Abstract

The description of the energy density associated with an electromagnetic field propagating through matter must treat two different phenomena: dispersion, the variation of the refractive index with frequency, and dissipation, the loss of field energy by absorption. In many cases, as in common dielectrics, the dispersive medium is essentially transparent, so dissipation can be neglected. For metals, however, both dispersion and dissipation must be taken into account, and their respective contributions vary significantly with the frequency of the electromagnetic field. Plasmonic structures such as slits, holes, and channel waveguides always involve surfaces between dielectrics and metals, and the energy density in the vicinity of the interface figures importantly in the dynamic response of these structures to light excitation in the visible and near-infrared spectral regions. Here we consider the electromagnetic energy density propagated on and dissipated at real metal–dielectric surfaces, including the important surface plasmon polariton, the wave guided by the interface. We show how the “stored energy” oscillates over an optical cycle between the plasmonic structure and the propagating surface mode, while the dissipated energy continues to accumulate over the same period. We calculate these energy densities for the case of the silver–air interface (using two datasets for silver permittivity commonly cited in the research literature) over a range of frequencies corresponding to the range of wavelengths from 200 to 2000nm.

© 2011 Optical Society of America

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References

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  1. V. Giannini, A. Berrier, S. M. Maier, J. A. Sanchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010).
    [CrossRef] [PubMed]
  2. M. Mansuripur, A. R. Zakharian, A. Lesuffleur, S. H. Oh, R. J. Jones, N. C. Lindquist, H. Im, A. Kobyakov, and J. V. Moloney, “Plasmonic nano-structures for optical data storage,” Opt. Express 17, 14001–14014 (2009).
    [CrossRef] [PubMed]
  3. S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
    [CrossRef]
  4. N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  8. P. G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601(2009).
    [CrossRef]
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    [CrossRef]
  10. S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
    [CrossRef]
  11. K. J. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220(2010).
    [CrossRef]
  12. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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  16. R. Loudon, “Propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245(1970).
    [CrossRef]
  17. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  18. D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).
  19. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  20. J. W. Nilsson and S. Riedel, Electric Circuits, 9th ed. (Prentice-Hall, 2010).
  21. E.Palik and G.Ghosh, eds., The Electronic Handbook of Optical Constants of Solids (Academic, 1999).
  22. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2010

S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
[CrossRef]

V. Giannini, A. Berrier, S. M. Maier, J. A. Sanchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010).
[CrossRef] [PubMed]

K. J. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220(2010).
[CrossRef]

E. Feigenbaum and H. A. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef] [PubMed]

2009

2008

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

2007

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

2006

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110(2006).
[CrossRef]

2005

S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
[CrossRef]

2002

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

1972

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

1970

R. Loudon, “Propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245(1970).
[CrossRef]

Atwater, H. A.

E. Feigenbaum and H. A. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Berrier, A.

Boardman, A. D.

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110(2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Brillouin, L.

L. Brillouin, Group Velocity and Wave Propagation (Academic, 1960).

Chen, H. S.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

Christy, R. W.

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

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Elezzabi, A. Y.

S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
[CrossRef]

Engheta, N.

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef] [PubMed]

Feigenbaum, E.

E. Feigenbaum and H. A. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef] [PubMed]

Giannini, V.

Griffiths, D.

D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

Grzegorczyk, T. M.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

Holmes, R. J.

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Huangfu, J. T.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

Im, H.

Johnson, P. B.

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

Jones, R. J.

Kobyakov, A.

Kong, J. A.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Lesuffleur, A.

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Lindquist, N. C.

M. Mansuripur, A. R. Zakharian, A. Lesuffleur, S. H. Oh, R. J. Jones, N. C. Lindquist, H. Im, A. Kobyakov, and J. V. Moloney, “Plasmonic nano-structures for optical data storage,” Opt. Express 17, 14001–14014 (2009).
[CrossRef] [PubMed]

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Loudon, R.

R. Loudon, “Propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245(1970).
[CrossRef]

Luan, P. G.

P. G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601(2009).
[CrossRef]

Luhman, W. A.

