Using Snell's law (i.e., \(n_{o} r_{o} \cos \theta_{0}=n_{1} r_{1} \cos \theta_{1}\) ), show that

\[
\left(\sin \left(\frac{\theta_{1}}{2}ight)ight)^{2}=\frac{r_{o}}{2 r_{1}}\left[2\left(\sin \left(\frac{\theta_{0}}{2}ight)ight)^{2}+\frac{r_{1}-r_{o}}{r_{o}}-\frac{\left(N_{o}-N_{1}ight) \cdot 10^{-6}}{n_{1}} \cos \theta_{0}ight]
\]