Metamaterials are man-made composites in mimicking natural crystals but with optical properties previously unattainable. These include negative refractive indices [1–3], extraordinary high-indices [4, 5], artificial magnetism [6–8] and invisibility cloaks [9–14]. In particular, in the technologically least explored regime of terahertz frequencies, metamaterials actually give us a plausible approach for wave manipulation. Active metamaterials [15–20], wave plates [21, 22], phase modulators [23], negative-index metamaterials [24–26], broadband invisibility cloak [27], perfect absorbers [28, 29] and high-index metamaterials [30], at THz frequencies are just some of the examples. There are also tremendous efforts in extending metamaterials into the nonlinear regime through embedding metallic resonating structures into a nonlinear dielectric host medium [31], inserting nonlinear electronic elements (e.g., diodes) into microwave metamaterials [32, 33] and inserting nonlinear optical materials [34] for optical frequencies into positions where local electric fields in the metamaterials are much enhanced.

Here, we would like to explore the possibility in using metamaterials to mimic in a similar way how a natural electro-optic crystal responds to an applied voltage with properties which can now be manipulated through careful design. Here, we are interested in mimicking the Kerr effect. In this case, an applied voltage across a piece of the electro-optic crystal distorts the crystal to introduce a change in refractive index, which is proportional to the square of the voltage. Accordingly, a phase difference results between the transmitted waves linearly polarized in two orthogonal directions [35]. Traditionally, the Kerr effect is relatively weak comparing to Pockels effect, making the Kerr effect less useful. For metamaterials at THz frequencies, we propose that a similar correspondence for the Kerr effect can be achieved using micromechanical systems [19, 20]. Here, we will show that by using high-index metamaterials with micromechanical systems, we can construct an actuated metamaterial exhibiting a behavior of phase difference induced by voltage similar to Kerr effect. In analogy to the original spirit of metamaterials, these artificial Kerr-media can give more significant Kerr effect which is unattainable by using natural Kerr crystals. We will show that the phase difference induced by the artificial Kerr effect exhibits anomalous dispersion near resonance as well. Interestingly, our approach is in a similar spirit in Ref. 36 in creating an artificial gyrotropic medium activated by an applied voltage instead of an actual static magnetic field. Instead of an artificial Faraday effect, here we are interested in an artificial Kerr effect.

Figure 1(a) shows the schematic of the artificial Kerr-medium based on micromechanical system. The medium is comprised of two similar kinds of meta-atoms, which are arranged orthogonally in a square lattice, as shown in Fig. 1(b). The meta-atoms along the x-axis are named with A, whereas those along the y-axis are named with B. Each meta-atom consists of two identical planar T-shaped metallic arms. The left arms of atoms A are anchored on the substrate, whereas all the rest metallic parts of the metamaterial can be moved by comb-drive actuators. Thus, the gap of atom A, g₁ can be dynamically tuned with a biasing voltage. In the simulation, a freestanding planar artificial Kerr-medium in air (without substrate) is chosen for simplicity without losing generality, as shown in Fig. 1(b). Here, the metal film thickness and the array period are fixed as t = 0.2 μm and p = 150 μm, respectively. The parameters of the atoms are fixed as a = b = 60 μm, w = 4 μm, and g₂ = 5 μm, respectively. Note that g₁ can be varied and has the same value as g₂ (always a constant) in the initial configuration before actuation. The square dashed line outlines the unit cell used in the simulation.

 

Fig. 1 (a) Schematic of the artificial Kerr-medium, and (b) the corresponding structure parameters. The metal film thickness and the array period are fixed as t = 0.2 μm and p = 150 μm, respectively. The parameters of the meta-atoms are fixed as a = b = 60 μm, w = 4 μm, and g₂ = 5 μm, respectively. Only g₁ will be changed.

