9.4 Energy flow in waveguides

Waveguide modes

Electromagnetic waves propagate through waveguides in distinct patterns called modes. Each mode has a unique field configuration and a cutoff frequency below which that mode cannot propagate. The two main categories are transverse electric (TE) and transverse magnetic (TM).

TE vs TM modes

In TE modes, the electric field is entirely transverse (perpendicular) to the propagation direction, while the magnetic field has a longitudinal component along the guide. Common examples: TE10, TE20, TE01.

In TM modes, the magnetic field is entirely transverse, and the electric field has a longitudinal component. Common examples: TM11, TM21, TM01.

The mode with the lowest cutoff frequency is the dominant mode. For rectangular waveguides, this is typically TE10.

Hybrid modes

Hybrid modes (labeled HE and EH) have both electric and magnetic field components along the propagation direction. They arise in waveguides with more complex cross-sections, such as circular or elliptical geometries. Examples include HE11 and EH11. In a circular dielectric waveguide, for instance, pure TE or TM solutions generally don't satisfy all boundary conditions, so hybrid modes become necessary.

Cutoff frequency

The cutoff frequency is the minimum frequency at which a given mode can propagate. Below cutoff, the mode's fields decay exponentially (evanescent behavior) and no net energy is carried.

Waveguide dimensions

The cross-sectional shape and size set the cutoff frequencies for all modes.

• Rectangular waveguide with width $a$ and height $b$ ($a > b$): The cutoff frequency for the TE$_{mn}$ or TM$_{mn}$ mode is

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

For the dominant TE10 mode this simplifies to $f_c = \frac{c}{2a}$.

• Circular waveguide with radius $r$: The cutoff frequency of the dominant TE11 mode is

$f_c = \frac{1.841\, c}{2\pi r}$

where 1.841 is the first zero of $J_1'$, the derivative of the Bessel function of order 1.

Energy flow and the Poynting vector in waveguides

This is the central topic for Unit 9. The time-averaged power flowing through a waveguide cross-section $S$ is found by integrating the Poynting vector over that surface:

$\langle P \rangle = \frac{1}{2}\operatorname{Re}\int_S \left(\mathbf{E} \times \mathbf{H}^*\right) \cdot d\mathbf{A}$

Only the transverse field components contribute to the axial ($\hat{z}$) component of $\mathbf{E} \times \mathbf{H}^*$, so the longitudinal field components carry no net power down the guide. This is why energy flows at the group velocity $v_g$, not the phase velocity.

For a propagating mode, the transmitted power can also be written as

$\langle P \rangle = \frac{1}{2} \frac{1}{Z_{\text{wg}}} \int_S |\mathbf{E}_t|^2 \, dA$

where $Z_{\text{wg}}$ is the wave impedance of the mode ($Z_{\text{TE}} = \eta / \sqrt{1 - (f_c/f)^2}$ for TE modes, $Z_{\text{TM}} = \eta \sqrt{1 - (f_c/f)^2}$ for TM modes, with $\eta$ the intrinsic impedance of the filling medium).

Below cutoff, the fields become evanescent and $\mathbf{E} \times \mathbf{H}^*$ is purely imaginary across the cross-section, meaning no time-averaged power is transmitted.

Power transmission and attenuation

The power carried by a mode decreases exponentially along the guide due to losses:

$P(z) = P_0 \, e^{-2\alpha z}$

The factor of 2 appears because $\alpha$ is the field attenuation constant, and power goes as the square of the fields.

Attenuation constant

The attenuation constant $\alpha$ quantifies power loss per unit length. It depends on the waveguide material, operating frequency, and mode. It is often expressed in dB/m via $\alpha_{\text{dB}} = 8.686\,\alpha$ (with $\alpha$ in Np/m).

A useful perturbation approach for computing $\alpha$ due to wall losses is the power-loss method:

$\alpha = \frac{P_\ell}{2\langle P \rangle}$

where $P_\ell$ is the time-averaged power dissipated per unit length in the walls (found by integrating the surface current density over the wall perimeter) and $\langle P \rangle$ is the transmitted power computed from the Poynting vector.

Sources of loss

• Conductor losses from finite wall conductivity. Surface currents flow in a thin skin depth $\delta_s = \sqrt{2/(\omega \mu \sigma)}$, and the resulting ohmic heating is usually the dominant loss mechanism.
• Dielectric losses if the waveguide is filled with a lossy material (characterized by a loss tangent $\tan\delta$).
• Radiation losses through gaps, slots, or other discontinuities in the guide walls.

Impedance matching

Mismatches between a waveguide and its load create reflected waves, setting up standing wave patterns and reducing the fraction of power delivered. The goal of impedance matching is to make the load impedance equal to the waveguide's characteristic impedance so that the reflection coefficient $\Gamma$ goes to zero.

Matching techniques

• Quarter-wave transformers: A waveguide section of length $\lambda_g/4$ with an intermediate impedance $Z_t = \sqrt{Z_0 Z_L}$ placed between the guide and the load. This works perfectly at one frequency and provides moderate bandwidth.
• Stub tuners: Adjustable short-circuited stubs inserted into the waveguide. A single stub can cancel a reactive mismatch; a double- or triple-stub tuner can match a wider range of loads.
• Tapered transitions: The waveguide cross-section is gradually changed over many wavelengths, producing a smooth impedance transformation with broad bandwidth.

