How does a film with some thickness without reflection alter a wave's phase

Question

How does the thickness of a film alter the phase of a beam of light passing directly through the film without reflection?

Answer 1

The phase of a wave varies by $2\pi$ over the distance of one wavelength, $\lambda$. So suppose you have some distance $d$, then the number of wavelengths in this distance is $d/\lambda$, and therefore the phase shift over the distance is:

$$ \varphi = 2\pi\frac{d}{\lambda} $$

But if the wave is passing through a material with refractive index $n$ this reduces the wavelength. If $\lambda_0$ is the wavelength of the light in a vacuum, then the wavelength of the light in the material is:

$$ \lambda_m = \frac{\lambda_0}{n} $$

So the phase shift travelling through a film of thickness $d$ with refractive index $n$ is:

$$ \varphi_m = 2\pi\frac{nd}{\lambda_0} $$

We are usually interested in the difference between the phase shift in the vacuum and the phase shift in the material, and this is just $\varphi_m - \varphi_0$ so:

$$\begin{align} \Delta\varphi &= \varphi_m - \varphi_0 \\ &= 2\pi\frac{nd}{\lambda_0} - 2\pi\frac{d}{\lambda_0} \\ &= 2\pi\frac{d}{\lambda_0} \left(n - 1\right) \end{align}$$