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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 17 — Aug. 26, 2013
  • pp: 20363–20375
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The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator

Jiawei Cong, Binfeng Yun, and Yiping Cui  »View Author Affiliations


Optics Express, Vol. 21, Issue 17, pp. 20363-20375 (2013)
http://dx.doi.org/10.1364/OE.21.020363


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Abstract

By introducing the frequency tuning sensitivity, an analytical model based on equivalent LC circuit is developed for the relative frequency tuning range of THz semiconductor split-ring resonator (SRR). And the model reveals that the relative tuning range is determined by the ratio of the kinetic inductance to the geometric inductance (RKG). The results show that under the same carrier density variation, a larger RKG results in a larger relative tuning range. Based on this model, a stacked SRR-dimer structure with larger RKG compared to the single SRR due to the inductive coupling is proposed, which improves the relative tuning range effectively. And the results obtained by the simple analytical model agree well with the numerical FDTD results. The presented analytical model is robust and can be used to analyze the relative frequency tuning of other tunable THz devices.

© 2013 OSA

1. Introduction

Electromagnetic metamaterials (MMs) are artificial media designed to offer a wide range of exotic phenomena, such as anomalous refractive index [1

1. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9(5), 407–412 (2010). [CrossRef] [PubMed]

5

5. P. Dardano, M. Gagliardi, I. Rendina, S. Cabrini, and V. Mocella, “Ellipsometric determination of permittivity in a negative index photonic crystal metamaterial,” Light: Sci. Appl. 1(12), e42 (2012). [CrossRef]

], optical beam self-collimation [6

6. V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102(13), 133902 (2009). [CrossRef] [PubMed]

, 7

7. V. Mocella, P. Dardano, I. Rendina, and S. Cabrini, “An extraordinary directive radiation based on optical antimatter at near infrared,” Opt. Express 18(24), 25068–25074 (2010). [CrossRef] [PubMed]

], perfect absorption [8

8. C. H. Lin, R. L. Chern, and H. Y. Lin, “Polarization-independent broad-band nearly perfect absorbers in the visible regime,” Opt. Express 19(2), 415–424 (2011). [CrossRef] [PubMed]

, 9

9. C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. 37(3), 308–310 (2012). [CrossRef] [PubMed]

] and sub-wavelength focusing [10

10. C. Ma and Z. Liu, “Focusing light into deep subwavelength using metamaterial immersion lenses,” Opt. Express 18(5), 4838–4844 (2010). [CrossRef] [PubMed]

12

12. T. Roy, E. T. Rogers, and N. I. Zheludev, “Sub-wavelength focusing meta-lens,” Opt. Express 21(6), 7577–7582 (2013). [CrossRef] [PubMed]

] etc. In particular, MMs are a promising candidate to construct functional components at the THz regime (0.1-10 THz), where rare natural materials effectively respond to the THz radiation. For such applications, it is of great importance to be able to actively control resonant frequencies of MMs. Real-time frequency tuning can be accomplished by applying external stimuli to the reconfigurable elements which are incorporated into the MMs, such as phase-change materials [13

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express 19(2), 1563–1568 (2011). [CrossRef] [PubMed]

, 14

14. M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express 17(20), 18330–18339 (2009). [CrossRef] [PubMed]

], superconductors [15

15. J. B. Wu, B. B. Jin, Y. H. Xue, C. H. Zhang, H. Dai, L. B. Zhang, C. H. Cao, L. Kang, W. W. Xu, J. Chen, and P. H. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express 19(13), 12021–12026 (2011). [CrossRef] [PubMed]

, 16

16. H. T. Chen, H. Yang, R. Singh, J. F. O’Hara, A. K. Azad, S. A. Trugman, Q. X. Jia, and A. J. Taylor, “Tuning the resonance in high-temperature superconducting terahertz metamaterials,” Phys. Rev. Lett. 105(24), 247402 (2010). [CrossRef] [PubMed]

], semiconductors [17

17. 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]

23

23. D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express 19(10), 9968–9975 (2011). [CrossRef] [PubMed]

] and micromechanical components [24

24. J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett. 11(5), 2142–2144 (2011). [CrossRef] [PubMed]

, 25

25. J. J. Li, C. M. Shah, W. Withayachumnankul, B. S.-Y. Ung, A. Mitchel, S. Sriram, M. Bhaskaran, S. J. Chang, and D. Abbott, “Mechanically tunable terahertz metamaterials,” Appl. Phys. Lett. 102(12), 121101 (2013). [CrossRef]

]. And the stimulus can be temperature [13

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express 19(2), 1563–1568 (2011). [CrossRef] [PubMed]

, 14

14. M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express 17(20), 18330–18339 (2009). [CrossRef] [PubMed]

, 21

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. 107(9), 093104 (2010). [CrossRef]

, 22

22. J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express 16(19), 14390–14396 (2008). [CrossRef] [PubMed]

], optical-pump [17

17. 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]

19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013). [CrossRef]

], voltage control [13

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express 19(2), 1563–1568 (2011). [CrossRef] [PubMed]

, 23

23. D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express 19(10), 9968–9975 (2011). [CrossRef] [PubMed]

] and so on.

