What is the physical explanation of Fresnel's equation?

Question

When EM radiation is travelling between the media of different refractive indices, parts of it are reflected and transmitted according to Fresnel's equations. The amount of reflectivity and transmittance depends on the polarization of incoming radiation. In the derivation of those equations we assume the continuity of components tangential to interface of electric and magnetic fields. Is there continuity only in the tangential or also in components perpendicular to interface? And also I would like to have a physical explanation in terms of rays or waves by which I can imagine the phenomena.

Answer 1

The continuities are:

1. The tangential components of the electric and magnetic fields are continuous across and interface. Physical meaning: form a rectangular, thin loop straddling the interface, in a plane normal to the interface, such that the two sides of the loop parallel to the interface are of significant length, but the sides normal to the interface are insignificant. Now apply Faraday's law to the loop. Unless the magnetic induction is infinite, the flux of $\partial_t\,\vec{B}$ through the loop is nought. Therefore the line integral around the loop is nought, by Faraday's law. Conclude therefore that the components of $vec{E}$ on either side of the loop must be equal. Do this for two loops in different planes to show all tangential components of $\vec{E}$ must be equal. Now do the same for the magnetic field $\vec{H}$ and apply Ampère's law to reach an analogous conclusion for the tangential components of $\vec{H}$.

2. The normal components of $\vec{B}$ and $\vec{D}$ are conntinuous across an interface. To see why, construct a Gaussian surface straddling the surface with significant area parallel to the interface but of negligible height. Unless the electric charge density on the surface is infinite, the flux of $\vec{D}$ through this surface must be nought. Conclude therefore that the normal components of $\vec{D}$ must be equal, by Gauss's law for electricity. Reason analogously for $\vec{B}$ using Gauss's law for magnetism.

These continuities are what govern the Fresnel equations. There aren't any simpler visualizations in terms of rays that I am aware of.

The above also show you how the idealizations of perfect conductors, with surface charge densities and surface current densities (i.e. idealizations where we "squash" a nonzero amount of charge / current into a zero thickness, 2D surface) will break these rules. The rules generalize to: the discontinuous jump across an interface of the normal component of $\vec{D}$ is the surface charge density. The discontinuous jump across an interface of a tangential component of $\vec{H}$ is equal to the surface current density.