We construct the first-principles theory for the scattering of the Coulomb field of a fast electron traveling along an array of nano-or microparticles. The electron's trajectory is arbitrarily oriented in the plane parallel to the surface. We show that Smith-Purcell radiation accompanying this process results in very rich diffraction patterns, which differ dramatically from the ones for conventional diffraction gratings; a numerical analysis was made for THz frequencies. The generalized Smith-Purcell dispersion relation has been derived; it describes the link between frequency, two periods of the structure, the electron's velocity, and the angle of radiation with maximal intensity. The unique spatial distribution of the generated light can form the basis for the generation of structured-light electron-driven photon sources.

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Transition radiation (TR) is widely used as a radiation source in a wide spectral range, from terahertz to x rays. Conventional flat surfaces are usually used, but with the development of applications using microscopically structured surfaces, periodic surface structures are beginning to be studied. The periodicity of the surface dramatically changes the characteristics of TR, so this type of radiation received its own name: grating transition radiation (GTR). In this work, we investigate the polarization properties of GTR from a two-dimensional (2D) photonic crystal consisting of small particles arranged in a flat lattice (a 2D photonic crystal slab). We show theoretically that the polarization properties of GTR differ significantly from those of the kindred types of radiation: conventional TR and Smith–Purcell radiation. Since we found that the asymptotic behavior depending on the electron velocity for GTR and classical TR diverges, we performed homogenization and show that the results for GTR after homogenization are in perfect agreement with those for classical TR. This means that different dependence on the electron velocity for TR from a slab and for GTR from a 2D photonic crystal slab is caused by the fundamental difference between a conventional slab and a 2D photonic crystal due to its microscopic structure. The constructed theory contains the coordinates of the particles the photonic crystal consists of, which allows considering structures of finite size, both symmetrical and asymmetric. For asymmetric targets, the polarization of the radiation proves to be very sensitive to the electron’s trajectory. This sensibility of polarization characteristics opens up good opportunities for studying fine fundamental effects connected with the electron trajectory, such as the effect of the quantum nature of free electrons which manifests itself in the properties of radiation generated by free electrons. Also, the obtained results may find application in the design of compact sources of polarized radiation based on microscopically structured surfaces.

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