Modified multipoles in photonics

Multipole decomposition method is a promising tool for investigation of radiating or scattering responses of electromagnetic sources or particles. It works even in the case of relatively complicated and compound scatterers like multilayer particles, clusters, or asymmetrical systems. Commonly, the radiation fields of point electric or magnetic sources are decomposed only into electric or magnetic dipole moments, while real sources can be described by a series of multipoles, including higher multipoles and toroidal moments. In this paper, we introduce the concept of modified multipoles describing real sources of electric, magnetic, and toroidal types. Using the analytical expressions of first-order multipoles, we discuss how they depend on the position of the center of radiation, as well as on the shift of the source, relative to the center of coordinates. We present results of multipoles for the sources with defects and asymmetry. The long-awaited question about distinguishing radiation patterns of electric and toroidal dipole moments in a far-field zone is discussed and solved in this paper. We show that radiation patterns of shifted electric and toroidal dipoles can be rotated to different angles relative to each other due to shifting. Moreover, we discuss modified anapoles. Our modified dipole approach will be useful for multipole analysis of complex systems in photonics such as nanoparticle clusters, metamaterials, and nanoantennas, as well as for better understanding issues of toroidal electrodynamics.