Abstract

In the context of electromagnetism and nonlinear optical interactions, damping is generally introduced as a phenomenological, viscous term that dissipates energy, proportional to the temporal derivative of the polarization. Here, we follow the radiation reaction method presented in [G. W. Ford, Phys. Lett. A 157, 217 (1991)], which applies to non-relativistic electrons of finite size, to introduce an explicit reaction force in the Newtonian equation of motion, and derive a hydrodynamic equation that offers new insight on the influence of damping in generic plasmas, metal-based and/or dielectric structures. In these settings, we find new damping-dependent linear and nonlinear source terms that suggest the damping coefficient is proportional to the local charge density and nonlocal contributions that stem from the spatial derivative of the magnetic field. We discuss the conditions that could modify both linear and nonlinear electromagnetic responses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Columbia University, 1909). The second edition appeared in 1915 and was reprinted by Dover.
  2. M. Abraham and A. Föppl, Theorie der Elektrizitat, Vol. II (Teubner, 1905).
  3. P. A. M. Dirac, “Classical Theory of Radiating Electrons,” Proc. R. Soc. Lond. A Math. Phys. Sci. 167(929), 148–169 (1938).
    [Crossref]
  4. R. F. O’Connell, “Radiation Reaction: General approach and applications, especially to electrodynamics,” Contemp. Phys. 53(4), 301–313 (2012).
    [Crossref]
  5. R. Hammond, “Charged Particle in a Constant, Uniform Electric Field with Radiation Reaction,” Adv. Studies Theor. Phys. 5, 275–282 (2011).
  6. J. D. Jackson, The Classical Electromagnetic Field (Wiley, 1999).
  7. F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65(11), 1051–1056 (1997).
    [Crossref]
  8. D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
    [Crossref]
  9. E. J. Moniz and D. H. Sharp, “Radiation reaction in nonrelativistic quantum electrodynamics,” Phys. Rev. D Part. Fields 15(10), 2850–2865 (1977).
    [Crossref]
  10. G. W. Ford and R. F. O’Connell, “Radiation reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A 157(4-5), 217–220 (1991).
    [Crossref]
  11. N. Bloembergen and Y. R. Shen, “Optical nonlinearity of a plasma,” Phys. Rev. 141(1), 298–305 (1966).
    [Crossref]
  12. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
    [Crossref]
  13. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
    [Crossref]
  14. M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011).
    [Crossref] [PubMed]
  15. M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
    [Crossref]
  16. M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
    [Crossref]
  17. A. D. Rakić, A. B. Djuriśić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).
    [Crossref] [PubMed]
  18. H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
    [Crossref]
  19. E. S. Sarachick and G. T. Schappert, “Classical theory of the scattering of intense laser radiation by free electrons,” Phys. Rev. D Part. Fields 1(10), 2738–2753 (1970).
    [Crossref]
  20. G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174(3), 182–184 (1993).
    [Crossref]

2016 (1)

D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
[Crossref]

2014 (1)

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

2012 (2)

M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
[Crossref]

R. F. O’Connell, “Radiation Reaction: General approach and applications, especially to electrodynamics,” Contemp. Phys. 53(4), 301–313 (2012).
[Crossref]

2011 (2)

2010 (1)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

1998 (1)

1997 (1)

F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65(11), 1051–1056 (1997).
[Crossref]

1993 (1)

G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174(3), 182–184 (1993).
[Crossref]

1991 (1)

G. W. Ford and R. F. O’Connell, “Radiation reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A 157(4-5), 217–220 (1991).
[Crossref]

1980 (1)

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

1977 (1)

E. J. Moniz and D. H. Sharp, “Radiation reaction in nonrelativistic quantum electrodynamics,” Phys. Rev. D Part. Fields 15(10), 2850–2865 (1977).
[Crossref]

1970 (1)

E. S. Sarachick and G. T. Schappert, “Classical theory of the scattering of intense laser radiation by free electrons,” Phys. Rev. D Part. Fields 1(10), 2738–2753 (1970).
[Crossref]

1966 (1)

N. Bloembergen and Y. R. Shen, “Optical nonlinearity of a plasma,” Phys. Rev. 141(1), 298–305 (1966).
[Crossref]

1962 (1)

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[Crossref]

1938 (1)

P. A. M. Dirac, “Classical Theory of Radiating Electrons,” Proc. R. Soc. Lond. A Math. Phys. Sci. 167(929), 148–169 (1938).
[Crossref]

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Bloembergen, N.

