Abstract

To understand the electronic mechanisms for the electric-field sensitive second-harmonic response of di-8-4-(2-[6-(dioctylamino)-2-naphthalenyl]ethneyl)-1-(3-sulfopropyl) pryridinium inner salt in biological membranes, we have measured its nonlinear optical hyperpolarizability using a combination of hyper-Rayleigh scattering and linear optical properties in solvent environments of various polarity. Changes in the first hyperpolarizabilities are discussed by modeling experimental inputs from the solvatochromic effect in linear absorption and appropriate dispersion relations near the two-photon resonance. We find that both frequency-dependent and frequency-independent effects make significant contributions to the response, and the strongest response is on the low-energy side of the two-photon resonance, while the weakest response is on the high-energy side of the two-photon resonance.

© 2007 Optical Society of America

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2006 (3)

L. Sacconi, D. A. Dombeck, and W. W. Webb, "Overcoming photodamage in second-harmonic generation microscopy: real-time optical recording of neuronal action potentials," Proc. Natl. Acad. Sci. U.S.A. 103, 3124-3129 (2006).
[CrossRef] [PubMed]

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

T. Pons and J. Mertz, "Membrane potential detection with second-harmonic generation and two-photon excited fluorescence: a theoretical comparison," Opt. Commun. 258, 203-209 (2006).
[CrossRef]

2005 (3)

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

J. A. N. Fisher, B. M. Salzberg, and A. G. Yodh, "Near infrared two-photon excitation cross-sections of voltage-sensitive dyes," J. Neurosci. Methods 148, 94-102 (2005).
[CrossRef] [PubMed]

D. A. Dombeck, L. Sacconi, M. Blanchard-Desce, and W. W. Webb, "Optical recording of fast neuronal membrane potential transients in acute mammalian brain slice: by second-harmonic generation microscopy," J. Neurophysiol. 94, 3628-3636 (2005).
[CrossRef] [PubMed]

2004 (3)

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

B. Kuhn, P. Fromherz, and W. Denk, "High sensitivity of stark-shift voltage-sensing dyes by one- or two-photon excitation near the red spectral edge," Biophys. J. 87, 631-639 (2004).
[CrossRef] [PubMed]

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

2003 (2)

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

L. Moreaux, T. Pons, V. Dambrin, M. Blanchard-Desce, and J. Mertz, "Electro-optic response of second-harmonic generation membrane potential sensors," Opt. Lett. 28, 625-627 (2003).
[CrossRef] [PubMed]

2002 (2)

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. Kelley, "Frequency-dependent first hyperpolarizabilities from linear absorption spectra," J. Opt. Soc. Am. B 19, 1890-1900 (2002).
[CrossRef]

2001 (2)

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,N′-diethylthiobarbituric 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]

2000 (3)

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]

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]

L. Moreaux, O. Sandre, and J. Mertz, "Membrane imaging by second-harmonic generation microscopy," J. Opt. Soc. Am. B 17, 1685-1694 (2000).
[CrossRef]

1999 (1)

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

1998 (1)

T. W. Chui and K. Y. Wong, "Study of hyper-Rayleigh scattering and two-photon absorption induced fluorescence from crystal violet," J. Chem. Phys. 109, 1391-1396 (1998).
[CrossRef]

1996 (2)

L. M. Loew, "Potentiometric dyes: imaging electrical activity of cell membranes," Pure Appl. Chem. 68, 1405-1409 (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]

1995 (1)

P. Fromherz, "Monopole dipole model for symmetrical solvatochromism of hemicyanine dyes," J. Phys. Chem. 99, 7188-7192 (1995).
[CrossRef]

1993 (3)

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

G. J. T. Heesink, A. G. T. Ruiter, N. F. Vanhulst, and B. Bolger, "Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh-scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

1992 (3)

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Rev. Sci. Instrum. 63, 3285-3289 (1992).
[CrossRef]

