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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 5, Iss. 10 — Jul. 19, 2010
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Asymmetric planar terahertz metamaterials

Ranjan Singh, Ibraheem A. I. Al-Naib, Martin Koch, and Weili Zhang  »View Author Affiliations


Optics Express, Vol. 18, Issue 12, pp. 13044-13050 (2010)
http://dx.doi.org/10.1364/OE.18.013044


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Abstract

We report an experimental observation of three distinct resonances in split ring resonators (SRRs) for both vertical and horizontal electric field polarizations at normal incidence by use of terahertz time-domain spectroscopy. Breaking the symmetry in SRRs by gradually displacing the capacitive gap from the centre towards the corner of the ring allows for an 85% modulation of the fundamental inductive-capacitive resonance. Increasing asymmetry leads to the evolution of an otherwise inaccessible high quality factor electric quadrupole resonance that can be exploited for bio-sensing applications in the terahertz region.

© 2010 OSA

1. Introduction

Here we demonstrate the excitation of three distinct resonance modes in SRRs for both horizontal and vertical incident electric field polarizations by introducing asymmetry in their structure. The three resonance modes are the LC, dipole and the quadrupole resonances. Quadrupole mode is known as a weak mode [21

21. R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express 16(9), 6537–6543 (2008). [CrossRef] [PubMed]

]. However, we show that a strong one can be achieved by crossing the symmetry of the structure. When the incident field is horizontal and parallel to the SRR gap, the LC resonance becomes weak but a strong evolution of quadrupole resonance is witnessed with increasing asymmetry. For the orthogonal incident polarization, we observe the simultaneous formation of a weak LC resonance and a strong quadrupole resonance as the degree of asymmetry is increased in a step wise fashion by displacing the split ring gap from the center towards the extreme corner of the gap arm in the SRR. An extremely sharp quadrupole resonance with a high quality factor (Q) is observed experimentally for both incident polarizations. Such narrow resonances can be exploited for highly efficient sensing and frequency selection in the terahertz domain.

2. Experiment

We employ broadband terahertz time-domain spectroscopy (THz-TDS) was employed to characterize the asymmetrical planar metamaterials [29

29. D. Grischkowsky, S. Keiding, M. Exter, and Ch. Fattinger, “Far infrared time domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7(10), 2006 (1990). [CrossRef]

]. The photoconductive-switch based THz-TDS system consists of four parabolic mirrors configured in an 8-F confocal geometry that enables a 3.5 mm diameter frequency independent beam waist for small sample characterization. Six sets of planar SRR metamaterials with 200 nm thick Al metal structures are fabricated by conventional photolithography on a silicon substrate (0.64-mm-thick, n-type resistivity 12 Ω cm). Asymmetry in the SRRs is introduced by displacing the gap gradually from the center, as shown in Figs. 1(a)
Fig. 1 (a) - (f) is the microscopic image of the six sample arrays; the gap in the SRR is moved in a step wise fashion by different values of ‘δx’ from the center (g) Unit cell with dimension parameters, t = 6 μm, d = 3 μm, l = 36 μm, Al metal film thickness is 200 nm. The periodicity of the unit cells in all samples is P = 50 μm
1(f) with δx = 0, 2.5, 4.5, 6.5, 8.5, and 10.5 μm, where ‘δx’ represents the gap displacement parameter. Figure 1(g) shows the diagram of a SRR unit cell with a minimum feature d = 3 μm in the splits of the rings and other dimensions t = 6 μm, l = 36 μm, and the lattice constant P = 50 μm. Each SRR array has a 1 cm × 1 cm clear aperture. In the first case the orientation of the incident terahertz field is perpendicular to the SRR gap which excites the weak LC resonance. In the second case the field is along the SRR gap in order to excite the regular LC and the dipole mode resonances. In the THz- TDS measurements, each metamaterial sample is placed midway between the transmitter and receiver modules in the far-field at the focused beam waist and the terahertz waves penetrate the SRRs at normal incidence.

