Abstract

We have developed a fluorescence saturation technique for accurate measurements of the absolute molecular two-photon absorption (TPA) cross-section of fluorescent dyes. We determine the TPA cross-section both from measurements at excitation intensities well below saturation onset (in the square power-law regime) and from data obtained near the onset of saturation. The two estimates have different sensitivities to potential sources of errors. Using the square power-law regime requires calibration of the overall collection efficiency of the detection channel, including the quantum yield of the dye. In the saturation regime, the two key requirements are a good knowledge of the excitation profile and an adequate model of the two-photon excitation transition. To fulfill the former requirement, we developed diagnostic tools to characterize the tightly focussed excitation beam. To satisfy the latter requirement, we included the correct polarization dependent averaging over molecular orientations in our model. We measured the TPA cross-section of Rhodamine B (RhB) and Rhodamine 6g (Rh6g) in methanol at 798 nm for linear and circular polarization. For RhB we observed excellent agreement between the TPA cross-section estimate 〈σ2〉 obtained from the square power-law regime and that obtained from the saturation regime, 〈σ2sat. For the case of linear polarization we found: 〈σ2〉 = 12 ± 2 GM and 〈σ2sat = 10.5 ± 2 GM. For the case of circular polarization we obtained: 〈σ2〉 = 8.4 ± 2 GM and 〈σ2sat = 7.5 ± 2 GM. The results obtained with linear polarization are in good agreement with previously published non-linear transmission data (δ = 2σ = 20.4 GM at 800nm). For Rh6g the difference between 〈σ2〉 and 〈σ2sat is larger, but still considerably smaller than the variance of σ2 values found in the literature.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  31. J. Demas and G. Crosby, "The measurement of photoluminescence quantum yields," J. Phys. Chem. 75, 991-1024 (1971).
    [CrossRef]
  32. V. Antonov and K. Hohla, "Dye stability under excimer-laser pumping II. Visible and uv dyes," Appl. Phys. 32, 9-14 (1983).
    [CrossRef]

2005

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

A. L. Dobryakov, S. A. Kovalenko, and N. P. Ernsting, "Coherent and sequential contributions to femtosecond transient absorption spectra of a rhodamine dye in solution," J. Chem. Phys. 123, 044502 (2005).
[CrossRef] [PubMed]

M. Wegmüller, M. Legré, N. Gisin, T. Hansen, C. Jakobsen, and J. Broeng, "Experimental investigation of the polarization properties of a hollow core photonic bandgap fiber for 1550 nm," Opt. Express 13, 1457-1467 (2005).
[CrossRef] [PubMed]

2004

Y. Wang, O.-H. Tai, C. Wang, and A. K.-Y. Jen, "One-, two-, and three-photon absorption induced fluorescence of a novel chromophore in chloroform solution," J. Chem. Phys. 121, 7901-7907 (2004).
[CrossRef] [PubMed]

S. S. Andrews, "Using rotational averaging to calculate the bulk response of isotropic and anisotropic samples from molecular parameters," J. Chem. Educ. 81, 877-885 (2004).
[CrossRef]

2003

2002

2001

D. Oulianov, I. Tomov, A. Dvornikov, and P. Rentzepis, "Observations on the measurement of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

2000

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

1999

J. Song, T. Inoue, H. Kawazumi, and T. Ogawa, "Determination of two photon absorption cross section of fluorescein using a mode locked titanium sapphire laser," Anal. Sci. 15, 601-603 (1999).
[CrossRef]

P. Kaatz and D. Shelton, "Two-photon fluorescence cross-section measurements calibrated with hyper-Rayleigh scattering," J. Opt. Soc. Am. B 16, 998-1006 (1999).
[CrossRef]

1998

1996

1995

1992

J. Lakowicz, I. Gryczynski, and E. Danielsen, "Anomalous differential polarized phase angles for two-photon excitation with isotropic depolarizing rotations," Chem. Phys. Lett. 191, 47-53 (1992).
[CrossRef]

1990

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1983

J. Rička, K. Amsler, and T. Binkert, "Flexibility of a labeled polymer chain: time resolved fluorescence depolarization measurements," Biopolymers 22, 1301-1318 (1983).
[CrossRef]

