Decoupling crossover in asymmetric broadside coupled split-ring resonators at terahertz frequencies

We investigate the electromagnetic response of asymmetric broadside coupled split-ring resonators (ABC-SRRs) as a function of the relative in-plane displacement between the two component SRRs. The asymmetry is defined as the difference in the capacitive gap widths ($\Delta g$) between the two resonators comprising a coupled unit. We characterize the response of ABC-SRRs both numerically and experimentally via terahertz time-domain spectroscopy. As with symmetric BC-SRRs ($\Delta g=0$ μm), a large redshift in the LC resonance is observed with increasing displacement, resulting from changes in the capacitive and inductive coupling. However, for ABC-SRRs, in-plane shifting between the two resonators by more than 0.375 $L_o$ ($L_o$ $=$ SRR sidelength) results in a transition to a response with two resonant modes, associated with decoupling in the ABC-SRRs. For increasing $\Delta g$, the decoupling transition begins at the same relative shift (0.375 $L_o$), though with an increase in the oscillator strength of the new mode. This strongly contrasts with symmetric BC-SRRs, which present only one resonance for shifts up to 0.75 $L_o$. Since all BC-SRRs are effectively asymmetric when placed on a substrate, an understanding of ABC-SRR behavior is essential for a complete understanding of BC-SRR based metamaterials.

Figure 1

(a) Photograph of unshifted ABC-SRR array. The front gap is $g$ $=$ 16 μm, while the back gap is $g_{\mathrm{back}}$ $=$ 2 μm. (b) Photograph of shifted ABC-SRR array. The dimensions are the same as in (a). (c) Top-down schematic of ABC-SRR unit cell, showing the top ring and relevant dimensions. (d) Perspective view of ABC-SRR unit cell showing the direction of lateral shifting and the polarization and direction of the terahertz signal used to excite the MMs.

Figure 2

(a) Experimental transmission spectra for single-layer SRRs for varying capacitative gap widths. (b) Experimental transmission spectra for ABC-SRRs with varying gap differences. In this unshifted case, only the fundamental electrical resonance of the ABC-SRR is excited. For comparison, the black dashed curve shows the experimental transmission spectrum for a single, uncoupled SRR with $g$ $=$ 2 μm. (c) Experimental transmission curves for the corresponding 25-μm shifted ABC-SRRs. Here, the electrical resonance has redshifted, consistent with Ref. 17. However, a new mode appears at higher frequencies and is strongly dependent on the asymmetry between the SRRs. For comparison, the black dashed curve shows the experimental transmission spectrum for a single, uncoupled SRR with $g$ $=$ 2 μm. Plots are scaled in dB for clarity, as some resonances are very wide when plotted on a linear scale.

Figure 3

Simulation results showing transmission (in dB) curves vs shift for (a) 2-μm, (b) 4-μm, (c) 10-μm, and (d) 14-μm gap differences. Shift distances are normalized to the side length of the SRRs, $L_o$. Asterisks denote the experimental $-$7 dB points and are included to show correspondence of simulation with experiment.

Figure 4

(a) Simulation results showing transmission curves (in dB) vs lateral shift for an ABC-SRR with a 14-μm gap difference in a rectangular unit cell lattice ($P_y$ $=$ 2 $P_x$ $=$ 120 μm). All other dimensions are the same as in the previous structures. The dotted lines at ${\omega }_1$ and ${\omega }_2$ correspond to the uncoupled resonance frequencies of the smaller gap SRR and the larger gap SRR, respectively. Shift distances are normalized to the side length of the SRR, $L_o$. In this large unit cell lattice the SRRs can be shifted far enough apart to completely decouple from each other and from SRRs in the nearest neighbor cells. As the SRRs are shifted, the two resonances trend to the uncoupled resonance frequencies. (b) Resonance surface current density in arbitrary units in both SRRs for $L_{\mathrm{shift}}$ $=$ 0 $L_o$, showing that at the 1.3 THz resonance the SRRs are equally excited and acting as one coupled resonator. (c) Resonance surface current density in arbitrary units in both SRRs for $L_{\mathrm{shift}}$ $=$ 1.5 $L_o$ and $f$ $=$ 0.8 THz, showing that the response is dominated by currents in the small gap SRR. (d) Resonance surface current density in arbitrary units in both SRRs for $L_{\mathrm{shift}}$ $=$ 1.5 $L_o$ and $f$ $=$ 1.1 THz, showing that the response is dominated by currents in the large gap SRR. Thus, for large shifts, the SRRs are excited separately, as uncoupled resonators.