Abstract

Expansion of the orientational distribution function f(θ, t) of molecular dipoles in terms of Legendre polynomials with spherical modified Bessel functions in(μE/kT) as coefficients yields an analytic relation between the steady-state birefringence Δnz(ω) and the electro-optic coefficient χxxz(2)(-ω;ω,0) for a poled nonlinear optical system. A rotational diffusion equation, with the diffusion constant D, for the distribution function describing the onset and the decay of the induced optical and electro-optic properties is solved, with the help of the recurrence relation for spherical modified Bessel functions. It is found that the onset of birefringence involves at least two time constants, with rise times of 1/2D and 1/6D, while the onset of the electro-optic effect is dominated by the rise time of 1/2D. After removal of the dc poling field, the birefringence and the electro-optic effect are found to relax in time with different decay time constants, 1/6D and 1/2D, respectively. This is due to the difference in the tensor rank describing the birefringence and the electro-optic effect.

© 1991 Optical Society of America

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References

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  1. S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media,” IEEE J. Quantum Electron. QE-5, 562 (1969).
    [CrossRef]
  2. G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
    [CrossRef]
  3. K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
    [CrossRef]
  4. K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
    [CrossRef]
  5. J. D. Le Grange, M. G. Kuzyk, K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567 (1987).
  6. K. D. Singer, M. G. Kuzyk, 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]
  7. M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, A. Dienes, “Second-harmonic generation and adsorption studies of polymer–dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
    [CrossRef]
  8. J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, R. Lytel, “Poled electro-optic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52, 1031 (1988).
    [CrossRef]
  9. C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
    [CrossRef]
  10. D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
    [CrossRef]
  11. B. F. Levine, C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989 (1976).
    [CrossRef]
  12. L. H. Hilary, J. Yang, G. K. Wong, J. M. Torkelson, “Orientation and second harmonic generation in doped polystyrene and poly (methyl methacrylate) films,” Macromolecules 21, 526 (1988).
    [CrossRef]
  13. C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
    [CrossRef]
  14. P. Debye, Polar Molecules (Dover, New York, 1929).
  15. H. Benoit, “Contribution a l’etute de l’effet Kerr presenté par les solutions dilutées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).
  16. L. D. Landau, E. M. Lifshitz, in Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
  17. L. I. Shiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).
  18. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1952).
  19. C. C. Wang, R. Pecora, “Time-correlation functions for restricted rotational diffusion,” J. Chem. Phys. 72, 5333 (1980).
    [CrossRef]
  20. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
    [CrossRef]
  21. A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

1989

1988

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

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

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

1987

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

K. D. Singer, M. G. Kuzyk, 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

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

1982

G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
[CrossRef]

1980

C. C. Wang, R. Pecora, “Time-correlation functions for restricted rotational diffusion,” J. Chem. Phys. 72, 5333 (1980).
[CrossRef]

1976

D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
[CrossRef]

B. F. Levine, C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989 (1976).
[CrossRef]

1969

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media,” IEEE J. Quantum Electron. QE-5, 562 (1969).
[CrossRef]

1959

C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
[CrossRef]

1951

H. Benoit, “Contribution a l’etute de l’effet Kerr presenté par les solutions dilutées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

1943

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[CrossRef]

Bauer, D. R.

D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
[CrossRef]

Benoit, H.

H. Benoit, “Contribution a l’etute de l’effet Kerr presenté par les solutions dilutées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Bethea, C. G.

B. F. Levine, C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989 (1976).
[CrossRef]

Brauman, J. I.

D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[CrossRef]

Debye, P.

P. Debye, Polar Molecules (Dover, New York, 1929).

Dienes, A.

Garito, A. F.

A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

Higgins, B. G.

Hilary, L. H.

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

Kielich, S.

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media,” IEEE J. Quantum Electron. QE-5, 562 (1969).
[CrossRef]

Knoesen, A.

Kowel, S. T.

Kuzyk, M. G.

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

K. D. Singer, M. G. Kuzyk, 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]

Lalama, S. J.

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

Lalama, S. L.

K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, in Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Le Grange, J. D.

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

Levine, B. F.

B. F. Levine, C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989 (1976).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, in Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Lipscomb, G. F.

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

A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

Lytel, R.

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

A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

Marks, T. J.

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

Meredith, G. R.

G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
[CrossRef]

Minami, N.

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

Mortazavi, M. A.

O’Konski, C. T.

C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
[CrossRef]

Orttung, W. H.

C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
[CrossRef]

Pecora, R.

C. C. Wang, R. Pecora, “Time-correlation functions for restricted rotational diffusion,” J. Chem. Phys. 72, 5333 (1980).
[CrossRef]

D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
[CrossRef]

Shiff, L. I.

