  Controlling free-electron momentum states is of high interest in electron
microscopy to achieve momentum and energy resolved probing and manipulation of
physical systems. Free-electron and light interactions have emerged as a
powerful technique to accomplish this. Here, we demonstrate both longitudinal
and transverse phase control of a slow electron wavepacket by extending the
Kapitza-Dirac effect to spatially-structured pulsed laser beams. This extension
enables both inelastic and elastic stimulated Compton scattering. The
interaction reveals the formation of distinct electron transverse momentum
orders, each demonstrating a comb-like electron energy spectrum. By exerting
complete control over light parameters, including wavelength, field intensity,
pulse duration, and spatial mode order, as well as their combinations, it is
possible to coherently control the population of these electron energy-momentum
states that are separated by a few meV energy and multiple photon momentum
orders. This free-space electron-light interaction phenomenon possesses the
capability to coherently control the energy and momentum of electron beams in
electron microscopes. Moreover, it has the potential to facilitate the
selective probing of various material excitations, including plasmons,
excitons, and phonons, and performing Talbot-Lau matter-wave interferometry
with transversely shaped electron beams.
