Abstract

We derive the electromagnetic energy density in a chiral metamaterial consisting of uncoupled single-resonance helical resonators. Both the lossless and absorptive cases are studied, and the energy density is shown to be positively definite. The key relation making the derivation successful is the proportionality between the magnetization and the rate of change of the electric polarization of the medium. The same time-domain formulation of energy density also applies to the bianisotropic medium proposed by Zhang et al. [Phys. Rev. Lett. 102, 023901 (2009)]. This work may provide insights for studying time-dependent phenomena in metamaterials.

© 2011 Optical Society of America

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References

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  1. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).
  2. B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
    [CrossRef]
  3. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd. ed. (Pergamon1984).
  4. R. Loudon, J. Phys. A 3, 233 (1970).
    [CrossRef]
  5. R. Ruppin, Phys. Lett. A 299, 309 (2002).
    [CrossRef]
  6. S. A. Tretyakov, Phys. Lett. A 343, 231 (2005).
    [CrossRef]
  7. A. D. Boardman and K. Marinov, Phys. Rev. B 73, 165110(2006).
    [CrossRef]
  8. P. G. Luan, Phys. Rev. E 80, 046601 (2009).
    [CrossRef]
  9. L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
    [CrossRef]
  10. I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
    [CrossRef]
  11. S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
    [CrossRef]
  12. R.-K. Zhao, T. Koschny, and C. M. Soukoulis, Opt. Express 18, 14553 (2010).
    [CrossRef] [PubMed]
  13. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).
  14. S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
    [CrossRef] [PubMed]

2010 (1)

2009 (4)

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

P. G. Luan, Phys. Rev. E 80, 046601 (2009).
[CrossRef]

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
[CrossRef]

2008 (1)

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

2006 (1)

A. D. Boardman and K. Marinov, Phys. Rev. B 73, 165110(2006).
[CrossRef]

2005 (1)

S. A. Tretyakov, Phys. Lett. A 343, 231 (2005).
[CrossRef]

2002 (1)

R. Ruppin, Phys. Lett. A 299, 309 (2002).
[CrossRef]

1996 (1)

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

1970 (1)

R. Loudon, J. Phys. A 3, 233 (1970).
[CrossRef]

Baena, J. D.

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

Boardman, A. D.

A. D. Boardman and K. Marinov, Phys. Rev. B 73, 165110(2006).
[CrossRef]

Heliot, J.-P.

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

Jelinek, L.

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

Kafesaki, M.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

Khakhomov, S. A.

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
[CrossRef]

Kharina, T. G.

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

Koschny, T.

R.-K. Zhao, T. Koschny, and C. M. Soukoulis, Opt. Express 18, 14553 (2010).
[CrossRef] [PubMed]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd. ed. (Pergamon1984).

Li, J.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd. ed. (Pergamon1984).

Loudon, R.

R. Loudon, J. Phys. A 3, 233 (1970).
[CrossRef]

Lu, X.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Luan, P. G.

P. G. Luan, Phys. Rev. E 80, 046601 (2009).
[CrossRef]

Marinov, K.

A. D. Boardman and K. Marinov, Phys. Rev. B 73, 165110(2006).
[CrossRef]

Mariotte, F.

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

Marqués, R.

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

Maslovski, S.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

Mesa, F.

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

Nefedov, I.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

Park, Y. S.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, Phys. Lett. A 299, 309 (2002).
[CrossRef]

Semchenko, I. V.

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
[CrossRef]

Shamonina, E.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

Sihvola, A.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

Simovski, C.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

Simovski, C. R.

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

Solymar, L.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

Soukoulis, C. M.

R.-K. Zhao, T. Koschny, and C. M. Soukoulis, Opt. Express 18, 14553 (2010).
[CrossRef] [PubMed]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

Tretyakov, S.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

Tretyakov, S. A.

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
[CrossRef]

S. A. Tretyakov, Phys. Lett. A 343, 231 (2005).
[CrossRef]

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

Wang, B.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

Zhang, S.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhang, W.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhao, R.-K.

