Obtain the equaitons for constructive and destructive interference for transmitted and reflected waves in thin films.
Interference in thin films:

Explanation

Let us consider a thin film of transparent material of refractive index `mu` (not to consuse with order of fringe n) and thickness d. A parallel bema of light is incident on the film at an angle i. The wave is divided into two parts at the upper surface, one is reflected and the other is refracted. The refracted part, which enters into the film, again gets divided at the lower surface into two parts, one is transmitted out of the film and the other is reflected back in to the film. Reflected as well as refracted waves are sent by the film as multiple reflectins take place inside the film. The interference is produced by both the reflected and transmitted light.

For transmitted light:
The light transmitted may interfere to produe a reslultant intensity. Let us consider the path difference between the two light waves transmitted from B and D. The two waves moved together and remained in phase up to B where spliting occured. The extra path travelled by the wave transmited from D is the path inside the film, BC + CD. If we approximate the incidence to be nearly normal (i = 0), then the points B and D are very close to each other. The extra distance travelled by the wave is approximately twice thickness of the film, BC + CD = 2d. As this extra path is travesed in a medium of refractive index `mu`, the optical path difference is `delta = 2mu d`.
Similarly, the condition for destructive interference in transmitted ray is,
`2mu d = (2n - 1) (lambda)/(2)`
For reflected light:
It is experimentally and theoretically proved that a wave while travelling in a rarer medium and getting relfected by a dener medium, undergoes a phase change of `pi`. Hence, an additional path difference of `lambda//2` should be considered.
Let us consider the path difference between the light waves reflected by the upper surface at A and the other wave coming out at C after passing through the film. The additional path travelled by were coming out from C is the path inside the film, AB + BC. For nearly normal incidence this refractive index `mu` the optical path difference is, `delta = 2mu d`. The condition for
`2mud+(lambda)/(2)=nlambda(or)2mud=(2n-1)(lambda)/(2)`
The additional path difference `lambda//2` is due to the phase change of `pi` in rarer to denser reflection taking place at A. The condition for destructive interference for reflected rays is,
`2mud+(lambda)/(2)=(2n+1)(or)2mud=nlamda.`