Transition Radiation in the Pre-wave Zone

The wave zone condition for TR produced on the boundary between a metal and vacuum is examined. It is shown that in both forward and backward directions the wave zone sets in at the same distance, that is of the order of the formation length for forward TR. The features of backward TR in the pre-wave zone are considered. PACS:41.60.-m; 41.75.Ht Submitted to Physics Letters A It is a widely accepted point of view that transition radiation (TR) in the forward direction is formed over the so-called formation length, that is the distance needed for the particle Coulomb field and radiation field to separate due to the difference in their velocities of propagation [1]. In the relativistic regime the formation length exceeds the radiation wavelength in orders of magnitude. For TR produced on the boundary between a metal and vacuum in the optical range, which is only the object under consideration in this paper, the formation length is roughly , where is a wavelength and is the relativistic factor. In the case of backward TR the particle and radiation fields overlap only within a wavelength from the target surface. Sometimes this argument is exploited to conclude that wave zone (or far-field) conditions are different for forward and backward TR [2]. Meanwhile, it is actually not true in regards to the radiation itself. Both backward and forward TR evolve , before reaching the wave zone, over the same distance determined by the transverse dimension of the electromagnetic field of a moving particle. In fact, while the particle itself can certainly be considered a point, its Coulomb field Fourier-component, involved in the radiation at the wavelength , occupies a finite space, outer border of which scales in the transverse ( to the particle trajectory) plane roughly as . Since, eventually, the source of TR is the atomic electron currents induced on the metallic surface by the incident particle Coulomb field, its size is that of the field. The last statement is, however, valid for the infinite boundary. For a target of the finite size this does not hold if the particle field exceeds the transverse dimension of the target. Then, strong variations in the radiation properties are expected [3] at wavelengths for which the parameter is larger than the target radius. Nevertheless, as long as the target size has no effect on the radiation, one can consider a portion of the target surface with an extension of the order of around the particle trajectory as the TR source. Though rather artificial for the first view, the concept of the source for TR was found very useful for making clear the physical arguments determining the wave zone condition. The term ”wave zone” is preferred in this paper to the widespread ”far-field” notation mainly to avoid use of the ”near-field” one, which is typically addressed to the domain of a close vicinity (within a wavelength) to the source, where quasistatical, i.e. of non-radiation nature, phenomena may occur. Instead, the term ”pre-wave zone” seemed more appropriate for our purpose, that was to give an outline on the problem of TR in the pre-wave zone with an emphasis on its physical side. We shall start from simple phase relations and consider an extended coherent source of a radiation. Let O and S be two points on the source surface separated by a distance ( Fig. 1). Generally, waves emitted by these points at the same phase arrive at an arbitrary