Imaging by Zernike phase plates in the TEM

• Rotationally symmetric systems can be analysed by 1D Fourier-Bessel transforms.
• Disc and spherical objects are modelled.
• Phase plates with step and ramp onset of phase change are modelled.
• The image from a disc object and a stepped plate can be found for any object phase.
• The monotonic range is larger for a plate phase change of 3π/2 than for π/2.

The images produced from simple phase objects, lenses and Zernike phase plates when all have rotational symmetry can be calculated by 1D Fourier–Bessel transforms. For a simple disc object producing a uniform phase shift over its diameter, the resulting image can be defined for any size of object phase change. The monotonic range of intensity variation with object phase is found to depend strongly on the phase change introduced by the phase plate; this property of the system is not well predicted by the weak phase approximation. The effect of spreading the phase transition at the plate over a range of radius is beneficial if the plate phase change is sufficiently small. Weak-phase calculations for a phase distribution more typical of a spherical object are also shown.