L. I. Shiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).

Singer, K. D.

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

K. D. Singer, M. G. Kuzyk, 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, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[CrossRef]

K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
[CrossRef]

Small, R. D.

K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
[CrossRef]

Sohn, J. E.

K. D. Singer, M. G. Kuzyk, 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, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[CrossRef]

K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
[CrossRef]

Stiller, M. A.

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

Thackara, J. I.

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

Torkelson, J. M.

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

VanDusen, J. G.

G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
[CrossRef]

Wang, C. C.

C. C. Wang, R. Pecora, “Time-correlation functions for restricted rotational diffusion,” J. Chem. Phys. 72, 5333 (1980).
[CrossRef]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1952).

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1952).

Williams, D. J.

G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
[CrossRef]

Wong, G. K.

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

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

Wu, J. W.

A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

Yang, J.

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

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

Ye, C.

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

Yoshioka, K.

C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
[CrossRef]

Ann. Phys. (Paris)

H. Benoit, “Contribution a l’etute de l’effet Kerr presenté par les solutions dilutées de macromolecules rigides,” Ann. Phys. (Paris) 6, 561 (1951).

Annu. Rev. Phys. Chem.

D. R. Bauer, J. I. Brauman, R. Pecora, “Depolarized light scattering from liquids,” Annu. Rev. Phys. Chem. 27, 443 (1976).
[CrossRef]

Appl. Phys. Lett.

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

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

IEEE J. Quantum Electron.

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media,” IEEE J. Quantum Electron. QE-5, 562 (1969).
[CrossRef]

J. Chem. Phys.

C. C. Wang, R. Pecora, “Time-correlation functions for restricted rotational diffusion,” J. Chem. Phys. 72, 5333 (1980).
[CrossRef]

B. F. Levine, C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989 (1976).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem.

C. T. O’Konski, K. Yoshioka, W. H. Orttung, “Electric properties of macromolecules. IV. Determination of electric and optical parameters from saturation of electric birefringence in solutions,” J. Phys. Chem. 63, 1558 (1959).
[CrossRef]

Macromolecules

G. R. Meredith, J. G. VanDusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385 (1982).
[CrossRef]

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

C. Ye, N. Minami, T. J. Marks, J. Yang, G. K. Wong, “Persistent, efficient frequency doubling by poled annealed films of a chromophore-functionalized poly (p-hydroxystyrene),” Macromolecules 21, 2899 (1988).
[CrossRef]

Mol. Cryst. Liq. Cryst.

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

Rev. Mod. Phys.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[CrossRef]

Other

A. F. Garito, J. W. Wu, G. F. Lipscomb, R. Lytel, “Nonlinear optical polymers: challenges and opportunities in photonics,” in Advanced Organic Solid State Materials, L. Y. Chiang, P. M. Chaikin, D. O. Cowan, eds., Vol. 173 of Materials Research Society Symposium Proceedings (Materials Research Society, Pittsburgh, Pa., 1990).

K. D. Singer, S. L. Lalama, J. E. Sohn, R. D. Small, “Electro-optic organic materials,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, New York, 1987), Vol. 1.
[CrossRef]

P. Debye, Polar Molecules (Dover, New York, 1929).

L. D. Landau, E. M. Lifshitz, in Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

L. I. Shiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1952).

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Figures (6)

Fig. 1
Fig. 1

Solid lines indicate the laboratory coordinates axes (x, y, z); dashed lines indicate the molecular principal axes (1, 2, 3), with the molecular symmetry axis along the 3-axis. The poling electric field is applied along the z axis. θ is the polar angle between the poling electric field and the molecular dipole; ϕ is the azimuthal angle. The macroscopic system has a rotational symmetry along the z axis.

Fig. 2
Fig. 2

Plot of spherical modified Bessel functions in(x) (n = 1, 2, 3, 4), used as coefficients in the expansion of the orientational distribution function f(θ, ϕ).

Fig. 3
Fig. 3

Onset of birefringence Δn(t) on the application of a dc poling field as a function of time, along with the individual fast and slow components. The long-dashed curve is the fast component, the short-dashed curve is the slow component, and the solid curve is the sum of the two.

Fig. 4
Fig. 4

Onset of the electro-optic coefficient χ z z z ( 2 ) ( t ) on the application of a dc poling field as a function of time. The contribution of the first-order Legendre polynomial P1(cos θ) is dominant.

