Unified Microscopic-Macroscopic Formulation of High-Order Difference-Frequency Mixing in Plasmas

We investigate high-order difference-frequency mixing in plasmas, taking into account the microscopic rescattering physics and propagation effects for the first time. We show that phase matching can occur over a broad frequency range, up to very high photon energies, and that it is confined to specific temporal and spatial windows. This gated phase matching mechanism is driven by the continuous phase slip between two driving fields and can be employed for manipulating the temporal, spatial, and spectral properties of high harmonic emission.

Figure 1

Longitudinal gated phase matching. (a) Harmonic spectral intensity of HDM (solid) and HHG (dotted). (b) Harmonic signal versus propagation distance for HDM (solid) and HHG (dotted), integrated over a 10% spectral window centered at 300, 500, and 800 eV. (c) Intensity temporal profile of the emitted harmonics via HDM for photon energies around 300 eV. The inset shows the pulse at 3.2 fs. (d) Local phase mismatch versus propagation distance at photon energies 300, 500, and 800 eV, generated via HDM (solid) and HHG (dotted). The vertical lines display the correspondence between the phase matching and enhancement regions. (e) Phase-matched trajectories (solid) released by the bichromatic field, that contribute to the first enhancement step at photon energies around 300 eV. The arrows show how the fields advance to the position of the second enhancement step (see Fig. 3), in which the dashed electron trajectory gives rise to phase matched HDM generation.

Figure 2

Temporal and spectral gated phase matching. (a) Harmonic spectral intensity for HDM (solid) and HHG (dotted). Plots (b)–(f) show the behavior of 800 eV harmonics emitted via HDM (b) Signal at $t\approx 3.1\mathrm{fs}$ (solid), $t\approx -0.8\mathrm{fs}$ (dashed), and integrated over all times (dotted) versus propagation distance. (c) Temporal profile of the intensity at $z=0.07\mathrm{mm}$ (upper) and $z=0.12\mathrm{mm}$ (lower). (d) Local phase mismatch versus propagation distance. (e) Intensity profile of the x-ray burst at $t\approx 3.1\mathrm{fs}$ at various distances. (f) FWHM of x-ray bursts versus propagation distance.

Figure 3

Phase matching dynamics over long distances. (a) Output harmonic spectral after propagating 2.3 mm. (b) Harmonic signal versus propagation distance for 10% spectral windows centered at 300, 500, and 800 eV. The inset shows the 300 eV curve on a linear scale. The first four enhancement steps are regular, while the fifth one is different. The intensity profiles of the HDM harmonics at 300 eV, after 4 (c) and 5 (d) enhancement steps.