V. Antonov and K. Hohla, "Dye stability under excimer-laser pumping II. Visible and uv dyes," Appl. Phys. 32, 9-14 (1983).
[CrossRef]

1972

J. Hermann and J. Ducuing, "Absolute Measurement of Two-Photon Cross Section," Phys. Rev. A 5, 2557-2568 (1972).
[CrossRef]

W. Carter, "Electromagnetic field of a Gaussian Beam with an elliptical cross section," J. Opt. Soc. Am. 62, 1195-1201 (1972).
[CrossRef]

1971

J. Demas and G. Crosby, "The measurement of photoluminescence quantum yields," J. Phys. Chem. 75, 991-1024 (1971).
[CrossRef]

W. McClain, "Excited state symmetry assignment through polarized two-photon absorption studies of fluids," J. Chem. Phys. 55, 2789-2796 (1971).
[CrossRef]

1960

L. Favro, "Theory of the rotational Brownian motion of a free rigid body," Phys. Rev. 119, 53-61 (1960).
[CrossRef]

1931

M. Goeppert-Mayer, "Ueber Elementarakte mit zwei Quantenspruengen," Ann. Phys. 9, 273-295 (1931).
[CrossRef]

Albota, M.

Amsler, K.

J. Rička, K. Amsler, and T. Binkert, "Flexibility of a labeled polymer chain: time resolved fluorescence depolarization measurements," Biopolymers 22, 1301-1318 (1983).
[CrossRef]

Andrews, S. S.

S. S. Andrews, "Using rotational averaging to calculate the bulk response of isotropic and anisotropic samples from molecular parameters," J. Chem. Educ. 81, 877-885 (2004).
[CrossRef]

Antonov, V.

V. Antonov and K. Hohla, "Dye stability under excimer-laser pumping II. Visible and uv dyes," Appl. Phys. 32, 9-14 (1983).
[CrossRef]

Balaji, J.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Banerjee, S.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Binkert, T.

J. Rička, K. Amsler, and T. Binkert, "Flexibility of a labeled polymer chain: time resolved fluorescence depolarization measurements," Biopolymers 22, 1301-1318 (1983).
[CrossRef]

Bouwmans, G.

Broeng, J.

Carter, W.

Chang, N.

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

Cremer, C.

Crosby, G.

J. Demas and G. Crosby, "The measurement of photoluminescence quantum yields," J. Phys. Chem. 75, 991-1024 (1971).
[CrossRef]

Danielsen, E.

J. Lakowicz, I. Gryczynski, and E. Danielsen, "Anomalous differential polarized phase angles for two-photon excitation with isotropic depolarizing rotations," Chem. Phys. Lett. 191, 47-53 (1992).
[CrossRef]

Demas, J.

J. Demas and G. Crosby, "The measurement of photoluminescence quantum yields," J. Phys. Chem. 75, 991-1024 (1971).
[CrossRef]

Dobryakov, A. L.

A. L. Dobryakov, S. A. Kovalenko, and N. P. Ernsting, "Coherent and sequential contributions to femtosecond transient absorption spectra of a rhodamine dye in solution," J. Chem. Phys. 123, 044502 (2005).
[CrossRef] [PubMed]

Ducuing, J.

J. Hermann and J. Ducuing, "Absolute Measurement of Two-Photon Cross Section," Phys. Rev. A 5, 2557-2568 (1972).
[CrossRef]

Dvornikov, A.

D. Oulianov, I. Tomov, A. Dvornikov, and P. Rentzepis, "Observations on the measurement of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Egelhaaf, S.

Ernsting, N. P.

A. L. Dobryakov, S. A. Kovalenko, and N. P. Ernsting, "Coherent and sequential contributions to femtosecond transient absorption spectra of a rhodamine dye in solution," J. Chem. Phys. 123, 044502 (2005).
[CrossRef] [PubMed]

Farr, L.

Favro, L.

L. Favro, "Theory of the rotational Brownian motion of a free rigid body," Phys. Rev. 119, 53-61 (1960).
[CrossRef]

Fischer, A.

Frenz, M.

F. Könz, J. Rička, and M. Frenz, "Dynamic light scattering in the vitreous: performance of the single-mode fiber technique," Opt. Eng. 34, 2390-2395 (1995).
[CrossRef]

Friend, C.

Gisin, N.

Gisler, T.

Goeppert-Mayer, M.

