Hypothetical CMB space telescope design problem, received power from extended thermal source versus receiver front end NEP?
The feed horn is an antenna which has an antenna pattern  Source
However, not all feeds are horns.  Here is an image of a parabolic dish with a Yagi feed:  Source
Ideally, you want the beam width of the horn antenna to cover the entire dish and only the dish.  So if the beam width of the horn antenna is, say, 30 degrees, you want the dish to subtend 30 degrees of the view from the center of the entrance of the feed.
Since the dish is parabolic, it can be described by the equation y=a(x-h)^2 Source.  Assuming the vertex of the parabola is at the origin, the feed antenna will be 1/(4a) units away from the vertex.  This is the focal length of the dish.  The aperture of the dish is its diameter.  Using the values from the above equation, the f-number of a dish antenna is (1/(4a))/x or 1/(4ax).
Now that we know where the focus is, we can compute the beam width of the feed required to cover the dish, then design a feed antenna meets that requirement.
Notice that the f-number doesn't contribute to the solution to the problem as we would expect if this were an optical system.  The most important thing is to match the beam width of the feed antenna to the radius of the dish, which is set by the value of &quot;a&quot;.  So a feed antenna will receive the same amount of power regardless of f-number as long as the feed antenna is at the focus of the aperture.  (Whether one can design a feed antenna for a given &quot;a&quot; is another question.)