  The complex range of interactions between electrons and electromagnetic
fields gave rise to countless scientific and technological advances. A prime
example is photon-induced nearfield electron microscopy (PINEM), enabling the
detection of confined electric fields in illuminated nanostructures with
unprecedented spatial resolution. However, PINEM is limited by its dependence
on strong fields, making it unsuitable for sensitive samples, and its inability
to resolve complex phasor information. Here, we leverage the nonlinear,
over-constrained nature of PINEM to present an algorithmic microscopy approach,
achieving far superior nearfield imaging capabilities. Our algorithm relies on
free-electron Ramsey-type interferometry to produce orders-of-magnitude
improvement in sensitivity and ambiguity-immune nearfield phase reconstruction,
both of which are optimal when the electron exhibits a fully quantum behavior.
Our results demonstrate the potential of combining algorithmic approaches with
novel modalities in electron microscopy, and may lead to various applications
from imaging sensitive biological samples to performing full-field tomography
of confined light.