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Maier, S. M.

Mansuripur, M.

Marinov, K.

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110(2006).
[CrossRef]

Moloney, J. V.

Nilsson, J. W.

J. W. Nilsson and S. Riedel, Electric Circuits, 9th ed. (Prentice-Hall, 2010).

Oh, S. H.

M. Mansuripur, A. R. Zakharian, A. Lesuffleur, S. H. Oh, R. J. Jones, N. C. Lindquist, H. Im, A. Kobyakov, and J. V. Moloney, “Plasmonic nano-structures for optical data storage,” Opt. Express 17, 14001–14014 (2009).
[CrossRef] [PubMed]

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Ran, L. X.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

Riedel, S.

J. W. Nilsson and S. Riedel, Electric Circuits, 9th ed. (Prentice-Hall, 2010).

Rivas, J. G.

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Sanchez-Gil, J. A.

Sederberg, S.

S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
[CrossRef]

Shivanand,

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Tretyakov, S. A.

S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
[CrossRef]

Van, V.

S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
[CrossRef]

Webb, K. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Zakharian, A. R.

Appl. Phys. Lett.

S. Sederberg, V. Van, and A. Y. Elezzabi, “Monolithic integration of plasmonic waveguides into a complimentary metal-oxide-semiconductor- and photonic-compatible platform,” Appl. Phys. Lett. 96, 121101 (2010).
[CrossRef]

N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

J. Appl. Phys.

H. S. Chen, L. X. Ran, J. T. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915 (2006).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A: Gen. Phys.

R. Loudon, “Propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245(1970).
[CrossRef]

Opt. Express

Phys. Lett. A

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
[CrossRef]

Phys. Rev. B

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

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110(2006).
[CrossRef]

Phys. Rev. E

P. G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601(2009).
[CrossRef]

Phys. Rev. Lett.

E. Feigenbaum and H. A. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef] [PubMed]

Science

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef] [PubMed]

Other

L. Brillouin, Group Velocity and Wave Propagation (Academic, 1960).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

J. W. Nilsson and S. Riedel, Electric Circuits, 9th ed. (Prentice-Hall, 2010).

E.Palik and G.Ghosh, eds., The Electronic Handbook of Optical Constants of Solids (Academic, 1999).

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Figures (9)

Fig. 1
Fig. 1

Brillouin composite field consisting of two “signal fields” with frequencies ω 1 , ω 2 displaced symmetrically above and below the “carrier” at ω 0 . Superposition produces a slowly varying envelope in which the amplitude of the rapidly varying carrier is modulated.

Fig. 2
Fig. 2

Solid curve shows electric-field behavior in time as expressed by Eq. (13). Dashed curve shows slowly varying envelope.

Fig. 3
Fig. 3

Plots of the integrands for the four constituent J n functions included in the power density expression, Eq. (29). J 1 4 correspond to (a)–(d), respectively.

Fig. 4
Fig. 4

Time behavior of the integrals J 1 4 ( t ) .

Fig. 5
Fig. 5

The “signal” sine wave sin 2 ν t is rapidly modulated by the “carrier” sine wave sin 2 ω t and can be replaced by the factor 1 / 2 sin 2 ( ν t ) . Averaging over the optical cycle period T results in the factor 1 / 4 in Eq. (32).

Fig. 6
Fig. 6

(a) Plot of the Palik data [21] for ϵ versus wavelength is shown in filled circles. Fits to these data points are indicated for the FEG HOM model by thick dashed curves and for the polynomial fit by the solid curve. (b) Similar plots of Palik data [21] for ϵ .

Fig. 7
Fig. 7

(a) Plot of the Johnson and Christy data [22] for ϵ versus wavelength is shown in filled circles. Fits to these data points are indicated for the FEG HOM model by thick dashed curves and for the polynomial fit by the solid curve. (b) Similar plots of Johnson and Christy data [22] for ϵ .

