  Recent technological advances allowed the coherent optical manipulation of
high-energy electron wavepackets with attosecond precision. Here we
theoretically investigate the collision of optically-modulated pulsed electron
beams with atomic targets and reveal a quantum interference associated with
different momentum components of the incident broadband electron pulse, which
coherently modulates both the elastic and inelastic scattering cross sections.
We show that the quantum interference has a high spatial sensitivity at the
level of Angstroms, offering potential applications in high-resolution
ultrafast electron microscopy. Our findings are rationalized by a simple model.
