω ) { M D M M M J ( δ ) R C T R ( δ ) A M M J ( n + 1 ) d δ } d Ω .
Δ χ T , M L ( n ) J = M M ( ) M L M M Δ T J , M L , M M A M M J ( n + 1 ) = τ C 1 Δ t M M , M ( ) M L M M C J , M L , M R C T , M , M M R , J A M M J ( n + 1 ) .
Δ T J M M = τ C 1 Δ t M C J M M R C T , M M R , J .

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