Interference from Thin Films — Question 10

Question 10

An imperfect coating retains reflection even at its optimum thickness

A quarter-wave film with $n_f=1.40$ coats a substrate $n_s=1.80$ in air at $\lambda_0=550\,\mathrm{nm}$. At this thickness the exact net amplitude coefficient is $r=(r_{0f}-r_{fs})/(1-r_{0f}r_{fs})$, where $r_{ij}=(n_i-n_j)/(n_i+n_j)$.

1. Calculate the thickness and exact residual reflectance $\mathcal R=|r|^2$.

2. Compare with the uncoated reflectance and calculate the film index that would eliminate reflection at the design wavelength.

Question 10 - Solution

An imperfect coating retains reflection even at its optimum thickness

(a) Method and calculation

$$\boxed{t=\lambda_0/(4n_f)=98.2143\,\mathrm{nm}};\quad r_{0f}=-1/6,\quad r_{fs}=-1/8.$$$$r=\frac{-1/6+1/8}{1-1/48}=-\frac2{47}\Longrightarrow\boxed{\mathcal R=4/2209=0.00181077}.$$

(b) Interpretation and further calculation

$$\boxed{\mathcal R_{\mathrm{bare}}=\left(\frac{1-1.8}{1+1.8}\right)^2=4/49=0.0816327}.$$$$\boxed{n_{f,\mathrm{ideal}}=\sqrt{n_0n_s}=\sqrt{1.80}=1.34164}.$$

Physical check

The coating greatly lowers reflection but does not cancel it, because its index does not supply the exact amplitude match.