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

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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!
 

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