Longitudinal wakefield for an axisymmetric collimator

We consider the longitudinal point-charge wakefield, w(s), for an axisymmetric collimator having inner radius b, outer radius d, inner length g, and taper length L. The taper angle alpha is defined by tan alpha = (d - b)/L. Using the electromagnetic simulation code ECHO, we explore the dependence of the wakefield on a collimator's geometric parameters over a wide range of profiles: from small-angle tapers to step-function transitions. The point-charge wakefield is determined using an approximation introduced by Podobedov and Stupakov. We have found it useful to exhibit the wakefield as a function of the scaled variable s/d alpha. For small taper angles, our results illustrate the satisfaction of the longitudinal scaling found by Stupakov, Bane, and Zagorodnov; and for larger taper angles, the breaking of this longitudinal scaling is clearly depicted. The use of the scaled variable s/d alpha turns out to be especially well suited to describing the wakefield for a collimator with step-function profile (alpha = pi/2).