Abstract

Real-time monitoring of the corona-poling process that is used to create a bulk second-order nonlinear-optical susceptibility was accomplished by observing electrochromic shifts and intensity decreases of charge-transfer absorption bands in both dye-doped and covalently functionalized polymer films. By measuring small changes in the refractive-index anisotropy, the optical waveguiding technique was demonstrated to be a sensitive measure of the poling-induced order and its relaxation. The guest–host systems were formed from the dyes N,N-dimethylaminonitrostilbene, N,N-dimethylindoaniline (Phenol Blue), and 4-(N-(2-hydroxyethyl)-N-ethyl)-amino-4-nitroazobenzene (Disperse Red 1), each dissolved in a poly(methyl methacrylate) matrix. The covalently functionalized polymers contained pendant para-nitroaniline (PNA) moieties. The first, poly(N-(4-nitrophenyl)allylamine), was formed from a poly(allylamine) derivative and is called PPNA. The second was based on poly(hydroxystyrene), with PNA attachment occurring between the phenol group and the PNA hydroxyethyl group; this polymer is named PHS-MENA. The final polymer is a linear epoxy (bisphenol A) with the PNA amino N atoms forming a link in the main chain; it is called Bis A-NA. A sample calculation demonstrated the use of experimental electrochromic spectral data to estimate the electro-optic coefficients.

© 1990 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  25. E. E. Havinga, P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices: influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979); “Intramolecular charge transfer, studied by electrochromism of organic molecules in polymeric matrices,” Mol. Cryst. Liq. Cryst. 52, 145 (1979).
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    [Crossref]
  27. H. Labhart, “Survey of the methods for the determination of charge distribution in electronically excited molecules,” Experientia 22, 65 (1966).
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  28. H. Labhart, “Electrochromism,” in Advances in Chemical Physics, I. Prigogine, ed. (Interscience, New York, 1967), Vol. XIII, p. 179.
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  31. S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media”, IEEE J. Quantum Electron QE-5, 562 (1960).
  32. H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
    [Crossref]
  33. W. M. Prest, D. J. Luca, “The origin of the optical anistropy of solvent cast polymeric films,” J. Appl. Phys. 50, 6067 (1979): “The alignment of polymers during the solvent-coating process,” J. Appl. Phys. 51, 5170 (1980).
    [Crossref]
  34. S. Matsuoka, T. K. Kwei, “Physical behavior of macromolecules” in Macromolecules, F. A. Bovey, F. H. Winslow, eds. (Academic, New York, 1979), p. 339.
    [Crossref]
  35. J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1961).
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    [Crossref]
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1989 (3)

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, A. Dienes, “Second-harmonic generation and absorption studies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
[Crossref]

1988 (5)

A. Peled, L. B. Schein, “Hole mobilities that decrease with increasing electric fields in a molecularly doped polymer,” Chem. Phys. Lett. 153, 422 (1988).
[Crossref]

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]

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]

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

A. P. Marchetti, M. Scozzafava, R. H. Young, “Electrochromism of an aggregating thiapyrylium dye,” J. Chem. Phys. 89, 1827 (1988).
[Crossref]

1987 (2)

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[Crossref]

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

1986 (1)

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

1979 (2)

W. M. Prest, D. J. Luca, “The origin of the optical anistropy of solvent cast polymeric films,” J. Appl. Phys. 50, 6067 (1979): “The alignment of polymers during the solvent-coating process,” J. Appl. Phys. 51, 5170 (1980).
[Crossref]

E. E. Havinga, P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices: influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979); “Intramolecular charge transfer, studied by electrochromism of organic molecules in polymeric matrices,” Mol. Cryst. Liq. Cryst. 52, 145 (1979).
[Crossref]

1977 (1)

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[Crossref]

1976 (2)

J. K. Fischer, D. M. von Bruning, H. Labhart, “Light modulation by electrochromism,” Appl. Opt. 15, 2812 (1976).
[Crossref] [PubMed]

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516 (1976).
[Crossref]

1973 (2)

J. L. Stevenson, S. Ayers, M. M. Faktor, “The linear electrochromic effect in meta-nitroaniline,” J. Phys. Chem. Solids 34, 235 (1973).
[Crossref]

