Abstract

In the previous paper [J. Opt. Soc. Am. B 26, 1057 (2009)], the theory of photoisomerization optical pumping cycles had been developed for three-dimensional molecules, and the evolution of tensorial properties had been simulated, with a particular attention to symmetry properties. Here different models of angular redistribution are compared—diffusion in the photoisomer or rotation in photoisomerization processes—in the case of axial molecules, with an axial symmetry of fields. The possibility of increasing χ(2), by destroying the anisotropy with a third pumping beam, is studied theoretically and experimentally. The failure of this experiment is explained by the too slow redistribution in DR1-MMA copolymers. While anisotropy measurements are unable to discriminate between the different models, the dynamics of second harmonic generation pleads for memoryless angular redistribution, with a very small probability.

© 2011 Optical Society of America

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  1. M. Dumont, “Dynamics of all-optical poling of photoisomerizable molecules. I: Symmetries of tensorial properties,” J. Opt. Soc. Am. B 26, 1057–1075 (2009).
    [CrossRef]
  2. X. Yu, X. Zhong, Q. Li, S. Luo, Y. C. Y. Sui, and J. Yin, “Method of improving optical poling efficiency in polymer films,” Opt. Lett. 26, 220–222 (2001).
    [CrossRef]
  3. Z. Sekkat and M. Dumont, “Photoassisted poling of azo dye doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (1992).
    [CrossRef]
  4. M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azo dyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188–199(1993).
    [CrossRef]
  5. 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]
  6. C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).
  7. M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).
  8. 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]
  9. This question is developed in many textbooks on LC. Many papers deal with photo-orientation of LC, with and without photoisomerisable dyes in the special topics issue, Mol. Cryst. Liq. Cryst. 282(1), 1–472 (1996).
  10. N. Nguyen, T. 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]
  11. 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]
  12. In our experiments the sample is much thinner than the coherence length, but the incidence angle is large so that there is a transversal averaging [formation of χ(2) fringes].
  13. T. G. Pedersen, P. M. Johansen, N. C. R. Holme, and P. S. Ramanujam, “Theorical model of photoinduced anisotropy in liquid-crystalline azobenzene side-chain polymers,” J. Opt. Soc. Am. B 15, 1120–1129 (1998).
    [CrossRef]
  14. M. Dumont, G. Froc, and S. Hosotte, “Alignment and orientation of chromophores by optical pumping,” Nonlinear Opt. 9, 327–338 (1995).
  15. 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]
  16. Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” doctoral thesis (Université Paris 11-Orsay, 1992).
  17. S. Yu. Grebenkin and B. V. Bol’shakov, “Photo orientation of azo dye molecules in glassy o-terphenyl,” J. Photochem. Photobiol. A 184, 155–168 (2006).
    [CrossRef]
  18. 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]
  19. S. P. Palto and G. Durand, “Friction model of photo-induced reorientation of optical axis in photo-oriented Langmuir–Blodgett films,” J. Phys. II 5, 963–978 (1995).
    [CrossRef]
  20. O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
    [CrossRef]
  21. O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
    [CrossRef]
  22. 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]
  23. This theory is developed for CW lasers. With pulsed lasers, the average power is considered. As the photo-orientation is a cumulative process, this is valid for the one-photon term (PIA) since the peak power at 2ω is far from the saturation of the electronic transition (the lifetime of the upper level is very short) and since the repetition rate is fast compared to τC. For the interference term and for the two-photon term, the efficiency of excitation is multiplied by sqrt(δ/Δ) and δ/Δ, respectively, where δ is the pulse width and Δ the pulse interval.
  24. A. El Osman and M. Dumont, “Dynamical and spectroscopic study of photoinduced orientation of dye molecules in polymer films,” Proc. SPIE 3417, 36–46 (1998).
    [CrossRef]
  25. N. N. T. Kim and M. Dumont, “3D characterization of molecular photo-orientation: application to all-optical poling,” in Organic Nanophotonics, F.Charra, ed., Vol.  100 of NATO Science Series (Kluwer Academic, 2003), pp 395–404.
  26. M. Dumont, “3D characterization of photo-induced anisotropy and all-optical poling of organic films,” Nonlinear Opt. Quantum Opt. (in press).
  27. The fitting procedure neglects the variation of Δnj′. In , Kjs are defined as Δn˜j=KjΔn˜iso, which is not correct if the dispersion is different in C and in T. The negligible numerical difference between both methods justifies the approximation .
  28. R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
    [CrossRef]
  29. Contour graphs (not presented here), as functions of Iω and I2ω, with Iinc=0, clearly showed a maximum of |χZZZ(2)| for ATI2ω=GTIω2, independent on I2ω, as far as ATI2ω<0.1τc−1 (Nc<0.05). This maximum is approximately proportional to B(AG)−1/2 (if B is taken small enough to avoid unrealistic negative pumping for θ=0). If diffusion is allowed in T, the maximum decreases when Iω and I2ω are decreased (ATI2ω≪τc−1).
  30. S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

