Asymmetric phase error in circular apertures

The scalar diffraction pattern of a circular aperture is determined under the condition that the aperture field exhibits an asymmetric phase error of the form
\psi(r,\phi') = -\xi(r) \cos \phi'
, where
r
and
\phi'
are polar coordinates in the aperture. The aperture phase varies as
\cos \phi'
and, therefore, is an odd function about the plane defined by
\phi' = \pi/2, 3\pi/2
. Function
\xi(r)
defines the radial variation of phase in the aperture. The radiation pattern is calculated by computer methods to determine the beam shift and pattern distortion which results from the asymmetric phase error. Radiation patterns are presented for the cases
\xi(r) = \beta r^{2}
, and
\xi(r) = \beta r^{3}
, where constant
\beta
takes on the values
\beta = 0, \pi/4
, and
\pi/2
. The aperture field is specified to be the parabolic squared on a pedestal distribution
f(r) = b + (1-b) (1-r^{2})^{2}
. Patterns are calculated for edge tapers of 0 db, -12 db, and -20 db. Although the patterns are presented for particular phase error functions, the diffraction integral is readily evaluated for arbitrary phase functions. The patterns conveniently illustrate the effects of asymmetric phase error in circular apertures. A paraboloidal reflector which is used as a satellite antenna can experience asymmetric distortion under certain conditions of sun illumination. The analysis presented here can be used to determine the reflector pattern under such conditions. The phase function and radiation pattern of a paraboloid which undergoes thermal distortion is given as an example.