Abstract

The orientational distribution function of a poled, doped polymer film under stress is measured using polarized second-harmonic generation. The four interface nonlinear boundary conditions are solved, and both the magnitude of the second-order susceptibility and the ratio of the tensor components of the second-order susceptibility, χ113(2)/χ333(2)(a), are shown to vary with poling field and stress. Further, the sign of the order parameter ratio, 〈P3〉/〈P1〉, as measured directly by polarized second-harmonic generation of these materials, shows that the materials fall into a regime of order unlike other material classes such as Langmuir–Blodgett films and liquid crystals. For a one-dimensional dopant molecule, the order parameters 〈P1〉, 〈P2〉, 〈P3〉, and 〈P4〉 can be estimated. Materials processed in this way are shown to result in a range of tensor ratios of a = 0.33 with a film under no stress to a = 0.7 with a film under moderate stress.

© 1989 Optical Society of America

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  1. G. T. Boyd, “Optical second-harmonic generation as an orientational probe in poled polymers,” Thin Solid Films 152, 295 (1987).
    [Crossref]
  2. P. Le Barny, “Chemistry of polymer molecules for ultrathin films,” Thin Solid Films 152, 99 (1987).
    [Crossref]
  3. C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
    [Crossref]
  4. H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
    [Crossref]
  5. J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
    [Crossref]
  6. C. S. Willand and D. J. Williams, “Nonlinear optical properties of polymeric materials,” Ber. Bunsenges Phys. Chem. 91, 1304 (1987).
  7. S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
    [Crossref]
  8. R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.
  9. A. C. Griffin, A. M. Bhatti, and G. A. Howell, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 115.
  10. T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.
  11. P. D. Calvert and B. D. Moyle, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 357.
  12. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
    [Crossref]
  13. G. R. Meridith, J. G. Vandusen, and D. J. Williams, in Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, ed., ACS Symp. Ser.233, 109 (1983).
    [Crossref]
  14. K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
    [Crossref]
  15. H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
    [Crossref]
  16. E. E. Havinga and P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices; influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979).
    [Crossref]
  17. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4, 968 (1987).
    [Crossref]
  18. J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.
  19. T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
    [Crossref] [PubMed]
  20. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1977), Vol. 1, p. 21.
  21. S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, Cambridge, 1980), p. 44.
  22. G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978), p. 630.
  23. L. E. Nielson, Mechanical Properties of Polymers (Van Nostrand Reinhold, New York, 1962), p. 238.
  24. J. Brandrup and E. H. Immergut, eds., Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), p. IV–380.
  25. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Orientationally ordered electro-optic materials,” in Nonlinear Optical and Electroactive Polymers, P. N. Prasad and D. R. Ulrich, eds. (Plenum, New York, 1988).
    [Crossref]
  26. J. F. Nye, Physical Properties of Crystals (Clarendon, London, 1957).
  27. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
    [Crossref]
  28. N. Bloembergen, Nonlinear Optics (Benjamin, Reading, Mass., 1965).
  29. S. Osaki, “A new method for quick determination of molecular orientation in poly(ethylene terephthalate) films by use of polarized microwaves,” Polym. J. 19, 821 (1987).
    [Crossref]
  30. M. G. Kuzyk, R. C. Moore, and L. A. King, “Second-harmonic generation measurements of the elastic constant of a molecule in a polymer matrix,” submitted to Phys. Rev. A.
  31. S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
    [Crossref]
  32. S. D. Durbin and Y. R. Shen, “Two-photon dichroism studies of molecular orientational distribution in a nematic liquid crystal,” Phys. Rev. A 30, 1419 (1984).
    [Crossref]
  33. K. Y. Wong and A. F. Garito, “Third-harmonic generation study of orientational order in nematic liquid crystals,” Phys. Rev. A 34, 5051 (1986).
    [Crossref] [PubMed]

1988 (3)

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

1987 (8)

C. S. Willand and D. J. Williams, “Nonlinear optical properties of polymeric materials,” Ber. Bunsenges Phys. Chem. 91, 1304 (1987).