M. Goeppert-Mayer, "Ueber Elementarakte mit zwei Quantenspruengen," Ann. Phys. 9, 273-295 (1931).
[CrossRef]

Gryczynski, I.

J. Lakowicz, I. Gryczynski, and E. Danielsen, "Anomalous differential polarized phase angles for two-photon excitation with isotropic depolarizing rotations," Chem. Phys. Lett. 191, 47-53 (1992).
[CrossRef]

Guild, J.

C. Xu, J. Guild, and W. Webb, "Two-photon excitation cross-sections for commonly used biological fluorophores," Biophys. J. 68, A197 (1995).

C. Xu, J. Guild, and W. Webb, "Determination of absolute two-photon cross sections by in situ second-order autocorrelation," Opt. Lett. 20, 2372-2374 (1995).
[CrossRef] [PubMed]

Hagan, D.

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Hansen, T.

Hermann, J.

J. Hermann and J. Ducuing, "Absolute Measurement of Two-Photon Cross Section," Phys. Rev. A 5, 2557-2568 (1972).
[CrossRef]

Hohla, K.

V. Antonov and K. Hohla, "Dye stability under excimer-laser pumping II. Visible and uv dyes," Appl. Phys. 32, 9-14 (1983).
[CrossRef]

Inoue, T.

J. Song, T. Inoue, H. Kawazumi, and T. Ogawa, "Determination of two photon absorption cross section of fluorescein using a mode locked titanium sapphire laser," Anal. Sci. 15, 601-603 (1999).
[CrossRef]

Jagatap, B.

Jakobsen, C.

Jen, A. K.-Y.

Y. Wang, O.-H. Tai, C. Wang, and A. K.-Y. Jen, "One-, two-, and three-photon absorption induced fluorescence of a novel chromophore in chloroform solution," J. Chem. Phys. 121, 7901-7907 (2004).
[CrossRef] [PubMed]

Kaatz, P.

Kapoor, R.

Kawazumi, H.

J. Song, T. Inoue, H. Kawazumi, and T. Ogawa, "Determination of two photon absorption cross section of fluorescein using a mode locked titanium sapphire laser," Anal. Sci. 15, 601-603 (1999).
[CrossRef]

Knight, J.

Könz, F.

F. Könz, J. Rička, and M. Frenz, "Dynamic light scattering in the vitreous: performance of the single-mode fiber technique," Opt. Eng. 34, 2390-2395 (1995).
[CrossRef]

Kovalenko, S. A.

A. L. Dobryakov, S. A. Kovalenko, and N. P. Ernsting, "Coherent and sequential contributions to femtosecond transient absorption spectra of a rhodamine dye in solution," J. Chem. Phys. 123, 044502 (2005).
[CrossRef] [PubMed]

Kumar, G. R.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Lakowicz, J.

J. Lakowicz, I. Gryczynski, and E. Danielsen, "Anomalous differential polarized phase angles for two-photon excitation with isotropic depolarizing rotations," Chem. Phys. Lett. 191, 47-53 (1992).
[CrossRef]

Legré, M.

Luan, F.

Maiti, S.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Mangan, B.

McClain, W.

W. McClain, "Excited state symmetry assignment through polarized two-photon absorption studies of fluids," J. Chem. Phys. 55, 2789-2796 (1971).
[CrossRef]

Meath, W.

Ogawa, T.

J. Song, T. Inoue, H. Kawazumi, and T. Ogawa, "Determination of two photon absorption cross section of fluorescein using a mode locked titanium sapphire laser," Anal. Sci. 15, 601-603 (1999).
[CrossRef]

Oulianov, D.

D. Oulianov, I. Tomov, A. Dvornikov, and P. Rentzepis, "Observations on the measurement of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Patra, A.

Philip, R.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Rentzepis, P.

D. Oulianov, I. Tomov, A. Dvornikov, and P. Rentzepis, "Observations on the measurement of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Rger, H.

Ricka, J.

