Abstract

The Lorentz model provides a simple and intuitive approach to linear optical dispersion. In this paper, we present a nonlinear formulation of the Lorentz model based on an analytical solution of a quantum-mechanical two-level model atom in the under-resonant limit and show how multiple nonlinear Lorentz equations can be combined to describe nonlinear optical dispersion in both time and frequency domain nanophotonics simulations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photonics Rev. 9, 577–603 (2015).
    [Crossref]
  2. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
    [Crossref]
  3. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
    [Crossref]
  4. J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
    [Crossref] [PubMed]
  5. S. I. Azzam and A. V. Kildishev, “Time-domain dynamics of saturation of absorption using multilevel atomic systems,” Opt. Mater. Express 8, 3829–3834 (2018).
    [Crossref]
  6. M. Scalora, M. A. Vincenti, D. de Ceglia, C. M. Cojocaru, M. Grande, and J. W. Haus, “Nonlinear Duffing oscillator model for third harmonic generation,” J. Opt. Soc. Am. B 32, 2129 (2015).
    [Crossref]
  7. D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
    [Crossref]
  8. C. Varin, G. Bart, R. Emms, and T. Brabec, “Saturable Lorentz model for fully explicit three-dimensional modeling of nonlinear optics,” Opt. Express 23, 2686–2695 (2015).
    [Crossref]
  9. C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
    [Crossref]
  10. W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem. 219, 272–282 (1871).
    [Crossref]
  11. B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23, 4477 (1984).
    [Crossref]
  12. H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, 1957).
    [Crossref]
  13. R. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).
  14. M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, 1997).
    [Crossref]
  15. L. Allen and J. Eberly, Optical Resonance and Two-level Atoms (Dover, 1975).
  16. A. Siegman, Lasers (University Science Books, 1986).
  17. W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
    [Crossref]

2018 (2)

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

S. I. Azzam and A. V. Kildishev, “Time-domain dynamics of saturation of absorption using multilevel atomic systems,” Opt. Mater. Express 8, 3829–3834 (2018).
[Crossref]

2015 (3)

2014 (1)

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

2013 (1)

D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
[Crossref]

2010 (1)

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

2000 (1)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[Crossref]

1990 (1)

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

1984 (1)

1871 (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem. 219, 272–282 (1871).
[Crossref]

Allen, L.

L. Allen and J. Eberly, Optical Resonance and Two-level Atoms (Dover, 1975).

Alù, A.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Amann, M.-C.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Argyropoulos, C.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Azzam, S. I.

Bart, G.

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

C. Varin, G. Bart, R. Emms, and T. Brabec, “Saturable Lorentz model for fully explicit three-dimensional modeling of nonlinear optics,” Opt. Express 23, 2686–2695 (2015).
[Crossref]

Belkin, M. A.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Bethe, H.

H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, 1957).
[Crossref]

Boehm, G.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Boyd, R.

R. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

Brabec, T.

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

C. Varin, G. Bart, R. Emms, and T. Brabec, “Saturable Lorentz model for fully explicit three-dimensional modeling of nonlinear optics,” Opt. Express 23, 2686–2695 (2015).
[Crossref]

Butet, J.

B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photonics Rev. 9, 577–603 (2015).
[Crossref]

Chen, P.-Y.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Cojocaru, C. M.

de Ceglia, D.

Demmerle, F.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Eberly, J.

L. Allen and J. Eberly, Optical Resonance and Two-level Atoms (Dover, 1975).

Emms, R.

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

C. Varin, G. Bart, R. Emms, and T. Brabec, “Saturable Lorentz model for fully explicit three-dimensional modeling of nonlinear optics,” Opt. Express 23, 2686–2695 (2015).
[Crossref]

Fennel, T.

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

Gallinet, B.

B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photonics Rev. 9, 577–603 (2015).
[Crossref]

Gordon, D.

D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
[Crossref]

Grande, M.

Hamm, J. M.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Haus, J. W.

Helle, M.

D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
[Crossref]

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Jiang, X.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[Crossref]

Kajzar, F.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Kildishev, A. V.

Leclerc, M.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Lee, J.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Lu, F.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Martin, O. J. F.

