Waveguides and Cavities

9.1 Rectangular Waveguides

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 $v_p \cdot v_g = c^2$.

9.2 Waveguide Impedance and Power Flow

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.

9.3 Resonant Cavities

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.