Photon Pathways and the Nonperturbative Scaling Law of High Harmonic Generation

High harmonic generation (HHG) has become a core pillar of attosecond science. Traditionally described with field-based models, HHG can also be viewed in a parametric picture, which predicts all properties of the emitted photons, but not the nonperturbative efficiency of the process. Driving HHG with two noncollinear beams and deriving analytically the corresponding yield scaling laws for any intensity ratio, we herein reconcile the two interpretations, introducing a generalized photonic description of HHG. It is in full agreement with field-based simulations and experimental data, opening the route to smart engineering of HHG with multiple driving beams.

Figure 1

(a) Schematic of the experimental setup. $\theta$ is in the $x\text{−}z$ plane. At the bottom are typical images observed on the microchannel plate ($x\text{−}y$ plane) for (b) low ($\alpha \ll 1$) and (c) high ($\alpha \approx 1$) perturbation regimes. The dashed yellow line marks the position of the zeroth order of diffraction, i.e., the position of the spectrum generated by beam 1 alone. The angle between the driving beams is 40 mrad.

Figure 2

(a) Experimental amplitude of the electric fields of beamlets $p=0$, $+1$, and $+2$ for harmonic $q=13$. These curves are fitted with one (dot-dashed), two (dashed lines), or three (solid lines) terms of Eq. (4). (b) Comparing the experimental fit coefficients (filled bars) to the analytical $a_p^n$ coefficients (square markers, signs were adjusted to have the first coefficient be positive). The signs of the first few successive coefficients are seen to alternate. (c) Illustration of the photon pathways contributing to beamlets $p=0$ and 1, for $q=3$.

Figure 3

Experimental (a), numerical (SFA) (b), and analytical (c) field amplitude of the beamlets of harmonic $q=15$ as a function of the amplitude ratio $\alpha$ of the two driving beams. The amplitudes were normalized by the maximum for each column to optimize readability over the whole $\alpha$ range. For the analytical plot, values of $n$ up to 15 were used in Eq. (3). The insets in the corners are lineouts of the field amplitude of order $p=1$ normalized to take into account the changing total field intensity as $\alpha$ is increased.