1. Interference in Thin films
What do soap bubbles and solar panels have in common?

2. What are thin films?
Thin films are optical devices that make use of the partial reflection and transmission of waves at the boundary between two mediums, and the interference patterns that result when said waves have specific path differences.
The two basic uses of thin films
Are a result of either reflection or
transmission effects.

3. Reflection in Thin Films
Recall that when light passes into a slower medium from a faster one, the partially reflected light at the boundary undergoes a phase change of 180˚. However, when the transition is from a slower medium to a faster one, the partially reflected light remains in phase with the incident light. Depending on the thickness of the film, and λ inside the film, we have different interference patterns.
Let us consider 3 thicknesses (t), when n1 < n2 & n2 > n3 :
t << λ
t = λ/4

4. Case 1: t<< λ
When the thickness is much, much less than the wavelength, the path difference between the initially reflected wave and the partially transmitted wave (the wave that enters n2 and gets reflected back at the n2 & n3 boundary) is very small. However, because the wave that was initially reflected was going from a faster medium to a slower one, it has undergone a phase change of 180˚. As a result, the negligible path difference and the phase change means these waves will interfere destructively.

5. Case 2: t = λ/4
When the thickness is t = λ/4, the total path difference is λ/2 for nearly normal paths. However, because the initially reflected wave was already 180˚ out of phase, when the transmitted wave is reflected back it will be in phase, producing constructive interference. And this is true whenever the thickness follows the relationship
t = (2n – 1) λ / 4 n = 1, 2, 3,…

6. Case 3: t = λ/2
When the thickness is t = λ/2, the total path difference is now λ nearly normal paths. However, again, because the initially reflected wave is 180˚ out of phase, when the transmitted wave is reflected back it will be in out phase, just like in Case 1; producing destructive interference. And this is true whenever the thickness follows the relationship
t = m λ / 2 m = 0, 1, 2,…

7. Transmission in Thin Films
In order to consider transmission in thin films, let us once again consider the same three cases:
t << λ
t = λ/4
λ is, once again, the wavelength inside the film
This time, instead of an initially reflected wave at n1 and a partially transmitted wave that’s been reflected back at the n2 & n3 boundary, this time we have a wave that’s initially transmitted right through the film and a wave that’s been partially reflected at the n2 & n3 boundary first followed by the n1 & n2 boundary, and then transmitted into n3.

8. Case 1: t << λ
Unlike it’s reflection counterpart, when t << λ the waves will interfere constructively. This is because the path difference traversed by the doubly reflected wave, which does not undergo a phase change, is very small, and therefore negligible. So, when the doubly reflected wave enters n3, it is roughly in phase with the initially transmitted wave.

9. Where these are the thicknesses for destructive interference
Case 2: t = λ/4
As we saw in the case of reflection, inside the film the path difference is always twice the thickness. The same is true for transmission. However, with transmission, we know the initially transmitted wave does not undergo a 180˚ phase change. As such, if the thickness is t = λ/4, that means that the total path difference is λ/2. This time, however, because of the lack of phase change, the waves will be out of phase with each other, and the thickness can be calculated using
t = (2n – 1) λ / 4 n = 1, 2, 3,…
Where these are the thicknesses for destructive interference

10. Case 3: t = λ/2
In this case, because the total path difference is λ and the initially transmitted wave does not undergo a 180˚ phase change, when the initially transmitted wave and the doubly reflected wave interfere they will do so constructively, and the thicknesses can be calculated using
t = m λ / 2 m = 0, 1, 2,…