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Abstract
Electromagnetic force actuated plasmonic nonlinear metamaterials have attracted a great deal of interest from the scientific community over the past several years, owing to the abundant interactions between the electromagnetically induced Ampère's force and the stored mechanical force within the meta-atoms. Despite this interest, a comprehensive study of such metamaterials is still lacking, especially for the nonlinear coupling states analysis. Here we fill this gap by extensively studying the physics of electromagnetic force actuated plasmonic nonlinear metamaterials and presenting a number of new significant findings. Our study will help physicists and engineers to better understand this hot topic and stimulate rapid developments of this promising nonlinear metamaterials.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metamaterials have been one of the most exciting research topics since the first reports on experimental demonstrations of negative refraction in 2000 [1] and 2001 [2]. Many kinds of metamaterials that are known today can operate over a broad frequency domain, ranging from low-frequency [3], microwave [4], and terahertz [5] up to infrared and even visible domains [6]. It is not surprising, therefore, that metamaterials have found many useful applications in such devices as invisibility cloaks [7], perfect absorbers [8], superlenses [9–11], superscatterers [12], and antennas [13–15], etc. But in spite of impressing success, metamaterial science still faces a number of fundamental and engineering challenges, including high absorption losses, narrow operating bandwidths, and non-tunable frequency bands. These have been addressed by different research groups, which came up with a number of performance-improved metamaterial designs with reduced losses [16], expanded operating bandwidth [17], and controllable frequency band [8,18].
Recent years, nonlinear metamaterials have shown the capacity to have controllable operating frequency and bandwidth, which can be tuned through the adjustment of strength of incident electromagnetic field [19,20]. This makes nonlinear metamaterials very useful for such applications as harmonic generation [21], wave mixing [22], phase conjugation [23], as well as amplification and squeezing of quantum noise [24]. The design of nonlinear metamaterials can include diodes and transistors [25,26], liquid crystals [27], graphene layers [28], and superconducting materials [29]. These materials are often incorporated into the tissue of individual meta-atoms, making it possible to alter the light–matter interactions within them. Most recently, due to the developments of micromechanically tunable metamaterials [30], the nonlinear interactions between electromagnetic wave induced forces and stored force within the meta-atoms expand the nonlinear metamaterial realization methods [31–37]. For examples, the nonlinear response of magnetoelastic metamaterials comes from the interaction of the electromagnetic induced Ampère’s force acting on meta-atoms and the elastic force induced in the substrate or meta-atoms themselves [31–33]. Although recently reported exciting works on the forces interactions have done a lot of analysis, there is still a lack of relevant detailed analysis for special states such as inaccessible stable state and compression limit. Here we provide a detailed theoretical analysis on the nonlinear coupling properties of the electromagnetic force actuated plasmonic nonlinear metamaterials, then provide a full understanding on such special states, and finally give a schematic experimental demonstration.
2. Theoretical analysis
Figure 1(a) shows one meta-atom of electromagnetic force actuated plasmonic nonlinear metamaterial considered in this paper. The meta-atom consists of two split-ring resonators (SRRs) deposited on an elastic substrate [32] of a finite stiffness (for example, with a stiffness coefficient kstiff = 0.44 mN/m). When an electromagnetic wave with its H-field along the axis of the SRRs incidents on this structure, the currents induced in the two rings are in-phase [32]. As a result, an attraction force (Ampère’s force) appears between the rings, bringing them closer to each other. Here we provide a detailed derivation of this Ampère’s force which can be found in Appendix, using both classical electrodynamics and magnetostatics.
Fig. 1 Description and characterization of electromagnetic force induced plasmonic nonlinear metamaterial. (a) Meta-atom in the form of two SRRs, (b) Ampère’s forces (curves 1–6) acting between them for different strengths of incident electromagnetic waves, and elastic force induced in the substrate (solid line). The horizontal axis in (b) is normalized by the radius r0 of the SRRs.
Based on the theoretical analysis in the Appendix section, we calculate Ampère’s force between two rings with increased and decreased strength of incident electromagnetic wave. The results of our calculation are shown in Fig. 1(b). The radius, width, and thickness of the rings are chosen to be 4, 0.4, and 0.018 mm, and the initial distance between the rings b0 = 1.2 mm. For these parameters, the resonance frequency of the SRR is ω0 = 2π × 5.565 GHz. In Fig. 1(b) it was therefore assumed that ω = 2π × 5.5 GHz. The elastic substrate is compressed by Ampère’s force according to the Hooke’s law Fs(b) = k(b-b0); the elastic force arising in the substrate is shown in Fig. 1(b) by solid line.