Fig. 8
Fig. 8

(a) Stored energy density and (b) dissipative energy density. Dashed curves trace the FEG HOM model [Eqs. (19, 25, 26)] normalized to ϵ 0 E 0 2 . Solid curves trace the Brillouin extended expressions [Eqs. (31, 32)] using fitted polynomials to the Palik data for ϵ and ϵ .

Fig. 9
Fig. 9

(a) Stored energy density and (b) dissipative energy density. Dashed curves traces the FEG HOM model [Eqs. (19, 25, 26)] normalized to ϵ 0 E 0 2 . Solid curve traces the Brillouin extended expressions [Eqs. (31, 32)] using fitted polynomials to the Johnson–Christy data for ϵ and ϵ .

Equations (44)

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

× E + B t = 0 ,
× H D t = J ,
· ( E × H ) + E · J = E · D t H · B t .
S ( E × H ) · n d A = V E · J d V V ( E · D t + H · B t ) d V .
S S · n d A = V u t d V ,
S ( E × H ) · n d A = S S · n d A = V ( E · D t + H · B t ) = V u t d V ,
· S = ( E · D t + H · B t ) = u t .
S = 1 2 ( E × H * ) ,
· S = 1 4 [ E · D * t + H · B * t ] = u t .
u = 1 4 [ E · D * + H · B * ] = 1 4 [ ϵ ( ω ) | E 2 | + μ 0 | H 2 | ] ,
| H | = ϵ ( ω ) μ 0 | E | ,
u = 1 2 ϵ ( ω ) | E | 2 = 1 2 E · D * .
W ( t ) = u t = 1 2 [ 1 2 ( E + E * ) · 1 2 ( D t + D * t ) ] .
ω 1 = ω + ν ω 2 = ω ν .
E = 1 2 E 0 ( e i ω 1 t e i ω 2 t ) E * = 1 2 E 0 ( e i ω 1 t e i ω 2 t ) .
1 2 ( E + E * ) = 1 2 E 0 [ cos ( ω 1 t ) cos ( ω 2 t ) ] = E 0 [ sin ( ω t ) sin ( ν t ) ] .
D = 1 2 ϵ 0 E 0 [ ϵ r ( ω 1 ) e i ω 1 t ϵ r ( ω 2 ) e i ω 2 t ] D * = 1 2 ϵ 0 E 0 [ ϵ r * ( ω 1 ) e i ω 1 t ϵ r * ( ω 2 ) e i ω 2 t ] ,
u = 1 2 ϵ 0 | E 0 | 2 [ ω ϵ r ( ω ) ω ] = 1 2 ϵ 0 | E 0 | 2 [ ϵ r ( ω ) + ω ϵ r ( ω ) ω ] .
E ( t ) = E 0 ( t ) e i ω t ,
E 0 ( t ) = η E η e i η t .
D t = i ω ϵ ( ω ) E + d ( ω ϵ ) d ω E t e i ω t .
u = 1 2 ϵ 0 | E 0 | 2 [ ω ϵ r ( ω ) ω ] = 1 2 ϵ 0 | E 0 | 2 [ ϵ r ( ω ) + ω ϵ r ( ω ) ω ] ,
d 2 P d t 2 + γ d P d t + ω 0 2 P = ω p 2 ϵ 0 E ,
u = 1 4 ϵ 0 | E 0 | 2 ( ϵ r + 2 ϵ r ω γ ) .
ϵ r = 1 + ω p 2 ( ω 0 2 ω 2 ) ( ω 0 2 ω 2 ) 2 + ( γ ω ) 2 ,
