Q. Prove that for thin films, the interference pattern in reflected and transmitted light are complementary to each other.

Also explain why the excessively thin films seen in reflected light appear dark.

When light from an extended source is incident on the thin film such as oil layer on water or soap bubble, light is reflected and refracted from upper and lower surface of thin film. When these light rays made to interact with each other, interference pattern produced.

By a thin film we produce two coherent sources by the way of division of amplitude. Let the beam of light AB of wavelength ¥ is incident on a thin film of refractive index 'µ' and thickness 't'. Incident light makes an incident angle I with the normal according to fig.

The incident light partially reflected along BR1 in same medium from upper surface at point B and partially refracted along BC. This ray again partially reflected along and partially transmiStted along CT1 at point C by the lower surface.

In this way incident light ray undergoes multiple reflections in this thin film and we get a set of parallel reflected rays BR1, DR2 .......... and a set of transmitted rays CT1, ET2...........

These parallel light rays when made to interact with each other produces interference pattern. Consider the case of reflected light, the path difference between reflected light BR1 and DR2 is:

According to Stoke's theorem, when a light ray suffers reflection at a denser medium, i.e., at B, a path difference of ¥/2 takes place.

For constructive interference it should be odd multiple of ¥/2 i.e., ¥.