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

Recap

Three waves at a plane interface, all along $\pm a_{x}$ with the same $ω$ . The reflected wave has $H^{r} = - (E_{0}^{r} / η_{1}) e^{j β_{1} x} a_{z}$ , since $H = \frac{1}{η} a_{k} \times E$ . Tangential $E$ and $H$ are continuous (no surface current for finite $σ$ ).
Coefficients, Eq. (6.3.10): $Γ = (η_{2} - η_{1}) / (η_{2} + η_{1})$ , $τ = 2 η_{2} / (η_{2} + η_{1}) = 1 + Γ$ , the transmission-line formula with $Z_{L} = η_{2}$ . Valid for lossy media with complex $η$ . Perfect conductor: $Γ = - 1$ , $τ = 0$ .
Power, Eq. (6.3.15): $R = ∣Γ ∣^{2}$ and $T = \frac{η _{1}}{η _{2}} ∣ τ ∣^{2}$ , with $R + T = 1$ . For nonmagnetic media $Γ = (n_{1} - n_{2}) / (n_{1} + n_{2})$ . For a lossy medium 2 $T = η_{1} ∣ τ ∣^{2} Re {1/ η_{2}^{*}}$ , and all of it is absorbed.
Standing waves, Eq. (6.3.19): maxima $∣ E_{0}^{i} ∣ (1 + ∣Γ∣)$ and minima $∣ E_{0}^{i} ∣ (1 - ∣Γ∣)$ , $λ_{1} /4$ apart, $S = (1 + ∣Γ∣) / (1 - ∣Γ∣)$ . A minimum on the interface means $Γ < 0$ .
Perfect conductor ( $Γ = - 1$ ): the pure standing wave Eq. (6.3.23) has $E_{1} \propto sin β_{1} x sin ω t$ and $H_{1} \propto cos β_{1} x cos ω t$ , it carries no average power, and the wall carries $J_{S} = (2 E_{0}^{i} / η_{1}) cos ω t a_{y}$ .
Three-part problem, Eqs. (6.3.32) and (6.3.33), and at the front face $Γ_{d} = (η_{in} - η_{1}) / (η_{in} + η_{1})$ with $η_{in} = η_{2} (η_{3} + j η_{2} tan β_{2} d) / (η_{2} + j η_{3} tan β_{2} d)$ . Slab in one medium: $∣Γ ∣^{2}$ of Eq. (6.3.35).
Layers: quarter wave, $η_{in} = η_{2}^{2} / η_{3}$ , matched by $n_{2} = n_{1} n_{3}$ (antireflection coatings). Half wave, $η_{in} = η_{3}$ , the layer disappears (radome walls). Both work at one frequency and its odd or whole multiples.

6.3.7 Recap