Reflection, refraction, and the Fresnel equations

The factor exp (icot) has been omitted for simplicity, but the amplitudes, Si, Ci, and So have been written explicitly. At y = 0, continuity of the tangenhal component, Ez, across the boundary requires 

Since this equahon must be valid for all values of x, all exponentials must be the same, which leads to two condihons  

The tangential component of H provides another set of equations for the amplitudes Si, Cl, and Sq. According to Eq. (1.3.32), the amplitude of H can be found by multiplying E by m. The right-hand rule for the vectors E, H, and S determines the choice of sign of m. 

For y = 0 the tangential components of H must be the same for both media, which leads to 

Now we consider the case of the magnetic vector normal to the plane of incidence i.e., only exists. E is orthogonal to H and, therefore, in the plane of incidence. The polarization of this wave is orthogonal to that of the first case. With similar considerations one finds Si + Ci = m ioSq and Si — Ci = So cos / o/ cos 4 i- Solving for Til = So/Si and R = Ci/Si yields in this case 

---