# Optics: Light ray goes through plate with transversely varying index of refraction

## Homework Statement

We have a parallel, transparent plate. The plate has an incidence of refraction that is dependent on a transverse coordinate. (i.e. the plate is perpendicular to z direction, and the refractive index changes along x direction)

n(x) = 1.5/(1 - x/0.13)

It is observed that when a light ray is perpendicularly incident on the plate (from the air) at x=0, it emerges on the other side of the plate, making a 30 degree angle from the normal.

Find the index of refraction where the ray exits. (Thickness of the plate is not known)

## Homework Equations

I'm not really sure about this. Snell's law tells us that the light ray should stay perpendicular to the normal across both boundaries, so there is something I am clearly missing.

Snell's Law: n1*sin(a1) = n2*sin(a2)

Where n1 and n2 are indices of refraction for the two materials
a1, a2 are the angles the ray makes from the normal of the material

## The Attempt at a Solution

Again, using Snell's Law, at the second boundary (going from plate back to air), we know that n1 = n(x), n2 = 1, a1 = ?, a2 = 30 deg

This is one equation with two unknowns, so I am stuck.

If for some reason, the light were to bend at the first boundary, then I would treat the plate as a bunch of thin plates, and integrate (as the light ray would be experiencing a different index of refraction as it travelled through the plate).

In order for the ray to be emerging at a 30 degree angle, something like this must be happening, but I don't know why!

Any help is appreciated!