Abstract

The first hyperpolarizabilities (β) of donor–acceptor-substituted push–pull molecules are generally described by a model in which the lowest excited electronic state dominates the optical response. It is shown that within the usual assumptions accompanying this two-state model, β(-2ω; ω, ω) can be expressed in terms of a Kramers–Kronig transform of the linear optical absorption spectrum. The method is applied to p–nitroaniline and several other push–pull chromophores, and results are compared with experimental data where available. Comparison of calculated and measured frequency dispersions is suggested as a purely experimental method, requiring no additional parameters, to test the assumptions of the two-state model.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2002 (1)

A. M. Moran, D. S. Egolf, M. Blanchard-Desce, and A. M. Kelley, “Vibronic effects on solvent dependent linear and nonlinear optical properties of push-pull chromophores: julolidinemalononitrile,” J. Chem. Phys. 116, 2542–2555 (2002).
[CrossRef]

2001 (4)

A. M. Moran, C. Delbecque, and A. M. Kelley, “Solvent effects on ground and excited electronic state structures of the push-pull chromophore julolidinyl-n-N, Ndiethylthiobarbituric acid,” J. Phys. Chem. A 105, 10208–10219 (2001).
[CrossRef]

A. M. Moran and A. M. Kelley, “Solvent effects on ground and excited electronic state structures of p-nitroaniline,” J. Chem. Phys. 115, 912–924 (2001).
[CrossRef]

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288, 119–122 (2001).
[CrossRef] [PubMed]

C.-C. Hsu, S. Liu, C. C. Wang, and C. H. Wang, “Dispersion of the first hyperpolarizability of a strongly charge-transfer chromophore investigated by tunable wavelength hyper-Rayleigh scattering,” J. Chem. Phys. 114, 7103–7108 (2001).
[CrossRef]

2000 (8)

V. M. Farztdinov, R. Schanz, S. A. Kovalenko, and N. P. Ernsting, “Relaxation of optically excited p-nitroaniline: Semiempirical quantum-chemical calculations compared to femtosecond experimental results,” J. Phys. Chem. A 104, 11486–11496 (2000).
[CrossRef]

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[CrossRef] [PubMed]

M. H. Davey, V. Y. Lee, L.-M. Wu, C. R. Moylan, W. Volksen, A. Knoesen, R. D. Miller, and T. J. Marks, “Ultrahigh-temperature polymers for second-order nonlinear optics. Synthesis and properties of robust, processable, chromophore-embedded polyimides,” Chem. Mater. 12, 1679–1693 (2000).
[CrossRef]

G. Rojo, F. Agulló-López, B. Cabezón, T. Torres, S. Brasselet, I. Ledoux, and J. Zyss, “Noncentrosymmetric triazolephthalocyanines as second-order nonlinear optical chromophores,” J. Phys. Chem. B 104, 4295–4299 (2000).
[CrossRef]

C. H. Wang, “Effects of dephasing and vibronic structure on the first hyperpolarizability of strongly charge-transfer molecules,” J. Chem. Phys. 112, 1917–1924 (2000).
[CrossRef]

G. Berkovic, G. Meshulam, and Z. Kotler, “Measurement and analysis of molecular hyperpolarizability in the two-photon resonance regime,” J. Chem. Phys. 112, 3997–4003 (2000).
[CrossRef]

B. Champagne, J. M. Luis, M. Duran, J. L. Andrés, and B. Kirtman, “Anharmonicity contributions to the vibrational second hyperpolarizability of conjugated oligomers,” J. Chem. Phys. 112, 1011–1019 (2000).
[CrossRef]

V. Chernyak, S. Tretiak, and S. Mukamel, “Electronic versus vibrational optical nonlinearities of push–pull polymers,” Chem. Phys. Lett. 319, 261–264 (2000).
[CrossRef]

1999 (5)

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfeld, S. Garner, 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]