2009 (1)

2006 (1)

S. Yu. Grebenkin and B. V. Bol’shakov, “Photo orientation of azo dye molecules in glassy o-terphenyl,” J. Photochem. Photobiol. A 184, 155–168 (2006).
[CrossRef]

2005 (1)

N. Nguyen, T. 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)

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

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

1999 (2)

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

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 (3)

1996 (1)

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 non centrosymmetry 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).

S. P. Palto and G. Durand, “Friction model of photo-induced reorientation of optical axis in photo-oriented Langmuir–Blodgett films,” J. Phys. II 5, 963–978 (1995).
[CrossRef]

1993 (5)

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azo dyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188–199(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, 941–943 (1993).
[CrossRef] [PubMed]

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[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]

1992 (2)

Z. Sekkat and M. Dumont, “Photoassisted poling of azo dye doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (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]

Bauer, S.

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

Bauer-Gogonea, S.

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

Boilot, J. P.

Bol’shakov, B. V.

S. Yu. Grebenkin and B. V. Bol’shakov, “Photo orientation of azo dye molecules in glassy o-terphenyl,” J. Photochem. Photobiol. A 184, 155–168 (2006).
[CrossRef]

Brasselet, S.

Brun, A.

Canva, M.

Chaput, F.

Charra, F.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry 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]

Delaire, J. A.

N. Nguyen, T. 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, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[CrossRef]

Dumont, M.

M. Dumont, “Dynamics of all-optical poling of photoisomerizable molecules. I: Symmetries of tensorial properties,” J. Opt. Soc. Am. B 26, 1057–1075 (2009).
[CrossRef]

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

A. El Osman and M. Dumont, “Dynamical and spectroscopic study of photoinduced orientation of dye molecules in polymer films,” Proc. SPIE 3417, 36–46 (1998).
[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]

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

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, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azo dyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188–199(1993).
[CrossRef]

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[CrossRef]

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

N. N. T. Kim and M. Dumont, “3D characterization of molecular photo-orientation: application to all-optical poling,” in Organic Nanophotonics, F.Charra, ed., Vol.  100 of NATO Science Series (Kluwer Academic, 2003), pp 395–404.

M. Dumont, “3D characterization of photo-induced anisotropy and all-optical poling of organic films,” Nonlinear Opt. Quantum Opt. (in press).

Durand, G.

S. P. Palto and G. Durand, “Friction model of photo-induced reorientation of optical axis in photo-oriented Langmuir–Blodgett films,” J. Phys. II 5, 963–978 (1995).
[CrossRef]

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]

A. El Osman and M. Dumont, “Dynamical and spectroscopic study of photoinduced orientation of dye molecules in polymer films,” Proc. SPIE 3417, 36–46 (1998).
[CrossRef]

Fiorini, C.

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

Froc, G.

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

Geoges, P.

Gerhard-Multhaupt, R.

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

Grebenkin, S. Yu.

S. Yu. Grebenkin and B. V. Bol’shakov, “Photo orientation of azo dye molecules in glassy o-terphenyl,” J. Photochem. Photobiol. A 184, 155–168 (2006).
[CrossRef]

Holme, N. C. R.

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]

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

Idiart, E.

Johansen, P. M.

Kajzar, F.

Kim, N. N. T.

N. N. T. Kim and M. Dumont, “3D characterization of molecular photo-orientation: application to all-optical poling,” in Organic Nanophotonics, F.Charra, ed., Vol.  100 of NATO Science Series (Kluwer Academic, 2003), pp 395–404.

