Liouville theory for fully analytic studies of longitudinal beam dynamics and bunch profile reconstruction in dispersive lines

A fully analytic formalism based on the exact solution of the Liouville equation expressed in terms of exponential operators describing the phase space of particles’ beams is adopted for studying the longitudinal dynamics in RF accelerators and magnetic compressors. A diagnostic solution is provided for the temporal reconstruction of bunch profiles in dispersive lines, solely relying on the measurement of energy spectra. Results obtained at the SOLARIS facility show pros and cons of the proposed approach, where the first bunch length measurement for the injector linac has been performed since the start of its operation.