Y.-C. Shu, Z.-H. Gong, C.-F. Shu, E. M. Breitung, R. J. McMahon, G.-H. Lee, and A. K.-Y. Jen, “Synthesis and characterization of nonlinear optical chromophores with conformationally locked polyenes possessing enhanced thermal stability,” Chem. Mater. 11, 1628–1632 (1999).
[CrossRef]

S. Thayumanavan, J. Mendez, and S. R. Marder, “Synthesis of functionalized organic second-order nonlinear optical chromophores for electro-optic applications,” J. Org. Chem. 64, 4289–4297 (1999).
[CrossRef]

V. Alain, L. Thouin, M. Blanchard-Desce, U. Gubler, C. Bosshard, P. Günter, J. Muller, A. Fort, and M. Barzoukas, “Molecular engineering of push–pull phenylpolyenes for nonlinear optics: improved solubility, stability, and nonlinearities,” Adv. Mater. 11, 1210–1214 (1999).
[CrossRef]

J. N. Woodford, C. H. Wang, A. E. Asato, and R. S. H. Liu, “Hyper-Rayleigh scattering of azulenic donor-acceptor molecules at 1064 and 1907 nm,” J. Chem. Phys. 111, 4621–4628 (1999).
[CrossRef]

1998 (6)

M. Del Zoppo, M. Tommasini, C. Castiglioni, and G. Zerbi, “A relationship between Raman and infrared spectra: the case of push–pull molecules,” Chem. Phys. Lett. 287, 100–108 (1998).
[CrossRef]

R. Cammi, B. Mennucci, and J. Tomasi, “Solvent effects on linear and nonlinear optical properties of donor-acceptor polyenes: Investigation of electronic and vibrational components in terms of structure and charge distribution changes,” J. Am. Chem. Soc. 120, 8834–8847 (1998).
[CrossRef]

E. Hendrickx, K. Clays, and A. Persoons, “Hyper-Rayleigh scattering in isotropic solution,” Acc. Chem. Res. 31, 675–683 (1998).
[CrossRef]

D. M. Bishop, B. Champagne, and B. Kirtman, “Relationship between static vibrational and electronic hyperpolarizabilities of π-conjugated push–pull molecules within the two-state valence-bond charge-transfer model,” J. Chem. Phys. 109, 9987–9994 (1998).
[CrossRef]

M. Cho, “Vibrational characteristics and vibrational contributions to the nonlinear optical properties of a push–pull polyene in solution,” J. Phys. Chem. A 102, 703–707 (1998).
[CrossRef]

F. L. Huyskens, P. L. Huyskens, and A. P. Persoons, “Solvent dependence of the first hyperpolarizability of p-nitroanilines: Differences between nonspecific dipole–dipole interactions and solute–solvent H-bonds,” J. Chem. Phys. 108, 8161–8171 (1998).
[CrossRef]

1997 (3)

H.-S. Kim, M. Cho, and S.-J. Jeon, “Vibrational contributions to the molecular first and second hyperpolarizabilities of a push–pull polyene,” J. Chem. Phys. 107, 1936–1940 (1997).
[CrossRef]

M. Blanchard-Desce, V. Alain, P. V. Bedworth, S. R. Marder, A. Fort, C. Runser, M. Barzoukas, S. Lebus, and R. Wortmann, “Large quadratic hyperpolarizabilities with donor–acceptor polyenes exhibiting optimum bond length alternation: correlation between structure and hyperpolarizability,” Chem.-Eur. J. 3, 1091–1104 (1997).
[CrossRef]

M. A. Pauley and C. H. Wang, “Hyper-Rayleigh scattering measurements of nonlinear optical chromophores at 1907 nm,” Chem. Phys. Lett. 280, 544–550 (1997).
[CrossRef]

1996 (7)

O. F. J. Noordman and N. F. van Hulst, “Time-resolved hyper-Rayleigh scattering: measuring first hyperpolarizabilities β of fluorescent molecules,” Chem. Phys. Lett. 253, 145–150 (1996).
[CrossRef]

