  We present a new and robust approach for interpreting the physics of Fano
resonances in planar oligomer structures of both metallic and dielectric
nanoparticles. We reveal a key mechanism for Fano resonances by demonstrating
that such resonances can be generated purely from the interference of
nonorthogonal collective eigenmodes, which are clearly identified based on the
coupled-dipole approximation. We prove analytically a general theorem to
identify the number of collective eigenmodes that can be excited in ring-type
nanoparticle oligomers and further demonstrate that no dark mode excitation is
necessary for existence of Fano resonances in symmetric oligomers. As a
consequence, we unify the understanding of Fano resonances for both plasmonic
and all-dielectric oligomers.
