Semi-Classical Cutoff Energies for Electron Emission and Scattering at Field-Enhancing Nanostructures with Large Ponderomotive Amplitudes

TL;DR

This paper generalizes semi-classical cutoff energies for electron emission at nanostructures by accounting for spatial field drop-off, revealing significant deviations from uniform field assumptions especially at large wavelengths and high fields.

Contribution

It introduces a method to calculate cutoff energies considering spatial field variations, extending classical models to more accurately describe electron dynamics at nanostructures.

Findings

Modified cutoff energies differ significantly from uniform field predictions.

Electron emission energy can be up to a factor of the ponderomotive energy higher.

Comparison with quantum simulations validates the generalized cutoff calculations.

Abstract

The uniform field assumption used to derive semi-classical cutoff energies of $10U_p$ for electron emission and $3.17U_p$ for high harmonic generation is applicable for ponderomotive amplitudes ($\propto E\lambda^2$) much smaller than the field drop-off scale. For large wavelength and high field experiments at nanoscale structures this assumption may break down by predicting energies beyond the true classical energy limits. Here we provide generalized calculations for these cutoff energies by taking into account the spatial field drop-off. The modified cutoff energies vary significantly from the uniform field results even with ponderomotive amplitudes still an order of magnitude below the field drop-off scale. Electron emission and scattering energy as a function of the time-of-ionization is considered for the nanotip ($\sim1/r^2$) field profile. The cutoff energies as a function of the adiabaticity parameter $\delta$, which may be easily calculated for given wavelength, apex field strength, and nanostructure scale, are then determined through maximization for nanotip, nanoblade ($\sim1/r$), and exponential field profiles. These profiles deviate from each other in electron emission energy by up to nearly a factor of the ponderomotive energy, indicating the importance of mid-field profile behavior. The electron emission energy cutoff also attains an additional factor of $U_p$ due to the smooth integrated ponderomotive force in the adiabatic drop-off and very long pulse regime. These results also provided as double-exponential fits for ease of use. We then compare the nanoblade electron emission cutoffs with a quantum simulation of the electron rescattering process. We also consider a short (few-cycle) pulsed field, focusing on a cosine-like pulse and overviewing the general carrier-envelope phase dependencies.