Multipole decompositions for directional light scattering

Applications of the multipole decomposition method for investigations of directional light scattering by a single nanoparticle and nanoparticle structures located in a finite spatial region are discussed. It is shown that, even in the case of relatively large scatterers, the multipole decomposition obtained in the long-wavelength approximation (LWA) may provide much better convergence than the multipole decomposition with the exact multipoles obtained from the spherical harmonics expansion. For an explanation of this seeming paradox, we derive in real space the exact multipole decomposition based on the spherical harmonics, presenting exact expressions for multipoles up to the electric 16-pole. Results obtained with the exact and approximate multipole expressions are discussed and compared. It is shown that for shape-anisotropic finite-size scatterers with different geometrical dimensions (like plates, rods, disks, rings, etc.), the required number of approximate multipoles providing accurate results may be much smaller than the required number of exact multipoles. For applicability of the LWA multipole decomposition, the only important parameter is the small ratio of the scatter size (its projection) in the scattering direction to the light wavelength. If this condition is fulfilled, the multipole decomposition with a small number of LWA multipoles is simpler than that based on the exact multipoles.