  The necessity for compact table-top x-ray sources with higher brightness,
shorter wavelength and shorter pulse duration has led to the development of
complementary sources based on laser-plasma accelerators, in contrast to
conventional accelerators. Relativistic interaction of short-pulse lasers with
underdense plasmas results in acceleration of electrons and in consequence in
the emission of spatially coherent radiation, which is known in the literature
as betatron radiation. In this article we report on our recent results in the
rapidly developing field of secondary x-ray radiation generated by high-energy
electron pulses. The betatron radiation is characterized with a novel setup
allowing to measure the energy, the spatial energy distribution in the
far-field of the beam and the source size in a single laser shot. Furthermore,
the polarization state is measured for each laser shot. In this way the emitted
betatron x-rays can be used as a non-invasive diagnostic tool to retrieve very
subtle information of the electron dynamics within the plasma wave. Parallel to
the experimental work, 3D particle-in-cell simulations were performed, proved
to be in good agreement with the experimental results.
