# An interference experiment in your bathroom

Optical interference is typically seen as an unusual phenomenon which you can only see with a laser and a specialized apparatus, such as Young’s slits or a four kilometre long gravitational wave detector.

What’s less well known as that anybody, without much effort or specialized scientific equipment can observe interference in their own bathroom! All you need is a bright light source, such as a flashlight or even just a match, and a dirty mirror. Normal domestic dust will do the trick – assuming there’s enough of it on the mirror. Now hold your light source next to your head and look directly into the mirror, and you will see multi-coloured stripes or rings: red on the outside, and blue on the inside. Figure 1: Interference patterns in a mirror covered in a layer of chalk dust.

This kind of interference pattern was first described in 1704 by Isaac Newton, but it was only in the nineteenth century that Thomas Young realized the crucial rôle light-scattering particles on the surface of the mirror play. Adolphe Quetelet, John Herschel, Ludwig Schläfli and others later succeeded in describing the phenomenon quantitatively.

The so-called Quetelet rings arise due to interference of light rays which have been scattered by dust particles: light can be scattered both on its way into the mirror and on its way out; these scattered light rays then interfere with each other. Light that reflects without being scattered does not contribute to the interference effect.

With the help of the scheme in figure 2 below, we can easily see where the interference comes from: if the angle of illumination, $$\alpha$$, and the angle of observation, $$\beta$$, are not equal, there will be a phase difference between two scattered light rays. Here we can use the well-known formula for interference in a thin film: the optical path length $$\Delta s_\alpha$$ added to a reflected ray is equal to $$2nd\cos\alpha'$$, where $$d$$ is the thickness of the mirror, and $$\alpha'$$ is the angle of the light ray in the mirror. This can be calculated using Snell's law, $$n\sin\alpha' = \sin\alpha$$.

This length is clearly much greater than the wavelength of the light. In this case, we must consider the difference in path length between the path the light takes before scattering (angle in the glass $$\alpha'$$) and the path it takes after scattering (angle $$\beta'$$):

$$\Delta s = 2 n d (\cos \alpha' - \cos \beta')$$

The interference pattern in the mirror thus depends on the thickness $$d$$ of the mirror and on two angles: the angle of illuminations, $$\alpha$$, and the angle of observation, $$\beta$$. Figure 2: Geometrical scheme of the phenomenon

In order to see the pattern particularly well, there are a number of things you can do. The most important component are obviously the scattering particles on the surface of the mirror: the more light they scatter forward, the brighter the interference pattern will be. A consistent cover of large, white microparticles, such as domestic dust, plaster dust or chalk dust works wonders. The size and shape of the interference fringes depends on the geometry: the rings are larger easier to see if $$\Delta s$$ is small, or if it doesn't change much depending on the angle.

Using the small angle approximation, we can see that $$\Delta s \approx \frac dn (\beta^2 - \alpha^2)$$ is smallest if you look into the mirror head-on at ninety degrees, with the light source as close to your as possible. The curvature of the fringes is determined by the relative distance from the mirror to the light source and to the eye.

If you want to get a good feeling for this effect, of course, there is but one thing for it: get some dirt on your own mirror!

This is a translation of the original article, “Een interferentie-effect in je badkamer”, by Thomas Jollans and Michel Orrit, published in the Nederlandse Tijdschrift voor Natuurkunde, volume 84, issue 7, pages 226–227, July 2018