# Spatial coherence of electron beams from field emitters and its effect on the resolution of imaged objects

## TL;DR

This paper investigates how the spatial coherence of electron beams from nano-tips affects imaging resolution, revealing that traditional methods may overestimate source size and providing a new formula for intrinsic resolution estimation.

## Contribution

The study critically examines the application of the van Cittert Zernike theorem to coherent electron sources and introduces a simple formula for estimating intrinsic resolution in electron holography.

## Key findings

- Effective source size is 2-3 times larger than predefined.
- Visibility in holograms is less than 1 due to Gaussian source distribution.
- Proposed formula aids in estimating intrinsic resolution for nano-tip electron sources.

## Abstract

Sub-nanometer and nanometer-sized tips provide high coherence electron sources. Conventionally, the effective source size is estimated from the extent of the experimental biprism interference pattern created on the detector by applying the van Cittert Zernike theorem. Previously reported experimental intensity distributions on the detector exhibit Gaussian distribution and our simulations show that this is an indication that such electron sources must be at least partially coherent. This, in turn means that strictly speaking the Van Cittert Zernike theorem cannot be applied, since it assumes an incoherent source. The approach of applying the van Cittert Zernike theorem is examined in more detail by performing simulations of interference patterns for the electron sources of different size and different coherence length, evaluating the effective source size from the extent of the simulated interference pattern and comparing the obtained result with the pre-defined value. The intensity distribution of the source is assumed to be Gaussian distributed, as it is observed in experiments. The visibility or the contrast in the simulated holograms is found to be always less than 1 which agrees well with previously reported experimental results and thus can be explained solely by the Gaussian intensity distribution of the source. The effective source size estimated from the extent of the interference pattern turns out to be of about 2-3 times larger than the pre-defined size, but it is approximately equal to the intrinsic resolution of the imaging system. A simple formula for estimating the intrinsic resolution, which could be useful when employing nano-tips in in-line Gabor holography or point-projection microscopy, is provided.

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Source: https://tomesphere.com/paper/1703.01582