  We theoretically demonstrate the ability of electron beams to probe the
nonlinear photonic response with nanometer spatial resolution, well beyond the
capabilities of existing optical techniques. Although the interaction of
electron beams with photonic modes is generally weak, the use of optical
pumping produces stimulated electron-light interactions that can reach
order-unity probabilities in photon-induded near field electron microscopy
(PINEM). Here, we demonstrate that PINEM can locally and quantitatively probe
the nonlinear optical response. Specifically, we predict a dependence of PINEM
electron spectra on the sample nonlinearity that can reveal the second-harmonic
(SH) response of optical materials with nanometer resolution, observed through
asymmetries between electron energy losses and gains. We illustrate this
concept by showing that PINEM spectra are sensitive to the SH near field of
centrosymmetric structures and by finding substantial spectral asymmetries in
geometries for which the linear interaction is reduced.
