Abstract
In this paper we introduce an optical approximation into the theory of impedance calculation, one valid in the limit of high frequencies. This approximation neglects diffraction effects in the radiation process, and is conceptually equivalent to the approximation of geometric optics in electromagnetic theory. Using this approximation, we derive equations for the longitudinal impedance for arbitrary offsets, with respect to a reference orbit, of source and test particles. With the help of the Panofsky-Wenzel theorem, we also obtain expressions for the transverse impedance (also for arbitrary offsets). We further simplify these expressions for the case of the small offsets that are typical for practical applications. Our final expressions for the impedance, in the general case, involve two-dimensional integrals over various cross sections of the transition. We further demonstrate, for several known axisymmetric examples, how our method is applied to the calculation of impedances. Finally, we discuss the accuracy of the optical approximation and its relation to the diffraction regime in the theory of impedance.
Highlights
The calculation of the impedance for the elements of a vacuum chamber system and the associated calculation of beam dynamics effects, such as beam instabilities or wakefield induced emittance growth, are important elements in the design of a modern accelerator
One example is the case of very short bunches, like those envisioned in future linear colliders and future light sources
Since numerical calculation of the short-range wake requires a spatial mesh size equal to a fraction of the bunch length, submillimeter bunches represent a challenging computational task. Another example where direct numerical calculation is difficult is related to long, small-angle tapers which are often used to minimize the abruptness of vacuum chamber transitions
Summary
Deutsches Elektronen-Synchrotron, Notkestrasse 85, 22603 Hamburg, Germany (Received 5 March 2007; published 7 May 2007). In this paper we introduce an optical approximation into the theory of impedance calculation, one valid in the limit of high frequencies. This approximation neglects diffraction effects in the radiation process, and is conceptually equivalent to the approximation of geometric optics in electromagnetic theory. Using this approximation, we derive equations for the longitudinal impedance for arbitrary offsets, with respect to a reference orbit, of source and test particles. With the help of the Panofsky-Wenzel theorem, we obtain expressions for the transverse impedance ( for arbitrary offsets). We discuss the accuracy of the optical approximation and its relation to the diffraction regime in the theory of impedance
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