Synthesizing free-electron wave functions by stimulated near-field interactions
Stimulated electron-optical-near-field interactions imprint a coherent, phase-coherent modulation on the wave function of a free electron, of which the free-space dispersion subsequently converts into a train of attosecond density peaks. Light thereby becomes a tool for synthesizing the electron wave function itself, setting when, where, and how narrowly the electron density concentrates. Here we study photon-induced near-field electron microscopy (PINEM) to gain control over that synthesis in two stages: first for a single PINEM interaction, whose design parameters we obtain in closed form, and then for multiple PINEM interactions acting in parallel, which extends the control across space and time. The design rules follow from decomposing the propagated density into temporal harmonics, which we classify into three: (i) the arrival time of the attosecond train, fixed by the optical phase of the coupling; (ii) the distance at which the train forms, set by diffraction at the point where all classical trajectories converge; and (iii) the pulse duration, fixed by the imprinted energy spread. We evaluate these parameters for a scanning-electron-microscope (SEM) configuration at 10~keV, where the compression completes within tens of micrometers from the interaction region, with results 20% better than previous approaches. Letting the electron then interact with several spatially separated near fields in parallel, normal to the trajectory, each pathway carrying its own coupling strength and optical phase, we show that retaining or erasing which-path information changes both the propagated density and the measurable energy spectrum, the latter providing in turn a measure of the mutual coherence of the driving fields. These results establish PINEM as a quantitative synthesis tool for free-electron wave functions in compact, low-voltage electron microscopes.