2. Structure design and simulation model

The two types of SRR structures studied in this article are presented in Fig. 1
Fig. 1 Schematic of an array of InSb metamaterial composed of (a) a single SRR and (b) two coaxially aligned SRRs in each unit cell. The geometric parameters are initially chosen as h = 2 μm, r = 15 μm, w = 4 μm, g = 6 μm, Px = Py = 54 μm and s = 3 μm.
. One structure consists of a single SRR in a unit cell while the other consists of a stacked SRR-dimer with a twist angle φ = 0° or φ = 180°, as shown in Figs. 1(a) and 1(b), respectively. And the material of SRRs is InSb, which can be grown on the GaAs or Si substrate by using molecular beam epitaxy (MBE) process [19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013). [CrossRef]

, 26

26. G. Singh, E. Michel, C. Jelen, S. Slivken, J. Xu, P. Bove, I. Ferguson, and M. Razeghi, “Molecular-beam epitaxial growth of high quality InSb for p-i-n photodetectors,” J. Vac. Sci. Technol. B 13(2), 782–785 (1995). [CrossRef]

]. We focus on InSb because its carrier density can be flexibly adjusted by the external stimuli including optical pump [19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013). [CrossRef]

] and ambient temperature [20

20. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. 56(4), 554–557 (2009). [CrossRef]

]. Moreover, InSb is suitable for sustaining strong localized plasmon in the THz region as a result of the low losses due to its high carrier mobility [19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013). [CrossRef]

]. The incident THz wave propagates in the direction normal to the SRR plane (along z-axis), with the electric field parallel to the SRR gap (along x-axis). With this configuration, the fundamental mode of the plasmon resonance is excited. At this resonance, the SRR can be viewed as a series RLC circuit, so this mode is also referred as the LC resonance [27

27. 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]

, 28

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) 3(2), 57–62 (2009). [CrossRef]

]. In order to understand the underlying mechanism of the frequency shift with carrier densities, an analytical treatment is made from the perspective of an equivalent RLC circuit. According to the Kirchhoff voltage rule, we obtain
(Lg+Lk)dIdt+RI+IdtC=0,
(1)
whereLgis the geometric inductance,Lk is the kinetic inductance, C is the capacitance and R is the resistance of the SRR. Assume that the current has the form of I = I0exp(-βt-iωt), where β accounts for the damping loss. Then the resonance frequency is obtained from Eq. (1) as
f=12π1(Lg+Lk)CR24(Lg+Lk)2.
(2)
Different from the commonly employed LC model [27

27. 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]

], where the resonance frequency solely depends on the inductance and capacitance of SRRs, the derived RLC model also takes into account the resistance of the semiconductor. For InSb with the low resistance (i.e.,R2/4(Lg+Lk)21/(Lg+Lk)C), the resistance has negligible influence on the resonance frequency. So in the subsequent analysis, the LC model (Eq. (3)) is employed for simplicity.
f=12π(Lg+Lk)C.
(3)
However, if SRRs are made of some highly resistive semiconductors (such as GaAs and Si), the effect of the resistance on the resonance frequency cannot be neglected. And the resistance is expected to reduce the resonance frequency relative to that calculated from the LC model. For the SRR shown in Fig. 1(a), the carrier kinetic inductanceLk, the geometric inductanceLg, the total capacitance C (the sum of the gap capacitanceCgand surface capacitanceCs) take the following form [28

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) 3(2), 57–62 (2009). [CrossRef]

]:
Lk=2πr-gε0whωp2=(2πr-g)m*whe21N,
(4a)
Lg=μ0r(Log(8rw+h)-0.5),
(4b)
Cg=ε0whg+ε0(w+h+g),
(4c)
Cs=2ε0π(w+h)Log(4rg),
(4d)
C=Cg+αCs,
(4e)
R=γε0ωp22πr-gwh.
(4f)
Hereε0is the vacuum permittivity,ωpis the plasma frequency andγis the damping constant of the semiconductor. α is a correction factor to the surface capacitance, since Eq. (4d) delivers a smaller result than the true value for a thin SRR [28

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) 3(2), 57–62 (2009). [CrossRef]

]. With regards to the SRR sizes under investigation, α=2.25ensures good agreement between the analytical and simulation results.

