Abstract

The optical nonlinearity of an aqueous colloid containing silver nanoparticles (NPs) was investigated using the Z-scan technique. Values for the third-, fifth-, seventh-, and ninth-order susceptibilities were obtained, and their dependence with the volume fraction occupied by the NPs was determined. The experiments were performed with intensities between 0.1 and 1.5GWcm2 using a laser at 532nm that delivers single 80ps pulses at a low repetition rate.

© 2007 Optical Society of America

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References

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  1. See, for example, M. Yamane, and Y. Asahara, Glasses for Photonics (Cambridge U. Press, 2000).
    [CrossRef]
  2. E. L. Falcão-Filho, C. A. C. Bosco, G. S. Maciel, L. H. Acioli, C. B. de Araújo, A. A. Lipovskii, and D. K. Tagantsev, "Third-order optical nonlinearity of a transparent glass ceramic containing sodium niobate nanocrystals," Phys. Rev. B 69, 134204 (2004).
    [CrossRef]
  3. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
    [CrossRef]
  4. H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, "Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation," Appl. Phys. Lett. 83, 4625-4627 (2003).
    [CrossRef]
  5. J. Bosbach, C. Hendrich, F. Stietz, T. Vartanyan, and F. Träger, "Ultrafast dephasing of surface plasmon excitation in silver nanoparticles: influence of particle size, shape, and chemical surrounding," Phys. Rev. Lett. 89, 257404 (2002).
    [CrossRef] [PubMed]
  6. W. Wenseleers, F. Stellacci, T. M. Friedrichsen, T. Mangel, C. A. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
    [CrossRef]
  7. S. I. Bozhevolnyi, J. Beermann, and V. Coello, "Direct observation of localized second-harmonic enhancement in random metal nanostructures," Phys. Rev. Lett. 90, 197403 (2003).
    [CrossRef] [PubMed]
  8. M. H. G. Miranda, E. L. Falcão-Filho, J. J. Rodrigues, Jr., C. B. de Araújo, and L. H. Acioli, "Ultrafast light-induced dichroism in silver nanoparticles," Phys. Rev. B 70, 161401(R) (2004).
    [CrossRef]
  9. V. P. Drachev, E. N. Khaliullin, W. Kim, F. Alzoubi, S. G. Rautian, V. P. Safonov, R. L. Armstrong, and V. M. Shalaev, "Quantum size effect in two-photon excited luminescence from silver nanoparticles," Phys. Rev. B 69, 035318 (2004).
    [CrossRef]
  10. C. Voisin, D. Chistofilos, N. Del Fatti, F. Valée, B. Prével, E. Cottancin, J. Lermé, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
  15. I. Antonov, F. Bass, Yu Kaganovskii, M. Rosenbluh, and A. A. Lipovskii, "Fabrication of microlenses in Ag-doped glasses by a focused continuous wave laser beam," J. Appl. Phys. 93, 2343-2348 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. M. I. Stockman, K. B. Kurlayev, and T. F. George, "Linear and nonlinear optical susceptibilities of Maxwell-Garnett composites: dipolar spectral theory," Phys. Rev. B 60, 17071-17083 (1999).
    [CrossRef]
  22. P. M. Hui, "Higher order nonlinear response in dilute random composites," J. Appl. Phys. 73, 4072-4073 (1993).
    [CrossRef]
  23. X. Liu and Z. Li, "High order nonlinear susceptibilities of composite medium," Solid State Commun. 96, 981-985 (1995).
    [CrossRef]
  24. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurements of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  25. P. C. Lee and D. Meisel, "Adsorption and surface-enhanced Raman of dyes on silver and gold sols," J. Phys. Chem. 86, 3391-3395 (1982).
    [CrossRef]
  26. See, for instance, Z. Peng, T. Walther, and K. Kleinermanns, "Photofragmentation of phase-transferred gold nanoparticles by intense pulsed laser light," J. Phys. Chem. B 109, 15735-15740 (2005), and references therein.
    [CrossRef]
  27. L. H. Acioli, M. H. G. Miranda, E. L. Falcão-Filho, J. J. Rodrigues, Jr., and C. B. de Araújo, "Time-resolved spectroscopy of metal nanoellipsoid," Proc. SPIE 6118, 61180S (2006).
  28. H. Ma, A. S. L. Gomes, and C. B. de Araújo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
    [CrossRef]
  29. P. B. Johnson and R. W. Christy, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  30. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe," J. Opt. Soc. Am. B 9, 405-414 (1992).
    [CrossRef]
  31. B. L. Lawrence, M. Cha, W. E. Torruellas, G. E. Stegeman, S. Etemad, G. Baker, and F. Kajzar, "Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064nm," Appl. Phys. Lett. 64, 2773-2775 (1994).
    [CrossRef]
  32. Y. Hamanaka, A. Nakamura, N. Hayashi, and S. Omi, "Dispersion curves of complex third-order optical susceptibilities around the surface plasmon resonance in Ag nanocrystal-glass composites," J. Opt. Soc. Am. B 20, 1227-1232 (2003).
    [CrossRef]

