Home exercise 6.4.2: The Fresnel formulas

and in the same way ${\tau }_{\perp }=\frac{2\sin {\theta }_t\cos {\theta }_i}{\sin ({\theta }_i+{\theta }_t)}$.

For $n_1\ne n_2$ and ${\theta }_i\ne 0$, ${\theta }_t\ne {\theta }_i$ and the numerator $\tan ({\theta }_i-{\theta }_t)$ is not zero. ${\Gamma }_{\parallel }$ can vanish only through $\tan ({\theta }_i+{\theta }_t)\to \infty$:

For S the denominator $\sin ({\theta }_i+{\theta }_t)$ is finite, and ${\Gamma }_{\perp }=0$ would need $\sin ({\theta }_i-{\theta }_t)=0$, that is ${\theta }_t={\theta }_i$, which by Snell's law means $n_1=n_2$: no interface at all.

Problem 6.4.1 (${\theta }_i=60^{\circ }$, ${\theta }_t=30^{\circ }$): ${\Gamma }_{\perp }=-\sin 30^{\circ }/\sin 90^{\circ }=-0.5$, ${\tau }_{\perp }=2\times 0.5\times 0.5/1=0.5$, ${\Gamma }_{\parallel }=-\tan 30^{\circ }/\tan 90^{\circ }=0$. At normal incidence, with $\theta \approx \sin \theta$, both $\Gamma$'s tend to $-({\theta }_i-{\theta }_t)/({\theta }_i+{\theta }_t)=(n_1-n_2)/(n_1+n_2)$.