Also the finite difference time domain (FDTD) numerical simulation [29

29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005)

] is implemented to testify the analytical results. In the simulation, the periodic boundary condition along the x and y axis and the absorption boundary in the z direction are adopted. A uniform grid spacing is set asΔx=Δy=Δz=0.02μmwhich is sufficient for the convergence of the FDTD results. The complex permittivity of InSb can be described by the Drude model: ε(f)=εωp2(2πf)2+i2πγf [19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013). [CrossRef]

21

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. 107(9), 093104 (2010). [CrossRef]

]. Here, εis the high-frequency permittivity and the plasma frequency ωp=Ne2/ε0m*depends on the carrier density (N), the electron charge (e), the vacuum permittivity (ε0) and the carrier effective mass (m*). The values of these parameters are taken asε=15.68, γ=2π×0.05THz, m* = 0.015me [20

20. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. 56(4), 554–557 (2009). [CrossRef]

, 21

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. 107(9), 093104 (2010). [CrossRef]

, 30

30. X. Y. Dai, Y. J. Xiang, S. C. Wen, and H. Y. He, “Thermally tunable and omnidirectional terahertz photonic bandgap in the one-dimensional photonic crystals containing semiconductor InSb,” J. Appl. Phys. 109(5), 053104 (2011). [CrossRef]

], where me is the electron’s rest mass. These parameters allow for a good fitting of the InSb permittivity around 1 THz when the carrier density is below N = 1 × 1017 cm−3 [31

31. S. C. Howells and L. A. Schlie, “Transient terahertz reflection spectroscopy of undoped InSb from 0.1 to 1.1 THz,” Appl. Phys. Lett. 69(4), 550–552 (1996). [CrossRef]

]. The substrate is taken as vacuum in the following analysis for simplicity without loss of generality, given that the material choice of these media would only slightly shift the resonances but not affect the tuning mechanism [22

22. J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express 16(19), 14390–14396 (2008). [CrossRef] [PubMed]

].

3. Results and discussion

We firstly explore the single-layer SRRs to capture the physics underlying the tuning effects and introduce the corresponding analytical model. Then we extend the analytical treatment to the case of stacked SRR-dimers. The resonance frequency of the single-layer SRRs undergoes a continuous blue shift from 0.28 THz to 0.97 THz when the carrier density N is increased from 5 × 1015 cm−3 to 1 × 1017 cm−3, as shown in Fig. 2(a)
Fig. 2 (a) Transmission spectra of the single-layer InSb SRRs with different carrier density N. The transmission dips indicated by triangles represent the LC resonance mode. (b) The resonance frequency and tuning sensitivity as a function of N. The red and black solid lines show the analytical results and the blue dots show the FDTD simulation results.
. For N = 5 × 1015 cm−3 and 1 × 1016 cm−3, the higher order modes are also excited which will not be discussed in this work. Two features can be seen in Fig. 2. First, the transmission dip becomes pronounced with the increment of N, which indicates that the resonance strength is gradually enhanced. This is because that InSb transits from an insulator to a conductor when considerable free carriers are thermally or optically excited. And a more conductive SRR is able to induce the stronger LC resonance [32

32. 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]

]. Second, the growth rate of resonance frequencies is gradually reduced with an increment of carrier densities, as shown in Fig. 2(b). The reason will be demonstrated below.

We can see from Eqs. (3) and (4) that only the kinetic inductance and the resistance vary with the carrier density. But the resistance of InSb SRR has negligible influence on the resonance frequency. So it is reasonable to ascribe the resonance shift with the carrier density to the variation of the carries kinetic inductance of SRRs. Based on the LC model, the increment of N will reduceLk, which in turn leads to the higher resonance frequency. An excellent agreement between the calculated resonance frequency from the LC model (red line) and the FDTD simulation results (blue dot line) verifies the validity of the LC model.

In order to evaluate the resonance frequency shift with the carrier density, the frequency tuning sensitivity S is introduced by us as:
S=fN.
(5)
Based on Eq. (3) and Eq. (4), the tuning sensitivity is derived as
S=Lk4πN(Lk+Lg)3/2(Cg+αCs)=f2(Lk+Lg)LkN.
(6)
Figure 2(b) shows that as N increases, the tuning sensitivity is reduced (black line), indicating that the kinetic inductance and hence the resonance frequency is less sensitive to the increasing carrier density. Therefore, the resonance frequency undergoes a slow growth with N and will finally saturate to the value 1/(2πLg(Cg+αCs))whenLkis negligible compared toLg.

