Chapter 14

Waveguides & Resonant Cavities

14.1 TE and TM Modes

In a hollow metallic waveguide, assume propagation as $e^{i(k_z z - \omega t)}$. The boundary conditions $\mathbf{E}_\parallel = 0$ at walls allow two families of modes:

TE modes ($E_z = 0$)

$$(\nabla_T^2 + \gamma^2)B_z = 0, \quad \frac{\partial B_z}{\partial n}\bigg|_S = 0$$

Transverse-electric; B has a longitudinal component

TM modes ($B_z = 0$)

$$(\nabla_T^2 + \gamma^2)E_z = 0, \quad E_z\big|_S = 0$$

Transverse-magnetic; E has a longitudinal component

14.1.1 Rectangular Waveguide: TE(m,n) Modes

For a rectangular guide with dimensions $a \times b$:

$$B_z = B_0 \cos\!\left(\frac{m\pi x}{a}\right)\cos\!\left(\frac{n\pi y}{b}\right)e^{i(k_z z-\omega t)}$$$$k_{c,mn} = \pi\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}, \qquad \omega_{c,mn} = c\,k_{c,mn}$$

The propagation constant and dispersion relation:

$$k_z = \sqrt{\left(\frac{\omega}{c}\right)^2 - k_{c,mn}^2}, \qquad \omega > \omega_{c,mn}$$

Below cutoff, $k_z$ is imaginary and the mode is evanescent. The dominant mode (lowest cutoff) in a rectangular waveguide is TE10: $\omega_c = \pi c/a$.

14.2 Phase & Group Velocity in Waveguides

$$v_p = \frac{\omega}{k_z} = \frac{c}{\sqrt{1 - (\omega_c/\omega)^2}} > c$$$$v_g = \frac{d\omega}{dk_z} = c\sqrt{1 - \left(\frac{\omega_c}{\omega}\right)^2} < c$$$$v_p \cdot v_g = c^2$$

The phase velocity exceeds $c$ but carries no information; the group velocity (signal speed) is always less than $c$.

14.3 Resonant Cavities

A closed metallic cavity supports discrete resonant modes (TE and TM). For a rectangular cavity of dimensions $a \times b \times d$:

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

The quality factor $Q = \omega_0 W/P_{\rm loss}$ measures sharpness of resonance. Microwave ovens operate at 2.45 GHz; particle accelerator cavities achieve $Q \sim 10^{10}$.

Simulation: Rectangular Waveguide Modes

Rectangular Waveguide Modes (WR-90)

Cutoff frequencies, dispersion, phase/group velocity, and TE10/TE11 field patterns for X-band waveguide.

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Rectangular waveguide: TE modes
# Cutoff frequencies, dispersion, field patterns
# ──────────────────────────────────────────────
c = 3e8
a = 0.0229   # broad dimension [m] (WR-90, X-band)
b = 0.0102   # narrow dimension [m]

# TE(m,n) cutoff wavenumber: k_mn = pi*sqrt((m/a)^2+(n/b)^2)
modes = [(1,0,'TE10','#60a5fa'), (2,0,'TE20','#f59e0b'),
         (0,1,'TE01','#34d399'), (1,1,'TE11','#a78bfa'),
         (2,1,'TE21','#ef4444')]

print("Mode    | k_c [1/m]  | f_c [GHz]")
print("-"*40)
for m, n, name, _ in modes:
    kc = np.pi * np.sqrt((m/a)**2 + (n/b)**2)
    fc = c*kc/(2*np.pi)
    print(f"{name:<7} | {kc:10.2f} | {fc/1e9:8.3f}")

# Dispersion curves
freq = np.linspace(5e9, 30e9, 500)
omega = 2*np.pi*freq

fig, axes = plt.subplots(2, 2, figsize=(13, 9))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

for m, n, name, color in modes:
    kc = np.pi * np.sqrt((m/a)**2 + (n/b)**2)
    fc = c*kc/(2*np.pi)
    k_prop = np.sqrt(np.maximum(0, (omega/c)**2 - kc**2))
    axes[0,0].plot(freq/1e9, k_prop, color=color, lw=2, label=name)
axes[0,0].set_xlabel('f [GHz]', color='#94a3b8'); axes[0,0].set_ylabel('k_z [1/m]', color='#94a3b8')
axes[0,0].set_title('Dispersion: k_z vs f (WR-90)', color='white', fontsize=11)
axes[0,0].tick_params(colors='#94a3b8')
axes[0,0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,0].grid(alpha=0.2, color='#334155')

# Phase and group velocity for TE10
kc10 = np.pi/a
fc10 = c*kc10/(2*np.pi)
f_above = np.linspace(fc10*1.01, 30e9, 400)
om_above = 2*np.pi*f_above
kz = np.sqrt((om_above/c)**2 - kc10**2)
vp = om_above/kz
vg = c**2*kz/om_above
axes[0,1].plot(f_above/1e9, vp/c, color='#60a5fa', lw=2, label='v_p/c (phase)')
axes[0,1].plot(f_above/1e9, vg/c, color='#f59e0b', lw=2, label='v_g/c (group)')
axes[0,1].axhline(1, color='white', lw=1, linestyle='--', alpha=0.5, label='c')
axes[0,1].set_xlabel('f [GHz]', color='#94a3b8'); axes[0,1].set_ylabel('v/c', color='#94a3b8')
axes[0,1].set_title('Phase & Group Velocity: TE10', color='white', fontsize=11)
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,1].grid(alpha=0.2, color='#334155')

# TE10 transverse field pattern (Ey component)
x = np.linspace(0, a, 80); y = np.linspace(0, b, 40)
X, Y = np.meshgrid(x, y)
Ey10 = np.sin(np.pi*X/a)   # TE10: E_y = E0 sin(πx/a)
Ex10 = np.zeros_like(Ey10)
im2 = axes[1,0].pcolormesh(x*100, y*100, Ey10, cmap='RdBu_r', shading='auto')
plt.colorbar(im2, ax=axes[1,0], label='$E_y$ [arb]')
axes[1,0].set_xlabel('x [cm]', color='#94a3b8'); axes[1,0].set_ylabel('y [cm]', color='#94a3b8')
axes[1,0].set_title('TE10 E-field Pattern ($E_y$)', color='white', fontsize=11)
axes[1,0].tick_params(colors='#94a3b8')

# TE11 field pattern
m11, n11 = 1, 1
Ey11 = -np.pi*n11/b * np.cos(np.pi*m11*X/a) * np.sin(np.pi*n11*Y/b)
Ex11 =  np.pi*m11/a * np.sin(np.pi*m11*X/a) * np.cos(np.pi*n11*Y/b)
skip = 4
axes[1,1].quiver(x[::skip]*100, y[::2]*100,
                 Ex11[::2,::skip], Ey11[::2,::skip],
                 color='#a78bfa', alpha=0.9)
axes[1,1].set_xlabel('x [cm]', color='#94a3b8'); axes[1,1].set_ylabel('y [cm]', color='#94a3b8')
axes[1,1].set_title('TE11 E-field Vectors', color='white', fontsize=11)
axes[1,1].tick_params(colors='#94a3b8')

plt.tight_layout()
plt.savefig('waveguide_modes.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())

Click Run to execute the Python code

First run will download Python environment (~15MB)

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