R. Ulrich, R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12, 2901 (1973).
[Crossref] [PubMed]

1972 (1)

K. Yamaoka, E. Charney, “Electric dichroism studies of macromolecules in solutions. I. Theoretical considerations of electric dichromism and electrochromism,” J. Am. Chem. Soc. 94, 8963 (1972).
[Crossref] [PubMed]

1970 (1)

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

1969 (1)

W. Liptay, “Electrochromism and solvatochromism,” Angew. Chem. Int. Ed. Eng. 8, 177 (1969).
[Crossref]

1966 (1)

H. Labhart, “Survey of the methods for the determination of charge distribution in electronically excited molecules,” Experientia 22, 65 (1966).
[Crossref]

1963 (1)

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

1960 (1)

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media”, IEEE J. Quantum Electron QE-5, 562 (1960).

Ayers, S.

J. L. Stevenson, S. Ayers, M. M. Faktor, “The linear electrochromic effect in meta-nitroaniline,” J. Phys. Chem. Solids 34, 235 (1973).
[Crossref]

Bjorklund, G. C.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Castellano, J. A.

J. A. Castellano, “Electro-optic light modulator,” U.S. Patent3,597,044 (August3, 1971).

Charney, E.

K. Yamaoka, E. Charney, “Electric dichroism studies of macromolecules in solutions. I. Theoretical considerations of electric dichromism and electrochromism,” J. Am. Chem. Soc. 94, 8963 (1972).
[Crossref] [PubMed]

Comizzoli, R. B.

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

Davies, G. J.

P. Pantelis, J. R. Hill, G. J. Davies, “Poled copoly(vinylidene fluoride-trifluorethylene) as a host for guest nonlinear optical molecules,” in P. N. Prasad, D. R. Ulrich, eds., Nonlinear Optical and Electroactive Polymers (Plenum, New York, 1988), p. 229.
[Crossref]

Dienes, A.

Dirk, C. W.

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

Donath, W. E.

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

Eich, M.

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Faktor, M. M.

J. L. Stevenson, S. Ayers, M. M. Faktor, “The linear electrochromic effect in meta-nitroaniline,” J. Phys. Chem. Solids 34, 235 (1973).
[Crossref]

Ferry, J. D.

J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1961).

Fischer, J.

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[Crossref]

Fischer, J. K.

Gordon, H. M.

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

Havinga, E. E.

E. E. Havinga, P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices: influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979); “Intramolecular charge transfer, studied by electrochromism of organic molecules in polymeric matrices,” Mol. Cryst. Liq. Cryst. 52, 145 (1979).
[Crossref]

Heller, W. R.

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

Higgins, B. G.

Hill, J. R.

P. Pantelis, J. R. Hill, G. J. Davies, “Poled copoly(vinylidene fluoride-trifluorethylene) as a host for guest nonlinear optical molecules,” in P. N. Prasad, D. R. Ulrich, eds., Nonlinear Optical and Electroactive Polymers (Plenum, New York, 1988), p. 229.
[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, M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Jerphagnon, J.

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

Jungbauer, D.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Katz, H. E.

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

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

Kielich, S.

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media”, IEEE J. Quantum Electron QE-5, 562 (1960).

King, L. A.

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

Knoesen, A.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, A. Dienes, “Second-harmonic generation and absorption studies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
[Crossref]

M. A. Mortazavi, A. Knoesen, S. T. Kowel, Department of Electrical Engineering and Computer Science, University of California, Davis, Davis, California 95616 (personal communication.)

Kowel, S. T.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, A. Dienes, “Second-harmonic generation and absorption studies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
[Crossref]

M. A. Mortazavi, A. Knoesen, S. T. Kowel, Department of Electrical Engineering and Computer Science, University of California, Davis, Davis, California 95616 (personal communication.)

Kumamoto, J.

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

Kurtz, S. K.

J. Jerphagnon, 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, M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

Kwei, T. K.

S. Matsuoka, T. K. Kwei, “Physical behavior of macromolecules” in Macromolecules, F. A. Bovey, F. H. Winslow, eds. (Academic, New York, 1979), p. 339.
[Crossref]

Labhart, H.

