effects with promising applications, like the optical bistability and hysteresis. In the present paper, however, we will only consider linear effects in 2D lattices of resonant atoms. We can rewrite Eqs. (1)–(3) in the following closed form, where we introduced the dimensionless strength Q of the dipole-dipole coupling between neighboring atoms, which is easily controlled through the normalized frequency detuning δ , The maximum normalized strength of interaction between neighboring atoms, is reached at zero detuning of the incident laser frequency from the atomic resonance ( δ = 0). Note that this is the point where our approach drastically departs from the conventional LF theory, where the LF is implied quasi-uniform on a scale of many l a , so that E$_{L}$ ( r ) ≈ E$_{L}$ ( r $^{′}$). More details on the general LF model that we use in this paper, including some quantitative estimates of Q a for realistic systems, can be found in our recent publication [2]. In the present paper we restrict our considerations to the nearest-neighbor approximation (NNA), in which only interactions between the closest atoms are taken into account in Eq. (4). As we demonstrated in [1, 2], this approximation leads to qualitatively similar results compared to the full solution, that takes into account interactions between all pairs of atoms in the system. The NNA allows us to derive simpler analytic expressions and undertake numerical simulations for reasonably large lattices. In this section we consider the emergence and properties of locsitons in infinite unbounded planar lattices of atoms. Here we aim at studying locsitons in their “simplest” form, without the system boundaries complicating the picture. At the same time, we will continue to assume the near-field character of the interatomic interaction, because our ultimate goal is III. LOCSITONS IN TRIANGULAR LATTICES: IN-PLANE POLARIZATION E$_{L}$ ( r ) = E$_{in}$ ( r ) − Q 4 $^{$_{r}$′}$̸ = r ∑ lattice l 3 a | r ′ − r | 3 × { 3 u [ E$_{L}$ ( r $^{′}$) · u ] − E$_{L}$ ( r $^{′}$) } , (4) Q = Q a δ + i . (5) Q a = 8 | d a | 2 ϵ ¯ h Γ l 3 a , (6) 7