We experimentally and numerically study the nature of coupling between laterally paired terahertz metamaterial split-ring resonators. Coupling is shown to modify the inductive–capacitive (LC) resonances resulting in either red or blue-shifting. Results indicate that tuning of the electric and magnetic coupling parameters may be accomplished not by changing the orientation or density of SRRs, but by a design modification at the unit cell level. These experiments illustrate additional degrees of freedom in tuning the electromagnetic response, which offers a path to more robust metamaterial designs.
© 2011 OSA
Recent developments in metamaterials (MMs) have demonstrated the possibility of manipulating light in compact and highly integrated micro and nanoscale structures [1–5]. These MM-based devices hold great promise for realizing next-generation photonic technologies applicable to the microwave, terahertz and the optical regimes. The operation of most MM devices relies on strategies employed to control the fundamental resonances in MM oscillators, which are often termed as split ring resonators (SRRs) [5–8]. These resonances accompany the high electric or magnetic polarizability of SRRs, resulting in extreme values of the electric permittivity (ε) and magnetic permeability (μ), hallmark properties of MMs. Recent work has studied lateral coupling between neighboring SRRs and its effect on these resonances [9–11]. Recent work in microwave MMs has shown that the relative orientation between coupled SRRs determines the nature of coupling [12,13]. Effects like resonance line narrowing, resonance mode splitting [14,15] and a classical analogue of electromagnetically induced transparency  were observed in coupled SRRs. Most of these effects were the result of changing the density and/or orientation of coupled SRRs. In one case, the SRRs were also modified by the addition of extra gaps . While these results nicely illustrated the increased complexity of coupled SRRs in MMs, there remains much opportunity to better understand the deliberate tuning of SRR coupling for maximum practical benefit. In terms of MM functionality, increased coupling leads to both opportunities and complications. For example, increasing the SRR density generally strengthens the response of a MM. But it also strengthens coupling effects, which may lead to superradiant damping, actually reducing the range of available ε and μ . Thus we become compelled to investigate more comprehensive MM design strategies, ones that simultaneously leverage the isolated SRR response as well as its coupling to neighbors. In this work, we show that coupling can be strongly modified by selective design at the unit cell level. In particular we show that opposing coupling effects may be balanced such that the isolated SRR response is approached, despite extremely close proximity between SRRs. We utilize a combination of numerical simulations and terahertz frequency measurements in this work. These offer a convenient platform for such studies since sample fabrication is straightforward, and terahertz time-domain spectroscopy (THz-TDS) provides a phase-coherent characterization technique.
Five sets of metamaterial samples were fabricated on an undoped GaAs wafer as shown in Figs. 1(a)-(f) . In samples MM1, MM2 and MM3 the distance between the two SRRs in the unit cell, s, was varied from 2 to 6 to 10 µm. In sample array MM4 and MM5, s is fixed at 2 μm but the SRRs gaps are oppositely offset by d from the center. The unit cell periodicity (or equivalently the SRR density) in all MM samples was kept constant at Px = 90 µm and Py = 52 µm. All samples were fabricated using photolithography, followed by e-beam deposition of 10 nm of titanium, and then 200 nm of gold. Optical images of the SRRs are shown in Fig. 1 along with detailed geometrical parameters. The samples were characterized in transmission using THz-TDS in the confocal geometry . The incident THz beam was linearly polarized with the electric field oriented parallel to the gap-bearing side, as in Fig. 1(f). The MM samples and a piece of bare GaAs identical to the MM substrates were all measured in the time domain, the last to serve as a transmission reference. All measurements were done at room temperature and in a dry atmosphere to mitigate water absorption. The time-domain data was converted into frequency-dependent amplitude and phase spectra via Fourier transformation. Transmission spectra (S21) normalized to the bare GaAs substrate were obtained for all the samples. Figure 2(a) shows the measured spectra for samples MM1, MM2 and MM3.
As the SRR spacing is reduced, we observe a 25 GHz blue shift in the LC resonance from MM1 to MM2 and an additional 35 GHz blue shift from MM2 to MM3. Only a small change is observed in the depth of the resonant transmission dip, along with a minor increase in the resonance quality factor (Q). Figure 2(b) shows the corresponding simulated data for samples MM1, MM2 and MM3 as well as the resonance frequency of the isolated ring (0.46 THz). All the measurement features have been reproduced well in the simulated spectrum. Figure 3(a) shows the response of samples MM3, MM4, and MM5. As the SRR gaps are displaced 4 µm from sample MM3 to MM4, the resonance red shifts 40 GHz and the transmitted field magnitude increases from 0.12 to 0.22. With the additional 6 µm offset in MM5 the resonance red shifts 30 GHz further and the field magnitude increases to 0.42. Thus, in this set of measurements we observe both red shifting and weakening of the response. The simulation data in Fig. 3(b) are in reasonable agreement with our measurements.
To understand these results, we note that the LC resonance mode arises from electric currents oscillating around the full circumference of the SRR loop. These are excited by the incident electric field due to the asymmetry of the SRR . The position of the anticipated LC resonance frequency can usually be described by a simple expression, where L and C are the effective inductance and capacitance of the individual SRRs. In the presence of coupling between neighboring resonators the situation is more complex.
