Abstract

We present an analytical model of the influence of a polymer matrix on the electro-optical and second-order nonlinear optical response of polymer films. The interaction between the chromophores and the matrix is characterized by a local molecular field with a randomly varying orientation throughout the polymer volume. The macroscopic properties of the polymer are subsequently obtained by averaging over the fluctuating direction of the field. We consider the influence of the polymer matrix on both static and dynamic properties. In both cases, our results are valid at arbitrarily large field strengths because we avoid resorting to usual expansion techniques. The mathematically demanding ac case is treated in a variational approach, and results for the frequency dependent electro-optic response are presented. The model is found to be in qualitative agreement with the observed temperature dependence of the frequency dependent electro-optic response of a Disperse Red 1/poly(methyl methacrylate) guest/host polymer if an exponentially decreasing molecular field is assumed.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Nonlinear electro-optic response. I. Steady-state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 106, 5824–5831 (1997).
    [CrossRef]
  21. J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 107, 508–523 (1997).
    [CrossRef]
  22. T. Verbiest and D. M. Burland, “Use of the Wagner function to describe poled-order relaxation processes in electrooptic polymers,” Chem. Phys. Lett. 236, 253–258 (1995).
    [CrossRef]
  23. E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
    [CrossRef]

1999 (3)

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245, 311–326 (1999).
[CrossRef]

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

1998 (2)

1997 (3)

F. Michelotti and E. Toussaere, “Pulse poling of side-chain and crosslinkable copolymers,” J. Appl. Phys. 82, 5728–5744 (1997).
[CrossRef]

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Nonlinear electro-optic response. I. Steady-state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 106, 5824–5831 (1997).
[CrossRef]

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 107, 508–523 (1997).
[CrossRef]

1996 (2)

Sandalphon, B. Kippelen, K. Meerholz, and N. Peygham-barian, “Ellipsometric measurements of poling birefringence, the Pockels effect, and the Kerr effect in high-performance photorefractive polymer composites,” Appl. Opt. 35, 2346–2354 (1996).
[CrossRef]

T. Goodson III and C. H. Wang, “Dispersion and dipolar orientational effects on the linear electroabsorption and electro-optic responses in model guest/host nonlinear optical systems,” J. Appl. Phys. 80, 6602–6609 (1996).
[CrossRef]

1995 (1)

T. Verbiest and D. M. Burland, “Use of the Wagner function to describe poled-order relaxation processes in electrooptic polymers,” Chem. Phys. Lett. 236, 253–258 (1995).
[CrossRef]

1994 (2)

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31–75 (1994).
[CrossRef]

W. E. Moerner, S. M. Silence, F. Hache, and G. C. Bjorklund, “Orientationally enhanced photorefractive effect in polymers,” J. Opt. Soc. Am. B 11, 320–330 (1994).
[CrossRef]

1991 (2)

K. D. Singer and L. A. King, “Relaxation phenomena in polymer nonlinear optical materials,” J. Appl. Phys. 70, 3251–3255 (1991).
[CrossRef]

J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142–151 (1991).
[CrossRef]

1990 (1)

1988 (1)

W. T. Coffey and S. G. McGoldrick, “Inertial effects in the theory of dielectric and Kerr effect relaxation on an assembly of non-interacting polar molecules in strong alternating fields,” Chem. Phys. 120, 1–35 (1988).
[CrossRef]

1987 (1)

1984 (1)

A. J. Nicastro and P. H. Keyes, “Electric-field-induced critical phenomena at the nematic–isotropic transition and the nematic–isotropic critical point,” Phys. Rev. A 30, 3156–3160 (1984).
[CrossRef]

1955 (1)

C. G. LeFevre and R. J. W. LeFevre, “The Kerr effect. Its measurement and application in chemistry,” Rev. Pure Appl. Chem. 5, 261–318 (1955).

1932 (1)

J. W. Beams, “Electric and magnetic double refraction,” Rev. Mod. Phys. 1, 133–160 (1932).
[CrossRef]

Aniszfeld, R.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Beams, J. W.

J. W. Beams, “Electric and magnetic double refraction,” Rev. Mod. Phys. 1, 133–160 (1932).
[CrossRef]

Bertolotti, M.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245, 311–326 (1999).
[CrossRef]

Bjorklund, G. C.

Burland, D. M.