R. S. Bedlack, M. D. Wei, and L. M. Loew, "Localized membrane depolarizations and localized calcium influx during electric field-guided neurite growth," Neuron 9, 393-403 (1992).
[CrossRef] [PubMed]

1991 (1)

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Phys. Rev. Lett. 66, 2980-2983 (1991).
[CrossRef] [PubMed]

1987 (1)

B. S. Brunschwig, S. Ehrenson, and N. Sutin, "Solvent reorganization in optical and thermal electron-transfer processes--solvatochromism and intramolecular electron-transfer barriers in spheroidal molecules," J. Phys. Chem. 91, 4714-4723 (1987).
[CrossRef]

1985 (2)

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

E. Fluhler, V. G. Burnham, and L. M. Loew, "Spectra, membrane-binding, and potentiometric responses of new charge shift probes," Biochemistry 24, 5749-5755 (1985).
[CrossRef] [PubMed]

1984 (1)

A. Hassner, D. Birnbaum, and L. M. Loew, "Charge-shift probes of membrane potential. Synthesis," J. Org. Chem. 49, 2546 (1984).
[CrossRef]

1981 (1)

L. M. Loew and L. L. Simpson, "Charge-shift probes of membrane-potential--probable electrochromic mechanism for para-aminostyrylpyridinium probes on a hemispherical lipid bilayer," Biophys. J. 34, 353-365 (1981).
[CrossRef] [PubMed]

1978 (1)

L. M. Loew, G. W. Bonneville, and J. Surow, "Charge shift optical probes of membrane potential. Theory," Biochemistry 17, 4065-4071 (1978).
[CrossRef] [PubMed]

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]

1969 (2)

D. B. Siano and D. E. Metzler, "Band shapes of the electronic spectra of complex molecules," J. Chem. Phys. 51, 1856-1861 (1969).
[CrossRef]

W. Liptay, "Electrochromism and solvatochromism," Angew. Chem., Int. Ed. 8, 177-188 (1969).
[CrossRef]

Bedlack, R. S.

R. S. Bedlack, M. D. Wei, and L. M. Loew, "Localized membrane depolarizations and localized calcium influx during electric field-guided neurite growth," Neuron 9, 393-403 (1992).
[CrossRef] [PubMed]

Berkovic, G.

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]

Bezanilla, F.

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

Birnbaum, D.

A. Hassner, D. Birnbaum, and L. M. Loew, "Charge-shift probes of membrane potential. Synthesis," J. Org. Chem. 49, 2546 (1984).
[CrossRef]

Blanchard-Desce, M.

D. A. Dombeck, L. Sacconi, M. Blanchard-Desce, and W. W. Webb, "Optical recording of fast neuronal membrane potential transients in acute mammalian brain slice: by second-harmonic generation microscopy," J. Neurophysiol. 94, 3628-3636 (2005).
[CrossRef] [PubMed]

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

L. Moreaux, T. Pons, V. Dambrin, M. Blanchard-Desce, and J. Mertz, "Electro-optic response of second-harmonic generation membrane potential sensors," Opt. Lett. 28, 625-627 (2003).
[CrossRef] [PubMed]

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]

Blunck, R.

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

Bolger, B.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Vanhulst, and B. Bolger, "Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh-scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Bonneville, G. W.

L. M. Loew, G. W. Bonneville, and J. Surow, "Charge shift optical probes of membrane potential. Theory," Biochemistry 17, 4065-4071 (1978).
[CrossRef] [PubMed]

Boudreau, D. M.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

Bouevitch, O.

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

Bredas, J. L.

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

Brunschwig, B. S.

B. S. Brunschwig, S. Ehrenson, and N. Sutin, "Solvent reorganization in optical and thermal electron-transfer processes--solvatochromism and intramolecular electron-transfer barriers in spheroidal molecules," J. Phys. Chem. 91, 4714-4723 (1987).
[CrossRef]

Burnham, V. G.

