Two-dimensional Effects on the Behavior of the Csr Force in a Bunch Compression Chicane *

The endeavour to reach higher brightness of electron beams in designs of future FEL is often challenged by the coherent synchrotron radiation (CSR) effect in bunch compression chicanes. Extensive studies on the CSR effects have shown that the 1D approximation of the CSR force is valid for a wide parameter regime. However, as the bunch gets increasingly compressed in the chicane, the behaviour of the CSR interaction force will be influenced by the evolution of the 2D bunch chargedensity distribution in the bending plane. Here we explore this 2D effect for over-compression cases using the previously developed semi-analytical study of the effective longitudinal CSR force for a bunch with evolving 4D Gaussian phase space distribution and an initial linear energy chirp. We will display the dependence of the 2D CSR force on the initial horizontal emittance and uncorrelated energy spread for path lengths around that of the minimum bunch length. We will also present and discuss the comparison of these results with their counterparts in the 1D rigid-line bunch model. INTRODUCTION The high-peak-current electron beams required by the short-wavelength high-gain FEL are often obtained by compressing the high-charge electorn bunches with lowtransverse emittance using one or more bunch compression chicanes. Inside a magnetic chicane, the electrons are transported through portions of circular orbits in the magnetic bends. The curvature of these circular orbits will induce additional electromagnetic interaction among the electrons via the acceleration term in the Lienard-Wiechert fields, which is closely related to the coherent synchrotron radiation (CSR) emitted by the bunch. Hence the curvature-induced additional EM interaction is often called the CSR interaction, and the corresponding forces are called the CSR forces. The CSR interaction in a bunch compression chicane can potentially cause degradation of the bunch phase space quality, and pose serious challenge to designs of future FEL. The bunch compression by a magnetic chicane is realized by imposing an initial energy chirp on the bunch prior to its entrance to the chicane and utilizing the correlation between path-length and energy in regions of non-zero dispersion. As a result, the bunch length changes during the bunch compression process, which is closely correlated to the evolution of bunch transverse size due to the non-zero dispersion in the chicane. The evolving 2D bunch distribution in the bending plane is the foundation for the calculation of retarded potentials and the collective Lienard-Wiechert fields experienced by the bunch. As the first-order approximation, many of previous analyses [1,2] and simulations [3] of the CSR interaction used 1D model for the calculation of the longitudinal CSR force, wherein the bunch is treated as a rigid-line bunch with the bunch length at all retarded time being fixed at the same value as the one when the CSR force is evaluated. The validity of the rigid-line bunch model was first discussed in Ref. [4], where it is stated that the effect of the transverse bunch size on the CSR force can be ignored when   1 / 3 / 1 2   z x R   . (1) Here R is the bend radius of the magnet, and x  and z  are the horizontal and longitudinal RMS bunch sizes. Even though this condition does not take the dependence of x  and z  on path length into account, it can still serve as an indicator of the occurrence of 2D CSR effect. The effects of the evolution of the 2D bunch distribution in the bending plane on the behaviour of CSR force, such as the delayed response of the longitudinal CSR force to the bunch-length variation and the dependence of longitudinal CSR force on the transverse coordinates of the particles, were studied extensively in Ref. [5] for path length around that of the minimum bunch length. In this paper, we apply the previously developed semi-analytical approach to further study the dependence of the 2D CSR force on the initial uncorrelated transverse and longitudinal phase space area, as well as on the initial energy chirp of the bunch. We will first describe the parameters used for our case study, including the parameters of the model chicane and the initial beam phase-space parameters for a bunch sent through the chicane. We then examine the results of the CSR force on the bunch at various path lengths along the chicane. Here we assume the bunch propagates through the chicane only following the linear optics (unperturbed by CSR). Comparison of the 2D results with their counterparts in the 1D rigid-line bunch model will be presented and discussed. BUNCH COMPRESSION IN THE MODEL CHICANE The model chicane we employed is illustrated in Fig. 1 [5]. The chicane consists of three rectangular dipole magnets with bending radius m 1  R . The length for the ___________________________________________ *Work supported by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. ... two side dipoles is m 3 . 0  b L and the length for the centre dipole is m 6 . 0  B L . The drift length between adjacent dipoles is m 4 . 0  d L . We now let an electron bunch with initial 4D Gaussian phase space distribution to be sent through the chicane. The initial bunch parameters are chosen as 5 0  x  m, 1 0  x  , E=70 MeV, and