In particular, what sorts of material correlate to a higher index of refraction? I'm not really seeing the correlation between let's say a diamond and a higher index of refraction
The index of refraction is related to how fast light moves through the material compared to how fast light moves in a vacuum. Unlike sound, light does not require a medium to travel and when not moving through a vacuum will move at a constant speed (the speed of light).
When light travels through mediums you can think of light traveling just as it would through a vacuum but it may hit an atom, be absorbed, and reemitted. This is why the denser an object the higher index of refraction. Of course I am sure there are more in depth quantum physics level explanations if you're interested but hopefully this will give you some basic intuition of which materials correlate to high or low index of refractions
My understanding is that photons are constantly absorbed and re-emitted by the atoms they encounter. Mediums that largely sparse act similarly to a vacuum because photons rarely collide with particles whereas in a densely populated medium like diamond, the photons constantly collide and interact with other particles regularly resulting in a scattering of photons as their directions change and thus resulting in refraction increased refraction.
Here is a video that explains Snell's law very clearly that I found super helpful: https://www.youtube.com/watch?v=M5EW15BOy-c
Is it possible there is a material, the exiting ray is along the surface of the two materials?