N. Bloembergen and Y. R. Shen, “Optical nonlinearity of a plasma,” Phys. Rev. 141(1), 298–305 (1966).
[Crossref]

Bloemer, M. J.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

de Ceglia, D.

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
[Crossref]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011).
[Crossref] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Dirac, P. A. M.

P. A. M. Dirac, “Classical Theory of Radiating Electrons,” Proc. R. Soc. Lond. A Math. Phys. Sci. 167(929), 148–169 (1938).
[Crossref]

Djurisic, A. B.

Eager, D.

D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
[Crossref]

Ehrenreich, H.

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[Crossref]

Elazar, J. M.

Ford, G. W.

G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174(3), 182–184 (1993).
[Crossref]

G. W. Ford and R. F. O’Connell, “Radiation reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A 157(4-5), 217–220 (1991).
[Crossref]

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

Grande, M.

Hammond, R.

R. Hammond, “Charged Particle in a Constant, Uniform Electric Field with Radiation Reaction,” Adv. Studies Theor. Phys. 5, 275–282 (2011).

Haus, J. W.

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
[Crossref]

Majewski, M. L.

Moniz, E. J.

E. J. Moniz and D. H. Sharp, “Radiation reaction in nonrelativistic quantum electrodynamics,” Phys. Rev. D Part. Fields 15(10), 2850–2865 (1977).
[Crossref]

O’Connell, R. F.

R. F. O’Connell, “Radiation Reaction: General approach and applications, especially to electrodynamics,” Contemp. Phys. 53(4), 301–313 (2012).
[Crossref]

G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174(3), 182–184 (1993).
[Crossref]

G. W. Ford and R. F. O’Connell, “Radiation reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A 157(4-5), 217–220 (1991).
[Crossref]

Pendrill, A.-M.

D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
[Crossref]

Philipp, H. R.

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[Crossref]

Rakic, A. D.

Reistad, N.

D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
[Crossref]

Rohrlich, F.

F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65(11), 1051–1056 (1997).
[Crossref]

Roppo, V.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011).
[Crossref] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Sarachick, E. S.

E. S. Sarachick and G. T. Schappert, “Classical theory of the scattering of intense laser radiation by free electrons,” Phys. Rev. D Part. Fields 1(10), 2738–2753 (1970).
[Crossref]

Scalora, M.

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
[Crossref]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011).
[Crossref] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Schappert, G. T.

E. S. Sarachick and G. T. Schappert, “Classical theory of the scattering of intense laser radiation by free electrons,” Phys. Rev. D Part. Fields 1(10), 2738–2753 (1970).
[Crossref]

Sharp, D. H.

E. J. Moniz and D. H. Sharp, “Radiation reaction in nonrelativistic quantum electrodynamics,” Phys. Rev. D Part. Fields 15(10), 2850–2865 (1977).
[Crossref]

Shen, Y. R.

N. Bloembergen and Y. R. Shen, “Optical nonlinearity of a plasma,” Phys. Rev. 141(1), 298–305 (1966).
[Crossref]

Sipe, J. E.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

So, V. C. Y.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

Stegeman, G. I.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

Vincenti, M. A.