Zhou, J.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

Eur. Phys. J. Appl. Phys. (1)

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, Eur. Phys. J. Appl. Phys. 46, 32607 (2009).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, IEEE Trans. Antennas Propagat. 44, 1006 (1996).
[CrossRef]

J. Opt. A (1)

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, J. Opt. A 11, 114003 (2009).
[CrossRef]

J. Phys. A (1)

R. Loudon, J. Phys. A 3, 233 (1970).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (2)

R. Ruppin, Phys. Lett. A 299, 309 (2002).
[CrossRef]

S. A. Tretyakov, Phys. Lett. A 343, 231 (2005).
[CrossRef]

Phys. Rev. B (2)

A. D. Boardman and K. Marinov, Phys. Rev. B 73, 165110(2006).
[CrossRef]

L. Jelinek, R. Marqués, F. Mesa, and J. D. Baena, Phys. Rev. B 77, 205110 (2008).
[CrossRef]

Phys. Rev. E (1)

P. G. Luan, Phys. Rev. E 80, 046601 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Other (3)

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” arXiv:cond-mat/0211012 (2002).

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd. ed. (Pergamon1984).

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Equations (26)

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D ω = ϵ 0 ϵ ( ω ) E ω + i κ ( ω ) c H ω ,
B ω = i κ ( ω ) c E ω + μ 0 μ ( ω ) H ω .
ϵ ( ω ) = 1 ω p 2 ω 2 ω 0 2 + i Γ ω ,
μ ( ω ) = 1 F ω 2 ω 2 ω 0 2 + i Γ ω ,
κ ( ω ) = A ω ω 2 ω 0 2 + i Γ ω ,
W = 1 4 V M 0 V ,
M 0 = ( ( ω ϵ ) ω i ( ω κ ) ω i ( ω κ ) ω ( ω μ ) ω ) , V = ( ϵ 0 E ω μ 0 H ω ) ,
( ω ϵ ) ω + ( ω μ ) ω > 0 , ( ω ϵ ) ω ( ω μ ) ω > ( ( ω κ ) ω ) 2 .
D = ϵ 0 E + P , B = μ 0 ( H + M ) ,
2 P t 2 + Γ P t + ω 0 2 P = ϵ 0 ω p 2 E + A c H t ,
M t + Γ M + ω 0 2 M d t = F H t A E μ 0 c .
L d I d t + R I + q C = V e d Φ d t
A = ± F ω p , P t = ω p 2 A c M .
· S = W 0 t + E · P t μ 0 M · H t ,
W 0 = ϵ 0 2 E 2 + μ 0 2 H 2 + μ 0 H · M
E · P t = 1 ϵ 0 ω p 2 ( P ˙ t + Γ P ˙ + ω 0 2 P A c H ˙ ) · P ˙ = t ( P ˙ 2 2 ϵ 0 ω p 2 + ω 0 2 P 2 ϵ 0 ω p 2 ) + Γ P ˙ 2 ϵ 0 ω p 2 + μ 0 M · H t ,
W = ϵ 0 E 2 2 + μ 0 H 2 2 + μ 0 H · M + P ˙ 2 + ω 0 2 P 2 2 ϵ 0 ω p 2 ,
= ϵ 0 E 2 2 + ω 0 2 P 2 2 ϵ 0 ω p 2 + μ 0 ( 1 F ) H 2 2 + μ 0 ( M + F H ) 2 2 F .
P loss = Γ ϵ 0 ω p 2 ( P t ) 2 = μ 0 Γ F M 2
W = 1 4 V M V , P loss = 1 2 V P V ,
M = ( 1 + ω p 2 ( ω 0 2 + ω 2 ) ( ω 0 2 ω 2 ) 2 + Γ 2 ω 2 A ω ( Γ ω + 2 i ω 0 2 ) ( ω 0 2 ω 2 ) 2 + Γ 2 ω 2 A ω ( Γ ω 2 i ω 0 2 ) ( ω 0 2 ω 2 ) 2 + Γ 2 ω 2 1 + F ω 2 ( 3 ω 0 2 ω 2 ) ( ω 0 2 ω 2 ) 2 + Γ 2 ω 2 ) ,
P = Γ ω 2 ( ω 0 2 ω 2 ) 2 + Γ 2 ω 2 ( ω p 2 i A ω i A ω F ω 2 ) .
2 P t 2 + Γ P t + ω 0 2 P = ϵ 0 ω p 2 E + A R c H t ,
M t + Γ M + ω 0 2 M d t = F H t A R 1 E μ 0 c .
P t = ω p 2 A c R ( α ) M .
A ( R H ˙ ) · P ˙ ϵ 0 ω p 2 c = μ 0 ( R H ˙ ) · ( R M ) = μ 0 H ˙ · ( R t R ) M = μ 0 H t · M .

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