Abstract

A general expression for the electromagnetic energy density in a lossy dispersive medium, applicable for a field having a narrow temporal frequency bandwidth, is derived and compared with exact results for an example dielectric constant. Consequently, the possibility of negative time-averaged stored field energy is shown to have physical meaning. This observation is of interest in the study of dispersive metamaterials, such as those which can exhibit a negative refractive index.

© 2010 Optical Society of America

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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. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
  4. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  5. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  6. V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).
  7. R. Ruppin, “Elecromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  8. S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
    [CrossRef]
  9. A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
    [CrossRef]
  10. 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]
  11. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001).
    [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  13. D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
    [CrossRef] [PubMed]
  14. J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
    [CrossRef]
  15. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).
  16. G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989).
    [CrossRef]
  17. 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]
  18. K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008).
    [CrossRef]

2008 (2)

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]

K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008).
[CrossRef]

2006 (2)

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

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 (1)

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]

2002 (1)

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

2001 (2)

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

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

2000 (1)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef] [PubMed]

1990 (1)

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

1989 (1)

G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989).
[CrossRef]

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]

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]

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef] [PubMed]

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]

Ludwig, A.

K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008).
[CrossRef]

Marinov, K.

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

Nedlin, G.

G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989).
[CrossRef]

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, “Elecromagnetic 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]

Seip, K.

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

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]

Skaar, J.

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

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]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef] [PubMed]

Thylén, L.

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]

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]

Webb, K. J.

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]

K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

Ziolkowski, R. W.

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

IEEE Trans. Circuits Syst. (1)

G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989).
[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. Phys. A (1)

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

J. Phys. D (1)

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

R. Ruppin, “Elecromagnetic 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 (3)

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]

K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (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. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

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Figures (4)

Fig. 1
Fig. 1

Real (solid curve) and imaginary (dotted curve) part of dielectric constant ϵ = 1 + 10 ( ω 1 2 ω 2 i 0.1 ω ) 1 for ω 1 = 1 . The three different carrier frequencies ( ω 0 = 8 9 , 1 , 8 7 ) are marked by thin vertical solid lines.

Fig. 2
Fig. 2

| u E | ϵ = 0 = w E obtained for ω 0 = 8 9 (dotted curve), ω 0 = 1 (dashed-dotted curve), and ω 0 = 8 7 (dashed curve). ϵ = Real { 1 + 10 ( ω 1 2 ω 2 i 0.1 ω ) 1 } and ω 1 = 1 . The model result (circles) plots Eq. (29), and the curves give exact results.

Fig. 3
Fig. 3

| u E t | ϵ = 0 = q t obtained for ω 0 = 8 9 (dotted curve), ω 0 = 1 (dashed-dotted curve), and ω 0 = 8 7 (dashed curve). ϵ = Imag { 1 + 10 ( ω 1 2 ω 2 i 0.1 ω ) 1 } and ω 1 = 1 . The model result is from Eq. (30), and the curves give exact results.

Fig. 4
Fig. 4

Exact u E obtained for ω 0 = 8 9 (dotted curve), ω 0 = 1 (dashed-dotted curve), and ω 0 = 8 7 (dashed curve). ϵ = 1 + 10 ( ω 1 2 ω 2 i 0.1 ω ) 1 and ω 1 = 1 .

Equations (39)

