More Bending Of Light

From the post "Split Cannot Mend" dated 10 Aug 2015,

$tan(\theta_2-\alpha_{2s})=\cfrac{\mu_2}{\mu_1}tan(\theta_1-\alpha)$

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 $\mu_1$.

When we consider,

$tan(\theta_2+\alpha_{2p})=\cfrac{\mu_2}{\mu_1}tan(\theta_1+\alpha)$

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 $\mu_2\lt\mu_1$,

$180^o+\theta_2+\alpha_{2p}\lt\theta_1+\alpha-180^o$

$\alpha_{2p}\lt-360^o-\Delta\theta+\alpha$

and when $\mu_2\gt\mu_1$,

$180^o+\theta_2+\alpha_{2p}\gt\theta_1+\alpha-180^o$

$\alpha_{2p}\gt-360^o-\Delta\theta+\alpha$

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.