Abstract

An exact separation of both electric and magnetic energies into stored and lost energies is shown to be possible in the special case when the wave impedance is independent of frequency. A general expression for the electromagnetic energy density in such a dispersive medium having a negative refractive index is shown to be accurate in comparison with numerical results. Using an example metamaterial response that provides a negative refractive index, it is shown that negative time-averaged stored energy can occur. The physical meaning of this negative energy is explained as the energy temporarily borrowed by the field from the material. This observation for negative index materials is of interest when approaching properties for a perfect lens. In the broader context, the observation of negative stored energy is of consequence in the study of dispersive materials.

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  3. K. J. Webb and L. Thylén, “A perfect lens material condition from adjacent absorptive and gain resonances,” Opt. Lett. 33, 747–749 (2008).
    [CrossRef] [PubMed]
  4. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
  5. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  6. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  7. V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).
  8. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  9. S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
    [CrossRef]
  10. A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
    [CrossRef]
  11. T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
    [CrossRef]
  12. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B 28, 1544–1552 (2011).
    [CrossRef]
  13. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001).
    [CrossRef]
  14. K. J. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220 (2010).
    [CrossRef]
  15. Y. Ben-Aryeh, “Energy dispersion relation for negative refraction (NR) materials,” Opt. Commun. 284, 5281–5283 (2011).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, 1999).
  17. A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).
  18. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  19. K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
    [CrossRef]

2011 (2)

2010 (1)

2008 (1)

2006 (1)

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

2005 (1)

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

2004 (2)

T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
[CrossRef]

2002 (1)

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

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1990 (1)

V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).

1970 (1)

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

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Ben-Aryeh, Y.

Y. Ben-Aryeh, “Energy dispersion relation for negative refraction (NR) materials,” Opt. Commun. 284, 5281–5283 (2011).
[CrossRef]

Bezerra, M.

Boardman, A. D.

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

Brillouin, L.

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

Cui, T. J.

T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, 1999).

Kong, J. A.

T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Loudon, R.

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

Marinov, K.

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

Nelson, K. A.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
[CrossRef]

Nunes, F. D.

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Polevoi, V. G.

V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).

Ruppin, R.

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

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Semchenko, I.

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).

Serdyukov, A.

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Shivanand,

Sihvola, A.

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Thylén, L.

Tretyakov, S.

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).

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]

Vasconcelos, T. C.

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Ward, D. W.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
[CrossRef]

Webb, K. J.

Weiner, J.

Yang, M.

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

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

Opt. Commun. (1)

Y. Ben-Aryeh, “Energy dispersion relation for negative refraction (NR) materials,” Opt. Commun. 284, 5281–5283 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

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 (2)

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

T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

Phys. Rev. E (2)

K. J. Webb, M. Yang, D. W. Ward, and K. A. Nelson, “Metrics for negative refractive index materials,” Phys. Rev. E 70, 035602 (2004).
[CrossRef]

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (4)

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

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, 1999).

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Publishing Group, 2001).

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Figures (8)

Fig. 1
Fig. 1

Real (solid line) and imaginary (dotted line) part of the dielectric constant and relative permeability { ɛ , μ } = 1 + 2.7 ( ω 1 2 ω 2 i 0.5 ω ) 1 for ω1 = 1. The three different carrier frequencies used (ω0 = 4/9, 8/7, 4/3) are marked by vertical solid lines.

Fig. 2
Fig. 2

uE〉|ε″=0 = 〈wE〉 and 〈uH〉|μ″=0 = 〈wH〉 obtained for ω0 = 4/9 (dotted line), ω0 = 8/7 (dashed-dotted line) and ω0 = 4/3 (dashed line). { ɛ , μ } = Real { 1 + 2.7 ( ω 1 2 ω 2 i 0.5 ω ) 1 } and ω1 = 1. The lines give exact numerical results and the model result (circles) plots Eq. (13) for the ω0 = 8/7 case.