J. K. Fischer, D. M. von Bruning, H. Labhart, “Light modulation by electrochromism,” Appl. Opt. 15, 2812 (1976).
[Crossref] [PubMed]

H. Labhart, “Survey of the methods for the determination of charge distribution in electronically excited molecules,” Experientia 22, 65 (1966).
[Crossref]

H. Labhart, “Electrochromism,” in Advances in Chemical Physics, I. Prigogine, ed. (Interscience, New York, 1967), Vol. XIII, p. 179.
[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, 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, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

Lei, D.

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[Crossref]

Levine, B. F.

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516 (1976).
[Crossref]

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]

Liptay, W.

W. Liptay, “Electrochromism and solvatochromism,” Angew. Chem. Int. Ed. Eng. 8, 177 (1969).
[Crossref]

Looser, H.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

Luca, D. J.

W. M. Prest, D. J. Luca, “The origin of the optical anistropy of solvent cast polymeric films,” J. Appl. Phys. 50, 6067 (1979): “The alignment of polymers during the solvent-coating process,” J. Appl. Phys. 51, 5170 (1980).
[Crossref]

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]

Marchetti, A. P.

A. P. Marchetti, M. Scozzafava, R. H. Young, “Electrochromism of an aggregating thiapyrylium dye,” J. Chem. Phys. 89, 1827 (1988).
[Crossref]

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]

Matsuoka, S.

S. Matsuoka, T. K. Kwei, “Physical behavior of macromolecules” in Macromolecules, F. A. Bovey, F. H. Winslow, eds. (Academic, New York, 1979), p. 339.
[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.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins, A. Dienes, “Second-harmonic generation and absorption studies of polymer-dye films oriented by corona-onset poling at elevated temperatures,” J. Opt. Soc. Am. B 6, 733 (1989).
[Crossref]

M. A. Mortazavi, A. Knoesen, S. T. Kowel, Department of Electrical Engineering and Computer Science, University of California, Davis, Davis, California 95616 (personal communication.)

Newnham, R. E.

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[Crossref]

Pantelis, P.

P. Pantelis, J. R. Hill, G. J. Davies, “Poled copoly(vinylidene fluoride-trifluorethylene) as a host for guest nonlinear optical molecules,” in P. N. Prasad, D. R. Ulrich, eds., Nonlinear Optical and Electroactive Polymers (Plenum, New York, 1988), p. 229.
[Crossref]

Peled, A.

A. Peled, L. B. Schein, “Hole mobilities that decrease with increasing electric fields in a molecularly doped polymer,” Chem. Phys. Lett. 153, 422 (1988).
[Crossref]

Powers, J. C.

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

Prest, W. M.

W. M. Prest, D. J. Luca, “The origin of the optical anistropy of solvent cast polymeric films,” J. Appl. Phys. 50, 6067 (1979): “The alignment of polymers during the solvent-coating process,” J. Appl. Phys. 51, 5170 (1980).
[Crossref]

Reck, B.

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Runt, J.

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[Crossref]

Safari, A.

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[Crossref]

Santo, R.

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[Crossref]

Schein, L. B.

A. Peled, L. B. Schein, “Hole mobilities that decrease with increasing electric fields in a molecularly doped polymer,” Chem. Phys. Lett. 153, 422 (1988).
[Crossref]

Schilling, M. L.

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

Scozzafava, M.

A. P. Marchetti, M. Scozzafava, R. H. Young, “Electrochromism of an aggregating thiapyrylium dye,” J. Chem. Phys. 89, 1827 (1988).
[Crossref]

Sen, A.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[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, M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53, 1800 (1988).
[Crossref]

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (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]

Sohn, J. E.

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

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (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]

Stevenson, J. L.

J. L. Stevenson, S. Ayers, M. M. Faktor, “The linear electrochromic effect in meta-nitroaniline,” J. Phys. Chem. Solids 34, 235 (1973).
[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]

Swalen, J. D.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[Crossref]

Tacke, M.

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[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]

Torge, R.

