Waveguides and Cavities

A rectangular waveguide with dimensions $a$ (width) and $b$ (height) supports electromagnetic waves propagating in the $z$-direction. Two families of modes exist: TE (transverse electric, $E_z = 0$) and TM (transverse magnetic, $B_z = 0$).

TE$_{mn}$ modes. The longitudinal field is $B_z = B_0\cos(m\pi x/a)\cos(n\pi y/b)\,e^{i(kz-\omega t)}$.

The transverse fields are determined from $B_z$ via:

$E_x = \frac{i\omega}{k_c^2}\frac{\partial B_z}{\partial y}, \quad E_y = -\frac{i\omega}{k_c^2}\frac{\partial B_z}{\partial x}$

$B_x = \frac{-ik}{k_c^2}\frac{\partial B_z}{\partial x}, \quad B_y = \frac{-ik}{k_c^2}\frac{\partial B_z}{\partial y}$

Where $k_c^2 = (m\pi/a)^2 + (n\pi/b)^2$ is the cutoff wavenumber.

Cutoff frequency: Waves propagate only when $\omega > \omega_{c,mn}$ where:

$f_{c,mn} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$

The dominant (lowest frequency) mode is TE$_{10}$ with $f_{c,10} = c/(2a)$ (for $a > b$).

Dispersion relation:

$k = \sqrt{\frac{\omega^2}{c^2} - k_c^2}, \quad v_{\text{phase} = \frac{\omega}{k} = \frac{c}{\sqrt{1 - (\omega_c/\omega)^2}} > c}$

$v_{\text{group} = \frac{d\omega}{dk} = c\sqrt{1 - \left(\frac{\omega_c}{\omega}\right)^2} < c}$

The product $v_p \cdot v_g = c^2$.

The wave impedance for TE modes:

$Z_{\text{TE} = \frac{E_x}{H_y} = \frac{\omega\mu_0}{k} = \frac{Z_0}{\sqrt{1 - (f_c/f)^2}}}$

Where $Z_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377\,\Omega$ is the impedance of free space.

The time-averaged power carried by TE$_{10}$ mode:

$\langle P \rangle = \frac{ab}{4}E_0^2\frac{\beta}{\omega\mu_0} = \frac{ab}{4Z_{\text{TE}}E_0^2}$

Where $\beta = k$ is the propagation constant and $E_0$ is the peak electric field.

A rectangular cavity of dimensions $a \times b \times d$ supports standing waves at resonant frequencies:

$f_{mnp} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{p}{d}\right)^2}$

Where $m, n, p$ are non-negative integers (not all zero). For TM modes, $p \geq 1$; for TE modes, $m$ and $n$ cannot both be zero.

Quality factor:

$Q = \frac{\omega \times \text{energy} stored}{\text{power} dissipated} = \frac{2\pi \times \text{energy} stored}{\text{energy} lost per cycle}$

For a cavity with conducting walls of conductivity $\sigma$:

$Q \approx \frac{V}{S\,\delta} \cdot \frac{3}{2}$

Where $V$ is the cavity volume, $S$ is the surface area, and $\delta$ is the skin depth.

• Assuming TEM modes exist in hollow waveguides: TEM modes require at least two separate conductors (e.g., coaxial cables). Hollow rectangular and circular waveguides cannot support TEM modes; they only support TE and TM modes.
• Confusing cutoff frequency with zero propagation: At $\omega = \omega_c$, $k = 0$ and the wave does not propagate. Below cutoff, $k$ becomes imaginary and fields decay exponentially (evanescent mode), carrying no net power.
• Forgetting that $m$ or $n$ can be zero in TE modes but not in TM modes: For TE$_{mn}$ modes, $m$ and $n$ cannot both be zero, but one may be zero. For TM$_{mn}$ modes, both $m$ and $n$ must be non-zero, meaning the lowest TM mode is TM$_{11}$.
• Misapplying the quality factor formula: The $Q$ of a cavity depends on the specific mode, as different field distributions produce different wall currents and hence different ohmic losses. The approximate formula $Q \approx (V/S\delta) \cdot 3/2$ is for the dominant mode only.

For a circular waveguide of radius $R$, the TE modes have cutoff wavenumbers $k_c = p'_{mn}/R$ where $p'_{mn}$ is the $n$-th root of $J'_m(x) = 0$. The TM modes have $k_c = p_{mn}/R$ where $p_{mn}$ is the $n$-th root of $J_m(x) = 0$.

The dominant mode in a circular waveguide is TE$_{11}$, with cutoff $f_c = \frac{cp'_{11}}{2\pi R} = \frac{1.841c}{2\pi R}$. Circular waveguides are used in rotating joints and polarisation-sensitive applications because TE$_{11}$ maintains polarisation orientation.

Problem. Design a rectangular cavity ($a = 3$ cm, $b = 1.5$ cm, $d = 2$ cm) that resonates at approximately 10 GHz. Which mode should be used?

Solution. The resonant frequency formula is $f_{mnp} = \frac{c}{2}\sqrt{(m/a)^2 + (n/b)^2 + (p/d)^2}$.

For TE$_{101}$: $f = \frac{3 \times 10^8}{2}\sqrt{(1/0.03)^2 + (1/0.02)^2} = 1.5 \times 10^8 \times \sqrt{1111.1 + 2500} = 1.5 \times 10^8 \times 60.09 = 9.01$ GHz.

For TE$_{102}$: $f = 1.5 \times 10^8 \times \sqrt{1111.1 + 10000} = 1.5 \times 10^8 \times 105.4 = 15.8$ GHz.

For TE$_{011}$: $f = 1.5 \times 10^8 \times \sqrt{(1/0.015)^2 + (1/0.02)^2} = 1.5 \times 10^8 \times \sqrt{4444.4 + 2500} = 1.5 \times 10^8 \times 83.33 = 12.5$ GHz.

TE$_{101}$ at 9.01 GHz is closest to 10 GHz. Fine-tuning the dimensions or inserting a dielectric can adjust the resonant frequency upward to exactly 10 GHz.

• Cutoff frequency determines single-mode operation: For a waveguide with $a > b$, the TE$_{10}$ mode has the lowest cutoff. Operating between $f_{c,10}$ and the next higher cutoff ensures only one mode propagates, avoiding modal dispersion.
• Phase velocity exceeds $c$ while group velocity is below $c$: This is consistent with special relativity because no information travels at the phase velocity; signal velocity is bounded by $v_g \leq c$.
• The product $v_p \cdot v_g = c^2$ is universal for all waveguide modes in a lossless rectangular guide, a direct consequence of the dispersion relation.
• Quality factor $Q$ increases with cavity size: Larger cavities store more energy relative to wall losses, giving higher $Q$. This is why microwave cavities in particle accelerators are large.
• Skin depth decreases with frequency: Higher frequency means thinner current-carrying layer on walls, reducing resistive losses but also reducing the effective conductor cross-section.

• Microwave communication: Rectangular waveguides (e.g., WR-90 for X-band) carry radar signals with low loss, as the confined mode avoids radiation losses.
• Particle accelerators: Resonant cavities (e.g., RF cavities in synchrotrons) accelerate charged particles by sustaining strong oscillating electric fields at precise frequencies.
• Microwave ovens: The magnetron generates microwaves at 2.45 GHz that propagate into the oven cavity, where standing waves heat food.
• Fibre optics: Although optical fibres are dielectric waveguides rather than metallic, the same concepts of modes, cutoff, and dispersion apply.
• Radar systems: Waveguide components (bends, twists, directional couplers) route microwave signals between the transmitter, antenna, and receiver with minimal loss.