Past exam of the materials science course of the University of Cambridge / 2015 / Paper 1 / 2 / a / Solution

The normal to the section is $[010]$, so its in-plane principal vibration directions are $[100]$ and $[001]$. Their refractive-index difference isThe slow direction is $[001]$. The optical path retardation isAt an angle $\alpha$ between a principal direction and the polarizer, ideal crossed-polar transmission has the formwhere $I_P$ is the intensity after the first polarizer. The two components recombine parallel to the incident polarization when $\Gamma =m\lambda$, and the crossed analyser removes them. ConsequentlyThe next order is260nm, outside ordinary visible illumination. At exact principal-axis alignment, $\sin 2\alpha =0$ and every wavelength is extinguished, explaining the black appearance rather than a special single-wavelength cancellation.