## The role of magnetic dipoles and non-zero-order Bragg waves in metamaterial perfect absorbers |

Optics Express, Vol. 21, Issue 3, pp. 3540-3546 (2013)

http://dx.doi.org/10.1364/OE.21.003540

Acrobat PDF (789 KB)

### 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

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]

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

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]

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]

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]

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]

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]

## 2. Grating theory for the metamaterial perfect absorber

*xy*plane. For simplicity, we assume its meta-atoms are arranged in a rectangle lattice with primitive lattice vectors

*d*

_{x}**e**

*and*

_{x}*d*

_{y}**e**

*. Here*

_{y}*d*and

_{x}*d*are the corresponding lattice constants. We further assume that the external illumination is a plane wave propagating along the

_{y}*z*direction with a wave vector

**k**

*. Under this external excitation polarization currents,*

^{i}**J**(

**r**,

*ω*) = −

*iωε*

_{0}[

*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 where the integration is performed over one unit cell,

*m*and

*n*are integers,

*λ*is the excitation wavelength, and

*k*

_{0}= 2

*π*/

*λ*is the free-space wave number. The wave vector

**k**

*is defined as*

_{mn}**g**

*= (2*

_{mn}*πm*/

*d*)

_{x}**e**

*+ (2*

_{x}*πn*/

*d*)

_{y}**e**

*are the reciprocal wave vectors, and*

_{y}*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,

**J**

_{mn,⊥}=

**J**−

**k**

*(*

_{mn}**k**

*·*

_{mn}**J**)/

*iωε*

_{0}[

*ε*(

**r**) − 1] [

**E**−

**k**

*(*

_{mn}**k**

*·*

_{mn}**E**)/

**J**

_{mn,⊥}contains information of all Bragg modes through the total electric field

**E**.

*κ*is imaginary except for

_{mn}*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 The backward wave bears a similar expression except that

**k**

*is replaced with*

^{i}### 2.1. Role of magnetic dipole

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]

*x*and

*y*mirror symmetries. Under normal incidence

**E**

*=*

^{i}*e*

^{ik0z′}

**e**

*, by symmetry*

_{x}*E*is an even function of

_{x}*x*, while

*E*and

_{y}*E*are odd functions of

_{z}*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 Here

*h*is the thickness of the MPA, and the integral function 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

*J*however contains contributions from all orders of Bragg waves within the structure since the quantity

_{x}**J**

_{00,⊥}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

*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

*e*

^{ik0z}as 1 +

*ik*

_{0}

*z*+ ⋯. The leading order gives the electric dipole and the first order of

*k*

_{0}

*z*gives the combination of magnetic dipole and electric quadrupole. Supposing these three multipoles dominate the above two integrations, we arrive at as well as where

*g*and

_{R}*g*are the real and imaginary parts of

_{I}*g*, respectively. Eqs. (8) immediately suggest that there exists an electric dipole moment with a value of 1/

*Z*

_{0}, and Eqs. (9) imply that the MPA possesses a magnetic dipole as well as an electric quadrupole, and their sum equals

*i/Z*

_{0}(see equation 9.31 of Ref. [24]). Moreover, as suggested by Eqs. (6) and (7), these three multipoles are destructive along the reflected direction, which leads to zero reflectance.

*ε*which is very different from the permittivity

_{d}*ε*of the metallic layers. For instance, the ratio (

_{m}*ε*− 1)/(

_{m}*ε*− 1) of the MPA studied below is about 1428

_{d}*e*

^{i0.94π}when

*λ*= 5.93

*μm*. Since

*E*is continuous crossing the spacer-ground interface, the polarization current

_{x}*J*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

_{x}*G*and

_{c}*G*, respectively, Eqs. (8) suggest In addition, Eqs. (9) imply |

_{g}*g*| is much bigger than |

_{I}*g*| because

_{R}*k*

_{0}|

*z*| ≪ 1. We therefore expect that

*G*and

_{c}*G*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

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

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]

*J*, which is usually carried out by full-wave simulations.