By analyzing the force acting on our metamaterial structure, one can see that the structure will be at equilibrium when Ampère’s force is equal to the elastic force. However, not all of the equilibrium points are stable. Here we divide the equilibrium points into three categories: a stable state, a critical state, and an inaccessible stable state, based on three possible situations. It should be noted that the definitions of states in this paper differ from the definitions in the available literature [31–33], but they have the same meanings.
3. Nonlinear coupling states study
Let us first assume that the incident magnetic field is increased from zero to a sufficiently large value and then decreased. There is no electromagnetic wave initially, so Ampère’s force and elastic force are both zero. As the magnetic field increases, the SRR structure remains in its stable state around the initial distance b, which is equal to 0.3 (normalized by r0) in this paper, in the right side of the curves. We call this state the right stable state or the ground state and mark it by red circles. When the amplitude of Ampère’s force increases to the condition of curve 5 in Fig. 1(b), there is a point of tangency in the right side (marked by the red cross). Although Ampère’s force is equal to the elastic force, this point is not an actual stable state. So such point is defined as a critical state and the corresponding strength of the magnetic field achieves the threshold point. In this state, the SRR structure jumps to an actual stable state upon a very small increase of the magnetic field strength.
According to Fig. 1(b), the SRR structure will be compressed suddenly and reach a new stable state, which is marked by red dashed circles on the left side of the curves (this is the left stable state). On another hand, when the incident magnetic field is decreased, SRR structure remains on the left side, as shown by black dots on curves 3, 4, 5, and 6 in Fig. 1(b). With the magnetic field decreased further, the structure comes to its critical state (marked by the black cross) again and then jumps to its right stable state (black circles).
Based on the above analysis, one can now study how the properties of the stable state vary upon an increase or decrease of the magnetic field and/or frequency of incident electromagnetic wave. In Figs. 2(a) and 2(b), we give the 2-D maps about b at increasing and decreasing magnetic field strengths and frequencies, respectively. Figures 2(c) and 2(d) describe the relationship between the distance b and the intensity of magnetic field at several certain frequencies, and the relationship between the distance b and frequency at several certain magnetic field strengths. Combining Figs. 2(a) to 2(c), it can be found that the hysteresis-like phenomenon occurs only in a range below a fixed frequency. This fixed frequency is the initial resonant frequency of the SRR structure. This phenomenon also occurs in the case of fixed H-field and tuned frequency, shown in Fig. 2(d). More interesting phenomena in this case worthy of our analysis, which will be described below.
Fig. 2 Theoretical responses of electromagnetic force induced plasmonic nonlinear metamaterial as functions of magnetic field strength and frequency in incident electromagnetic wave. Two-D maps of distance b with (a) increasing H and f and (b) decreasing H and f. (c) The variation of distance b with increasing and decreasing H at different incident frequencies, blue: 5.4 GHz, black: 5.45 GHz, red: 5.5 GHz, and green: 5.54 GHz. (d) The variation of distance b with increasing and decreasing f at different magnetic field strengths, red: 0.6 A/m, blue: 0.2 A/m, green: 0.025 A/m, and black: 0.002 A/m.
We now fix the strength of the incident magnetic field and analyze the performance of the SRR structure for increasing and decreasing exciting frequencies. When increasing the exciting frequency, the peaks of Ampère’s force shift in the positive direction of the b-axis and their amplitudes reduce, as shown in Figs. 3(a) and 3(b) for two different strengths of the incident magnetic field. As the curves of the Ampère’s force are shifted to the right, their amplitudes decrease nonlinearly so that the curve that fits the peak decay is concave. In the process of decreasing the exciting frequency, the elastic force curve shows that the elastic substrate cannot be compressed to zero thickness, so in this paper we assume that the minimal thickness of the elastic substrate is 0.1. As shown in Figs. 3(a) and 3(b), this makes the elastic force curves stopped by this point when b is reduced to 0.1, and we draw a vertical curve to represent the acting force related to the Ampère’s force.
