on a scale of many wavelengths. The “ ⊥ ” configuration presents a much closer analogy to the 1D problems considered in [1, 2], although some details inevitably differ. We will discuss it below in Sec. V. Further in this section, we only consider the “ ‖ ” case, where both E$_{in}$ and E$_{L}$ lie in the lattice plain. Optical response of a triangular lattice of atoms is, in general, anisotropic: it depends on the orientation of E$_{in}$ in the lattice plane with respect to the symmetry directions of the lattice. We have demonstrated, however, that one can use the near-ring approximation (NRA) to describe the behavior of long-wavelength locsitons in the lattice [1] in an isotropic fashion. In this approximation, the contribution from the surrounding dipoles to the LF E$_{L}$ [the second term in Eq. (1)] is substituted with a field from an effective dipole ring with the radius l a , such that the polarizability of the nearest six atoms is evenly redistributed along the ring. The NRA is thus a further simplification of the NNA, making the model isotropic in the lattice plane. Within the NRA we then replace the summation in Eq. (4) with an integration over the imaginary ring, so that the equation for the LF becomes where u is a unit vector in the direction from the center r of the effective ring to a point at the ring, and θ is the polar angle of u counted from the direction of E$_{in}$ in the lattice plane. The strength of the dipole-dipole coupling between neighboring atoms is still given by the dimensionless parameter Q [Eq. (5)]. Equation (7) has a uniform ( Lorentz ) solution where is the normalized frequency detuning at which the Lorentz-Lorenz resonance is achieved in the triangular lattice of atoms within the NRA. As we will see in Sec. V, δ$_{LL}$ is different for the “ ‖ ” and “ ⊥ ” configurations, but to simplify the formulas, we omit the index “ ‖ ” everywhere in this section and in Sec. IV, except for Eq. (9). From Eq. (8) one can see that, 9 ¯ E$_{L}$ = δ + i δ − δ$_{LL}$ + i E$_{in}$ , (8) δ ‖ LL = − 3 4 Q a (9) E$_{L}$ ( r ) = E$_{in}$ − 3 4 π Q ∫ 2 π 0 E$_{L}$ ( r + l a u ) (3 cos 2 θ − 1) dθ, (7) A. Near-ring approximation