Interference from Thin Films — Question 4

Question 4

Oblique film interference depends on cosine of the internal angle

A plane-parallel film has thickness t and index n. The external index is $n_0$. Let i and r be incidence and internal angles, with $n_0\sin i=n\sin r$. For the two reflected rays compared on a common outgoing wavefront, use $D=2nt/\cos r-n_0(2t\tan r)\sin i$.

1. Simplify D and explain why using only the internal round-trip length gives the wrong phase.

2. For a film in air with one relative reflection reversal, $n=1.50$, $i=30^\circ$, and $\lambda_0=600\,\mathrm{nm}$, find the least thickness for constructive reflection.

Question 4 - Solution

Oblique film interference depends on cosine of the internal angle

(a) Method and calculation

$$D=\frac{2nt}{\cos r}-2nt\tan r\sin r=\frac{2nt(1-\sin^2r)}{\cos r}.$$$$\boxed{D=2nt\cos r};\quad\text{the external wavefront correction is essential}.$$

(b) Interpretation and further calculation

$$\sin r=1/3\Longrightarrow\cos r=2\sqrt2/3;\quad 2nt\cos r=\lambda_0/2.$$$$\boxed{t=\frac{\lambda_0}{4n\cos r}=106.066\,\mathrm{nm}}.$$

Physical check

The phase comparison must refer to the same outgoing wavefront. The increased internal geometric travel is partly offset by the external path difference.