The Fresnel Equations
At a boundary between two media, the tangential components of the electric field E are conserved. If there are no surface currents, the tangential components of the magnetic field H are also conserved.
If the incident ray is s-polarized, meaning the electric field is perpendicular to the plane of incidence, then from the conservation of the tangential components of E and H,
(3-15)
(3-16)
Substitution of Equation 3-16 into Equation 3-15 yields the ratios
Similarly, if the incident ray is p-polarized, meaning that the electric field lies in the plane of incidence and the magnetic field lies perpendicular to it, then from the conservation of the tangential components of E and H,
(3-17)
(3-18)
Substitution of Equation 3-18 into Equation 3-17 yields the ratios
Finally, we define the reflection coefficient as the ratio of the reflected electric field amplitude to the incident electric field amplitude, and similarly define the transmission coefficient as the ratio of the refracted electric field amplitude to the incident electric field amplitude.
The above definition of rp is consistent with Ref. 9 but shows the opposite sign compared to Ref. 10. This is because Ref. 9 defines the direction of the in-plane field component such that the s-polarized component, p-polarized component, and ray direction vector (in that order) always comprise a right-handed coordinate system; whereas in Ref. 10 the vectors written in this order would comprise a left-handed coordinate system for the reflected ray.