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To investigate the THz electromagnetic response of the artificial Kerr-medium, simulations have been carried out using commercial software CST Microwave Studios. In the simulations, the metal is modeled as gold with a conductivity of $7\times {10}^6$S m⁻¹ [16], and the permittivity of the background materials is modeled as vacuum for simplicity. Normal incident light polarized along the x- and y-direction is considered. Figure 2(a) shows the transmittance spectra of the artificial Kerr-medium for both x-(dashed curve) and y-(solid curve) polarizations. Here, to distinguish the response of the two orthogonal polarizations, the gap g₁ / g₂ is set as 2.5 / 5 μm. The two transmission dips come from the excited resonance modes of the meta-atoms. For the initial configuration g₁ = g₂ = 5 μm, the light of the two orthogonal polarizations see no difference and the two transmission dips will overlap at the same frequency (1.309 THz). Figures 2(b) and 2(c) show the surface current profile of the artificial Kerr-medium for the two orthogonal polarizations at 1.309 THz, marked with a vertical dashed line in Fig. 2(a). The currents are confined independently on atoms A and B for the x and y-polarization respectively, showing that only atom A/B has been excited by the x/y-polarized incident light. Therefore, the transmission dip (associated to a resonance mode) for x/y-polarization can be tuned independently by adjusting g₁/g₂. Here, the difference between Fig. 2(b) and 2(c) except the orientation is not large. It is due to the broad nature of our resonance. Using the above approach, the phase difference for two orthogonal polarizations is individually modulated for simplicity of design. We note that it is also possible to control the properties (both amplitudes and phases) of the two polarizations by cross-coupling them with each other using plasmonic effects [37].

 

Fig. 2 (a) Simulated transmittance spectra for x- and y-polarizations, respectively. (b) For x- and (c) y-polarization, the simulated surface current patterns at 1.309 THz (marked by the vertical dashed line in Fig. 2(a)), respectively. The solid arrows show the polarization of the incident light. Here, g₁ is 2.5 μm.

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Here, we define$\Delta g_1=g_2-g_1$ to denote the extent of deviation from the initial configuration to be driven by the micromechanical system with bias voltage V. It has a characteristics$\Delta g_1=\alpha V^2$where $\alpha$is called the actuation coefficient which depends on the actual form factor of the system and is taken here as 0.04 μm V⁻² [19, 20]. Figure 3(a) shows the transmittance dip being red-shifted as V is gradually increasing. As atom A is very similar to the structure of a high-refractive-index material [30] which can change the propagation phase dramatically, we can accordingly change the phase difference between the two polarizations very efficiently through the actuation of g₁ in affecting the x-polarization without affecting the y-polarization as explained. Figure 3(b) shows the corresponding phase difference between the transmitted waves with x- and y-polarizations (the one in x-direction minus the one in y-direction), as V increases. Far away from the resonance frequency, the phase difference on both the left and right hand sides of the spectrum is positive and its valuegradually increases as V increases. In other words, this change in phase is related to the change of the refractive index of the metamaterial. Near the resonance, the phase difference becomes negative and its absolute value increases abruptly. This complicated evolution behavior of the phase difference is different from that of the traditional electric-optical crystals, e.g., Kerr media, because of the additional anomalous dispersion introduced by the resonating nature of the metamaterials. We will investigate these different regimes of phase shift in the following.

 

Fig. 3 (a) Simulated transmittance spectra for the x-polarization as a function of applied voltage. (b) The corresponding phase difference of the transmitted waves between x- and y-polarizations.

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Figure 4 shows the voltage (V) dependence of the phase difference between the two orthogonal polarizations at various frequencies. Figures 4(a) and 4(b) present the results for frequencies far away from the resonance. The phase difference is positive and increases approximately linearly with V² at small voltage. It increases faster with V² at a larger frequency on the left side of the resonance, whereas the phase difference increases slower with V² at a larger frequency, as shown in Fig. 4(b), on the right side of the resonance. On the other hand, there is a larger deviation of the curves from linearity with V² when it is working nearer to the resonance. It is significant that near resonance, the phase difference becomes negative with a much larger amplitude (see Fig. 4(c)). Actually, from the simulated lines we can find a frequency regime very near to the resonance so that the phase difference still varies linearly with V² when V is smaller than 6 V. In all the three regimes, the quadratic dependence of the phase difference against applied voltage shows that our metamaterial under actuation mimics a natural Kerr cell. We note that our artificial Kerr-medium is activated by an applied voltage to give the same effect of a conventional Kerr-medium but without actual static electric field inside the medium to invoke nonlinearity.

