Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum

We show how bright, fully coherent, hard x-ray beams can be generated through nonlinear upconversion of femtosecond laser light. By using longer-wavelength mid-infrared driving lasers of moderate peak intensity, full phase matching of the high harmonic generation process can extend, in theory, into the hard x-ray region of the spectrum. We identify the dominant phase matching mechanism for long wavelength driving lasers, and verify our predictions experimentally by demonstrating phase-matched up-conversion into the soft x-ray region of the spectrum around 330 eV using an extended, high-pressure, gas medium that is weakly ionized by the laser. Scaling of the overall conversion efficiency is surprisingly favorable as the wavelength of the driving laser is increased, making useful, fully coherent, multi-keV x-ray sources feasible. Finally, we show that the rapidly decreasing microscopic single-atom yield at longer driving wavelengths is compensated macroscopically by an increasing optimal pressure for phase matching and a rapidly decreasing reabsorption of the generated light at higher photon energies.