V. LOCSITONS IN TRIANGULAR LATTICES: NORMAL POLARIZATION The “ ⊥ ” configuration, where the incident field is polarized at the normal to the lattice plane, may be realized, for example, by creating a standing wave by two counter-propagating laser beams with the beam axes lying in the lattice plane. For systems much smaller than the laser wavelength, the incident field may be then assumed nearly uniform. Locsitons emerging in the “ ⊥ ” configuration are similar in many respects to 1D locsitons discussed in detail in [2], so here we only provide a brief overview of the “ ⊥ ” case and outline the most distinctive features appearing in this geometry. Let us start again with describing locsitons in unbounded lattices. The most general equation for the LF in the “ ⊥ ” case is obtained from Eq. (4) by setting E$_{in}$ ( r ) ‖ E$_{L}$ ( r ) ⊥ u : As in the “ ‖ ” case, we will assume here that the incident optical field is uniform, E$_{in}$ ( r ) ≡ E$_{in}$ , which will help us to build a clearer understanding of the locsiton behavior in 2D lattices. For an equilateral triangular lattice, Eq. (31) can be simplified using the NNA as Equation (32) has a uniform, Lorentz, solution, which is given by Eq. (8) with δ$_{LL}$ = δ ⊥ LL where which is three times the NNA value for δ$_{LL}$ in a 1D array of atoms if E$_{in}$ is perpendicular to the array (and the dipoles are aligned “side-to-side”) [2]. This is a consequence of each atom having now 6 instead of 2 neighbors. To obtain the corresponding dispersion relation δ ( q ) for “ ⊥ ” locsitons in the lattice, we substitute the LF E$_{L}$ ( r ) in Eq. (32) as a sum of the Lorentz solution given by Eqs. (8) and (33) and 2D plane-wave excitations (12). This dispersion relation can be written as E$_{L}$ ( r ) = E$_{in}$ ( r ) + Q 4 $^{$_{r}$′}$̸ = r ∑ lattice l 3 a | r ′ − r | $_{3}$E$_{L}$ ( r $^{′}$) . (31) E$_{L}$ ( r ) = E$_{in}$ + Q 4 ∑ $^{u}$K E$_{L}$ ( r + l a u$_{K}$ ) . (32) δ ⊥ LL = 3 2 Q a , (33) D NNA 2 ⊥ ( q ) ≡ 1 3 2 ∑ n =0 cos[ q cos( θ n − ψ )] = δ + i δ ⊥ LL , (34) D NNA 2 ⊥ ( q ) ≈ 1 − 1 4 q 2 + O ( q $^{4}$) , (35) where, as in Sec. III B, θ n = θ$_{0}$ + nπ/ 3. In the long-wavelength limit, 22