Past exam of the physics B course of the University of Cambridge / 2015 / Paper 2 / Section B / B6 / d

Reflected and transmitted waves arise because the fields must satisfy the electromagnetic boundary conditions. With no free surface charge or current, tangential $E,H$ and normal $D,B$ are continuous. At normal incidence between lossless media, define the real wave impedance $Z_j=\sqrt{{\mu }_j/{ϵ}_j}$. Writing $r=E_r/E_i$ and $t=E_t/E_i$ in a common transverse electric-field direction givesFor the power normalization of Fresnel amplitudes, power fractions are $R=|r{|}^2$ and $T=(Z_1/Z_2)|t{|}^2$, so $R+T=1$. An electric-field amplitude coefficient alone is not a transmitted power fraction.

At oblique incidence, phase matching along the interface gives Snell's law, $n_1\sin {\theta }_i=n_2\sin {\theta }_t$. Applying the same boundary conditions separately to the two linear polarizations gives the Fresnel equations. For nonmagnetic dielectrics the parallel-polarized reflection vanishes at $\tan {\theta }_B=n_2/n_1$. Above the critical angle into a lower-index lossless medium, the transmitted field is evanescent and reflection is total. A perfect conductor instead has zero tangential electric field and unit reflected power; finite conductivity introduces penetration and absorption.^[ref]