  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, nonlinear switching, and sensing.
