# Thin Film Interference formula

1. Nov 3, 2008

### Dr. Jekyll

Hi everyone!

Here's the problem related to thin film interference. I mean, it's not quite a problem. I've been looking for the formulae related to this type of interference and fount two different things in two different textbooks.

It's the formula for the optical difference of the wave reflected from the top (air-film) surface and the wave reflected from the bottom (film-air) surface. Now, I ran into these two formulae (for the same thing):

$$\delta=2nt - \frac{\lambda}{2}$$ (1)

and

$$\delta=2nt + \frac{\lambda}{2}$$ (2).

Now, I'm somehow sure that formula (2) is the right one but I can't find any mathematical proofs for it.

Now, I can get the (1) formula but I'm probably wrong somwhere.
This is how I would do it:

We have a thin film (t is the thickness) and the light beams vertically on the film. I also took a spot somewhere above the film (x will be distance from the film). It's actually not relevant but makes things a bit more clearer (at least for me).

The wave 1 reflects from the upper surface and has a phase shift $\frac{\lambda}{2}$. The second, wave 2, reflects from the lower surface (without any phase shift). Now, this would be the difference when they both reach the spot:

$$\delta=x+2nt+x - (x+\frac{\lambda}{2}+x) \Rightarrow \delta = 2nt - \frac{\lambda}{2}$$.

Please tell me where did I go wrong.

Sorry for maybe the bad english. It's not my native language.