The Axial choke conical horn is a very popular reflector feed as it provides an almost-flat wavefront in the main beam with a sharp roll-off which is ideal for uniform dish illumination with little spillover.
This antenna can be seen as a small aperture horn with corrugations placed in the flanges to design for a wide beamwidth with good pattern symmetry and low cross-polarisation. The Classical choke horn is the extreme of a scalar horn which has been 'opened' completely - thus the corrugations of the choke horn remain directed along the main axis of the horn, while those of the scalar horn remain perpendicular to the flare wall.
The design algorithm in Antenna Magus can adjust the design based on the degree of required flatness. The following graph shows pattern cuts for three different main beam designs, with the same gain, but different required levels of flatness in the main beam. ...Axial-choke pattern cuts for three different main beam designs.