A Semi-Analytical Modelling of Multistage Bunch Compression with Collective Effects

In this paper we introduce an analytical solution (up to the third order) for a multistage bunch compression and acceleration system without collective effects. The solution for the system with collective effects is found by an iterative procedure based on this analytical result. The developed formalism is applied to the FLASH facility at DESY. Analytical estimations of RF tolerances are given.


I. INTRODUCTION
Free-electron lasers require an electron beam with high peak current and low transverse emittance. In order to meet these requirements several bunch compressors are usually planned in the beam line [1], [2].
The nonlinearities of the radio frequency (RF) fields and of the bunch compressors (BC's) can be corrected with a higher harmonic RF system [3]. An analytical solution for cancellation of RF and BC's nonlinearities for a one stage bunch compressor system was presented in [3]. The second order treatment of multistage bunch compressor systems was done in [4], where the difficulty to extend the third-order analysis to multistage systems was pointed out as well.
In this paper we present, for the first time, an analytical solution for the nonlinearity correction up to the third order in a multistage bunch compression and acceleration system without collective effects for an arbitrary number of stages. A more general solution for a system with collective effects (space charge forces, wakefields, a coherent synchrotron radiation (CSR) within the chicane magnets) is found by an iterative tracking procedure based on this analytical result. We apply the developed formalism to study the two stage bunch compression scheme at FLASH [1]. The analytical estimations of RF tolerances are given for two and three stage bunch compression as well.

A. Problem formulation
Let us consider the transformation of the longitudinal phase space distribution in a multistage bunch compression and accelerating system shown in Fig.1. The system has bunch compressors ( ,…, ) and accelerating modules ( ,…, ). The first module consists of the fundamental harmonic module and of the higher harmonic module placed as shown in Fig. 1.
where i ϕ is a phase, is the on crest voltage and k is a wave number.
i V The energy change in the high harmonic module is given by 1 Let us suggest that we know the desired energies In order to find settings of RF parameters , 2N + 2 , of the accelerating modules we have to solve the non-linear system of 2,3,..., In the next section we describe the analytical solution of this system for arbitrary number of stages . Then in Section II.C the explicit forms of the solution for two and three stage bunch compression systems are given.

B. Analytical solution of the multistage bunch compression problem
In order to simplify the form of the solution and to generalize it for arbitrary number of stages we split system (1) in two independent problems.
To simplify the notation let us introduce the new variables ( ) 2 ,..., Then the first problem for variables reads 2N +1 ain difficulty which remains is to find the solution of non-linear system (2). In order The m to write explicitly the last two equations in system (2) we need to find the first three derivatives of functions ( ) i s s and ( ) i s δ . In the following we omit argument s . In this simplified notation the firs e derivatives at 0 s t thre = read From the n equations, It is easy to check that the solution of the problem (10) can be found as where y x N x x and N are solutions of the particular homogeneous and inhomogeneous problems 1 1 1 , The unknowns N x and N x can be found straightforwardly fr m the recurrence relations (12). o Finally, the last equation, , allows to find 3 α . This equation can be rewritten in a system of linear difference equations like (10) with some of the coefficients being different: Hence, we have found a unique solution of the original problem (1) for any number of tages . We will use this analytical solution in section IV to define a bunch compression orking point for the FLASH facility [1].
In this section we present the above derived analytical solution explicitly for two and three sta 2 BC bun co ss e mpre ion system can be written xplicitly: The European X-ray r (XFEL) will use a three stage b compression schem harmonic module for the longitudin h Free Electron Lase e with third al p ase space linearization.
In this  case  we  have  to  define  8  RF  parameters  ( 1,1 1,1 1,3 1,3 2 2 3 3 , , , , , , , X Y X Y X Y X Y ). In order to define 8 equations in system (1) Other RF param (s eters can be found by the same relation as for two bunch compression system ee Eq. (13)).
final bunch leng hirp and thus to ise values of the e compression due to a

D. Analytical estimation of RF tolerances
The th and the peak current are sensitive to the energy c RF parameters. Let us calculate a change of th the prec change of the RF parameters.
To simplify the notation we define is an initial energy chirp. Additionally we introduce RF parameter ectors where sy lu o mbol " " stays for the RF parameters as obtained in Section II.C from the analytical ti n. b 0 so tain a stab In order to o le bunch compression and to estimate the acceptable change in the RF parameters we require that the relative change of compression s the last inequality can be rewritten in the form Neglecting the second order term . Applying the Cauchy-Bunyakovsky inequality we obtain the admissible in order to estim te the RF tolera e Hence, a nces we need to estimate the partial derivatives relative to the RF param ters. Let us denote by a point over the symbol the partial derivative with respect to a RF parameter. Then the partial derivative of compression i Z after stage i can be found by relations X Y Finally, the partial derivatives of the compression with respect to RF parameters mediately after compressor 3 BC im can be found from relations It follows from Eq. (15) that the partial derivatives with respect to the RF parameters in odules and third harmonic module 1,3 M m 1,1 M are given by relations The partial derivatives of the compression ith respect to RF para w meters can be und as follows he partial derivatives of the compression with respect to RF rameters read T pa order to estimate th ivati e phases we use the relations In e partial der ves of the compression with respect to the voltages or th cos sin where we have used relation [5].
If we neglect the non-linear compression terms and use Eqs. (7) Without additional calculations it is easy to write the partial derivatives with respect to the itial parameters: initial energy in and initial chirp 1 ξ . From q. (15) we obtain

LT AG UNCH COMPRESSION WITH COLLECTIVE EFFECTS
The analytical solutio effects in the main beam ne. In order to take them into account we do tracking simulations taking into account the ensembles through beam lines with arbitrary arge forces. The free space longitudinal space charge impedance and

