  Using three-dimensional particle-in-cell simulations we show that a strongly
nonlinear plasma wave excited by an ultrahigh intensity laser pulse works as a
compact high-brightness source of X-ray radiation. It has been recently
suggested by A. Pukhov and J. Meyer-ter-Vehn, Appl. Phys. B \textbf{74}, 355
(2002), that in a strongly nonlinear regime the plasma wave transforms to a
``bubble'', which is almost free from background electrons. Inside the bubble,
a dense bunch of relativistic electrons is produced. These accelerated
electrons make betatron oscillations in the transverse fields of the bubble and
emit a bright broadband X-ray radiation with a maximum about 50 keV. The
emission is confined to a small angle of about 0.1 rad. In addition, we make
simulations of X-ray generation by an external 28.5-GeV electron bunch injected
into the bubble. $\gamma$-quanta with up to GeV energies are observed in the
simulation in a good agreement with analytical results. The energy conversion
is efficient, leading to a significant stopping of the electron bunch over 5 mm
interaction distance.
