Spherical Surface Refraction -- from Eric Weisstein's World of Physics

Optics "> Refraction v

Spherical Surface Refraction

Denote the optical path length , then

Take a point A a distance from the sphere. Let a ray strike the mirror at I, a distance from A. Take a point B in the sphere, a distance from the interface. Let the ray pass through the sphere and come to a focus a distance from the interface at F. Define the angle as the angle from the center of curvature to the point I.

so
Applying Fermat's principle,

In the first order approximation,

so (2) and (3) simplify to

so (9) becomes

Taking terms to third order,

and

where h is the vertical off-axis distance.