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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 24 — Nov. 24, 2008
  • pp: 19504–19511
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Coherently controlling metamaterials

Sangeeta Chakrabarti, S. Anantha Ramakrishna, and Harshawardhan Wanare  »View Author Affiliations


Optics Express, Vol. 16, Issue 24, pp. 19504-19511 (2008)
http://dx.doi.org/10.1364/OE.16.019504


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Abstract

Two independent significant developments have challenged our understanding of light-matter interaction, one, involves the artificially structured materials known as metamaterials, and the other, relates to the coherent control of quantum systems via the quantum interference route. We theoretically demonstrate that one can engineer the electromagnetic response of composite metamaterials using coherent quantum interference effects. In particular, we predict that these composite materials can show a variety of effects ranging from dramatic reduction of losses to switchable ultraslow-to-superluminal pulse propagation. We propose parametric control of the metamaterials by active tuning of the capacitance of the structures, which is most efficiently engineered by embedding the metamaterial structures within a coherent atomic/molecular medium. This leads to dramatic frequency dependent features, such as significantly reduced dissipation accompanied by enhanced filling fraction. For a Split-ring resonator medium with magnetic properties, the associated splitting of the negative permeability band can be exploited for narrow band switching applications at near infrared frequencies involving just a single layer of such composite metamaterials.

© 2008 Optical Society of America

Fig. 1. Schematic pictures highlighting the principle behind the proposed controllable metamaterials realized by embedding a resonant medium within the metamaterial design. (a) The composite Split-ring resonators with resonant material embedded within the capacitive gaps can be thought of as resonantly driven LC circuits as shown in (c), where the capacitance can be manipulated by external applied fields. Examples of such controllable resonant media are given at the top: Λ-level structure of a resonant atomic medium on the left and the level structure of a molecule with an appropriate Raman transition on the right; (b) The dielectric permittivity experienced by the probe field for these cases of an embedded medium with a single resonance (Raman medium) and an embedded EIT medium (with zero absorption at the line center) are shown in the top right panel as a function of Δ/γ=(ω-ω0)/γ.

We now show that the properties of a coherent atomic or molecular medium can be utilized to manipulate the metamaterial properties by embedding the metamaterial structural units in the coherent medium. The capacitance of the metamaterial units depends on the dielectric permittivity of the embedding coherent medium, which can be actively manipulated by a control field applied at an appropriate frequency where the metamaterial is transparent due to a pass band (see Fig. 1). Each metamaterial unit can be modeled as a resonant L-C circuit and its resonance frequency can be accurately tuned via its capacitance. The properties of the metamaterial can be most effectively controlled via the dielectric properties of the resonant atomic medium driven to coherence. Here we demonstrate two magnetic metamaterial designs that operate in two distinct frequency regimes and offer enormous control over radiation: from complete tunability of the metamaterial parameters and drastic reduction of losses, with the possibilities of ultra-slow and superluminal light to extremely narrow band metamaterial switches.

Fig. 2. (a) The real (black line) and imaginary (dashed-blue line) parts of the effective magnetic permeability obtained when the capacitive gaps of the SRR are embedded with a resonant Raman medium. The dashed red curve shows Re(µ eff) for the bare SRR medium, to be contrasted with the transformed response due to the resonant Raman medium (black curve). (b) The computed band-structure for the bare (red circles) and the composite SRR medium with an embedded Raman medium (black squares) whose γ 13=2.4 THz. The bandgap associated with µ eff<0 for the bare SRR is shown as cross-hatched regions on the right, whereas the bandgaps due to the inclusion of the Raman medium in the SRR are shown as the two hatched regions on the left. The blue lines indicate the dispersion predicted by the (approximate) analytic formulae. (c) The real (black line) and imaginary (dotted-red line) parts of µ eff of the composite SRR medium when an EIT medium is embedded in the capacitive gaps. For the EIT medium, ω 1=ω m=74.9THz, γ 13=0.24GHz and Ωc=2.4THz. The resonance linewidths are much narrower than that of the bare SRR medium indicating reduced dissipation. (d) The dissipation parameter Γm (black), and the filling fraction fm (dash-dotted red line) for the µ eff shown in (c). The reduced dissipation as well as the filling fraction can be contrasted with the bare SRR levels indicated by the dotted blue and dashed dark green lines, respectively.