E. Fluhler, V. G. Burnham, and L. M. Loew, "Spectra, membrane-binding, and potentiometric responses of new charge shift probes," Biochemistry 24, 5749-5755 (1985).
[CrossRef] [PubMed]

Campagnola, P. J.

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

Chemla, D. S.

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]

Choi, H.

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

Chui, T. W.

T. W. Chui and K. Y. Wong, "Study of hyper-Rayleigh scattering and two-photon absorption induced fluorescence from crystal violet," J. Chem. Phys. 109, 1391-1396 (1998).
[CrossRef]

Clays, K.

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Rev. Sci. Instrum. 63, 3285-3289 (1992).
[CrossRef]

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Phys. Rev. Lett. 66, 2980-2983 (1991).
[CrossRef] [PubMed]

Cohen, L. B.

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

Dambrin, V.

Delbecque, C.

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,N′-diethylthiobarbituric acid," J. Phys. Chem. A 105, 10208-10219 (2001).
[CrossRef]

Denk, W.

B. Kuhn, P. Fromherz, and W. Denk, "High sensitivity of stark-shift voltage-sensing dyes by one- or two-photon excitation near the red spectral edge," Biophys. J. 87, 631-639 (2004).
[CrossRef] [PubMed]

Dhenaut, C.

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

Dombeck, D. A.

L. Sacconi, D. A. Dombeck, and W. W. Webb, "Overcoming photodamage in second-harmonic generation microscopy: real-time optical recording of neuronal action potentials," Proc. Natl. Acad. Sci. U.S.A. 103, 3124-3129 (2006).
[CrossRef] [PubMed]

D. A. Dombeck, L. Sacconi, M. Blanchard-Desce, and W. W. Webb, "Optical recording of fast neuronal membrane potential transients in acute mammalian brain slice: by second-harmonic generation microscopy," J. Neurophysiol. 94, 3628-3636 (2005).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1998).

Egolf, D. S.

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]

Ehrenson, S.

B. S. Brunschwig, S. Ehrenson, and N. Sutin, "Solvent reorganization in optical and thermal electron-transfer processes--solvatochromism and intramolecular electron-transfer barriers in spheroidal molecules," J. Phys. Chem. 91, 4714-4723 (1987).
[CrossRef]

Eisenthal, K. B.

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

Fisher, J. A. N.

J. A. N. Fisher, B. M. Salzberg, and A. G. Yodh, "Near infrared two-photon excitation cross-sections of voltage-sensitive dyes," J. Neurosci. Methods 148, 94-102 (2005).
[CrossRef] [PubMed]

Fluhler, E.

E. Fluhler, V. G. Burnham, and L. M. Loew, "Spectra, membrane-binding, and potentiometric responses of new charge shift probes," Biochemistry 24, 5749-5755 (1985).
[CrossRef] [PubMed]

Fromherz, P.

B. Kuhn, P. Fromherz, and W. Denk, "High sensitivity of stark-shift voltage-sensing dyes by one- or two-photon excitation near the red spectral edge," Biophys. J. 87, 631-639 (2004).
[CrossRef] [PubMed]

P. Fromherz, "Monopole dipole model for symmetrical solvatochromism of hemicyanine dyes," J. Phys. Chem. 99, 7188-7192 (1995).
[CrossRef]

Ghoneim, N.

P. Suppan and N. Ghoneim, Solvatochromism (Royal Society of Chemistry, 1997).

Hassner, A.

A. Hassner, D. Birnbaum, and L. M. Loew, "Charge-shift probes of membrane potential. Synthesis," J. Org. Chem. 49, 2546 (1984).
[CrossRef]

Heath, J. R.

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

Heesink, G. J. T.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Vanhulst, and B. Bolger, "Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh-scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Jiang, J.

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

Jin, L.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

Kaatz, P.

P. Kaatz and D. P. Shelton, "Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores," J. Chem. Phys. 105, 3918-3929 (1996).
[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. Kelley, "Frequency-dependent first hyperpolarizabilities from linear absorption spectra," J. Opt. Soc. Am. B 19, 1890-1900 (2002).
[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]

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,N′-diethylthiobarbituric acid," J. Phys. Chem. A 105, 10208-10219 (2001).
[CrossRef]

Klessinger, M.

