Impedance scaling for small angle transitions

Based on the parabolic equation approach to Maxwell's equations we have derived scaling properties of the high frequency impedance/short bunch wakefields of structures. For the special case of small angle transitions we have shown the scaling properties are valid for all frequencies. Using these scaling properties one can greatly reduce the calculation time of the wakefield/impedance of long, small angle, beam pipe transitions, like one often finds in insertion regions of storage rings. We have tested the scaling with wakefield simulations of 2D and 3D models of such transitions, and found that the scaling works well. In modern ring-based light sources one often finds insertion devices having extremely small vertical apertures (on the order of millimeters) to allow for maximal undulator fields reaching the beam. Such insertion devices require that there be beam pipe transitions from these small apertures to the larger cross-sections (normally on the order of centimeters) found in the rest of the ring. The fact that there may be many such transitions, and that these transitions introduce beam pipe discontinuities very close to the beam path, means that their impedance will be large and, in fact, may dominate the impedance budget of the entire ring. To reducemore » their impact on impedance, the transitions are normally tapered gradually over a long distance. The accurate calculation of the impedance or wakefield of these long transitions, which are typically 3D objects (i.e. they do not have cylindrical symmetry), can be quite a challenging numerical task. In this report we present a method of obtaining the impedance of a long, small angle transition from the calculation of a scaled, shorter one. Normally, the actual calculation is obtained from a time domain simulation of the wakefield in the structure, where the impedance can be obtained by performing a Fourier transform. We shall see that the scaled calculation reduces the computer time and memory requirements significantly, especially for 3D problems, and can make the difference between being able to solve a problem or not. The method is based on the parabolic equation approach to solving Maxwell's equation developed in Refs. [1, 2].« less