_{x}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]

*ε*= 2.44,

_{∞}*ω*= 93.77 THz,

_{p}*γ*= 173.73 THz and

*ω*

_{0}= 3.1 THz, which results in a permittivity

*ε*≈ 2.28 + 0.091

_{d}*i*, approximately constant in the relevant frequency regime, and almost identical to the one used in Ref. [16

**104**, 207403 (2010). [CrossRef] [PubMed]

*ε*does not alter the underlying physics of MPA. The permittivity of the metal is described by a Drude model,)

_{d}*ε*(

_{m}*ω*) = 1 −

*ω*

^{2}+

*iωγ*), with

_{m}*ω*= 1.37 × 10

_{pm}^{4}THz and

*γ*= 40.8 THz. Using a finite-difference time-domain method [27], 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

_{m}*μm*, the absorption is found to be nearly 100%. Note that this wavelength is bigger than the lattice constant

*d*= 2

*μ*m, 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 0.94

*π*appears at the spacer-ground interface. Furthermore, we obtain

*G*

_{c}Z_{0}= (−0.15 − 6.77

*i*) and

*G*

_{g}Z_{0}= (1.1 + 6.78

*i*), in perfect agreement with the discussions above.

### 2.2. Role of non-zero-order Bragg waves

**J**

*·*

**E**[24]. Consequently we define to measure the relative contribution from a specific

*z*plane. Here

*ε*=

*ε*(

**r**,

*ω*) describes the permittivity of the whole MPA structure. In general, the electric field intensity inside a meta-material is highly inhomogeneous, which is an indication of the appearance of high-order Bragg waves, since the zero-order mode only gives a homogeneous field distribution. Therefore one can conclude that high-order Bragg waves definitely consume electromagnetic energy inside any nanostructure, in particular a MPA.

*η*for the structure shown in Fig. 1, and plot the result in Fig. 2(b). Under normal incidence, the two approximate theories proposed in Ref. [1

**100**, 207402 (2008). [CrossRef] [PubMed]

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

*m*=

*n*= 0), which is a plane wave propagating along the

*z*direction. The corresponding EM field of the

*m*=

*n*= 0 mode is parallel to the interface and must be continuous across the spacer-ground interface. If only this mode consumes electromagnetic energy, we expect that

*η*of the spacer layer is much smaller than that of the ground layer because Im(

*ε*)/Im(

_{m}*ε*) ≈ 3000, which however does not agree with our full-wave numerical result (a similar calculation was reported in Ref. [16

_{d}**104**, 207403 (2010). [CrossRef] [PubMed]

*E*/

_{z}*E*| inside the spacer layer ranges from 0.84 to nearly 1.0, suggesting that non-zero-order waves have a dominant contribution to the total field in the spacer region. As discussed in the previous paragraph, one can alternatively infer the existence of high-order Bragg waves from the highly localized electric field distribution inside the MPA, which is shown in figure 4 of Ref. [16

**104**, 207403 (2010). [CrossRef] [PubMed]

## 3. Conclusions

## Acknowledgments

## 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. |

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. |

3. | C. Hägglund and S. Peter Apell, “Plasmonic near-field absorbers for ultrathin solar cells,” J. Phys. Chem. Lett. |

4. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

5. | N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. |

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. |

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. |

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. |

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. |

10. | C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. |

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. |

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. |

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. |

14. | G. Dayal and S. A. Ramakrishna, “Design of highly absorbing metamaterials for infrared frequencies,” Opt. Express |

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 |

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. |

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 |

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. | 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 |

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. |

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 |

23. | See, for example, M. Born and E. Wolf, |

24. | J. D. Jackson, |

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. |

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, |

**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

- 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]
- 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]
- C. Hägglund and S. Peter Apell, “Plasmonic near-field absorbers for ultrathin solar cells,” J. Phys. Chem. Lett.3, 1275–1285 (2012). [CrossRef]
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett.37, 308–310 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- G. Dayal and S. A. Ramakrishna, “Design of highly absorbing metamaterials for infrared frequencies,” Opt. Express20, 17503–17508 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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.
- H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express20, 7165–7172 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- See, for example, M. Born and E. Wolf, Principles of Opticss, 7th ed. (Cambridge, Cambridge, 2011).
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
- 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]
- 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.
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, Boston, 2000).

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