Optical vortices, which carry orbital angular momentum (OAM), can be flexibly produced and\nmeasured with infrared and visible light. Their application is an important research topic for\nsuper-resolution imaging, optical communications and quantum optics. However, only a few\nmethods can produce OAM beams in the extreme ultraviolet (XUV) or X-ray, and controlling\nthe OAM on these beams remains challenging. Here we apply wave mixing to a tabletop\nhigh-harmonic source, as proposed in our previous work, and control the topological charge\n(OAM value) of XUV beams. Our technique enables us to produce first-order OAM beams\nwith the smallest possible central intensity null at XUV wavelengths. This work opens a route\nfor carrier-injected laser machining and lithography, which may reach nanometre or even\nangstrom resolution. Such a light source is also ideal for space communications, both in the\nclassical and quantum regimes.
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