Optical Magnetic Multipolar Resonances in Large Dynamic Metamolecules

TL;DR

This study develops and applies a full T-matrix framework to map high-order optical magnetic resonances in large, randomly packed dynamic metamolecules formed from a hydrogel core and plasmonic beads. By combining FDTD simulations with T-matrix analyses up to the magnetic hexadecapole ($n=4$), the authors show that increasing bead size and overall structure size, and decreasing inter-bead gaps, activate higher-order magnetic modes and induce strong intermodal mixing, with observable Fano-like interference in angular scattering. They validate these findings against experimental extinction data for gold DMMs and provide a practical recipe to identify magnetic multipoles via forward and backward scattering, polarization control, and directional measurements. The work offers a detailed, transferable methodology for designing and verifying high-order optical magnetic resonances in complex nanostructures, with potential implications for negative-index materials, sensing, and nonlinear optics.

Abstract

Dynamic metamolecules (DMMs) are composed of a dielectric core made of hydrogel surrounded by randomly-packed plasmonic beads that can display magnetic resonances when excited by light at optical frequencies. Their optical properties can be controlled by controlling their core diameter through temperature variations. We have recently shown that DMMs display strong optical magnetism, including magnetic dipole and magnetic quadrupole resonances, offering significant potential for novel applications. Here, we use a T-matrix approach to characterize the magnetic multipole resonance modes of model metamolecules and explore their presence in experimental data. We show that high-order multipole resonances become prominent as the bead size and the overall structure sizes are increased, and when the the inter-bead gap is decreased. In this limit, mode mixing among high-order magnetic multipole modes also become significant, particularly in the directional scattering spectra. We discuss trends in magnetic scattering observed in both experiments and simulations, and provide suggestions for experimental design and verification of high-order optical magnetic resonances in the forward or backward scattering spectra. In addition, angular scattering of higher-order magnetic modes can display Fano-like interference patterns that should be experimentally detectable.

Figures (29)

• Figure 1: a) Simulation region set up in Lumerical FDTD package with a cubic field monitor. b) The cubic shell field monitors set up in the Lumerical FDTD package. The grey box in (a) represents Lumerical's built-in TFSF source and the yellow boxes in both figures represent field monitors.
• Figure 2: (a-d) Temperature-dependent experimental extinction spectra of gold DMMs with nanobeads of average number $N$ and size $D$; (a) $N=57\pm3$, $D=35\pm3$ nm, (b) $N=61 \pm 4$, $D=45\pm2$ nm (c) $N=59 \pm 4$, $D=49\pm2$ nm, and (d) $N=41 \pm 4$, $D=60\pm3$ nm. Other structural parameters are summarized in Table S1 and Figures S1 and S2. The dashed arrow in each figure shows the direction of increasing temperature from 20 °C to 55 °C, resulting in shrinking values of the core diameter, $Z$.
• Figure 3: Scattering cross-section ($C_{scat}$), calculated based on the full T-matrix with the order $n=4$ (modal sum, light blue circles), along with its relevant modal contributions; the sum of all electric modes; ($E_{tot}$, orange), magnetic dipole ($H_{dip}$, green), magnetic quadrupole ($H_{quad}$, purple), magnetic octopole ($H_{oct}$, red), and magnetic hexadecapole ($H_{hex}$, blue) modes, for a $Z=415$ nm diameter MM with $N=126$ gold nanobeads of $D=70$ nm diameter. The solid line is the total scattering cross-section directly obtained from FDTD simulations. The simulated structure is shown in the inset.
• Figure 4: Vector plots of far field displacement currents contributing to the total magnetic scattering modes ($H_{tot}$, top row) as well as the contribution the magnetic dipole ($H_{dip}$, row 2), quadrupole ($H_{quad}$, row 3), and octupole mode ($H_{oct}$, row 4), respectively, at the a-d) magnetic dipole resonance ($\lambda_{Hdip}=1280$ nm), e-h) magnetic quadrupole resonance ($\lambda_{Hquad}=1020$ nm), and i-l) magnetic octupole resonance ($\lambda_{Hoct}=860$ nm) wavelengths, for the structure shown in Figure \ref{['Fig:Csct']}. For each condition, the figure on the left shows the plane of $\vec{H}$ and $\vec{k}$ (side view) and the figure on the right shows the plane of $\vec{E}$ and $\vec{k}$ (top view). The length of the arrows in all figures are plotted on the same scale and are proportional to the value of the displacement current.
• Figure 5: The amplitude of the magnetic elements of the full T-matrix for the structure shown in Figure \ref{['Fig:Csct']} at the (a) magnetic dipole ($\lambda_{Hdip}=1280$ nm), (b) magnetic quadrupole ($\lambda_{Hquad}=1020$ nm), (c) magnetic octupole ($\lambda_{Hoct}=860$ nm), and (d) the magnetic hexadecapole ($\lambda_{Hhex}=780$ nm) resonance wavelengths. The labels indicate $(n, m)$ where $n$ is the multipole mode order and $m$ is the submode ($-n \leq m \leq n$). All elements with magnitude $\leq 0.01$ are shown as white squares.