Loss Factor of Axisymmetric and Rectangular Tapered Collimators from Optical to Inductive Regimes

Abstract

In this work, we investigate the loss factor of axisymmetric and rectangular tapered collimators over a broad range of bunch lengths and geometrical parameters using the electromagnetic simulation codes ECHO and GdfidL. The calculations are performed for Gaussian bunches of rms length $σ_s$, with the collimator geometry characterized by the minimum and maximum apertures $b$ and $d$, taper length $L$, and chamber width $W$ for rectangular structures. For short bunches and tapered collimators with $d-b>b$, we extend the phenomenological formula previously obtained for axisymmetric to rectangular tapered collimators, describing the dependence of the loss factor on bunch length and taper geometry through an arctangent function of the scaling parameter $μd^2/(σ_s L)$. We demonstrate that this parameter is related to a Fresnel number and, by combining diffraction theory with the electrodynamics of the beam-induced image charge, analytically explain the factor of two between the coefficients $μ=0.2$ and $μ=0.4$ for axisymmetric and rectangular collimators, respectively. The resulting expression is applicable to rectangular collimators with $W/(2d) \geq 1.8$, extending its validity beyond the idealized flat geometry. For longer bunches, we identify the transition from the optical to the inductive regime and derive an analytical expression for the loss factor based on the cutoff wavenumber of the lowest relevant electromagnetic mode.