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

M. Scalora, M. A. Vincenti, D. de Ceglia, M. Grande, and J. W. Haus, “Raman scattering near metal nanostructures,” J. Opt. Soc. Am. B 29(8), 2035–2045 (2012).
[Crossref]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011).
[Crossref] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Adv. Studies Theor. Phys. (1)

R. Hammond, “Charged Particle in a Constant, Uniform Electric Field with Radiation Reaction,” Adv. Studies Theor. Phys. 5, 275–282 (2011).

Am. J. Phys. (1)

F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65(11), 1051–1056 (1997).
[Crossref]

Appl. Opt. (1)

Contemp. Phys. (1)

R. F. O’Connell, “Radiation Reaction: General approach and applications, especially to electrodynamics,” Contemp. Phys. 53(4), 301–313 (2012).
[Crossref]

Eur. J. Phys. (1)

D. Eager, A.-M. Pendrill, and N. Reistad, “Beyond velocity and acceleration: jerk, snap and higher derivatives,” Eur. J. Phys. 37(6), 065008 (2016).
[Crossref]

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

Opt. Express (1)

Phys. Lett. A (2)

G. W. Ford and R. F. O’Connell, “Radiation reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A 157(4-5), 217–220 (1991).
[Crossref]

G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174(3), 182–184 (1993).
[Crossref]

Phys. Rev. (2)

N. Bloembergen and Y. R. Shen, “Optical nonlinearity of a plasma,” Phys. Rev. 141(1), 298–305 (1966).
[Crossref]

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[Crossref]

Phys. Rev. A (2)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

M. Scalora, D. de Ceglia, M. A. Vincenti, and J. W. Haus, “Nonlocal and Quantum Tunneling Contributions to Harmonic Generation in Nanostructures: Electron Cloud Screening Effects,” Phys. Rev. A 90(1), 013831 (2014).
[Crossref]

Phys. Rev. B (1)

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21(10), 4389–4402 (1980).
[Crossref]

Phys. Rev. D Part. Fields (2)

E. S. Sarachick and G. T. Schappert, “Classical theory of the scattering of intense laser radiation by free electrons,” Phys. Rev. D Part. Fields 1(10), 2738–2753 (1970).
[Crossref]

E. J. Moniz and D. H. Sharp, “Radiation reaction in nonrelativistic quantum electrodynamics,” Phys. Rev. D Part. Fields 15(10), 2850–2865 (1977).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

P. A. M. Dirac, “Classical Theory of Radiating Electrons,” Proc. R. Soc. Lond. A Math. Phys. Sci. 167(929), 148–169 (1938).
[Crossref]

Other (3)

H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Columbia University, 1909). The second edition appeared in 1915 and was reprinted by Dover.

M. Abraham and A. Föppl, Theorie der Elektrizitat, Vol. II (Teubner, 1905).

J. D. Jackson, The Classical Electromagnetic Field (Wiley, 1999).

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Figures (1)

Fig. 1
Fig. 1 Complex dielectric constant plotted with (dashed lines) and without (solid lines) the extra factor in the second term of Eq. (17) for a gold-like material modeled by one Drude and one Lorentz oscillator with resonance frequency near 400nm.

Equations (18)