Fig. 5
Fig. 5

Weight of two decay terms in the electro-optic coefficient χ z z z ( 2 ) ( t ) plotted as a function of x. The fast component, L3(x) − 3L1(x)/5, with time constant 1/D3, grows slowly as a function of x, while the slow component, 3L1(x)/5, with time constant 1/D1, grows rapidly as a function of x.

Fig. 6
Fig. 6

Semilogarithmic plot of the decay of the induced birefringence Δnz(t) and the induced electro-optic coefficients χ z z z ( 2 ) ( t ) and χ x x z ( 2 ) ( t ) on the removal of the poling electric field. It is seen that the birefringence decays more rapidly than the electro-optic coefficients, which is due to the difference in the ranks of susceptibility tensors.

Equations (67)

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

α 11 ( ω ) = α 22 ( ω ) α ( ω ) ,             α 33 ( ω ) α ( ω ) .
p i ω = α i i ω E i ω .
P z ω = N [ p 3 ω cos ( 3 , z ) + p 1 ω cos ( 1 , z ) + p 2 ω cos ( 2 , z ) ] f ( Ω ) d Ω = { N [ α ω + ( α ω - α ω ) cos 2 ( 3 , z ) ] f ( Ω ) d Ω } E z ω = χ z z ( 1 ) ( - ω ; ω ) E z ω ,
D = ɛ E = E + 4 π P ,             P = χ ( 1 ) E ,             n 2 = ɛ ,
n z 2 ( ω ) = ɛ z z ( ω ) = 1 + 4 π χ z z ( 1 ) ( - ω ; ω ) = 1 + 4 π N [ α ω + ( α ω - α ω ) cos 2 ( 3 , z ) ] f ( Ω ) d Ω ,
n x 2 ( ω ) = ɛ x x ( ω ) = 1 + 4 π χ x x ( 1 ) ( - ω ; ω ) = 1 + 4 π N [ α ω + ( α ω - α ω ) cos 2 ( 3 , x ) ] f ( Ω ) d Ω .
Δ [ n z 2 ( ω ) ] = 4 π N ( α ω - α ω ) × [ cos 2 ( 3 , z ) f ( Ω ) d Ω - cos 2 ( 3 , z ) f 0 ( Ω ) d Ω ] = 4 π N ( α ω - α ω ) ( cos 2 θ - 1 3 ) ;
Δ [ n x 2 ( ω ) ] = 4 π N ( α ω - α ω ) × [ cos 2 ( 3 , x ) f ( Ω ) d Ω - cos 2 ( 3 , x ) f 0 ( Ω ) d Ω ] = 4 π N ( α ω - α ω ) [ ( 1 2 - 1 2 cos 2 θ ) - 1 3 ] = - 1 2 4 π N ( α ω - α ω ) ( cos 2 θ - 1 3 ) .
Δ [ n x 2 ( ω ) ] = - 1 2 Δ [ n z 2 ( ω ) ] .
Δ [ n x ( ω ) ] = Δ [ n y ( ω ) ] = - 1 2 Δ [ n z ( ω ) ] ,
Δ [ n z ( ω ) ] = 2 π N n ( α ω - α ω ) ( cos 2 θ - 1 3 ) .
Δ [ n z ( ω ) ] ( 2 π N / n ) ( α ω - α ω ) = cos 2 θ - 1 3 = 2 3 P 2 ( cos θ ) .
p i ω 1 + ω 2 = β i j k ( - ω 1 - ω 2 ; ω 1 , ω 2 ) E j ω 1 E k ω 2 ,
p i ω = β i j k ( - ω ; ω , 0 ) E j ω E k 0 .
p 3 ω = β 333 ( - ω ; ω , 0 ) E 3 ω E 3 0 .
P z ω = N p 3 ω cos ( 3 , z ) f ( Ω ) d Ω = N β 333 ( - ω ; ω , 0 ) cos 3 ( 3 , z ) f ( Ω ) d Ω E z ω E z 0 .
P x ω = N p 3 ω cos ( 3 , x ) f ( Ω ) d Ω = N β 333 ( - ω ; ω , 0 ) cos 2 ( 3 , x ) cos ( 3 , z ) f ( Ω ) d Ω E x ω E z 0 .
χ z z z ( 2 ) ( - ω ; ω , 0 ) = N β 333 ( - ω ; ω , 0 ) cos 3 ( 3 , z ) f ( Ω ) d Ω ,
χ x x z ( 2 ) ( - ω ; ω , 0 ) = N β 333 ( - ω ; ω , 0 ) cos 2 ( 3 , x ) cos ( 3 , z ) f ( Ω ) d Ω .
