text: The zone plate microscope uses a zone plate (that acts roughly like as a classical lens) instead of a pinhole to focus the atom beam into a small focal spot. This means that the beam width equation changes significantly (see below).
  Here,  is the zone plate magnification and  is the width of the smallest zone. Note the presence of chromatic aberrations (). The approximation sign indicates the regime in which the distance between the zone plate and the skimmer is much bigger than its focal length.
  The first term in this equation is similar to the geometric contribution  in the pinhole case: a bigger zone plate (taken all parameters constant) corresponds to a bigger focal spot size. The third term differs from the pinhole configuration optics as it includes a quadratic relation with the skimmer size (which is imaged through the zone plate) and a linear relation with the zone plate magnification, which will at the same time depend on its radius.
  The equation to maximise, the intensity, is the same as the pinhole case with the substitution . By substitution of the magnification equation:
  where  is the average de-Broglie wavelength of the beam. Taking a constant , which should be made equal to the smallest achievable value, the maxima of the intensity equation with respect to the zone plate radius and the skimmer-zone plate distance  can be obtained analytically. The derivative of the intensity with respect to the zone plate radius can be reduced the following cubic equation (once it has been set equal to zero):
  Here some groupings are used:  is a constant that gives the relative size of the smallest aperture of the zone plate compared with the average wavelength of the beam and  is the modified beam width, which is used through the derivation to avoid explicitly operating with the constant airy term: .
  This cubic equation is obtained under a series of geometrical assumptions and has a closed-form analytical solution that can be consulted in the original paper or obtained through any modern-day algebra software. The practical consequence of this equation is that zone plate microscopes are optimally designed when the distances between the components are small, and the radius of the zone plate is also small. This goes in line with the results obtained for the pinhole configuration, and has as its practical consequence the design of smaller scanning helium microscopes.