Search for a chirp of a synchrotron x-ray beam at the European synchrotron radiation facility: II. Analysis and theory.

In an earlier paper(1) the emission process of synchrotron radiation was studied theoretically for the current induced by a single electron interacting with a bending magnet showing showing that a chirp of the hard x-ray could be expected that could be used for a compression of x-ray bunches using dispersive optics. In this paper we will present a detailed analysis of the results of an experimental study(2-3) that was conducted on a bending magnet beamline (BM5) at the ESRF to learn if such a chirp exists for an electron bunch as a function of the x-ray energy and of the emission angle below and above the electron orbit plane. Fourier transforms of the observed time signals were performed that indicated a phase shift of the xrays with energy and emission angle. Using a classical method to describe the propagation of quasi-Gaussian beams and introducing partial time and space derivatives of the wave packet drift, we established a new relationship to describe analytically this particular behaviour of synchrotron x-rays that involves a relativistic Doppler effect.