\(\alpha\to\alpha_2\)
For the case of \(\mu_2\gt\mu_1\),
\(\alpha_{2s}\lt\alpha_2+\Delta\theta\)
and
\(\alpha_{2p}\gt\alpha_2-\Delta\theta\)
For the case of \(\mu_2\lt\mu_1\),
\(\alpha_{2s}\gt\alpha_2+\Delta\theta\)
and
\(\alpha_{2p}\lt\alpha_2-\Delta\theta\)
When \(\Delta \theta=0\), the equations collapse to
\(\alpha_{2s}=\alpha_{2p}=\alpha_2\)
as we observe by Snell's Law for refraction alone. (We take the intersection of the two regions defining \(\alpha_{2s}\) and \(\alpha_{2p}\) after admitting the case of \(\mu_1=\mu_2\). )
It might appear that the split is due to the change in polarization \(\theta\) alone. In fact, both polarization, \(\theta\) and boundary conditions for the \(B\) field at the interface, contribute to refraction.
Note: \(\mu\ne n\)
