An annealing algorithm to correct positioning errors in ptychography

Ptychography offers the possibility of improving the resolution of atomic-scale (electron and X-ray) transmission microscopy without any of the demands of high quality lenses: its resolution is in theory only limited by the effective synthetic numerical aperture determined by the angular size of the detector. However, it has been realised experimentally that a major weakness of the approach is that the obtainable resolution is only as good as the accuracy to which the illuminating beam can be moved relative to the specimen. This can be catastrophic in the electron case because of thermal drift and hysteresis in the probe scan coils. We present here a computationally efficient extension of the 'ePIE' ptychographic reconstruction algorithm for correcting these errors retrospectively. We demonstrate its effectiveness using simulations and results from visible light and electron beam experiments that show it can correct positioning errors tens of times larger than the pixel size in the resulting image.