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

G. T. Boyd, “Optical second-harmonic generation as an orientational probe in poled polymers,” Thin Solid Films 152, 295 (1987).
[Crossref]

P. Le Barny, “Chemistry of polymer molecules for ultrathin films,” Thin Solid Films 152, 99 (1987).
[Crossref]

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.

S. Osaki, “A new method for quick determination of molecular orientation in poly(ethylene terephthalate) films by use of polarized microwaves,” Polym. J. 19, 821 (1987).
[Crossref]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4, 968 (1987).
[Crossref]

1986 (3)

K. Y. Wong and A. F. Garito, “Third-harmonic generation study of orientational order in nematic liquid crystals,” Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

1985 (1)

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

1984 (1)

S. D. Durbin and Y. R. Shen, “Two-photon dichroism studies of molecular orientational distribution in a nematic liquid crystal,” Phys. Rev. A 30, 1419 (1984).
[Crossref]

1979 (1)

E. E. Havinga and P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices; influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979).
[Crossref]

1977 (1)

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

1970 (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[Crossref]

Armstrong, R. C.

R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1977), Vol. 1, p. 21.

Baur, G.

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Bhatti, A. M.

A. C. Griffin, A. M. Bhatti, and G. A. Howell, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 115.

Bird, R. B.

R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1977), Vol. 1, p. 21.

Bloembergen, N.

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

Bock, J.

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

Box, G. E. P.

G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978), p. 630.

Boyd, G. T.

G. T. Boyd, “Optical second-harmonic generation as an orientational probe in poled polymers,” Thin Solid Films 152, 295 (1987).
[Crossref]

Calvert, P. D.

P. D. Calvert and B. D. Moyle, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 357.

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, Cambridge, 1980), p. 44.

Chatani, Y.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

Clark, N. A.

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

Comizzoli, R. B.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Durbin, S. D.

S. D. Durbin and Y. R. Shen, “Two-photon dichroism studies of molecular orientational distribution in a nematic liquid crystal,” Phys. Rev. A 30, 1419 (1984).
[Crossref]

Fichou, D.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

Garito, A. F.

K. Y. Wong and A. F. Garito, “Third-harmonic generation study of orientational order in nematic liquid crystals,” Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Griffin, A. C.

A. C. Griffin, A. M. Bhatti, and G. A. Howell, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 115.

Haas, D.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Hampsch, H. L.

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

Hassager, O.

R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1977), Vol. 1, p. 21.

Havinga, E. E.

E. E. Havinga and P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices; influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979).
[Crossref]

Holland, W. R.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Howell, G. A.

A. C. Griffin, A. M. Bhatti, and G. A. Howell, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 115.

Hunter, J. S.

G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978), p. 630.

Hunter, W. G.

G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978), p. 630.

Jen, S.

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[Crossref]

Kaino, T.

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

Katz, H. E.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Khanarian, G.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Kiefer, R.

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Kim, M. W.

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

King, L. A.

M. G. Kuzyk, R. C. Moore, and L. A. King, “Second-harmonic generation measurements of the elastic constant of a molecule in a polymer matrix,” submitted to Phys. Rev. A.

Kubodera, K.

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

Kurihara, T.

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[Crossref]

Kuzyk, M. G.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4, 968 (1987).
[Crossref]

M. G. Kuzyk, R. C. Moore, and L. A. King, “Second-harmonic generation measurements of the elastic constant of a molecule in a polymer matrix,” submitted to Phys. Rev. A.

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Orientationally ordered electro-optic materials,” in Nonlinear Optical and Electroactive Polymers, P. N. Prasad and D. R. Ulrich, eds. (Plenum, New York, 1988).
[Crossref]

Lalama, S. J.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

Le Barny, P.

P. Le Barny, “Chemistry of polymer molecules for ultrathin films,” Thin Solid Films 152, 99 (1987).
[Crossref]

LeGrange, J. D.

J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.

Leslie, T.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Lipscomb, G. F.

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

Lytel, R.

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

Man, H. T.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Marks, T. J.

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

Martino, R. D.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Matsumoto, S.

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

Meridith, G. R.

G. R. Meridith, J. G. Vandusen, and D. J. Williams, in Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, ed., ACS Symp. Ser.233, 109 (1983).
[Crossref]

Miyata, S.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

Moore, R. C.