As in previous work [10,11,17], the dipole-dipole interaction model is generally sufficient to explain the coupling effects. Each oscillating SRR may be thought to behave like two dipoles, one electric dipole oriented across the gap, and one magnetic dipole oriented out of the plane of the SRR and located somewhere inside its perimeter. In order to identify these dipoles, we simulated the electric and magnetic field distributions using a numerical electromagnetic solver . In all cases the magnetic field distributions for the coupled resonators remain similar to the uncoupled SRRs. This suggests that magnetic dipole coupling is a relatively weak mechanism in this work. This result is consistent with previous observations of SRRs with facing gaps studied in the near-infrared and microwave [10,11,17].
However, the electric fields within the SRR pairs clearly go through significant changes for the various coupled resonators. The numerically simulated electric field distributions at the LC resonance for three cases – MM1, MM3 and MM5 – are shown in the Figs. 4(a) , 4(b) and 4(c), respectively. When the SRRs are far apart and SRR gaps are aligned, as in case of MM1, fringing fields around the gap-bearing side contribute significantly to the overall capacitance of the individual resonators. As the separation between SRRs is reduced these fringing fields increasingly couple the individual responses of the rings. Coulomb repulsion occurs through the fringing fields between like charges in the paired rings, thereby opposing charge accumulation near the gap.
The resulting decrease in electric flux density reduces the fringing capacitance in both rings, as shown in Figs. 4(a) and 4(b). In this case, therefore, increased coupling gives rise to a blue shift of the resonance frequency (Fig. 2(a)). This is also understood as two parallel side-by-side electric dipoles brought into close proximity. As they become closer the coupled system resonance is driven to higher frequency. An important observation in Fig. 2 is that there is little change in depth of the LC resonance and only a minor increase in the quality factor (Q), suggesting that the increased SRR coupling has little effect on the excitation of the SRRs by the incoming wave or their radiation damping. This is contrast to a previous result in the near-infrared where closer SRR spacing always resulted in a smaller extinction cross section per SRR .
When the gaps are displaced towards opposite corners, as in MM4 and MM5, the Coulomb interaction between charges in paired rings becomes attractive, thereby increasing the net electric flux density, as shown in Fig. 4(c). This produces additional capacitance in both rings, causing a red shift in the resonance, Fig. 3(a). This is also understood as two parallel side-by-side electric dipoles being displaced from one another longitudinally. As the longitudinal displacement increases, the dipoles approach an end-to-end configuration, which is energetically weaker, thereby red-shifting the resonance. We note that this mechanism opposes the effect of brining the SRRs into close proximity thereby cancelling some of the previous effects of coupling. The observed red shift is also accompanied by a weakening of the resonance, which results in enhanced transmission amplitude. This is not likely a significant result of SRR coupling because external electric fields are known to more weakly excite the LC resonance in SRRs with displaced gaps . This interpretation is supported by the fact that samples MM1-MM3 exhibit little to no resonance weakening. It is expected that other SRR designs may exist that preserve strong external excitation while still favorably tuning SRR coupling. Thus, our experiments reveal two apparently independent and simultaneous effects: strong capacitive (electric dipole) coupling between SRRs causes resonance frequency shifts, and symmetry breaking in the individual rings causes variation of the response strength.
Previously it was observed that coupling effects arising in SRR arrays prevent the macroscopic metamaterial response from being a simple function of unit cell polarizability and density . Thus high-density, high-polarizability unit cells may not create the strongest metamaterial response once coupling is accounted for . Our experiments show that the metamaterial unit cell may be engineered to create coupling effects that enhance, reverse, or compensate responses due to other design features, such as orientation or packing density. Of course this adds complication to metamaterial design but it also suggests a more comprehensive metamaterial design path where coupling works in concert with the high polarizability of unit cells, allowing metamaterials to achieve larger macroscopic effective parameters. In addition, it suggests a method to use coupling to stabilize a metamaterial resonance frequency against design variances. Here, properly designed unit cells may use coupling to compensate the functional effects of other design features when design variances occur, such is the case with fabrication tolerances. This would be particularly useful in high-frequency metamaterials where fabrication consistency is much more difficult.
To illustrate, if an uncoupled SRR array was undercut during fabrication it is expected to reduce the ring capacitance and drive up the resonance frequency. With a properly designed unit cell, this same undercut could simultaneously cause an opposite effect through the SRR coupling, thereby preserving the original resonance frequency as much as possible. This idea was tested by simulating an uncoupled SRR array and an array of coupled SRRs very similar to MM3. Undercuts of both 0.5 um and 1.0 um were simulated. Figure 5a shows that a 1.0 um undercut on the uncoupled array results in a 16 GHz blue shift to the resonance, a shift of 3.5% of the intended resonance. In the coupled SRRs, undercuts from 0 to 1 um resulted in a (worst case) 6 GHz red shift, which is within about 1.1% of the intended resonance. Additional design iterations could likely improve this.
In summary, we report passive tuning of the strength and frequency of the LC resonance by exploiting the lateral coupling mechanism in terahertz MMs. Due to the changes in fringing capacitance between the resonators, a blue shift or red shift of the resonance is observed depending upon the spacing between SRR gaps and the placement of the SRR gaps. Resonance strength appears to be independently affected in the asymmetric gap case due to weaker excitation from the external electric field. All the measured features were reproduced well by the numerical calculations and explained in terms of electric dipole coupling effects. The work shows how coupling may be designed in parallel with the polarizability and densities of the unit cells in order to more comprehensively or robustly design the macroscopic metamaterial response.
We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD Program for this work. We gratefully acknowledge the cleanroom facilities of Center for Integrated NanoTechnologies (CINT) located at Sandia National Laboratory for the fabrication of the metamaterial samples.
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