The choice among these depends on required bandwidth, power handling, and mechanical constraints.

Waveguide junctions

Waveguide junctions split or combine signals. They are classified by the plane in which the branching occurs.

E-plane vs H-plane junctions

• E-plane junctions branch in the plane of the electric field (the narrow wall of a rectangular guide). In an E-plane tee, the signal splits with a 180° phase difference between the two output arms.
• H-plane junctions branch in the plane of the magnetic field (the broad wall). In an H-plane tee, the signal splits in phase between the two output arms.

T-junctions

A T-junction has three ports arranged in a T-shape. Power division and relative phase between the output ports depend on whether it is an E-plane or H-plane tee and on the port dimensions. A magic tee combines both E-plane and H-plane branching into a four-port device that can simultaneously provide sum and difference outputs.

Waveguide discontinuities

Any abrupt change in cross-section or direction creates a discontinuity. Discontinuities scatter the incident mode into reflected waves and possibly higher-order (often evanescent) modes, and they can radiate if the guide is open.

Irises and posts

• Irises are thin metallic plates with shaped openings inserted across the guide. A narrow opening (small gap in the broad wall) acts as a shunt capacitive element; a wide opening (small gap in the narrow wall) acts as a shunt inductive element. A resonant iris combines both effects and can be used to build bandpass filters.
• Posts are metallic rods extending into the guide. Depending on their diameter and penetration depth, they produce localized capacitive or inductive susceptances, useful for tuning and filter design.

Coupling through apertures

An aperture is an opening in the waveguide wall that allows energy to couple between adjacent guides or cavities. The coupling strength depends on the aperture's size, shape, and position relative to the field maxima. Bethe's small-aperture theory relates the coupling to equivalent electric and magnetic dipole moments at the aperture location. Applications include directional couplers and cavity-coupled filters.

Excitation methods

To launch a particular mode in a waveguide, you need to create a field pattern that overlaps well with that mode's transverse field distribution.

Waveguide probes

A probe is typically a short monopole antenna inserted through the broad wall of the guide at a point of maximum electric field for the desired mode. For TE10 excitation in a rectangular guide, the probe is placed at the center of the broad wall ($x = a/2$), roughly a quarter guide wavelength from a short-circuit end wall, so that the reflected wave adds constructively with the direct excitation.

Coupling from transmission lines

Energy can also be coupled in from coaxial cables or microstrip lines using:

• Aperture coupling: an opening in a shared wall between the transmission line and the waveguide.
• Probe coupling: a coaxial center conductor extending into the guide (essentially the probe method above).
• Tapered transitions: a gradual geometric transformation from the transmission line cross-section to the waveguide cross-section, providing broadband coupling.

Dispersion characteristics

Waveguides are inherently dispersive: different frequency components travel at different speeds. This matters whenever you transmit pulses or modulated signals.

Group vs phase velocity

The phase velocity is the speed at which a constant-phase surface moves along the guide:

$v_p = \frac{\omega}{\beta} = \frac{c}{\sqrt{1 - (f_c/f)^2}}$

This exceeds $c$ for every propagating frequency, but no information travels at $v_p$, so special relativity is not violated.

The group velocity is the speed at which the envelope of a wave packet (and therefore energy and information) propagates:

$v_g = \frac{d\omega}{d\beta} = c\sqrt{1 - (f_c/f)^2}$

Notice that $v_g < c$ always, and the two velocities satisfy $v_p \, v_g = c^2$. Near cutoff ($f \to f_c$), $v_g \to 0$ and $v_p \to \infty$, meaning energy barely moves while the phase fronts race ahead. Far above cutoff, both velocities approach $c$ and the guide behaves more like free space.

Waveguide applications

Waveguides are used extensively at microwave and millimeter-wave frequencies where their low loss and high power-handling capacity outweigh their bulk compared to planar transmission lines.

Microwave components

• Filters: bandpass, bandstop, and diplexers built from cascaded irises or coupled cavities.
• Couplers: directional couplers, hybrid junctions, and power dividers for signal sampling and combining.
• Isolators and circulators: ferrite-based devices that allow transmission in one direction while absorbing or redirecting reverse-traveling waves, protecting sources from reflections.
• Resonant cavities: used in oscillators, amplifiers, and frequency meters. A cavity's quality factor $Q$ can reach tens of thousands, far exceeding lumped-element resonators.

Antenna feed systems

Waveguides commonly feed high-gain antennas:

• Horn antennas: rectangular or conical horns that flare the waveguide aperture to produce a directive beam.
• Reflector antennas: parabolic dishes illuminated by a waveguide-fed horn at the focal point.
• Slot antennas: arrays of precisely cut slots in the waveguide wall, where each slot radiates a fraction of the guided power.

Waveguide feeds provide low loss, well-controlled illumination patterns, and straightforward impedance matching to the antenna elements.