Δff(N1)=N1N2S(N)dNf(N1)=f(N2)-f(N1)f(N1)=(Lg+Lk1)(Cg+αCs)(Lg+Lk2)(Cg+αCs)1=1+N2N1Lk2Lg1+Lk2Lg1.
(7)

Based on the LC model, RKG is derived as a function of the thickness (h) and radius (r) of SRR at N = 1 × 1016 cm−3 (Fig. 3(a)
Fig. 3 Contour plot of RKG (Lk /Lg) as a function of SRR thickness h and radius r at (a) N = 1 × 1016 cm−3 and (b) N = 1 × 1017 cm−3. The red to blue colors represent high to low values. The relative tuning range in dependence with (c) thickness and (d) radius. The resonance frequency as a function of N at different (e) thickness and (f) radius. In (c)-(e), the dots represent the simulation results and solid curves show the analytical results from LC model.
) and 1 × 1017 cm−3 (Fig. 3(b)). As shown from a comparison between Figs. 3(a) and 3(b), the dependence of RKG on the geometric sizes (h and r) is similar at the two carrier densities. But RKG is larger at the lower N because the kinetic inductance is inversely proportional to the carrier density according to Eq. (4a). Figures 3(a) and 3(b) further show that for a fixed carrier density, RKG is reduced with an increasing h. So under the same variation of N, a lower relative tuning range is expected at a larger h. This is verified in Fig. 3(c), where the relative tuning range (N1 = 1 × 1016 cm−3 and N2 = 1 × 1017 cm−3) as a function of h is obtained. The analytical result (red solid line) shows that the relative tuning range decreases from 170% to 130% as h increases from 1 μm to 3 μm, which agrees well with the simulation results (blue dots). The decrease of RKG and hence the relative tuning range with h can be explained by the LC model. An examination of Eqs. (4a) and (4b) reveals that the inductanceLkandLgare both reduced as h is increased. Nonetheless,Lkundergoes a more substantial change thanLg. Therefore, RKG is reduced as the thickness of SRRs grows, which then leads to a reduction of the relative tuning range. Besides, the resonance shifts to higher frequencies with increasing h at the same time, as shown in Fig. 3(e). Although the capacitanceCgandCsboth increase with h, which is expected to reduce the resonance frequency according to Eq. (3), the simultaneous reduction of the inductance makes a more significantly competing effect, eventually increasing the resonance frequency.

In addition to the altering of the SRR thickness, Fig. 3 shows that tailoring the radius r provides another means to adjust the RKG. Similar to the situation of h, an increment of r also results in reduced RKG, which then reduces the relative tuning range. However, Lk/Lgis less sensitive to the radius r than to the thickness h, as shown in the contour plots. And another notable difference can be seen by comparing Fig. 3(e) with Fig. 3(f). Figure 3(e) shows that at a given N the resonance frequency increases with the thickness, as mentioned above. While the resonance frequency is reduced with an increasing radius because both the inductance and capacitance are increased. In other words, by reducing h, an increased relative tuning range is achieved and accompanied by the reduction of the resonance frequency. In contrast, by reducing r, the relative tuning range Δf / f(N1) and the resonance frequencies f(N1) are both increased. Consequently, as r is reduced the absolute resonance shift Δf is also increased. For example, as N rises from 1 × 1016 cm−3 to 1 × 1017 cm−3, the resonance frequency of SRRs with r = 17 μm increases from 0.36 THz to 0.86 THz, with a relative tuning range of 139% and the absolute resonance shift Δf = 0.50 THz. While for r = 9 μm, the resonance shifts from 0.58 THz to 1.47 THz, with a relative tuning range of 153% and Δf = 0.89 THz. So the adjustment of the radius of SRRs can result in the simultaneous increment of the relative tuning range and absolute resonance shift. In Fig. 3(f), there is a slight discrepancy between the analytical and simulated resonance frequency. The reason is that Eq. (4b) takes the expression of the geometric inductance of a closed ring, while the simulated SRR structure has a gap. Furthermore, the geometric inductance of a SRR can be approximated with Eq. (4b) only when the condition 2πr>>gis well satisfied [28

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) 3(2), 57–62 (2009). [CrossRef]

]. So with the same gap size g, the larger the radius r, the smaller discrepancy between the analytical and simulation results can be achieved, which is shown as in Fig. 3(f).

Although the relative tuning range can be increased by reducing the thickness and/or radius of SRRs, there are some constraints at the same time. First, as the thickness is reduced, the LC resonance strength is weakened [34

34. 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]

]. Consequently, the transmission dip at resonance is greatly decreased and even vanishes (not shown here), which is undesirable for practical applications. Second, since the RKG is not sensitive to the radius, increasing the tuning range via reducing the radius is not very effective [Figs. 3(b) and 3(d)]. Therefore, an alternative approach is in demand which can effectively increase the relative tuning range without weakening the resonance strength.

