# Thin-film interference

Thin-film interference occur when light waves reflect on the upper and lower boundaries of a thin film. This causes interference and it is responsible for the rainbow of colours that you see in soap bubbles and in puddles that have a thin layer of oil.

When the light strikes the thin film, it is both reflected and refracted. The refracted ray reflects off the bottom and refracts through the top to come out parallel with the original reflected ray:

When a wave reflects is in a fast medium and reflects on the interface to a slower medium, it inverts. This means the first wave inverts when it reflects, but the second wave does not. All other things being equal, they should be out of phase. Of course, other things *aren’t* equal because the second wave travels an extra distance. For small angles of incidence, this extra distance is about $\displaystyle 2 t$ where
$\displaystyle t$ is the thickness of the film.

There are three interesting cases for the thickness of the film. When $\displaystyle t ≪ \lambda$, the extra distance is so small that the interference is destructive for all colours. When $\displaystyle t = 1 / 4 \lambda$, the two waves are in phase. Why? If we don’t consider the extra distance, the waves are out of phase, meaning a phase delay of $\displaystyle 1 / 2 \lambda$; when we do consider it, we have $\displaystyle 1 / 2 \lambda + 2 t = \lambda$. Since they are in phase, they interfere constructively—all other colours (with different values of $\displaystyle \lambda$) are blocked because the interference is destructive. When $\displaystyle t = 1 / 2 \lambda$, the waves are out of phase and therefore destructive. This means that only that colour is blocked—the others are not.

phase delay | type of interference | the $\displaystyle \lambda$ colour | other colours | |
---|---|---|---|---|

$\displaystyle t ≪ \lambda$ | small | destructive | blocked | blocked |

$\displaystyle t = \frac{1}{4} \lambda$ | $\displaystyle \lambda$ | constructive | reflected | blocked |

$\displaystyle t = \frac{1}{2} \lambda$ | $\displaystyle \frac{3}{2} \lambda$ | destructive | blocked | reflected |