A note on the Fourier series representation of the dispersion curves for circular iris-loaded waveguides

The dispersion curve for the lowest pass band in a uniform corrugated circular waveguide, such as is used in travelling-wave linear accelerators, may be expressed as a Fourier series of period 2¿/(corrugation pitch). Under the slowest conditions of convergence in the waveguides investigated, an accuracy of ±1 part in 15000 in the determination of frequency corresponding to a given phase velocity can be obtained by using only six terms of this series. For the majority of structures considered, three terms only are required. An empirical expression for the dispersion curve can thus be obtained from only a small number of experimental results, and is of considerable use in the design of suitable slow-wave structures for accelerators. The equation of the dispersion curve can be readily differentiated to give the group velocity at all points within the pass band. The validity of the method is illustrated using a selection of experimental results which have been obtained in the course of an extensive programme of measurements on corrugated slow-wave structures for linear accelerators operating at wavelengths around 10cm.