Optical characteristics associated with magnetic resonance in toroidal metamaterials of vertically coupled plasmonic nanodisks

Magnetic resonance in metamaterials is basically formed by an optically induced current loop that, when arranged in a specific order, can exhibit exotic features. In this work, we numerically study the magnetic-resonanceassociated optical characteristics in strongly coupled plasmonic nanodisks, which are shown to support various magnetic resonances for vertical polarization (E-field out of the disk plane). To identify these modes, which are essentially relevant to the optically induced current loops, we calculate the optical spectrum, the radiation powers from multipole moments, and the resonant magnetic field pattern. It is shown that the lowest-order resonances are sequentially magnetic dipole mode, magnetic quadrupole mode, and toroidal mode. The surface charge density, the induced current density, and the magnetic near-field are carefully examined for these modes, which show that these modes are quite differently located in both spectrum and real space, bearing distinct symmetry nature. In view of that, we introduce a concentric and nonconcentric air hole in the disks, as a small perturbation, to tune their resonance. It is found that both the radii of the air hole and its position are capable to shift the resonant frequencies. However, their impacts are interestingly distinct with respect to the mode order. These results can provide a useful way to adjust the optical properties of the metamaterials constructed by double disks.