more detail on the local field distribution in the simplest “magic shape” that can be cut out of a triangular lattice—a six-point star with an atom at the center. In the case where the incident field is polarized normally to the lattice, we found that locsitons bear more analogy to locsitons in 1D arrays of atoms, compared to the case of an in-plane polarization. Finally, we illustrated the role of lattice defects in supporting localized locsitons. While this paper does not elaborate on nonlinear effects involving 2D locsitons, to be addressed in our future publications, we note that, similarly to the 1D case [1, 2], our numerical simulations have shown optical bistability and hysteresis, which may be especially important for potential applications of 2D locsitons in designing all-dielectric nanoscale logic elements, devices for signal processing, etc. This work is supported by US AFOSR. Acknowledgments [1] A. E. Kaplan and S. N. Volkov, Phys. Rev. Lett. 101 , 133902 (2008). [2] A. E. Kaplan and S. N. Volkov, Phys. Rev. A 79 , 053834 (2009). [3] M. Born and E. Wolf, Principles of Optics , (Pergamon, Oxford, 1980), Ch. 2 and references therein. [4] H. A. Lorentz, Ann. Phys. Chem. 9 , 641 (1880). [5] L. Lorenz, Ann. Phys. Chem. 11 , 70 (1881). [6] A. Aharoni, Introduction to the Theory of Ferromagnetism (Oxford Univ. Press, Oxford, 2001). [7] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996). [8] V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30 , 3356 (2005). [9] V. A. Markel and A. K. Sarychev, Phys. Rev. B 75 , 085426 (2007). [10] V. B. Sandomirskii, Sov. Phys. JETP 25 , 101 (1967). [11] V. Chernyak, S. N. Volkov, and S. Mukamel, Phys. Rev. Lett. 86 , 995 (2001). 25