Kim, T.

N. Nguyen, T. 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]

Kiselev, A. D.

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

Le Saux, G.

Li, Q.

Lindau, J.

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Loucif-Saibi, R.

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[CrossRef]

Luo, S.

Nakatani, K.

N. Nguyen, T. 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, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[CrossRef]

Nguyen, N.

N. Nguyen, T. 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.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry 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]

Palto, S. P.

S. P. Palto and G. Durand, “Friction model of photo-induced reorientation of optical axis in photo-oriented Langmuir–Blodgett films,” J. Phys. II 5, 963–978 (1995).
[CrossRef]

Pedersen, T. G.

Puchkovs’ka, G.

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Raimond, P.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry 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]

Ramanujam, P. S.

Reshentnyak, V.

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Sekkat, Z.

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 and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azo dyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188–199(1993).
[CrossRef]

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[CrossRef]

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

Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” doctoral thesis (Université Paris 11-Orsay, 1992).

Shan’ky, L.

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Stumpe, J.

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

Sui, Y. C. Y.

Tereeshchenko, A.

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Wintle, H. J.

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

Wirges, W.

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

Yaroshchuk, O.

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Yin, J.

Yu, X.

Zakrevskyy, Yu.

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

Zhong, X.

Zyss, J.

Appl. Phys. B (1)

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

Chem. Mater. (1)

R. Loucif-Saibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, “Photoisomerization and second harmonic generation in disperse red one-doped and -functionalized Poly(methyl methacrylate) films,” Chem. Mater. 5, 229–236 (1993).
[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]

Eur. Phys. J. E (1)

O. Yaroshchuk, A. D. Kiselev, Yu. Zakrevskyy, J. Stumpe, and J. Lindau, “Spatial reorientation of azobenzene side groups of liquid crystalline polymer induced by linearly polarized light,” Eur. Phys. J. E 6, 57–67 (2001).
[CrossRef]

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

J. Photochem. Photobiol. A (1)

S. Yu. Grebenkin and B. V. Bol’shakov, “Photo orientation of azo dye molecules in glassy o-terphenyl,” J. Photochem. Photobiol. A 184, 155–168 (2006).
[CrossRef]

J. Phys. II (1)

S. P. Palto and G. Durand, “Friction model of photo-induced reorientation of optical axis in photo-oriented Langmuir–Blodgett films,” J. Phys. II 5, 963–978 (1995).
[CrossRef]

Mater. Sci. Eng. C (1)

O. Yaroshchuk, V. Reshentnyak, A. Tereeshchenko, L. Shan’ky, G. Puchkovs’ka, and J. Lindau, “Main-chain ordering and stability of the light induced anisotropy in the films of comb-like azo-polymer,” Mater. Sci. Eng. C 8–9, 211–216 (1999).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

N. Nguyen, T. 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]

Nonlinear Opt. (3)

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

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

M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani, and J. A. Delaire, “Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polimeric films,” Nonlinear Opt. 5, 395–406 (1993).

Nonlinear Opt. Quantum Opt. (1)

M. Dumont, “3D characterization of photo-induced anisotropy and all-optical poling of organic films,” Nonlinear Opt. Quantum Opt. (in press).

Opt. Lett. (3)

Proc. SPIE (3)

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]

M. Dumont and Z. Sekkat, “Dynamical study of photoinduced anisotropy and orientational relaxation of azo dyes in polymeric films poling at room temperature,” Proc. SPIE 1774, 188–199(1993).
[CrossRef]

A. El Osman and M. Dumont, “Dynamical and spectroscopic study of photoinduced orientation of dye molecules in polymer films,” Proc. SPIE 3417, 36–46 (1998).
[CrossRef]

Synth. Met. (1)

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]

Other (8)

This theory is developed for CW lasers. With pulsed lasers, the average power is considered. As the photo-orientation is a cumulative process, this is valid for the one-photon term (PIA) since the peak power at 2ω is far from the saturation of the electronic transition (the lifetime of the upper level is very short) and since the repetition rate is fast compared to τC. For the interference term and for the two-photon term, the efficiency of excitation is multiplied by sqrt(δ/Δ) and δ/Δ, respectively, where δ is the pulse width and Δ the pulse interval.

