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

The TE $_{011}$ mode is the TE $_{01}$ mode of the circular guide with one half guide wavelength between the end plates. With $h = x_{01}^{'} / a$ ,

H_{z} E_{ϕ} H_{ρ} = H_{0} J_{0} (\frac{x _{01}^{'} ρ}{a}) sin \frac{π z}{d}, \propto J_{1} (\frac{x _{01}^{'} ρ}{a}) sin \frac{π z}{d}, \propto J_{1} (\frac{x _{01}^{'} ρ}{a}) cos \frac{π z}{d} . (7.8.34)

Three properties make it special.

Its wall currents flow only along $a_{ϕ}$ : on the side wall $J_{s} = a_{n} \times H$ with $H = H_{z} a_{z}$ there, and on the end plates with $H = H_{ρ} a_{ρ}$ . None of them crosses the joint between the side wall and an end plate, so a sliding non-contacting plate can tune it without spoiling it. This is the mode of wavemeters.
Its $Q$ is very high, $2$ to $3 \times 1 0^{4}$ for copper in the X band (Worked example 7.8.2).
It is degenerate with TM $_{111}$ , because $x_{01}^{'} = x_{11} = 3.832$ (from $J_{0}^{'} = - J_{1}$ ).

The mode chart

Multiply Eq. (7.8.29) by $2 a$ and square:

(2 a f)^{2} = (\frac{v x}{π})^{2} + (\frac{p v}{2})^{2} (\frac{2 a}{d})^{2}, (7.8.35)

where $x$ is $x_{mn}$ or $x_{mn}^{'}$ . Plotted against $(2 a / d)^{2}$ , every mode is a straight line, with intercept $(v x / π)^{2}$ and slope $(p v /2)^{2}$ (Figure 7.8.13, with $v = 30 GHz cm$ for air). A vertical line at the $2 a / d$ of a given cavity lists all its resonances in order, with $f = y /2 a$ . A tuning plunger, which changes $d$ , moves the cavity along the horizontal axis. The $p = 0$ lines (TM $_{010}$ , TM $_{110}$ ) are horizontal, and the point $(2 a / d)^{2} = 0.97$ , where TE $_{111}$ crosses TM $_{010}$ , is the limit $d = 2.03 a$ of Eq. (7.8.30).

Mode chart of the air-filled circular cylindrical cavity: (2af)^2 in (\mathrm{GHz\,cm})^2 against (2a/d)^2, from Eq. (7.8.35) with v=30\,\mathrm{GHz\,cm}. TM_{010} is the lowest mode to the right of the dot (d\lt 2.03a), TE_{111} to the left of it. — **Figure 7.8.13.** Mode chart of the air-filled circular cylindrical cavity: $(2 a f)^{2}$ in $(GHz cm)^{2}$ against $(2 a / d)^{2}$ , from Eq. (7.8.35) with $v = 30 GHz cm$ . TM $_{010}$ is the lowest mode to the right of the dot ( $d < 2.03 a$ ), TE $_{111}$ to the left of it.

Quick check

An air-filled pillbox has $a = 1 cm$ and $d = 1 cm$ . Which mode is the lowest, and at what frequency?

$d = a < 2.03 a$ , so TM $_{010}$ : $f = 4.775 \times 2.405/1 = 11.48 GHz$ . On the chart $(2 a / d)^{2} = 4$ , where the TM $_{010}$ line ( $527.4$ ) lies far below TE $_{111}$ ( $309.1 + 225 \times 4 = 1209$ ).

Cavity versus circuit

A cavity is a resonant circuit in which $L$ and $C$ are no longer separate elements: $E$ fills the middle, $H$ the regions near the walls, and the energy moves between them. It has many resonances instead of one, and a $Q$ about a hundred times higher. As an order of magnitude, $Q \approx (size / δ) \times (a geometry factor near \frac{1}{3})$ , about $1 0^{4}$ for copper at $10 GHz$ .

