# 8.1 Thin film interference

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We look at thin film interference.

## Thin films

Suppose there is a very thin film of dielectric and light is incident on it normally. Lets consider single reflections. (We make the small angle ofincidence approximation)

We will assume ${n}_{3}>{n}_{2}>{n}_{1}$ . The physical path length difference of the reflected light is $\Delta r=2d$ . We will get maxima in the interference when: $\Delta r=2d=m{\lambda }_{2}\text{ }m=1,2,3\dots$ where ${\lambda }_{2}$ is the wavelength in the film. Now ${\lambda }_{i}{\nu }_{i}=c/{n}_{i}\text{.}$ In our example we have ${\nu }_{1}={\nu }_{2}={\nu }_{3}$ , that is the frequency does not change moving between the media. So we have ${\lambda }_{1}{n}_{1}={\lambda }_{2}{n}_{2}={\lambda }_{3}{n}_{3}\text{.}$ Thus constructive interference will happen when $\begin{array}{c}{\lambda }_{2}={\lambda }_{1}\frac{{n}_{1}}{{n}_{2}}\\ 2d=m{\lambda }_{2}\text{ }m=1,2,3.\text{.}\\ 2d=m{\lambda }_{1}\frac{{n}_{1}}{{n}_{2}}\\ 2d=m{\lambda }_{air}\frac{{n}_{air}}{{n}_{film}}\\ 2d=m{\lambda }_{1}\frac{1}{{n}_{film}}\\ \left(2d\right){n}_{film}=m{\lambda }_{air}\text{ }m=1,2,3\dots \end{array}$ where ${n}_{film}={n}_{2}$ . Destructive interference will happen when $\left(2d\right){n}_{film}=m{\lambda }_{air}/2\text{ }m=1,3,5\dots$

When destructive interference occurs then that value of $\lambda$ is not reflected. Note that this is a function of both $d$ and $\lambda$ . The next effect is that different colours of light get reflected at differentthicknesses of the film. This is why soap films or oil films on water give rainbow effects.

Note I have assumed that ${n}_{3}>{n}_{film}>{n}_{air}$ in the above, where ${n}_{3}$ is the material that the film sits upon.

Consider an interface between two materials with indices of refraction ${n}_{1}$ and ${n}_{2}$ . If ${n}_{2}>{n}_{1}$ . Then lets examine what happens to the phase of an electromagnetic wave uponreflection. For a transverse electric field, there is a phase change of $\pi \text{.}$ For the transverse magnetic field (or ${E}_{\parallel }$ ) there is not, if the light ray is close to the normal. However if ${n}_{1}>{n}_{2}$ then and the situation is reversed and the transverse electric field does not undergo a phase change and the transverse magnetic field does. In the exampleabove, their will be no relative phase change between the rays in either case. Either both will change by $\pi$ or neither will change, depending on the orientation of the E field.

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