N. N. T. Kim and M. Dumont, “3D characterization of molecular photo-orientation: application to all-optical poling,” in Organic Nanophotonics, F.Charra, ed., Vol.  100 of NATO Science Series (Kluwer Academic, 2003), pp 395–404.

The fitting procedure neglects the variation of Δnj′. In , Kjs are defined as Δn˜j=KjΔn˜iso, which is not correct if the dispersion is different in C and in T. The negligible numerical difference between both methods justifies the approximation .

Contour graphs (not presented here), as functions of Iω and I2ω, with Iinc=0, clearly showed a maximum of |χZZZ(2)| for ATI2ω=GTIω2, independent on I2ω, as far as ATI2ω<0.1τc−1 (Nc<0.05). This maximum is approximately proportional to B(AG)−1/2 (if B is taken small enough to avoid unrealistic negative pumping for θ=0). If diffusion is allowed in T, the maximum decreases when Iω and I2ω are decreased (ATI2ω≪τc−1).

S. Bauer-Gogonea, S. Bauer, W. Wirges, R. Gerhard-Multhaupt, and H. J. Wintle, “Physical aging after photo-induced or thermally assisted poling for enhancing the stability of polymeric dipole glasses,” in Organic Thin films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), pp. 133–136.

This question is developed in many textbooks on LC. Many papers deal with photo-orientation of LC, with and without photoisomerisable dyes in the special topics issue, Mol. Cryst. Liq. Cryst. 282(1), 1–472 (1996).

In our experiments the sample is much thinner than the coherence length, but the incidence angle is large so that there is a transversal averaging [formation of χ(2) fringes].

Z. Sekkat, “Création d’anisotropie et d’effets non linéaires du second ordre par photoisomérisation de dérivés de l’azobenzène dans des films de polymères,” doctoral thesis (Université Paris 11-Orsay, 1992).

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Figures (7)

Fig. 1
Fig. 1

Comparison of the different redistribution models, for rodlike molecules (with a small isotropic part, α r / α i = 10 ) pumped by linearly polarized fields, switched on at t = 0 . χ X X ( 1 ) , χ Z Z ( 1 ) , and χ Z Z Z ( 2 ) are tensorial components of the lower level T. N C is the total population of C. For all graphs one has [see Eq. (18)] τ C = 1 , I ω = 1000 , I 2 ω = 0.1 , A = 1 , B = 10 4 , G = 10 7 , so that τ C 1 = 10 A I 2 ω = 10 G I ω 2 = 31.62 B I ω I 2 ω . All curves (except AHB) tend to the same asymptotes (dashed horizontal lines) calculated from the stationary solution (Table 1, line 1, the C T pumping is neglected). The decrease of | χ Z Z Z ( 2 ) | and of N C , after a first growing, is due to the accumulation of molecules in the { x , y } plane, perpendicular to the polarization of light. D. ML n n , ML diffusion in C [Eq. (9)] with n n = τ C / τ D , C = D C τ C . D. BM n n , Brownian motion diffusion in C [Eq. (14)] with n n = D C τ C . I. ML n n , ML isomerization from C to T [Eq. (18)] with n n = L . I. C n n , conical isomerization from C to T [Eq. (A12)] with n n = ζ 0 . I. GA n n or I. GB n n , Gaussian isomerization from C to T [Eq. (A13a) or Eq. (A13b)] with n n = w = width . I. VL n n , diffusion in a virtual state during C to T isomerization [Eq. (A3)] with n n = D V τ V .

Fig. 2
Fig. 2

3D experimental setup, with all possible pump and probe directions. The sample, spin coated on a glass slide, is put in optical contact with the two prisms, with the help of index matching oil. In AOP experiments, the film is pumped simultaneously by 1064 and 532 nm pulses. 1 s every 30 s , the IR filter cuts the 532 nm pump, while the shutter is opened, in front of the photomultiplier (PM), for measuring SHG from the sample. For 3D pumping, both pulsed pumping beams are linearly z polarized, while the CW laser beam is circularly polarized, in order to pump molecules in the ( x , y ) plane. The two laser beams cross each other at a right angle in the sample, while the probe beams are not exactly orthogonal, but their angles are taken into account for the interpretation of measurements. The three probe beams are made of a periodic succession ( 25 ms cycles) of 2 ms pulses of different wavelengths (up to 6), produced by a multislit monochromator and a rotating slit wheel [24]. The Nd:YAG laser pulses ( 100 ps , 10 Hz ) are synchronized within the probe pulses cycles (1 over 4). Each probe beam is split in two perpendicularly polarized components, labeled horizontal (in the x, z plane) and vertical (y), which are detected by photodiodes (D).

