On detector linearity and precision of beam shift detection for quantitative differential phase contrast applications

Differential phase contrast is a STEM imaging mode where minute sideways deflections of the electron probe are monitored, usually by using a position sensitive device (Chapman, 1984 [1]; Lohr et al., 2012 [2]) or, alternatively in some cases, a fast camera (Müller et al., 2012 [3,4]; Yang et al., 2015 [5]; Pennycook et al., 2015 [6]) as a pixelated detector. While traditionally differential phase contrast electron microscopy was mainly focused on investigations of micro-magnetic domain structures and their specific features, such as domain wall widths, etc. (Chapman, 1984 [1]; Chapman et al., 1978, 1981, 1985 [7-9]; Sannomiya et al., 2004 [10]), its usage has recently been extended to mesoscopic (Lohr et al., 2012, 2016 [2,12]; Bauer et al., 2014 [11]; Shibata et al., 2015 [13]) and nano-scale electric fields (Shibata et al., 2012 [14]; Mueller et al., 2014 [15]). In this paper, the various interactions which can cause a beam deflection are reviewed and expanded by two so far undiscussed mechanisms which may be important for biological applications. As differential phase contrast microscopy strongly depends on the ability to detect minute beam deflections we first treat the linearity problem for an annular four quadrant detector and then determine the factors which limit the minimum measurable deflection angle, such as S/N ratio, current density, dwell time and detector geometry. Knowing these factors enables the experimenter to optimize the set-up for optimum performance of the microscope and to get a clear figure for the achievable field resolution error margins.