Fig. 1: Measurement of arbitrary amplitude and phase responses by free-electron homodyne detection.

Optical excitation of an investigated sample at frequency ω, inducing a response at the fundamental driving frequency and its harmonics. A transmitted or diffracted electron beam experiences modulation in its amplitude and/or phase, which traces the response. For example, a modulation of the magnitude of the structure factor \(f(\bf{k})\) of a material leads to an amplitude modulation of a diffracted electron wavefunction (left), whereas localized optical fields and polarizations typically result in a phase modulation (right). A second interaction with a local oscillator—serving as a reference or mixer with variable phase—yields antisymmetric and symmetric signals, respectively, in the final electron kinetic energy spectrum, which is measured in this homodyne detection scheme.