T. Verbiest and D. M. Burland, “Use of the Wagner function to describe poled-order relaxation processes in electrooptic polymers,” Chem. Phys. Lett. 236, 253–258 (1995).
[CrossRef]

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31–75 (1994).
[CrossRef]

Chang, C.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Chen, A.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Cline, J. A.

Coffey, W. T.

W. T. Coffey and S. G. McGoldrick, “Inertial effects in the theory of dielectric and Kerr effect relaxation on an assembly of non-interacting polar molecules in strong alternating fields,” Chem. Phys. 120, 1–35 (1988).
[CrossRef]

Dalton, L. R.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Déjardin, J. L.

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Nonlinear electro-optic response. I. Steady-state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 106, 5824–5831 (1997).
[CrossRef]

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 107, 508–523 (1997).
[CrossRef]

Déjardin, P. M.

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 107, 508–523 (1997).
[CrossRef]

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Nonlinear electro-optic response. I. Steady-state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 106, 5824–5831 (1997).
[CrossRef]

Dirk, C. W.

Dureiko, R. D.

Garder, S.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Goodson III, T.

T. Goodson III and C. H. Wang, “Dispersion and dipolar orientational effects on the linear electroabsorption and electro-optic responses in model guest/host nonlinear optical systems,” J. Appl. Phys. 80, 6602–6609 (1996).
[CrossRef]

Guenther, B. D.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

Hache, F.

Harper, A. W.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Hendrickx, E.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

Herman, W. N.

Houbrecht, S.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Jen, A. K. Y.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Kalmykov, Yu. P.

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Nonlinear electro-optic response. I. Steady-state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 106, 5824–5831 (1997).
[CrossRef]

J. L. Déjardin, P. M. Déjardin, and Yu. P. Kalmykov, “Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field,” J. Chem. Phys. 107, 508–523 (1997).
[CrossRef]

Keyes, P. H.

A. J. Nicastro and P. H. Keyes, “Electric-field-induced critical phenomena at the nematic–isotropic transition and the nematic–isotropic critical point,” Phys. Rev. A 30, 3156–3160 (1984).
[CrossRef]

King, L. A.

K. D. Singer and L. A. King, “Relaxation phenomena in polymer nonlinear optical materials,” J. Appl. Phys. 70, 3251–3255 (1991).
[CrossRef]

Kippelen, B.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

Sandalphon, B. Kippelen, K. Meerholz, and N. Peygham-barian, “Ellipsometric measurements of poling birefringence, the Pockels effect, and the Kerr effect in high-performance photorefractive polymer composites,” Appl. Opt. 35, 2346–2354 (1996).
[CrossRef]

Kuzyk, M. G.

Ledoux, I.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Lee, M.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

LeFevre, C. G.

C. G. LeFevre and R. J. W. LeFevre, “The Kerr effect. Its measurement and application in chemistry,” Rev. Pure Appl. Chem. 5, 261–318 (1955).

LeFevre, R. J. W.

C. G. LeFevre and R. J. W. LeFevre, “The Kerr effect. Its measurement and application in chemistry,” Rev. Pure Appl. Chem. 5, 261–318 (1955).

Marder, S. R.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

McGoldrick, S. G.

W. T. Coffey and S. G. McGoldrick, “Inertial effects in the theory of dielectric and Kerr effect relaxation on an assembly of non-interacting polar molecules in strong alternating fields,” Chem. Phys. 120, 1–35 (1988).
[CrossRef]

Meerholz, K.

Michelotti, F.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245, 311–326 (1999).
[CrossRef]

F. Michelotti and E. Toussaere, “Pulse poling of side-chain and crosslinkable copolymers,” J. Appl. Phys. 82, 5728–5744 (1997).
[CrossRef]

Miller, R. D.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31–75 (1994).
[CrossRef]

Moerner, W. E.

Nicastro, A. J.

A. J. Nicastro and P. H. Keyes, “Electric-field-induced critical phenomena at the nematic–isotropic transition and the nematic–isotropic critical point,” Phys. Rev. A 30, 3156–3160 (1984).
[CrossRef]

Nicolao, G.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245, 311–326 (1999).
[CrossRef]

Persoons, A.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Peyghambarian, N.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

Peygham-barian, N.

Ren, A.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Robinson, B. H.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Sandalphon,

Schuele, D. E.

Silence, S. M.