S. Stadler, G. Bourhill, and C. Bräuchle, “Problems associated with hyper-Rayleigh scattering as a means to determine the second-order polarizability of organic chromophores,” J. Phys. Chem. 100, 6927–6934 (1996).
[CrossRef]

P. Kaatz and D. P. Shelton, “Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores,” J. Chem. Phys. 105, 3918–3929 (1996).
[CrossRef]

M. Ahlheim, M. Barzoukas, P. V. Bedworth, M. Blanchard-Desce, A. Fort, Z.-Y. Hu, S. R. Marder, J. W. Perry, C. Runser, M. Staehelin, and B. Zysset, “Chromophores with strong heterocyclic acceptors: A poled polymer with a large electro-optic coefficient,” Science 271, 335–337 (1996).
[CrossRef]

A. Otomo, M. Jäger, G. I. Stegeman, M. C. Flipse, and M. Diemeer, “Key trade-offs for second harmonic generation in poled polymers,” Appl. Phys. Lett. 69, 1991–1993 (1996).
[CrossRef]

B. Champagne, “Vibrational polarizability and hyperpolarizability of p-nitroaniline,” Chem. Phys. Lett. 261, 57–65 (1996).
[CrossRef]

C. Castiglioni, M. Del Zoppo, and G. Zerbi, “Molecular first hyperpolarizability of push–pull polyenes: Relationship between electronic and vibrational contribution by a two-state model,” Phys. Rev. B 53, 13319–13325 (1996).
[CrossRef]

1995 (2)

M. Kauranen, T. Verbiest, C. Boutton, M. N. Teerenstra, K. Clays, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Supramolecular second-order nonlinearity of polymers with orientationally correlated chromophores,” Science 270, 966–969 (1995).
[CrossRef]

M. Blanchard-Desce, R. Wortmann, S. Lebus, J.-M. Lehn, and P. Krämer, “Intramolecular charge transfer in elongated donor-acceptor conjugated polyenes,” Chem. Phys. Lett. 243, 526–532 (1995).
[CrossRef]

1992 (2)

M. Stähelin, D. M. Burland, and J. E. Rice, “Solvent dependence of the second order hyperpolarizability in p-nitroaniline,” Chem. Phys. Lett. 191, 245–250 (1992).
[CrossRef]

A. Willetts, J. E. Rice, D. M. Burland, and D. P. Shelton, “Problems in the comparison of theoretical and experimental hyperpolarizabilities,” J. Chem. Phys. 97, 7590–7599 (1992).
[CrossRef]

1991 (3)

D. M. Bishop and B. Kirtman, “A perturbation method for calculating vibrational dynamic dipole polarizabilities and hyperpolarizabilities,” J. Chem. Phys. 95, 2646–2658 (1991).
[CrossRef]

D. Yaron and R. Silbey, “Vibrational contributions to third-order nonlinear optical susceptibilities,” J. Chem. Phys. 95, 563–568 (1991).
[CrossRef]

S. R. Marder, D. N. Beratan, and L.-T. Cheng, “Approaches for optimizing the first electronic hyperpolarizability of conjugated organic molecules,” Science 252, 103–106 (1991).
[CrossRef] [PubMed]

1986 (1)

K. T. Schomacker and P. M. Champion, “Investigations of spectral broadening mechanisms in biomolecules: cytochrome-c,” J. Chem. Phys. 84, 5314–5325 (1986).
[CrossRef]

1984 (1)

K. T. Schomacker, O. Bangcharoenpaurpong, and P. M. Champion, “Investigations of the Stokes and anti-Stokes resonance Raman scattering of cytochrome-c,” J. Chem. Phys. 80, 4701–4717 (1984).
[CrossRef]

1983 (4)

C. C. Teng and A. F. Garito, “Dispersion of the nonlinear second-order optical susceptibility of an organic system: p-nitroaniline,” Phys. Rev. Lett. 50, 350–352 (1983).
[CrossRef]