M. G. Kuzyk, R. C. Moore, and L. A. King, “Second-harmonic generation measurements of the elastic constant of a molecule in a polymer matrix,” submitted to Phys. Rev. A.

Moyle, B. D.

P. D. Calvert and B. D. Moyle, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 357.

Nielson, L. E.

L. E. Nielson, Mechanical Properties of Polymers (Van Nostrand Reinhold, New York, 1962), p. 238.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, London, 1957).

Osaki, S.

S. Osaki, “A new method for quick determination of molecular orientation in poly(ethylene terephthalate) films by use of polarized microwaves,” Polym. J. 19, 821 (1987).
[Crossref]

Pershan, P. S.

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

Priestly, E. B.

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

Rasing, T.

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

Riggs, J.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Ringsdorf, H.

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Sansone, M.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Schilling, M. L.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Schmidt, H. W.

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Shen, Y. R.

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

S. D. Durbin and Y. R. Shen, “Two-photon dichroism studies of molecular orientational distribution in a nematic liquid crystal,” Phys. Rev. A 30, 1419 (1984).
[Crossref]

Singer, K. D.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4, 968 (1987).
[Crossref]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Orientationally ordered electro-optic materials,” in Nonlinear Optical and Electroactive Polymers, P. N. Prasad and D. R. Ulrich, eds. (Plenum, New York, 1988).
[Crossref]

Sohn, J. E.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4, 968 (1987).
[Crossref]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Orientationally ordered electro-optic materials,” in Nonlinear Optical and Electroactive Polymers, P. N. Prasad and D. R. Ulrich, eds. (Plenum, New York, 1988).
[Crossref]

Stamitoff, J.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Stiller, M. A.

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

Teng, C.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Thackara, J. I.

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

Ticknor, A. J.

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

Torkelson, J. M.

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

Valint, P.

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

van Pelt, P.

E. E. Havinga and P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices; influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979).
[Crossref]

Vandusen, J. G.

G. R. Meridith, J. G. Vandusen, and D. J. Williams, in Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, ed., ACS Symp. Ser.233, 109 (1983).
[Crossref]

Watanabe, T.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

Willand, C. S.

C. S. Willand and D. J. Williams, “Nonlinear optical properties of polymeric materials,” Ber. Bunsenges Phys. Chem. 91, 1304 (1987).

Williams, D. J.

C. S. Willand and D. J. Williams, “Nonlinear optical properties of polymeric materials,” Ber. Bunsenges Phys. Chem. 91, 1304 (1987).

G. R. Meridith, J. G. Vandusen, and D. J. Williams, in Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, ed., ACS Symp. Ser.233, 109 (1983).
[Crossref]

Windscheid, F.

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Wong, G. K.

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

Wong, K. Y.

K. Y. Wong and A. F. Garito, “Third-harmonic generation study of orientational order in nematic liquid crystals,” Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Yang, J.

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

Ye, C.

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

Yoon, H.

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

Yoshinaga, K.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

Appl. Phys. Lett. (4)

S. Matsumoto, K. Kubodera, T. Kurihara, and T. Kaino, “Nonlinear optical properties of an azo dye attached polymer,” Appl. Phys. Lett. 51, 1 (1987).
[Crossref]

J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electrooptic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
[Crossref]

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

Ber. Bunsenges Phys. Chem. (1)

C. S. Willand and D. J. Williams, “Nonlinear optical properties of polymeric materials,” Ber. Bunsenges Phys. Chem. 91, 1304 (1987).

Ber. Bunsenges. Phys. Chem. (1)

E. E. Havinga and P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices; influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979).
[Crossref]

J. Appl. Phys. (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[Crossref]

J. Chem. Phys. (1)

S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestly, “Polarized Raman scattering studies of orientational order in uniaxial liquid crystals,” J. Chem. Phys. 66, 4635 (1977).
[Crossref]

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

Liq. Cryst. (GB) (1)

H. Ringsdorf, H. W. Schmidt, G. Baur, R. Kiefer, and F. Windscheid, “Orientational ordering of dyes in the glassy state of liquid-crystalline side group polymers,” Liq. Cryst. (GB) 1, 319 (1986).
[Crossref]

