Interference in thin films

Thin films (e.g. soap bubbles,oil on water) often display brilliant coloration when reflecting white light and show fringes when in monochromatic light.
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      A thin film is a transparent medium whose thickness is comparable with the wavelength of light. Brilliant and beautiful colors in soap bubbles and oil film on the surface of water are due to interference of light reflected from the two surfaces of the film as explained below:

       Consider a thin film of a reflecting medium. A beam AB of monochromatic light of wavelength λ is is incident on its upper surface. It is partly reflected along BC and partly refracted into the medium along BD. At D it is again partly reflected inside the medium along DE and then at E refracted along EF as shown in the figure given below:

   Fig  . Geometrical construction of interference of light due to a thin oil film

        The beams BC and EF, being the parts of the same beam have a phase coherence. As the film is thin, so the separation between the beam BC and EF will be very small, and they will superpose and the result of their interference will be detected by the eye. It can be seen in the above figure, that the original beam splits into two parts at the point B and they inter the eye after covering different lengths of paths. Their path difference depends upon (i) thickness and nature of the film (ii) angle of incidence. If the two reflected waves reinforce each other, then the film as seen with help of a parallel beam of monochromatic light will look bright however, if the thickness of the film and angle of incidence are such that the two reflected waves cancel each other, the film will look dark.
        If white light is incident on a film of irregular thickness at all possible angles, we should consider the interference pattern due to each spectral color separately. It is quite possible that at a certain place on the film, its thickness and the angle of incidence of light are such that the condition of destructive interference of one color is being satisfied. Hence, that portion of the film will exhibit the remaining constituent colors of the white light as shown in the above figure.

        From the above figure incident light first reflects from upper surface gives Part I (pink ray) and also refracts  into the film which again reflects from the bottom surface and comes to the eye as Part II (violet ray).
         Part I of light has phase change of 180° as it is reflected from a surface beyond which there is medium of higher refractive index. But Part II of light has no phase change as it is reflected from a surface beyond which there is a medium of lower index. Therefore the condition for constructive and destructive interference are reversed then the Young's double slit experiment. For nearly normal incidence the path difference between the two interfering rays is twice the thickness of the film i.e equal to 2t where t is the thickness of the film. If n is the refractive index of medium of the film then,

Path difference = 2tn

Hence condition for the maxima or constructive interference is,

                  m = 0,1,2,....

similarly condition for the minima or destructive interference is,

                           m = 0,1,2,....

     In case of varying thickness of film, there will be a pattern of alternate dark and bright fringes. 

   Fig  . Interference pattern produced by thin soap bubbles