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

One formula covers both families. With $v = 1/ μ ε$ ,

Cutoff frequency and cutoff wavelength

f_{co, mn} = \frac{v}{2} (\frac{m}{a})^{2} + (\frac{n}{b})^{2}, λ_{c, mn} = \frac{v}{f _{co, mn}}, (7.3.28)

TE $_{mn}$ with $m, n = 0, 1, 2, \dots$ not both zero, TM $_{mn}$ with $m, n = 1, 2, 3, \dots$ A mode propagates when $f > f_{co, mn}$ , with $τ$ , $λ_{g}$ , $v_{p}$ , $v_{g}$ and the wave impedance of Eqs. (7.3.4) and (7.3.5), and is evanescent below.

In plain words

A mode can travel only if the half-wavelengths of the filling fit its pattern into the cross-section: for TE $_{10}$ the width must exceed half a wavelength. Every finer pattern needs a shorter wavelength, that is a higher frequency.

For the common proportion $a = 2 b$ the cutoffs in units of $f_{co, 10} = v /2 a$ are $m^{2} + 4 n^{2}$ :

mode	$f_{co} / f_{co, 10}$
TE $_{10}$	$1$
TE $_{20}$ , TE $_{01}$	$2$
TE $_{11}$ , TM $_{11}$	$5 = 2.236$
TE $_{21}$ , TM $_{21}$	$8 = 2.828$
TE $_{30}$	$3$

A mode chart shows the same information graphically (Figure 7.3.3). Each bar starts at the cutoff of its mode and extends to the right, over all the frequencies at which the mode propagates. A vertical line at the operating frequency crosses the bars of all the propagating modes. Below $2 f_{co, 10}$ only TE $_{10}$ propagates: when $b \leq a /2$ the guide has a full octave of single-mode band, from $v /2 a$ to $v / a$ .

Mode chart of the air-filled 6\times3\,\mathrm{cm} guide of Problem 7.3.1 (a=2b). Each bar starts at the cutoff of its mode: orange for TE, orange and blue halves for a degenerate TE_{mn}/TM_{mn} pair. Between 2.5 and 5\,\mathrm{GHz} only TE_{10} propagates. — **Figure 7.3.3.** Mode chart of the air-filled $6 \times 3 cm$ guide of Problem 7.3.1 ( $a = 2 b$ ). Each bar starts at the cutoff of its mode: orange for TE, orange and blue halves for a degenerate TE $_{mn}$ /TM $_{mn}$ pair. Between $2.5$ and $5 GHz$ only TE $_{10}$ propagates.

Quick check

Which modes propagate at $20 GHz$ in an air-filled guide with $a = 2 cm$ and $b = 1 cm$ ?

$f_{co, 10} = c /2 a = 7.5 GHz$ , $f_{co, 20} = f_{co, 01} = 15 GHz$ , $f_{co, 11} = 1.5 \times 1 0^{8} 5 0^{2} + 10 0^{2} = 16.77 GHz$ , $f_{co, 21} = 1.5 \times 1 0^{8} 10 0^{2} + 10 0^{2} = 21.21 GHz$ . Five modes propagate: TE $_{10}$ , TE $_{20}$ , TE $_{01}$ , TE $_{11}$ and TM $_{11}$ .

7.3.4 Cutoff frequencies and the mode chart