# Interference on curved film

Let's say I have a curved film in the XY plane, delimited by $y=0$, $y=\sqrt{x}$, $x=L$. Now I light the thin film from the top and I try to see the interference pattern by transmision.

I'm stuck with this problem. I've tried some things. First, I solved the problem for an inclined plane with constant angle $\alpha$ (i.e with equation $y=x\sin\alpha$). In this problem, you use the difference of optical path from a flat thin film with thickness $d$, which is $2nd\cos\theta _t$, and simply let $d=x\sin\alpha\approx x\alpha$. But when you try to do this on the curved film, you find that you cannot use the approximations used in the previous cases, because angle is given by $dy/dx = 1/(2\sqrt{x})$, and the angle becomes larger in $x\simeq 0$.

I also I've tried to analyze the difference of optical paths between the two first trasmitted rays, but I'm not able to use a correct aproximation to find a good solution.

My bet is that the interference patron will be lines, increasing the distance between max and mins. However, as I said, I couldn't demonstrate it.

My question is, in fact, to find a formula for the difference of optical path which produces the interference.

Any ideas on this? Thank you!