2005 (2)

See, for instance, Z. Peng, T. Walther, and K. Kleinermanns, "Photofragmentation of phase-transferred gold nanoparticles by intense pulsed laser light," J. Phys. Chem. B 109, 15735-15740 (2005), and references therein.
[CrossRef]

E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, "Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide," J. Opt. Soc. Am. B 22, 2444-2449 (2005).
[CrossRef]

2004 (4)

J. Qiu, X. Jiang, C. Zhu, H. Inouye, J. Si, and K. Hirao, "Optical properties of structurally modified glasses doped with gold ions," Opt. Lett. 29, 370-372 (2004).
[CrossRef] [PubMed]

E. L. Falcão-Filho, C. A. C. Bosco, G. S. Maciel, L. H. Acioli, C. B. de Araújo, A. A. Lipovskii, and D. K. Tagantsev, "Third-order optical nonlinearity of a transparent glass ceramic containing sodium niobate nanocrystals," Phys. Rev. B 69, 134204 (2004).
[CrossRef]

M. H. G. Miranda, E. L. Falcão-Filho, J. J. Rodrigues, Jr., C. B. de Araújo, and L. H. Acioli, "Ultrafast light-induced dichroism in silver nanoparticles," Phys. Rev. B 70, 161401(R) (2004).
[CrossRef]

V. P. Drachev, E. N. Khaliullin, W. Kim, F. Alzoubi, S. G. Rautian, V. P. Safonov, R. L. Armstrong, and V. M. Shalaev, "Quantum size effect in two-photon excited luminescence from silver nanoparticles," Phys. Rev. B 69, 035318 (2004).
[CrossRef]

2003 (5)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, "Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation," Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

S. I. Bozhevolnyi, J. Beermann, and V. Coello, "Direct observation of localized second-harmonic enhancement in random metal nanostructures," Phys. Rev. Lett. 90, 197403 (2003).
[CrossRef] [PubMed]

Y. Hamanaka, A. Nakamura, N. Hayashi, and S. Omi, "Dispersion curves of complex third-order optical susceptibilities around the surface plasmon resonance in Ag nanocrystal-glass composites," J. Opt. Soc. Am. B 20, 1227-1232 (2003).
[CrossRef]

I. Antonov, F. Bass, Yu Kaganovskii, M. Rosenbluh, and A. A. Lipovskii, "Fabrication of microlenses in Ag-doped glasses by a focused continuous wave laser beam," J. Appl. Phys. 93, 2343-2348 (2003).
[CrossRef]

2002 (2)

J. Bosbach, C. Hendrich, F. Stietz, T. Vartanyan, and F. Träger, "Ultrafast dephasing of surface plasmon excitation in silver nanoparticles: influence of particle size, shape, and chemical surrounding," Phys. Rev. Lett. 89, 257404 (2002).
[CrossRef] [PubMed]

W. Wenseleers, F. Stellacci, T. M. Friedrichsen, T. Mangel, C. A. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

2001 (2)

2000 (1)

C. Voisin, D. Chistofilos, N. Del Fatti, F. Valée, B. Prével, E. Cottancin, J. Lermé, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[CrossRef] [PubMed]

1999 (2)

S. Link and M. A. El-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods," J. Phys. Chem. 103, 8410-8426 (1999).
[CrossRef]

M. I. Stockman, K. B. Kurlayev, and T. F. George, "Linear and nonlinear optical susceptibilities of Maxwell-Garnett composites: dipolar spectral theory," Phys. Rev. B 60, 17071-17083 (1999).
[CrossRef]

