 Solution

Choose the interface as $x=0$ and write the incident, reflected, and transmitted fields as plane waves with wavevectors $k_I,k_R,k_T$. The boundary conditions must hold for every tangential coordinate and every time. The phase matching at a planar wave interface therefore requiresand equality of tangential wavevectors:

The incident and reflected waves are in the same medium, so
$|k_I|=|k_R|$. Equality of their tangential components giveshence

At fixed frequency, $|k_{\pm }|=\omega /v_{\pm }$ is proportional to the refractive index $n_{\pm }$. Tangential phase matching between incident and transmitted waves gives the Snell law for electromagnetic waves:orHere $n_{\pm }=c/v_{\pm }=c\sqrt{{ϵ}_{\pm }\mu }$ if the vacuum speed $c$ is used to normalize the index.