T. Gisler, H. Rger, S. Egelhaaf, J. Tschumi, P. Schurtenberger, and J. Rička, "Mode-selective dynamic light scattering: theory versus experimental realization," Appl. Opt. 34, 3546-3553 (1995).
[CrossRef] [PubMed]

F. Könz, J. Rička, and M. Frenz, "Dynamic light scattering in the vitreous: performance of the single-mode fiber technique," Opt. Eng. 34, 2390-2395 (1995).
[CrossRef]

J. Rička, K. Amsler, and T. Binkert, "Flexibility of a labeled polymer chain: time resolved fluorescence depolarization measurements," Biopolymers 22, 1301-1318 (1983).
[CrossRef]

Russel, P. S. J.

Sabert, H.

Said, A.

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Schurtenberger, P.

Sengupta, P.

P. Sengupta, J. Balaji, S. Banerjee, R. Philip, G. R. Kumar, and S. Maiti, "Sensitive measurement of absolute two-photon absorption cross sections," J. Chem. Phys. 112, 9201-9205 (2000).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Shelton, D.

Song, J.

J. Song, T. Inoue, H. Kawazumi, and T. Ogawa, "Determination of two photon absorption cross section of fluorescein using a mode locked titanium sapphire laser," Anal. Sci. 15, 601-603 (1999).
[CrossRef]

Stelzer, E.

Stryland, E. V.

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Tai, O.-H.

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

Y. Wang, O.-H. Tai, C. Wang, and A. K.-Y. Jen, "One-, two-, and three-photon absorption induced fluorescence of a novel chromophore in chloroform solution," J. Chem. Phys. 121, 7901-7907 (2004).
[CrossRef] [PubMed]

Tian, P.

Tomov, I.

D. Oulianov, I. Tomov, A. Dvornikov, and P. Rentzepis, "Observations on the measurement of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Tsai, T.

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

Tschumi, J.

Wang, C.

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

Y. Wang, O.-H. Tai, C. Wang, and A. K.-Y. Jen, "One-, two-, and three-photon absorption induced fluorescence of a novel chromophore in chloroform solution," J. Chem. Phys. 121, 7901-7907 (2004).
[CrossRef] [PubMed]

Wang, Y.

C. Wang, O.-H. Tai, Y. Wang, T. Tsai, and N. Chang, "Non-quadratic-intensity dependence of two-photon absorption induced fluorescence of organic chromophores in solution," J. Chem. Phys. 122, 084509 (2005).
[CrossRef]

Y. Wang, O.-H. Tai, C. Wang, and A. K.-Y. Jen, "One-, two-, and three-photon absorption induced fluorescence of a novel chromophore in chloroform solution," J. Chem. Phys. 121, 7901-7907 (2004).
[CrossRef] [PubMed]

Warren, W.

Webb, W.

Wegmüller, M.

Wei, T.

M. Sheik-Bahae, A. Said, T. Wei, D. Hagan, and E. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Xu, C.

Anal. Sci.

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Figures (6)

Fig. 1.
Fig. 1.

Excitation transition dipole moments 0i and i1 in the molecular coordinate system X’,Y, Z’ and incident polarization ê E in the laboratory coordinate system X, Y, Z. Euler angles θ , ϕ and ψ relate the two coordinate systems.

Fig. 2.
Fig. 2.

Left: Correction factor r(α) = 〈ω2〉/ 〈ω〉2 for linear- and circular polarization used for the determination of 〈σ2 sat = p 2/r(α). Right: First and second-order polarization ratios γ 1 (α) = 〈ω〉 CP /〈ω〉 LP and γ 2(α) = [〈ω2 CP /〈ω〉 LP ]. The two points plotted in the right-hand graph show the measured values of γ 1 (circle) and γ 2 (square) for Rhodamine B, together with the associated error bars. These results are discussed in Section 4.

Fig. 3.
Fig. 3.

Experimental setup used for two-photon cross section measurements: A: Pump laser oscillator, B: Femtosecond oscillator, C: Pump laser regenerative amplifier, D: Femtosecond regenerative amplifier, E: Beam attenuator (half-wave plate, Glan-laser polarizer and half-wave plate), F: Hollow core photonic bandgap fiber, G: Wavelength spectrometer, H: Autocorrelator, J: Beam profiling camera system, K: Integrating sphere, L: Dye flow chamber, M: Fiber assisted detection with single-photon counting unit

Fig. 4.
Fig. 4.

Information about the beam waist radius a0 of a tightly focussed beam can be obtained by taking several far-field images at different z-positions

Fig. 5.
Fig. 5.

