Abstract
We examine the transverse impedance of a periodic array of cavities in a beam pipe at high frequency. The calculation is an extension of a previous one for the longitudinal impedance of a periodic array of azimuthally symmetric pillboxes, for which only TM modes were needed. In the present case, we must include TE modes as well. In addition, we extend the applicability of the previous calculation by including an extra term in the coupling kernel so that the results are valid for all values of the ratio of the cavity length to the period of the structure (all values of the ratio of iris thickness to structure period). In spite of the presence of TE modes, we find that the high frequency limit of the transverse impedance is simply $(2/{\mathrm{ka}}^{2})$ times the corresponding limit of the longitudinal impedance, just as it is for the resistive wall impedances, a relation which occurs frequently for azimuthally symmetric structures. Finally, we present numerical results as well as approximate expressions for the impedance per period, valid for all ratios of cavity length to structure period.
Highlights
The acceleration of charged particles in periodic structures leads to wakefields which are capable of interacting adversely with particles in the same bunch, or in following bunches
With the increasing use of short bunches, it becomes necessary to evaluate these coupling impedances at high frequencies. This has been done for the longitudinal impedance of small periodic obstacles in an azimuthally symmetric structure [1,2], where we have derived an integral equation for the axial electric field at the inner bore radius
By taking the high frequency limit of the kernel in the integral equation, it was found that the local frequency average of the real part of the longitudinal impedance per period of the structure varies as the 23͞2 power of the frequency
Summary
The acceleration of charged particles in periodic structures leads to wakefields which are capable of interacting adversely with particles in the same bunch, or in following bunches. With the increasing use of short bunches, it becomes necessary to evaluate these coupling impedances at high frequencies (wavelength of the order of the bunch length). This has been done for the longitudinal impedance of small periodic obstacles in an azimuthally symmetric structure [1,2], where we have derived an integral equation for the axial electric field at the inner bore radius. We first derive the integral equation for the electric field at the bore radius of the structure and obtain the result for the transverse impedance of a single small obstacle at high frequency by way of the high frequency limit of the kernels.
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