Differential phase contrast: An integral perspective

Differential phase contrast (DPC) in a scanning transmission electron microscope is a widely employed technique for probing electromagnetic fields on the nanoscale. We show that the DPC signal corresponds to the averaged lateral probability current of the scattered electron probe. Based on this result we discuss the interpretation of DPC in terms of the projected electric and magnetic fields and the influence of experimental parameters thereon. We further show that DPC can be interpreted as a quantum weak measurement and that the reciprocal broad beam illumination technique is given by an astigmatic transport of intensity reconstruction.

Figure 1

DPC setup in one lateral dimension. The shift of the momentum center of mass recorded in the far field of the specimen is indicated by the thick (green) arrow.

Figure 2

Detector geometries employed in STEM DPC: (a) conventional dark-field annular four-quadrant detector, (b) advanced segmented annular detector, (c) Cartesian pixel detector, and (d) speculative Cartesian stripe detector illustrating the binning degree of freedom of (c). The labels at the bottom indicate the type of averaged probability current measured with this detector.

Figure 3

Astigmatic TIE setup in one lateral dimension. The defocus step is denoted $\delta z$. The differential shift of the intensity with respect to the neighboring image point is indicated by the thick (green) arrow. Note the (generalized) reciprocal beam path in comparison to the DPC setup (Fig. 1).