Michelson Interferometer, Optics Derivation

[Traius]

Homework Statement

Derive the exact expression for the index of refraction n of a glass plate as a function of the
fringe shift m, the plate thickness t and the angle of deflection of the plate (hint: find the
optical path difference for an incident vs a tilted incident beam, and solve for n ; don’t forget refraction inside the plate!). What approximation is involved?

The final equation I'm to get is:
n =(2t − mF)[1 − cos(C)] /[ 2t(1 − cos(C)) − mF]

F = lambda, C = alpha, not sure why the forum translated the symbols to be those symbols...

It's from a michelson interferometer experiment, using the interferometer to determine the index of refraction of a glass plate.

Homework Equations

Optical path difference = delta L = mF (?) (not sure about m*lambda, perhaps this is the source of my error?)
In one arm, light travels 2[ d1 + d2 + t]
in the other, 2[d1 + d2 +nt (the glass plate)]
But the nt part is dependent upon the angle which it enters the glass after reflecting the
mirror.
Therefore, nt = nt(cosC)

The Attempt at a Solution

delta L = 2 [ d1 + d2 + t] - 2 [ d1 + d2 + nt]
delta L = m = 2t- 2ntcosC
n = [2t - m] / 2tcosC
Obviously this is not right, but I can't figure out what is wrong with the model... I now have an idea, though, after doing this work, that perhaps my second arm path is wrong; perhaps only after bouncing off the mirror is it ntcos, and before the mirror it is nt? That would change my equations...
Any help would be much appreciated.

 

[Traius]

Tried another idea, ended up with n = [mF +2t] / [t{1 + cos (theta)}], which is still not right...