Problem 5.185

Natural monochromatic light of intensity. $I_{0}$ falls on a system of two Polaroids between which a crystalline plate is inserted, cut parallel to its optical axis. The plate introduces a phase difference $\delta$ between the ordinary and extraordinary rays. Demonstrate that the intensity of light transmitted through that system is equal to $$I=\frac{1}{2} I_{0}\left[\cos ^{2}\left(\varphi-\varphi^{\prime}\right)-\sin 2 \varphi \cdot \sin 2 \varphi^{\prime} \sin ^{2}(\delta / 2)\right]$$ where $\varphi$ and $\varphi^{\prime}$ are the angles between the optical axis of the crystal and the principal directions of the Polaroids. In particular, consider the cases of crossed and parallel Polaroids.