  Accessing the low-energy non-equilibrium dynamics of materials and their
polaritons with simultaneous high spatial and temporal resolution has been a
bold frontier of electron microscopy in recent years. One of the main
challenges lies in the ability to retrieve extremely weak signals while
simultaneously disentangling amplitude and phase information. Here, we present
Free-Electron Ramsey Imaging (FERI), a microscopy approach based on
light-induced electron modulation that enables coherent amplification of
optical near-fields in electron imaging. We provide simultaneous time-, space-,
and phase-resolved measurements of a micro-drum made from a hexagonal boron
nitride membrane visualizing the sub-cycle dynamics of 2D polariton wavepackets
therein. The phase-resolved measurements reveals vortex-anti-vortex
singularities on the polariton wavefronts, together with an intriguing
phenomenon of a traveling wave mimicking the amplitude profile of a standing
wave. Our experiments show a 20-fold coherent amplification of the near-field
signal compared to conventional electron near-field imaging, resolving peak
field intensities in the order of ~W/cm2, corresponding to field amplitudes of
a few kV/m. As a result, our work paves the way for spatio-temporal electron
microscopy of biological specimens and quantum materials, exciting yet delicate
samples that are currently difficult to investigate.
