FIG. 1: (a) Geometry of an equilateral triangular lattice of resonant atoms. (b) The corresponding reciprocal lattice, shown with open circles, and some high-symmetry points and directions in the first Brillouin zone. The dashed lines in (b) illustrate constructing the first Brillouin zone as the Wigner-Seitz cell of the reciprocal lattice. to study systems that are much smaller than λ . For the same reason, we will assume the incident field to be uniform throughout the lattice, E$_{in}$ ( r ) ≡ E$_{in}$ , although our formalism is not limited to this case. As we have proven in [1, 2] for linear (1D) systems of interacting atoms, such an approach is indeed of a great help for understanding the locsitons’ behavior. We consider an equilateral triangular lattice (also known as a hexagonal lattice) of atoms, which has a six-fold rotation symmetry and belongs to the plane symmetry group (wallpaper group) p6m . [See Fig. 1(a).] This lattice type has a remarkable property of providing the most close-packed configuration of identical circular objects in a plane. Each atom in the lattice has six neighbors at the distances of l a . The “second layer” of neighbors is removed by √$_{3}$$_{l}$$_{a}$ or 2 l a ; considering the fast decrease of the dipole-dipole interaction force with the distance ( ∝ 1 /r $^{3}$), the NNA which ignores interaction with the second and further layers of neighbors is expected to work well. As in the case of 1D arrays of atoms [1, 2], two major cases can be studied separately, where the incident laser field E$_{in}$ lies in plane (the “ ‖ ” case) or is normal to the plane of the lattice (the “ ⊥ ” case). The linear optical response in the case of any other incident polarization can be obtained using a superposition of these two configurations. Of the two major cases, the “ ‖ ” configuration is by far more interesting, exhibiting richer locsiton behavior and differing significantly from the 1D case due to the crucial in-plane anisotropy of the dipole-dipole interaction. It is also easier to implement, as the laser beam may be incident at the normal to the lattice of atoms, with the incident field being virtually uniform 8