Position dependent refractive index

Question

How to determine the angle of refraction when the medium has position dependent refraction index? Lets say that the angle of incidence on the glassy slab is $\theta_1$ and the ray goes through the slab and emerges on the other side. Refractive index of glass changes with position $n(x)$. The Length of the slab is $H$. How to determine the angle of refraction on the other side?

Some modification of Snell's law is needed?

As long as the index of refraction varies slowly on the scale of one wavelength Snell's law is fine. If you had a material (like a meta-material) with a variation that's fast on that spatial scale, then you would have to deal with the problem differently. — CuriousOne, May 13 '15 at 16:53
Very closely related: How to calculate the refracted light path when refraction index continuously increasing — 299792458, May 13 '15 at 17:44
^ I mean, so closely related that my answer there answers this as well. :) — 299792458, May 13 '15 at 17:45

score 1 · Accepted Answer · answered May 14 '15 at 01:41

The angle at which the light ray exits the material actually only depends on the index of refraction of the material at the last point at which it is in the material. You can demonstrate this if you consider a series of thin layers of materials (possibly all of which have different indexes of refraction), and when you perform the Snel's law calculation, you'll notice that the refracted angle when it exits is actually still the same as the incident angle. However the position when the ray exits will be affected. All assuming you meant the index varied with the "depth-into" the material.

Position dependent refractive index

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