Charge Density Wave bending observed by Xfel source acting as a tunable electronic lens for hard x-rays

Ultrafast X-ray diffraction by the LCLS free-electron laser has been used to probe Charge Density Wave (CDW) systems under applied external currents. At sufficiently low currents, CDW wavefronts bend in the direction transverse to the 2k$_F$ wave vector. We show that this shear effect has the ability to focus or defocus hard X-ray beams, depending of the current direction, making it an electronic lens of a new kind, tunable at will from the Fraunhofer to the Fresnel regime. The effect is interpreted using the fractional Fourier transform showing how the macroscopic curvature of a nanometric modulation can be beneficially used to modify the propagation of X-ray beams.