1995 (1)

X. Liu and Z. Li, "High order nonlinear susceptibilities of composite medium," Solid State Commun. 96, 981-985 (1995).
[CrossRef]

1994 (1)

B. L. Lawrence, M. Cha, W. E. Torruellas, G. E. Stegeman, S. Etemad, G. Baker, and F. Kajzar, "Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064nm," Appl. Phys. Lett. 64, 2773-2775 (1994).
[CrossRef]

1993 (1)

P. M. Hui, "Higher order nonlinear response in dilute random composites," J. Appl. Phys. 73, 4072-4073 (1993).
[CrossRef]

1992 (1)

1991 (1)

H. Ma, A. S. L. Gomes, and C. B. de Araújo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

1990 (2)

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

N. C. Kothari, "Effective-medium theory of a nonlinear composite medium using the T-matrix approach: exact results for spherical grains," Phys. Rev. A 41, 4486-4492 (1990).
[CrossRef] [PubMed]

1988 (1)

G. S. Agarwal and S. Dutta Gupta, "T-matrix approach to the nonlinear susceptibilities of heterogeneous media," Phys. Rev. A 38, 5678-5687 (1988).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

1982 (1)

P. C. Lee and D. Meisel, "Adsorption and surface-enhanced Raman of dyes on silver and gold sols," J. Phys. Chem. 86, 3391-3395 (1982).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Appl. Phys. Lett. (3)

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, "Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation," Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

B. L. Lawrence, M. Cha, W. E. Torruellas, G. E. Stegeman, S. Etemad, G. Baker, and F. Kajzar, "Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064nm," Appl. Phys. Lett. 64, 2773-2775 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (2)

P. M. Hui, "Higher order nonlinear response in dilute random composites," J. Appl. Phys. 73, 4072-4073 (1993).
[CrossRef]

I. Antonov, F. Bass, Yu Kaganovskii, M. Rosenbluh, and A. A. Lipovskii, "Fabrication of microlenses in Ag-doped glasses by a focused continuous wave laser beam," J. Appl. Phys. 93, 2343-2348 (2003).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. Chem. (2)

P. C. Lee and D. Meisel, "Adsorption and surface-enhanced Raman of dyes on silver and gold sols," J. Phys. Chem. 86, 3391-3395 (1982).
[CrossRef]

S. Link and M. A. El-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods," J. Phys. Chem. 103, 8410-8426 (1999).
[CrossRef]

J. Phys. Chem. B (3)

See, for instance, Z. Peng, T. Walther, and K. Kleinermanns, "Photofragmentation of phase-transferred gold nanoparticles by intense pulsed laser light," J. Phys. Chem. B 109, 15735-15740 (2005), and references therein.
[CrossRef]

W. Wenseleers, F. Stellacci, T. M. Friedrichsen, T. Mangel, C. A. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

G. S. Agarwal and S. Dutta Gupta, "T-matrix approach to the nonlinear susceptibilities of heterogeneous media," Phys. Rev. A 38, 5678-5687 (1988).
[CrossRef] [PubMed]

N. C. Kothari, "Effective-medium theory of a nonlinear composite medium using the T-matrix approach: exact results for spherical grains," Phys. Rev. A 41, 4486-4492 (1990).
[CrossRef] [PubMed]

Phys. Rev. B (5)

M. I. Stockman, K. B. Kurlayev, and T. F. George, "Linear and nonlinear optical susceptibilities of Maxwell-Garnett composites: dipolar spectral theory," Phys. Rev. B 60, 17071-17083 (1999).
[CrossRef]

M. H. G. Miranda, E. L. Falcão-Filho, J. J. Rodrigues, Jr., C. B. de Araújo, and L. H. Acioli, "Ultrafast light-induced dichroism in silver nanoparticles," Phys. Rev. B 70, 161401(R) (2004).
[CrossRef]

V. P. Drachev, E. N. Khaliullin, W. Kim, F. Alzoubi, S. G. Rautian, V. P. Safonov, R. L. Armstrong, and V. M. Shalaev, "Quantum size effect in two-photon excited luminescence from silver nanoparticles," Phys. Rev. B 69, 035318 (2004).
[CrossRef]

