  We present a comprehensive study of the polarization and spatial coherence
properties of the lasing modes supported by a 4-fold symmetric plasmonic
lattice. By modifying only, the scattering properties of the individual
particles while keeping the lattice geometry constant, we are able to
distinguish the scattering induced effects from the lattice geometry induced
effects. Customized interferometric measurements reveal that the lasing
emission undergoes a drastic change from 1D to 2D spatial coherence with
increasing particle size, accompanied with dramatic changes in the far field
polarization and beaming properties. By utilizing T-matrix scattering
simulations, we reveal the physical mechanism governing this transition. In
particular, we find that there exists increased radiative coupling in the
diagonal directions at the plane of the lattice when the particle diameter is
increased. Finally, we demonstrate that the x- and y-polarized (degenerate)
lasing modes become phase locked with sufficiently large particles.