III. MU IST E B
A. Collective effects and tracking codes.
n introduced before neglects the collective li collective effects through analytical estimations (space charge forces, wakefields), or through direct numerical solution with tracking codes.
To take into account coherent synchrotron radiation (CSR) in bunch compressors we use code CSRtrack [6]. This code tracks particle geometry. It offers different algorithms for the field calculation: from the fast "projected" 1-D method [7] to the most rigorous one, the three-dimensional integration over 3D Gaussian subbunch distributions [8].
For high peak currents the compression is affected by wakefields from the vacuum chamber and by space ch the corresponding wake function for bunch with Gaussian transverse profile are given by [9] where is the transverse RMS size of the beam, ( ) s θ is the Heaviside step function, 0 Z σ ⊥ is the free space impedance, is the vacuum light velocity. y c Let us consider the bunch accelerated from energ 0 γ to the energy 1 γ along distanc .
Then we use an adiabatic approximation which takes in e L to account the ow change of t RMS size of the bunch during the acceleration: sl he 2 ( ) 2 0 ( , , ) e (0, ( ) ) ( ) here is the averaged optical beta function along distance L , n ε w β is the normalized rse e g with code ASTRA [10]. This program tracks particles through user ions) to transve mittance. Along with the above analytical estimations we use an alternative approach based on the straightforward trackin defined external fields taking into account the space charge field of the particle cloud.
The both codes, CSRtrack and ASTRA, do tracking in free space neglecting the impact of the vacuum chamber on the self fields. We use coupling impedances (or wake funct take into account interactions of the bunch with the boundary. The wakefield code ECHO [11] was used to estimate the wake functions of different beam line elements.
The FLASH facility contains 56 TESLA accelerating cavities. Their wake function is given by [12]  l solution for RF parameters given in Section II will not prod c compressed bunch. In order to take the collective effects into account we have to carry out the tracking simulations. For the adjustment of the RF parameters we use an iterative procedure, which starts from the values of the RF parameters obtained through the analytical solution introduced in Section II.
The problem without self fields can be written in operator form tor for a given vector of the m Section II.B describes the inversion of this opera f . We w acroparameters rite the solution of the problem formally in the operator form is the in t form, where one iteration includes wing steps:  The iterative schem is robust tion. We apply this iterative a xt rder to find the working point for two stage bunch compressor system in FLASH.

IV. MODELLING OF TWO STAGE BUNCH COMPRESSION IN FLASH FACILITY
The Free-Electron Laser FLASH at DESY is the first user facility for VUV and soft X-ray laser like radiation using the SASE scheme. Since summer 2005, it provides coherent femtosecond light pulses to user experiments with impressive brilliance [1,14]. It includes two bunch compressors, a C-chicane and an S-chicane. These two chicanes have to compress the electron bunches to achieve the peak current of 2500 A. After the recent upgrade in 2010 the third harmonic module was installed and the linearized bunch compression is now possible. In the following we describe a way to define a working point in the current technical constrains for a special case of bunch with charge of 1 nC. The results from tracking simulations will be presented as well.

A. Definition of the working point
Before to look for the RF parameters settings we have to define 12 macroparameters (see Section II.C). These parameters define operator and vector in Eq. (33), which is an operator form of system (1).
The initial conditions are obtained from numerical simulations of the gun with code ASTRA [3]. The code is used to model the self-consistent beam dynamics for the bunch with charge of 1 nC The initial energy from the gun The current profile and the longitudinal phase space after the RF gun, before the booster 1,1 M , are shown in Fig.2. The initial peak current after the gun is about 52 A. Hence, in order to reach the peak current of 2.5 kA we need the total compression given by .
(37) After the recent upgrade the FLASH facility has the following technical constrains on the achieved voltages: , , The deflecting radii in the bunch compressors have to fulfill the restrictions 1 . In order to reduce the space charge forces between the bunch compressors we ai use only a weak compression in . Hence the deflecting radius of the first bunch compressor is fixed at the maximum m.
This solution has two additional benefits: small CSR fields in compressor itself and a possibility of a larger energy chirp after it. The last feature reduces the ent on RF module It means that we aim to produce the large chirp with the RF system ( ) , V 2 2 ϕ . It means that for the fixed comp ion factor C the energy chirp at trance of B ress en C will be as rge as only possible. Such solution uses a larger deflecting radius and it weaker CSR fields in the last chicane. In order to find the deflecting radius 2 r we have to solve the system Bunch compressor BC is of S-type and 2 the deflecting radius is given by [5] Fig. 4 Finally, we would like to fix the last parameter 2 0 Z′ . W n in Fig  (43) symmetrize the current.
The same tolerances are shown for the third harmonic module at the right plot in Fig. 6. Table  II presents all RF tolerances for the working point defined in this section. Finally, we show in Fig. 7 the dependence of the strongest tolerance in the booster 1  The left plot shows current profile ( ) x s ε , vertical slice emittance ( ) y s , and RMS slice energy spread ( ) E s σ ε . The right plot presents the longitudinal phase space. It can be seen that the iterative procedure described in section III.B, Eq. (36), indeed has found the solution for the RF parameters which produces the desired longitudinal bunch compression. The found RF parameters are listed in the second row of Table I. We have checked with the tracking that the tolerances are left approximately the same as described in Table II for the situation without self fields.

V. SUMMARY
In this paper we have derived an analytical solution for multistage bunch compressor system with high harmonic module at the first stage. On the basis alytical solution we have proposed an iterative procedure to find the working point from tracking simulations with c ef of this an ollective fects included. The introduced formalism was applied to study the bunch ompression in FLASH facility. The derivation of the analytical solution is quite general and an be generalized to Phys. Rev. ST Accel. Beams .