Nonlinear optical methods have become ubiquitous in many scientific areas, from fundamental studies of time-resolved electron dynamics to microscopy and spectroscopy applications. They are, however, often limited to a certain range of parameters such as pulse energy and average power. Restrictions arise from, for example, the required field intensity as well as from parasitic nonlinear effects and saturation mechanisms. Here, we identify a fundamental principle of nonlinear light–matter interaction in gases and show that paraxial nonlinear wave equations are scale-invariant if spatial dimensions, gas density, and laser pulse energy are scaled appropriately. As an example, we apply this principle to high-order harmonic generation and provide a general method for increasing peak and average power of attosecond sources. In addition, we experimentally demonstrate the implications for the compression of short laser pulses. Our scaling principle extends well beyond those examples and includes many nonlinear processes with applications in different areas of science.

Significance: In most optics textbooks, one writes the electric field describing an optical wave as a product of temporal and spatial amplitudes. This approximation often breaks down for short optical pulses. An example of such spatiotemporal coupling is chromatic aberrations, where the focal properties of the radiation vary with frequency over the pulse bandwidth. In this work, we point out significant chromatic aberrations of attosecond pulses, which depend on the geometry of the generation process. These aberrations are intrinsic to the generation process and need to be eliminated in applications requiring attosecond pulses to be focused over a small region.

We present a demonstration of two-photon double ionization of neon using an intense extreme ultraviolet (XUV) attosecond pulse train (APT) in a photon energy regime where both direct and sequential mechanisms are allowed. For an APT generated through high-order harmonic generation (HHG) in argon we achieve a total pulse energy close to $1\mu \mathrm{J}$, a central energy of 35 eV, and a total bandwidth of $\sim 30$ eV. The APT is focused by broadband optics in a neon gas target to an intensity of $3\times {10}^{12}\mathrm{W}{\mathrm{cm}}^{-2}$. By tuning the photon energy across the threshold for the sequential process the double ionization signal can be turned on and off, indicating that the two-photon double ionization predominantly occurs through a sequential process. The demonstrated performance opens up possibilities for future XUV-XUV pump-probe experiments with attosecond temporal resolution in a photon energy range where it is possible to unravel the dynamics behind direct versus sequential double ionization and the associated electron correlation effects.

We present an optical system based on two toroidal mirrors in a Wolter configuration to focus broadband extreme ultraviolet (XUV) radiation. Optimization of the focusing optics alignment is carried out with the aid of an XUV wavefront sensor. Back-propagation of the optimized wavefront to the focus yields a focal spot of 3.6 × 4.0 µm² full width at half maximum, which is consistent with ray-tracing simulations that predict a minimum size of 3.0 × 3.2 µm². This work is important for optimizing the intensity of focused high-order harmonics in order to reach the nonlinear interaction regime.

We perform wavefront measurements of high-order harmonics using an extreme-ultraviolet (XUV) Hartmann sensor and study how their spatial properties vary with different generation parameters, such as pressure in the nonlinear medium, fundamental pulse energy and duration as well as beam size. In some conditions, excellent wavefront quality (up to $\lambda /11$) was obtained. The high throughput of the intense XUV beamline at the Lund Laser Centre allows us to perform single-shot measurements of both the full harmonic beam generated in argon and individual harmonics selected by multilayer mirrors. We theoretically analyze the relationship between the spatial properties of the fundamental and those of the generated high-order harmonics, thus gaining insight into the fundamental mechanisms involved in high-order harmonic generation (HHG).