E. L. Falcão-Filho, C. A. C. Bosco, G. S. Maciel, L. H. Acioli, C. B. de Araújo, A. A. Lipovskii, and D. K. Tagantsev, "Third-order optical nonlinearity of a transparent glass ceramic containing sodium niobate nanocrystals," Phys. Rev. B 69, 134204 (2004).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. Lett. (3)

J. Bosbach, C. Hendrich, F. Stietz, T. Vartanyan, and F. Träger, "Ultrafast dephasing of surface plasmon excitation in silver nanoparticles: influence of particle size, shape, and chemical surrounding," Phys. Rev. Lett. 89, 257404 (2002).
[CrossRef] [PubMed]

C. Voisin, D. Chistofilos, N. Del Fatti, F. Valée, B. Prével, E. Cottancin, J. Lermé, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, J. Beermann, and V. Coello, "Direct observation of localized second-harmonic enhancement in random metal nanostructures," Phys. Rev. Lett. 90, 197403 (2003).
[CrossRef] [PubMed]

Solid State Commun. (1)

X. Liu and Z. Li, "High order nonlinear susceptibilities of composite medium," Solid State Commun. 96, 981-985 (1995).
[CrossRef]

Other (2)

See, for example, M. Yamane, and Y. Asahara, Glasses for Photonics (Cambridge U. Press, 2000).
[CrossRef]

L. H. Acioli, M. H. G. Miranda, E. L. Falcão-Filho, J. J. Rodrigues, Jr., and C. B. de Araújo, "Time-resolved spectroscopy of metal nanoellipsoid," Proc. SPIE 6118, 61180S (2006).

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Figures (6)

Fig. 1
Fig. 1

Absorption spectrum of the silver colloid after photofragmentation (Sample length: 1.0 mm ; volume fraction: f = 0.83 × 10 4 ). The inset illustrates the size distribution of the NPs as determined using a transmission electronic microscope.

Fig. 2
Fig. 2

Normalized Z-scan traces obtained at 532 nm for different laser peak intensities; (a)–(d) closed aperture scheme ( S = 0.002 ) and (e)–(h) open aperture scheme ( S = 1 ) . The volume fraction is f = 0.70 × 10 4 .

Fig. 3
Fig. 3

Intensity dependence of Δ T I , for NL refraction (a)–(d) and NL absorption (e)–(h). The solid curves are guides to the eye. The sample length is 1 mm for f = 1.47 × 10 4 , 2 mm for f = 0.83 × 10 4 , and 5 mm for f = 0.70 × 10 4 and f = 0.46 × 10 4 .

Fig. 4
Fig. 4

Dependence of the effective third- and fifth-order nonlinearity with the NP filling fraction; (a), (b) refractive indices and (c), (d) absorption coefficients.

Fig. 5
Fig. 5

Z-scan profiles calculated through Eq. (7) as a function of intensity and volume fraction. From the bottom to the top, the curves correspond to 1.2, 0.9, 0.6, and 0.3 GW cm 2 , respectively. All the data were normalized to unit and shifts in the baselines were introduced to prevent overlap among the curves. The sample length is 1 mm for f = 1.47 × 10 4 , 2 mm for f = 0.83 × 10 4 , and 5 mm for f = 0.70 × 10 4 and f = 0.46 × 10 4 .

Fig. 6
Fig. 6

Z-scan traces, calculated considering only the third-order susceptibility, as a function of intensity and volume fraction. From the bottom to the top, the curves correspond to 1.2, 0.9, 0.6, and 0.3 GW cm 2 , respectively. All the data were normalized to unit and shifts in the baselines were introduced to prevent overlap among the curves. The sample length is 1 mm for f = 1.47 × 10 4 , 2 mm for f = 0.83 × 10 4 , and 5 mm for f = 0.70 × 10 4 and f = 0.46 × 10 4 .

