Problem 6.4.5: Parallel polarisation on polystyrene

Snell's law gives ${\theta }_t$, the Fresnel formulas for P polarisation give ${\Gamma }_{\parallel }$ and ${\tau }_{\parallel }$. Reflectivity and transmissivity are power ratios across the interface, $R_{\parallel }=|{\Gamma }_{\parallel }{|}^2$ and $T_{\parallel }=\frac{{\eta }_1\cos {\theta }_t}{{\eta }_2\cos {\theta }_i}|{\tau }_{\parallel }{|}^2$. The cosines appear because the three beams share one footprint $A$ on the interface but have the cross-sections $A\cos {\theta }_i$ and $A\cos {\theta }_t$ (Figure 6.4.5), while each beam carries $\mathcal{P}=|E_0{|}^2/2\eta$ through its own cross-section.