Fig. 3 Performances of SRR structure under the acting force induced by external electromagnetic field. Ampère’s force curves at excitation frequencies ranging from 4.3 to 5.5 GHz with a step of 0.2 GHz for magnetic field strengths of (a) 0.035 A/m and (b) 0.025 A/m. (c) The zoom-in plot of panel (a) with more Ampère’s force curves. (d) The schematically representation for explaining the inaccessible stable state. (e) and (f) Variations of magnetization M with increasing and decreasing excitation frequency for magnetic field strengths 0.035 A/m and 0.025 A/m.
Spectacular phenomena originate from the considered responses of distance b and magnetization M (M = (π/r0a2)·(I/b), where a is the lattice constant of the SRR array [32]). Moreover, as was concluded from Figs. 3(a) and 3(b), we can find how the curve that fits the peaks behave in two cases. In one case, when the incident magnetic field is stronger enough (we assume it to be 0.035 A/m), there is no intersection point between the elastic force curve and the curve fitting Ampère’s force peaks, as shown in Fig. 3(a). In this situation the structure can be compressed to its minimal thickness easily and a peak appears at this limit, as shown Figs. 3(e). Let us consider the situation at bmin separately. As can be concluded from the zoom-in plot around bmin shown in Fig. 3(c), the elastic force is fixed at bmin but Ampère’s force varies with excitation frequency. If Ampère’s force is greater than the elastic force when excitation frequency is decreased (beginning from the cyan curve, f = 4.339 GHz), the structure remains in its stable state. However, according to the equation M = (π/r0a2)·(I/b), the magnetization is determined by current I, and so does Ampère’s force. Thus, when the rest of parameters are fixed and only the excitation frequency is varied, Ampère’s force and magnetization depend solely on current I and behave similar to this current at the minimal b. As a consequence, a peak appears in Fig. 3(e). Specially, three stable points—shown in Fig. 3(c) by cyan, blue, and red circles— correspond to the three jumping points in Fig. 3(e).
In another case, the incident magnetic field is in the range shown Fig. 2(b) (the blue part below the white dashed line), the structure cannot be compressed to its limit and there are intersection points between the fitted curve and the elastic force curve, as shown in Fig. 3(b). It means that there is only one stable state (ground or right stable state) in some range of frequency, where the peak value of Ampère’s force is smaller than the elastic force. In the process of further increasing the excitation frequency, the structure will jump to its left stable state. The trace of equilibrium points corresponding to this process is shown in Fig. 3(f) by blue circles. However, in the process of decreasing incident frequency, there is no distance b making Ampère’s force equal to the elastic force other than that corresponding to the ground state. So if the excitation frequency is reduced to this range, the SRR structure jumps into its ground state. As the excitation frequency is decreased further, Ampère’s force becomes greater than the elastic force again, the structure can come into a special stable state if some external factor is given. The respective magnetization responses for increasing and decreasing excitation frequencies are shown in Fig. 3(f).
We define such special stable state mentioned previously as the third equilibrium state, namely, the inaccessible stable state. It means that the structure cannot get to this theoretical state, unless the external force is given or temporarily increase magnetic field, although Ampère’s force is equal to the elastic force in this state. As shown in Fig. 3(d), if a temporarily external force between F1 and F2 is given, the structure comes to circle 1, which is called as inaccessible state 1. If the external is over F2, the structure will jump from ground state to inaccessible state 2. In another case, temporarily increasing the intensity makes the structure jump to state 3 and then be stable at inaccessible state 2.
4. Experimental demonstration
Finally, to verify the proposed electromagnetic force actuated plasmonic nonlinear metamaterial, we carry out experiments to investigate the nonlinear response of the SRR structure. The measurement setup is schematically shown in Fig. 4(a). In our experimental demonstrations, the SRRs of thicknesses of 0.018 mm are initially fabricated on a 0.02-mm-thick flexible printed circuit (FPC) substrate. The fabricated sample is shown in Fig. 4(b). The size of the SRR is the same as taken in the previous section for theoretical analysis. Then we cut two of the rings and hang them in a C-band rectangular waveguide with a Teflon rod, as shown in Fig. 4(b). This system has a similar function to investigate the nonlinear response of SRR, as the elastic force is achieved by the gravity component along the central axis of the SRR.
Fig. 4 Experimental demonstration of nonlinear response of electromagnetic force induced plasmonic nonlinear metamaterial. (a) Schematic view of the measurement setup. (b) Rectangular waveguide with magnetoelastic metamaterial inside it. (c) 2D plot for the metamaterial transmission spectra under different incident power levels from 10 dBm to 30 dBm. The inset of panel (c) is the corresponding numerical results. (d) The resonance frequency shift and the corresponding acting force with increasing powers ranged from 10 dBm to 30 dBm.