Macromolecules (1)

H. L. Hampsch, J. Yang, G. K. Wong, and J. M. Torkelson, “Orientation and second-harmonic generation in doped polystyrene and poly(methyl methacrylate) films,” Macromolecules 21, 526 (1988).
[Crossref]

Marcromolecules (1)

C. Ye, T. J. Marks, J. Yang, and G. K. Wong, “Synthesis of molecular arrays with nonlinear optical properties. Second-harmonic generation by covalently functionalized glassy polymers,” Marcromolecules 20, 2322 (1987).
[Crossref]

Mol. Cryst. Liq. Cryst. (1)

J. D. LeGrange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987), and references therein.

Phys. Rev. A (3)

T. Rasing, Y. R. Shen, M. W. Kim, P. Valint, and J. Bock, “Orientation of surfactant molecules at a liquid-air interface measured by optical second-harmonic generation,” Phys. Rev. A 31, 537 (1985).
[Crossref] [PubMed]

S. D. Durbin and Y. R. Shen, “Two-photon dichroism studies of molecular orientational distribution in a nematic liquid crystal,” Phys. Rev. A 30, 1419 (1984).
[Crossref]

K. Y. Wong and A. F. Garito, “Third-harmonic generation study of orientational order in nematic liquid crystals,” Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Polym. J. (1)

S. Osaki, “A new method for quick determination of molecular orientation in poly(ethylene terephthalate) films by use of polarized microwaves,” Polym. J. 19, 821 (1987).
[Crossref]

Thin Solid Films (2)

G. T. Boyd, “Optical second-harmonic generation as an orientational probe in poled polymers,” Thin Solid Films 152, 295 (1987).
[Crossref]

P. Le Barny, “Chemistry of polymer molecules for ultrathin films,” Thin Solid Films 152, 99 (1987).
[Crossref]

Other (14)

R. D. Martino, D. Haas, G. Khanarian, T. Leslie, H. T. Man, J. Riggs, M. Sansone, J. Stamitoff, C. Teng, and H. Yoon, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 65.

A. C. Griffin, A. M. Bhatti, and G. A. Howell, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 115.

T. Watanabe, K. Yoshinaga, D. Fichou, Y. Chatani, and S. Miyata, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 339.

P. D. Calvert and B. D. Moyle, in Nonlinear Optical Properties of Polymers, A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds. (Materials Research Society, Pittsburgh, Pa., 1988), p. 357.

R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids (Wiley, New York, 1977), Vol. 1, p. 21.

S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, Cambridge, 1980), p. 44.

G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978), p. 630.

L. E. Nielson, Mechanical Properties of Polymers (Van Nostrand Reinhold, New York, 1962), p. 238.

J. Brandrup and E. H. Immergut, eds., Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), p. IV–380.

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Orientationally ordered electro-optic materials,” in Nonlinear Optical and Electroactive Polymers, P. N. Prasad and D. R. Ulrich, eds. (Plenum, New York, 1988).
[Crossref]

J. F. Nye, Physical Properties of Crystals (Clarendon, London, 1957).

M. G. Kuzyk, R. C. Moore, and L. A. King, “Second-harmonic generation measurements of the elastic constant of a molecule in a polymer matrix,” submitted to Phys. Rev. A.

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

G. R. Meridith, J. G. Vandusen, and D. J. Williams, in Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, ed., ACS Symp. Ser.233, 109 (1983).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

The dashed curve represents the form of the electric potential energy as a function of orientation, the dotted curve represents the stress potential as a function of orientation, and the solid curve represents the sum. Note that the poling potential energy is minimized with orientation along the field direction θ = 0, while the energy minimum for the stress potential is perpendicular to the field direction. The equilibrium orientation for the sum of the two potentials is between θ = 0 and θ = 90°.

Fig. 2
Fig. 2

Stress potential parameter as a function of the order parameter ratio.

Fig. 3
Fig. 3

Sample geometry for second-harmonic generation. The glass substrate has a thin transparent ITO-conducting film at the doped polymer film surfaces.

Fig. 4
Fig. 4

Angular dependence of pp- (PP) and sp- (SP) polarized second-harmonic intensities.

