  Structured light has gained significant attention in recent years, especially
in the generation and application of vector beams. These beams, characterized
by a spatially varying polarization state, are a powerful tool to enhance our
capacity to control light-matter interactions. In this study, we demonstrate
the synthesis of extreme-ultraviolet (EUV) vector beams in a chip through
high-order harmonic generation (HHG). Our findings showcase the chip's ability
to transfer the laser polarization state into the EUV beam despite the extended
interaction length. This approach not only outperforms conventional free-space
methods but also paves the way for a multitude of on-chip investigations in the
realms of EUV and soft-X-ray spectroscopy.
