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

Recap

Equation: $E_{z}$ (TM) or $H_{z}$ (TE) $= Ψ e^{- j τ z}$ obeys $\nabla_{T}^{2} Ψ + h^{2} Ψ = 0$ , $h^{2} = k^{2} - τ^{2}$ , Eq. (7.4.5). Separation $Ψ = R (ρ) Φ (ϕ)$ gives $cos m ϕ$ or $sin m ϕ$ with integer $m$ , and Bessel's equation for $R$ . In a hollow pipe $Ψ = C J_{m} (h ρ) cos m ϕ$ , Eq. (7.4.10), since $Y_{m}$ is infinite on the axis.
Transverse fields, Eq. (7.4.13): the Cartesian formulas with $x \to ρ$ , $y \to ρϕ$ . Wave impedances, $τ$ , $λ_{g}$ , $v_{p}$ , $v_{g}$ as in every guide.
Cutoffs: TM $_{mn}$ from the zeros of $J_{m}$ , $h = x_{mn} / a$ , TE $_{mn}$ from the zeros of $J_{m}^{'}$ , $h = x_{mn}^{'} / a$ , $f_{c} = xv / (2 π a)$ , Eq. (7.4.21). No TEM, no TM $_{m 0}$ or TE $_{m 0}$ .
Mode order: TE $_{11}$ ( $1.841$ , dominant, $λ_{c} = 3.41 a$ ), TM $_{01}$ ( $2.405$ ), TE $_{21}$ ( $3.054$ ), TE $_{01}$ and TM $_{11}$ ( $3.832$ , degenerate). Single-mode band only $1.31 : 1$ . Every mode with $m \geq 1$ is a pair of polarisations.
Patterns: TE $_{11}$ like a rounded TE $_{10}$ , TM $_{01}$ and TE $_{01}$ rotationally symmetric. Two TE $_{11}$ copies in quadrature make a circularly polarised wave.
Loss, Eq. (7.4.23): $α_{c}^{T M} = R_{s} / (a η s)$ , minimum at $3 f_{c}$ . TE $_{01}$ falls as $f^{- 3/2}$ (purely azimuthal wall current). Applications: TM $_{01}$ rotary joints, TE $_{01}$ long-haul pipes.

7.4.10 Recap