A Numerical Analysis of Interaction-free Measurement With a Shot-Noise-Limited Electron Source and Conditional Sample Re-illumination

Interaction-free measurement (IFM) has been proposed as a means of high-resolution, low-damage imaging of radiation-sensitive samples, such as proteins and biomolecules. The basic setup for IFM is a Mach-Zehnder interferometer (MZI). Recent progress in nanofabricated electron diffraction gratings has made it possible to incorporate an MZI in a transmission-electron microscope (TEM). Therefore, analysis of IFM with an MZI and a shot-noise-limited electron source (such as that in a TEM) is of interest. In this work, we numerically compared the error probability and number of scattered electrons for IFM and classical imaging schemes, with and without an additional detector for scattered electrons, for a black and white sample. The additional detector reduced error for the same number of scattered electrons by an order of magnitude, for both IFM and classical schemes. We also investigated a sample re-illumination scheme based on updating priors after each round of illumination until a stopping criterion is met. This scheme further reduced error by a factor of two. Implementation of these methods is likely achievable with existing instrumentation and would result in improved resolution in low-dose electron microscopy.