How to measure refractive index of plastic

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I have a plastic disk 0.5" in diameter whose radius of refraction is supposedly non-uniform. It will vary by 0.015 at most from the center to the outside.

I need to come up with some method of measuring the refractive index locally. I have access to:
helium neon laser
a photodetector and software
optical table with moving mounts
CNC machine shop
lots of plastic/glass

Every way I try seems to not work very well. I'll need some ideas from someone experienced please.

 

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Does anyone know if this is too difficult to pull off?

 

[xts]

Have you tried to measure Brewster's angle using very narrow laser beam pointing at small spot on your disc?

 

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I'll try it.
But the angle doesn't seem to change significantly if the index varies by, say 0.002 which is the precision I desire.

 

[xts]

If your disk has well defined thickness of about 1mm (with accuracy of the same order as you require), you may build an interferometer with disc in one arm.

Other idea:
If its thickness is about 1mm, such change in refr.index makes it a lens of focal length of about 1m. You may easily measure that.
Then use reflected light t0 check if both surfaces are really flat.

 

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Can you tell me a little more about the second idea? I need to find refractive index n as a function of radius n(r). Do I measure the focal length of said "lens" or what? I'm not sure I'm understanding.