Loss factor for short bunches in azimuthally symmetric tapered structures

We use the program ECHO to study numerically the loss factor of axisymmetric cavities and collimators with and without tapering. We consider a Gaussian driving bunch of length sigma, and structures with inner radius b, outer radius d, cavity gap (or collimator inner length) g, and taper length L. In all cases we consider a short bunch with sigma << b. For the untapered structures (L = 0), in certain regimes, we describe our numerical results using the known expressions for a step transition and for the diffraction model. In addition, we identify new regimes where g < sigma and/or d - b < sigma, for which the data must be described by new phenomenological expressions. For a tapered collimator with d - b > b, we present a phenomenological formula which accurately describes the dependence of the loss factor on the parameters sigma, b, d, and L. This formula also holds for a tapered cavity with g > d(2)/sigma and d - b > b.