Optimum cavity height for radiating a pencil beam at broadside from a Fabry–Perot cavity antenna

Fabry–Perot cavity antennas are widely used to produce a pencil beam at broadside. Over the years, various numerical and analytical techniques have been proposed to optimize the structure parameters to maximize the radiated power or the directivity at broadside. Most of these techniques are structure-specific or based on approximations. Here, we rigorously demonstrate that a fully analytical expression exists to set the cavity height to have maximum radiated power density at broadside. A new formula is also found to maximize the directivity at broadside. Connections with previous interpretations and approximate analytical formulas based on ray optics and leaky waves are reported. A practical case study is finally investigated to demonstrate the consistency of the results with full-wave simulations.