Abstract

We develop the time-domain discrete dipole approximation (DDA), describing the temporal evolution of electric field in plasmonic nanostructures. The main equation is obtained by taking the inverse Fourier transform of the Taylor expansion of the frequency-domain DDA in terms of frequency deviation from the central frequency. Thus we assume that incident wavefronts of different frequencies accumulate relatively small phase difference when passing the particle. This assumption is always valid for nanoparticles much smaller than the wavelength. Being the time-domain method, the proposed approach also requires an analytic frequency dependence of electric permittivity, e.g. the Drude model. We present numerical results of application of the time-domain DDA to silver nanosphere, rod, and disk, which agree well with that obtained with its frequency-domain counterpart and the finite-difference time-domain method. Moreover, the time-domain DDA is the fastest of the three methods for incident pulses of several-femtoseconds width. Thus, it can effectively be applied for modeling the temporal responses of plasmonic nanostructures.

© 2015 Optical Society of America

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References

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  1. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
    [Crossref]
  2. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
    [Crossref]
  3. Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).
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    [Crossref] [PubMed]
  6. 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(12), 127401 (2010).
    [Crossref] [PubMed]
  7. D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
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    [Crossref]
  21. M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
    [Crossref]
  22. K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010).
    [Crossref] [PubMed]
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2013 (2)

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

P. C. Chaumet, T. Zhang, A. Rahmani, B. Gralak, and K. Belkebir, “Discrete dipole approximation in time domain through the Laplace transform,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(6), 063303 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (4)

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljacić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express 19(2), 1539–1562 (2011).
[Crossref] [PubMed]

M. S. Dhoni and W. Ji, “Extension of discrete-dipole approximation model to compute nonlinear absorption in gold nanostructures,” J. Phys. Chem. C 115(42), 20359–20366 (2011).
[Crossref]

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Discrete dipole approximation for time-domain computation of optical forces on magnetodielectric scatterers,” Opt. Express 19(3), 2466–2475 (2011).
[Crossref] [PubMed]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

2010 (4)

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
[Crossref]

K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010).
[Crossref] [PubMed]

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(12), 127401 (2010).
[Crossref] [PubMed]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

2009 (1)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

2005 (1)

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B 72(19), 195429 (2005).
[Crossref]

1994 (1)

1991 (1)

1988 (1)

Arsenin, A. V.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Belkebir, K.

Bravo-Abad, J.

Chaumet, P. C.

Chong, Y.

Chua, S.-L.

de Kanter, D.

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
[Crossref]

Dhoni, M. S.

M. S. Dhoni and W. Ji, “Extension of discrete-dipole approximation model to compute nonlinear absorption in gold nanostructures,” J. Phys. Chem. C 115(42), 20359–20366 (2011).
[Crossref]

Draine, B. T.

Fedyanin, D. Y.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Flatau, P. J.

Fu, Y.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Goedecke, G. H.

Gong, Q.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Goodman, J. J.

Gralak, B.

P. C. Chaumet, T. Zhang, A. Rahmani, B. Gralak, and K. Belkebir, “Discrete dipole approximation in time domain through the Laplace transform,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(6), 063303 (2013).
[Crossref] [PubMed]

Griebner, U.

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(12), 127401 (2010).
[Crossref] [PubMed]

Herrmann, J.

Herz, E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

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(12), 127401 (2010).
[Crossref] [PubMed]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Hohenester, U.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B 72(19), 195429 (2005).
[Crossref]

Hu, X.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Husakou, A.

Ji, W.

M. S. Dhoni and W. Ji, “Extension of discrete-dipole approximation model to compute nonlinear absorption in gold nanostructures,” J. Phys. Chem. C 115(42), 20359–20366 (2011).
[Crossref]

Kauranen, M.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Kim, K.-H.

Krasavin, A. V.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Krenn, J.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B 72(19), 195429 (2005).
[Crossref]

Narimanov, E. E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Noginov, M. A.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Novotny, L.

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

O’Brien, S. G.

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(12), 127401 (2010).
[Crossref] [PubMed]

Quidant, R.

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

Rahmani, A.

Renger, J.

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

Shalaev, V. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Soljacic, M.

Stone, A. D.

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

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(12), 127401 (2010).
[Crossref] [PubMed]

van Hulst, N.

