Structurally Tunable Nonlinear Terahertz Metamaterials using Broadside Coupled Split Ring Resonators

We present an experimental and numerical study of a terahertz metamaterial with a nonlinear response that is controllable via the relative structural positioning of two stacked split ring resonator arrays. The first array is fabricated on an n-doped GaAs substrate and the second array is fabricated vertically above the first using a polyimide spacer layer. Due to GaAs carrier dynamics, the on-resonance terahertz transmission at 0.4 THz varies in a nonlinear manner with incident terahertz power. The second resonator layer dampens this nonlinear response. In samples where the two layers are aligned, the resonance disappears and total nonlinear modulation of the on-resonance transmission decreases. The nonlinear modulation is restored in samples where an alignment offset is imposed between the two resonator arrays. Structurally tunable metamaterials can therefore act as a design template for tunable nonlinear THz devices by controlling the coupling of confined electric fields to nonlinear phenomena in a complex material substrate or inclusion.

and design novel nonlinear optical elements for THz radiation. 8,9 Combining THz excitation with Xray probes allows for time-resolved measurements of nonlinear structural changes in complex materials. 10 Within the field of nonlinear THz science, plasmonic metamaterials and metasurfaces (MMs) play a critical role. MMs are engineered composites with optical properties that are determined by the geometry and layout of the component sub-wavelength inclusions. 11 MMs have been applied across the electromagnetic spectrum and may be used to precisely control light through manipulation of phase, intensity, and polarization. 12 . This unprecedented level of control over light has led to notable MM demonstrations of left-handed materials 13 , electromagnetic cloaking 14 , and perfect absorption 15,16 .
Additionally, MMs are interesting from a materials science standpoint due to their enhancement of lightmatter interaction 2,7 . This interaction can be seen in a magnified nonlinear MM response, making MMs an ideal platform to study THz nonlinear behavior and design nonlinear devices.
Nonlinear MM devices have been created at microwave and terahertz frequencies through a variety of methods including incorporating nonlinear lumped circuit elements into the design [17][18][19] , and by using the inherent nonlinear response of the subwavelength inclusions that make up the MM 20 . Another approach used in nonlinear MMs devices makes use of resonant inclusions, such as the split ring resonator (SRR), which confine electric fields to localized regions in the unit cell. Resonant field confinement (FC) enhances the local peak electric field intensity, and may excite a nonlinear response in the MM inclusions or substrate. 7,21 Many demonstrations of nonlinear MMs based on FC exist, including devices that couple confined fields to charge carrier dynamics in VO2, GaAs, and InAs 21,22 , and devices that couple fields to superconducting phase transitions in both low and high Tc superconductors. [23][24][25] Dynamic control over the MM response dramatically extends device capability; and integration of control and tuning is an active area of study for linear and nonlinear MM devices. Two of the most commonly used methods for electrical control of MM properties are via modulation of substrate conductivity 26 and structural design or actuation, for instance via microelectrical mechanical (MEMS) based devices 27,28 .
We recently have shown that the nonlinear response of a THz MM can be modulated through control of substrate conductivity 9 . Yet to date there has been no report on how the nonlinear response of MMs responds to structural manipulation of the unit cell.
In this paper, we report an experimental study outlining how the nonlinear response of a THz MM can be controlled solely through manipulation of the structural design of the MM inclusions. Our MM design is based on the common broadside-coupled SRR (BCSRR) 29 and is shown schematically in figure 1a. This design is composed of two layered SRR arrays, oriented to maximize electromagnetic interactions between the two arrays. More details on device design are discussed below. We show that by altering the lateral positions of the two component split ring resonators in the unit cell, one can control the field confinement (FC) in the capacitive gaps of the resonators and thus control the coupling of incident THz fields to the nonlinear carrier dynamics of an n-doped GaAs substrate. When the two resonator layers are aligned to overlap as shown in figure 1b, very little nonlinear behavior is seen in the device response at the design frequency of 0.4 THz. When a lateral offset is placed between the upper and lower SRR array (shown in Figure 1c) the FC in the capacitive gap of the lower resonator increases, enabling a nonlinear response from the carrier dynamics of the n-doped epilayer at a relative low incident THz power. The nonlinear carrier dynamics result in a drop in the epilayer conductivity, turning onthe lowest order SRR resonance at 0.4 THz. The resonance is thus highly dependent on incident THz power and on resonance THz transmission is modulated by approximately 3dB as the incident THz power is increased. Below, we present nonlinear THz spectroscopy measurements to characterize device response and numerical simulations to provide insight into the physical mechanisms for device behavior. The MM studied in this work consists of two stacked, planar arrays of gold SRRs as shown in Figure 1.