In the aforementioned discussion of the geometric inductance, only the self-inductance is considered while mutual inductance (M) arising from the inductive coupling between SRRs [35

35. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B 76(7), 073101 (2007). [CrossRef]

37

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

] is not taken into the LC model. Under a large separation between neighboring SRRs (20 μm as chosen above), such an approximation is reasonable because the inter-SRR inductive coupling is sufficiently weak compared with the self-inductance. It is verified by the good agreement between the analytical and simulation results. However, if SRRs are densely spaced, the inductive coupling among SRRs is substantially enhanced. So the mutual inductance has to be taken into account. Subsequently, an extended LC model which includes the effects of the mutual inductance will be introduced. And it is shown that the relative tuning range can be increased by adjusting the mutual inductance.

The sign of mutual inductance can be either positive or negative, depending on the relative direction of the magnetic field induced by coupled SRRs. Mutual inductance is negative if the magnetic field generated by the coupled SRRs cancels each other, and becomes positive otherwise. In the former case with M<0, the net geometric inductance is reduced fromLgtoLg=Lg|M|. Correspondingly, RKG is increased fromLk/LgtoLk/(Lg|M|). In the latter case with a positive mutual inductance, the net geometric inductance increases fromLgtoLg=Lg+|M|and RKG drops fromLk/LgtoLk/(Lg|M|). In addition to the inductive coupling, there also exists capacitive coupling between two neighboring SRRs, which results in the mutual capacitance. Assume the total capacitance is Ctot, which takes into account both the self capacitance and the mutual capacitance. Then the resonance frequency is obtained as:
f=12π(Lg|M|+Lk)Ctot.
(8)
After a similar mathematical deduction to obtain Eq. (7), the relative tuning range taking mutual inductance into account is derived as Eq. (9). It takes the similar form with Eq. (7), but withLg.replaced byLg, which is defined asLg=Lg|M|for M<0 andLg=Lg+|M|for M>0.
Δff(N1)=N1N2S(N)dNf(N1)=(Lg+Lk1)Ctot(Lg+Lk2)Ctot1=1+N2N1Lk2Lg/1+Lk2Lg1.
(9)
Equation (9) reveals that the mutual inductance may affect the relative tuning range by altering RKG. While the mutual capacitance, no matter how large it is, has no effect on the relative tuning range. A comparison between Eq. (7) and (9) shows that under the same carrier density variation, an introduction of negative mutual inductance will increase RKG and hence improve the relative tuning range. On the contrary, a positive mutual inductance is expected to reduce the relative tuning range.

In order to test this hypothesis, a stacked SRR-dimer structure is proposed due to the strong inductive coupling between the two identical SRRs, which are coaxially aligned in the same unit cell. The structure is shown in Fig. 1(b), with the same SRR size parameters as those in Fig. 1(a), except for an introduction of separation s and twist angle φ. This so-called stereo-metamaterial has been investigated in terms of the plasmon coupling between the two stacked SRRs [37

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

]. But as we know, it has not yet been employed for the frequency tuning. It is known that when the two SRRs are brought in close proximity, the original LC resonance of each SRR splits into two new plasmon modes: the bonding mode with the loop currents of two SRRs oscillating in phase and the anti-bonding mode with the loop currents 180° out of phase [37

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

]. Figure 4
Fig. 4 The magnetic field Hz of the stacked SRR-dimer at different resonance modes. (a) Hz of the anti-boning mode for the stacked SRR-dimer MM with φ = 180°. (b) Hz of the boning mode for the stacked SRR-dimer MM with φ = 0°. The white arrows represent the direction of loop currents. The color scale represents the direction and the magnitude of Hz, with red color meaning the maximum value in the z direction and blue color the maximum value in the -z direction.
displays the spatial distribution of the magnetic field Hz at the anti-bonding mode for φ = 180° (Fig. 4(a)) and the bonding mode for φ = 0° (Fig. 4(b)), respectively. The magnetic fields induced by the two SRRs interferes destructively (constructively) at the anti-bonding (bonding) mode, which gives rise to the negative (positive) mutual inductance.

