The Cramér-Rao lower bound for the precision of scan position refinement in momentum-resolved STEM

Accurate knowledge of probe positions in ptychographic object retrieval using momentum-resolved scanning transmission electron microscopy (STEM) is vital for reconstructing phase gratings with utmost precision and spatial resolution. When measuring the entire 2D diffraction pattern, ptychography can not only reconstruct the specimen's potential landscape but also refine the scan positions via various methods. However, due to Poisson shot noise, the achievable precision limit for each refinement strategy is constrained by the measurement itself. Using the concept of Fisher information, we derive a method to calculate the highest achievable precision for the scan positions that any unbiased estimator could attain, called the Cramér-Rao lower bound (CRLB). We investigate the impacts of different simulation, experimental, and reconstruction parameters on the CRLB, namely dose, defocus, partial coherence, and specimen structure. Using the Wirtinger Flow, the results are compared to the effectively achieved precisions in gradient-based reconstructions. Based on these findings, the CRLB is worked out as an important tool for both the choice of experimental settings in a STEM acquisition and especially as a guideline for any scan position refinement strategy, in that it offers a robust criterion to decide whether further optimization can be expected to improve or even worsen the result.