First, we present coherent control of the ubiquitous metallic Split-ring resonator (SRR) medium [3

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

] at mid-infrared frequencies. For incoming radiation with the magnetic field along the axis of the SRR, the induced currents in the SRR have a resonance at a frequency ωm given by ωm=1/LC, where L is the net inductance and C is the net capacitance of the SRR units. The effective magnetic permeability of the SRR medium is reasonably described by [19

19. S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys.: Condens. Matter 14, 6383–6394 (2002). [CrossRef]

]

μ(ω)=1+fmω2ωm2ω2iΓmω,
(1)

A coherent atomic medium with a Λ-level structure [see Fig. 1(a)] when driven to EIT, offers a dielectric permittivity [14

14. M. O. Scully and M. S. Zubairy, Quantum OpticsCambridge Univ. Press, Cambridge, U.K. (1997).

]

ε(ω)=1+κ(ω1ω)(ω1ω)2(Ωc24)iγ13(ω1ω),
(2)

to the probe radiation at frequency ω close to the atomic resonance frequency ω 1, where Ωc is the Rabi frequency of the control field and γ 13 is the decay rate between the atomic levels |1〈 and |3〉 while κ is the oscillator strength that can be controlled via the atomic number density. The capacitance of the SRR, when such a coherent atomic medium is placed within the capacitive gaps, is approximately given by C(ω)=ε0ε(ω)A/d, where A and d are the area and width of the capacitive gaps in the SRR, respectively. The resonant interaction of the magnetically driven SRR with the driven dielectric material within the capacitive gaps gives rise to entirely new resonant features. Further, each new resonance has an effective width Γ(ω) and a renormalized filling fraction f (ω) that differ significantly from the bare SRR medium.

Consider the dispersion of µ eff shown in Fig. 2(a) for the composite medium. The embedding dielectric medium is assumed to have a Lorentz dispersion centered close to the magnetic resonance frequency of the bare SRR structure (ω 1=73.6THz,ωm=74.9THz, γ 13=0.24GHz and Γm=4.8THz). This Lorentz dispersion corresponds to Ωc→0 in Eq. (2). Such a dispersion can be readily realized by either using resonant quantum dots or choosing a Raman transition (see for example [20

20. N. G. Kalugin and Y. V. Rostovtsev, “Efficient generation of short terahertz pulses via stimulated Raman adiabatic passage,” Opt. Lett. 31, 969–971 (2006). [CrossRef] [PubMed]

]) with a strong pump field such that the Raman resonant probe frequency lies in the vicinity of the magnetic resonance frequency. Strong modulation of µ eff is obtained when the dielectric resonance frequency gets close to the bare SRR resonance frequency. The large dispersion, with the consequent large group indices obtainable shows the possibility of a new class of media for ultra-slow or superluminal light, wherein the magnetic interaction with the medium determines the passage of light.

Similarly in Fig. 2(c), we show the behaviour of µ eff for SRR with an embedding medium exhibiting EIT, where the control field at a lower frequency propagates in the metamaterial due to a passband. For the parameters of Fig. 2(c), the single broad magnetic resonance peak has split into three peaks, where all the peaks have significantly narrower widths indicating drastically reduced dissipation. The spacing of the peaks is mainly determined by the strength of the control field, while the extent of modulation in µ eff depends on the proximity of the EIT line center with the bare SRR resonance frequency. Thus, complete control over the dispersion and dissipation of the metamaterial can be attained via the control field.

The most well-known counterintuitive aspect of EIT, namely zero dissipation at the EIT line center, manifests here as drastically reduced dissipation (Γeff) in the metamaterial [Fig. 2(d)]. Such reduction of the dissipation occurs primarily due to vanishing currents in the SRR loops and is also reflected in the reduction of the filling fraction [Fig. 2(d)]. It appears as if the EIT aided effective magnetic medium mimics vacuum at these frequencies, where µ eff→1 around the EIT line center. To further validate these effects, we present the results of photonic bandstructure calculations based on the transfer matrix method [21

21. J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992). [CrossRef] [PubMed]

] in Fig. 2(b) for two-dimensional SRR with the cross-sections shown in Fig. 1(a) with a=600nm, b=312nm, L=144nm, d= 24nm, and D=24nm. The band dispersions [Fig. 2(b)] agree reasonably with those predicted by the approximate analytic formulae [Eq. (1) & Eq. (2)] although the exact locations of the computed bandgaps, due to µ eff<0 are shifted.

Fig. 3. Left: The level structure diagram of the relevant levels involved in EIT for metastable helium. Right: The unit cell of the SRR metamaterial made of silver and resonant (ω 1) at about 272 THz and immersed within a near-resonant controllable EIT medium (metastable helium gas). The metamaterial response shown in Fig. 4 corresponds to the composite SRR structures of the dimensions shown above.

