Fourier transform of a polygonal shape function and its application in electromagnetics

The shape function
s(x, y)
of a planar aperture
\Sigma
is defined ashaving value one inside
\Sigma
and zero outside. The two-dimensional Fourier transform of
s(x, y)
when
\Sigma
is an
N
-sided polygon is considered, and a closed-form exact solution is presented. Applications of these results are described for 1) the calculation of the secondary pattern of a reflector antenna with a noncircular aperture, and 2) high-frequency transmission through a periodic frequency selective surface.