Fig. 5
Fig. 5

Polar plot of the distribution function, G(θ), as a function of the polar angle, θ. The length of the vector from the origin to the curve shows the magnitude of the distribution function in that direction. The curves shown are for the isotropic film (short-dashed curve), the poled film with no stress (long-dashed curve) and the poled film under stress (solid curve).

Fig. 6
Fig. 6

Survey of liquid-crystalline order parameters 〈P2〉 and 〈P4〉. Lines represent constant values of a under poling. The material systems are the following: 4-cyano-4′-pentylbiphenyl as measured by two-photon dichroism (filled circles), N-(p′-methoxy-benzylidene)-p-cyanoaniline (MBBA) as measured by Raman scattering (crosses) and the third harmonic (asterisks), N-(p′-butoxy-benzylidene)-p-cyanoaniline (BBCA) MMBA as measured by Ra-man scattering (open circles), N-(p′-butoxybenzylidene)-p-n-octylaniline (40.8) as measured by Raman scattering in the smectic B phase (plus), the smectic A phase (open circles with pluses), and the nematic phase (open circles with crosses).18

Fig. 7
Fig. 7

Survey of Langmuir–Blodgett film order parameters 〈P1〉 and 〈P3〉. Lines represent constant values of a. The films are monolayers of sodium dodecylnaphthalene (SDNS) (filled circle), menocyanine (MC) (plus), hemicyanine (HC) (cross), nitrostilbene (NS) (asterisk), and HC/NS (open circle) and multilayers of COOH (open circle with plus), SO2NH2 (open circle with cross), SO2N(C2H5)2 (open circle with asterisk), and piperidinyl sulfamide (PS) (open circle with dot).18

Fig. 8
Fig. 8

Order parameters for poled polymer films under stress, where solid lines show constant values for a. The hatched region is the area defined by the two extremes of films measured with and without stress. The case a = 1/3 corresponds to a poled film with no stress.

Tables (3)

Tables Icon

Table 1 Results of Second-Harmonic Measurementsa

Tables Icon

Table 2 PMMA–DR1 Properties Used in Calculating the Local-Field-Corrected Dipole Moment, m*

Tables Icon

Table 3 Film Processing Parameters for PMMA Doped with DR1

Equations (73)