The lower array is fabricated directly onto a GaAs substrate with a 1 µm thick n-doped GaAs epilayer (n = 2x10 16 cm -3 ) using standard photolithographic techniques. The SRRs of this layer are inter-connected with metallic wires for use in applying an electrical bias (not used in this study). A 2 µm thick polyimide spacer layer is then deposited on the epilayer followed by a second SRR array. The orientation of the second array is rotated by 180 o relative to the first array, which maximizes electromagnetic coupling between the two layers. 29 The dimensions of the component SRRs are chosen to maximize coupling between the two resonator layers and to place the device resonance near the peak of the TPFG THz signal at ~0. To characterize the nonlinear response of the two samples, THz pulses with field strengths between 50 and 400 kV/cm were generated using TPFG in LiNO3 and focused onto the MM at normal incidence with the THz electric field polarized perpendicular to the SRR capacitive gaps, as shown in figure 1a. The  For the sample with Lshift = 0 µm ( fig. 2a) only a small nonlinear response can be seen in the data. As the THz field strength is increased from 50 -400 kV/cm, the overall transmission through the sample slightly increases, but no clear resonance is seen, regardless of incident field strength. Figure 2b shows noticeably different behavior exhibited by the structure in which the SRRs are offset. The MM now exhibits a strong resonance near approximately 0.4 THz with a larger nonlinear modulation. As the THz field strength is increased from 50 -400 kV/cm, the on-resonance transmission is modulated by approximately 3dB.
Thus, the presence of a strong nonlinear modulation in the MM resonance can be controlled solely by structural positioning of the two MM layers. As another comparison of the stark difference in modulation range, figure 3 plots the transmission at 0.4THz for both samples as a function of incident THz field strength. Not only is the modulation range noticeably increased in the case for Lshift = 48 µm, the direction of modulation is also reversed compared to the unshifted structure. Using numerical simulations, we investigate the physical origins of the nonlinear response of the BC-SRR MM discussed above. In order to show the mode structure of the MM, we simulate the THz transmission spectra of the BC-SRR structure for Lshift = 0 and 48 µm using commercial solvers based on finite difference time domain techniques. 30 The SRR gold patterning is modeled as a lossy metal, while the GaAs substrate is modeled with a 1 µm lossy semiconductor epilayer on a loss-free semi-insulating GaAs substrate. These simulated spectra are shown together in Figure 4a. Due to the limited frequency resolution in the tilted pulse front THz-TDS system, the resonances in simulation are narrower than in the experimental results.
The blue curve in figure 4a shows the spectra for the sample with Lshift = 0 µm. The incident THz electric field polarized perpendicular to the capacitive gap of the SRRs excites two modes. The resonance at 0.73THz (resonance A) corresponds to the electrically active coupled mode of the BC-SRR. 31 The frequency position and oscillator strength of mode A has been shown in previous work to be highly dependent on the electromagnetic coupling between the two component SRRs. The frequency position of mode A can be approximately described using an LC oscillator model: where L is the BC-SRR total inductance, and C the BC-SRR total capacitance. 32,33 .
The red curve in figure 4a shows  For Lshift = 48 µm (fig 2b), resonance A is clearly visible in the experimental results since the overall oscillator strength, and thus resonance depth, of resonance A is now much greater. Here, the decrease in carrier mobility decreases the on-resonance loss of the BC-SRR structure, leading to a stronger resonance and explaining the difference in modulation direction discussed above in figure 3. The where Qo is the peak charge across the SRR capacitive gap. With no lateral offset, the out of phase electric fields from the two resonators superimpose and distructively interfere, leading to a low net electric field strength in the unit cell of the MM. As the two resonators are laterally offset, the resonant electric fields of the two SRRs no longer superimpose spatially, leading to less destructive interference and a higher net electric field strength inside the MM unit cell. The end result is an increase in the strength of the local electric fields in the lower SRR gap region. This results in a stronger resonance for the sample with Lshift = 48 µm and a larger nonlinear modulation of the resonance as the incident THz field strength is increased.
We can confirm this explanation of tunable local FC by performing simulations of the local electric field distributions within the MM unit cell. Figure 5 shows