text: For the pinhole configuration the width of the beam (which we aim to minimise) is largely given by geometrical optics. The size of the beam at the sample plane is given by the lines connecting the skimmer edges with the pinhole edges.  When the Fresnel number is very small (), the beam width is also affected by Fraunhofer diffraction (see equation below).
  In this equation  is the Full Width at Half Maximum of the beam,  is the geometrical projection of the beam and  is the Airy diffraction term.  is the Heaviside step function used here to indicate that the presence of the diffraction term depends on the value of the Fresnel number. Note that there are variations of this equation depending on what is defined as the "beam width" (for details compare  and ). Due to the small wavelength of the helium beam, the Fraunhofer diffraction term can usually be omitted.
  The intensity of the beam (which we aim to maximise) is given by the following equation (according to the Sikora and Andersen model):
  Where  is the total intensity stemming from the supersonic expansion nozzle (taken as a constant in the optimisation problem),  is the radius of the pinhole, S is the speed ratio of the beam,  is the radius of the skimmer,  is the radius of the supersonic expansion quitting surface (the point in the expansion from which atoms can be considered to travel in a straight line), is the distance between the nozzle and the skimmer and  is the distance between the skimmer and the pinhole. There are several other versions of this equation that depend on the intensity model, but they all show a quadratic dependency on the pinhole radius (the bigger the pinhole, the more intensity) and an inverse quadratic dependency with the distance between the skimmer and the pinhole (the more the atoms spread, the less intensity).
  By combining the two equations shown above, one can obtain that for a given beam width  for the geometrical optics regime the following values correspond to intensity maxima:
  In here,  represents the working distance of the microscope and  is a constant that stems from the definition of the beam width. Note that both equations are given with respect to the distance between the skimmer and the pinhole, a. The global maximum of intensity can then be obtained numerically by replacing these values in the intensity equation above. In general, smaller skimmer radii coupled with smaller distances between the skimmer and the pinhole are preferred, leading in practice to the design of increasingly smaller pinhole microscopes.