B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photonics Rev. 9, 577–603 (2015).
[Crossref]

Neher, D.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Peñano, J.

D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
[Crossref]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Salpeter, E.

H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, 1957).
[Crossref]

Scalora, M.

Scully, M.

M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[Crossref]

Sellmeier, W.

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem. 219, 272–282 (1871).
[Crossref]

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Soukoulis, C. M.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[Crossref]

Stegeman, G.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Tatian, B.

Torruellas, W.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Tymchenko, M.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Varin, C.

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

C. Varin, G. Bart, R. Emms, and T. Brabec, “Saturable Lorentz model for fully explicit three-dimensional modeling of nonlinear optics,” Opt. Express 23, 2686–2695 (2015).
[Crossref]

Vincenti, M. A.

Wuestner, S.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Zanoni, R.

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Zubairy, M.

M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[Crossref]

Ann. Phys. Chem. (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem. 219, 272–282 (1871).
[Crossref]

Appl. Opt. (1)

Chem. Phys. Lett. (1)

W. Torruellas, D. Neher, R. Zanoni, G. Stegeman, F. Kajzar, and M. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Comput. Phys. Commun. (1)

C. Varin, R. Emms, G. Bart, T. Fennel, and T. Brabec, “Explicit formulation of second and third order optical nonlinearity in the FDTD framework,” Comput. Phys. Commun. 222, 70–83 (2018).
[Crossref]

J. Comput. Phys. (1)

D. Gordon, M. Helle, and J. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” J. Comput. Phys. 250, 388–402 (2013).
[Crossref]

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

Laser Photonics Rev. (1)

B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics: standard problems and future challenges,” Laser Photonics Rev. 9, 577–603 (2015).
[Crossref]

Nature (1)

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511, 65–69 (2014).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Mater. Express (1)

Phys. Rev. Lett. (2)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[Crossref]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref]

Other (5)

H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, 1957).
[Crossref]

R. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[Crossref]

L. Allen and J. Eberly, Optical Resonance and Two-level Atoms (Dover, 1975).

A. Siegman, Lasers (University Science Books, 1986).

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Figures (3)

Fig. 1
Fig. 1 Normalized spectral signature of the nonlinear Lorentz model [Eq. (13)] for (a) linear and (b) nonlinear scattering (third-harmonic) for reference media with optical properties comparable to air (n0 = 1.0003 and n2 = 5 × 10−19 cm2/W) and BK-7 glass (n0 = 1.5 and n2 = 3.4 × 10−16 cm2/W) [13]. Comparison with numerical solutions of the quantum-mechanical two-level model equations in the Schrödinger representation (see text for details) shows agreement for third-harmonic generation when ωLω0/3.
Fig. 2
Fig. 2 Using two nonlinear Lorentz model equations [see Eq. (20)] to fit nonlinear (third-harmonic) scattering predicted by the quantum-mechanical two-level model for a reference medium with optical properties comparable to BK-7 glass (n0 = 1.5 and n2 = 3.4× 10−16 cm2/W) [13]. With appropriate fit parameters [ω1 = ω0, χ ¯ 1 ( 3 ) = 3.118 × 10 22 m 2 / V 2, γ1 = 0, ω2 = 3ω0, χ ¯ 2 ( 3 ) = 3.4 × 10 23 m 2 / V 2, and γ2 = 4.5 × 1014 s−1], it is seen in (a) that the quantum solution and the two-band model fit are barely distinguishable. With damping in (b), the average error is in the 3% range, which can be further reduced by using more nonlinear Lorentz equations.
Fig. 3
Fig. 3 Four-band nonlinear-Lorentz fit [“fit (nonlinear Lorentz)”] of the experimental data (exp.) for χ(3)(3ω) in poly(3-decylthiophene) [17]. With optimized parameters (see Table 1), our model reproduces the best fit proposed by Torruellas et al. [“fit (original)” in (a)] with a maximum error of 4.2% [in (b)] (average error is 0.8%). Fit parameters were chosen to fit both the amplitude of χ(3)(3ω) [see (a)] and its phase (not shown).

