Predicting the Performances of Coherent Electron Cooling with Plasma Cascade Amplifier

Abstract

Recently, we proposed a new type of instability, Plasma Cascade Instability (PCI), to be used as the amplification mechanism of a Coherent Electron Cooling (CeC) system, which we call Plasma Cascade Amplifier (PCA). In this work, we present our analytical estimate of the cooling force as expected from a PCAbased CeC system. As an example, we apply our analysis to a planned PCA-based CeC test system and investigate the evolution of the circulating ion bunch in the presence of cooling. INTRODUCTION Cooling high energy, high intensity hadron beams remains one of the serious challenges in modern accelerator physics. Such cooling of natural emittances, while overcoming and mitigating other limitations, guarantees longer, more efficient stores that would result in significantly higher integrated luminosity in a hadron collider such as the Large Hadron Collider (LHC) at CERN, the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory (BNL) or a future Electron-Ion Collider (EIC). One of the candidates to provide effective cooling for a high luminosity EIC is Coherent electron Cooling (CeC)[1]. CeC belongs to the family of stochastic coolers, but with the amplifier’s bandwidth extending into the optical region, e.g. beyond the THz range. Several possible broadband CeC amplifier, based on instabilities in the electron beam, have been suggested including high-gain free-electron lasers (FEL), microbunching instability (MBI) [2-4] and the plasma cascade instability (PCI) [5]. In this work, we have developed an analytical tool to estimate the performance of the plasma-cascade amplifier (PCA) based CeC system. Section II shows our derivation of the electrons’ line density modulation induced by a moving ion. In section III, we use the results in reference [5] to obtain the amplified line density perturbation with the initial condition derived in section II. From the amplified line density perturbation, we derived the longitudinal cooling field in section IV. We implement the one turn energy kick that ions receive from the cooling section into a tracking code and section V consists of our prediction for a test system of the PCA-based CeC. LINE DENSITY MODULATION AT MODULATOR Regardless of the amplification mechanism, a CeC system consists of a modulator, an amplifier and a kicker. To derive the longitudinal cooling force at the kicker section, we start with deriving electrons’ line density modulation at the exit of the modulator. For a uniform electron beam with 2 \uf06b \uf02d velocity distribution, the 3-D density modulation in the wave-vector domain is given by[6] \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 1 2 2 , 1 cos sin k t i p p p p p k Z n k t e t t k \uf06c \uf06c \uf077 \uf077 \uf077 \uf077 \uf077 \uf06c \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0ea \uf0fa \uf0e7 \uf0f7 \uf03d \uf02d \uf02d \uf0e7 \uf0f7 \uf0ea \uf0fa \uf02b \uf0e8 \uf0f8 \uf0eb \uf0fb \uf072 \uf072 \uf072 \uf025 \uf072 , (1) where p \uf077 is the angular plasma frequency in the comoving frame, k \uf072 is the wave vector, i Z is the charge number of the ion, 0 v \uf072 is the velocity of the ion, x \uf062 , y \uf062 , and z \uf062 are the velocity spread of electrons, and \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 0 x x y y z z k ik v k k k \uf06c \uf062 \uf062 \uf062 \uf03d \uf0d7 \uf02d \uf02b \uf02b \uf072 \uf072 \uf072 . (2) The line number density modulation in the longitudinal wave vector, z k , domain is given by \uf028 \uf029 1 0,0, , z n k t \uf025 , i.e. \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 m 1 m m 2 1 cos sin 1 z z k i z z z z z Z e k e k k \uf06c \uf079 \uf072 \uf079 \uf06c \uf079 \uf06c \uf0e9 \uf0f9 \uf03d \uf02d \uf02d \uf0eb \uf0fb \uf02b \uf025 ,(3) where \uf028 \uf029 \uf028 \uf029 0 / z z z z z z p k ik v k \uf06c \uf062 \uf077 \uf03d \uf02d , , 2 / m m p lab L \uf079 \uf070 \uf06c \uf03d is the phase advance of plasma oscillation in the modulator, m L is the length of the modulator section and , p lab \uf06c is the plasma wavelength in the lab frame. The line number density in space-time domain is given by the inverse Fourier transformation of Eq. (3):