Fig. 3
Fig. 3

∂uE/∂t〉|ε′=0 = 〈∂qE/∂t〉 and 〈∂uH/∂t|μ′=0 = 〈∂qH/∂t〉 obtained for ω0 = 4/9 (dotted line), ω0 = 8/7 (dashed-dotted line) and ω0 = 4/3 (dashed line). { ɛ , μ } = Imag { 1 + 2.7 ( ω 1 2 ω 2 i 0.5 ω ) 1 } and ω1 = 1. The lines give exact numerical results and the model result (circles) is from Eq. (14) for the ω0 = 8/7 case.

Fig. 4
Fig. 4

Exact time-averaged energy densities, 〈uE〉 and 〈uH〉, obtained for ω0 = 4/9 (dotted line), ω0 = 8/7 (dashed-dotted line) and ω0 = 4/3 (dashed line). { ɛ , μ } = 1 + 2.7 ( ω 1 2 ω 2 i 0.5 ω ) 1 and ω1 = 1.

Fig. 5
Fig. 5

Real (solid line) and imaginary (dotted line) part of the dielectric constant ɛ = 1 + 1.5 ( ω 1 2 ω 2 i 0.1 ω ) 1 1.5 ( ω 2 2 ω 2 i 0.1 ω ) 1 for ω1 = 1 and ω 2 = α 2 ω 0 2 ω 1 2 with α = 1. The carrier frequency is chosen to be ω0 = 1 and is marked by the vertical solid line.

Fig. 6
Fig. 6

uE|ε″=0 = 〈wE〉 obtained for ω0 = 1. ɛ = Real { 1 + 1.5 ( ω 1 2 ω 2 i 0.1 ω ) 1 1.5 ( ω 2 2 ω 2 i 0.1 ω ) 1 } with ω1 = 1 and ω 2 = α 2 ω 0 2 ω 1 2. α = 1, 1.0001 and 0.9999 for the lossless, small loss and small gain cases shown in solid, dashed and dotted lines, respectively. The three line types overlap in this case because of negligible change in ε′ for the different cases. The lines give exact numerical results and the model result (circles) plots Eq. (13).

Fig. 7
Fig. 7

∂uE/∂t|ε′=0 = 〈∂qE/∂t〉 obtained for ω0 = 1. ɛ = Imag { 1 + 1.5 ( ω 1 2 ω 2 i 0.1 ω ) 1 1.5 ( ω 2 2 ω 2 i 0.1 ω ) 1 } with ω1 = 1 and ω 2 = α 2 ω 0 2 ω 1 2. α = 1, 1.0001 and 0.9999 for the lossless, small loss and small gain cases shown in solid, dashed and dotted lines respectively. The lines give exact numerical results and the model result (circles) plots Eq. (14).

Fig. 8
Fig. 8

Exact time-averaged energy densities 〈uE〉 obtained for ω0 = 1. ɛ = 1 + 1.5 ( ω 1 2 ω 2 i 0.1 ω ) 1 1.5 ( ω 2 2 ω 2 i 0.1 ω ) 1 with ω1 = 1 and ω 2 = α 2 ω 0 2 ω 1 2. α = 1, 1.0001 and 0.9999 for the lossless, small loss and small gain cases shown in solid, dashed and dotted lines respectively.

Equations (24)