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m 0 v ˙ 2 e 2 3 c 3 v ¨ = F ext ,
m 0 dv dt = F ext + F rad = F ext + τ 0 d F ext dt ,
m 0 ( v t +( v )v )=( eE+ e c v×H p n ), + τ 0 [ t ( eE+ e c v×H p n )+( v )( eE+ e c v×H p n ) ]
J ˙ n ˙ n J+( J ) J ne = n e 2 m 0 E+ e m 0 c J×H e m 0 p + τ 0 m 0 [ n e 2 E t + e c ( J ˙ n ˙ n J )×H+ e c J× H t e p t + ep n n t +e( J )E+ e c ( J )( J ne ×H )( J ) p n ].
ep m 0 τ 0 e m 0 p t + τ 0 n ˙ n e m 0 p 5 E F 3 m 0 ( P )+ τ 0 5 E F 3 m 0 ( P ˙ ).
P ¨ + 1 n 0 e [ P ˙ P ˙ +( P ˙ ) P ˙ ]= e 2 m 0 ( n 0 1 e P )E+ e m 0 c P ˙ ×H + 5 E F 3 m 0 ( P )+ τ 0 e 2 m 0 ( n 0 1 e P ) E t + τ 0 5 E F 3 m 0 ( P ˙ ).
( ×H 4π c P t )= 1 c E t
P ¨ + ω p 2 τ 0 ( 1 P n 0 e ) P ˙ = ω p 2 4π [ E+c τ 0 ×H ] e m 0 P[ E+c τ 0 ×H ], + 5 E F 3 m 0 ( P )+ τ 0 5 E F 3 m 0 ( P ˙ )+ e m 0 c P ˙ ×H+ 1 n 0 e [ P ˙ P ˙ +( P ˙ ) P ˙ ]
P ¨ +γ P ˙ = ω p 2 4π E e m 0 ( P )E+ 5 E F 3 m 0 ( P ) + e m 0 c P ˙ ×H+ 1 n 0 e [ P ˙ P ˙ +( P ˙ ) P ˙ ],
m b d 2 r d t 2 =( kr+eE+ e c r ˙ ×H )+ τ 0,b [ ( k r ˙ +e E t + e c r ¨ ×H+ e c r ˙ × H ˙ ) +( v )( kr+eE+ e c r ˙ ×H ) ],
P ¨ b + τ 0,b ω 0,b 2 P ˙ b = ω 0,b 2 P b + n b e 2 m b E+ τ 0,b n b e 2 m b E t ,
×H 4π c P Total t = 1 c E t .
P ¨ b + τ 0,b ( ω 0,b 2 + ω p,b 2 ) P ˙ b + τ 0,b ω p,b 2 P ˙ f + ω 0,b 2 P b = n b e 2 m b ( E+c τ 0,b ×H ). P ¨ f + τ 0 ω p 2 P ˙ f + τ 0 ω p 2 P ˙ b = n 0 e 2 m 0 ( E+c τ 0 ×H )
( ω 2 i2 γ b ω+ ω 0 2 ) P b i γ b ω P f n 0 e 2 m 0 E ( ω 2 i γ f ω ) P f i γ f ω P b n 0 e 2 m 0 E
P b = n 0 e 2 m 0 ( ω 2 i2 γ b ω+ ω 0 2 ) ( 1+ i γ b ω ( ω 2 i γ f ω ) ) ( 1+ γ b γ f ω 2 ( ω 2 i γ f ω )( ω 2 i2 γ b ω+ ω 0 2 ) ) E,
P f = n 0 e 2 m 0 ( ω 2 iγω ) ( 1+ i γ f ω ( ω 2 i2 γ b ω+ ω 0 2 ) ) ( 1+ γ b γ f ω 2 ( ω 2 i γ f ω )( ω 2 i2 γ b ω+ ω 0 2 ) ) E.
P total =[ n 0 e 2 m 0 ( ω 2 i γ f ω ) + n 0 e 2 m 0 ( ω 2 i2 γ b ω+ ω 0 2 ) ( 1+ i( γ b + γ f )ω ( ω 2 i γ f ω ) γ f γ b ω 2 ( ω 2 i γ f ω ) 2 1+ γ f γ b ω 2 ( ω 2 i γ f ω )( ω 2 i2 γ b ω+ ω 0 2 ) ) ]E,
ε(ω)=1 4π n 0 e 2 m 0 ( ω 2 i γ f ω ) + 4π n 0 e 2 m 0 ( ω 2 i2 γ b ω+ ω 0 2 ) ( 1+ i( γ b + γ f )ω ( ω 2 i γ f ω ) γ f γ b ω 2 ( ω 2 i γ f ω ) 2 1+ γ f γ b ω 2 ( ω 2 i γ f ω )( ω 2 i2 γ b ω+ ω 0 2 ) ).

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