Focusing of Smith-Purcell radiation from a two-dimensional particle array in the prewave zone

A generalized theory of Smith-Purcell radiation (SPR) from a two-dimensional (2D) array of subwavelength particles is presented. Unlike most theories of SPR, this one is valid in the prewave zone and the near zone. We have obtained a generalized dispersion relation that determines the points of concentration of radiation in the prewave zone rather than in the ordinary directions of propagation of the most intense radiation; in the wave zone, the dispersion relation transforms into the ordinary one. Analytically, we derived that it is the parabolic law that describes how the individual elements should be arranged for the most intense radiation to be achieved. Interestingly, for the ultrarelativistic limit, the parabolic law turns into the hyperbolic one. Therefore, to focus the radiation from ultrarelativistic electrons in the prewave zone, a hyperbolically arranged 2D grating should be used instead of a rectangular one. Our findings suggest that the radiation intensity at a point of focus in the prewave zone can be sensitive to numerically very small changes in the locations of the individual constituent elements of the array determining the topology-the way the array's elements are arranged. The theory is constructed for particles with arbitrary dielectric properties, including metal ones; numerical analysis has been performed for dielectric particles.