Re: What are the exact formulas for refraction of light through glass spheres?

As is the case in most of science, the exact "formula" is very simple, but 
the application of the formula requires that you concentrate quite a bit and do 
some accurate trigonometry/geometry.

The formula is just Snell's Law:

  n1 * sin(theta1) = n2 * sin(theta2),

where n1 and n2 are the refractive indices of the materials on both sides of  
each glass-to-air interface (usually air is taken as having a refractive index 
of 1 and glass as having a refractive index about 1.5).  So, all you have to do 
is to figure out where the ray you want to trace hits the sphere on its way in, 
then calculate the angle to the normal to the surface at that point (that's 
theta1) and apply the above formula above to get theta 2 (the angle to the 
normal that the outgoing ray -- now within the glass -- makes to the normal.

Knowing the position and angle of the ray at the inside first surface of the 
sphere, propagate the ray in a straight line in space and determine its 
intersection with the second surface of the sphere.  Then apply Snell's Law 
again, only now n1 = 1.5 and n2 = 1... and you have to figure out what theta 1 
based on the ray path in space and the radius of the sphere.  Once you have 
this, you will have calculated theta2 -- the angle of the outgoing ray from the 
sphere in air, relative to the normal to the surface at the point of emergence.

From that, you can just propagate the ray in space as far as you like.

As you probably will notice when you first start this, it's a geometrical 
mess to try to do this for an arbitrary incoming ray angle and position, if you 
plan to trace a bunch of rays at starting at different angles.  In fact, I doubt 
that it's even something that can be expressed in closed form analytically -- I 
sure wouldn't try it.  For a bunch of rays coming in parallel to each other, 
it's a lot easier, because the problem is reduced to a 2 dimensional one...  
This would be a good exercise to do, using a computer.  By hand, it would be an 
agony.

By the way, since you're trying to figure distortion effects, you should 
avoid trying to use simple approximations like the lensmaker's formula and such 
-- they just won't work.

Sorry it's seems so complicated and messy, but sometimes life's just like that!