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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3540–3546
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The role of magnetic dipoles and non-zero-order Bragg waves in metamaterial perfect absorbers

Yong Zeng, Hou-Tong Chen, and Diego A. R. Dalvit  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3540-3546 (2013)
http://dx.doi.org/10.1364/OE.21.003540


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Abstract

We develop a simple treatment of a metamaterial perfect absorber (MPA) based on grating theory. We analytically prove that the condition of MPA requires the existence of two currents, which are nearly out of phase and have almost identical amplitude, akin to a magnetic dipole. Furthermore, we show that non-zero-order Bragg modes within the MPA may consume electromagnetic energy significantly.

© 2013 OSA

1. Introduction

According to the grating theory [23

23. See, for example, M. Born and E. Wolf, Principles of Opticss, 7th ed. (Cambridge, Cambridge, 2011).

], the electric field inside a MPA can be expanded as a superposition of Bragg waves. On the other hand, since both theoretical approaches mentioned above disregard the periodic nature of the metamaterial structure, they thereby restrict to zero-order Bragg waves. In this paper we go beyond this approximation by including all orders of Bragg waves inside the MPA. We analytically show that the condition of MPA requires the existence of two currents within the metamaterial, which are nearly out of phase and have almost identical amplitude. Furthermore, using a combination of analytical and numerical arguments, we show that non-zero-order Bragg waves within the MPA consume electromagnetic energy.

2. Grating theory for the metamaterial perfect absorber

Let us consider a metamaterial membrane, surrounded by vacuum, which is periodic in the xy plane. For simplicity, we assume its meta-atoms are arranged in a rectangle lattice with primitive lattice vectors dxex and dyey. Here dx and dy are the corresponding lattice constants. We further assume that the external illumination is a plane wave propagating along the z direction with a wave vector ki. Under this external excitation polarization currents, J(r,ω) = −iωε0[ε (r,ω) − 1]E(r,ω), will appear inside the metamaterial. Here ε (r,ω) is the permittivity of the metamaterial, a periodic function of r. Using the free space Green’s function, we can express the scattered field in terms of the current
E(r,ω)=mnEmn(r,ω)=mnπZ0eikmnrdxdyλκmndrJmn,(r,ω)eikmnr,
(1)
where the integration is performed over one unit cell, m and n are integers, Z0=μ0/ε0 is the free-space intrinsic impedance, λ is the excitation wavelength, and k0 = 2π/λ is the free-space wave number. The wave vector kmn is defined as kmn=kigmn±κmnez with, κmn=k02|kmn,|2, where gmn = (2πm/dx)ex + (2πn/dy)ey are the reciprocal wave vectors, and ki is the projection of the incident wave vector onto the xy plane. Here the positive sign corresponds to forward scattering (propagating along the positive z direction), and the negative sign corresponds to backward scattering (propagating along the negative z direction). Moreover, Jmn,⊥ = Jkmn(kmn·J)/ k02 = − iωε0 [ε (r) − 1] [Ekmn(kmn·E)/ k02]. It is important to emphasize that Jmn,⊥ contains information of all Bragg modes through the total electric field E.

Under certain circumstances κmn is imaginary except for m = n = 0, so that only the zero order waves can survive in the far-field zone, e.g. at normal incidence and for λ bigger than the lattice constants. The forward wave is then given by
E00f(r,ω)=πZ0dxdyλkzieikirdrJ00,(r,ω)eikir.
(2)
The backward wave bears a similar expression except that ki is replaced with kikziez. This equation suggests that the tangential component of the incident wave vector is conserved.

2.1. Role of magnetic dipole

We now apply these equations to a metamaterial perfect absorber. We use a typical MPA proposed in Ref. [16

