Figure 5: Size-dependent emission parameters.

Comparison of measured (symbols) and calculated (lines) critical final emission angles (a), critical phases (b), and cutoff energies E_c (c) as function of nanosphere size. Energies are given in units of the free space ponderomotive potential U_p. The experimental and theoretical intensities are (3.0±0.5) × 10¹³ and 3.0 × 10¹³ W cm⁻², respectively. Differences between solid and broken red lines in (a,b) indicate the small dependence of the M³C results on the threshold energy E_th. M³C electron trajectories with E=E_c±1% are used in (c) to calculate the selective energy gain from different field contributions . Compared with the SMM result, the energy gains calculated with Mie fields (blue lines) show the enhancement due to the local trapping potential. The difference between gains for radial and full Mie fields (dotted versus dashed blue line) quantifies the tangential field contribution. Space-charge repulsion is included when using the full self-consistent near-field for the energy gain integration, recovering the cutoff energy of the full M³C runs. For details see Methods. Note that because of low overall count rate for d=550 nm particles (low aerodynamic lens transmission) we might underestimate the cutoff energy for this size. The slope of the critical SMM phase in (b) mainly reflects the small trivial CEP shift, that is, the propagation induced CEP offset of the effective radial field relative to the incoming pulse. Horizontal error bars in (a–c) indicate the standard deviation of the particle diameter. Vertical error bars in (a) and (b) are composed by the standard deviations of upper and lower detector sides combined with systematic errors estimated to be 4° for and 0.1 π for , respectively. The error for E_c in (c) is dominated by the uncertainty associated with laser intensity which is approximated to be ±15%. Uncertainties (≲5%) in the refractive index and work function because of surface contamination modify the simulation results by less than the experimental error bars. (d,e) Time evolution of the energy gains for two selected sphere sizes (as indicated); note the different scaling of the time axes before and after the black vertical lines. The pulse peak (in free space and at x=0) is set to t=15 fs.