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E × H d s = [ E D t + H B t ] d v ,
D ( t ) = ɛ 0 2 π ɛ ( ω ) E ( ω ) e i ω t d ω
B ( t ) = μ 0 2 π μ ( ω ) H ( ω ) e i ω t d ω ,
u E t E D t
= ɛ 0 E ( t ) 2 π i ω [ ɛ ( ω ) + i ɛ ( ω ) ] [ E ( ω ) + i E ( ω ) ] [ cos ( ω t ) i sin ( ω t ) ] d ω ,
u E t = ɛ 0 E ( t ) 2 π ω ɛ ( ω ) [ E ( ω ) sin ( ω t ) + E ( ω ) cos ( ω t ) ] + ω ɛ ( ω ) [ E ( ω ) cos ( ω t ) + E ( ω ) sin ( ω t ) ] d ω u E t | ɛ = 0 w E / t + u E t | ɛ = 0 q E / t .
u H t H B t
= μ 0 H ( t ) 2 π i ω [ μ ( ω ) + i μ ( ω ) ] [ H ( ω ) + i H ( ω ) ] [ cos ( ω t ) i sin ( ω t ) ] d ω ,
u H t = μ 0 H ( t ) 2 π ω μ ( ω ) [ H ( ω ) sin ( ω t ) + H ( ω ) cos ( ω t ) ] + ω μ ( ω ) [ H ( ω ) cos ( ω t ) + H ( ω ) sin ( ω t ) ] d ω u H t | μ = 0 w H / t + u H t | μ = 0 q H / t .
H ( t ) = 1 2 π H ( ω ) e i ω t d ω
= 1 2 π E ( ω ) η ( ω ) e i ω t d ω ,
u t = w E t + q E t + w H t + q H t .
w E 1 4 ɛ 0 [ ω ɛ ( ω ) ] ω | ω 0 e 2 ( t )
q E t 1 2 ω 0 ɛ 0 ɛ e 2 ( t ) .
w H 1 4 μ 0 [ ω μ ( ω ) ] ω | ω 0 h 2 ( t )
q H t 1 2 ω 0 μ 0 μ h 2 ( t ) ,
H ( t ) 1 2 [ u ( ω 0 ) e ( t ) + u ω | ω 0 ( i e ( t ) t ) ] e i ω 0 t + c . c . = e ( t ) [ u ( ω 0 ) cos ( ω 0 t ) + u ( ω 0 ) sin ( ω 0 t ) ] + e ( t ) t [ u ω | ω 0 sin ( ω 0 t ) u ω | ω 0 cos ( ω 0 t ) ] ,
B ( t ) t ω 0 e ( t ) [ v ( ω 0 ) cos ( ω 0 t ) v ( ω 0 ) sin ( ω 0 t ) ] + e ( t ) t [ ( ω v ) ω | ω 0 cos ( ω 0 t ) + ( ω v ) ω | ω 0 sin ( ω 0 t ) ] ,
u H t 1 2 { ω 0 e 2 ( t ) [ u ( ω 0 ) v ( ω 0 ) u ( ω 0 ) v ( ω 0 ) ] + e ( t ) e ( t ) t [ u ( ω 0 ) ( ω v ) ω | ω 0 + u ( ω 0 ) ( ω v ) ω | ω 0 ω 0 v ( ω 0 ) u ω | ω 0 ω 0 v ( ω 0 ) u ω | ω 0 ] + [ e ( t ) t ] 2 [ u ω | ω 0 ( ω v ) ω | ω 0 u ω | ω 0 ( ω v ) ω | ω 0 ] } .
u H t 1 2 [ ω 0 e 2 ( t ) u ( ω 0 ) v ( ω 0 ) + e ( t ) e ( t ) t u ( ω 0 ) ( ω v ) ω | ω 0 ] = 1 2 [ μ 0 e ( t ) e ( t ) t ( u ( ω 0 ) ) 2 ( ω μ ( ω ) ) ω | ω 0 + ω 0 μ 0 e 2 ( t ) ( u ( ω 0 ) ) 2 μ ( ω 0 ) ] ,
u E + u H = q E + q H + w E + w H 0 ,
χ E ( ω ) = a 1 ω 1 2 ω 2 i γ 1 ω = χ M ( ω ) .
E ( t ) = ( σ 2 π ) 1 exp [ ( t t c ) 2 ( 2 σ 2 ) 1 ] cos [ ω 0 ( t t c ) ] ,
χ E ( ω ) = 1.5 ω 1 2 ω 2 i 0.1 ω 1.5 ω 2 2 ω 2 i 0.1 ω ,

Metrics