M. Klessinger and J. Michl, Excited States and Photochemistry of Organic Molecules (VCH, 1995).

Kotler, Z.

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]

Kuhn, B.

B. Kuhn, P. Fromherz, and W. Denk, "High sensitivity of stark-shift voltage-sensing dyes by one- or two-photon excitation near the red spectral edge," Biophys. J. 87, 631-639 (2004).
[CrossRef] [PubMed]

Ledoux, I.

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

Lewis, A.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

Li, H.

H. Li, "Nonlinear optical studies of potential-sensitive dyes," Ph.D. dissertation (Case Western Reserve University, 2007).

Liptay, W.

W. Liptay, "Electrochromism and solvatochromism," Angew. Chem., Int. Ed. 8, 177-188 (1969).
[CrossRef]

Loew, L. M.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

L. M. Loew, "Potentiometric dyes: imaging electrical activity of cell membranes," Pure Appl. Chem. 68, 1405-1409 (1996).
[CrossRef]

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

R. S. Bedlack, M. D. Wei, and L. M. Loew, "Localized membrane depolarizations and localized calcium influx during electric field-guided neurite growth," Neuron 9, 393-403 (1992).
[CrossRef] [PubMed]

E. Fluhler, V. G. Burnham, and L. M. Loew, "Spectra, membrane-binding, and potentiometric responses of new charge shift probes," Biochemistry 24, 5749-5755 (1985).
[CrossRef] [PubMed]

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

A. Hassner, D. Birnbaum, and L. M. Loew, "Charge-shift probes of membrane potential. Synthesis," J. Org. Chem. 49, 2546 (1984).
[CrossRef]

L. M. Loew and L. L. Simpson, "Charge-shift probes of membrane-potential--probable electrochromic mechanism for para-aminostyrylpyridinium probes on a hemispherical lipid bilayer," Biophys. J. 34, 353-365 (1981).
[CrossRef] [PubMed]

L. M. Loew, G. W. Bonneville, and J. Surow, "Charge shift optical probes of membrane potential. Theory," Biochemistry 17, 4065-4071 (1978).
[CrossRef] [PubMed]

Mertz, J.

T. Pons and J. Mertz, "Membrane potential detection with second-harmonic generation and two-photon excited fluorescence: a theoretical comparison," Opt. Commun. 258, 203-209 (2006).
[CrossRef]

L. Moreaux, T. Pons, V. Dambrin, M. Blanchard-Desce, and J. Mertz, "Electro-optic response of second-harmonic generation membrane potential sensors," Opt. Lett. 28, 625-627 (2003).
[CrossRef] [PubMed]

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

L. Moreaux, O. Sandre, and J. Mertz, "Membrane imaging by second-harmonic generation microscopy," J. Opt. Soc. Am. B 17, 1685-1694 (2000).
[CrossRef]

Meshulam, G.

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]

Metzler, D. E.

D. B. Siano and D. E. Metzler, "Band shapes of the electronic spectra of complex molecules," J. Chem. Phys. 51, 1856-1861 (1969).
[CrossRef]

Meyers, F.

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

Michl, J.

M. Klessinger and J. Michl, Excited States and Photochemistry of Organic Molecules (VCH, 1995).

Millard, A. C.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1998).

Mongin, O.

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

Moran, 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,N′-diethylthiobarbituric 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]

Moreaux, L.

Nemet, B.

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

Nuriya, M.

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

Obaid, A. L.

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

Oudar, J. L.

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]

Persoons, A.

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Rev. Sci. Instrum. 63, 3285-3289 (1992).
[CrossRef]

K. Clays and A. Persoons, "Hyper-Rayleigh scattering in solution," Phys. Rev. Lett. 66, 2980-2983 (1991).
[CrossRef] [PubMed]

Pierce, B. M.