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

Wiesner, U.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

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(12), 127401 (2010).
[Crossref] [PubMed]

Yang, H.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Zhang, T.

P. C. Chaumet, T. Zhang, A. Rahmani, B. Gralak, and K. Belkebir, “Discrete dipole approximation in time domain through the Laplace transform,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(6), 063303 (2013).
[Crossref] [PubMed]

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Zhu, Y.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Appl. Opt. (1)

J. Nanophoton. (1)

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4(1), 041585 (2010).
[Crossref]

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

J. Phys. Chem. C (1)

M. S. Dhoni and W. Ji, “Extension of discrete-dipole approximation model to compute nonlinear absorption in gold nanostructures,” J. Phys. Chem. C 115(42), 20359–20366 (2011).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (2)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

Nano Lett. (1)

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Nature (1)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. B (1)

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B 72(19), 195429 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

P. C. Chaumet, T. Zhang, A. Rahmani, B. Gralak, and K. Belkebir, “Discrete dipole approximation in time domain through the Laplace transform,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(6), 063303 (2013).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

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(12), 127401 (2010).
[Crossref] [PubMed]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104(4), 04683 (2010).
[Crossref]

Sci. Rep. (1)

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication,” Sci. Rep. 3, 2338 (2013).

Other (3)

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE, New York, 1998).

E. D. Palik, Handbook of Optical Properties of Solids (Academic, New York, 1985).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

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Figures (4)

Fig. 1
Fig. 1

Validation the second-order approximation: absolute values of g, g1, and g2 (a) and real (b) and imaginary (c) parts of f, f1, and f2 versus the wavelength for the values of parameters R = r = 50 nm, δ = 1, s = 0.1, and λ0 = 2πс/ω0 = 400 nm.

Fig. 2
Fig. 2

Temporal evolution of enhanced field in Ag nanosphere with a diameter 70 nm surrounded by air. Grid size is 2 nm. The central wavelength and the duration of incident pulse are 390 nm and 1.88 fs, respectively. Figure 2(a) presents the x-component of real part of enhanced field calculated by using TDDDA in comparison with the result of FDTD at the central position of the nanosphere. Figure 2(b) illustrates the amplitudes of enhanced field, averaged over the space occupied by the nanosphere, for TDDDA and FDTD.

Fig. 3
Fig. 3

Extinction spectra for the same nanosphere as in in Fig. 2 calculated from the data obtained by the TDDDA, in comparison with that directly computed with the Mie theory and the frequency-domain DDA.

Fig. 4
Fig. 4

TDDDA simulations of time- and frequency-domain responses of silver nanorod with a diameter of 40 nm and a length of 70 nm (a, b) and nanodisk with a diameter of 70 nm and a thickness of 42 nm (c, d), embedded in silica glass. The time-domain response (a, c) is given in terms of the space-averaged amplitude of enhanced field in comparison with the incident one. The frequency-domain response (c, d) is given in terms of extinction spectra in comparison with the direct DDA simulations.

Equations (29)

