where J m ( q ) is the Bessel function of the first kind, we evaluate the integral in Eq. (13) in an explicit form and write Eq. (13) as By substituting Q from Eqs. (5) and (9), we can rewrite the last formula as an explicit dispersion relation connecting the normalized detuning δ with the normalized locsiton wave vector q which is represented by its polar coordinates q and ψ : In the limit of low absorption, i. e., | δ$_{LL}$ | ≫ 1, the r. h. s. of Eq. (16) may be replaced with δ/δ$_{LL}$ . A more detailed analysis of the dispersion relation shows that spatial oscillations— locsitons with almost real q —emerge in a limited frequency band around ω$_{0}$ with a normal- ized bandwidth ∼ | δ$_{LL}$ | . It is exactly the frequency range where the interaction between resonant atomic dipoles may become comparable to or much stronger than the effects of the external field E$_{in}$ . The dispersion dependence δ/δ$_{LL}$ = D NRA 2 ( q ) in this case is shown in Fig. 2(a), where the external field is assumed to be aligned with the x axis. At every given laser frequency ω , locsitons with a whole range of wave vectors q may be excited. This set of q , all having different orientations, is represented by an isoline for the respective δ/δ$_{LL}$ = D NRA 2 ( q ) in Fig. 2(a). Recalling that δ$_{LL}$ < 0 [see Eq. (9)], we may notice that solid isolines (red shaded areas) correspond to red-shifted laser frequencies ( ω < ω$_{0}$ ), while dashed isolines (blue shaded areas) correspond to blue-shifted frequencies. The non-circular shape of the isolines reflects the highly anisotropic dispersion dependence for locsitons in the 2D lattice in the case of short-wave locsitons. In particular, at any given normalized frequency detuning δ , locsitons with different orientations of q with respect to E$_{in}$ may have very different q and, consequently, different wavelengths. At the same time, because the NRA describes the lattice in an averaged way, the orientation of E$_{in}$ with respect to the lattice does not affect the result. In the general case, especially when looking at the limitations on the size of the structure, one needs to consider complex locsiton wave vectors q = q ′ + i q ′′ in the dispersion relations (13) or (16), like it was done for 1D arrays of resonant atoms in [2]. This would allow to describe the dissipation of 2D locsitons, which is most prominent near the Lorentz resonance, and evanescent locsitons, which exist outside the locsiton band. One can also calculate the 1 + 3 4 Q [ J$_{0}$ ( q ) − 3 J$_{2}$ ( q ) cos(2 ψ )] = 0 . (15) D NRA 2 ( q ) ≡ J$_{0}$ ( q ) − 3 J$_{2}$ ( q ) cos(2 ψ ) = δ + i δ$_{LL}$ . (16) 11