The time domain data were taken for all the six MM samples in a sequential order for both orientations of the electric terahertz field. Figure 2(a)
Fig. 2 (a)–(b) Measured sub-picosecond transmitted pulses through the symmetric (δx = 0 µm) and extreme asymmetric (δx = 10.5 µm) MM samples with electric field vertical and horizontal to the SRR gap respectively. The inset shows the blow up of the later time delay pulses.
shows the measured terahertz pulses transmitted through the symmetric SRR (δx = 0 µm) and the extreme asymmetric SRR (δx = 10.5 µm) while the terahertz field is vertical to the SRR gap. For asymmetric SRR, a strong ringing in the pulse is observed for late times as shown in the top inset of Fig. 2(a). Figure 2(b) shows the pulse measured for the other orientation when the field is along the SRR gap. We see an increase in the pulse oscillation towards later times for the asymmetric sample. As we will see below that this is due to strengthening of a quadrupole resonance at around 1.72 THz.

3. Measurement and Simulation

The transmission is extracted from the ratio of the Fourier-transformed amplitude spectra of the samples to the reference, defined as|Es(ω)/Er(ω)|, where Es(ω)and Er(ω) are Fourier-transformed time traces of the transmitted electric fields of the signal and the reference pulses, respectively. Figure 3(a)
Fig. 3 Measured (a) and simulated (b) amplitude transmission spectra of with varying asymmetry in SRR for vertical E field polarization. (c), (d) show the measured and simulated spectra for horizontal polarization.
shows the measured transmission spectra through all the six samples with varying asymmetry. The field orientation with respect to the SRR gap is shown in the inset. For the perfect symmetric SRR when the gap is right at the center of the arm (δx = 0 µm), there is excitation of only a typical dipole resonance at 1.36 THz due to linear oscillating currents in the SRR arms parallel to the incident field. Gradually, as we introduce asymmetry in the structure by altering the gap displacement, we observe the evolution of a weak LC resonance at 0.55 THz where the transmission gradually changes from 100% to 75%. The formation of the resonance is most pronounced for δx = 10.5 µm when the SRR gap is pushed all the way to the extreme corner. The Q-factor for the LC resonance of this sample is 7. The increase in asymmetry of the MM also gives rise to another sharp quadrupole resonance feature with Q-factor as high as 35 at 1.72 THz for the SRR with δx = 10.5. The dipole resonance feature blue shifts by 160 GHz for extreme asymmetry. Figure 3(b) shows the simulation results for identical experimental conditions and most of the transmission spectra are found in good agreement with the measurements [30

30. CST Microwave Studio®, (http://www.cst.com).

].

As the incident terahertz field is aligned along the SRR gap, it usually excites the two regular resonance modes, the lower frequency LC resonance and the higher frequency dipole resonance. Figure 3(c) reveals the change in both these regular resonance modes as the SRR gap is swept along the arm. The LC resonance undergoes slight shrinking in the line width and the transmission at the resonance frequency gets enhanced from 12% (δx = 0 µm) to almost up to 35% (δx = 10.5 µm). The dipole resonance does not shift in frequency but undergoes significant broadening by about 15 GHz. As the asymmetry is increased, there is appearance of another sharp quadrupole resonance feature at 1.72 THz. Thus the weak LC mode resonance for vertical polarization and the high Q quadrupole resonance features at 1.72 THz for both orientations appear entirely due to the introduction of asymmetry in the SRRs. It should be noted that the weak LC mode resonance and the regular strong LC resonance occurs at the same frequency but for orthogonal electric field polarizations. Figure 3(d) represents the simulation for the horizontal polarization and it reproduces all of the transmission spectrum features identical with the measurement.

4. Discussion

The gradual evolution of this resonance can be observed in Figs. 5(b) and 5(c) where the Q factor reaches as high as 55 for vertical polarization and 93 for horizontal polarization when the SRR gap is displaced by δx = δy = 8.5 μm. The Q factor is extracted from the simulated transmission curves. The high Q factor of the electric quadrupole resonances can be easily exploited for sensing and narrow band filtering purposes.

5. Conclusion

In conclusion, we have characterized asymmetrical planar terahertz SRRs and excited the LC, dipole and the quadrupole mode resonances for vertical and horizontal electric field polarizations at normal incidence. Shifting the gap position allows us to engineer the transmission properties of metamaterials across a large portion of electromagnetic spectrum. A very high passive modulation of the LC resonance has been achieved by varying the asymmetry parameter. This property can find applications in the design of terahertz modulators in which the SRR gap could be actively displaced by external pumping. The symmetry breaking also gives access to the high Q quadrulpole resonances where the concentrated field in a much smaller volume would open up avenues for efficient bio-sensing applications in the terahertz domain.