The effect of the negative mutual inductance, which arises from the anti-bonding resonance (see Fig. 4(a)), on the frequency tuning is explored in an array of stacked SRR-dimers with φ = 180°. Under the incident configuration as exhibited in Fig. 1, the bonding mode cannot be effectively excited and only the anti-bonding mode couples to the incident wave, which is manifested by the dip in the transmission spectrum. The simulated transmission spectra of the stacked SRR-dimers (solid lines) and single-layer SRRs (dash lines) are presented in Fig. 5
Fig. 5 The transmission spectra of a single-layer (dashed lines) and stacked SRR-dimers with φ = 180° (solid lines) at different carrier densities. The black to pink color represents an increasing carrier density from 1 × 1016 cm−3 to 1 × 1017 cm−3. The structure sizes are the same with those in Fig. 1.
. The carrier density ranges from 1 × 1016 cm−3 (black) to 1 × 1017 cm−3 (pink). For the same carrier density, the stacked SRR-dimer structure exhibits the higher resonance frequency than the single-layer SRRs. It can also be explained with the LC model. On one hand, a negative mutual inductance is induced in the stacked SRR-dimer MM. So the stacked SRRs have a lower geometric inductance(Lg|M|) than the single-layer SRRs. On the other hand, the capacitive coupling results in reduced total capacitance than that of single-layer SRRs. Therefore, the resonance frequency of the stacked SRRs blue shifts relative to the single-layer SRR. As N is increased from 1 × 1016 cm−3 to 1 × 1017 cm−3, the resonance frequency of single-layer SRR (stacked SRR-dimer) shifts from 0.39 THz (0.41 THz) to 0.97 THz (1.12 THz), with a relative tuning range of 149% (172%). So the improvement of relative tuning range due to the negative mutual inductance is observed, which meets the expectation from the analytical model as proposed above. Moreover, different from reducing the SRR thickness, the SRR-dimer structure effectively increases the relative tuning range with no expense of weakening the resonance strength.

Since the strength of the inductive coupling is dependent on the separation between SRRs [35

35. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B 76(7), 073101 (2007). [CrossRef]

], the mutual inductance can be tailored via adjusting the separation s. The mutual inductance is evaluated from Eq. (10), which is widely used to calculate the mutual inductance between two coaxially aligned circular coils [38

38. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon Press, 1984).

]. Because the circumference of the SRR is much longer than the gap width, i.e., 2πr>>g, Eq. (10) can describe the mutual inductance of stacked SRRs quite well.
|M|=2μ0r(K(x)E(x))x.
(10)
E and K are the complete elliptic integrals of the first and second kind with the geometry-dependent parameterx=4r2+(h+s)2(h+s)4r2+(h+s)2+(h+s). The mutual inductance, which is normalized to the self-inductance for convenience of comparison, is calculated as a function of separation s and shown in the red line of Fig. 6(a)
Fig. 6 (a) RKG and the normalized mutual inductance (red line) at different separation s. The blue (black) line presents RKG of stacked SRR-dimers at the anti-bonding (bonding) mode under the carrier density N = 1 × 1017 cm−3. The green dashed line shows RKG of the single-layer SRRs. (b) The s dependent relative tuning range of stacked SRR-dimers at the anti-bonding mode (upper branch) and the bonding-mode (lower branch). Also the green dashed line shows the relative tuning range of the single-layer SRRs.
. Not surprisingly, M decreases with increased s due to weakened inductive coupling between SRRs. And the decreased mutual inductance leads to the decline ofRKG=Lk/(Lg|M|)when the two SRRs of a dimer are gradually separated (blue line in Fig. 6(a)). Combining the expression of mutual inductance with Eq. (9), the relative tuning range of the stacked SRR-dimers as a function of the separation is derived and shown in the upper branch of Fig. 6(b). The stacked SRR-dimer structure shows larger relative tuning range than the single-layer SRRs of ~150% (green dashed line) under the same variation of carrier density (N increases from 1 × 1016 cm−3 to 1 × 1017 cm−3). And the improvement of the relative tuning range is especially pronounced at a smaller separation due to the stronger inductive coupling, which is responsible for a larger negative mutual inductance. As the separation s grows, the relative tuning range of the stacked SRR-dimers is reduced as a result of the decreased mutual inductance. But even at a large separation of s = 8 μm, the relative tuning range of stacked SRRs is still larger than that of the single-layer structure. The analytical results are well reproduced by the simulation data, indicating that the LC model effectively captures the main physics involved in the frequency tuning.

Having demonstrated that the negative mutual inductance improves the relative tuning range, we proceed to study the effect of the positive mutual inductance on the relative tuning range. The positive mutual inductance originates from the bonding resonance mode of a stacked SRR-dimer structure, as shown in Fig. 4(b). Note that the bonding mode is considered in the stacked SRR-dimers with φ = 0°, rather than the structure with φ = 180°.The reason is that the bonding mode is a dark mode for φ = 180°, which weakly couples to the incident beam and so can hardly be discerned in the transmission spectrum [39

39. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]

]. However for φ = 0°, the bonding mode can be effectively excited by the incident THz wave. At the bonding resonance, the magnetic field along z axis (Hz) induced by the two SRRs of the dimer oscillates in phase, giving rise to a positive mutual inductance and a net geometric inductanceLg=Lg+|M|. Therefore, the SRR-dimer has a larger geometric inductance and hence lower RKG than the single-layer SRR, as shown in Fig. 6(a). Based on Eqs. (9) and (10), the relative tuning range is calculated and shown as the lower red line in Fig. 6(b), which is well supported by the simulation results. As we expected above, the stacked SRR-dimer at the bonding mode actually exhibits a lower relative tuning range than the single-layer SRRs. When the separation s grows, the mutual inductance becomes smaller due to the gradually weakened inductive coupling. Consequently, RKG increases with s until it approaches the value of the single-layer SRRs. Correspondingly, the relative tuning range increases with s and asymptotically approaches the relative tuning range of the single-layer SRRs.