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Ε m = Ε m in + nm G mn p n ,
G mn = exp(ik R mn ) R mn [ k 2 (1 r ^ mn r ^ mn )+ ik R mn 1 R mn 2 (13 r ^ mn r ^ mn ) ],
G mn (k)= G mn ( k 0 )+ G mn k | k= k 0 (k k 0 )+ 1 2 2 G mn k 2 | k= k 0 (k k 0 ) 2 +,
E m in = E in (ω)exp(ik r m ),
E m in = E in (ω)exp(i k 0 r m )[ 1+i(k k 0 ) r m 1 2 (k k 0 ) 2 r m 2 + ].
E m = E in (ω)exp(i k 0 r m )( 1+i ε h 1/2 c 0 r m Δω ε h 2 c 0 2 r m 2 Δ ω 2 )+ nm ( G mn (0) + G mn (1) Δω+ G mn (2) Δ ω 2 ) p n ,
G mn (1) = ε h 1/2 c 0 exp(i k 0 R mn ) R mn [ k 0 (2+i k 0 R mn )(1 r ^ mn r ^ mn ) k 0 (13 r ^ mn r ^ mn ) ],
G mn (2) = ε h 2 c 0 2 exp(i k 0 R mn ) R mn [ ( k 0 2 R mn 2 +4i k 0 R mn +2)(1 r ^ mn r ^ mn )(i k 0 R mn +1)(i k 0 R mn +1) ].
g= exp(ikR) R [ k 2 (δs)+ R 2 (ikR1)(δ3s) ]
g 1 = g| k= k 0 + ε h 1/2 c 0 exp(i k 0 R) R [ k 0 (2+i k 0 R)(δs) k 0 (δ3s) ]Δω,
g 2 = g 1 + ε h 2 c 0 2 exp(i k 0 R) R [ ( k 0 2 R 2 +4i k 0 R+2)(δs)(i k 0 R+1)(δ3s) ]Δ ω 2 .
f 1 =exp(i k 0 r )( 1+i ε h 1/2 с 0 r m Δω ),
f 2 =exp(i k 0 r )( 1+i ε h 1/2 с 0 r m Δω ε h 2 c 0 2 r m 2 Δ ω 2 ).
A m =( 1 ε h 1/2 c 0 r m t + ε h 2 c 0 2 r m 2 2 t 2 ) A in exp(i k 0 r m )+ nm ( G mn (0) +i G mn (1) t G mn (2) 2 t 2 ) q n ,
ε n (ω)= ε ω p 2 ω 2 +iγω ,
p n0 (ω)= α 0 E n (ω)= 3v 4π ε ε h ε +2 ε h E n (ω),
p n1 (ω)= 3v 4π w E n (ω) ω 2 +iωγ ω p 2 / ( ε +2 ε h ) ,
( ω 2 +iωγ ω p 2 ε +2 ε h ) p n1 (ω)= 3v 4π w E n (ω).
q ¨ n1 +c q ˙ n1 +b q n1 =a A n ,
A m =( 1 ε h 1/2 c 0 r m t + ε h 2 c 0 2 r m 2 2 t 2 ) A in e i k 0 r m + nm ( G mn (0) +i G mn (1) t G mn (2) 2 t 2 )( α 0 A n + q n1 ) .
q n1,N = aΔ t 2 A n,N +(2+cΔt) q n1,N1 q n1,N2 1+cΔt+bΔ t 2 ,
q ˙ n1,N = q n1,N q n1,N1 Δt = aΔ t 2 A n,N (bΔ t 2 1) q n1,N1 q n1,N2 Δt(1+cΔt+bΔ t 2 ) ,
q ¨ n1,N = q n1,N 2 q n1,N1 + q n1,N2 Δ t 2 = aΔt A n,N (c+2bΔt) q n1,N1 +(c+bΔt) q n1,N2 Δt(1+cΔt+bΔ t 2 ) ,
A m,N ( α 0 + aΔ t 2 l ) nm ( G mn (0) + i Δt G mn (1) 1 Δ t 2 G mn (2) ) A n,N = =[ ( 1 ε h 1/2 r m c 0 Δt + ε h r m 2 2 c 0 2 Δ t 2 ) A m,N in +( ε h 1/2 r m c 0 Δt ε h r m 2 c 0 2 Δ t 2 ) A m,N1 in + ε h r m 2 2 c 0 2 Δ t 2 A m,N2 in ]exp(i k 0 r m ) + nm ( G mn (0) (2+cΔt) q n1,N1 q n1,N2 l i G mn (1) { α 0 Δt A n,N1 + 1 lΔt [ (bΔ t 2 1) q n1,N1 + q n1,N2 ] } G mn (2) { α 0 Δ t 2 (2 A n,N1 + A n,N2 )+ 1 lΔt [ (c+2bΔt) q n1,N1 +(c+bΔt) q n1,N2 ] } ),
E m enh (ω)= f 0 ( ω 2 +iωγ) ω 2 +iωγ ω p 2 / ( ε +2 ε h ) E m (ω),
A ¨ m enh +c A ˙ m enh +b A m enh = f 0 ( A ¨ m +c A ˙ m h A m ),
A m,N enh = f 0 l ( 1+cΔthΔ t 2 ) A m,N 2+cΔt l ( f 0 A m,N1 A m,N1 enh )+ 1 l ( f 0 A m,N2 A m,N2 enh ).
A enh = Re( E enh ) 2 + Im( E enh ) 2 ,
dΔ ε n dt = Δ ε n τ ep + χ n (3) τ ee τ ep t d t A n enh ( t ) 2 exp( t t τ ee ) ,

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