Acknowledgements

The authors thank H. T. Chen, J. F. O’Hara and J. Zhou for their support and discussions. This work was supported by the U.S. National Science Foundation.

References and links

1.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

2.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]

3.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004). [CrossRef] [PubMed]

4.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005). [CrossRef] [PubMed]

5.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]

6.

A. K. Azad, J. M. Dai, and W. Zhang, “Transmission properties of terahertz pulses through subwavelength double split-ring resonators,” Opt. Lett. 31(5), 634–636 (2006). [CrossRef] [PubMed]

7.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006). [CrossRef] [PubMed]

8.

C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14(19), 8827–8836 (2006). [CrossRef] [PubMed]

9.

P. Markoš and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036622 (2002). [CrossRef] [PubMed]

10.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006). [CrossRef] [PubMed]

11.

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,” Nat. Photonics 2(5), 295–298 (2008). [CrossRef]

12.

H. T. Chen, W. Padilla, M. Cich, A. Azad, R. Averitt, and A. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009). [CrossRef]

13.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008). [CrossRef]

14.

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(10), 7181–7188 (2008). [CrossRef] [PubMed]

15.

J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16(3), 1786–1795 (2008). [CrossRef] [PubMed]

16.

C. Debus and P. H. Bolivar, “Frequency selective surfaces for high sensitivity terahertz sensing,” Appl. Phys. Lett. 91(18), 184102 (2007). [CrossRef]

17.

I. A. I. Al-Naib, C. Jansen, and M. Koch, “Thin film sensing with planar asymmetric metamaterial resonators,” Appl. Phys. Lett. 93(8), 083507 (2008). [CrossRef]

18.

I. A. I. Al-Naib, C. Jansen, and M. Koch, “Applying the Babinet principle to asymmetric resonators,” Electron. Lett. 44(21), 1228 (2008). [CrossRef]

19.

H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable terahertz metamaterials,” Phys. Rev. Lett. 103(14), 147401 (2009). [CrossRef] [PubMed]

20.

I. A. I. Al-Naib, C. Jansen, and M. Koch, “High Q-factor metasurfaces based on miniaturized asymmetric single split resonators,” Appl. Phys. Lett. 94(15), 153505 (2009). [CrossRef]

21.

R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express 16(9), 6537–6543 (2008). [CrossRef] [PubMed]

22.

O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Opt. Express 16(9), 6736–6744 (2008). [CrossRef] [PubMed]

23.

R. Singh, A. K. Azad, J. F. O’Hara, A. J. Taylor, and W. Zhang, “Effect of metal permittivity on resonant properties of terahertz metamaterials,” Opt. Lett. 33(13), 1506–1508 (2008). [CrossRef] [PubMed]

24.

M. Walther, A. Ortner, H. Meier, U. Loffelmann, P. J. Smith, and J. G. Korvink, “Terahertz metamaterials fabricated by inkjet printing,” Appl. Phys. Lett. 95(25), 251107 (2009). [CrossRef]

25.

R. Singh, C. Rockstuhl, F. Lederer, and W. Zhang, “The impact of nearest neighbor interaction on the resonances in terahertz metamaterials,” Appl. Phys. Lett. 94(2), 021116 (2009). [CrossRef]

26.

S. Y. Chiam, R. Singh, J. Gu, J. Han, W. Zhang, and A. A. Bettiol, “Increased frequency shifts in high aspect ratio terahertz split ring resonators,” Appl. Phys. Lett. 94(6), 064102 (2009). [CrossRef]

27.

R. Singh, C. Rockstuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B 79(8), 085111 (2009). [CrossRef]

28.

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in terahertz metamaterial,” Phys. Rev. B 80(15), 153103 (2009). [CrossRef]

29.

D. Grischkowsky, S. Keiding, M. Exter, and Ch. Fattinger, “Far infrared time domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7(10), 2006 (1990). [CrossRef]

30.

CST Microwave Studio®, (http://www.cst.com).

31.