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

G ( Ω ) = exp ( - U T k T ) d Ω exp ( - U T k T ) ,
G ( θ ) = l = 0 2 l + 1 2 A l P l ( cos θ ) ,
P 0 ( x ) = 1 , P 1 ( x ) = x , P 2 ( x ) = ( 3 x 2 - 1 ) / 2 , P 3 ( x ) = ( 5 x 3 - 3 x ) / 2 , P 4 ( x ) = ( 35 x 4 - 30 x 2 + 3 ) / 8.
P l A l = - 1 + 1 d ( cos θ ) G ( θ ) P l ( cos θ ) ,
P i ( t ) = χ i ( 0 ) + χ i j ( 1 ) ( t ) E j ( t ) + χ i j k ( 2 ) ( t ) E j ( t ) E k ( t ) + ,
p I = μ I + α I J ( t ) F J ( t ) + β I J K ( t ) F J ( t ) F K ( t ) + ,
P i ( t ) = N p I ( t ) i ,
β I J K i j k = 0 2 π d ϕ 0 π sin θ d θ 0 2 π d ψ a i I a j J a k K β I J K G ( θ ) ,
χ i j k ( 2 ) = N β I J K * i j k ,
G ( Ω , E p ) = ( 1 + a 3 l m l * E 3 / k T ) exp ( - U P / k T ) exp ( - U P / k T ) d Ω ,
P i = - 1 + 1 d ( cos θ ) P i ( cos θ ) exp ( - U P / k T ) - 1 + 1 d ( cos θ ) exp ( - U P / k T ) ,
χ 333 ( 2 ) = N β z z z * ( P 1 + P 3 )
χ 113 ( 2 ) = χ 131 ( 2 ) = χ 311 ( 2 ) = N β z z z * ( ¹ / P 1 - ¹ / P 3 ) ,
χ 333 ( 2 ) = N β z z z * m z * E p k T ( 1 5 + 4 7 P 2 + 8 35 P 4 )
χ 113 ( 2 ) = χ 131 ( 2 ) = χ 311 ( 2 ) = N β z z z * m z * E p k T ( 1 15 + 1 21 P 2 - 4 3 P 4 ) ,
m z * = ( n 2 + 2 ) n 2 + 2 μ ,
P 3 P 1 = 1 - 3 a 1 + 2 a .
P 2 = 1 2 [ P 1 ( m * E p 3 k T ) - 1 ]
P 4 = 1 4 [ P 3 ( m * E p 7 k T ) - 3 P 2 ] .
G p ( θ ) = 1 l 0 l G p ( z , θ ) d z .
U p = n = 1 b 2 n P 2 n ( cos θ ) ,
U T = U P + U E = b P 2 ( cos θ ) - m * E p P 1 ( cos θ ) ,
exp [ - ( U P + U E ) k T ] - U E k T + U E k T U P k T - 1 2 U E k T ( U P k T ) 2 + 1 6 U E k T ( U P k T ) 3 - 1 24 U E k T ( U P k T ) 4 + 1 120 U E k T ( U P k T ) 5 .
exp [ - ( U P + U E ) k T ] ( m * E k T ) P 1 ( cos θ ) [ 1 - 2 5 ( b k T ) + 11 70 ( b k T ) 2 - 4 105 ( b k T ) 3 + 73 9240 ( b k T ) 4 - 79 60060 ( b k T ) 5 ] + P 3 ( cos θ ) [ - 3 5 ( b k T ) + 1 5 ( b k T ) 2 - 7 110 ( b k T ) 3 + 19 1430 ( b k T ) 4 - 41 17160 ( b k T ) 5 ] .
α P 3 P 1 = 3 5 a 3 a 1 ,
a 3 = - 3 5 ( b k T ) + 1 5 ( b k T ) 2 - 7 110 ( b k T ) 3 + 19 1430 ( b k T ) 4 - 41 17160 ( b k T ) 5
a 1 = 1 - 2 5 ( b k T ) + 11 70 ( b k T ) 2 - 4 105 ( b k T ) 3 + 73 9240 ( b k T ) 4 - 79 60060 ( b k T ) 5 .
P 1 = 2 3 m * E p k T a 1 A n ,
A n = - 1 + 1 d ( cos θ ) exp [ - 1 k T ( U P - m * · E p ) ] .
A n = ( 2 π k T 3 b ) 1 / 2 exp { b 2 k T [ 1 - 1 3 ( m * E p b ) 2 ] } × { g [ ( 3 b k T ) 1 / 2 ( m * E p 3 b - 1 ) ] - g [ ( 3 b k T ) 1 / 2 ( m * E p 3 b + 1 ) ] } ,
g ( x ) = 1 ( 2 π ) 1 / 2 x exp ( - x 2 / 2 ) d x .
α z z 11 = 1 4 π 2 d Ω α z z a 1 z a 1 z G ( θ ) ,
exp ( μ * E p k T cos θ ) ~ 1 + 1 2 ( μ * E p k T cos θ ) 2 .
α z z 11 = [ 1 - P 2 ] α z z ,
