Interface conditions for electromagnetic fields

Across an interface with unit normal $\mathbf{n}$ from medium one to medium two, Maxwell equations imply $\mathbf{n}\times ({\mathbf{E}}_2-{\mathbf{E}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{B}}_2-{\mathbf{B}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{D}}_2-{\mathbf{D}}_1)={\sigma }_f$, and $\mathbf{n}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_f$. Here ${\sigma }_f$ and ${\mathbf{K}}_f$ are the free surface electric charge and electric current densities.

Without free surface charge or current, the tangential components of $E$ and $H$ and the normal components of $D$ and $B$ are continuous across an 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.

The Fresnel equations give reflection and transmission amplitudes for both transverse-electric and transverse-magnetic polarization of an electromagnetic wave at a planar interface. They follow by applying interface conditions for electromagnetic fields to incident, reflected and refracted plane electromagnetic waves.

When the permeabilities agree, the signed reflected-to-incident electric-field ratio for polarization normal to the plane of incidence isFor incidence from lower to higher refractive index it never vanishes.

For equal permeabilities on the two sides of an interface, tangential-field continuity and Snell law for electromagnetic waves givefor consistently signed electric amplitudes normal to the plane of incidence.