cos 3 ( 3 , z ) 0 = cos 2 ( 3 , x ) cos ( 3 , z ) 0 = 0.
χ z z z ( 2 ) ( - ω ; ω , 0 ) = N β 333 ( - ω ; ω , 0 ) cos 3 θ ,
χ x x z ( 2 ) ( - ω ; ω , 0 ) = N β 333 ( - ω ; ω , 0 ) 1 2 ( cos θ - cos 3 θ ) .
χ x x z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) = 1 2 ( cos θ - cos 3 θ ) = 1 5 P 1 ( cos θ ) - 1 5 P 3 ( cos θ ) .
f ( Ω ) exp ( - U / k T ) .
U = - μ · E - 1 2 p · E = - μ E 3 0 - 1 2 ( p 1 0 E 1 0 + p 2 0 E 2 0 + p 3 0 E 3 0 ) = - μ E cos ( 3 , z ) - 1 2 [ α 0 + ( α 0 - α 0 ) cos 2 ( 3 , z ) ] E 2 = const - μ E cos θ - 1 2 ( α 0 - α 0 ) E 2 cos 2 θ = const - x k T cos θ - y k T cos 2 θ ,
x = μ E k T ,             y = ( α 0 - α 0 ) E 2 2 k T ,
y x .
f ( θ ) exp ( - U / k T ) = exp ( x cos θ ) .
exp ( x cos θ ) = n = 0 ( 2 n + 1 ) i n ( x ) P n ( cos θ ) .
i n - 1 ( x ) - i n + 1 ( x ) = [ ( 2 n + 1 ) / x ] i n ( x ) .
f ( θ ) = n = 0 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) .
P n ( cos θ ) = i n ( x ) / i 0 ( x ) .
P n - 1 ( cos θ ) - P n + 1 ( cos θ ) = [ ( 2 n + 1 / x ) ] P n ( cos θ ) .
Δ [ n z ( ω ) ] ( 2 π N / n ) ( α ω - α ω ) / χ x x z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) = 2 x 3 F 0 F 2 ω
Δ [ n z ( ω ) ] χ x x z ( 2 ) ( - ω ; ω , 0 ) = ( 2 π / n ) ( α ω - α ω ) β 333 ( - ω ; ω , 0 ) 2 x 3 F 0 F 2 ω ,
1 D f ( θ , t ) t = 1 sin θ θ { ( sin θ ) [ θ f ( θ , t ) + 1 k T U θ f ( θ , t ) ] } ,
U = - μ · E = - μ E cos θ = - x k T cos θ .
θ f ( θ ) + 1 k T U θ f ( θ ) = 0 ,
f ( θ ) exp ( - U / k T ) = exp [ μ E ( cos θ ) / k T ] = exp ( x cos θ ) .
U θ = x k T sin θ ,
1 D f ( θ , t ) t = 1 sin θ θ { ( sin θ ) [ θ f ( θ , t ) + ( x sin θ ) f ( θ , t ) ] } .
f ( θ , t ) = n = 0 b n ( t ) 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) ,
b n ( t ) = 0 at t = 0 b n ( t ) = 1 as t
b 0 ( t ) = 1             for all t
P n ( cos θ ) t = b n ( t ) i n ( x ) i 0 ( x ) .
n = 0 [ 1 D b n ( t ) θ + n ( n + 1 ) b n ( t ) ] 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) = n = 0 b n ( t ) 2 n + 1 2 i n ( x ) i 0 ( x ) x 1 sin θ θ [ ( sin 2 θ ) P n ( cos θ ) ] = n = 0 b n ( t ) 2 n + 1 2 i n ( x ) i 0 ( x ) x 1 2 n + 1 × [ ( n + 2 ) ( n + 1 ) P n + 1 ( cos θ ) - n ( n - 1 ) P n + 1 ( cos θ ) ] .
1 D b n ( t ) t + n ( n + 1 ) b n ( t ) = x [ b n - 1 ( t ) i n - 1 ( x ) i n ( x ) - b n + 1 ( t ) i n + 1 ( x ) i n ( x ) ] n ( n + 1 ) 2 n + 1 .
b 1 ( t ) = 1 - exp ( - D 1 t ) + 1 6 x i 2 ( x ) i 1 ( x ) × [ exp ( - D 1 t ) - exp ( - D 2 t ) ] ,
b 2 ( t ) = 1 - exp ( - D 2 t ) - 3 10 x i 1 ( x ) i 2 ( x ) × [ exp ( - D 1 t ) - exp ( - D 2 t ) ] + 1 5 x i 3 ( x ) i 2 ( x ) [ exp ( - D 2 t ) - exp ( - D 3 t ) ] ,
