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

Abstract

We show how bright, tabletop, fully coherent hard X-ray beams can be generated through nonlinear upconversion of femtosecond laser light. By driving the high-order harmonic generation process using longer-wavelength midinfrared light, we show that, in theory, fully phase-matched frequency upconversion can extend into the hard X-ray region of the spectrum. We verify our scaling predictions experimentally by demonstrating phase matching in the soft X-ray region of the spectrum around 330 eV, using ultrafast driving laser pulses at 1.3-microm wavelength, in an extended, high-pressure, weakly ionized gas medium. We also show through calculations that scaling of the overall conversion efficiency is surprisingly favorable as the wavelength of the driving laser is increased, making tabletop, fully coherent, multi-keV X-ray sources feasible. The rapidly decreasing microscopic single-atom yield, predicted for harmonics driven by longer-wavelength lasers, is compensated macroscopically by an increased optimal pressure for phase matching and a rapidly decreasing reabsorption of the generated X-rays.

Fig. 1.

Extreme nonlinear upconversion of a femtosecond laser light to shorter wavelengths. Phase-matched (coherent) addition of the high harmonic X-ray fields emitted by many atoms in the medium is shown.

Fig. 2.

Full phase matching of HHG. (A) Theoretical predictions for scaling of the phase matching cutoffs in Ar, Ne, and He gases, showing that full-phase matching of HHG emission can extend into the multi-keV X-ray region. (B) Experimentally, phase-matched emission (linear intensity scale) extends to 50 eV in Ar for 0.8-μm driving lasers. (C–E) Experimental data verifying significant extension of the phase matching cutoffs as λ_L increased from 0.8 to 1.3 μm in Ar (C), Ne (D), and He (E). In He, phase matching extends into the water window. (Above B–E, the filter transmission curves of the filters used to eliminate the driving laser light are shown.)

Fig. 3.

Scaling of phase matching parameters. (A) Experimental macroscopic HHG signal growth versus pressure at photon energies close to the phase-matching cutoffs for 1.3-μm lasers. (B) Theoretical predictions of the pressure required for phase matching by using 0.8- and 1.3-μm lasers, based on a simple semianalytical model (solid lines). Quantum calculations for He are shown in dashed lines. (C) Evolution of the laser field in He, represented as a temporal overlap of the field at different distances along the wave guide for a pressure of 1,500 torr and for a propagation distance of 1 cm. The macroscopic in-phase HHG emission is confined within the central cycle of the laser pulse.

Fig. 4.

Characteristics of phase-matched HHG. Experimental phase-matched HHG emission from He, Ne (Ag/Zr filters, solid/dashed line), and Ar driven by λ_L = 1.3 μm, illustrating the sharp HHG cutoffs. (Inset) Raw image of the near-perfect Gaussian soft X-ray beam generated in He, for a broad band of photon energies (200–330 eV), spanning into the water window.

Fig. 5.

Scaling of the pressures-length products required for efficient phase-matched HHG emission at the phase-matching cutoffs for laser wavelengths up to λ_L = 6 μm and corresponding phase-matching limits up to 3 keV. (A) Predicted optimal pressure showing quadratic growth with λ_L. (B) Absorption-limited medium length for reaching 90% of the asymptotic limit of the macroscopic harmonic signal, determined by reabsorption. (C) Absorption-limited intensity of phase-matched HHG in a line width of λ/Δλ = 100 at the phase-matching cutoffs for laser wavelengths between 0.8 and 10.0 μm. The curves are normalized to the phase-matched HHG emission at λ_L = 0.8 μm. For high photon energies, group velocity mismatch and magnetic field effects reduce the HHG flux below these predictions (see SI Text for details).