We study theoretically an optical trapping of many nanoparticles by a single focused laser. The optical trapping of nanoparticles at an interface between a water solvent and a glass substrate forms a two-dimensional assembly of particles. Since the trapping force depends on the polarization of incident light, the shape and structure of assembly depends on the polarization. To understand an origin of the assembly formation, we examine numerical simulations of the light scattering and optical force with ordered structures of polystyrene nanoparticles by using the T-matrix method. The electric fields give the optical force on the polystyrene nanoparticles. The simulated results follow a formation of two-dimensional ordered structures beyond the focal size and their polarization-dependence. If the linear polarization is applied, the assembly shows a tetragonal regularly ordered structure. The scattering directions of light from the center of assembly and focal spot are four according to the symmetry of tetragonal structure. The particles swarming around the assembly are attracted to the scattering light by the gradient force, and would form four horns evolved from the assembly. In the case of circular polarization, the assembly of optically trapped particles shows a hexagonal structure. Then, the number of directions of the light scattering becomes six according to the symmetry of assembly. The number of scattering directions corresponds to horns. Our results for both polarizations agree with experiment by Kudo et al. [Nano Lett., 16, 3058 (2016)].
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