16. X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104, 207403 (2010). [CrossRef] [PubMed]

] (depicted schematically in Fig. 1), which consists of three thin layers. The first layer is a perforated metallic membrane which is referred to as the cross layer, the second layer, called the spacer layer, is filled with a homogeneous dielectric medium with weak absorption, and the last layer is generally a metallic ground plane. Furthermore, the meta-atoms of the cross layer are arranged periodically in a square lattice. The whole structure possesses both x and y mirror symmetries. Under normal incidence Ei = eik0z′ex, by symmetry Ex is an even function of x, while Ey and Ez are odd functions of x. Furthermore, the lattice constant d is smaller than the incident wavelength, so that only the zero-order waves survive in the far field. The forward and backward scattered fields in the far-field zone are hence given by
E00f(r,ω)=Z0ex2eik0zh/2h/2g(z,ω)eik0zdz,
(3)
E00b(r,ω)=Z0ex2eik0zh/2h/2g(z,ω)eik0zdz.
(4)
Here h is the thickness of the MPA, and the integral function
g(z,ω)=1d2d/2d/2d/2d/2Jx(x,y,z,ω)dxdy
(5)
stands for the current density at a specific z plane. Note that although the forward and backward waves in the far field are determined by the zero-order Bragg mode, the current Jx however contains contributions from all orders of Bragg waves within the structure since the quantity J00,⊥ defined in Eq. (2) equals to J, the transverse component of the total current inside the metamaterial. As a direct result, the reflected wave is E00b, and the transmitted wave is Ei+E00f.

Fig. 1 Reflection and absorption spectra of the metamaterial perfect absorber under normal incidence. The inset shows the geometry of the metamaterial. The metallic cross consists of two 0.4 × 1.7 metallic bars. The thicknesses of the cross, spacer and ground layer are 0.1, 0.09 and 0.2, respectively. The total thickness h equals 0.39, and the lattice constant d = 2. All dimensions are in micrometers.

A metamaterial with unity absorption neither reflects nor transmits the incident EM wave to the far field, which is equivalent to requiring
h/2h/2g(z)eik0zdz=0,
(6)
h/2h/2g(z)eik0zdz=2Z0.
(7)
To simplify the notation, we dropped the ω dependence of g here and in the following. The first equation imposes no reflection, and the second one leads to zero transmission. Because the thickness h is much smaller than λ, we can employ multipolar analysis by expanding eik0z as 1 + ik0z + ⋯. The leading order gives the electric dipole and the first order of k0z gives the combination of magnetic dipole and electric quadrupole. Supposing these three multipoles dominate the above two integrations, we arrive at
h/2h/2gR(z)dz=1Z0,h/2h/2gI(z)dz=0,
(8)
as well as
h/2h/2gR(z)zdz=0,k0h/2h/2gI(z)zdz=1Z0,
(9)
where gR and gI are the real and imaginary parts of g, respectively. Eqs. (8) immediately suggest that there exists an electric dipole moment with a value of 1/Z0, and Eqs. (9) imply that the MPA possesses a magnetic dipole as well as an electric quadrupole, and their sum equals i/Z0 (see equation 9.31 of Ref. [24

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

]). Moreover, as suggested by Eqs. (6) and (7), these three multipoles are destructive along the reflected direction, which leads to zero reflectance.

It is important to notice that the dielectric layer generally has a permittivity εd which is very different from the permittivity εm of the metallic layers. For instance, the ratio (εm − 1)/(εd − 1) of the MPA studied below is about 1428ei0.94π when λ = 5.93 μm. Since Ex is continuous crossing the spacer-ground interface, the polarization current Jx is strongly concentrated inside the metallic layers. We therefore can neglect the polarization current within the dielectric layer. Denoting the total current of the cross and ground layer as Gc and Gg, respectively, Eqs. (8) suggest
Z0×Re(Gc+Gg)=1,Im(Gc+Gg)=0.
(10)
In addition, Eqs. (9) imply |gI| is much bigger than |gR| because k0|z| ≪ 1. We therefore expect that Gc and Gg are nearly purely imaginary. In other words, the currents in the two metallic layers are nearly out of phase and have almost identical amplitude. This fact, here proved analytically, was reported in earlier numerical simulations [1

1. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008). [CrossRef] [PubMed]

, 14

14. G. Dayal and S. A. Ramakrishna, “Design of highly absorbing metamaterials for infrared frequencies,” Opt. Express 20, 17503–17508 (2012). [CrossRef] [PubMed]

17

17. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16, 7181–7188 (2008). [CrossRef] [PubMed]

, 25

25. Y. Ma, Q. Chen, J. Grant, S. C. Saha, A. Khalid, and D. R. S. Cumming, “A terahertz polarization insensitive dual band metamaterial absorber,” Opt. Lett. 36, 945–947 (2011). [CrossRef] [PubMed]

]. It is worth emphasizing that our proof does not require the computation of the current Jx, which is usually carried out by full-wave simulations.

To support the statements above, we will now perform full-wave simulations of a MPA whose geometrical and optical parameters are almost identical to the ones used in Ref. [16

16. X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104, 207403 (2010). [CrossRef] [PubMed]

]. The geometrical parameters are specified in Fig. 1. The permittivity of the dielectric is described by a Lorentz model [26

26. Because our finite-difference time-domain approach cannot handle a permittivity with a nondispersive imaginary part, we adapt a dispersive Lorentz model for the dielectric.

