  Free-electron beams serve as uniquely versatile probes of microscopic
structure and composition, and have repeatedly revolutionized atomic-scale
imaging, from solid-state physics to structural biology. Over the past decade,
the manipulation and interaction of electrons with optical fields has seen
significant progress, enabling novel imaging methods, schemes of near-field
electron acceleration, and culminating in 4D microscopy techniques with both
high temporal and spatial resolution. However, weak coupling strengths of
electron beams to optical excitations are a standing issue for existing and
emerging applications of optical free-electron control. Here, we demonstrate
phase matched near-field coupling of a free-electron beam to optical whispering
gallery modes of dielectric microresonators. The cavity-enhanced interaction
with these optically excited modes imprints a strong phase modulation on
co-propagating electrons, which leads to electron-energy sidebands up to
hundreds of photon orders and a spectral broadening of 700 eV. Mapping the
near-field interaction with ultrashort electron pulses in space and time, we
trace the temporal ring-down of the microresonator following a femtosecond
excitation and observe the cavity's resonant spectral response. Resonantly
enhancing the coupling of electrons and light via optical cavities, with
efficient injection and extraction, can open up novel applications such as
continuous-wave acceleration, attosecond structuring, and real-time all-optical
electron detection.