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

Pinevsky, I.

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

Pons, T.

T. Pons and J. Mertz, "Membrane potential detection with second-harmonic generation and two-photon excited fluorescence: a theoretical comparison," Opt. Commun. 258, 203-209 (2006).
[CrossRef]

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

L. Moreaux, T. Pons, V. Dambrin, M. Blanchard-Desce, and J. Mertz, "Electro-optic response of second-harmonic generation membrane potential sensors," Opt. Lett. 28, 625-627 (2003).
[CrossRef] [PubMed]

Ries, R. S.

R. S. Ries, H. Choi, R. Blunck, F. Bezanilla, and J. R. Heath, "Black lipid membranes: visualizing the structure, dynamics, and substrate dependence of membranes," J. Phys. Chem. B 108, 16040-16049 (2004).
[CrossRef]

Ruiter, A. G. T.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Vanhulst, and B. Bolger, "Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh-scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Sacconi, L.

L. Sacconi, D. A. Dombeck, and W. W. Webb, "Overcoming photodamage in second-harmonic generation microscopy: real-time optical recording of neuronal action potentials," Proc. Natl. Acad. Sci. U.S.A. 103, 3124-3129 (2006).
[CrossRef] [PubMed]

D. A. Dombeck, L. Sacconi, M. Blanchard-Desce, and W. W. Webb, "Optical recording of fast neuronal membrane potential transients in acute mammalian brain slice: by second-harmonic generation microscopy," J. Neurophysiol. 94, 3628-3636 (2005).
[CrossRef] [PubMed]

Salzberg, B. M.

J. A. N. Fisher, B. M. Salzberg, and A. G. Yodh, "Near infrared two-photon excitation cross-sections of voltage-sensitive dyes," J. Neurosci. Methods 148, 94-102 (2005).
[CrossRef] [PubMed]

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

Sandre, O.

Shelton, D. P.

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

Siano, D. B.

D. B. Siano and D. E. Metzler, "Band shapes of the electronic spectra of complex molecules," J. Chem. Phys. 51, 1856-1861 (1969).
[CrossRef]

Simpson, L. L.

L. M. Loew and L. L. Simpson, "Charge-shift probes of membrane-potential--probable electrochromic mechanism for para-aminostyrylpyridinium probes on a hemispherical lipid bilayer," Biophys. J. 34, 353-365 (1981).
[CrossRef] [PubMed]

Suppan, P.

P. Suppan and N. Ghoneim, Solvatochromism (Royal Society of Chemistry, 1997).

Surow, J.

L. M. Loew, G. W. Bonneville, and J. Surow, "Charge shift optical probes of membrane potential. Theory," Biochemistry 17, 4065-4071 (1978).
[CrossRef] [PubMed]

Sutin, N.

B. S. Brunschwig, S. Ehrenson, and N. Sutin, "Solvent reorganization in optical and thermal electron-transfer processes--solvatochromism and intramolecular electron-transfer barriers in spheroidal molecules," J. Phys. Chem. 91, 4714-4723 (1987).
[CrossRef]

Van, T. C.

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

Vanhulst, N. F.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Vanhulst, and B. Bolger, "Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh-scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Wang, C. H.

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]

Webb, W. W.

L. Sacconi, D. A. Dombeck, and W. W. Webb, "Overcoming photodamage in second-harmonic generation microscopy: real-time optical recording of neuronal action potentials," Proc. Natl. Acad. Sci. U.S.A. 103, 3124-3129 (2006).
[CrossRef] [PubMed]

D. A. Dombeck, L. Sacconi, M. Blanchard-Desce, and W. W. Webb, "Optical recording of fast neuronal membrane potential transients in acute mammalian brain slice: by second-harmonic generation microscopy," J. Neurophysiol. 94, 3628-3636 (2005).
[CrossRef] [PubMed]

Wei, M. D.