C. C. Teng and A. F. Garito, “Dispersion of the nonlinear second-order optical susceptibility of organic systems,” Phys. Rev. B 28, 6766–6773 (1983).
[CrossRef]

C. K. Chan and J. B. Page, “Temperature effects in the time-correlator theory of resonance Raman scattering,” J. Chem. Phys. 79, 5234–5250 (1983).
[CrossRef]

B. R. Stallard, P. M. Champion, P. R. Callis, and A. C. Albrecht, “Advances in calculating Raman excitation profiles by means of the transform theory,” J. Chem. Phys. 78, 712–722 (1983).
[CrossRef]

1977 (1)

J. L. Oudar and D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[CrossRef]

1971 (1)

B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. 20, 513–526 (1971).
[CrossRef]

Agulló-López, F.

G. Rojo, F. Agulló-López, B. Cabezón, T. Torres, S. Brasselet, I. Ledoux, and J. Zyss, “Noncentrosymmetric triazolephthalocyanines as second-order nonlinear optical chromophores,” J. Phys. Chem. B 104, 4295–4299 (2000).
[CrossRef]

Ahlheim, M.

M. Ahlheim, M. Barzoukas, P. V. Bedworth, M. Blanchard-Desce, A. Fort, Z.-Y. Hu, S. R. Marder, J. W. Perry, C. Runser, M. Staehelin, and B. Zysset, “Chromophores with strong heterocyclic acceptors: A poled polymer with a large electro-optic coefficient,” Science 271, 335–337 (1996).
[CrossRef]

Alain, V.

V. Alain, L. Thouin, M. Blanchard-Desce, U. Gubler, C. Bosshard, P. Günter, J. Muller, A. Fort, and M. Barzoukas, “Molecular engineering of push–pull phenylpolyenes for nonlinear optics: improved solubility, stability, and nonlinearities,” Adv. Mater. 11, 1210–1214 (1999).
[CrossRef]

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S. R. Marder, D. N. Beratan, and L.-T. Cheng, “Approaches for optimizing the first electronic hyperpolarizability of conjugated organic molecules,” Science 252, 103–106 (1991).
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Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288, 119–122 (2001).
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B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfeld, S. Garner, 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).
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M. H. Davey, V. Y. Lee, L.-M. Wu, C. R. Moylan, W. Volksen, A. Knoesen, R. D. Miller, and T. J. Marks, “Ultrahigh-temperature polymers for second-order nonlinear optics. Synthesis and properties of robust, processable, chromophore-embedded polyimides,” Chem. Mater. 12, 1679–1693 (2000).
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M. Del Zoppo, M. Tommasini, C. Castiglioni, and G. Zerbi, “A relationship between Raman and infrared spectra: the case of push–pull molecules,” Chem. Phys. Lett. 287, 100–108 (1998).
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A. M. Moran, C. Delbecque, and A. M. Kelley, “Solvent effects on ground and excited electronic state structures of the push-pull chromophore julolidinyl-n-N, Ndiethylthiobarbituric acid,” J. Phys. Chem. A 105, 10208–10219 (2001).
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B. Champagne, J. M. Luis, M. Duran, J. L. Andrés, and B. Kirtman, “Anharmonicity contributions to the vibrational second hyperpolarizability of conjugated oligomers,” J. Chem. Phys. 112, 1011–1019 (2000).
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A. M. Moran, D. S. Egolf, M. Blanchard-Desce, and A. M. Kelley, “Vibronic effects on solvent dependent linear and nonlinear optical properties of push-pull chromophores: julolidinemalononitrile,” J. Chem. Phys. 116, 2542–2555 (2002).
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M. Blanchard-Desce, V. Alain, P. V. Bedworth, S. R. Marder, A. Fort, C. Runser, M. Barzoukas, S. Lebus, and R. Wortmann, “Large quadratic hyperpolarizabilities with donor–acceptor polyenes exhibiting optimum bond length alternation: correlation between structure and hyperpolarizability,” Chem.-Eur. J. 3, 1091–1104 (1997).
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M. Ahlheim, M. Barzoukas, P. V. Bedworth, M. Blanchard-Desce, A. Fort, Z.-Y. Hu, S. R. Marder, J. W. Perry, C. Runser, M. Staehelin, and B. Zysset, “Chromophores with strong heterocyclic acceptors: A poled polymer with a large electro-optic coefficient,” Science 271, 335–337 (1996).
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V. Alain, L. Thouin, M. Blanchard-Desce, U. Gubler, C. Bosshard, P. Günter, J. Muller, A. Fort, and M. Barzoukas, “Molecular engineering of push–pull phenylpolyenes for nonlinear optics: improved solubility, stability, and nonlinearities,” Adv. Mater. 11, 1210–1214 (1999).
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B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfeld, S. Garner, 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).
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E. Hendrickx, K. Clays, and A. Persoons, “Hyper-Rayleigh scattering in isotropic solution,” Acc. Chem. Res. 31, 675–683 (1998).
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B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfeld, S. Garner, 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).
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A. Otomo, M. Jäger, G. I. Stegeman, M. C. Flipse, and M. Diemeer, “Key trade-offs for second harmonic generation in poled polymers,” Appl. Phys. Lett. 69, 1991–1993 (1996).
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Y.-C. Shu, Z.-H. Gong, C.-F. Shu, E. M. Breitung, R. J. McMahon, G.-H. Lee, and A. K.-Y. Jen, “Synthesis and characterization of nonlinear optical chromophores with conformationally locked polyenes possessing enhanced thermal stability,” Chem. Mater. 11, 1628–1632 (1999).
[CrossRef]

