  Toroidal dipole moments, which consist of enclosed circulating currents
aligned along paths within a torus shape, can be experimentally achieved in
metamaterials using various geometrical configurations. Here, we investigate
the excitation of toroidal dipole moments in a core-shell nanosphere composed
of dispersive materials within the framework of the Lorenz-Mie theory. By
isolating the contributions related to volumetric oscillations of electric
charges from those of transverse and radial current oscillations, we derive
closed-form analytic expressions for the scattering coefficients associated
with electric and magnetic toroidal dipole moments. These analytic expressions,
obtained beyond the Rayleigh scattering regime and previously concealed within
the Lorenz-Mie theory, agree with the Cartesian toroidal dipoles determined
from the radial and transverse current oscillations. By exploring the
resonances in these calculated coefficients, we demonstrate that the electric
and magnetic toroidal dipole excitations can be manipulated and tuned for near-
and far-field applications involving plasmonic core-shell nanoparticles and
engineered metamaterials. We believe that our analytic results could shed light
on some of the extraordinary effects obtained in Lorenz-Mie theory that depend
on the interference of multipole coefficients, such as toroidal dipole-induced
transparency, Fano resonances and superscattering of light.
