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 of incidence approximation)

We will assume n 3 > n 2 > n 1 n 3 > n 2 > n 1 . The physical path length difference of the reflected light is Δ r = 2 d Δ r = 2 d . We will get maxima in the interference when: Δ r = 2 d = m λ 2    m = 1 , 2 , 3 … Δ r = 2 d = m λ 2    m = 1 , 2 , 3 … where λ 2 λ 2 is the wavelength in the film. Now λ i ν i = c / n i . λ i ν i = c / n i . In our example we have ν 1 = ν 2 = ν 3 ν 1 = ν 2 = ν 3 , that is the frequency does not change moving between the media. So we have λ 1 n 1 = λ 2 n 2 = λ 3 n 3 . λ 1 n 1 = λ 2 n 2 = λ 3 n 3 . Thus constructive interference will happen when λ 2 = λ 1 n 1 n 2 2 d = m λ 2   m = 1 , 2 , 3 . . 2 d = m λ 1 n 1 n 2 2 d = m λ a i r n a i r n f i l m 2 d = m λ 1 1 n f i l m ( 2 d ) n f i l m = m λ a i r   m = 1 , 2 , 3 … λ 2 = λ 1 n 1 n 2 2 d = m λ 2   m = 1 , 2 , 3 . . 2 d = m λ 1 n 1 n 2 2 d = m λ a i r n a i r n f i l m 2 d = m λ 1 1 n f i l m ( 2 d ) n f i l m = m λ a i r   m = 1 , 2 , 3 … where n f i l m = n 2 n f i l m = n 2 . Destructive interference will happen when ( 2 d ) n f i l m = m λ a i r / 2   m = 1 , 3 , 5 … ( 2 d ) n f i l m = m λ a i r / 2   m = 1 , 3 , 5 …

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

Note I have assumed that n 3 > n f i l m > n a i r n 3 > n f i l m > n a i r in the above, where n 3 n 3 is the material that the film sits upon.

Consider an interface between two materials with indices of refraction n 1 n 1 and n 2 n 2 . If n 2 > n 1 n 2 > n 1 . Then lets examine what happens to the phase of an electromagnetic wave upon reflection. For a transverse electric field, there is a phase change of π . π . For the transverse magnetic field (or E ∥ E ∥ ) there is not, if the light ray is close to the normal. However if n 1 > n 2 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 example above, their will be no relative phase change between the rays in either case. Either both will change by π π or neither will change, depending on the orientation of the E field.