B. H. Robinson, L. R. Dalton, A. W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfeld, S. Garner, 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]

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H.-S. Kim, M. Cho, and S.-J. Jeon, “Vibrational contributions to the molecular first and second hyperpolarizabilities of a push–pull polyene,” J. Chem. Phys. 107, 1936–1940 (1997).
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M. Kauranen, T. Verbiest, C. Boutton, M. N. Teerenstra, K. Clays, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Supramolecular second-order nonlinearity of polymers with orientationally correlated chromophores,” Science 270, 966–969 (1995).
[CrossRef]

Kelley, A. M.

A. M. Moran, D. S. Egolf, M. Blanchard-Desce, and A. M. Kelley, “Vibronic effects on solvent dependent linear and nonlinear optical properties of push-pull chromophores: julolidinemalononitrile,” J. Chem. Phys. 116, 2542–2555 (2002).
[CrossRef]

A. M. Moran, C. Delbecque, and A. M. Kelley, “Solvent effects on ground and excited electronic state structures of the push-pull chromophore julolidinyl-n-N, Ndiethylthiobarbituric acid,” J. Phys. Chem. A 105, 10208–10219 (2001).
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A. M. Moran and A. M. Kelley, “Solvent effects on ground and excited electronic state structures of p-nitroaniline,” J. Chem. Phys. 115, 912–924 (2001).
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H.-S. Kim, M. Cho, and S.-J. Jeon, “Vibrational contributions to the molecular first and second hyperpolarizabilities of a push–pull polyene,” J. Chem. Phys. 107, 1936–1940 (1997).
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B. Champagne, J. M. Luis, M. Duran, J. L. Andrés, and B. Kirtman, “Anharmonicity contributions to the vibrational second hyperpolarizability of conjugated oligomers,” J. Chem. Phys. 112, 1011–1019 (2000).
[CrossRef]

D. M. Bishop, B. Champagne, and B. Kirtman, “Relationship between static vibrational and electronic hyperpolarizabilities of π-conjugated push–pull molecules within the two-state valence-bond charge-transfer model,” J. Chem. Phys. 109, 9987–9994 (1998).
[CrossRef]

D. M. Bishop and B. Kirtman, “A perturbation method for calculating vibrational dynamic dipole polarizabilities and hyperpolarizabilities,” J. Chem. Phys. 95, 2646–2658 (1991).
[CrossRef]

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M. H. Davey, V. Y. Lee, L.-M. Wu, C. R. Moylan, W. Volksen, A. Knoesen, R. D. Miller, and T. J. Marks, “Ultrahigh-temperature polymers for second-order nonlinear optics. Synthesis and properties of robust, processable, chromophore-embedded polyimides,” Chem. Mater. 12, 1679–1693 (2000).
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Figures (9)

Fig. 1
Fig. 1

Structures of the four molecules studied.

