Waveguides

A structure that contains a propagating electromagnetic wave is called a waveguide. Consideration is limited to metal structures with uniform cross section and infinite extent in the z direction. In particular, we will concentrate on the cylindrical waveguide, which is simply a hollow tube.

Waveguides transport electromagnetic energy. Waveguides are often used in accelerators to couple power from a microwave source to resonant cavities. Furthermore, it is possible to transport particle beams in a waveguide in synchronism with the wave phase velocity so that they continually gain energy. Waveguides used for direct particle acceleration must support slow waves with phase velocity equal to or less than the speed of light. Slow-wave structures have complex boundaries that vary periodically in the axial direction. This complex boundaries are made inserting metallic pieces in the waveguide.

In contrast to transmission lines, waveguides do not have a center conductor. This difference influences the nature of propagating waves in the following ways:

1. The phase velocity in a waveguide varies with frequency. A structure with frequency-dependent phase velocity exhibits dispersion. Propagation in transmission lines is dispersionless.
2. Waves of any frequency can propagate in a transmission line. In contrast, low-frequency waves cannot propagate in a waveguide. The limiting frequency is called the cutoff frequency.
3. The phase velocity of waves in a waveguide is greater than the speed of light.

Waveguide with phase velocity greater than $c$ cannot be used for particle acceleration. It is necessary to generate slow waves with phase velocity less than $c$ .