  Laser-plasma accelerators have become compact sources of ultrashort electron
bunches at energies up to the gigaelectronvolt range thanks to the remarkable
progress made over the past decade. A direct application of these electron
bunches is the production of short pulse x-ray radiation sources. In this
letter, we study a fully optically driven x-ray source based on the combination
of a laser-plasma accelerator and a plasma wave undulator. The longitudinal
electric field of a laser-generated plasma wave is used to wiggle electrons
transversally. The period of this plasma undulator being equal to the plasma
wavelength, tunable photon energies in the 10 keV range can be achieved with
electron energies in the 100-200 MeV range. Considering a 10s TW class
femtosecond laser system, undulators with a strength parameter K~0.5 and with
about ten periods can be combined with a laser-plasma accelerator, resulting in
several 10^-2 emitted x-ray photons per electron.