Twieg, R.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

Twieg, R. J.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Ulrich, R.

van Pelt, P.

E. E. Havinga, P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices: influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979); “Intramolecular charge transfer, studied by electrochromism of organic molecules in polymeric matrices,” Mol. Cryst. Liq. Cryst. 52, 145 (1979).
[Crossref]

von Bruning, D. M.

Willson, C. G.

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

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]

Yamaoka, K.

K. Yamaoka, E. Charney, “Electric dichroism studies of macromolecules in solutions. I. Theoretical considerations of electric dichromism and electrochromism,” J. Am. Chem. Soc. 94, 8963 (1972).
[Crossref] [PubMed]

Yang, 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]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 329.

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]

Yoon, D. Y.

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

Young, R. H.

A. P. Marchetti, M. Scozzafava, R. H. Young, “Electrochromism of an aggregating thiapyrylium dye,” J. Chem. Phys. 89, 1827 (1988).
[Crossref]

Angew. Chem. Int. Ed. Eng. (1)

W. Liptay, “Electrochromism and solvatochromism,” Angew. Chem. Int. Ed. Eng. 8, 177 (1969).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, 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, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248 (1986).
[Crossref]

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]

Ber. Bunsenges. Phys. Chem. (1)

E. E. Havinga, P. van Pelt, “Electrochromism of substituted polyalkenes in polymer matrices: influence of chain length on charge transfer,” Ber. Bunsenges. Phys. Chem. 83, 816 (1979); “Intramolecular charge transfer, studied by electrochromism of organic molecules in polymeric matrices,” Mol. Cryst. Liq. Cryst. 52, 145 (1979).
[Crossref]

Chem. Phys. Lett. (2)

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516 (1976).
[Crossref]

A. Peled, L. B. Schein, “Hole mobilities that decrease with increasing electric fields in a molecularly doped polymer,” Chem. Phys. Lett. 153, 422 (1988).
[Crossref]

Experientia (1)

H. Labhart, “Survey of the methods for the determination of charge distribution in electronically excited molecules,” Experientia 22, 65 (1966).
[Crossref]

IBM J. Res. Dev. (1)

J. D. Swalen, R. Santo, M. Tacke, J. Fischer, “Properties of polymeric thin films by integrated optical techniques,” IBM J. Res. Dev. 21, 168 (1977).
[Crossref]

IEEE J. Quantum Electron (1)

S. Kielich, “Optical second-harmonic generation by electrically polarized isotropic media”, IEEE J. Quantum Electron QE-5, 562 (1960).

J. Am. Chem. Soc. (3)

H. E. Katz, K. D. Singer, J. E. Sohn, C. W. Dirk, L. A. King, H. M. Gordon, “Greatly enhanced second-order nonlinear optical susceptibilities in donor-acceptor organic molecules,” J. Am. Chem. Soc. 109, 6561 (1987).
[Crossref]

K. Yamaoka, E. Charney, “Electric dichroism studies of macromolecules in solutions. I. Theoretical considerations of electric dichromism and electrochromism,” J. Am. Chem. Soc. 94, 8963 (1972).
[Crossref] [PubMed]

J. C. Powers, W. R. Heller, J. Kumamoto, W. E. Donath, “Stark effect of phenol blue (electrochromism),” J. Am. Chem. Soc. 86, 1004 (1963).
[Crossref]

J. Appl. Phys. (4)

W. M. Prest, D. J. Luca, “The origin of the optical anistropy of solvent cast polymeric films,” J. Appl. Phys. 50, 6067 (1979): “The alignment of polymers during the solvent-coating process,” J. Appl. Phys. 51, 5170 (1980).
[Crossref]

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

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, G. C. Bjorklund, “Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field, J. Appl. Phys. 66, 3241 (1989).
[Crossref]

M. Eich, A. Sen, H. Looser, G. C. Bjorklund, J. D. Swalen, R. Twieg, D. Y. Yoon, “Corona poling and real time second harmonic generation study of a novel covalently functionalized amorphous NLO-polymer,” J. Appl. Phys. 66, 2559 (1989).
[Crossref]

