Polarization by Reflection — Question 5

Question 5

Angular uncertainty becomes an index uncertainty in a Brewster measurement

A transparent sample in air has measured Brewster angle 57.0 degrees with small uncertainty 0.2 degrees. Assume the air index is exactly one and the interface obeys the nonmagnetic lossless model.

1. Infer the sample index and propagate the angular uncertainty to first order.

2. Find the relative index uncertainty and state the required angular unit in the derivative.

Question 5 - Solution

Angular uncertainty becomes an index uncertainty in a Brewster measurement

(a) Method and calculation

$$\boxed{n=\tan\theta_B=\tan57.0^\circ=1.53986}.$$$$\sigma_n\simeq\left|\frac{dn}{d\theta_B}\right|\sigma_\theta=\sec^2\theta_B\,\sigma_\theta.$$

(b) Interpretation and further calculation

$$\sigma_\theta=0.2\pi/180=0.00349066\,\mathrm{rad}\Rightarrow\boxed{\sigma_n=0.0117677}.$$$$\boxed{\sigma_n/n=0.00764200\simeq0.764\%};\quad n\simeq1.540\pm0.012.$$

Physical check

The derivative of tangent is taken with its angle in radians; substituting a degree value directly would overestimate the uncertainty.