Imaging by Zernike phase plates in the TEM

Abstract

Abbe lens theory can be used to calculate the images produced by a system made up of a simple phase object, a round lens and a Zernike phase plate. When all these have rotational symmetry, 1D Fourier‐Bessel transforms are suitable. Given a disc object producing a uniform phase shift over its diameter and a phase plate providing a simple step in phase by α at a defined spatial frequency q 0 , the resulting image intensity can be defined for any size of object phase change. The distortion of the image by the central hole in the Zernike plate depends on a parameter B which is proportional to the product of the diameter of the object and q 0 and also depends on the lens focal length; when B is less than 1, the distortion is small (Fig. 1). The range of object phase that produces a monotonic variation of intensity is found to depend strongly on the phase change α introduced by the phase plate, being greatest when α is between π and 2π (Fig. 2). For this object and plate, the weak phase approximation (WPA) is useful in giving an indication of the radial distribution of intensity. It does not predict well the range of monotonic variation of intensity with object phase. In the search to reduce the ‘ringing’ in the image produced by large objects with a stepped plate, we analysed also the behaviour when the phase change at the plate increases to its maximum value α over a range of radius (instead of a step). When the phase change is proportional to radius for q < q 0 , we find that as object diameter increases, the intensity is reduced gracefully when α ∼ π/2, but that contrast reversal can occur when α ∼ 3π/2. The intensity has also been calculated in WPA for an object phase distribution typical of a spherical object. With this object, the image intensity varies continuously with object radius, as expected, and is reduced as B increases above 1. The behaviour when the phase profile of the plate is ramped is similar to that for the disc object with the same phase profile for the plate (Fig. 3). The advantages for different kinds of objects offered by plates with phase shifts α either less than or greater than π suggest that in practical use it will be desirable for phase plate holders to be fitted with plates of both thicknesses.