Tables (1)

Tables Icon

Table 1 Fit parameters for third-order dispersion in poly(3-decylthiophene) [17].

Equations (21)

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d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 ( 1 + Δ μ ω 0 E ) p = ω 0 2 ( 1 + Δ μ ω 0 E ) ( μ ¯ Δ μ 2 w ) κ E w ,
d w d t + w w ( eq ) T 1 = ( 2 ω 0 ) E d p d t ,
d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 ( 1 + Δ μ ω 0 E ) p = ω 0 2 ( 1 + Δ μ ω 0 E ) μ g + κ E ,
d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 p = κ E ,
d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 p = f ( E ) ,
p ( μ ¯ w 2 Δ μ ) κ w E ω 0 2 ( 1 + a E ) ,
d w d t + w w ( eq ) T 1 ( 2 ω 0 ) E d d t [ ( μ ¯ w 2 Δ μ ) κ w E ω 0 2 ( 1 + a E ) ] .
d w d t [ a E + b E 2 ( 1 + a E ) ] d w d t [ b E ( 1 + a E ) 2 ] w d E d t ,
d w w = b E [ ( 1 + a E ) 2 + b E 2 ] ( 1 + a E ) d E .
w = C ( 1 + a E ) [ ( 1 + a E ) 2 + b E 2 ] 1 / 2 = ( 1 + a E ) [ ( 1 + a E ) 2 + b E 2 ] 1 / 2 .
d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 ( 1 + a E ) p ω 0 2 ( 1 + a E ) μ ¯ + [ ω 0 2 ( 1 + a E ) f + κ E ] ( 1 + a E ) [ ( 1 + a E ) 2 + b E 2 ] 1 / 2
d 2 p d t 2 + 2 T 2 d p d t + ω 0 2 p ω 0 2 { μ g + a 1 E + a 2 E 2 + a 3 E 3 + } ,
d 2 P d t 2 + γ d P d t + ω 0 2 P = ω 0 2 0 { χ ¯ ( 1 ) E + χ ¯ ( 2 ) E 2 + χ ¯ ( 3 ) E 3 + } ,
P ( 1 ) ( ω L ) = 0 χ ¯ ( 1 ) ( 1 ω L 2 / ω 0 2 ) E 0 , P ( 3 ) ( 3 ω L ) = 0 χ ¯ ( 3 ) ( 1 ω L 2 / ω 0 2 ) 3 ( 1 9 ω L 2 / ω 0 2 ) E 0 3 ,
P ˜ ± ( 1 ) ( ω ) = 0 ( ω 0 2 ω 0 2 ω 2 i γ ω ) χ ¯ ( 1 ) E 0 2 G ^ 1 ( ω ± ω L ) ,
P ˜ ± ( 3 ) ( ω ) = 0 ( ω 0 2 ω 0 2 ω 2 i γ ω ) χ ¯ ( 3 ) E 0 3 8 [ 3 G ^ 3 ( ω ± ω L ) + G ^ 3 ( ω ± ω L ) ] .
G ^ n ( ω ) = 1 2 π e n t 2 / T 2 e i ω t d t = T 2 n e T 2 ω 2 / 4 n .
P ˜ ( 1 ) ( ω L ) = 1 2 0 χ ¯ ( 1 ) ( 1 ω L 2 / ω 0 2 ) E 0 , P ˜ ( 3 ) ( 3 ω L ) = 1 8 0 χ ¯ ( 3 ) ( 1 9 ω L 2 / ω 0 2 ) E 0 3 .
P ˜ ( ω ) = 0 ξ ( ω 0 2 ω 0 2 ω 2 i γ ω ) χ ¯ ( ξ ) FT { E ξ } ξ P ˜ ( ξ ) ( ω ) ,
d 2 P ( ξ ) d t 2 + γ d P ( ξ ) d t + ω 0 2 P ( ξ ) = ω 0 2 0 χ ¯ ( ξ ) E ξ .
χ ( ξ ) ( ω ) = k χ k ( ξ ) ( ω ) = k ( ω k 2 ω k 2 ω 2 i γ k ω ) χ ¯ k ( ξ ) .

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