Fig. 3
Fig. 3

3D AOP experimental curves as functions of time, with a 120 nm thick film of DR1-MMA (70/30) copolymer. (a) SHG signal and the average intensity of the three laser beams (arbitrary units). The energy of pumping pulses ( 100 ps , 10 Hz ) is 0.7 mJ at 1.06 μm and 0.3 μJ at 532 nm , which corresponds roughly to average powers of 2 W / cm 2 and 1 mW / cm 2 , respectively (by comparison with the efficiency of the CW laser alone). The power of the CW 532 nm beam is 0.5 mW / cm 2 . (b) Optical densities measured at 446 nm , with vertical and horizontal polarization components of probes 1 and 2 (see Fig. 3). (c) Results of the fitting procedure, applied to the OD experimental curves. K x , K y , and K z are the normalized components of the susceptibility corresponding to the three principal axes imposed by pumping beams. K x and K y are not exactly equal, probably because of weak reflected pump beams, propagating along z. S 0 is proportional to N C [Eq. (22)], while S 2 χ 0 ( 1 ) 2 represents the anisotropy [Eq. (23)]. It is clear that all three pumping periods are too short to lead to the photostationary states.

Fig. 4
Fig. 4

Stationary solution of differential equations as a function of I 2 ω and I inc , for quasi rodlike molecules. I ω and I 2 ω are z polarized; I inc is circularly polarized in { x , y } . Each graph is a contour plot of a tensorial component of χ T ( 2 ) (solid curves, bold labels) superposed with the anisotropy χ T , 0 ( 1 ) 2 (dotted curves) and the population of C (dashed lines, italic labels). For convenience, χ T ( 1 ) and χ T ( 2 ) have been multiplied by 100 and 1000, respectively. In all graphs presented here, the ML diffusion model is used (line 3 in Table 1) with ( a r / a i ) T = ( g r / g i ) T = 10 , A T τ C = 1 , B T τ C = 10 4 , G T τ C = 10 8 , and I ω = 1000 ( G T I ω 2 τ C = 0.01 ). For C molecules the choice is A C = 3 A T , G C = 3 G T (which corresponds to σ C = σ T / 2 ), B C = 0 , ( a r / a i ) C = 3 ), g r = 0 . The choice of the value of G T I ω 2 is experimentally reasonable since it corresponds to N C 0.005 for I 2 ω = I inc = 0 . The thick dotted curve [ χ T , 0 ( 1 ) 2 = 0 ] corresponds to the exact cancellation of the PIA. Points A ( I 2 ω = 0.2 ) and A ( I inc = 0.067 ) correspond to experimental values of S 0 , measured from Fig. 3, for two-beam and three-beam pumping, respectively (on the x axis, I inc = 10 5 , can be neglected). The values of χ 0 ( 1 ) 2 and χ Z Z Z ( 2 ) at points A and A are compared with experimental values. Points B and B have been obtained from the experiment presented in [26]. By drawing vertical lines on Fig. 4c, one sees that I inc should exceed largely the inversion of anisotropy to produce a decrease of | χ Z Z Z ( 2 ) | by reduction of N T .