By means of a stacked SRR-dimer metamaterial, we have verified the proposed hypotheses, i.e., the relative tuning range of SRRs can be adjusted through the incorporation of the inductive coupling. The relative tuning range is improved by the negative mutual inductance originating from the destructive inductive coupling, but reduced by the positive mutual inductance arising from the constructive inductive coupling. We would like to mention that the contribution from the inductive coupling to the tuning performance is not limited to the multilayer vertically-stacked SRRs. Similar phenomena are also expected in the planar SRR-dimer configuration [40

40. B. Sauivac, C. R. Simovski, and S. Tretyakov, “Double split-ring resonators: Analytical modeling and numerical simulations,” Electromagnetics 24(5), 317–338 (2004). [CrossRef]

], where the inductive coupling exists as well. It will be investigated in the future work.

4. Conclusion

Acknowledgment

This work was supported by the National Science Foundation of China under Grant No.60907025 and the Fundamental Research Funds for the Central Universities.

References and links

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2.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106(6), 067402 (2011). [CrossRef] [PubMed]

3.

J. W. Cong, B. F. Yun, and Y. P. Cui, “Negative-index metamaterial at visible frequencies based on high order plasmon resonance,” Appl. Opt. 51(13), 2469–2476 (2012). [CrossRef] [PubMed]

4.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470(7334), 369–373 (2011). [CrossRef] [PubMed]

5.

P. Dardano, M. Gagliardi, I. Rendina, S. Cabrini, and V. Mocella, “Ellipsometric determination of permittivity in a negative index photonic crystal metamaterial,” Light: Sci. Appl. 1(12), e42 (2012). [CrossRef]

6.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102(13), 133902 (2009). [CrossRef] [PubMed]

7.

V. Mocella, P. Dardano, I. Rendina, and S. Cabrini, “An extraordinary directive radiation based on optical antimatter at near infrared,” Opt. Express 18(24), 25068–25074 (2010). [CrossRef] [PubMed]

8.

C. H. Lin, R. L. Chern, and H. Y. Lin, “Polarization-independent broad-band nearly perfect absorbers in the visible regime,” Opt. Express 19(2), 415–424 (2011). [CrossRef] [PubMed]

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C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. 37(3), 308–310 (2012). [CrossRef] [PubMed]

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D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012). [CrossRef] [PubMed]

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T. Roy, E. T. Rogers, and N. I. Zheludev, “Sub-wavelength focusing meta-lens,” Opt. Express 21(6), 7577–7582 (2013). [CrossRef] [PubMed]

13.

F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express 19(2), 1563–1568 (2011). [CrossRef] [PubMed]

14.

M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express 17(20), 18330–18339 (2009). [CrossRef] [PubMed]

15.

J. B. Wu, B. B. Jin, Y. H. Xue, C. H. Zhang, H. Dai, L. B. Zhang, C. H. Cao, L. Kang, W. W. Xu, J. Chen, and P. H. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express 19(13), 12021–12026 (2011). [CrossRef] [PubMed]

16.

H. T. Chen, H. Yang, R. Singh, J. F. O’Hara, A. K. Azad, S. A. Trugman, Q. X. Jia, and A. J. Taylor, “Tuning the resonance in high-temperature superconducting terahertz metamaterials,” Phys. Rev. Lett. 105(24), 247402 (2010). [CrossRef] [PubMed]

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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]

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J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. 56(4), 554–557 (2009). [CrossRef]

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Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. 107(9), 093104 (2010). [CrossRef]

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J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express 16(19), 14390–14396 (2008). [CrossRef] [PubMed]

23.

D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express 19(10), 9968–9975 (2011). [CrossRef] [PubMed]

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J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett. 11(5), 2142–2144 (2011). [CrossRef] [PubMed]

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J. J. Li, C. M. Shah, W. Withayachumnankul, B. S.-Y. Ung, A. Mitchel, S. Sriram, M. Bhaskaran, S. J. Chang, and D. Abbott, “Mechanically tunable terahertz metamaterials,” Appl. Phys. Lett. 102(12), 121101 (2013). [CrossRef]

26.

G. Singh, E. Michel, C. Jelen, S. Slivken, J. Xu, P. Bove, I. Ferguson, and M. Razeghi, “Molecular-beam epitaxial growth of high quality InSb for p-i-n photodetectors,” J. Vac. Sci. Technol. B 13(2), 782–785 (1995). [CrossRef]

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33.

