Transverse Emittance Reduction with Tapered Foil

The idea of reducing transverse emittance with tapered energy-loss foil is proposed by J.M. Peterson in 1980s and recently by B. Carlsten. In this paper, we present the physical model of tapered energy-loss foil and analyze the emittance reduction using the concept of eigen emittance. The study shows that, to reduce transverse emittance, one should collimate at least 4% of particles which has either much low energy or large transverse divergence. The multiple coulomb scattering is not trivial, leading to a limited emittance reduction ratio. Small transverse emittances are of essential importance for the accelerator facilities generating free electron lasers, especially in hard X-ray region. The idea of reducing transverse emittance with tapered energy-loss foil is recently proposed by B. Carlsten (1), and can be traced back to J.M. Peterson's work in 1980s (2). Peterson illustrated that a transverse energy gradient can be produced with a tapered energy-loss foil which in turn leads to transverse emittance reduction, and also analyzed the emittance growth from the associated multiple coulomb scattering. However, what Peterson proposed was rather a conceptual than a practical design. In this paper, we build a more complete physical model of the tapered foil based on Ref. (2), including the analysis of the transverse emittance reduction using the concept of eigen emittance and confirming the results by various numerical simulations. The eigen emittance equals to the projected emittance when there is no cross correlation in beam's second order moments matrix (3). To calculate the eigen emittances, it requires only to know the beam distribution at the foil exit. Thus, the analysis of emittance reduction and the optics design of the subsequent beam line section can be separated. In addition, we can combine the effects of multiple coulomb scattering and transverse energy gradient together in the beam matrix and analyze their net effect. We find that,when applied to an electron linac or electron beam line, the energy spread increase and angular growth due to multiple scattering are not trivial; as a result, the transverse emittance can only be reduced with a limited ratio, e.g. down to about 65% the original value. The contents of this paper are arranged as follows. In Sec. II, we build the physical model of the tapered foil, derive the transverse eigen emittance and discuss the emittance reduction criteria. In Sec. III, we implement numerical simulations to verify the physical model; and in Sec. IV, we present numerical experiments and subsequent beam line to remove the transverse energy gradient to demonstrate the applicability of such method. Conclusions are given in the last section.