Fig. 5
Fig. 5

Some examples of transients of S 0 = N C ( 1 σ C / σ T ) , S 2 = e T 3 / 2 χ 0 ( 1 ) 2 , and χ Z Z Z ( 2 ) obtained by solving numerically Eqs. (1), with τ C = 1 , A T = 1 , B T = 10 4 , G T = 10 8 , A C = 3 A T , G C = 3 G T , I ω = 1000 , ( a r / a i ) T = ( g r / g i ) T = 5 , ( a r / a i ) C = 3 . Four cases of AR have been used: ML diffusion in C with τ D , C = 10 (label D.ML), Brownian motion diffusion, with D C = 0.02 (D.BM), ML C T isomerization, with L = 0.08 (I.ML), and conical C T isomerization, with ζ 0 = 12 ° (I.C). The parameters have been chosen so that the transients of anisotropy, S 2 , are as similar as possible. I inc = 0 in intervals ( 0 500 τ C ) and ( 750 1000 τ C ); I inc = 2 I 2 ω in ( 500 750 τ C ). Horizontal lines, on the right, are the asymptotic values at t = , for I inc = 0 (long lines) and for I inc = 2 I 2 ω (short lines). For I inc = 0 , the common asymptotes of both diffusion models are labeled D with the value of I 2 ω (line 3 in Table 1), and the common asymptotes of RiI models are labeled I (line 4 in Table 1). Asymptotes for I inc = 2 I 2 ω are identical for all models, since the pumping is isotropic. S 0 is drawn for D.ML only since the curves are practically the same with all models.

Fig. 6
Fig. 6

Definition of the rotation angles ζ and η. Z and Z are the initial and the final orientation of the molecular axis, respectively.

Fig. 7
Fig. 7

Test of the redistribution models. Curves represent the angular distribution n f ( θ ) obtained after one isomerization process, with the different models of AR, for a Gaussian initial distribution n i ( θ ) = exp ( ( ( θ 60 ) / 10 ) 2 ) . The labels are “cone_ ζ 0 ” for the conical redistribution, “ GA _ w ” or “ GB _ w ” for the Gaussian models [w is the width in Eq. (A13a) or Eq. (A13b), respectively], and “ VL _ n n ” for the virtual level model, with n n = ( D V τ V ) 1 . It is noticeable that VL _ 100 GA _ 10 and VL _ 10 GB _ 50 . VL_1 appears as almost ML.

Tables (1)

Tables Icon

Table 1 Seven Main Cases Where the Stationary Solution of Eqs. (1) Can Be Expressed Analytically a

Equations (39)

