Symmetry analysis of trimer-based all-dielectric metasurfaces with toroidal dipole modes

Abstract

Herein, we discuss the conditions for excitation of symmetry-protected toroidal dipole modes in all-dielectric metasurfaces composed of trimer or twin-trimer clusters of dielectric disks. Such metasurfaces permit enhanced light–matter interaction due to spatially confined light in resonant systems with a high-quality factor. To describe characteristics of toroidal modes existing in the clusters, we use the magnetic dipole moments approximation, group-theoretical methods, group representation theory, symmetry-adapted linear combination method, and circuit theory. To validate the obtained theoretical results, we fulfill both full-wave numerical simulations and microwave experiments. In particular, we have shown that the toroidal dipole mode appears as a quasi-dark state of the trimer. It can be excited in the metasurface by the field of a linearly polarized wave, providing the symmetry of the trimer is properly reduced. In the metasurface, the properties of the toroidal dipole mode are determined primarily by the parameters of a single trimer and are not a consequence of the periodicity of the array. The coupling of the toroidal dipole modes in the twin-trimers can appear in both bonding and anti-bonding fashion resulting in different orders of the net toroidal dipole moment. Due to the unique field configuration of these modes, the proposed metasurfaces can be considered as a platform for efficient light–matter interaction for enhanced absorption, non-linear switching, and sensing.

Toroidic and antitoroidic orders in hexagonal arrays of dielectric trimers: magnetic group approach

“…Thus, we have the following interaction hierarchy: the eigenmodes of three individual disks are coupled into the trimer eigenmodes, which are then coupled into the hexagonal cluster supermode. We are interested in a particular eigenmode of the trimer, which exhibits a significant contribution from the toroidal dipole moment [62,66,67]. Then we consider the coupling of these toroidal eigenmodes within the hexagonal cluster which can be realized as a bonding (antitoroidal) or antibonding (toroidal) state.…”

“…Since the center of the equilateral triangle of the underlying trimer and the center of the minicell are the same, the minicell geometry corresponds to the C 3v symmetry. To get access to the toroidal dipole mode, the minicell symmetry must be reduced [67]. This symmetry reduction can be realized in several ways.…”

“…In the eigenmode of unperturbed trimer, the toroidal dipole moment T has only single component T z [67]. In a fixed moment of time, the normalized vector T can acquire only two states being oriented either "up" (T z = +1) or "down" (T z = −1).…”

“…This mechanism is studied both theoretically and experimentally in Refs. [62,66,67], so we omit the details here (for a brief description, see Appendix D). In the present consideration, we concentrate on solving the second subproblem, i.e., defining the selection rules for symmetries of the hexagonal supercell which allow or forbid excitation of toroidal dipole modes in the metamaterial by the field of a normally incident linearly polarized wave.…”

“…Due to low inherent losses in constituent materials, the symmetry-protected toroidal modes in alldielectric metamaterials demonstrate a very high quality factor (high-Q) resonant response accompanied by the near-surface confinement of the strong electromagnetic field [64][65][66]. Remarkably, the conditions for the appearance of toroidal dipole modes in cluster-based alldielectric metamaterials can be expressed in the explicit electromagnetic formulation and described in the grouptheoretical language [67].…”