  Bright sources of high energy electromagnetic radiation are widely employed
in fundamental research as well as in industry and medicine. This steadily
growing interest motivated the construction of several facilities aiming at the
realisation of sources of intense X- and gamma-ray pulses. To date, free
electron lasers and synchrotrons provide intense sources of photons with
energies up to 10-100 keV. Facilities under construction based on incoherent
Compton back scattering of an optical laser pulse off an electron beam are
expected to yield photon beams with energy up to 19.5 MeV and peak brilliance
in the range 10$^{20}$-10$^{23}$ photons s$^{-1}$ mrad$^{-2}$ mm$^{-2}$ per
0.1% bandwidth. Here, we demonstrate a novel mechanism based on the strongly
amplified synchrotron emission which occurs when a sufficiently dense electron
beam interacts with a millimetre thickness solid target. For electron beam
densities exceeding approximately $3\times10^{19}\text{ cm$^{-3}$}$
filamentation instability occurs with the self-generation of 10$^{7}$-10$^{8}$
gauss magnetic fields where the electrons of the beam are trapped. This results
into a giant amplification of synchrotron emission with the production of
collimated gamma-ray pulses with peak brilliance above $10^{25}$ photons
s$^{-1}$ mrad$^{-2}$ mm$^{-2}$ per 0.1% bandwidth and photon energies ranging
from 200 keV up to several hundreds MeV. These findings pave the way to
compact, high-repetition-rate (kHz) sources of short (30 fs), collimated (mrad)
and high flux ($>10^{12}$ photons/s) gamma-ray pulses.