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

R. S. Bedlack, M. D. Wei, and L. M. Loew, "Localized membrane depolarizations and localized calcium influx during electric field-guided neurite growth," Neuron 9, 393-403 (1992).
[CrossRef] [PubMed]

Wong, K. Y.

T. W. Chui and K. Y. Wong, "Study of hyper-Rayleigh scattering and two-photon absorption induced fluorescence from crystal violet," J. Chem. Phys. 109, 1391-1396 (1998).
[CrossRef]

Wuskell, J. P.

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

Yodh, A. G.

J. A. N. Fisher, B. M. Salzberg, and A. G. Yodh, "Near infrared two-photon excitation cross-sections of voltage-sensitive dyes," J. Neurosci. Methods 148, 94-102 (2005).
[CrossRef] [PubMed]

Yuste, R.

M. Nuriya, J. Jiang, B. Nemet, K. B. Eisenthal, and R. Yuste, "Imaging membrane potential in dendritic spines," Proc. Natl. Acad. Sci. U.S.A. 103, 786-790 (2006).
[CrossRef] [PubMed]

Zyss, J.

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

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

W. Liptay, "Electrochromism and solvatochromism," Angew. Chem., Int. Ed. 8, 177-188 (1969).
[CrossRef]

Biochemistry (2)

E. Fluhler, V. G. Burnham, and L. M. Loew, "Spectra, membrane-binding, and potentiometric responses of new charge shift probes," Biochemistry 24, 5749-5755 (1985).
[CrossRef] [PubMed]

L. M. Loew, G. W. Bonneville, and J. Surow, "Charge shift optical probes of membrane potential. Theory," Biochemistry 17, 4065-4071 (1978).
[CrossRef] [PubMed]

Biophys. J. (6)

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

O. Bouevitch, A. Lewis, I. Pinevsky, J. P. Wuskell, and L. M. Loew, "Probing membrane-potential with nonlinear optics," Biophys. J. 65, 672-679 (1993).
[CrossRef] [PubMed]

L. M. Loew, L. B. Cohen, B. M. Salzberg, A. L. Obaid, and F. Bezanilla, "Charge-shift probes of membrane-potential--characterization of aminostyrylpyridinium dyes on the squid giant-axon," Biophys. J. 47, 71-77 (1985).
[CrossRef] [PubMed]

L. M. Loew and L. L. Simpson, "Charge-shift probes of membrane-potential--probable electrochromic mechanism for para-aminostyrylpyridinium probes on a hemispherical lipid bilayer," Biophys. J. 34, 353-365 (1981).
[CrossRef] [PubMed]

A. C. Millard, L. Jin, M. D. Wei, J. P. Wuskell, A. Lewis, and L. M. Loew, "Sensitivity of second harmonic generation from styryl dyes to transmembrane potential," Biophys. J. 86, 1169-1176 (2004).
[CrossRef] [PubMed]

B. Kuhn, P. Fromherz, and W. Denk, "High sensitivity of stark-shift voltage-sensing dyes by one- or two-photon excitation near the red spectral edge," Biophys. J. 87, 631-639 (2004).
[CrossRef] [PubMed]

Chem. Phys. (1)

J. Zyss, T. C. Van, C. Dhenaut, and I. Ledoux, "Harmonic Rayleigh-scattering from nonlinear octupolar molecular media--the case of crystal violet," Chem. Phys. 177, 281-296 (1993).
[CrossRef]

J. Am. Chem. Soc. (1)

J. L. Bredas, F. Meyers, B. M. Pierce, and J. Zyss, "On the 2nd-order polarizability of conjugated pi-electron molecules with octupolar symmetry--the case of triaminotrinitrobenzene," J. Am. Chem. Soc. 114, 4928-4929 (1992).
[CrossRef]

J. Biomed. Opt. (1)

T. Pons, L. Moreaux, O. Mongin, M. Blanchard-Desce, and J. Mertz, "Mechanisms of membrane potential sensing with second-harmonic generation microscopy," J. Biomed. Opt. 8, 428-431 (2003).
[CrossRef] [PubMed]

