Bright X-ray radiation from plasma bubbles in an evolving laser wakefield accelerator

We show that the properties of the electron beam and bright x-rays produced by a laser wakefield accelerator can be predicted if the distance over which the laser self-focuses and compresses prior to self-injection is taken into account. A model based on oscillations of the beam inside a plasma bubble shows that performance is optimised when the plasma length is matched to the laser depletion length. With a 200~TW laser pulse this results in an x-ray beam with median photon energy of unit[20]{keV}, $> 6times 10^{8}$ photons above unit[1]{keV} per shot and a peak brightness of $unit[3 times 10^{22}]{photons~s^{-1}mrad^{-2}mm^{-2} (0.1% BW)^{-1}}$.