], εd(ω)=ε[1+ωp2/(ω02ω2iωγ)] with ε = 2.44, ωp = 93.77 THz, γ = 173.73 THz and ω0 = 3.1 THz, which results in a permittivity εd ≈ 2.28 + 0.091i, approximately constant in the relevant frequency regime, and almost identical to the one used in Ref. [16

16. X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104, 207403 (2010). [CrossRef] [PubMed]

]. It should be emphasized that this specific Lorentz model for εd does not alter the underlying physics of MPA. The permittivity of the metal is described by a Drude model,) εm(ω) = 1 − ωpm2 /(ω2 + iωγm), with ωpm = 1.37 × 104 THz and γm = 40.8 THz. Using a finite-difference time-domain method [27

27. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, Boston, 2000).

], where the size of spatial grid cell is fixed at 5 nm, we calculate the linear spectra at normal incidence and plot the results in Fig. 1. Around a wavelength of 5.93 μm, the absorption is found to be nearly 100%. Note that this wavelength is bigger than the lattice constant d = 2 μm, so that only zero-order Bragg wave propagates to the far field. We further calculate the current function g(z) at this wavelength and plot the result in Fig. 2(a). To check our numerical results, we computed the integrations in Eqs. (8) and (9), and verified that they are nearly identically satisfied. As discussed above, we find that the polarization current are strongly localized inside the two metallic layers, and a phase jump of 0.94π appears at the spacer-ground interface. Furthermore, we obtain GcZ0 = (−0.15 − 6.77i) and GgZ0 = (1.1 + 6.78i), in perfect agreement with the discussions above.

Fig. 2 (a) Amplitude and phase of the function g(z), and (b) the function η (z), when λ = 5.93μm. The dielectric layer is highlighted.

2.2. Role of non-zero-order Bragg waves

3. Conclusions

To sum up, we studied the metamaterial perfect absorber in the framework of grating theory. We proved analytically that there always exists one circulating current loop (akin to a magnetic dipole), together with an electric dipole as well as an electric quadrupole. We further showed that non-zero-order Bragg waves may contribute significantly to the dissipation of the electromagnetic energy inside the perfect absorber. Such an understanding of the process in the microscopic scale is important in exploring potential applications of metamaterial absorbers.

Acknowledgments

We acknowledge support from the LANL LDRD program. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.

References and links

1.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008). [CrossRef] [PubMed]

2.

J. A. Schuller, E. S. Barnard, W. Cai, Y. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010). [CrossRef] [PubMed]

3.

C. Hägglund and S. Peter Apell, “Plasmonic near-field absorbers for ultrathin solar cells,” J. Phys. Chem. Lett. 3, 1275–1285 (2012). [CrossRef]

4.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef] [PubMed]

5.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010). [CrossRef] [PubMed]

6.

M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater. 23, 5410–5414 (2011). [CrossRef] [PubMed]

7.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011). [CrossRef] [PubMed]

8.

R. Taubert, D. Dregely, N. Liu, H. Giessen, A. Tittl, and P. Mai, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11, 4366–4369 (2011). [CrossRef] [PubMed]

9.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011). [CrossRef] [PubMed]

10.

C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. 37, 308–310 (2012). [CrossRef] [PubMed]

11.

J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, and P. Sheng, “Dark acoustic metamaterials as super absorbers for low-frequency sound,” Nat. Commun. 3, 756 (2012). [CrossRef] [PubMed]

12.

Y. Cui, K. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012). [CrossRef] [PubMed]

13.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Commun. 3, 969 (2012). [CrossRef] [PubMed]

14.

G. Dayal and S. A. Ramakrishna, “Design of highly absorbing metamaterials for infrared frequencies,” Opt. Express 20, 17503–17508 (2012). [CrossRef] [PubMed]

15.

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78, 241103(R) (2008). [CrossRef]

16.

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104, 207403 (2010). [CrossRef] [PubMed]

17.

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16, 7181–7188 (2008). [CrossRef] [PubMed]

18.

J. Zhou, H.-T. Chen, T. Koschny, A. K. Azad, A. J. Taylor, C. M. Soukoulis, and J. F. O’Hara, “Application of metasurface description for multilayered metamaterials and an alternative theory for metamaterial perfect absorber,” arXiv:1111.0343v1.