J. Chem. Phys. (8)

T. W. Chui and K. Y. Wong, "Study of hyper-Rayleigh scattering and two-photon absorption induced fluorescence from crystal violet," J. Chem. Phys. 109, 1391-1396 (1998).
[CrossRef]

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]

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]

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]

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

D. B. Siano and D. E. Metzler, "Band shapes of the electronic spectra of complex molecules," J. Chem. Phys. 51, 1856-1861 (1969).
[CrossRef]

J. Membr. Biol. (1)

A. C. Millard, L. Jin, J. P. Wuskell, D. M. Boudreau, A. Lewis, and L. M. Loew, "Wavelength- and time-dependence of potentiometric non-linear optical signals from styryl dyes," J. Membr. Biol. 208, 103-111 (2005).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Structure of di-8-ANEPPS.

Fig. 2
Fig. 2

Experimental setup for HRS measurements as described in the text.

Fig. 3
Fig. 3

Linear absorption spectra of di-8-ANEPPS in solutions A, B, C, D, and E. Absorption spectrum shifts toward the red as the polarity decreases from A to E.

Fig. 4
Fig. 4

Solvent polarity- ( F ) dependent measured quantities. (a) Absorption frequency, (b) transition moment, (c) spectral half-width, and (d) first hyperpolarizability. (d) Solid square 800 nm ; open square, 900 nm ; triangle, 976 nm ; solid circle, 1020 nm ; and open circle, 1064 nm .

Fig. 5
Fig. 5

Hyperpolarizability β of di-8-ANEPPS in DMSO solution (dots), F = 0.220 , and fit to Eq. (5) (solid curve) and (6) (dashed curve).

Fig. 6
Fig. 6

Linear absorption spectra of di-8-ANEPPS in DMSO solution (solution A, solid curve). Dashed curve is the lognormal fit for the first electronic absorption band. Dotted–dashed curve is the CDHO line-shape fit.

Fig. 7
Fig. 7

Plot of Δ μ versus F and the corresponding linear fit.

Fig. 8
Fig. 8

Relative second-harmonic sensitivity as a function of frequency. Solid curve is the model of Eq. (7) (right-hand side), and data are the left-hand side of Eq. (7).

Fig. 9
Fig. 9

Contributions of different terms as a function of frequency. The solid curve is the ω 0 term, and the dashed curve is the γ term, both of which are frequency dependent. The dashed–dotted line is the μ g e term, and the dotted line is the Δ μ term, both of which are frequency independent.

Tables (1)

Tables Icon

Table 1 Solution Properties a

Equations (19)

