#

# Interference From Thin Films

"Everything we call real is made of things that cannot be regarded as real"
Niels Bohr

- Double slit interference, described on the previous page, is rarely observed in nature. On the other hand, interference due to thin films is quite frequently observed - swirling colours on an oil slick, colours on a soap bubble, the purple tinge on an expensive camera lens - are all examples of thin film interference.

- Consider a thin film, separating two regions of the same refractive index, for example a soap bubble. Incident light undergoes reflection and refraction at every interface as shown at right. There will be multiple "rays" reflected back into the air (only the first two are shown in the diagram) and multiple "rays" transmitted through the film (only the first of these is shown).

- The reflected rays will interfere with each other, their phase difference being determined by their path difference.
- Similarly the transmitted rays will interfere with each other.

- Note that coherence of the interfering rays is ensured since there is only one source - the incident ray coloured blue in the diagram at right.

> Assuming a monochromatic source and normal incidence, we can expect that the two reflected rays will interfere constructively if their path difference is an integer multiple of wavelengths,
> where d is the thickness of the film.

- First Correction

> The path difference 2d is measured inside the film, but the wavelength is the wavelength in air... Therefore the condition for a maximum should be,
> However, we know that,
> and using the definition of refractive index,
> This means the condition for constructive interference can be written,
> where n is the refractive index of the film and λ is the wavelength of the light in air.

- Second Correction

> This result implies that when the film becomes very thin (d approximately zero), we should expect constructive interference (equivalent to m = 0). That is, just before a soap bubble bursts, as its thickness gets smaller and smaller, we should expect the swirling colours to become brighter. Actually the opposite is observed - bubbles becomes dull just before they burst - in other words destructive interference must be taking place.
> The explanation to this apparent conflict is that the first reflected ray (green in the above diagram) is phase changed by 1800 by the act of reflection. In the above diagram, only this ray undergoes such a phase change. In fact the rule (which can be verified theoretically from Maxwell's equations) is,
> "A 1800 phase change on reflection occurs when light incident from a less dense medium (smaller n) reflects off the boundary with a more dense (larger n) medium"
> Reflection from more dense to less dense (large n to small n) causes no phase change and transmitted rays never undergo phase changes.
> Incorporating this phase change leads to the condition for constructive interference,
> and for destructive interference,

- IMPORTANT !!

> The second correction applies specifically to the air-soap-air configuration, or more generally to the circumstance of a film with greater refractive index than the media on either side. This correction may not be needed for other thin film configurations; for example, a film of water on a glass surface, shown at right. In this case, since nair < nwater < nglass , there is a 1800 phase change for the reflected rays from both the air-water and water-glass interface. That is the condition for constructive interference reverts to

> and for destructive interference,

- All of the above analysis assumes a single wavelength of light. With incident white light on a thin film maxima for different wavelengths will occur at different film thicknesses. Thus, in nature, where a thin film will not typically have a fixed thickness, we often observe swirling colours, as the thickness of the film changes.

- Finally, why thin films ? A thin film implies path differences (thicknesses) of no more than a few wavelengths. Films thicker than this will not necessarily exhibit this type of interference due to re-emission and scattering of light within the film.

" Military intelligence is a contradiction in terms."
Groucho Marx

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu