  We present 3D calculations for dielectric haloscopes such as the currently
envisioned MADMAX experiment. For ideal systems with perfectly flat, parallel
and isotropic dielectric disks of finite diameter, we find that a geometrical
form factor reduces the emitted power by up to $30\,\%$ compared to earlier 1D
calculations. We derive the emitted beam shape, which is important for antenna
design. We show that realistic dark matter axion velocities of $10^{-3} c$ and
inhomogeneities of the external magnetic field at the scale of $10\,\%$ have
negligible impact on the sensitivity of MADMAX. We investigate design
requirements for which the emitted power changes by less than $20\,\%$ for a
benchmark boost factor with a bandwidth of $50\,{\rm MHz}$ at $22\,{\rm GHz}$,
corresponding to an axion mass of $90\,\mu{\rm eV}$. We find that the maximum
allowed disk tilt is $100\,\mu{\rm m}$ divided by the disk diameter, the
required disk planarity is $20\,\mu{\rm m}$ (min-to-max) or better, and the
maximum allowed surface roughness is $100\,\mu{\rm m}$ (min-to-max). We show
how using tiled dielectric disks glued together from multiple smaller patches
can affect the beam shape and antenna coupling.
