  We investigate the collective plasmonic modes of metal nano-particles in
periodic two-dimensional (2D) arrays within a point-dipole description. As an
open system, the full-dynamic dispersion relations of the 2D arrays are
obtained through an efficient method which gives an effective polarizability
describing the collective response of a system. Both the dispersion relations
and mode qualities are simultaneously related to the imaginary part of the
effective polarizability, which has contributions from the single-particle
response as well as the inter-particle coupling. The transversal long-range
dipolar interaction is dominated by a wave term together with a purely
geometrical constant representing the static geometrical contribution to
resonant frequencies. As concrete examples, we considered small Ag spheres
arranged in a square lattice. We find that inside the light-cone, the
transverse quasi-mode has a reasonably high mode quality while the two in-plane
modes show significant radiation damping. Near the light-line, we observe
strong coupling with free photons for the bands of the transverse mode and the
transversal in-plane mode, and the longitudinal in-plane mode exhibits a
negative group-velocity inside the light-cone. Vanishing group velocities in
the light-cone for all the quasi-modes are found to be intrinsic properties of
the 2D metal nano-sphere dense arrays.
