Designing localized electromagnetic fields in a source-free space.
An approach to characterizing and designing localized electromagnetic fields, based on the use of differentiable manifolds, differentiable mappings, and the group of rotation, is presented. By way of illustration, novel families of exact time-harmonic solutions to Maxwell's equations in the source-free space--localized fields defined by the rotation group--are obtained. The proposed approach provides a broad spectrum of tools to design localized fields, i.e., to build-in symmetry properties of oscillating electric and magnetic fields, to govern the distributions of their energy densities (both size and form of localization domains), and to set the structure of time-average energy fluxes. It is shown that localized fields can be combined as constructive elements to obtain a complex field structure with desirable properties, such as one-, two-, or three-dimensional field gratings. The proposed approach can be used in designing localized electromagnetic fields to govern motion and state of charged and neutral particles. As an example, motion of relativistic electrons in one-dimensional and three-dimensional field gratings is treated.