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.

 

[m2003]

I can explain the phenomenon of dark fringe at contact lenses in simpler terms. When light passes through a contact lens, it encounters a small air gap at the point of contact with the lens. This air gap acts as a thin film, which can reflect some of the light. When this reflected light reaches the central point of contact, it interferes with the direct light passing through the lens. This interference can result in a dark spot, or dark fringe, at the central point of contact.

The reason for this dark spot is due to the difference in path length between the reflected light from the top and bottom of the air gap. The light reflected from the top of the air gap does not undergo any phase change, meaning its wavelength remains the same. However, the light reflected from the bottom of the air gap, which is a denser medium, undergoes a phase change of 180 degrees or a difference in path length of half a wavelength (lambda/2). This difference in path length causes the two waves to interfere destructively, resulting in a dark spot.

In simple terms, the dark spot is formed because the light reflected from the bottom of the air gap cancels out the light passing through the lens, creating a dark spot. I hope this explanation helps in understanding the concept of dark fringe at contact lenses.