Tables (1)

Tables Icon

Table 1 NL parameters from Fig. 3 for Different Values of the NP Filling Factor a

Equations (30)

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

ε e f f = ε h [ 1 + 3 β f 1 β f ] ,
Δ T = 0.406 k L e f f n 2 I ,
Δ T = ( 2 ) 3 2 L e f f α 2 I ,
Δ Φ 0 = Δ Φ 0 ( 3 ) + Δ Φ 0 ( 5 ) + Δ Φ 0 ( 7 ) + Δ Φ 0 ( 9 ) ,
Δ Φ 0 ( 3 ) = k n 2 I 0 [ 1 exp ( α 0 L ) α 0 ] ,
Δ Φ 0 ( 5 ) = k n 4 I 0 2 [ 1 exp ( 2 α 0 L ) 2 α 0 ] ,
Δ Φ 0 ( 7 ) = k n 6 I 0 3 [ 1 exp ( 3 α 0 L ) 3 α 0 ] ,
Δ Φ 0 ( 9 ) = k n 8 I 0 4 [ 1 exp ( 4 α 0 L ) 4 α 0 ] .
n 2 ( m 2 W ) = 3 4 ε 0 n 0 2 c Re [ χ ( 3 ) ] ,
n 4 ( m 4 W 2 ) = 5 4 ε 0 3 n 0 3 c 3 Re [ χ ( 5 ) ] ,
n 6 ( m 6 W 3 ) = 35 16 ε 0 3 n 0 4 c 3 Re [ χ ( 7 ) ] ,
n 8 ( m 8 W 4 ) = 63 16 ε 0 4 n 0 5 c 4 Re [ χ ( 9 ) ] .
T ( z , Δ Φ 0 ) 1 + 4 Δ Φ 0 ( 3 ) z z 0 [ ( z z 0 ) 2 + 9 ] [ ( z z 0 ) 2 + 1 ] + 8 Δ Φ 0 ( 5 ) z z 0 [ ( z z 0 ) 2 + 25 ] [ ( z z 0 ) 2 + 1 ] 2 + 12 Δ Φ 0 ( 7 ) z z 0 [ ( z z 0 ) 2 + 49 ] [ ( z z 0 ) 2 + 1 ] 3 + 16 Δ Φ 0 ( 9 ) 0 z z 0 [ ( z z 0 ) 2 + 81 ] [ ( z z 0 ) 2 + 1 ] 4 ,
Δ T p , v 0.396 Δ Φ 0 ( 3 ) + 0.198 Δ Φ 0 ( 5 ) + 0.102 Δ Φ 0 ( 7 ) + 0.056 Δ Φ 0 ( 9 ) .
α 2 ( m W ) = 3 ω 2 ε 0 n 0 2 c 2 Im [ χ ( 3 ) ] ,
α 4 ( m 3 W 2 ) = 5 ω 2 ε 0 2 n 0 3 c 3 Im [ χ ( 5 ) ] ,
α 6 ( m 5 W 3 ) = 35 ω 8 ε 0 3 n 0 4 c 4 Im [ χ ( 7 ) ] ,
α 8 ( m 7 W 4 ) = 63 ω 8 ε 0 4 n 0 5 c 5 Im [ χ ( 9 ) ] .
Δ T = ( 2 ) 3 2 L e f f ( α 2 + α 4 I 0 + α 6 I 0 2 + α 8 I 0 3 ) I 0 .
ε t o t a l = ε e f f + 4 π χ e f f ( 3 ) P 2 E L 2 1 + a 0 E L 2 ,
χ e f f ( 3 ) = f χ i ( 3 ) P 2 P 2 ,
χ t o t a l ( 3 ) = χ e f f ( 3 ) + χ ¯ ( 5 ) E 2 + χ ¯ ( 7 ) E 4 + χ ¯ ( 9 ) E 6 + ,
χ ¯ ( 5 ) = η 0 χ e f f ( 3 ) ,
χ ¯ ( 7 ) = ( η 1 + η 0 2 ) χ e f f ( 3 ) ,
χ ¯ ( 9 ) = ( η 2 + 2 η 1 η 0 + η 0 3 ) χ e f f ( 3 ) ,
η 0 = 2 a 0 + a 0 * P 2 ,
η 1 = a 0 2 + a 0 2 + ( a 0 * ) 2 P 4 ,
η 2 = 2 a 0 3 + 3 a 0 2 ( a 0 + a 0 * ) + 2 ( a 0 * ) 3 P 6 .
ε t o t a l = ε e f f + 4 π [ χ t o t a l ( 3 ) E 2 + χ e f f ( 5 ) E 4 ] ,
χ e f f ( 5 ) = f χ i ( 5 ) P 2 P 4 .

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