Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Recap

Two polarisations: the plane of incidence contains the incident ray and the normal. In S (perpendicular, TE) polarisation $E$ is normal to it, in P (parallel, TM) polarisation $H$ is. They do not mix at the interface, and every wave is a sum of the two.
Phase matching: the boundary conditions hold for all points of the interface, hence $θ_{r} = θ_{i}$ and Snell's law $β_{1} sin θ_{i} = β_{2} sin θ_{t}$ , or $n_{1} sin θ_{i} = n_{2} sin θ_{t}$ , Eq. (6.4.10).
Fresnel coefficients: $Γ_{⊥}$ , $τ_{⊥} = 1 + Γ_{⊥}$ , Eq. (6.4.13), and $Γ_{∥}$ , $τ_{∥}$ , Eq. (6.4.29), the normal-incidence formulas with the impedances $η / cos θ$ (S) and $η cos θ$ (P). Perfect conductor: $Γ = - 1$ .
Power: $R = ∣Γ ∣^{2}$ , $T = \frac{η _{1} c o s θ _{t}}{η _{2} c o s θ _{i}} ∣ τ ∣^{2}$ , $R + T = 1$ , Eq. (6.4.15). The cosines account for the different beam cross-sections.
Total reflection: from a denser medium, beyond $θ_{c} = arcsin (n_{2} / n_{1})$ , Eq. (6.4.17), $∣ Γ_{⊥} ∣ = ∣ Γ_{∥} ∣ = 1$ and the field in medium 2 is evanescent, $e^{- α_{2} x}$ with $α_{2} = β_{1}^{2} sin^{2} θ_{i} - β_{2}^{2}$ , Eq. (6.4.19), carrying no power across. Prisms, fibres, $NA = n_{1}^{2} - n_{2}^{2}$ .
Brewster angle: Eq. (6.4.23) and Eq. (6.4.30). Between nonmagnetic media only P has one, $tan θ_{B} = n_{2} / n_{1}$ , $θ_{B} + θ_{t} = 9 0^{\circ}$ , and the light reflected there is purely S. Laser windows, polarisers, sunglasses.

6.4.12 Recap