Bright Compton source based on the interaction of an ultraintense laser pulse with a relativistically transparent foil and a plasma mirror

We propose and numerically study an x-ray source in which a single laser pulse first accelerates electrons while passing through a relativistically transparent thin overdense foil and then, after reflection from a plasma mirror, collides head-on with them. Two-dimensional kinetic simulations show that for a laser pulse with an intensity of the order of $10^{21}$~W/cm$^2$ and optimal foil parameters, electrons are accelerated to energies of the order of 100~MeV, and the source can deliver up to $10^{7}$ photons per 0.1\% bandwidth per joule of laser energy in the 10--250~keV range, two orders of magnitude more than a similar gas-jet scheme. This also exceeds the spectral yield per joule of the best gas-target betatron sources by about an order of magnitude and is comparable to the best experimental results obtained with near-critical-density targets.