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Since the new millennium coherent extreme ultra-violet and soft x-ray radiation has revolutionized the understanding of dynamical physical, chemical and biological systems at the electron’s natural timescale. Unfortunately, coherent laser-based upconversion of infrared photons to vacuum-ultraviolet and soft x-ray high-order harmonics in gaseous, liquid and solid targets is notoriously inefficient. In dense nonlinear media, the limiting factor is strong re-absorption of the generated high-energy photons. Here we overcome this limitation by generating high-order harmonics from a periodic array of thin one-dimensional crystalline silicon ridge waveguides. Adding vacuum gaps between the ridges avoids the high absorption loss of the bulk and results in a ~ 100-fold increase of the extraction depth. As the grating period is varied, each high harmonic shows a different and marked modulation, indicating their waveguiding in the vacuum slots with reduced absorption. Looking ahead, our results enable bright on-chip coherent short-wavelength sources and may extend the usable spectral range of traditional nonlinear crystals to their absorption windows. Potential applications include on-chip chemically-sensitive spectro-nanoscopy.