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d n T ( θ ) d t = Φ T C Pr T ( θ ) n T ( θ ) + Φ C T R C T P ( θ θ ) Pr C ( θ ) n C ( θ ) d ( cos θ ) + 1 τ C R C T R ( θ θ ) n C ( θ ) d ( cos θ ) + ( d n T ( θ ) d t ) Diff ,
d n C ( θ ) d t = Φ T C R T C P ( θ θ ) Pr T ( θ ) n T ( θ ) d ( cos θ ) Φ C T Pr C ( θ ) n C ( θ ) 1 τ C n C ( θ ) + ( d n C ( θ ) d t ) Diff ,
[ n T ( θ ) + n C ( θ ) ] d ( cos θ ) = N T + N C = N ,
R T C P ( θ θ ) d ( cos θ ) = 1 ,
E ω = z ^ E ω cos ( ω t k ω x ) , E 2 ω = z ^ E 2 ω cos ( 2 ω t k 2 ω x ϕ ) .
Pr ( θ ) = 1 2 α z z ( 2 ω ) E 2 ω 2 + 1 β z z z ( 2 ω = ω + ω ) E 2 ω E ω 2 cos Ψ + 1 4 γ z z z z ( ω = ω + ω ω ) E ω 4 .
β z z z = I , J , K R z I ( θ , φ ) R z J ( θ , φ ) R z K ( θ , φ ) β I J K .
α z z = α X X + ( α Z Z α X X ) cos 2 θ , β z z z = cos 3 θ β Z Z Z + 3 cos θ sin 2 θ β Z X X , γ z z z z = 3 γ Z Z X X + ( γ Z Z Z Z 3 γ Z Z X X ) cos 4 θ + ( γ X X X X 3 γ Z Z X X ) sin 4 θ .
R C T P ( Ω Ω ) = R C T R ( Ω Ω ) = f T ( Ω ) , R T C P ( Ω Ω ) = f C ( Ω ) ,
( d n A ( Ω ) d t ) Dif = 1 τ D , A ( N A f A ( Ω ) n A ( Ω ) ) ,
f A ( Ω ) exp ( W A ( Ω ) / k T ) ,
W A ( Ω ) = μ A · E 0 1 2 α A ( E 0 E 0 ) .
( d n A ( Ω ) d t ) Dif = D A ( Δ n A ( Ω ) + 1 k T · ( n A ( Ω ) W A ( Ω ) ) ) .
W A ( θ ) = μ A E 0 cos θ 1 2 ( α α ) A E 0 2 cos 2 θ ,
( d n A ( θ ) d t ) Dif = D A sin θ θ ( sin θ n A ( θ ) θ + μ A E 0 k T sin 2 θ n A ( θ ) ) .
R A B P , R ( θ θ ) d ( cos θ ) = 1.
R C T R ( θ θ ) n C ( θ ) d ( cos θ ) = ( 1 L ) n C ( θ ) + 1 2 L N C .
α = α ¯ a , γ = γ ¯ g .
Φ Pr ( θ ) = A I 2 ω ( a i + a r cos 2 θ ) + B I ω I 2 ω ( cos 3 θ + 3 b cos θ sin 2 θ ) + G I ω 2 ( g i + g r cos 4 θ + g d sin 4 θ ) ,
a z z θ = a i + a r / 3 = ( 2 a + a ) / 3 = 1 , g z z z z θ = g i + g r / 5 + ( 8 / 15 ) g d = 1.
n ˜ j = n j + i n j = n 0 2 + χ ˜ j j n 0 + ( 2 n 0 ) 1 χ ˜ j j = n 0 + Δ n ˜ j .
Δ n j = K j Δ n iso K j = C T ( a i + a r cos 2 θ j ) T + C C ( σ C / σ T ) λ ( a i + a r cos 2 θ j ) C .
S 0 = 1 ( K x + K y + K z ) / 3 = N C ( 1 σ C / σ T ) ,
S 2 = ( 2 K z K x K y ) / 6 = N T e T P 2 ( cos 2 θ z ) T + N C ( σ C / σ T ) λ e C P 2 ( cos 2 θ z ) C ,
Pr inc ( θ ) = A I inc [ a i + ( a r / 2 ) sin 2 θ ] .
x 0 ( θ ) = n C ( θ , t ) δ t / τ C ,
d x d t = D V Δ x x τ V ,
δ n T ( θ ) = ( δ t / τ C ) R C T R ( θ θ ) n C ( θ ) d ( cos θ ) = ( 1 / τ V ) 0 x ( θ , t ) d t .
A = ( d n f ( θ , φ ) d t ) = 0 π d ( cos θ ) 0 2 π n i ( θ , φ ) P ( ζ ) d φ ,
A = ( d n f ( θ ) d t ) = 0 π n i ( θ ) d ( cos θ ) 0 2 π P ( ζ ) d Φ .
cos ζ = cos θ cos θ + sin θ sin θ cos Φ , sin ζ d ζ = sin θ sin θ sin Φ d Φ ,
A = 2 0 π P ( ζ ) sin ζ d ζ D = real D 1 n i ( θ ) d ( cos θ ) = 2 0 π P ( ζ ) I ( ζ ) sin ζ d ζ ,
D = sin θ sin θ sin Φ = ( cos ( θ ζ ) cos θ ) ( cos θ cos ( θ + ζ ) ) .
u = cos θ , C + = cos ( θ + ζ ) , C = cos ( θ ζ ) , y | C C + | = 2 u ( C + C + ) , n ˜ i ( y ) = n i ( arccos ( u ( y ) ) ) ,
I ( ζ ) = 1 + 1 n ˜ i ( y ) d y 1 y 2 = [ n ˜ i ( y ) arcsin ( y ) ] 1 + 1 1 + 1 arcsin ( y ) n ˜ i ( y ) y d y = π 2 ( n i ( arccos C ) + n i ( arccos C + ) ) + arccos C + arccos C arcsin ( y ( θ ) ) n i ( θ ) θ d θ .
P ( ζ ) = ( 2 π sin ζ 0 ) 1 δ ( ζ ζ 0 ) ,
A = ( d n f ( θ ) d t ) = 1 π I ( ζ 0 ) .
P A ( ζ ) = N 1 exp ( ( ζ / w ) 2 ) ,
P B ( ζ ) = ( N sin ζ ) 1 exp ( ( ζ / w ) 2 ) .

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