  Herein, we investigate the symmetry-protected toroidal dipole resonances and
conditions of their excitation in a new type of electromagnetic metamaterials.
These metamaterials are all-dielectric planar periodic arrays of dielectric
disks disposed on a dielectric substrate. The elementary building blocks of the
array are trimers which are distributed in hexagonal unit supercells. The
highest geometrical symmetry of the unit supercell is C6v. The analysis is
fulfilled by using the representation theory of groups with the application of
the magnetic group theory, which is a new approach in solving such problems. We
have shown that to get access to the toroidal supermodes of the array, the unit
supercell symmetry must be broken twice: firstly, the C3v symmetry of the
trimer, and secondly, the C6v symmetry of the unit supercell needs to be
reduced. Selection rules for the symmetric and antisymmetric orders of the
toroidal dipole moments in the arrays are defined. In particular, we have shown
that with the reduction of the unit supercell symmetry to the C2v group, the
array exhibits the toroidal dipole resonance with an antitoroidic order. The
arrays with the lower Cs symmetry can provide the resonances with both toroidic
and antitoroidic orders. It is also shown that these arrays are always
polarization sensitive. Full-wave simulations and experiments confirm the
theoretical predictions. The suggested metamaterials can provide an enhanced
light-matter interaction due to the spatially and temporally confined light in
resonant systems with very high quality factors.