b 3 ( t ) = 1 - exp ( - D 3 t ) - 2 7 x i 2 ( x ) i 3 ( x ) × [ exp ( - D 2 t ) - exp ( - D 3 t ) ] + 3 14 x i 4 ( x ) i 3 ( x ) [ exp ( - D 3 t ) - exp ( - D 4 t ) ] ,
D n = n ( n + 1 ) D .
Δ [ n z ( ω ) ] ( 2 π N / n ) ( α ω - α ω ) ( t ) = 2 3 i 2 ( x ) i 0 ( x ) b 2 ( t ) = 2 3 i 2 ( x ) i 0 ( x ) [ 1 - exp ( - D 2 t ) ] - 1 5 x i 1 ( x ) i 0 ( x ) × [ exp ( - D 1 t ) - exp ( - D 2 t ) ] + 2 15 x i 3 ( x ) i 0 ( x ) [ exp ( - D 2 t ) - exp ( - D 3 t ) ] .
χ z z z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) ( t ) = 3 5 i 1 ( x ) i 0 ( x ) b 1 ( t ) + 2 5 i 3 ( x ) i 0 ( x ) b 3 ( t ) = 3 5 { i 1 ( x ) i 0 ( x ) [ 1 - exp ( - D 1 t ) ] + 1 6 x i 2 ( x ) i 0 ( x ) × [ exp ( - D 1 t ) - exp ( - D 2 t ) ] } + 2 5 { i 3 ( x ) i 0 ( x ) [ 1 - exp ( - D 3 t ) ] - 2 7 x i 2 ( x ) i 0 ( x ) [ exp ( - D 2 t ) - exp ( - D 3 t ) ] + 3 14 x i 4 ( x ) i 0 ( x ) [ exp ( - D 3 t ) - exp ( - D 4 t ) ] } ,
χ x x z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) ( t ) = 1 5 i 1 ( x ) i 0 ( x ) b 1 ( t ) - 1 5 i 3 ( x ) i 0 ( x ) b 3 ( t ) = 1 5 { i 1 ( x ) i 0 ( x ) [ 1 - exp ( - D 1 t ) ] + 1 6 x i 2 ( x ) i 0 ( x ) × [ exp ( - D 1 t ) - exp ( - D 2 t ) ] } - 1 5 { i 3 ( x ) i 0 ( x ) [ 1 - exp ( - D 3 t ) ] - 2 7 x i 2 ( x ) i 0 ( x ) [ exp ( - D 2 t ) - exp ( - D 3 t ) ] + 3 14 x i 4 ( x ) i 0 ( x ) [ exp ( - D 3 t ) - exp ( - D 4 t ) ] } .
1 sin θ θ { ( sin θ ) [ θ P n ( cos θ ) ] } + n ( n + 1 ) P n ( cos θ ) = 0 ,
f ( θ , t ) = n = 0 a n ( t ) 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) ,
a n ( t ) = 1 at t = 0 , a n ( t ) = 0 as t
a 0 ( t ) = 1             for all t
1 D a n ( t ) t + n ( n + 1 ) a n ( t ) = 0.
a n ( t ) = exp [ - n ( n + 1 ) D t ] ;
f ( θ , t ) = n = 0 exp [ - n ( n + 1 ) D t ] 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) .
f ( θ , 0 ) = n = 0 2 n + 1 2 i n ( x ) i 0 ( x ) P n ( cos θ ) ,
P n ( cos θ ) t = i n ( x ) i 0 ( x ) exp [ - n ( n + 1 ) D t ] = i n ( x ) i 0 ( x ) exp ( - D n t ) ,
Δ [ n z ( ω ) ] ( 2 π N / n ) ( α ω - α ω ) ( t ) = 2 3 i 2 ( x ) i 0 ( x ) exp [ - 2 ( 2 + 1 ) D t ] = [ L 2 ( x ) - 1 3 ] exp ( - D 2 t ) ,
τ = 1 / D 2 = 1 / 6 D .
χ z z z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) ( t ) = cos 3 θ t = 3 5 P 1 ( cos θ ) t + 2 5 P 3 ( cos θ ) t = 3 5 L 1 ( x ) exp ( - D 1 t ) + [ L 3 ( x ) - 3 5 L 1 ( x ) ] exp ( - D 3 t ) ,
χ x x z ( 2 ) ( - ω ; ω , 0 ) N β 333 ( - ω ; ω , 0 ) ( t ) = 1 2 ( cos θ t - cos 3 θ t ) = 1 5 L 1 ( x ) exp ( - D 1 t ) - [ 1 2 L 3 ( x ) - 3 10 L 1 ( x ) ] exp ( - D 3 t ) .

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