Understanding Dark Fringe at Contact Lenses

[shehri]

Homework Statement

I'm unable to understand the production of dark fringe in the central point of contact of lens.The text says "the path difference between the rays reflected at the top & bottom of air gap at the point of contact is zero.Actually the ray reflected from the top of air film doesn't undergo any change in phase but the ray reflected from bottom of air film i.e denser medium suffers a phase change of 180 degrees or additional P.D of lambda/2.Hence dark spot is formed".Plz. explain in simple words.Thanks.

Homework Equations

Nill.

The Attempt at a Solution

 

[exec]

Suppose the radius of curvature of the lens(plano-convex, I studied that before, couldn't quite remember) = R, and the incident light is directed towards the lens upright.

We're now concerned with the radii of circular inteference,r.

As you've said, we're concerned with the phase change(180) caused by the reflection at the denser medium UNDER the air film.

So, the condition for maxima is given by 2d = (m + 1/2)lambda for m = 0,1,2,..

We now find the relationship of d, thickness of air film, corresponding to the distance,r away from the center of the lens.

d = R - (R^2 - r^2)^1/2
= R - R[1 - (r/R)^2]^1/2

by binomial expansion, d approximates to (r^2)/2R

Put it in d = (m+ 1/2)lambda,
r = [(m+1/2)*lambda*R]^1/2 for m = 0,1,2,...

That means, for m = 0, r must be something larger than 0, and we're talking about maxima now. Simply put, for m = 0, the nearest maxima is not on the center where r = 0 because we've calculated that r>0.

Thus, center should be dark.