For any feed, one has to know how an incoming radiation in the
direction 
is mapped in the focal plane at the position of the
feed. This corresponds to a Jones matrix for each direction in the
sky. To define the Jones matrix, one has to give oneself reference
frames, both for the incoming radiation, in the plane tangent to the
celestial sphere in the direction , and in the focal
plane
,
for the radiation entering
the feed.
There are usual conventions for this, some of which are described in
Ludwig [Ludwig1973] (see figure 1 for the Ludwig III
convention). In the particular case of antennas, one only needs to know
the Co and Cross amplitudes, corresponding to the exact
direction of the antenna, in other word the first line of the Jones
matrix in the Co-Cross reference frame of the antenna. But, for any
change in the direction of the antenna, one needs the
full Jones matrix, or equivalently, the Co and Cross amplitudes for two
different directions of the antenna. In addition, for bolometric
observations, the antenna is replaced by a polarimeter which is never
perfect and will leak some energy in the orthogonal
direction to the bolometer. Therefore, the radiation pattern in the
orthogonal direction is also required, although a rough knowledge will
often be sufficient.
Figure 1: Co and Cross basis vector in the Ludwig III
convention