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E × H d s = [ E D t + H B t ] d v ,
u E t = E D t
= ϵ 0 E t [ 1 2 π ϵ ( ω ) E ( ω ) e i ω t d ω ]
= ϵ 0 E 2 π i ω [ ϵ ( ω ) + i ϵ ( ω ) ] [ E ( ω ) + i E ( ω ) ] [ cos ( ω t ) i sin ( ω t ) ] d ω ,
u E t = ϵ 0 E 2 π { ω ϵ ( ω ) [ E ( ω ) sin ( ω t ) + E ( ω ) cos ( ω t ) ] + ω ϵ ( ω ) [ E ( ω ) cos ( ω t ) + E ( ω ) sin ( ω t ) ] } d ω = | u E t | ϵ = 0 + | u E t | ϵ = 0 ,
D = 1 2 π D ( ω ) e i ω t d ω
= ϵ 0 2 π ϵ ( ω ) E ( ω ) e i ω t d ω ,
E ( ω ) = 1 2 [ e ( ω ω 0 ) + e ( ω + ω 0 ) ] ,
D = ϵ 0 4 π ϵ ( ω ) e ( ω ω 0 ) e i ω t d ω + ϵ 0 4 π ϵ ( ω ) e ( ω + ω 0 ) e i ω t d ω .
ϵ 0 4 π ϵ ( ω ) e ( ω ω 0 ) e i ω t d ω = ϵ 0 4 π ϵ ( ω + ω 0 ) e ( ω ) e i ( ω + ω 0 ) t d ω .
ϵ 0 4 π ϵ ( ω ) e ( ω + ω 0 ) e i ω t d ω = ϵ 0 4 π ϵ [ ( ω + ω 0 ) ] e ( ω ) e i ( ω + ω 0 ) t d ω .
ϵ 0 4 π ϵ ( ω ) e ( ω + ω 0 ) e i ω t d ω = ϵ 0 4 π ϵ * ( ω + ω 0 ) e * ( ω ) e i ( ω + ω 0 ) t d ω
= [ ϵ 0 4 π ϵ ( ω + ω 0 ) e ( ω ) e i ( ω + ω 0 ) t d ω ] * .
D = ϵ 0 4 π ϵ ( ω + ω 0 ) e ( ω ) e i ( ω + ω 0 ) t d ω + c.c. ,
D t = ϵ 0 4 π e i ω 0 t [ i ( ω + ω 0 ) ϵ ( ω + ω 0 ) ] e ( ω ) e i ω t d ω + c.c.
( ω + ω 0 ) ϵ ( ω + ω 0 ) = ω 0 ϵ ( ω 0 ) + ω | [ ( ( ω + ω 0 ) ϵ ( ω + ω 0 ) ) ω ] | ω = 0 + .
D t ϵ 0 4 π e i ω 0 t [ i ( ω 0 ϵ ( ω 0 ) + ω | [ ( ( ω + ω 0 ) ϵ ( ω + ω 0 ) ) ω ] | ω = 0 ) ] e ( ω ) e i ω t d ω + c.c.
| ( ( ω + ω 0 ) ϵ ( ω + ω 0 ) ) ω | ω = 0 = | ( ω ϵ ( ω ) ) ω | ω = ω 0 ,
D t ϵ 0 4 π e i ω 0 t [ i ω 0 ϵ ( ω 0 ) i ω | ( ω ϵ ( ω ) ) ω | ω = ω 0 ] e ( ω ) e i ω t d ω + c.c. = ϵ 0 2 e i ω 0 t [ i ω 0 ϵ ( ω 0 ) 2 π e ( ω ) e i ω t d ω + | ( ω ϵ ( ω ) ) ω | ω = ω 0 1 2 π i ω e ( ω ) e i ω t d ω ] + c.c.
= ϵ 0 2 [ i ω 0 ϵ ( ω 0 ) e ( t ) + | ( ω ϵ ( ω ) ) ω | ω = ω 0 e ( t ) t ] e i ω 0 t + c.c.
D t = ω 0 ϵ 0 e ( t ) [ ϵ ( ω 0 ) cos ( ω 0 t ) ϵ ( ω 0 ) sin ( ω 0 t ) ] + ϵ 0 e ( t ) t [ | ( ω ϵ ( ω ) ) ω | ω = ω 0 cos ( ω 0 t ) + | ( ω ϵ ( ω ) ) ω | ω = ω 0 sin ( ω 0 t ) ] .
w E t | u E t | ϵ = 0
ω 0 ϵ 0 ϵ ( ω 0 ) e 2 ( t ) 2 sin ( 2 ω 0 t ) + ϵ 0 | ( ω ϵ ( ω ) ) ω | ω = ω 0 e ( t ) e ( t ) t cos 2 ( ω 0 t ) ,
q t | u E t | ϵ = 0
ω 0 ϵ 0 ϵ ( ω 0 ) e 2 ( t ) cos 2 ( ω 0 t ) + ϵ 0 e ( t ) 2 e ( t ) t | ( ω ϵ ( ω ) ) ω | ω = ω 0 sin ( 2 ω 0 t ) .
u E t = ω 0 ϵ 0 e 0 2 cos ( ω 0 t ) [ ϵ ( ω 0 ) sin ( ω 0 t ) ϵ ( ω 0 ) cos ( ω 0 t ) ] .
f ( t ) = t f ( t ) t d t .
f ( t ) ( t n ) = 1 t 0 t n t 0 2 t n + t 0 2 f ( t ) d t ,
w E 1 4 ϵ 0 | ( ω ϵ ( ω ) ) ω | ω = ω 0 e 2 ( t ) ,
q t 1 2 ω 0 ϵ 0 ϵ e 2 ( t ) .
u E + u H = q + w E + u H 0 ,
ζ ( ω ) 1 = 2 π 0 ω ζ ( ω ) ω 2 ω 2 d ω ,
ζ ( ω ) = 2 ω π 0 ζ ( ω ) 1 ω 2 ω 2 d ω ,
χ ( ω ) = e 2 ϵ 0 m j Δ N j f j ω j 2 ω 2 i γ j ω ,
ϵ ( ω ) = 1 + a 1 ( ω 1 2 ω 2 ) ( ω 1 2 ω 2 ) 2 + γ 1 2 ω 2 ,
| ( ω ϵ ) ω | ω = ω 1 + Δ ω = 1 2 a 1 γ 1 2 ( 1 Δ ω ω 1 ) .
w E = e 2 ( t ) ϵ 0 4 [ 1 2 a 1 γ 1 2 ( 1 Δ ω ω 1 ) ] { 1 + cos [ 2 ( ω 1 + Δ ω ) t ] } ,
w E = e 2 ( t ) ϵ 0 4 [ 1 2 a 1 γ 1 2 ( 1 Δ ω ω 1 ) ] .
E = ( σ 2 π ) 1 exp [ ( t t c ) 2 ( 2 σ 2 ) 1 ] cos [ ω 0 ( t t c ) ] ,

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