(c) E$_{in}$ ‖ ΓK, q ‖ ΓK ‖ E$_{in}$ [solid line in Fig. 3(b)], cos q 2 = 1 16   1 ± √ 1 + 32 ( 4 + 3 δ + i δ$_{LL}$ )   , (25) − 1 33 96 ≤ δ δ$_{LL}$ ≤ 1 2 3 ; (26) cos $_{q}$√$_{3}$ 2 = 1 5 ( 2 + 3 δ + i δ$_{LL}$ ) , (27) − 2 1 3 ≤ δ δ$_{LL}$ ≤ 1 . (28) where the maximum of δ/δ$_{LL}$ is reached outside the first Brillouin zone, which point more appropriately belongs to the case (a); (d) E$_{in}$ ‖ ΓM, q ‖ ΓM ‖ E$_{in}$ [long-dashed line in Fig. 3(b)], The inequalities (24) and (26) also represent the edges of the locsiton bands for the cases of E$_{in}$ ‖ ΓM and E$_{in}$ ‖ ΓK, respectively (which explains our desire not to restrict the range of q to the first Brillouin zone when obtaining these inequalities). Note that locsitons with the largest red shift ( δ/δ$_{LL}$ > 0) can be achieved with E$_{in}$ ‖ ΓK, while locsitons with the largest blue shift ( δ/δ$_{LL}$ < 0) can be achieved with E$_{in}$ ‖ ΓM. Therefore, re-orienting the lattice with respect to the polarization of the incident laser beam may assist in controlling the type of locsitons excited in the lattice. In this section, like in Sec. III, we only consider the case where E$_{in}$ is spatially uniform and lies in the lattice plane (the “ ‖ ” case), which is easier to achieve if the laser beam is incident normally to the lattice plane. The presence of boundaries and defects in 2D lattices of resonant atoms can cause various locsitonic effects, including giant LF resonances, formation of dipole strata, and “magic” cancellation of the resonant LF suppression. These effects are similar in their nature to their counterparts in 1D arrays of resonant atoms [1, 2], but their manifestations are much more diverse because of the inherent anisotropy of the dipole-dipole interaction in the 2D case, especially in the “ ‖ ” geometry. Because of this, an all-encompassing study of finite 2D lattices is hardly possible within this pilot study on the subject, so we will restrict ourselves to providing some of the most characteristic results, which emphasize distinctions from the 1D problem. IV. FINITE LATTICES: IN-PLANE POLARIZATION 17