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Newton rings: spherical droplet with thin film interference

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Homework Statement


So I have a similar situation with the traditional newton rings. Instead of having however a convex lens, I have a (partial) spherical surface with height d with sitting on top of a thin film of thickness t, with d >>> t, and refractive index n and below the thin film is a mirror. I have to find what is the condition for constructive/destructive interference for the resulting newton rings.

Homework Equations





The Attempt at a Solution


I know that: i have optical length inside the spherical surface of the would be [tex]d[/tex], so the optical path would be [tex]2d[/tex]. Also, I have an optical length inside the thin film, where there would also be a phase change of [tex]\pi[/tex]. So total change of phase would be [tex]2d+2t+\pi[/tex].

Also, from the geometry You can infer that:[tex] r_m^2 = t*(2R-t)[/tex] where [tex]r_m[/tex] would be the radius of the mth dark ring and [tex]R[/tex] the radius the circle (curvature of the lense).

Any advice welcomed. Many thanks
 

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