When a ray of light hits a lens at angle of incidence \(\theta_{i},\) some of
the light is refracted (bent) as it passes through the lens, and some is
reflected by the lens. In the diagram, \(\theta_{r}\) is the angle of reflection
and \(\theta_{t}\) is the angle of refraction. Fresnel equations describe the
behaviour of light in this situation.
a) Snells's law states that \(n_{1} \sin \theta_{i}=n_{2} \sin \theta_{\ell}\)
where \(n_{1}\) and \(n_{2}\) are the refractive indices of the mediums. Isolate
\(\sin \theta_{t}\) in this equation.
b) Under certain conditions, a Fresnel equation to find the fraction, \(R,\) of
light reflected is \(R=\left(\frac{n_{1} \cos \theta_{i}-n_{2} \cos
\theta_{t}}{n_{1} \cos \theta_{i}+n_{2} \cos \theta_{t}}\right)^{2}\)
Use identities to prove that this can be written as \(R=\left(\frac{n_{1} \cos
\theta_{i}-n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}{n_{1} \cos \theta_{i}+n_{2}
\sqrt{1-\sin ^{2} \theta_{t}}}\right)^{2}\)
c) Use your work from part a) to prove that
$$
\begin{array}{l}
\left(\frac{n_{1} \cos \theta_{i}-n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}{n_{1}
\cos \theta_{i}+n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}\right)^{2} \\
=\left(\frac{n_{1} \cos \theta_{i}-n_{2}
\sqrt{1-\left(\frac{n_{1}}{n_{2}}\right)^{2} \sin ^{2} \theta_{i}}}{n_{1} \cos
\theta_{i}+n_{2} \sqrt{1-\left(\frac{n_{1}}{n_{2}}\right)^{2} \sin ^{2}
\theta_{i}}}\right)^{2}
\end{array}
$$