J. Chem. Phys. (1)

A. P. Marchetti, M. Scozzafava, R. H. Young, “Electrochromism of an aggregating thiapyrylium dye,” J. Chem. Phys. 89, 1827 (1988).
[Crossref]

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

J. Phys. Chem. Solids (1)

J. L. Stevenson, S. Ayers, M. M. Faktor, “The linear electrochromic effect in meta-nitroaniline,” J. Phys. Chem. Solids 34, 235 (1973).
[Crossref]

Macromolecules (2)

D. Lei, J. Runt, A. Safari, R. E. Newnham, “Dielectric properties of azo dye-poly(methyl methacrylate) mixtures,” Macromolecules 20, 1797 (1987).
[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]

Other (13)

P. Pantelis, J. R. Hill, G. J. Davies, “Poled copoly(vinylidene fluoride-trifluorethylene) as a host for guest nonlinear optical molecules,” in P. N. Prasad, D. R. Ulrich, eds., Nonlinear Optical and Electroactive Polymers (Plenum, New York, 1988), p. 229.
[Crossref]

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

D. S. Chemla, J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, New York, 1987).

A. J. Heeger, J. Orenstein, D. R. Ulrich, eds., Nonlinear Optical Properties of Polymers, Mater. Res. Soc. Symp. Proc.109 (1988).

Digest of Topical Meeting on Nonlinear Optical Properties of Materials (Optical Society of America, Washington, D.C., 1988).

P. N. Prasad, D. R. Ulrich, eds., Nonlinear Optical and Electroactive Polymers (Plenum, New York, 1988).
[Crossref]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Cross-linked epoxy polymers with large and stable optical susceptibilities,” in Nonlinear Optical Properties of Organic Materials II, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147 (to be published).

M. A. Mortazavi, A. Knoesen, S. T. Kowel, Department of Electrical Engineering and Computer Science, University of California, Davis, Davis, California 95616 (personal communication.)

H. Labhart, “Electrochromism,” in Advances in Chemical Physics, I. Prigogine, ed. (Interscience, New York, 1967), Vol. XIII, p. 179.
[Crossref]

J. A. Castellano, “Electro-optic light modulator,” U.S. Patent3,597,044 (August3, 1971).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 329.

S. Matsuoka, T. K. Kwei, “Physical behavior of macromolecules” in Macromolecules, F. A. Bovey, F. H. Winslow, eds. (Academic, New York, 1979), p. 339.
[Crossref]

J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1961).

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Figures (9)

Fig. 1
Fig. 1

Chemical formulas of dyes used to produce doped films of PMMA. Each has been mentioned in the literature as a potential E-O material, displays a charge-transfer absorption band, and has electron donor (alkyl-amino) and acceptor (nitro in two cases) groups at opposite ends of the molecule. (a) N,N-dimethylamino-nitrostilbene (DANS), (b) N,N-dimethylindoaniline (Phenol Blue), (c) 4-(N-(2-hydroxyethyl)-N-ethyl)-amino-4-nitroazobenzene (Disperse Red 1).

Fig. 2
Fig. 2

Formulas of nonlinear-optical polymers synthesized in our laboratory. Each incorporates a derivative of PNA molecule (a) PPNA [poly(para-nitroaniline)], (b) PHS-MENA [poly(hydroxystyrene-methyl-ethyl-mtroanihne)], (c) bis A-NA (bisphenol A-nitroaniline). Glass transition temperatures, as determined by differential scanning calorimetry, are 125, 133, and 84°C, respectively. See text for literal names of the polymers and descriptions of syntheses.

Fig. 3
Fig. 3

Apparatus for heating and poling films. An ITO-coated substrate, onto which the sample has been spin coated, is heated with current from a variable autotransformer. When the substrate is of conductive (e.g., soda lime) glass, the film can be coated onto either side. A thermocouple pressed against the substrate is used for temperature measurement and as part of the feedback circuit for temperature control. The metered high-voltage supply is connected to a scanning–tunneling microscope (STM) needle 1–2 cm from the substrate, to produce a discharge current of a few microamperes. The substrate and needle fit inside the sample chamber of a spectrophotometer.

