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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: $$\Delta r=2d=m{\lambda}_{2}\text{\hspace{1em}\hspace{1em}}m=1,2,3\dots $$ 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 $$\begin{array}{c}{\lambda}_{2}={\lambda}_{1}\frac{{n}_{1}}{{n}_{2}}\\ 2d=m{\lambda}_{2}\text{\hspace{1em}}m=1,2,3.\text{.}\\ 2d=m{\lambda}_{1}\frac{{n}_{1}}{{n}_{2}}\\ 2d=m{\lambda}_{air}\frac{{n}_{air}}{{n}_{film}}\\ 2d=m{\lambda}_{1}\frac{1}{{n}_{film}}\\ (2d){n}_{film}=m{\lambda}_{air}\text{\hspace{1em}}m=1,2,3\dots \end{array}$$ where ${n}_{film}={n}_{2}$ . Destructive interference will happen when $$(2d){n}_{film}=m{\lambda}_{air}/2\text{\hspace{1em}}m=1,3,5\dots $$
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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