  We describe a scheme that characterizes the main beam and sidelobe in all
Stokes parameters employing parameters that allow reconstruction of the
complete beam patterns and, also, afford an easy way to see how the beam
changes with azimuth, zenith angle, and time. For the main beam in Stokes I the
parameters include the beam width, ellipticity and its orientation, coma and
its orientation, the point-source gain, the integrated gain (or, equivalently,
the main beam efficiency); for the other Stokes parameters the beam parameters
include beam squint and beam squash. For the first sidelobe ring in Stokes I
the parameters include an 8-term Fourier series describing the height, radius,
and radial width; for the other Stokes parameters they include only the
sidelobe's fractional polarization.
  We illustrate the technique by applying it to the Arecibo telescope. The main
beam width is smaller and the sidelobe levels higher than for a
uniformly-illuminated aperture of the same effective area. These effects are
modeled modestly well by a blocked aperture, with the blocked area equal to
about 10% of the effective area (this corresponds to 5% physical blockage). In
polarized emission, the effects of beam squint (difference in pointing
direction between orthogonal polarizations) and squash (difference in beamwidth
between orthogonal polarizations) do not correspond to theoretical expectation
and are higher than expected; these effects are almost certainly caused by the
blockage. The first sidelobe is highly polarized because of blockage.
