The cut-off energy was shown to scale linearly with both field amplitude and wavelength using quantum mechanical calculations, which agree with pioneering experiments 8 and the momentum-space trajectory picture 7, 13, 16, 17, 18, 19, 20. For HHG in crystalline solids, an analogous electron-trajectory picture can, in principle, be applied in momentum space (after applying Bloch’s theorem), which includes both same-site recollision 7 and coherent scattering from the nearest neighbour atoms 8, 9, 10, 11, 12, 13, 14, 15. A hallmark of the model is that it correctly predicts the HHG cut-off and its quadratic dependence on both the electric-field amplitude and wavelength, as these depend on the trajectory of the most energetic returning electron trajectory 5, 6. The results of this approach are usually in good agreement with those of full ab-initio calculations, and can be used to extract dynamical information from high-harmonic spectroscopy 3, 4. In the gas phase, this understanding is often based on the semi-classical three-step model (TSM) 1 or its quantum mechanical extension 2, which describes high-harmonic generation (HHG) in terms of a set of electron trajectories initiated by a tunnelling process. Our work also indicates the possibility of resolving subfemtosecond electron dynamics in liquids offering an all-optical approach to attosecond spectroscopy of chemical processes in their native liquid environment.Ī prerequisite for accurately interpreting the underlying dynamics of a system using measured high-harmonic spectra lies in formulating a broadly applicable theoretical or conceptual model. This regime is extremely difficult to access with other methodologies, but is critical for understanding radiation damage to living tissues. Our results propose high-harmonic spectroscopy as an all-optical approach for determining the effective mean free paths of slow electrons in liquids. This is further confirmed by measurements performed with elliptically polarized light and with ab-initio time-dependent density functional theory calculations. We explain these observations with a semi-classical model based on electron trajectories that are limited by the electron scattering. By studying high-harmonic generation over a broad range of wavelengths and intensities, we show that the cut-off energy is independent of the wavelength beyond a threshold intensity and that it is a characteristic property of the studied liquid. Here we extend its use to liquid samples. It has been applied to a variety of systems in the gas phase and solid state. High-harmonic spectroscopy is an all-optical nonlinear technique with inherent attosecond temporal resolution.
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