  In this paper we discuss a method for nondestructive measurements of the
longitudinal profile of sub-picosecond electron bunches for X-Ray Free Electron
Lasers (XFELs). The method is based on the detection of the Coherent
Synchrotron Radiation (CSR) spectrum produced by a bunch passing a dipole
magnet system. This work also contains a systematic treatment of synchrotron
radiation theory which lies at the basis of CSR. Standard theory of synchrotron
radiation uses several approximations whose applicability limits are often
forgotten: here we present a systematic discussion about these assumptions.
Properties of coherent synchrotron radiation from an electron moving along an
arc of a circle are then derived and discussed. We describe also an effective
and practical diagnostic technique based on the utilization of an
electromagnetic undulator to record the energy of the coherent radiation pulse
into the central cone. This measurement must be repeated many times with
different undulator resonant frequencies in order to reconstruct the modulus of
the bunch form-factor. The retrieval of the bunch profile function from these
data is performed by means of deconvolution techniques: for the present work we
take advantage of a constrained deconvolution method. We illustrate with
numerical examples the potential of the proposed method for electron beam
diagnostics at the TESLA Test Facility (TTF) accelerator. Here we choose, for
emphasis, experiments aimed at the measure of the strongly non-Gaussian
electron bunch profile in the TTF femtosecond-mode operation. We demonstrate
that a tandem combination of a picosecond streak camera and a CSR spectrometer
can be used to extract shape information from electron bunches with a narrow
leading peak and a long tail.