The fabricated SRRs were initially placed inside a C-band rectangular waveguide with a Teflon rod. There was an open window on the top face of the waveguide, as shown in Fig. 4(b), which was used to properly position the Teflon rod and the SRRs. Then a metallic plate of the window size was used to cover the window and prevent scattering from it. Two C-band rectangular waveguide to coaxial transitions were connected to the waveguide and to the coaxial cables of Agilent N5230A vector network analyzer, as shown in Fig. 4(a). The applied power levels were changed by setting the vector network analyzer in the allowed range (up to 5 dBm) and by adding a low noise power amplifier (Mini-circuits ZVE-3W-183, Gain: 35 dB), and capture the obtained transmissions. The entire system was calibrated using the thru-reflect-line (TRL) calibration procedure to eliminate the system errors before the measurements [8, 38].
The measured transmissions of the SRR under different incident power levels are shown in Fig. 4(c), and the collected resonance frequency shift is presented in Fig. 4(d). The corresponding numerical results are plotted as an inset of Fig. 4(c) and the theoretical fitting at the same structural parameters and the same experimental condition is plotted in Fig. 4(d). As can be seen, when the incident power is increased from 10 to 30 dBm, the shifted range of the transmission dip (the resonance frequency) is about 40 MHz. The resonance frequency of the SRR structure changes with an increase or decrease of the incident magnetic field due to changes of the distance between the SRRs, which directly verified the proposed electromagnetic force actuated plasmonic nonlinear metamaterial. From the theoretical fitting result, it is seen that the measured resonance frequency shift feature is matching to the theoretical results. Finally, we calculated the Ampère’s force for each sample point of resonance frequency shift and obtained the red curve in Fig. 4(d). It can be seen that the induced magnetization and electromagnetic force of the sample with maximum input power are respectively 0.39 A/m and 34 pN.
5. Conclusions
The conducted theoretical and experimental studies have revealed the detailed interactions between Ampère’s force and the elastic force under different intensities and frequencies of incident electromagnetic radiation. They also demonstrated unusual nonlinear responses of the electromagnetic force actuated plasmonic nonlinear metamaterial. Our results will help engineers of nonlinear metamaterials to better understand the origin of electromagnetic force actuated nonlinear response. As a concluding remark, we would like to note that the SRRs used in this paper can be replaced by other kind of magnetic-resonance metamaterial unit cells. In line with recent developments in the fast-growing field of optomechanics, we next intend to investigate nonlinear electromagneto-mechanical metamaterials driven into mechanical resonance/oscillation state, focusing on their mechanical-modulation properties.
Appendix
As shown in Fig. 1(a), an electromagnetic wave incident on the SRR induces an electrodynamic potential, which can be written in the form [39]
here μ0 = 4π × 107 H/m is the vacuum permeability, ω and H0 are the frequency and magnetic field strength of the incident electromagnetic wave, and r0 is the radius of the SRR. The complete impedance equation for the SRR unit is given bywhere b is the distance between the rings. Here equivalent inductance L, capacitance C, and resistance R can be determined using the equivalent circuit method [39], as detailed below,where c is the width of the SRR, and a = r0-c/2, b = r0 + c/2. And A(x) is defined ashere Sn(x) and Jn(x) are the nth order Struve and Bessel functions. The inductance L = 11.494 nH was calculated using Wolfram Research Mathematica based on the size chosen in this paper.The equivalent capacitance was calculated using its definition C = πr0Cpul, where
is the capacitance per unit length of the SRR structure, andAnd the resistance was calculated as
where δ = 6.62/f1/2 (in cm) is the approximate value of the skin depth. Notice that the radiation resistance of the SRR is not considered here for simplicity.Then the induced current I(b) can be calculated from Eq. 2 and Ampère’s force can be obtained using the Biot-Savart law. Let us introduce the Cartesian coordinates as shown in Fig. 1(a). The magnetic field at point P generated by the lower SRR with current I is given by
where Idl is the current element on the ring and R is the radius vector from the current element to point P. Equation (8) can be conveniently rewritten using spherical coordinates asAccording to this equation, By = 0 whereas the other two components of the magnetic field are given by whereare the complete elliptic integrals of the second and first kind andWe next consider the upper SRR of the same radius r0 and current I. The structural element of the upper ring will experience Ampère’s force
The force acting on the entire ring is given by the line integralNote that the second term in Eq. (13) does not contribute to the force between the two rings of the meta-atom.In the above theoretical analysis, we have assumed that the currents induced on the rings are uniform, because the frequency of the incident electromagnetic wave is much higher than the mechanical eigen-frequency of the SRR. It should also be noted that these currents depend on the distance b between the two rings. Using Eq. (2), we can write the current in the form
where ω0 = (LC)1/2 is the resonance frequency of the SRR. If the radius of the rings is fixed, ω0 scales in inverse proportion to b [39].Funding
National Natural Science Foundation of China (No. 61371047, 61601093, 61701082, 51777168, and 61701303); Guangdong Provincial Science and Technology Planning Program (Industrial High-Tech Field) of China (No. 2016A010101036); Sichuan Provincial Science and Technology Planning Program (Technology Supporting Plan) of China (No.2016GZ0061); Natural Science Foundation of Shanghai (17ZR1414300); and Shanghai Pujiang Program (17PJ1404100); Science of the Russian Federation Scholarship (MD-1975.2016.1) of the President of the Russian Federation for Young Scientists.
