EC-2.5 Waveguides

What this is and why it exists

A hollow metal pipe carries a wave that a pair of conductors could not. The price is that it refuses to carry anything below a frequency set by its own dimensions.

That cutoff is worth understanding rather than memorising. The wave travels by bouncing between the walls, and below cutoff the bouncing cannot satisfy the boundary conditions at any angle.

The vocabulary

• Transverse electromagnetic wave — a wave with both fields entirely across the direction of travel.
• Transverse electric mode — a mode with no electric field along the direction of travel.
• Transverse magnetic mode — a mode with no magnetic field along the direction of travel.
• Mode — one standing pattern the guide can support across its cross-section.
• Mode indices — the two integers counting half-cycles along each cross-section dimension.
• Cutoff frequency — the frequency below which a given mode will not propagate.
• Dominant mode — the mode with the lowest cutoff frequency.
• Guide wavelength — the wavelength measured along the guide.

The mental model

Start with what a hollow guide cannot do. A wave with both fields entirely across the direction of travel needs two separate conductors. One carries the going current and one the return. A single pipe has only one conductor, so that simplest wave cannot exist inside it.

What propagates instead has a field component along the direction of travel. If the electric field has none, the mode is transverse electric. If the magnetic field has none, it is transverse magnetic. Those two families are all a hollow guide offers.

Each mode is a standing pattern across the cross-section, labelled by two integers. They count how many half-cycles of the pattern fit along each of the two dimensions of a rectangular guide. So the dimensions decide which modes can exist at a given frequency.

Cutoff falls out of that. A pattern with a given number of half-cycles across a given width fixes a wavelength. Below the corresponding frequency the wave will not fit, and the mode does not propagate at all. It does not merely attenuate a little; it decays away without travelling.

The mode with the lowest cutoff frequency is the dominant one, and it is the mode a guide is normally used in. Its cutoff is set by the broad dimension of the cross-section, and its cutoff wavelength is twice that dimension. Operating between that cutoff and the cutoff of the next mode keeps the guide carrying one mode only, which is what a designer wants.

Guide wavelength is where the bouncing picture repays itself. The wave is not travelling straight down the pipe. It is zig-zagging between the walls at an angle. So it advances along the guide more slowly than it moves through space. Measured along the axis, the distance between successive equal phases is longer than a free-space wavelength.

The same picture explains the velocities. The phase velocity along the guide is faster than light, and the group velocity is slower. Neither is a contradiction. The phase pattern sweeps along the axis quickly because the wavefront is tilted, and no energy travels with it. The energy takes the zig-zag path, so it makes less progress along the axis than a straight wave would.

A circular guide is the same physics with different mode labels, because the cross-section is round rather than rectangular. Its symmetry suits rotating joints and some low-loss runs. A rectangular guide is preferred where a fixed polarization must be held, because the two unequal dimensions pin the orientation.

Finally, why use a guide at all. It has no centre conductor to heat and no dielectric to lose energy in. So it handles far more power with far less attenuation than a cable at the same frequency. That advantage is what pays for its bulk and for its cutoff.

What you should now be able to explain or do

Say why a hollow guide cannot carry a transverse electromagnetic wave. Identify a mode from its two indices and compute its cutoff frequency. Name the dominant mode of a rectangular guide and relate its cutoff wavelength to the broad dimension. Explain guide wavelength and the two velocities using the bouncing picture. Say when a circular guide is preferred and why guides are used despite their bulk.

Check yourself