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

Every problem of this section, and of Section 6.3, follows the same few steps.

Recipe · Reflection and transmission at a plane interface

Geometry. Normal or oblique incidence? Identify the interface, its normal and the plane of incidence, and measure every angle from the normal. Split the incident field into its S part ( $E$ normal to the plane of incidence) and its P part ( $H$ normal to it).
Snell first. $θ_{r} = θ_{i}$ and $n_{1} sin θ_{i} = n_{2} sin θ_{t}$ . If $n_{1} > n_{2}$ , compare $θ_{i}$ with $θ_{c} = arcsin (n_{2} / n_{1})$ . Beyond it use $cos θ_{t} = - j sin^{2} θ_{t} - 1$ , and the field in medium 2 is evanescent.
Impedances. $η = η_{0} μ_{r} / ε_{r}$ , that is $η = η_{0} / n$ for nonmagnetic media. Dividing every impedance by $η_{0}$ keeps the numbers simple. For a lossy medium $η$ is complex, for a perfect conductor $η = 0$ .
Coefficients. $Γ_{⊥}$ , $τ_{⊥} = 1 + Γ_{⊥}$ from Eq. (6.4.13), $Γ_{∥}$ , $τ_{∥}$ from Eq. (6.4.29). Then the reflected and the transmitted amplitudes, and the fields from $H = \frac{1}{η} a_{n} \times E$ .
Power and checks. $R = ∣Γ ∣^{2}$ , $T = \frac{η _{1} c o s θ _{t}}{η _{2} c o s θ _{i}} ∣ τ ∣^{2}$ and $R + T = 1$ . Check the limits: normal incidence, a perfect conductor, grazing incidence, the Brewster and the critical angles. Check the tangential fields on the interface.

6.4.11 A recipe for oblique incidence