A Bloch wave analysis of optical sectioning in aberration-corrected STEM

The reduction in the focal depth of field that occurs through the use of larger apertures in aberration-corrected STEM allows three-dimensional information to be retrieved by optical depth sectioning. This paper explores depth sectioning in zone-axis crystals using Bloch wave calculations. By decomposing the calculation into the contribution from individual states and from individual partial plane waves in the convergent cone of illumination, we explain the form of the electron intensity in the crystal as a function of depth. Two separate effects are found that can cause the intensity maximum to deviate from that of the expected defocus value. Firstly it is found that the unbound, high angle excited states give rise to a behaviour similar to that of the probe focusing in the vacuum, but with a prefocusing effect due to the lensing effect of the potential of the atomic column. Superimposed upon this prefocused peak is an oscillation due to interference between the channelling 1s state and the rest of the wavefunction. This oscillation can actually prevent an intensity maximum being formed at certain depths in the crystal, and will complicate the interpretation of optical sectioning data.