The unsolved mathematical light beam problem

The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at
the interface of two media(like water and oil). And the position(the
heigth) of the interface to the center of the sphere is fixed and known(I
drew a picture for two different situations). Now think about this: A beam
of light (parallel rays) that enclose an angle alpha to the interface
enters (the direction is supposed to be: they first enter the lower medium
and then go to the upper one) hits this interface with the sphere. Now the
question is: Can we find an analytical expression for the maximum area of
the sphere perpendicular to the direction of the rays that is defined by
the rays that enter the sphere without going first into the other medium?
If this is unclear look at the picture: In situation 1 I drew 2 rays (of
the infinitely many that enter at this angle) that fulfill the condition
that they first hit the sphere without going in the upper medium. Now
especially the left one is crucial since if I had chosen one that was even
more slightly shifted to the left side, this one would have been in the
upper medium, so no longer a reasonable candidate. The right one is not a
restriction. The right picture is poorly drawn, since I wanted to have
one, where both rays restrict the accessible area. But I think you have
the idea now, but what do I mean by area?
I am looking for the biggest area(=PROJECTION OF THE SURFACE AREA of
points that fulfill this on a plane) perpendicular to the direction of the
rays inside the sphere, that consists of rays that enter the sphere and
fulfill the property above . So in the first picture this would probably
the area going through the center and "somewhat enclosed by the two
arrays" and in the second part, this one should be the "imaginary
interface" inside the sphere.
If you have any questions, please do not hesitate to post them