The bandwidths of the gaps and the dispersion of the allowed bands are all controllable by the amplitude and the frequency of the control radiation. For example, a blue-shifted control laser would require red-shifting the probe laser in order to maintain the two-photon resonance condition, thus providing the possibility to align the EIT resonance at any desired location in/around the bandgap. Wider bandgaps would be obtained as the EIT line center gets closer to the bare SRR resonance frequency. In Fig. 4(c), we show the reflectance and the transmittance of light incident on a slab of SRR medium composed of just one unit cell layer, with and without the EIT control field. Note the sharp switching of reflectance and transmittance at specific frequencies that are entirely governed by the control field parameters. In particular, around 277 THz, the transmittance drops dramatically from nearly 95% to a few percent within about 10 GHz bandwidth indicating its potential use for an extremely narrow-band switching applications. Recently non-resonant rescaling of capacitance resulting in the tuning of the resonance frequency by up to 20% has been demonstrated at about 1THz [25

25. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nature Photon. 2, 295–298 (2008). [CrossRef]

]. We would like to note the following: first, we have assumed that the EIT effect is itself not hampered by the metamaterial. This assumption is well justified as we have chosen the EIT control field to lie within a propagating band and well within the first Brillouin zone, hence the control field would be uniform across the medium. The introduction of the metastable atomic gas within the metamaterial structure will not affect the atomic decay rates significantly. Second, we have not attempted to homogenize the structure that is less than a quarter of the free space wavelength. We have calculated and presented only the photonic bandstructure and transmission/reflection properties of this metamaterial in Fig. 4.

Before we conclude, we discuss the effects of inhomogeneous broadening caused by imperfect fabrication of the metamaterial units. We do require the nanostructures to have high quality (and low surface roughness). Present day FIB based nanotechnology does have the technical capability to meet such requirements. However, the thickness of the capacitive gaps is extremely critical and could be slightly different in different units resulting in a distribution of the values of the resonant frequency. This is a particularly important issue for the nanostructures discussed here. We have analyzed this by including inhomogeneous broadening due to the imperfect manufacturing by assuming a Gaussian distribution for the resonant frequency ω m, with a width comparable to the bare SRR linewidth (Γm), and found that all the above narrow band effects continue to remain nearly unaltered. This robust response we believe is due to the enhanced frequency dependent filling fraction that remains independent of ω m, this can be easily seen by substituting Eq. (2) in Eq. (1) and identifying the renormalized filling fraction. This effect far out-weighs the effects of inhomogeneous broadening and the narrow peak due to EIT remains unaffected. We have also undertaken similar calculations involving cut wire metamaterials that rely on plasmonic resonances for their dielectric or magnetic response, these can also be coherently controlled (whose details will be published separately elsewhere). Collections of silver or gold nanorods can be accurately fabricated with little size dispersion and such metamaterials will hence be even less prone to the effects of inhomogeneous broadening.

Fig. 4. Computed bandstructures (a, b) of a SRR metamaterial with the unit cell shown in Fig. 3 (right) and submersed in metastable helium gas. The reflectance (black) and transmittance (dashed red) are shown in (c, d) from a slab of one layer of the composite SRR metamaterial. (a) and (c) are for zero control field, while (b) and (d) show the response when the EIT control field has been switched on (Ωc=10γ21). Note the sharp changes in the transmittance around 277 THz in (d) and the nearly dispersion-free band in (b) due to EIT.

We have demonstrated here a new paradigm of parametric control of metamaterials. The metamaterial response is completely transformed by introducing coherently controllable resonant media into the design. Our results provide a new mechanism to reducing dissipation inherent to metamaterials, which has been the main hindrance in developing applications. Despite the control scheme being inherently dispersive, we stress that it is entirely tunable and can be used effectively for a variety of narrow band applications. Our results bring together two powerful, yet disparate ideas to manipulate on-demand the optical properties of such composite materials.

Acknowledgment

The authors thank Srihari Kesavamurthy for discussions, and Sir John Pendry for critical comments, particularly pointing out the issue of inhomogeneous broadening. SAR thanks the Department of Science and Technology (India) for funding.

References and links

1.

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005). [CrossRef]

2.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

3.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

4.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

5.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

6.

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

7.

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

8.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in in magnetic resonance imaging,” Science 291, 849–851 (2001). [CrossRef] [PubMed]

9.

F. Magnus, B. Wood, J. Moore, K. Morrison, G. Perkins, J. Fyson, M. C. K. Wiltshire, D. Caplin, L. F. Cohen, and J. B. Pendry, “A d.c magnetic metamaterial,” Nature Materials 7, 295–297 (2008). [CrossRef] [PubMed]

10.

V. M. Shalaev, “Optical negative index metamaterials,” Nature Photon. 1, 41–48 (2007). [CrossRef]

11.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997). [CrossRef]

12.

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). [CrossRef] [PubMed]

13.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991). [CrossRef] [PubMed]

14.

M. O. Scully and M. S. Zubairy, Quantum OpticsCambridge Univ. Press, Cambridge, U.K. (1997).

15.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

16.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]

17.

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991). [CrossRef] [PubMed]

18.

G. S. Agarwal and Harshawardhan Wanare, “Inhibition and enhancement of two photon absorption,” Phys. Rev. Lett. 77, 1039–1042 (1996). [CrossRef] [PubMed]

19.