ϵ r = γ ω p 2 ω ( ω 0 2 ω 2 ) 2 + ( γ ω ) 2 .
2 ϵ r ω γ 2 ω p 2 ω 2 ( ω 0 2 ω 2 ) 2 .
lim γ 0 d ϵ r d ω = 2 ω ω p 2 ( ω 0 2 ω 2 ) 2 ,
lim γ 0 u = u s = 1 4 ϵ 0 | E 0 | 2 ( ϵ r + ω d ϵ r d ω ) ,
u d = 1 2 ϵ 0 | E 0 | 2 ϵ r ω γ .
( γ ω 2 ϵ 0 ω p 2 ) | P | 2 = ( ϵ 0 ϵ r ω ) | E | 2 ,
u ¯ d = 1 2 ϵ 0 ϵ r | E 0 | 2 ,
ϵ r 1 ω p 2 ω 2 ,
ϵ r γ ω p 2 ω 3 .
d ϵ r d ω = 2 ω p 2 ω 3 and ω d ϵ r d ω = 2 ϵ r ω γ ,
W ( t ) = [ ϵ 0 | E 0 | 2 4 ν 2 π ] × [ 2 ν [ ω ϵ ( ω ) ] ω 0 2 π / ν [ sin 2 ( ω t ) sin ( 2 ν t ) ] d t + 4 ω ϵ ( ω ) 0 2 π / ν [ sin 2 ( ω t ) sin 2 ( ν t ) ] d t + 2 ω ϵ ( ω ) 0 2 π / ν [ sin ( 2 ω t ) sin 2 ( ν t ) ] d t ν [ ω ϵ ( ω ) ] ω 0 2 π / ν [ sin ( 2 ω t ) sin ( 2 ν t ) ] d t ] .
W ( t ) = [ ϵ 0 | E 0 | 2 4 ν 2 π ] × [ 2 ν [ ω ϵ ( ω ) ] ω J 1 ( 2 π ν ) + 4 ω ϵ ( ω ) J 2 ( 2 π ν ) + 2 ω ϵ ( ω ) J 3 ( 2 π ν ) ν [ ϵ ( ω ) ] ω J 4 ( 2 π ν ) ] ,
J 1 ( t ) = 1 4 ν [ 1 cos ( 2 ν t ) + ν 2 ( 1 cos ( 2 ω t ) cos ( 2 ν t ) ) ν ω sin ( 2 ω t ) sin ( 2 ν t ) ( ω 2 ν 2 ) ] 0 t J 2 ( t ) = 1 4 [ t sin ( 2 ω t ) 2 ω sin ( 2 ν t ) 2 ν + ω sin ( 2 ω t ) cos ( 2 ν t ) ν cos ( 2 ω t ) cos ( 2 ν t ) 2 ( ω 2 ν 2 ) ] 0 t J 3 ( t ) = 1 4 ω [ 1 cos ( 2 ω t ) ω 2 ( 1 cos ( 2 ω t ) cos ( 2 ν t ) ) ν ω sin ( 2 ω t ) sin ( 2 ν t ) ( ω 2 ν 2 ) ] 0 t J 4 ( t ) = 1 2 ν [ ν 2 sin ( 2 ω t ) cos ( 2 ν t ) ω ν cos ( 2 ω t ) cos ( 2 ν t ) ( ω 2 ν 2 ) ] 0 t .
J 1 ( π 2 ν ) = 1 2 ν + ν cos 2 ( ω π / 2 ν ) 2 ( ω 2 ν 2 ) J 2 ( π 2 ν ) = 1 4 [ π 2 ν 1 2 ω sin ( ω π / ν ) ( ω 2 ( ω 2 ν 2 ) ) × sin ( ω π / ν ) ] J 3 ( π 2 ν ) = 1 2 [ sin 2 ( ω π / 2 ν ) ω ( ω ω 2 ν 2 ) cos 2 ( ω π / 2 ν ) ] J 4 ( π 2 ν ) = 1 2 ν sin ( ω π / ν ) ω 2 ν 2 .
J 1 ( π 2 ν ) 1 2 ν J 2 ( π 2 ν ) π 8 ν J 3 ( 2 π ν ) 0 J 4 ( 2 π ν ) 0 ,
u ( t ) = W ( t ) T = W ( t ) ( 2 π ν ) .
u s = ϵ 0 | E 0 | 2 4 [ [ ω ϵ r ( ω ) ] ω ] ,
u d = 1 4 ϵ 0 | E 0 | 2 ω ϵ r ( ω ) ( 2 π ν ) .
u d = 1 4 ϵ 0 | E 0 | 2 γ ω p 2 ω 2 ( 2 π ν ) .

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