  We explore finite size 3D effects in open axion haloscopes such as a dish
antenna, a dielectric disk and a minimal dielectric haloscope consisting of a
mirror and one dielectric disk. Particularly dielectric haloscopes are a
promising new method for detecting dark matter axions in the mass range above
$40\,\mu{\rm eV}$. By using two specialized independent approaches - based on
finite element methods and Fourier optics - we compute the electromagnetic
fields in these settings expected in the presence of an axion dark matter
field. This allows us to study diffraction and near field effects for
realistically sized experimental setups in contrast to earlier idealized 1D
studies with infinitely extended mirrors and disks. We also study axion
velocity effects and disk tiling. Diffraction effects are found to become less
relevant towards larger axion masses and for the larger disk radii for example
aimed at in full size dielectric haloscopes such as MADMAX. The insights of our
study not only provide a foundation for a realistic modelling of open axion
dark matter search experiments in general, they are in particular also the
first results taking into account 3D effects for dielectric haloscopes.