References and links
1. 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(18), 4184–4187 (2000). [CrossRef] [PubMed]
2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
3. Y. J. Yoo, H. Y. Zheng, Y. J. Kim, J. Y. Rhee, J.-H. Kang, K. W. Kim, H. Cheong, Y. H. Kim, and Y. P. Lee, “Flexible and elastic metamaterial absorber for low frequency, based on small-size unit cell,” Appl. Phys. Lett. 105(4), 041902 (2014). [CrossRef]
4. J. Wang, Z. Xu, Z. Yu, X. Wei, Y. Yang, J. Wang, and S. Qu, “Experimental realization of all-dielectric composite cubes/rods left-handed metamaterial,” J. Appl. Phys. 109(8), 084918 (2011). [CrossRef]
5. J. Liu, K. Zhang, X. Liu, Z. Zhang, Z. Jin, X. He, and G. Ma, “Switchable metamaterial for enhancing and localizing electromagnetic field at terahertz band,” Opt. Express 25(13), 13944–13952 (2017). [CrossRef] [PubMed]
6. M. Giloan and R. Gutt, “Optical negative index metamaterial based on hexagonal arrays of metallic meta-atoms with threefold rotational symmetry,” J. Opt. Soc. Am. B 33(1), 27–34 (2016). [CrossRef]
7. Z. Su, J. Yin, K. Song, Q. Lei, and X. Zhao, “Electrically controllable soft optical cloak based on gold nanorod fluids with epsilon-near-zero characteristic,” Opt. Express 24(6), 6021–6033 (2016). [CrossRef] [PubMed]
8. Y. Huang, G. Wen, W. Zhu, J. Li, L.-M. Si, and M. Premaratne, “Experimental demonstration of a magnetically tunable ferrite based metamaterial absorber,” Opt. Express 22(13), 16408–16417 (2014). [CrossRef] [PubMed]
9. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008). [CrossRef] [PubMed]
10. H. X. Xu, G. M. Wang, M. Q. Qi, L. Li, and T. J. Cui, “Three-dimensional super lens composed of fractal left-handed materials,” Adv. Opt. Mater. 1(7), 495–502 (2013). [CrossRef]
11. H. X. Xu, G. M. Wang, K. Ma, and T. J. Cui, “Superscatterer illusions without using complementary media,” Adv. Opt. Mater. 2(6), 572–580 (2014). [CrossRef]
12. H. X. Xu, G. M. Wang, M. Q. Qi, Y. Y. Lv, and X. Gao, “Metamaterial lens made of fully printed resonant-type negative-refractive index transmission lines,” Appl. Phys. Lett. 102(19), 193502 (2013). [CrossRef]
13. E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater. 10(3), 216–222 (2011). [CrossRef] [PubMed]
14. Y. Huang, L. Yang, J. Li, Y. Wang, and G. Wen, “Polarization conversion of metasurface for the application of wide band low-profile circular polarization slot antenna,” Appl. Phys. Lett. 109(5), 054101 (2016). [CrossRef]
15. H. X. Xu, G. M. Wang, M. Q. Qi, and Z. M. Xu, “A metamaterial antenna with frequency-scanning omnidirectional radiation patterns,” Appl. Phys. Lett. 101(17), 173501 (2012). [CrossRef]
16. A. Boltasseva and H. A. Atwater, “Materials science. Low-loss plasmonic metamaterials,” Science 331(6015), 290–291 (2011). [CrossRef] [PubMed]
17. G. Deng, J. Yang, and Z. Yin, “Broadband terahertz metamaterial absorber based on tantalum nitride,” Appl. Opt. 56(9), 2449–2454 (2017). [CrossRef] [PubMed]
18. Y. Zhao, B. Li, C. Lan, K. Bi, and Z. Qu, “Tunable silicon-based all-dielectric metamaterials with strontium titanate thin film in terahertz range,” Opt. Express 25(18), 22158–22163 (2017). [CrossRef] [PubMed]
