Longitudinal coupling impedance of a hole in the accelerator beam pipe

In the design of modern accelerators, an accurate estimate of coupling impedance is very important. The sources which give rise to coupling impedance are the geometric discontinuities in the accelerator beam pipe. In various discontinuities such as RF cavities, bellows, and collimators, the coupling impedance of the holes has not been well understood. Although coupling impedance can be obtained in general from the Fourier transform of the corresponding wake potential which may be obtained numerically, this is time consuming and requires a large amount of computer storage when applied to a small dimension of a discontinuity in a typical beam pipe, often imposing a fundamental limitation of the numerical approach. More fundamentally, however, numerical calculation does not have the predictive power because of limited understanding of how the coupling impedance of a hole should behave over a wide frequency range. This question was studied by developing a theoretical analysis based on a variational method. An analytical formula for the coupling impedance of a hole is developed in this work using a variational method. The result gives good qualitative agreements with the coupling impedances evaluated numerically from the Fourier transform of the wake potential which is obtained from the computer code MAFIA-T3. The author shows that the coupling impedance of a hole behaves quite similar to the impedance of an RLC-resonator circuit. Important parameters used to describe such a resonator circuit are the resonant frequency and bandwidth. The author provides a theoretical insight on how to parameterize properly the numerical impedance of a hole when data exhibit complicated dependence on frequency. This is possible because one can show that the parameters are a function of the dimensionless quantity kd alone, with k the free-space wave number and d the radius of hole