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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 isapplicable for ponderomotive amplitudes ($\propto E\lambda^2$) much smallerthan the field drop-off scale. For large wavelength and high field experimentsat nanoscale structures this assumption may break down by predicting energiesbeyond the true classical energy limits. Here we provide generalizedcalculations for these cutoff energies by taking into account the spatial fielddrop-off. The modified cutoff energies vary significantly from the uniformfield results even with ponderomotive amplitudes still an order of magnitudebelow the field drop-off scale. Electron emission and scattering energy as afunction 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 fieldstrength, and nanostructure scale, are then determined through maximization fornanotip, nanoblade ($\sim1/r$), and exponential field profiles. These profilesdeviate from each other in electron emission energy by up to nearly a factor ofthe ponderomotive energy, indicating the importance of mid-field profilebehavior. The electron emission energy cutoff also attains an additional factorof $U_p$ due to the smooth integrated ponderomotive force in the adiabaticdrop-off and very long pulse regime. These results also provided asdouble-exponential fits for ease of use. We then compare the nanoblade electronemission cutoffs with a quantum simulation of the electron rescatteringprocess. We also consider a short (few-cycle) pulsed field, focusing on acosine-like pulse and overviewing the general carrier-envelope phasedependencies.

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