Fig. 2
Fig. 2

Linear absorption spectrum of p-nitroaniline in 1,4-dioxane (solid curve) and calculated fit to resonance-Raman intensities and absorption spectrum (dashed curve) using parameters from Ref. 23. The experimental spectrum has been extrapolated from 35 000 to 45 000 cm-1 by adding an exponential tail.

Fig. 3
Fig. 3

β values for p-nitroaniline in 1,4-dioxane as a function of incident laser frequency. Experimental values are taken from Refs. 45 (squares), 47 (circles), and 46 (triangles), rescaled as discussed in Ref. 26. Curves are calculated from Eq. (18) and the experimental (thick solid curve) and fitted (thin solid curve) absorption spectra of Fig. 1. The curve labeled “2-level model” is the result of the nonresonant two-level model, Eq. (20), with Δμ=8 D, μge=6.17 D, and ϖe=28169 cm-1. 1 esu=1 g-1/2 cm7/2 s=4.192×10-10 m4 V-1.

Fig. 4
Fig. 4

First hyperpolarizability of p-nitroaniline as a function of wave number calculated from Eq. (18) and the experimental linear absorption spectra in four different solvents. The calculated values at 1064 nm, for comparison with the experimental values in Table 1, are 56, 53, 62, and 73×10-30 esu in dioxane, methylene chloride, acetonitrile, and methanol, respectively.

Fig. 5
Fig. 5

Linear absorption spectra of JTB in chloroform and JM in acetone.

Fig. 6
Fig. 6

Solid curves, calculated hyperpolarizabilities of JTB in chloroform and JM in acetone calculated from Eq. (18) and the absorption spectra of Fig. 5, with Δμ values given in the text. Dashed curves, hyperpolarizabilities calculated from two-level model of Eq. (20).

Fig. 7
Fig. 7

Thin solid curve and left-hand axis, linear absorption spectrum of GATB (Fig. 1) plotted against half the absorption wave number. Thick solid curve and right-hand axis, modulus of hyperpolarizability β calculated from Kramers–Kronig transform of absorption spectrum with Δμ=5 D. Solid circles and right-hand axis, experimental β values from Ref. 51 scaled as discussed in the text. Dashed curve and right-hand axis, modulus of β calculated from two-level model [Eq. (20)] in the preresonant region with Δμ=5 D, μge=13.7 D, and ϖe=15250 cm-1.

Fig. 8
Fig. 8

Solid curve, linear absorption spectrum of PNA in methanol calculated from the best-fit modeling parameters of Ref. 23. Dashed curve, spectrum calculated from the same parameters but with the inhomogeneous distribution of electronic zero–zero energies (750 cm-1 standard deviation) omitted.

Fig. 9
Fig. 9

Effective hyperpolarizability of PNA in methanol calculated from Eq. (18) in two different ways. The solid curve was obtained by calculating β from the single-molecule absorption spectrum of Fig. 8, ensemble-averaging |β|2 over the inhomogeneous distribution of electronic energies (Gaussian of 750 cm-1 standard deviation), and then taking the square root to get |βeffective|. The dashed curve was obtained by directly transforming the ensemble-averaged absorption spectrum of Fig. 8.