19.

H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express 20, 7165–7172 (2012). [CrossRef] [PubMed]

20.

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009). [CrossRef]

21.

H.-T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105, 073901 (2010). [CrossRef] [PubMed]

22.

D. Yu. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B 82, 205117 (2010). [CrossRef]

23.

See, for example, M. Born and E. Wolf, Principles of Opticss, 7th ed. (Cambridge, Cambridge, 2011).

24.

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

25.

Y. Ma, Q. Chen, J. Grant, S. C. Saha, A. Khalid, and D. R. S. Cumming, “A terahertz polarization insensitive dual band metamaterial absorber,” Opt. Lett. 36, 945–947 (2011). [CrossRef] [PubMed]

26.

Because our finite-difference time-domain approach cannot handle a permittivity with a nondispersive imaginary part, we adapt a dispersive Lorentz model for the dielectric.

27.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, Boston, 2000).

OCIS Codes
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: September 7, 2012
Revised Manuscript: October 24, 2012
Manuscript Accepted: January 16, 2013
Published: February 5, 2013

Citation
Yong Zeng, Hou-Tong Chen, and Diego A. R. Dalvit, "The role of magnetic dipoles and non-zero-order Bragg waves in metamaterial perfect absorbers," Opt. Express 21, 3540-3546 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3540


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References

  1. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett.100, 207402 (2008). [CrossRef] [PubMed]
  2. J. A. Schuller, E. S. Barnard, W. Cai, Y. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010). [CrossRef] [PubMed]
  3. C. Hägglund and S. Peter Apell, “Plasmonic near-field absorbers for ultrathin solar cells,” J. Phys. Chem. Lett.3, 1275–1285 (2012). [CrossRef]
  4. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010). [CrossRef] [PubMed]
  5. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett.10, 2342–2348 (2010). [CrossRef] [PubMed]
  6. M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater.23, 5410–5414 (2011). [CrossRef] [PubMed]
  7. X. Liu, T. Tyler, T. Starr, A. F. Starr, N. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett.107, 045901 (2011). [CrossRef] [PubMed]
  8. R. Taubert, D. Dregely, N. Liu, H. Giessen, A. Tittl, and P. Mai, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett.11, 4366–4369 (2011). [CrossRef] [PubMed]
  9. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun.2, 517 (2011). [CrossRef] [PubMed]
  10. C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett.37, 308–310 (2012). [CrossRef] [PubMed]
  11. J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, and P. Sheng, “Dark acoustic metamaterials as super absorbers for low-frequency sound,” Nat. Commun.3, 756 (2012). [CrossRef] [PubMed]
  12. Y. Cui, K. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett.12, 1443–1447 (2012). [CrossRef] [PubMed]
  13. T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Commun.3, 969 (2012). [CrossRef] [PubMed]
  14. G. Dayal and S. A. Ramakrishna, “Design of highly absorbing metamaterials for infrared frequencies,” Opt. Express20, 17503–17508 (2012). [CrossRef] [PubMed]
  15. H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incidence terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B78, 241103(R) (2008). [CrossRef]
  16. X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104, 207403 (2010). [CrossRef] [PubMed]
  17. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express16, 7181–7188 (2008). [CrossRef] [PubMed]
  18. J. Zhou, H.-T. Chen, T. Koschny, A. K. Azad, A. J. Taylor, C. M. Soukoulis, and J. F. O’Hara, “Application of metasurface description for multilayered metamaterials and an alternative theory for metamaterial perfect absorber,” arXiv:1111.0343v1.
  19. H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express20, 7165–7172 (2012). [CrossRef] [PubMed]
  20. C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials,” Metamaterials3, 100–112 (2009). [CrossRef]
  21. H.-T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett.105, 073901 (2010). [CrossRef] [PubMed]
  22. D. Yu. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82, 205117 (2010). [CrossRef]
  23. See, for example, M. Born and E. Wolf, Principles of Opticss, 7th ed. (Cambridge, Cambridge, 2011).
  24. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  25. Y. Ma, Q. Chen, J. Grant, S. C. Saha, A. Khalid, and D. R. S. Cumming, “A terahertz polarization insensitive dual band metamaterial absorber,” Opt. Lett.36, 945–947 (2011). [CrossRef] [PubMed]
  26. Because our finite-difference time-domain approach cannot handle a permittivity with a nondispersive imaginary part, we adapt a dispersive Lorentz model for the dielectric.
  27. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, Boston, 2000).

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