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β = 3 Δ μ μ g e 2 2 ω 0 2 ω 0 4 ( ω 0 2 ω 2 ) ( ω 0 2 4 ω 2 ) .
β = β ( 2 ω ; ω 0 , Δ μ , μ g e , γ ) ,
1 β d β d F = [ 1 β Δ μ F β Δ μ ] μ ¯ g e , ω ¯ 0 , γ ¯ + [ 1 β μ g e F β μ g e ] Δ μ ¯ , ω ¯ 0 , γ ¯ + [ 1 β ω 0 F β ω 0 ] Δ μ ¯ , μ ¯ g e , γ ¯ + [ 1 β γ F β γ ] Δ μ ¯ , μ ¯ g e , ω ¯ 0 ,
β = Δ μ μ g e 2 2 { 1 ( ω 0 + i γ + 2 ω ) ( ω 0 + i γ + ω ) + 1 ( ω 0 i γ 2 ω ) ( ω 0 i γ ω ) + 1 ( ω 0 + i γ + ω ) ( ω 0 i γ ω ) } .
β = 3 Δ μ μ g e 2 2 ( ω 0 i γ ) 2 [ ( ω 0 i γ ) 2 ω 2 ] [ ( ω 0 i γ ) 2 4 ω 2 ] .
β ( 2 ω , ω , ω ) = Δ μ 2 π c 2 ( ω e ω ) [ μ g e 2 ( 2 ω e + ω ) 2 π c ( ω e + ω ) ( ω e + 2 ω ) + χ ( 2 ω ) ] .
μ g e 2 = 9.185 × 10 3 n ε [ ω ¯ ] ω ¯ d ω ¯ ,
ε ( ω ) = A 2 γ ( ω ω 0 ) 2 + γ 2 ,
ε ( ω ) = ε 0 b ω a exp ( c 2 ) exp { 1 2 c 2 [ ln ( ω a b ) ] 2 } for ω > a ,
ε ( ω ) = 0 , for ω a ,
ω 0 = a + b exp ( c 2 ) ,
γ = b exp ( c 2 ) { exp [ 2 c ( 2 ln 2 ) 1 2 ] 1 } 2 exp [ c ( 2 ln 2 ) 1 2 ] .
Re [ χ ( ω ) ] = 2 π P 0 ω Im [ χ ( ω ) ] ω 2 ω 2 d ω .
1 β d β d F = 1 Δ μ ¯ Δ μ F + 2 μ ¯ g e μ g e F + A ω ¯ 0 , γ ¯ ω 0 F + B ω ¯ 0 , γ ¯ γ F ,
μ e = μ g Δ E emission Δ E absorption ,
β = 3 Δ μ μ g e 2 2 ( ω 0 i γ ) 2 [ ( ω 0 i γ ) 2 ω 2 ] [ ( ω 0 i γ ) 2 4 ω 2 ] ,
β = 3 Δ μ μ g e 2 ( γ 2 + ω 0 2 ) 2 × ( ( γ 4 + 5 γ 2 ω 2 + 4 ω 4 ) 2 + 2 ( 2 γ 6 + 5 γ 4 ω 2 + γ 2 ω 4 20 ω 6 ) ω 0 2 + ( 6 γ 4 10 γ 2 ω 2 + 33 ω 4 ) ω 0 4 + 2 ( 2 γ 2 5 ω 2 ) ω 0 6 + ω 0 8 ) 0.5 .
A = [ 2 ω 0 ( γ 8 5 γ 6 ω 2 32 γ 4 ω 4 60 γ 2 ω 6 16 ω 8 + ( 4 γ 6 15 γ 4 ω 2 + 32 γ 2 ω 4 + 20 ω 6 ) ω 0 2 + 3 ( 2 γ 4 5 γ 2 ω 2 ) ω 0 4 + ( 4 γ 2 5 ω 2 ) ω 0 6 + ω 0 8 ) ] [ ( γ 2 + ω 0 2 ) ( ( γ 4 + 5 γ 2 ω 2 + 4 ω 4 ) 2 + 2 ( 2 γ 6 + 5 γ 4 ω 2 + γ 2 ω 4 20 ω 6 ) ω 0 2 + ( 6 γ 4 10 γ 2 ω 2 + 33 ω 4 ) ω 0 4 + 2 ( 2 γ 2 5 ω 2 ) ω 0 6 + ω 0 8 ) ] ,
B = [ 2 γ ( γ 8 + 5 γ 6 ω 2 20 γ 2 ω 6 16 ω 8 + ( 4 γ 6 + 15 γ 4 ω 2 + 32 γ 2 ω 4 + 60 ω 6 ) ω 0 2 + ( 6 γ 2 + 15 γ 2 ω 2 32 ω 4 ) ω 0 4 + ( 4 γ 2 + 5 ω 2 ) ω 0 6 + ω 0 8 ) ] [ ( γ 2 + ω 0 2 ) ( ( γ 4 + 5 γ 2 ω 2 + 4 ω 4 ) 2 + 2 ( 2 γ 6 + 5 γ 4 ω 2 + γ 2 ω 4 20 ω 6 ) ω 0 2 + ( 6 γ 4 10 γ 2 ω 2 + 33 ω 4 ) ω 0 4 + 2 ( 2 γ 2 5 ω 2 ) ω 0 6 + ω 0 8 ) ]

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