Spatial coherence in the transition radiation spectrum

The formal expression of the spectral distribution of the transition radiation intensity will be here derived in the case of a relativistic three-dimensional charged beam. Charged beams with a particle density such as is typically encountered in a particle accelerator will be considered. In particular, a sufficiently high particle density will be supposed so that a continuous spatial distribution function can be reliably attributed to the charged bunch. The formula of the spectral distribution of the transition radiation intensity originated by a relativistic three-dimensional charged beam - already presented in a previous work - will be here submitted to a formal check and interpreted in the physical consequences. The present work contains an additional mathematical derivation of the radiation energy spectrum consisting in a different method to implement the continuous limit in the distribution function of the particle coordinates. In the former derivation of the formula, the average operation with respect to the continuous distribution function of the particle coordinates was applied to the radiation intensity of a N electron bunch. In the present one, it is applied to the radiation electric field of a N electron bunch. The comparison of the two alternative but in any case equivalent formal routes to the spectral distribution of the transition radiation intensity will offer the possibility to directly cross-check the mathematical self-consistency of the presented results within the limits of applicability of the continuous limit approximation. According to such results, both the flux and the angular distribution of the photons emitted at a given wavelength - even shorter than the longitudinal length of the bunch - are expected to undergo a modification as the beam transverse size is varied with respect to the observed wavelength. As a function of the beam transverse size the spatial coherence degree of the transition radiation source is thus expected to change. The physical consistency of such an effect occurring in the transition radiation emission by a charged beam can be argued on the basis of a compatibility criterion with other similar relativistic electromagnetic radiative phenomena and interpreted in the framework of the temporal causality and the Huygens-Fresnel principles. Finally, the aspect of the applicability of the continuous limit approximation to the case of a charged beam in a particle accelerator is treated in terms of a practical quantitative criterion.