Tables (1)

Tables Icon

Table 1 Solvent-Dependent Hyperpolarizabilities (β) of p-Nitroaniline

Equations (31)

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

μind=μ0+αE+βE2+.
βzzz(-2ω; ω, ω)
=12 uρumwgunvgugu|μ|mwmw|μ¯|nvnv|μ|gu×1(ωmw,gu+ω)(ωnv,gu+2ω)+1(ωmw,gu-2ω-iγmw,gu)(ωnv,gu-ω-iγnv,gu)+1(ωmw,gu+ω)(ωnv,gu-ω-iγnv,gu).
μ¯=μ-gu|μ|gu.
u|g|μ|e|ww|e|μ¯|e|νν|e|μ|g|u,
u|μge(Q)|ww|Δμ(Q)|νν|μge(Q)|u,
μge(Q)=μge(0)+i=13N-6μgeqiqi+ ,
Δμ(Q)=Δμ(0)+i=13N-6Δμqiqi+ .
βzzz(-2ω; ω, ω)
=μge2Δμ2 u,vρu|u|v|2×1(ω0+ωvu+ω)(ω0+ωvu+2ω)+1(ω0+ωvu-2ω-iγ)(ω0+ωvu-ω-iγ)+1(ω0+ωvu+ω)(ω0+ωvu-ω-iγ),
βzzz(-2ω; ω, ω)
=μge2Δμ2 u,vρu|u|v|2×1(ωe+ω)(ωe+2ω)+1(ωe+ω)(ωe-ω)+1(ω0+ωvu-2ω-iγ)(ωe-ω).
β(-2ω; ω, ω)
=μge2Δμ2(ωe-ω)2ωe+ω(ωe+ω)(ωe+2ω)+uρuv |v|u|2(ω0+ωv-ωu-2ω)-iγ.
σa(ω)=4π2μge2ω3cn uρuv γπ×|v|u|2(ω0+ωv-ωu-ω)2+γ2,
σa(ω)=4πω3cn Im [χ(ω)],
β(-2ω; ω, ω)
=Δμ2(ωe-ω) μge2(2ωe+ω)(ωe+ω)(ωe+2ω)+χ(2ω).
Im [χ(ω)]=3cn4π σa(ω)ω,
σA=103 ln(10)ε/NA.
Re[χ(ω)]=1πP- Im [χ(ω)](ω-ω) dω
|μge|2=9.185×10-3n[ε(ϖ)/ϖ]dϖ,
Im[χ(ϖ)]=3×103 ln(10)nε(ϖ)8π2NAϖ=1.532×10-49 nε(ϖ)ϖ,
β(-2ϖ; ϖ, ϖ)
=Δμ2πc2(ϖe-ϖ)×μge2(2ϖe+ϖ)2πc(ϖe+ϖ)(ϖe+2ϖ)+χ(2ϖ)=4.773×1024Δμ(ϖe-ϖ) 5.309×10-48μge2(2ϖe+ϖ)(ϖe+ϖ)(ϖe+2ϖ)+Re[χ(2π)]+i Im[χ(2ϖ)],
β(-2ω; ω, ω)=3μge2Δμωe22(ωe2-ω2)(ωe2-4ω2),
β(-2ϖ, ϖ, ϖ)=7.602×10-23μge2Δμϖe2(ϖe2-ϖ2)(ϖe2-4ϖ2),
μge2(0)Δμ(0)|u|v|2+i=13N-6μge(0)Δμ(0)μgeqi
×(u|qi|vv|u+u|vv|qi|u)+μge2(0)Δμqiu|ww|qi|vv|u.
β(-2ω; ω, ω)=12(ωe-ω) μge2Δμ(2ωe+ω)(ωe+ω)(ωe+2ω)+uρuvμge2Δμ|v|u|2+iμgeΔμ(μge/qi)(u|qi|vv|u+u|vv|qi|u)(ω0+ωv-ωu-2ω)-iγ+iΔμqi μge2u|vv|qi|u(ω0+ωv-ωu-2ω)-iγ,
σa(ω)=4π2ω3cn uρuv γπμge2|v|u|2+i(μge/qi)μge(u|vv|qi|u+u|qi|vv|u)(ω0+ωv-ωu-ω)2+γ2.

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