Singer, K. D.

Sohn, J. E.

Staub, K.

E. Hendrickx, B. D. Guenther, Y. Zhang, J. F. Wang, K. Staub, Q. Zhang, S. R. Marder, B. Kippelen, and N. Peyghambarian, “Ellipsometric determination of the electric-field-induced birefringence of photorefractive dyes in a liquid carbazole derivative,” Chem. Phys. 245, 407–415 (1999).
[CrossRef]

Steier, W. H.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Tesi, F.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245, 311–326 (1999).
[CrossRef]

Todorova, G.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Toussaere, E.

F. Michelotti and E. Toussaere, “Pulse poling of side-chain and crosslinkable copolymers,” J. Appl. Phys. 82, 5728–5744 (1997).
[CrossRef]

Verbiest, T.

T. Verbiest and D. M. Burland, “Use of the Wagner function to describe poled-order relaxation processes in electrooptic polymers,” Chem. Phys. Lett. 236, 253–258 (1995).
[CrossRef]

Walsh, C. A.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94, 31–75 (1994).
[CrossRef]

Wang, C. H.

T. Goodson III and C. H. Wang, “Dispersion and dipolar orientational effects on the linear electroabsorption and electro-optic responses in model guest/host nonlinear optical systems,” J. Appl. Phys. 80, 6602–6609 (1996).
[CrossRef]

Wang, F.

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Chang, G. Todorova, M. Lee, R. Aniszfeld, S. Garder, A. Chen, W. H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss, and A. K. Y. Jen, “The molecular and supramolecular engineering of polymeric electro-optic materials,” Chem. Phys. 245, 35–50 (1999).
[CrossRef]

Wang, J. F.

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of a polymer sample with planar geometry. The applied field is along the z axis with the chosen coordinate system.

Fig. 2
Fig. 2

Vector diagram illustrating the angles among applied field E0, molecular field EM, and dipole moment μ. The coordinate system corresponds to the one used for Fig. 1.

Fig. 3
Fig. 3

Reduction factors r1 and r2 for the two lowest order parameters as a function of the normalized strength of the molecular field.

Fig. 4
Fig. 4

Frequency dependence of real and imaginary parts of the first reduction factor r1 (ε, ω). As indicated, different curves correspond to different values of the normalized molecular field.

Fig. 5
Fig. 5

Same as Fig. 4 but for the second reduction factor r2(ε, ω). Inset, frequency dependence of the absolute value.

Fig. 6
Fig. 6

Predicted frequency dependence of the first reduction factor for various temperatures in the case of a vanishing molecular field. The spectra are averaged over the experimental distribution of rotational diffusion times for a Disperse Red 1/poly(methyl methacrylate) guest/host polymer.

Fig. 7
Fig. 7

Same as Fig. 6 but with a nonvanishing normalized molecular field ε=3 exp[(Tg-T)/ΔT], where Tg96 °C is the glass-transition temperature and ΔT20 K. Notice the pronounced temperature dependence of the maxima in the imaginary part.

Fig. 8
Fig. 8

Comparison of Eq. (5) of the present theory with experimental birefringence data from Ref. 9. The data are for the low-Tg polymer DMNPAA:PVK:ECZ:TNF.

Equations (52)

