# University of Wisconsin Green Bay

If you blow soap bubbles in sunlight, you will often see bands of color in them. What are the three smallest thicknesses of soap that will show up as red bands?

• You may immediately recognize this as a thin film interference problem based on similar examples in your text or from lecture. But remember interference results from superposition of waves and so it is very useful to recognize this problem in the broader context of superposition.

You know that sunlight is composed of light of many different wavelengths, and each wavelength in the visible portion of the spectrum is interpreted by our brains as color. So something must happen when light hits the soap bubble to cause only some wavelengths to be seen. All colors reflect, so that "something" must be happening with the waves in the soap itself. The fact that some colors are seen and others damped out should remind you of musical instruments--some frequencies are heard and others damped out, and indeed the root physics is the same in both cases. You see (or hear) frequencies that correspond to constructively interfering waves, and you do not see or hear the frequencies corresponding to waves that cancel out.

To approach this problem, then, you need look at which wavelengths of light superpose to interfere constructively.

• We realized above that all colors of light will reflect from the surface of the soap, and therefore realized that we need to think about interference and what happens to the light as it is in the soap itself. Therefore, your picture needs to allow room to look inside the film of soap. Furthermore, because the physics behind this problem is superposition of waves, a useful picture will show that multiple waves are present. At each interface, waves both reflect and refract, and so the lower figure shows that the incoming wave partially reflects off of the outer surface of the soap and partially refracts into the soap layer. At the inner surface there is also both reflection and refraction. This continues at each interface, although I only drew enough to show the general idea.

• We know that waves interfere constructively--that we will see the light--when they are in-phase.

The waves start out in phase, and so there are two factors that can change that relationship. The first is the thickness of the soap film. If one wave travels half a wavelength further than the other wave, they can end up completely out of phase and will always interfere destructively. If it travels an integer number of wavelengths further, the phase relationship (e.g. in phase) is maintained. The other factor is reflection. When a wave reflects off of an interface into a more dense medium with a higher index of refraction, the reflected wave will flip. This is the same as taking the wave 1800 out of phase.

Both effects are present in this case, and so the equation that gives an interference maximum--a bright spot--is

• We now know the three thinnest films of soap that will give interference maxima for red light. No further solution is necessary

• The colored bands on soap bubbles result from the interference of light. Different colors correspond to different wavelengths, and so conditions that give constructive interference for some colors will give destructive interference for other colors. In this case, the interfering waves arise because of reflection--part of the incident wave reflects off of the outer surface, part enters the soap bubble and reflects off the inner surface. This is commonly called "thin film interference."

There are two effects that affect the relative phase of the interfering waves in the case of thin films. The first is the relative path length between the waves, and the second is the possibility that the waves will be inverted on reflection from one or both interfaces. For that reason, you can't memorize an equation that gives inteference minima and one that gives interference maxima. Instead, you need to think carefully about the required path difference for your case of interest. For the soap bubble, only one of the interfaces led to inversion of the reflected wave, and so an additional half wavelength path difference is required to bring the waves into phase.

Finally, make sure that your numbers make sense. The wavelength of light is shorter inside soap than inside air, and so you expect less than half of 7 x 10-7 m for the smallest path difference, and you expect to add a number smaller than 7 x 10-7 m for each subsequent case. Don't forget that the thickness is the path difference divided by two, since the light goes across the soap film going in and again coming back out.