Traditional tomographic methods based on Radon transformation require a full set projections covering full 180-degrees. This technique is applicable only to a stationary distribution of electrons, which do not evolve. In addition, this method can’t work with incomplete sub-set of data such as a few projections covering total angle of few degrees. We present novel method of tomography working with a limited number of non-degenerated linear projections. We present the description of the method, discuss its advantages as well as limitations. We compare the method with the Radon transformation. We present the application of this method to the study the dynamics of e-beam in longitudinal phase space using dual sweep streak-camera with psec resolution. This method allowed us to restore the evolution of e-beam during a giant pulse in a storage ring FEL when the distribution of the electrons changes completely during one synchrotron oscillation. We discuss another possible applications of this method in advanced FEL systems, where effects of space charge, nonlinearities or coherent synchrotron radiation impair traditional diagnostics methods.

In this paper we present the experimental results on the storage ring free electron laser operating in the giant pulse mode with variable repetition rate. The experiments were conducted in the wide range of the electron beam energies from 270 MeV to 600 MeV with the giant pulses generated using a gain modulator. Dependence of the peak and average power, and the other properties and parameters of giant pulses on the pulse repetition rate are studied. In particular, it is found that the average lasing power in the giant pulse mode reaches levels of 70-90% of that in the CW power mode. Applications of such mode of operation are discussed.