  Time-resolved electron microscopy aims at tracking nanoscale excitations and
dynamic states of matter with a temporal resolution ultimately reaching the
attosecond regime. Periodically time-varying fields in an illuminated specimen
cause free-electron inelastic scattering, which enables the spectroscopic
imaging of near-field intensities. However, access to the evolution of
nanoscale fields and structures within the light cycle requires a sensitivity
to the optical phase. Here, we introduce Free-Electron Homodyne Detection
(FREHD) as a universally applicable approach to electron microscopy of
phase-resolved optical responses at high spatiotemporal resolution. In this
scheme, a phase-controlled reference interaction serves as the local oscillator
to extract arbitrary sample-induced modulations of a free-electron wave
function. We demonstrate this principle through the phase-resolved imaging of
plasmonic fields with few-nanometer spatial and sub-cycle temporal resolutions.
Due to its sensitivity to both phase- and amplitude-modulated electron beams,
FREHD measurements will be able to detect and amplify weak signals stemming
from a wide variety of microscopic origins, including linear and nonlinear
optical polarizations, atomic and molecular resonances and attosecond-modulated
structure factors.
