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An optics-free method is proposed to generate X-ray radiation with spatiallyvariant states of polarization via an afterburner extension to a Free ElectronLaser (FEL). Control of the polarization in the transverse plane is obtainedthrough the overlap of different coherent transverse light distributionsradiated from a bunched electron beam in two consecutive orthogonally polarisedundulators. Different transverse profiles are obtained by emitting at a higherharmonic in one or both of the undulators. This method enables the generationof beams structured in their intensity, phase, and polarization - so-calledPoincar\'e beams - at high powers with tunable wavelengths. Simulations areused to demonstrate the generation of two different classes of light withspatially inhomogeneous polarization - cylindrical vector beams and fullPoincar\'e beams.

Free Electron Laser Generation of X-Ray Poincar\'e Beams