Direct observation of defect modes in molecular aggregate analogs

In this work we investigate defect modes localized at the ends and within the bulk of 1D metamaterial analogs of molecular aggregates. The study is undertaken in the microwave regime, where the cm scale of the metamaterial analog provides an opportunity to directly probe the modes deep within their near fields in ways not easily achieved in molecular systems. To demonstrate the power of this approach we compare our observations to predictions from a simple Su, Schrieffer, and Heeger (SSH) model and find good qualitative agreement.

Figure 1

SSH structure. (a) A schematic of a chain of eight resonators, each with a resonance frequency $\omega$. Two resonators within a ‘cell’ are coupled together via an intracell coupling term $k_A$ (red ellipse), with each cell in the chain being coupled to neighboring cells via a second intercell coupling term $k_B$. When $k_A\ne k_B$ the system deviates from the monoatomic, single banded, case (b) and a band gap will emerge in the dispersion of the chain. (c) In the dimer limit when $k_B=0$ and $k_A\ne 0$ the chain consists of four isolated dimers. Each dimer will support two modes with frequencies that will lie above and below $\omega$. The difference in frequency between them is determined by the magnitude of $k_A$. (d) In the other dimer limit, when $k_A=0$ and $k_B\ne 0$, there are three isolated dimers, and in this case the difference in frequency between their modes is determined by $k_B$. In addition, there are two isolated resonators at each end of the chain. These isolated resonators will have resonance frequencies of $\omega$, which lie between those of the dimers, i.e., within the band gap. It is not just in the dimer limit that these ‘edge’ modes appear, rather it is only necessary that $k_A<k_B$.

Figure 2

Boundary mode eigenvectors. Top: Calculated eigenvector of a fifty resonator long chain in which the relative strengths of $k_A$ and $k_B$ are switched midway along the chain such that the resonator at the boundary is strongly coupled to both its nearest neighbors. This boundary mode has an asymmetric form. Bottom: The same as the top figure, but in this case the position of the switch occurs such that the resonator at the boundary is weakly coupled to its neighbors. In this case the boundary mode is symmetric. Boundaries in both cases are marked by the vertical black dashed line.

Figure 3

The metamolecule. The metamolecules used in this study are based on the split-ring resonator, a loop of wire with a split that acts as a resonant circuit. Dark regions correspond to copper-coated areas of a dielectric substrate, light regions correspond to the bare substrate. The structure comprises two double split rings so as to remove any bianisotropy. $r=16$ mm, $w_1=w_2=1.2\mathrm{mm}$, and $g=1.5$ mm.

Figure 4

Chain formations. Exploded schematics of the four chain formations where A refers to copper (orange) and a dielectric spacer (white) while B refers to copper and three dielectric spacers. The top row shows two units cells each of the ABAB (left) and BABA (right) formations. The bottom row shows the formations of the two defect structures going from AB to BA periodicity (left) and from BA to AB periodicity (right).

Figure 5

Scanning methods. A schematic of the two methods used to detect localized modes within a chain of resonators. In both probe 1 is the exciting antenna and probe 2 is the receiving antenna. Black arrows indicate which probes are scanned as part of the measurement. (Top) The scanning method for probing edge modes uses a loop antenna (probe 1) placed at a split on the outer ring of the first resonant element so as to excite the modes of the chain. The loop was orientated in the plane of the resonator. A second loop antenna (probe 2) was scanned along the edge of the chain with the same loop orientation. The transmission between the two antennas was measured. (Bottom) The scanning method for probing localized modes within the bulk of the chain. Two loop antennas (probes 1 and 2) are scanned in tandem along opposite edges of the chain and the transmission between them measured.

Figure 6

Termination defect modes. (a) The magnitude of the transmission between the exciting and probing antennas as a function of frequency and distance along a 60 mm section at one end of a chain with $k_B>k_A$. Collective modes of the entire chain are evident as oscillations in the transmission for frequencies above and below the band gap (between 1.38 and 1.65 GHz). No modes are evident within the band gap. (b) The same as (a) but with $k_A>k_B$. In this case a localized mode confined to the end of the chain within the band gap occurs at 1.52 GHz. In both plots light green is used to show areas of high transmission while dark blue shows areas of low transmission. For both a change from maximum to minimum represents a $100\%$ change in magnitude.

Figure 7

Boundary defect modes. (a) The magnitude of the transmission between the exciting and probing antennas as a function of frequency and distance along a chain that supports a symmetric localized mode at the boundary between subchains with $k_A>k_B$ and $k_A<k_B$. The mode is observed as a localized oscillation in the transmission at the boundary at a frequency of 1.42 GHz. (b) The same as (a) but for a chain that supports an asymmetric localized mode. In this case the localized mode occurs at a frequency of 1.526 GHz. In both plots, light green is used to show areas of high transmission while dark blue shows areas of low transmission. For (a), a change from maximum to minimum represents a $100\%$ change in magnitude while for (b) the same color shift is only a $62\%$ change in magnitude. (c) The absolute value of the change in the transmission for the two modes (symmetric at 1.42 GHz, asymmetric at 1.526 GHz) demonstrating the different symmetries. [Note that background slopes in the data for (c) have been removed for clarity.]