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

Recap

Geometry: plates at $y = 0$ , $y = b$ , $\partial / \partial x = 0$ . TE modes have $E_{x}, H_{y}, H_{z}$ , TM modes $E_{y}, E_{z}, H_{x}$ , Eq. (7.2.1).
TE $_{n}$ , Eq. (7.2.12): $H_{z} = C cos (nπ y / b) e^{- j τ_{n} z}$ , from $\partial H_{z} / \partial y = 0$ on the plates. TM $_{n}$ , Eq. (7.2.23): $E_{z} = C sin (nπ y / b) e^{- j τ_{n} z}$ , from $E_{z} = 0$ on the plates. Both have $h_{n} = nπ / b$ , $τ_{n} = k^{2} - (nπ / b)^{2}$ and $f_{co, n} = n v /2 b$ (degenerate pairs).
TEM = TM $_{0}$ , Eq. (7.2.28): $E_{y} = - η A_{0} e^{- j k z}$ , $H_{x} = A_{0} e^{- j k z}$ , no cutoff: the dominant mode. As a line: $Z_{0} = η b / w$ , $L^{'} = μ b / w$ , $C^{'} = εw / b$ , Eq. (7.2.29).
Zig-zag: every mode is two plane waves at $sin θ = f_{co, n} / f$ , $τ_{n} = k cos θ$ , $v_{g} = v cos θ$ , $v_{p} = v / cos θ$ . Transverse resonance gives $h_{n} b = nπ$ .
Modes: TEM plus a TE $_{n}$ , TM $_{n}$ pair for every $n < 2 b f / v$ . TEM only if $b < λ /2$ .
Power, Eq. (7.2.34): $P^{'} = b E_{0}^{2} / (2 η)$ (TEM), $b E_{0}^{2} / (4 Z)$ (TM $_{n}$ , TE $_{n}$ ). $P^{'} / W^{'} = v_{g}$ .
Losses, Eqs. (7.2.35)–(7.2.37): $α = P_{L}^{'} / (2 P^{'})$ , $α_{d} = σ η /2$ divided by $1 - (f_{co} / f)^{2}$ for the higher modes, $α_{c}^{T E M} = R_{s} / (η b)$ , $α_{c}^{T M_{n}} = 2 R_{s} / (η b s)$ , $α_{c}^{T E_{n}} = 2 R_{s} (f_{co, n} / f)^{2} / (η b s)$ , falling as $f^{- 3/2}$ .

7.2.11 Recap