19. M. Lapine, I. Shadrivov, and Y. Kivshar, “Wide-band negative permeability of nonlinear metamaterials,” Sci. Rep. 2(1), 1–4 (2012). [CrossRef] [PubMed]
20. M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Colloquium: Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093–1123 (2014). [CrossRef]
21. W. Wu, L. Fan, W. Zang, X. Yang, P. Zhan, Z. Chen, and Z. Wang, “Second harmonic generation enhancement from a nonlinear nanocrystal integrated hyperbolic metamaterial cavity,” Opt. Express 25(18), 21342–21348 (2017). [CrossRef] [PubMed]
22. S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2012). [CrossRef] [PubMed]
23. K. Kim, “Enhanced optical phase conjugation in nonlinear metamaterials,” Opt. Express 22(S7), A1744–A1752 (2014). [CrossRef] [PubMed]
24. M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable josephson metamaterial,” Nat. Phys. 4(12), 929–931 (2008). [CrossRef]
25. D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95(8), 084102 (2009). [CrossRef]
26. J. P. Barrett, A. R. Katko, and S. A. Cummer, “Transistor-based metamaterials with dynamically tunable nonlinear susceptibility,” Appl. Phys. Lett. 109(6), 061901 (2016). [CrossRef]
27. A. Minovich, J. Farnell, D. N. Neshev, I. McKerracher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100(12), 121113 (2012). [CrossRef]
28. A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. De Angelis, S. A. Boden, E. Di Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012). [CrossRef]
29. C. Kurter, P. Tassin, A. P. Zhuravel, L. Zhang, T. Koschny, A. V. Ustinov, C. M. Soukoulis, and S. M. Anlage, “Switching nonlinearity in a superconductor-enhanced metamaterial,” Appl. Phys. Lett. 100(12), 121906 (2012). [CrossRef]
30. A. Q. Liu, W. M. Zhu, D. P. Tsai, and N. I. Zheludev, “Micromachined tunable metamaterials: a review,” J. Opt. 14(11), 114009 (2012). [CrossRef]
31. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1(1), 138 (2011). [CrossRef] [PubMed]
32. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2012). [CrossRef] [PubMed]
33. M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87(23), 235126 (2013). [CrossRef]
34. A. P. Slobozhanyuk, M. Lapine, D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, R. C. McPhedran, and P. A. Belov, “Flexible helices for nonlinear metamaterials,” Adv. Mater. 25(25), 3409–3412 (2013). [CrossRef] [PubMed]
35. J. Y. Ou, E. Plum, J. Zhang, and N. I. Zheludev, “Giant nonlinearity of an optically reconfigurable plasmonic metamaterial,” Adv. Mater. 28(4), 729–733 (2016). [CrossRef] [PubMed]
36. N. I. Zheludev and E. Plum, “Reconfigurable nanomechanical photonic metamaterials,” Nat. Nanotechnol. 11(1), 16–22 (2016). [CrossRef] [PubMed]
37. M. Liu, D. A. Powell, R. Guo, I. V. Shadrivov, and Y. S. Kivshar, “Polarization-induced chirality in metamaterials via optomechanical interaction,” Adv. Opt. Mater. 5(16), 1600760 (2017). [CrossRef]
38. Y. Huang, Y. Tian, G. Wen, and W. Zhu, “Experimental study of absorption band controllable planar metamaterial absorber using asymmetrical snowflake-shaped configuration,” J. Opt. 15(5), 055104 (2013). [CrossRef]
39. R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (New Jersey: Wiley, 2007).
