\(tan(\theta_2-\alpha_{2s})=\cfrac{\mu_2}{\mu_1}tan(\theta_1-\alpha)\)
when
\(\theta_1-\alpha\lt0\)
ie incident angle \(\alpha\) large,
\(tan(\theta_2-\alpha_{2s})=-\cfrac{\mu_2}{\mu_1}tan(\alpha-\theta_1)\)
\(tan(\theta_2-\alpha_{2s})=\cfrac{\mu_2}{\mu_1}tan(180^o-\alpha+\theta_1)\)
When \(\mu_2\lt\mu_1\),
\(\theta_2-\alpha_{2s}\lt180^o-\alpha+\theta_1\)
\(\alpha_{2s}\gt\Delta \theta-180^o+\alpha\)
But if \(\alpha_{2s}\) is to make a turn of \(-180^o\),
\(\theta_2=\theta_1\)
\(\Delta \theta=0\)
And so,
\(\alpha_{2s}\gt-180^o+\alpha\)
Also consider when \(\mu_2\gt\mu_1\),
\(\theta_2-\alpha_{2s}\gt180^o-\alpha+\theta_1\)
\(\alpha_{2s}\lt\Delta \theta-180^o+\alpha\)
But if \(\alpha_{2s}\) is to make a turn of \(-180^o\),
\(\theta_2=\theta_1\)
\(\Delta \theta=0\)
And so,
\(\alpha_{2s}\lt-180^o+\alpha\)
So, when \(\mu_2=\mu_1\)
\(\alpha_{2s}=-180^o+\alpha\)
\(\alpha_{2s}\) is reflected back along \(\alpha\), as \(\alpha_{2s}\) is measured anticlockwise positive. In all cases, \(\alpha_{2s}\) is reflected back into in medium \(\mu_1\).
\(tan(\theta_2+\alpha_{2p})=\cfrac{\mu_2}{\mu_1}tan(\theta_1+\alpha)\)
for large incident angle \(\alpha\), such that
\(\theta_1+\alpha\gt180^o\)
This happens with \(\small{EMW}\) where \(\theta\) is measured towards the positive \(E\) direction.
Let \(x+180^o=\theta_1+\alpha\) then
\(tan(\theta_1+\alpha)=tan(x+180^o)=tan(x)\)
So,
\(tan(\theta_2+\alpha_{2p})=\cfrac{\mu_2}{\mu_1}tan(\theta_1+\alpha-180^o)\)
\(tan(180^o+\theta_2+\alpha_{2p})=\cfrac{\mu_2}{\mu_1}tan(\theta_1+\alpha-180^o)\)
When \(\mu_2\lt\mu_1\),
\(180^o+\theta_2+\alpha_{2p}\lt\theta_1+\alpha-180^o\)
\(\alpha_{2p}\lt-360^o-\Delta\theta+\alpha\)
\(180^o+\theta_2+\alpha_{2p}\gt\theta_1+\alpha-180^o\)
\(\alpha_{2p}\gt-360^o-\Delta\theta+\alpha\)
In these cases, \(\alpha_{2p}\) is not in the same medium.
The two beams \(\alpha_{2s}\) and \(\alpha_{2p}\) behave differently for large incident angle \(\alpha\). \(\alpha_{2s}\) is reflected back along \(\alpha\), the incident ray and \(\alpha_{2p}\) is displaced from the extrapolated path of \(\alpha\) by \(-\Delta \theta\).
Total internal reflection due to velocity changes as photons pass through the two mediums is a distinct phenomenon apart from these.