α z z 33 = 1 4 π 2 d Ω α z z a 3 z a 3 z G ( θ )
α z z 33 = ( 1 + 2 P 2 ) α z z .
α z z 11 = 1 4 π 2 d Ω α z z a 1 z a 1 z G ( θ ) 1 2 ( μ * E p k T cos θ ) 2
α z z 33 = 1 4 π 2 d Ω α z z a 3 z a 3 z G ( θ ) 1 2 ( μ * E p k T cos θ ) 2 ,
α z z 11 = α z z ( μ * E p k T ) 2 ( + 1 30 + 1 42 P 2 - 2 35 P 4 )
α z z 33 = α z z ( μ * E p k T ) 2 ( + 1 10 + 2 7 P 2 + 4 35 P 4 ) .
χ 11 ( 1 ) = χ 22 ( 1 ) = χ iso + N α z z * 11
χ 33 ( 1 ) = χ iso + N α z z * 33 ,
n 33 2 - n 11 2 2 Δ n n ¯ = 4 π N ( α z z * 33 - α z z * 11 ) ,
Δ n = 2 π N α z z * n ¯ [ P 2 + ( m * E p k T ) 2 × ( 1 15 + 11 42 P 2 + 6 35 P 4 ) ] ,
Δ n = K σ .
K = 2 π ( n 2 + 2 ) 2 ( a 1 - a 2 ) 45 n k T .
P x 2 ω = 2 d 15 E x E z , P y 2 ω = 2 d 15 E y E z , P z 2 ω = d 31 ( E x 2 + E y 2 ) + d 33 E z 2 ,
E g ω = t i E ω ,
x ^ · e ^ r E R = x ^ · e ^ f E f + x ^ · e ^ b Q
x ^ · ( k R × e ^ R ) E R = x ^ · ( k f × e ^ f ) E f + x ^ · ( k b × q ^ ) Q ,
E 2 ω = e ^ f E f + e ^ b Q ,
P 2 ω ( r , t ) = q ^ P 2 ω ( r , t )
e ^ b = q ^ - [ k b ( k b · q ^ ) / k f 2 ] ,
Q = 4 π P 2 ω n ω 2 - n 2 ω 2 ,
cos θ 2 ω E f exp ( i k f · z ^ l ) + b x Q exp ( i k b · z ^ l ) - cos θ 2 ω E f R exp ( - i k f · z ^ l ) = cos θ o E 2 ω exp ( 2 i k ω · z ^ l )
n 2 ω E f exp ( i k f · z ^ l ) + n ω Q ( cos θ ω p x - sin θ ω p z ) exp ( i k b · z ^ l ) + n 2 ω E f R exp ( i k f · z ^ l ) = n o E 2 ω exp ( 2 i k ω · z ^ l ) ,
1 n 2 ω 2 = sin 2 θ 2 ω n e 2 ( 2 ω ) + cos 2 θ 2 ω n o 2 ( 2 ω ) ,
tan θ 2 ω = n e ( 2 ω ) n i sin θ i n o ( 2 ω ) [ n e 2 ( 2 ω ) - n i 2 sin 2 θ i ] 1 / 2 .
1 n ω 2 = sin 2 θ ω n e 2 ( ω ) + cos 2 θ ω n o 2 ( ω ) ,
tan θ ω = n e ( ω ) n i sin θ i n o ( ω ) [ n e 2 ( ω ) - n i 2 sin 2 θ i ] .
E T = t o E 2 ω ,
t o = 2 n o cos θ o cos θ o + n o cos θ
P T = 512 π 3 A d 33 2 t i 4 t ω 4 T 2 ω t o 2 p 2 ( θ ) I ω 2 ( 1 n ω 2 - n 2 ω 2 ) 2 sin 2 ψ ,
T 2 ω = 2 n 2 ω cos θ 2 ω × ( n ω cos θ i + n i cos θ ω ) ( n 2 ω cos θ ω + n ω cos θ 2 ω ) ( n 2 ω cos θ 2 ω + n i cos θ i ) ( n 2 ω cos θ 2 ω + n o cos θ o ) 2 .
t i = 2 cos θ n i cos θ + cos θ i ,
t ω = 2 n i cos θ i n ω cos θ i + n i cos θ ω ,
p ( θ ) = ( a cos 2 θ ω + sin 2 θ ω ) sin θ 2 ω + 2 a cos θ ω sin θ ω cos θ 2 ω ,
t i = 2 cos θ n i cos θ i + cos θ ,
t ω = 2 n i cos θ i n ω cos θ ω + n i cos θ i ,
p ( θ ) = a sin θ 2 ω .
sin 2 ψ ~ ( π 2 l l c n ¯ ( N 2 - sin 2 θ ) 1 / 2 ) 2 ,
a = sin 2 θ ω f ( θ ) ( P T p P T s ) 1 / 2 - 3 cos 2 θ ω ,
f ( θ ) = [ ( n i cos θ p + cos θ i p ) ( n ω cos θ i p + n i cos θ ω p ) ( n i cos θ i s + cos θ s ) ( n ω cos θ ω s + n i cos θ i s ) ] 2

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