Fluorescence decay curves of Rhodamine 6g and Rhodamine B taken at low incident intensity (square power-law regime) and high incident intensity (onset of saturation)

Fig. 6.
Fig. 6.

Plot of the dead-time corrected fluorescence signal as a function of laser power. The inset shows a log-log representation of the data. The dashed line indicates the square power-law whereas the solid line is the fit according to Eq. (23). The dye concentrations are 1.0 ∙ 10-6 M for Rhodamine B and 4.3 ∙ 10-7 M for Rhodamine 6g. Note also the change in the power axis

Tables (1)

Tables Icon

Table 1. Compilation of the data for Rhodamine 6g and Rhodamine B at an excitation wavelength of 798 nm. The first column contains the orientationally averaged TPA cross-sections from the square power-law regime. The last column is model-based estimates from the saturation regime based on the polarization ratios γ 1 (α) = p 1CP /P 1LP = 〈σ2 CP / 〈σ2 LP , and γ 2(α) = P 2CP /p 2LP . r(α) is a correction factor discussed in text.

Equations (27)

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P ( r ) = P E ( r ) P D ( r )
N F = c ̅ P E ( r ) P D ( r ) Ω d 3 r .
P E ( r ) = 1 exp [ σ 2 I 2 r t dt ]
I r t = Q π a E 0 2 Y E ( r ) X ( t ) ,
Y E ( r ) = a E 0 2 a E 2 ( z ) exp [ ρ 2 a E 2 ( z ) ] ,
P E ( r ) = 1 exp [ σ 2 I 2 ̅ τ Y E 2 ( r ) ] ,
I 2 ̅ = Q 2 π 2 a E 0 4 1 τ 2 .
τ = X ( t ) dt 2 X 2 ( t ) dt .
P E ( r ) = σ 2 I 2 ̅ τ Y E 2 ( r ) 1 2 [ σ 2 I 2 ̅ τ Y E 2 ( r ) ] 2 + O { [ σ 2 I 2 ̅ τ Y E 2 ( r ) ] 3 }
P D ( r ; e D ) = η q 3 2 ( a ̂ . e ̂ D ) 2 Φ O Y O ( r ) .
Φ O = 1 4 π n 2 Ω 0 ,
P D ( r ) = η q 3 2 [ ( a ̂ e ̂ x ) 2 + ( a ̂ e ̂ y ) 2 ] Φ O Y O ( r ) .
Y O ( r ) = a O 0 2 a 0 2 ( z ) exp [ ρ 2 a 0 2 ( z ) ] ,
N F = c ̅ η q Φ O ω O σ 2 I 2 ̅ τ V 0 1 2 ω O σ 2 2 ( I 2 ̅ τ ) 2 V 1 + O [ ω O σ 2 3 ( I 2 ̅ τ ) 3 V 2 ]
V n = Y E ( 2 n + 2 ) ( r ) Y O ( r ) d 3 r .
V n = Γ ( 2 n + 1 2 ) ( 2 n + 2 ) Γ ( 2 n + 1 ) π 3 2 a E 0 4 k E
ω O Ω = 3 2 ( a ̂ e ̂ x ) 2 + ( a ̂ e ̂ y ) 2 Ω = 1 + 1 2 τ R τ R + τ F ,
P D ( r ) Ω = ηq Φ O Y O ( r ) .
σ 2 n Ω = μ 2 n ( e ̂ E . d ̂ 0 i ) ( d ̂ i 1 . e ̂ E ) 2 n Ω = μ 2 n ω n Ω .
N F = c ̅ η q Φ O ω α μ 2 I 2 ̅ τ V 0 { 1 1 2 V 1 V 0 ω 2 α ω α μ 2 I 2 ̅ τ + O [ V 2 V 0 ω 3 α ω α μ 2 2 I ̅ 4 τ 2 ] } .
p 1 = μ 2 ω α = σ 2
p 2 = μ 2 ω 2 α ω α .
N F ( P ) = C 1 p 1 P 2 { 1 C 2 p 2 P 2 + O [ P 4 ] }
< N F > = ln ( 1 < M F > ) ,
η = S F ( λ ) η M ( λ ) η D ( λ ) ,
P E = η D q c ̅ σ 1 Φ O V O P π a E 0 2 ,
V O = Y E ( r ) Y O ( r ) d 3 r .

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