Phase matching at a planar wave interface

Equality of the boundary fields for every tangential position and time forces equal frequency and equal tangential wavevector for incident, reflected, and transmitted plane waves.

For refractive indices $n_{-}$ and $n_{+}$,

Total internal reflection occurs when a wave travels from higher to lower refractive index above the critical incidence angle, leaving no propagating transmitted plane wave normal to the interface.

For refractive indices $n_{-}>n_{+}$, the critical angle satisfies $\sin {\theta }_c=n_{+}/n_{-}$. Above it, the transmitted normal wavenumber is imaginary and the transmitted field is an evanescent wave.