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Vigorous efforts to harness the topological properties of light have enabled a multitude of novel applications. Translating the applications of structured light to higher spatial and temporal resolutions mandates their controlled generation, manipulation, and thorough characterization in the short-wavelength regime. Here, we resort to high-order harmonic generation (HHG) in a noble gas to upconvert near-infrared (IR) vector, vortex, and vector-vortex driving beams that are tailored, respectively, in their spin angular momentum (SAM), orbital angular momentum (OAM), and simultaneously in their SAM and OAM. We show that HHG enables the controlled generation of extreme-ultraviolet (EUV) vector beams exhibiting various spatially dependent polarization distributions, or EUV vortex beams with a highly twisted phase. Moreover, we demonstrate the generation of EUV vector-vortex beams (VVB) bearing combined characteristics of vector and vortex beams. We rely on EUV wavefront sensing to unambiguously affirm the topological charge scaling of the HHG beams with the harmonic order. Interestingly, our work shows that HHG allows for a synchronous controlled manipulation of SAM and OAM. These EUV structured beams bring in the promising scenario of their applications at nanometric spatial and sub-femtosecond temporal resolutions using a table-top harmonic source. 

Phased-necklace driving beams enable precise coherent control over the line spacing and divergence of EUV/soft x-ray harmonics.