 

Fig. 4 Applied voltage dependence of the phase difference between the two orthogonal polarizations at different frequencies in the regimes far from resonance ((a) and (b)) and in the regime very near to the resonance ((c)). Solid lines correspond to the simulated results; dashed, dotted and dash-dot lines correspond to the analytic results using Lorentzian model.

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To have a more detailed understanding, we use a Lorentzian model to explain the phase difference with the change of V. The complex transmission coefficient t for x-polarization for our structure can be well fitted by the model using

The parameters$A$, and$T$are the absorption and transmission coefficients with $T={\left|t\right|}^2$. This is the same phase difference spectrum shown in Fig. 3(b). For convenience, we divide the phase difference spectra into two parts: near resonance and far from resonance. Far from resonance, the term $1/a-1/aT$ can be neglected due to large transmission. Thus we obtain $\Delta \phi \approx \beta A\alpha V^2/2\gamma$, i.e., $\Delta \phi \propto A\alpha V^2$. It clearly shows that the phase difference increases with V in a quadratic dependence for a specific frequency, mimicking the same response of a Kerr cell. The analytic results are shown in Figs. 4(a)-4(b). Note that the analytic results agree well with the simulated especially for smaller voltage. Furthermore, in this case, the absorption A increases as the frequency approaches the resonating frequency and therefore the slope of the phase difference against V² increases as well. This also agrees well with the simulated results shown in Figs. 4(a)-4(b). The discrepancy of the theory from the simulation at large voltages comes mainly from the linear relationship between$\Delta {\omega }_0$and$\Delta g_1$which is a good approximation at low voltage only. Actually, our metamaterial Kerr cell only requires a far smaller voltage than a conventional modulator with Kerr media to obtain the same phase difference. For example, at 1.1 THz, to get a phase difference of 1 degree a small bias voltage of 4.67 V in our design is needed. Although we do not have a THz Kerr cell for comparison, a Kerr cell made of Nitrobenzene (already a very strong Kerr medium for visible and infrared light) with a length of 2 cm (in the light propagating direction) and a transverse thickness of 0.1 mm [35], 1 degree of phase difference requires a biasing voltage of 21.5 V which is already 4 times larger than that of our case yet our sample has a thickness much thinner than a wavelength.

From the application point of view, a larger phase difference is desirable. For this purpose, we can choose multilayer structure or work near the resonating frequency. From Fig. 3(b), we can see that larger phase difference appears near the resonance, with a smaller transmission $T$ as a compromise. In this case, the term of $-1/aT$in Eq. (2) becomes dominant, leading to an anomalous phase difference $\Delta \phi \approx -(\alpha \beta A/2aT)V^2<0$. This agrees well with the results in Fig. 3(b). For example, when the operating frequency (1.300-1.304 THz) is chosen to be near the original resonance, $\Delta \phi$ changes very sharply with $V^2$, as shown in Fig. 4(c). Here, the analytic results also agree well with the simulated for smaller voltage. The same bias voltage of 4.67 V is already enough to make a phase difference of 21.6 degrees (equivalently 0.99°/V²) between the two polarizations.

Actually if the metamaterial atoms are geometrically scaled down (e.g. by 30 times), we can even obtain a smaller voltage to generate the same phase difference according to $\Delta g_1=\alpha V^2$. Here, the actuation coefficient$\alpha$of the micromechanical system is remained unchanged by assuming the gap size between comb drive fingers is kept constant [38]. Then, the voltage required to generate the same phase change is expected to be scaled down by $\sqrt{30}$ times. Full-wave simulations have been performed and we have found that at 20 THz far away from the resonance (now scaled to around 33THz), 1 degree of phase difference requires a bias voltage of 1.62 V. It is a bit larger than our expected value due to the larger absorption loss of gold at infrared frequencies. Nevertheless, it is smaller than the previous voltage in the THz case and even much smaller than the 21.5V for Nitrobenzene.

In conclusion, we have shown a micromechanical-actuated THz metamaterial to mimic strong Kerr effect by applying voltage across the metamaterial. In our design, the effective Kerr effect is achieved through a high index metamaterial where the index for one of the polarizations can be tuned with micromechanical system without affecting the other polarization. These investigations can be useful for metamaterials to mimic natural Kerr-media with well-controlled and stronger response and provide an efficient way to dynamically control electromagnetic waves.