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E0(t)=00E0(t),EM=EMsin u0cos u,
μ=μsin θ cos φsin θ sin φcos θ,
F(θ, t)=120π f(θ, φ, u, t)sin u du,
F(θ, t)=n=0 an(t)2n+12Pn(cos θ),
Δn=ΔnBR+ΔnEO,
ΔnBR(t)=Nα332nε0P2(cos θ)=Nα332nε0a2(t),
ΔnEO(t)=Nβ333(ω; ω, 0)2nε0×cos3 θ-12cos θ sin2 θE0(t)
Nβ333(ω; ω, 0)5nε0a1(t)E0(t),
χ33(2)(t)=Nβ333(2ω; ω, ω)cos3 θ3N5β333(2ω; ω, ω)a1(t),
χ31(2)(t)=N2β333(2ω; ω, ω)[cos θ-cos3 θ]N5β333(2ω; ω, ω)a1(t).
an=4π Pn(cos θ)exp(α cos θ)dΩ4π exp(α cos θ)dΩ,
U=-μ·(E0+EM)/kT=-α cos θ-ε cos δ,
exp(ε cos δ)=n=0(2n+1)in(ε)Pn(cos δ),
Pn(cos δ)=Pn(cos θ)Pn(cos u)+2 k=1(-1)kPn-k(cos θ)Pnk(cos u)cos(kφ),
i0(ε)=sinh εε,i1(ε)=-sinh εε2+cosh εε,
i2(ε)=3ε3+1εsinh ε-3 cosh εε2.
f(θ, φ, u)Z-11+α cos θ+α22cos2 θexp(ε cos δ),
Z-114πi0(ε)1+α23i12(ε)i02(ε)-12-αi1(ε)i0(ε) cos u+α232i12(ε)i02(ε)-i2(ε)i0(ε)P2(cos u).
an=12 4π0πPn(cos θ)f(θ, φ, u)sin u dudΩ.
a1α3r1(ε),a2α215r2(ε),
r1(ε)=1-i12(ε)i02(ε),
r2(ε)=1-2 i12(ε)i02(ε)+2 i12(ε)i2(ε)i03(ε)-i22(ε)i02(ε).
a1=α6[3-4 exp(-3/ε)+exp(-4/ε)],
a2=α2240{15-40 exp(-3/ε)+5 exp(-4/ε)+56 exp(-7/ε)-23 exp(-8/ε)-16 exp(-9/ε)+3 exp(-12/ε)}.
τft2f+·(fU),
U=-α cos ωt cos θ-ε cos δ.
f=Z-1 exp(ηα cos θ+ε cos δ),
η(t)=½[w(ε, ω)exp(-iωt)+w*(ε, ω)exp(iωt)].
S=02π/ω0π4πτft-2f-·(fU)2×dΩ sin u du dt.
a1(t)=α6r1(ε)[w(ε, ω)exp(-iωt)+w*(ε, ω)exp(iωt)],
a2(t)=α260r2(ε)[w(ε, ω)exp(-iωt)+w*(ε, ω)exp(iωt)]2.
τft-2f-·(fU)
=τdηdts1(Ω, u)+[η-cos(ωt)]s2(Ω, u).
s1(Ω, u)=cos θ-i1(ε)i0(ε) cos uexp(ε cos δ),
s2(Ω, u)=[2 cos θ+ε sin θ(-sin θ cos u+cos θ sin u cos φ)]exp(ε cos δ).
w(ε, ω)=I22(ε)+iωτI12(ε)I22(ε)+(ωτ)2I11(ε),
Iij(ε)=320π4πsi(Ω, u)sj(Ω, u)dΩ sin u du.
I11(ε)=12i0(2ε)1+i12(ε)i02(ε)-i1(2ε)i1(ε)i0(ε),
I12(ε)=i0(2ε)1+ε3i1(ε)i0(ε)-i1(2ε)i1(ε)i0(ε)-ε3i2(2ε)i1(ε)i0(ε),
I22(ε)=i0(2ε)2+ε23-ε23i2(2ε).
a1(t; ε=0)
=α312-iωτ exp(-iωτ)+12+iωτ exp(iωτ),
r1(ε, ω)=r1(ε)w(ε, ω),
r2(ε, ω)=r2(ε)w2(ε, ω).
exp[-(t/τ˜)β]=0g(τ)exp(-t/τ)dτ
G(Ωτ˜)0 g(τ)11-iΩτ dτ=-0exp(iΩt)ddt exp(-(t/τ˜)β)dt.
G(Ωτ˜)=β n=1(-1)n-1(n-1)!Γ(nβ)iΩτ˜nβ.
w(ε, ω)=121+I12/(I11I22)1/21-iωτ(I11/I22)1/2+1-I12/(I11I22)1/21+iωτ(I11/I22)1/2.
w˜(ε, w)=121+I12(I11I22)1/2G[ωτ˜(I11/I22)1/2]+121-I12(I11I22)1/2G[-ωτ˜(I11/I22)1/2],
H(Ωτ˜)0g(τ)1(1-iΩτ)2 dτ=β n=1(-1)n-1(n-1)!Γ(nβ)(1-nβ)iΩτ˜nβ,
ΔnBR(t)=Nα3330nε0μkT2Ep2(t),
Ep(t)=½E0{[r2(ε, ω)]1/2 exp(-iωt)+[r2*(ε, ω)]1/2 exp(iωt)}.

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