  Plasmonic nanoarrays which support collective surface lattice resonances
(SLRs) have become an exciting frontier in plasmonics. Compared with the
localized surface plasmon resonance (LSPR) in individual particles, these
collective modes have appealing advantages such as angle-dependent dispersions
and much narrower linewidths. Here, we investigate systematically how the
geometry of the lattice affects the SLRs supported by metallic nanoparticles.
We present a general theoretical framework from which the various SLR modes of
a given geometry can be straightforwardly obtained by a simple comparison of
the diffractive order (DO) vectors and orientation of the nanoparticle dipole
given by the polarization of the incident field. Our experimental measurements
show that while square, hexagonal, rectangular, honeycomb and Lieb lattice
arrays have similar spectra near the $\Gamma$-point ($k=0$), they have
remarkably different SLR dispersions. Furthermore, their dispersions are highly
dependent on the polarization. Numerical simulations are performed to elucidate
the field profiles of the different modes. Our findings extend the diversity of
SLRs in plasmonic nanoparticle arrays, and the theoretical framework provides a
simple model for interpreting the SLRs features, and vice versa, for designing
the geometrical patterns.
