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

For a TM mode $H_{z} = 0$ , and by Eq. (7.4.10)

E_{z} = C J_{m} (h ρ) cos m ϕ e^{- j τ z} . (7.4.15)

The wall condition $E_{z} (a, ϕ) = 0$ must hold for every $ϕ$ , and $cos m ϕ$ is not zero for every $ϕ$ . Therefore $J_{m} (ha) = 0$ : the product $ha$ must be one of the zeros $x_{mn}$ of $J_{m}$ .

TM modes of the circular guide

h_{mn} = \frac{x _{mn}}{a} (zeros of J_{m}), m = 0, 1, 2, \dots, n = 1, 2, \dots (7.4.16)

The second wall condition, $E_{ϕ} (a, ϕ) = 0$ , is then satisfied by itself. With $H_{z} = 0$ , Eq. (7.4.13) gives $E_{ϕ} = - (j τ / h^{2} ρ) \partial E_{z} / \partial ϕ$ , which is proportional to $J_{m} (h ρ)$ and vanishes at $ρ = a$ together with $E_{z}$ . One condition is enough. The transverse components follow from Eq. (7.4.13) by differentiating Eq. (7.4.15). For instance $\partial E_{z} / \partial ρ = C h J_{m}^{'} (h ρ) cos m ϕ$ , so that $E_{ρ} = - (j τ / h^{2}) C h J_{m}^{'} (h ρ) cos m ϕ$ . Altogether (the factor $e^{- j τ z}$ omitted)

E_{ρ} E_{ϕ} H_{ρ} H_{ϕ} = - \frac{j τ}{h} C J_{m}^{'} (h ρ) cos m ϕ, = \frac{j τ m}{h ^{2} ρ} C J_{m} (h ρ) sin m ϕ, = - \frac{j ω ε m}{h ^{2} ρ} C J_{m} (h ρ) sin m ϕ, = - \frac{j ω ε}{h} C J_{m}^{'} (h ρ) cos m ϕ . (7.4.17)

Note that $H_{ρ} \propto J_{m} (h ρ)$ also vanishes at the wall, as the condition on the normal magnetic field requires: the TM fields satisfy all the wall conditions with the single choice $J_{m} (ha) = 0$ .

Definition · Mode indices of the circular guide

In TM $_{mn}$ and TE $_{mn}$ , the first index $m$ counts the full periods of the field around the guide ( $cos m ϕ$ ), the second index $n$ counts the zeros of the radial function along a radius, from the axis (excluded) to the wall: for a TM mode the zeros of $E_{z}$ , the wall included. Hence $n \geq 1$ always, and there is no TM $_{m 0}$ mode.

Figure 7.4.3a shows the radial profiles of $E_{z}$ for TM $_{01}$ and TM $_{02}$ . The wall is placed exactly at the first, or the second, zero of $J_{0}$ . A TM $_{m 0}$ mode would need the wall at the zeroth zero of $J_{m}$ , which does not exist.

(a) TM: E_z(\rho). — (a) TM: $E_{z} (ρ)$ .

(b) TE: H_z(\rho). — (a) TM: $E_{z} (ρ)$ .

7.4.5 TM modes