Nonperturbative transverse mode coupling in high-order harmonic generation

This paper is a joint publication with the paper by Vimal et al. [Phys. Rev. Lett. 131, 203402 (2023)]. High-order harmonic generation (HHG) is an extreme nonlinear optical phenomenon at the heart of modern ultrafast physics. When HHG is driven by two noncollinear beams, one being significantly weaker than the other, the yields of the harmonic emission channels obey a perturbative power law with respect to the drivers' intensity ratio, even though HHG has a nonperturbative efficiency in general. Here, we drive HHG with two different transverse modes and show this power law to be directly imprinted in the spatial structure of the harmonic beamlets. As the driving beam intensities are brought closer, changes in the beamlet profiles reflect a transition from a perturbative to a nonperturbative behavior, understood by generalizing the photon-pathway formalism introduced in the companion paper. This work provides a visual method to study the efficiency of nonlinear optical processes and paves the way for finer all-optical shaping of extreme ultraviolet light.

Figure 1

Principle of noncollinear HHG with different transverse modes in the driving beams. Here, the main IR beam is a Gaussian mode, and the weak secondary beam is a ${\mathcal{H}}_{1,0}$ mode. At the crossing point, the intensity and phase of the IR field exhibit a gratinglike structure. The generated harmonic beamlets are separated angularly, as dictated by conservation of the linear momentum of the absorbed photons. Each beamlet is observed to have a specific transverse profile.

Figure 2

(a) Experimental beamlet intensities for harmonic $q=11$ driven by an intense Gaussian beam and a weak ${\mathcal{H}}_1$ mode in a noncollinear geometry for $\overline{\alpha }\approx 0.17$. The inset shows the intensity profile of the driving field. (b) Horizontal lineouts of the beamlets in (a) (top) and corresponding HG mode spectra (bottom) obtained by a fit of the intensity profiles. The best fit with a superposition of the first five HG modes is shown with red curves in the top row. The error bars correspond to the standard deviation associated with each of the five parameters of the fit. (c) Thin-slab simulation [Eq. (2)] of the experiment in (a) ($\overline{\alpha }\approx 0.17$). The inset shows the intensity profile of the driving field. (d) Horizontal intensity profiles (top) and HG mode spectra (bottom) of the first four beamlets, according to the perturbative law, Eq. (8).

Figure 3

(a) Far-field image of harmonic $q=11$ as a function of the power ratio of the drivers, ranging from $\overline{\alpha }\approx 0.05$ to $\overline{\alpha }\approx 0.4$. (b) Horizontal lineouts of three beamlets ($p=1,0,-1$) in (a). (c) Normalized intensity profiles of the beamlets obtained by thin-slab simulations based on Eq. (2) for the same parameters as in the experiment. (d) HG mode decomposition of the profiles in (c) as a function of $\overline{\alpha }$.

Figure 4

Generalization of the photon-pathway formalism to spatial modes. Equation (3) formulates each beamlet as a power series, conveniently interpreted as a natural continuation of the parametric picture of HHG: photon pathways involving the exchange of $\left|p\right|+2n$ photons with beam 2 add contributions beyond the leading perturbative term, with profiles scaling as $|E_2/E_1{|}^{\left|p\right|+2n}(x,y)$. Here, the total profiles are those observed for $\overline{\alpha }\approx 0.4$. Note that as $\overline{\alpha }$ is increased, the validity condition $|E_2/E_1|(x,y)<1$ for using Eq. (3) is violated in more and more regions. However, the mirror formula for $|E_2/E_1|(x,y)>1$ still has the same form of a series expansion, and the photon-pathway understanding remains valid, albeit with less trivial spatial envelopes, which become functions of $\overline{\alpha }$.

Figure 5

Raw image recorded at the back of the MCP for a Gaussian main driving beam and a ${\mathcal{H}}_1$ secondary beam. It corresponds to $\overline{\alpha }\approx 0.8$ (the highest amplitude ratio that we could reach with our setup) to show many beamlets with the expected parities and clear deviations from the perturbative-regime profiles. The white horizontal line indicates the position of the $p=0$ beamlet for all harmonics.

Figure 6

(a) Intensity profile of harmonic $q=11$ with a ${\mathcal{H}}_1$ main driving beam and Gaussian secondary beam. The digits indicate the order $p$ of each beamlet. The inset image displays the IR intensity at focus. (b) Top: lineouts along the horizontal lines in (a) (blue) and the best fit with the sum of the first ten HG modes (red lines). Bottom: corresponding HG mode decompositions. The vertical dashed line indicates the barycenter of the spectrum. (c) Intensity profile of harmonic $q=11$ with the same ${\mathcal{H}}_1$ mode in both the main beam and the secondary beam. The inset displays the IR intensity at focus. (d) Top: lineouts along the horizontal lines in (c) (blue) and the best fit with the sum of the first ten HG modes (red lines). Bottom: corresponding HG mode decomposition.

Figure 7

Comparison between the exact near-field amplitude $J_p\left(X\right)e^{-a{X}^2}$ of Eq. (B3) (red lines) and the approximation using Eq. (B3) (blue lines) for $\overline{\alpha }=0.05, q_{\text{eff}}=3.5$, and $q=11$ for different beamlets $p$.