We demonstrate that a mm-scale free electron laser can operate in the X-ray range, in the interaction between a moderately relativistic electron bunch, and a transverse high intensity optical lattice. The corrugated light-induced ponderomotive potent
ial acts simultaneously as a guide and as a low-frequency wiggler, triggering stimulated Raman scattering. The gain law in the small signal regime is derived in a fluid approach, and confirmed from Particle-In-Cell simulations. We describe the nature of bunching, and discuss the saturation properties. The resulting all-optical Raman X-ray laser opens perspectives for ultra-compact coherent light sources up to the hard X-ray range.