Application 7.8.2 · Wavemeters and cavity filters

THE ABSORPTION WAVEMETER

A TE $_{011}$ cavity hangs on a guide through a small coupling hole (Figure 7.8.14a). Off resonance it takes no power. At resonance it absorbs a little, and the detector after the guide shows a dip. The plunger is turned until the dip appears, and the frequency is read from the micrometer that sets $d$ . The non-contacting plunger does not spoil $Q$ , because the TE $_{011}$ currents do not cross the gap around it, and the degenerate TM $_{111}$ mode, whose currents do, has its $Q$ destroyed and shows no dip:

f_{011} Δ f = \frac{c}{2 π} (\frac{3.832}{a})^{2} + (\frac{π}{d})^{2}, = \frac{f _{0}}{Q _{L}} . (7.8.36)

CAVITY FILTERS

A transmission cavity, with two couplings, passes a band $Δ f = f_{0} / Q_{L}$ around $f_{0}$ and reflects the rest. Base stations, satellites and radar receivers chain several of them to obtain steep skirts. The loss at the centre follows from Eq. (7.8.27),

I L = - 20 lo g_{10} (1 - \frac{Q _{L}}{Q _{0}}) [dB], (7.8.37)

so a narrow band (large $Q_{L}$ ) costs loss unless $Q_{0}$ is much larger still.

(b) The dip seen by the detector. — (a) The absorption wavemeter.

	Wavemeter (X band)	Filter ( $2 GHz$ )
$Q_{0}$	$\approx 2 \times 1 0^{4}$ (TE $_{011}$ , copper)	$4000$
$Q_{L}$	close to $Q_{0}$ (weak coupling)	$400$
$Δ f$	$\approx 0.5 MHz$	$5 MHz$
Centre loss	a small dip	$0.92 dB$

A $2 GHz$ filter with $Q_{0} = 4000$ loaded to $Q_{L} = 400$ passes $Δ f = 5 MHz$ with $I L = - 20 lo g_{10} (1 - 0.1) = 0.92 dB$ . No lumped $L C$ filter could do this: its $Q_{0}$ of about $100$ is below the $Q_{L}$ required. Narrow-band filters therefore use cavities, often silver-plated, or low-loss ceramic dielectric resonators. Worked example 7.8.2 designs an X-band wavemeter.

Application 7.8.3 · Accelerator and klystron cavities

THE TM $_{010}$ PILLBOX

On the axis of a TM $_{010}$ cavity the electric field is axial and largest. A bunch of particles moving at $v \approx c$ that crosses a cell of length $d = λ /2$ in half a period always sees $E$ pointing forward and gains energy all the way. Chains of such cells, with the field alternating from cell to cell (the " $π$ mode", Figure 7.8.15), form linear accelerators (linacs) for medical radiotherapy, synchrotron light sources and free-electron lasers. A klystron uses the same field to bunch an electron beam in a first cavity and a second cavity to extract microwave power from the bunched beam, and klystrons power most radars and accelerators. Because the field changes while the particle crosses the gap, the energy gain is reduced by the transit-time factor $T$ :

f_{010} T = \frac{2.405 c}{2 π a}, V = E_{0} d T, = \frac{sin ( π d / λ )}{π d / λ} . (7.8.38)

A particle entering at the field maximum sees $E_{0} cos (ω z / c)$ while it moves from $z = - d /2$ to $d /2$ , and the average of this over the cell is $T$ . For $d = λ /2$ , $T = 2/ π = 0.64$ .

Five cells of a linear accelerator at one instant (\pi mode). E_z (orange) alternates from cell to cell. The bunches (red) sit in every second cell and ride with the field: half a period later the field has reversed and each bunch has moved into the next cell. — **Figure 7.8.15.** Five cells of a linear accelerator at one instant ( $π$ mode). $E_{z}$ (orange) alternates from cell to cell. The bunches (red) sit in every second cell and ride with the field: half a period later the field has reversed and each bunch has moved into the next cell.

	Copper, S band	Niobium, $2 K$
$f_{010}$	$2.856 GHz$	$1.3 GHz$
Radius $a$	$4.02 cm$	$8.83 cm$
$Q_{0}$	$1.8 \times 1 0^{4}$	$\sim 1 0^{10}$
$Δ f = f_{0} / Q_{0}$	$155 kHz$	$0.13 Hz$

A copper cell at $20 MV/m$ stores about $0.13 J$ and loses $124 kW$ in its walls (Worked example 7.8.3). Copper linacs are therefore pulsed, with pulses a few microseconds long.
Superconducting niobium has $R_{s}$ in the n $Ω$ range, so $Q_{0}$ is about six orders of magnitude higher and the cavities can run continuously. The loaded $Q$ is then set by the power coupler.

7.8.13 The TE₀₁₁ mode and the mode chart

The mode chart