H. R. Stuart and D. G. Hall, “Enhanced dipole-dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80(25), 5663–5666 (1998). [CrossRef]

OCIS Codes
(260.5740) Physical optics : Resonance
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: April 19, 2010
Revised Manuscript: May 27, 2010
Manuscript Accepted: May 27, 2010
Published: June 2, 2010

Virtual Issues
Vol. 5, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Ranjan Singh, Ibraheem A. I. Al-Naib, Martin Koch, and Weili Zhang, "Asymmetric planar terahertz metamaterials," Opt. Express 18, 13044-13050 (2010)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-12-13044


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References

  1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
  2. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]
  3. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004). [CrossRef] [PubMed]
  4. H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005). [CrossRef] [PubMed]
  5. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]
  6. A. K. Azad, J. M. Dai, and W. Zhang, “Transmission properties of terahertz pulses through subwavelength double split-ring resonators,” Opt. Lett. 31(5), 634–636 (2006). [CrossRef] [PubMed]
  7. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006). [CrossRef] [PubMed]
  8. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14(19), 8827–8836 (2006). [CrossRef] [PubMed]
  9. P. Markoš and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036622 (2002). [CrossRef] [PubMed]
  10. H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006). [CrossRef] [PubMed]
  11. 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,” Nat. Photonics 2(5), 295–298 (2008). [CrossRef]
  12. H. T. Chen, W. Padilla, M. Cich, A. Azad, R. Averitt, and A. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009). [CrossRef]
  13. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008). [CrossRef]
  14. 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(10), 7181–7188 (2008). [CrossRef] [PubMed]
  15. J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16(3), 1786–1795 (2008). [CrossRef] [PubMed]
  16. C. Debus and P. H. Bolivar, “Frequency selective surfaces for high sensitivity terahertz sensing,” Appl. Phys. Lett. 91(18), 184102 (2007). [CrossRef]
  17. I. A. I. Al-Naib, C. Jansen, and M. Koch, “Thin film sensing with planar asymmetric metamaterial resonators,” Appl. Phys. Lett. 93(8), 083507 (2008). [CrossRef]
  18. I. A. I. Al-Naib, C. Jansen, and M. Koch, “Applying the Babinet principle to asymmetric resonators,” Electron. Lett. 44(21), 1228 (2008). [CrossRef]
  19. H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable terahertz metamaterials,” Phys. Rev. Lett. 103(14), 147401 (2009). [CrossRef] [PubMed]
  20. I. A. I. Al-Naib, C. Jansen, and M. Koch, “High Q-factor metasurfaces based on miniaturized asymmetric single split resonators,” Appl. Phys. Lett. 94(15), 153505 (2009). [CrossRef]
  21. R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express 16(9), 6537–6543 (2008). [CrossRef] [PubMed]
  22. O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Opt. Express 16(9), 6736–6744 (2008). [CrossRef] [PubMed]
  23. R. Singh, A. K. Azad, J. F. O’Hara, A. J. Taylor, and W. Zhang, “Effect of metal permittivity on resonant properties of terahertz metamaterials,” Opt. Lett. 33(13), 1506–1508 (2008). [CrossRef] [PubMed]
  24. M. Walther, A. Ortner, H. Meier, U. Loffelmann, P. J. Smith, and J. G. Korvink, “Terahertz metamaterials fabricated by inkjet printing,” Appl. Phys. Lett. 95(25), 251107 (2009). [CrossRef]
  25. R. Singh, C. Rockstuhl, F. Lederer, and W. Zhang, “The impact of nearest neighbor interaction on the resonances in terahertz metamaterials,” Appl. Phys. Lett. 94(2), 021116 (2009). [CrossRef]
  26. S. Y. Chiam, R. Singh, J. Gu, J. Han, W. Zhang, and A. A. Bettiol, “Increased frequency shifts in high aspect ratio terahertz split ring resonators,” Appl. Phys. Lett. 94(6), 064102 (2009). [CrossRef]
  27. R. Singh, C. Rockstuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B 79(8), 085111 (2009). [CrossRef]
  28. S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in terahertz metamaterial,” Phys. Rev. B 80(15), 153103 (2009). [CrossRef]
  29. D. Grischkowsky, S. Keiding, M. Exter, and Ch. Fattinger, “Far infrared time domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7(10), 2006 (1990). [CrossRef]
  30. CST Microwave Studio®, ( http://www.cst.com ).
  31. H. R. Stuart and D. G. Hall, “Enhanced dipole-dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80(25), 5663–5666 (1998). [CrossRef]

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