We consider use of a variable polarizing undulator for generating brilliant x-ray beams carrying orbital angular momentum. We find that higher harmonics of the radiation correspond to Laguerre-Gaussian modes with azimuthal mode indices $l$ equal to one less than the harmonic number when the undulator is operated to produce circularly polarized light. Beams with nonzero $l$ carry orbital angular momentum quantized in units of $l\hslash$ per photon. When operated to produce linear polarization, the harmonics correspond to Hermite-Gaussian modes. Selection of these modes with conventional monochromator optics opens the door for new research with x-ray synchrotron and free-electron laser sources.

• Received 9 September 2007

DOI:https://doi.org/10.1103/PhysRevLett.100.124801