R. S. Penciu, M. Kafesaki, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Magnetic response of nanoscale left-handed metamaterials,” Phys. Rev. B 81(23), 235111 (2010). [CrossRef]

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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]

35.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B 76(7), 073101 (2007). [CrossRef]

36.

N. T. Tung, D. T. Viet, B. S. Tung, N. V. Hieu, P. Lievens, and V. D. Lam, “Broadband Negative Permeability by Hybridized Cut-Wire Pair Metamaterials,” Appl. Phys. Express 5(11), 112001 (2012). [CrossRef]

37.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

38.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon Press, 1984).

39.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]

40.

B. Sauivac, C. R. Simovski, and S. Tretyakov, “Double split-ring resonators: Analytical modeling and numerical simulations,” Electromagnetics 24(5), 317–338 (2004). [CrossRef]

OCIS Codes
(130.5990) Integrated optics : Semiconductors
(260.5740) Physical optics : Resonance
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Metamaterials

History
Original Manuscript: June 17, 2013
Revised Manuscript: August 2, 2013
Manuscript Accepted: August 7, 2013
Published: August 22, 2013

Citation
Jiawei Cong, Binfeng Yun, and Yiping Cui, "The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator," Opt. Express 21, 20363-20375 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-20363


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References

  1. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater.9(5), 407–412 (2010). [CrossRef] [PubMed]
  2. C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett.106(6), 067402 (2011). [CrossRef] [PubMed]
  3. J. W. Cong, B. F. Yun, and Y. P. Cui, “Negative-index metamaterial at visible frequencies based on high order plasmon resonance,” Appl. Opt.51(13), 2469–2476 (2012). [CrossRef] [PubMed]
  4. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011). [CrossRef] [PubMed]
  5. P. Dardano, M. Gagliardi, I. Rendina, S. Cabrini, and V. Mocella, “Ellipsometric determination of permittivity in a negative index photonic crystal metamaterial,” Light: Sci. Appl.1(12), e42 (2012). [CrossRef]
  6. V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett.102(13), 133902 (2009). [CrossRef] [PubMed]
  7. V. Mocella, P. Dardano, I. Rendina, and S. Cabrini, “An extraordinary directive radiation based on optical antimatter at near infrared,” Opt. Express18(24), 25068–25074 (2010). [CrossRef] [PubMed]
  8. C. H. Lin, R. L. Chern, and H. Y. Lin, “Polarization-independent broad-band nearly perfect absorbers in the visible regime,” Opt. Express19(2), 415–424 (2011). [CrossRef] [PubMed]
  9. C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett.37(3), 308–310 (2012). [CrossRef] [PubMed]
  10. C. Ma and Z. Liu, “Focusing light into deep subwavelength using metamaterial immersion lenses,” Opt. Express18(5), 4838–4844 (2010). [CrossRef] [PubMed]
  11. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun.3, 1205 (2012). [CrossRef] [PubMed]
  12. T. Roy, E. T. Rogers, and N. I. Zheludev, “Sub-wavelength focusing meta-lens,” Opt. Express21(6), 7577–7582 (2013). [CrossRef] [PubMed]
  13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express19(2), 1563–1568 (2011). [CrossRef] [PubMed]
  14. M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express17(20), 18330–18339 (2009). [CrossRef] [PubMed]
  15. J. B. Wu, B. B. Jin, Y. H. Xue, C. H. Zhang, H. Dai, L. B. Zhang, C. H. Cao, L. Kang, W. W. Xu, J. Chen, and P. H. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express19(13), 12021–12026 (2011). [CrossRef] [PubMed]
  16. H. T. Chen, H. Yang, R. Singh, J. F. O’Hara, A. K. Azad, S. A. Trugman, Q. X. Jia, and A. J. Taylor, “Tuning the resonance in high-temperature superconducting terahertz metamaterials,” Phys. Rev. Lett.105(24), 247402 (2010). [CrossRef] [PubMed]
  17. 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. Photonics2(5), 295–298 (2008). [CrossRef]
  18. N. H. Shen, M. Massaouti, M. Gokkavas, J. M. Manceau, E. Ozbay, M. Kafesaki, T. Koschny, S. Tzortzakis, and C. M. Soukoulis, “Optically implemented broadband blueshift switch in the terahertz regime,” Phys. Rev. Lett.106(3), 037403 (2011). [CrossRef] [PubMed]
  19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater.1(2), 128–132 (2013). [CrossRef]
  20. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt.56(4), 554–557 (2009). [CrossRef]
  21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys.107(9), 093104 (2010). [CrossRef]
  22. J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express16(19), 14390–14396 (2008). [CrossRef] [PubMed]
  23. D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express19(10), 9968–9975 (2011). [CrossRef] [PubMed]
  24. J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett.11(5), 2142–2144 (2011). [CrossRef] [PubMed]
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