S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys.: Condens. Matter 14, 6383–6394 (2002). [CrossRef]

20.

N. G. Kalugin and Y. V. Rostovtsev, “Efficient generation of short terahertz pulses via stimulated Raman adiabatic passage,” Opt. Lett. 31, 969–971 (2006). [CrossRef] [PubMed]

21.

J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992). [CrossRef] [PubMed]

22.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

23.

W. Vassen, “Laser cooling and trapping of metastable helium: towards Bose-Einstein condensation,” Comptes Rendus de l’Academie des Sciences Series IV - Physics 2, 613–618 (2001).

24.

F. S. Pavone, G. Bianchini, F. S. Cataliotti, T.W. Hänsch, and M. Inguscio, “Birefringence in electromagnetically induced transparency,” Opt. Lett. 22, 736–738 (1997). [CrossRef] [PubMed]

25.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nature Photon. 2, 295–298 (2008). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: September 18, 2008
Revised Manuscript: October 31, 2008
Manuscript Accepted: October 31, 2008
Published: November 10, 2008

Citation
Sangeeta Chakrabarti, S. Anantha Ramakrishna, and Harshawardhan Wanare, "Coherently controlling metamaterials," Opt. Express 16, 19504-19511 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19504


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References

  1. S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449 - 521 (2005). [CrossRef]
  2. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773 - 4776 (1996). [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 - 2084 (1999). [CrossRef]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184 - 4187 (2000). [CrossRef] [PubMed]
  5. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 - 1782 (2006). [CrossRef] [PubMed]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 - 79 (2001). [CrossRef] [PubMed]
  7. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 - 3969 (2000). [CrossRef] [PubMed]
  8. M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, "Microstructured magnetic materials for RF flux guides in in magnetic resonance imaging," Science 291, 849 - 851 (2001). [CrossRef] [PubMed]
  9. F. Magnus, B. Wood, J. Moore, K. Morrison, G. Perkins, J. Fyson, M. C. K. Wiltshire, D. Caplin, L. F. Cohen, and J. B. Pendry, "A d.c magnetic metamaterial," Nature Materials 7, 295 - 297 (2008). [CrossRef] [PubMed]
  10. Q1. V. M. Shalaev, "Optical negative index metamaterials," Nature Photon. 1, 41- 48 (2007). [CrossRef]
  11. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36 - 42 (1997). [CrossRef]
  12. S. E. Harris, J. E. Field, and A. Imamoglu, "Nonlinear optical processes using electromagnetically induced transparency," Phys. Rev. Lett. 64, 1107 - 1110 (1990). [CrossRef] [PubMed]
  13. K. J. Boller, A. Imamoglu, and S. E. Harris,"Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593 - 2596 (1991). [CrossRef] [PubMed]
  14. M. O. Scully and M. S. Zubairy, Quantum Optics Cambridge Univ. Press, Cambridge, U.K. (1997).
  15. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594 - 598 (1999). [CrossRef]
  16. L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain assisted superluminal light propagation," Nature 406, 277 - 279 (2000). [CrossRef] [PubMed]
  17. M. O. Scully, "Enhancement of the index of refraction via quantum coherence," Phys. Rev. Lett. 67, 1855 - 1858 (1991). [CrossRef] [PubMed]
  18. Q2. G. S. Agarwal and Harshawardhan Wanare, "Inhibition and enhancement of two photon absorption," Phys. Rev. Lett. 77, 1039 - 1042 (1996). [CrossRef] [PubMed]
  19. S. O’Brien and J. B. Pendry,"Magnetic activity at infrared frequencies in structured metallic photonic crystals," J. Phys.: Condens. Matter 14, 6383 - 6394 (2002). [CrossRef]
  20. N. G. Kalugin and Y. V. Rostovtsev, "Efficient generation of short terahertz pulses via stimulated Raman adiabatic passage," Opt. Lett. 31, 969 - 971 (2006). [CrossRef] [PubMed]
  21. J. B. Pendry and A. Mackinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772 - 2775 (1992). [CrossRef] [PubMed]
  22. P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals" Phys. Rev. B 6, 4370 - 4379 (1972). [CrossRef]
  23. W. Vassen, "Laser cooling and trapping of metastable helium: towards Bose-Einstein condensation," Comptes Rendus de l’Academie des Sciences Series IV - Physics 2, 613-618 (2001).
  24. F. S. Pavone, G. Bianchini, F. S. Cataliotti, T.W. H¨ansch, and M. Inguscio, "Birefringence in electromagnetically induced transparency," Opt. Lett. 22, 736 - 738 (1997). [CrossRef] [PubMed]
  25. Q3. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, "Experimental demonstration of frequency-agile terahertz metamaterials," Nature Photon. 2, 295 - 298 (2008). [CrossRef]

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