In many papers, researchers mentioned the formula below for the optical path length in corner cube reflector, however they didn’t show any proof. I want to understand how they reached to this formula.
The total optical path length of a ray in corner cube reflector is
OPL=2*n*D*sec(theta)
where n is the refraction index of the glass, D is the height of the corner cube (the distance from the apex to the glass face), and theta is the angle to the normal of the corner cube glass face. By snell's formula the sec(theta)=n/sqrt(n^2-sin^2(theta)).
below is some ref mentioned the formula above without proof.
ref:
1- theorem 7 in "Theory of the Corner-Cube Interferometer", EDsoN R. PECK
2- Eq (7), "Simultaneous Measurement Method and Error Analysis of the Six Degrees-of-Freedom Motion Errors of a Rotary Axis",Chuanchen Bao
3- Eq(1), "Laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect”, Enzheng Zhang
Thank you very much for your valuable input. You are correct, the optical path difference is found by subtracting the optical path when we have a ray hit the retroreflector at angle from the OPL when the ray hit the retroreflector perpendicularly. Now showing the perpendicular case is easy to proof as 2*D and since the light is propagating throw glass then that distance will multiplied by n. The formula for the ray at angle is actually for the incident angle. So yes I think it might be an approximation and just multiply by 1/cos(r) where r is the refracted angle, and then we convert it to incident angle by snell’s law. Thanks again.
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