Fig. 4
Fig. 4

Absorption spectra of films doped with dyes in Fig. 1, before and during (after) poling. The poling field causes reorientation of the dye molecules, reducing the absorbance of light passing normally through the film. Also, the Stark shifts induced by the field cause a red shift of the absorption band. The absorbance decrease and red shift can be quantitatively analyzed to find the order parameter and poling field if the molecular dipole moments in the ground and excited states are known. The strong absorption near 300 nm is due to the substrate. (a) DANS, (b) Phenol Blue, (c) Disperse Red 1. This last system is unusual in that the poled region of the film shows an obvious color change from reddish orange to purple, since the transmission of blue light is enhanced. The color change disappears in a matter of hours as the surface potential of the film decays from the value that it had during poling.

Fig. 5
Fig. 5

Same as in Fig. 4, for the covalently functionalized polymers (a) PPNA, (b) PHS-MENA, and (c) bis A-NA.

Fig. 6
Fig. 6

Poling geometry as it affects absorption and waveguiding measurements. The angle θ measures the inclination of the ground-state dipole moments with respect to the poling field, which is normal to the film. During poling, θ = 0 is the favored alignment and decreases the absorbance A for light that is propagating normal to the film, as in the spectrophotometer. If the absorbance of light that is polarized parallel to the poling field (A) could be measured, it would be found to be greater than the unpoled value. When waveguide measurements are made of the indices of refraction, TE and TM polarizations are used, closely approximating the perpendicular and parallel directions.

Fig. 7
Fig. 7

Index of refraction of PPNA as a function of wavelength; data replotted from the paper of Eich et al.13 Our three-parameter Sellmeier fit is shown as a smooth curve through the points; it includes a nonresonant background index and one resonance of adjustable strength centered at 383 nm. See the text for the explicit formula. Fits such as this are used to determine the resonant (chromophoric) contribution to the index of refraction at the 633-nm wavelength used in the guided-wave spectroscopy. In this case the value is 0.232, which is unusually large because the number density of PNA groups is high.

Fig. 8
Fig. 8

Guided-wave mode spectra of a Disperse Red 1–PMMA film. (a) Spectra before poling, showing two guided modes of TE and TM polarization. The inset shows how the coupling angle ϕ, which increases from right to left, is defined. Larger coupling angles correspond to higher longitudinal wave vectors and higher effective indices of refraction. The knee at the right-hand side of the figure defines the onset of total internal reflection at the film–substrate interface. In these spectra the film thickness is 1.4 μm and the birefringence is negligible, as is evident from the overlapping mode positions. (b) Spectra of a freshly poled film, showing that the TE modes have moved to the right (a lower refractive index) and the TM modes have shifted to a higher index. Also, the appreciable red shift of dye molecules that are nearly aligned with the film normal has caused the tail of the absorption band to reach to the 633-nm probe wavelength, resulting in loss that broadens the modes. The film birefringence is measured as 0.024. (c) Spectra taken 2 weeks after poling, showing that the modes are again sharp and have converged considerably for a birefringence of 0.0075.

Fig. 9
Fig. 9

Order parameter Φ versus time for the various samples, as determined from the waveguide measurements. (a) Disperse Red 1–PMMA: Relaxation by 50% occurs in ~100 h for this guest–host system. (b) PPNA: Relaxation is only ~10% in 100 h. (c) PHS-MENA. (d) Bis A-NA. The quasi-logarithmic decay with time is reminiscent of other sorts of viscoelastic decay observed in ordinary polymers and described with standard free-volume theories.

Tables (3)

Tables Icon

Table 1 Bis A-NA and PHS-MENA Refractive-Index Dataa

Tables Icon

Table 2 Disperse Red 1–PMMA Waveguide Data versus Time after Polinga

Tables Icon

Table 3 PPNA Waveguide Data versus Time after Polinga

Equations (37)

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u = μ E p k T .
Δ = δ μ E p ћ c ,
A A 0 = sin 2 θ exp ( u cos θ ) d Ω / sin 2 θ d Ω exp ( u cos θ ) d Ω / d Ω ,
A A 0 = cos 2 θ exp ( u cos θ ) d Ω / cos 2 θ d Ω exp ( u cos θ ) d Ω / d Ω .
Φ A A A + 2 A = A A 3 A 0 = 1 A A 0 .
δ ν = ( Δ cos θ ) sin 2 θ exp ( u cos θ ) d Ω sin 2 θ exp ( u cos θ ) d Ω ,
δ ν = ( Δ cos θ ) cos 2 θ exp ( u cos θ ) d Ω cos 2 θ exp ( u cos θ ) d Ω ,
L n ( u ) cos n ( θ ) exp ( u cos θ ) d Ω exp ( u cos θ ) d Ω ,
L 1 ( u ) = coth u 1 u u 3 for u 1 ,
L 2 ( u ) = 1 + 2 u 2 2 u coth u 1 3 ( 1 + 2 u 2 15 ) for u 1 ,
L 3 ( u ) = ( 1 + 6 u 2 ) coth u 3 u ( 1 + 2 u 2 ) u 5 for u 1.
A A 0 = 3 L 2 ( u ) = 3 ( 1 + 2 u 2 2 u coth u ) 1 + 2 u 2 15 for u 1 ,
A A 0 = 3 2 [ 1 L 2 ( u ) ] = 3 u coth u 3 u 2 1 u 2 15 for u 1 ,
Φ = 1 + 3 u 2 3 u coth u u 2 15 for u 1.
δ ν Δ = L 3 ( u ) L 2 ( u ) = ( 1 + 6 u 2 ) coth u 3 u ( 1 + 2 u 2 ) 1 + 2 u 2 2 u coth u 3 5 u for u 1
δ ν Δ = L 1 ( u ) L 3 ( u ) 1 L 2 ( u ) = coth u 1 u + 3 u ( 1 + 2 u 2 ) ( 1 + 6 u 2 ) coth u 2 u coth u 2 u 2 = 1 coth u 1 u 3 u u 5 for u 1.
u = 24.2 μ E p T
D = 16.8 δ μ E p .
E local E applied = n 2 + 2 ( n 2 / ε ) + 2 ,
n ( ν ) = n 0 + δ n ( ν ) , δ n ( ν ) 1 ν 0 2 ν 2 .
δ n + 2 δ n = 3 δ n 0 .
Φ = δ n δ n δ n + 2 δ n .
δ n = ( 1 + 2 Φ ) δ n 0 , δ n = ( 1 Φ ) δ n 0 .
Φ = δ n TM δ n TE δ n TM + 2 δ n TE .
n ( ν ) = 1.52 + 10 8 ( 26 110 ) 2 ν 2 .
n ( ν ) = 1.5402 + 0.4449 × 10 8 ( 25 641 ) 2 ν 2 ,
n ( ν ) = 1.5379 + 0.4428 × 10 8 ( 25 000 ) 2 ν 2
r d ( n 2 ) d E = 2 n 3 d n d E = 2 n 3 d n d ν 0 d ν 0 d E ,
r = 2 n 3 δ n 0 ( 1 + 2 Φ ) 2 ν 0 ν 0 2 ν 2 δ ν E p
r = 2 n 3 δ n 0 ( 1 Φ ) 2 ν 0 ν 0 2 ν 2 δ ν E p .
r r = 1 + 2 Φ 1 Φ δ ν δ ν .
δ ν δ ν = ( 1 + 6 u 2 ) coth u 3 u ( 1 + 2 u 2 ) ( 1 + 2 u 2 2 u coth u ) ( 1 coth u 1 u 3 u ) .
E local E mod = n 2 + 2 3 ~ 1.5.
r = 2 ( 1.51 ) 3 ( 0.021 ) ( 0.74 ) 720 4610 1 4.8 cm MV = 2.9 × 10 12 m / V ,
r = 1.52 0.74 ( 1.84 ) ( 2.9 × 10 12 ) = 11 × 10 12 m / V ,
r = 2 ( 1.82 ) 3 ( 0.30 ) ( 0.82 ) 270 7320 1 4.0 cm MV = 7.5 × 10 12 m / V ;
r = ( 1.66 ) ( 2.10 ) ( 7.5 ) × 10 12 m / V = 26 + 10 12 m / V ,

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