The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

An analysis of the coherent synchrotron radiation effectin an energy-recovery linac by first-order transfer matrix
R. Hajima∗
Japan Atomic Energy Research Institute, Tokai, Ibaraki 319–1195 Japan
Abstract
We propose a novel approach to the analysis of the coherent synchrotron radiation (CSR) effect on beam dynamics in energy-recovery linacs (ERL). Since an ERL returnarc consists of a large number of dipole and quadrupole
magnets, it is not a practical way to search an optimum design by time-consuming particle simulations. We show thata first-order transfer matrix can be applied to calculate theemittance growth due to the CSR in the linear regime. Theoptimum design of ERL arcs such as triple-bend achromat
lattice is easily obtained by matrix calculation. As an example, optimization of a 3GeV ERL arc is presented.

INTRODUCTION
Dilution of electron beam emittance caused by coherent
synchrotron radiation (CSR) at a bending path has been acritical problem in the design of bunch compressors for Xray free-electron lasers (XFEL) and return arcs for energyrecovery linacs (ERL) since the problem was pointed out
by Derbenev and Carlsten[1]. Many studies have been
devoted to this problem by experimental[2], analytical[3],and numerical approach[4] .
2010∆E0-10-20-30

bunch-40-50-10-50s

510

Figure 1: Energy modulation induced by CSR wake potential results in emittance dilution at the bend exit
Figure 1 is a schematic view to show the emittance dilution by CSR. When an electron bunch travels in a bendingpath, CSR wake potential induces energy modulation in thebunch. Since the energy modulation for each electron depends on its longitudinal position in the bunch, displacement of electrons appears at the bending exit, which is thedilution of beam emittance. It should be noted that the longitudinal CSR wake depends little on the transverse position of electrons, and the emittance of each beamlet, that isslice emittance, is reserved, while the projection emittanceis diluted by CSR. In the ERL light sources, the propertyof synchrotron radiation and undulator radiation is a strong∗ hajima@popsvr.tokai.jaeri.go.jp

function of the projection emittance, thus, the dilution of
the projection emittance is a critical phenomenon.
Analyses of the CSR-induced emittance dilution havebeen conducted by numerical simulation. Several particle
tracking codes have been developed for this purpose andcontributed to the design of bunch compressors for SASEFELs [4]. Since these simulation codes require long calculation time, they are not suitable to search an optimum
achromatic lattice of an ERL-loop by scanning a numberof parameters such as angle and radius of dipoles, positionand strength of quadrupoles, sextupoles, beam envelope
matching to straight sections. We, therefore, propose analternative method for the CSR analysis based on a matrix
calculation, which provides a simple method to scan large
number of parameters and to optimize an ERL-loop[5].

LINEAR ANALYSIS BY TRANSFER
MATRIX
A first-order equation of electron motion in a uniform
field of a dipole magnet is1xx = − 2 + (δ0 + δcsr + κ[s − s0 ]) ,ρρ(1)where the variables are x the deviation of an electron fromthe ideal path in the bending plane, x second-order derivative of x with respect to a coordinate along the ideal paths, ρ the bending radius, and δ0 the initial momentum deviation at s = 0 normalized by the reference momentum. The last two terms on the right-hand side have been
added to calculate the CSR effect: δcsr the normalized momentum deviation caused by CSR in the upstream path
(0 < s < s0 ), s = s0 the entrance of the bending magnet, κ = W/E0 the normalized CSR wake potential in thebending path defined by CSR wake potential W and thereference energy E0 .
We assume in the following discussion that (1) all thebending magnet have the same bending radius, (2) an electron bunch keeps constant longitudinal profile, (3) transient CSR effect at the entrance and exit of magnets is notlarge. These assumptions are valid for high-energy ERL
light sources.
Under the above assumptions we can attribute the normalized CSR wake potential to individual electrons, andthe electron motion is uniquely specified by a vector
x(s0 ) = (x, x , δ0 , δCSR , κ)T .

(2)

Tracking this vector according to Eq.(1) can be made byusing first-order matrices.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

Given an initial vector at the entrance of a bending magnet, s = s0 , we can solve the electron trajectory throughthe bending magnet and find a vector at the bending exit,s = s1 :(3)x(s1 ) = Rbend x(s0 )The transfer matrix of a sector bending magnet is derived
by Green’s function method[6]:⎛cos θ
ρ sin θ ρ(1 − cos θ)⎜ − 1 sin θ cos θsin θ⎜ ρ
Rbend = ⎜001⎜⎝000000⎞
2ρ(1 − cos θ) ρ (θ − sin θ)sin θ
ρ(1 − cos θ) ⎟⎟
⎟
00(4)
⎟⎠
1ρθ
01
This is an extension of a well-known 3x3 R-matrix used inbeam transport analyses with momentum dispersion. Similar matrices are obtained for a quadrupole and a drift, andthe electron motion along a beam transport system can becalculated from these R-matrices.
The deviation of electrons from the design trajectory due
to initial momentum error is described by momentum dispersion function:(ηx (s1 ) ηx (s1 ) 1 0 0)=

Unnormalized emittance at an arbitrary position along abeam transport is obtained by2

ε2 = (ε0 βx + D2 )(ε0 γx + D ) − (ε0 αx − DD )2 , (10)where (αx , βx , γx ) are Courant-Snyder parameters, ε0 is
the initial emittance. Terms related to the initial momentum
error have been dropped from eq.(10), because the emittance is usually evaluated where the momentum dispersionis zero, η = η  = 0.

EMITTANCE COMPENSATION IN THELINEAR REGIME
The linear analysis derived in the previous section gives
good approximation of electrons motion in high-energy
ERLs, where x/ρ  1 and δ  1. In this linear regime, all
the bunch slices align on a single line at the exit of achromatic cell including bending paths as shown in Fig.2, and3. These figures indicate that the projection emittance depends on the orientation of the CSR kick and the phase ellipse, and has a minimum value, when the achromatic cellis designed so that the CSR kick coincides with the orientation of the phase ellipse at the cell exit. We show thatan achromatic cell for such minimum emittance can be designed by matrix calculation.

x’T

R0→1 (ηx (s0 ) ηx (s0 ) 1 0 0)

T.

(5)

ζ xx’-ζ’x x=0

The deviation of each electron in the (x, x ) phase space
due to the initial momentum error is expressed by d =(δ0 ηx , δ0 ηx ).
Following this style, we introduce a CSR wake dispersion function, ζx (s), to track off-axis motion due to theCSR effect:(ζx (s1 ) ζx (s1 ) 0 Lb (s1 ) 1)
=tan2φ=2α/(γ-β)φxTR0→1 (ζx (s0 ) ζx (s0 ) 0 Lb (s0 ) 1)

T, (6)

where Lb (s1 ) is the total bending path length for 0 <s < s1 , and has a relation δCSR = κLb . The deviation of each electron due to the CSR effect is expressedby d = (κζx , κζx ).
Energy spread caused by CSR is calculated for a Gaussian bunch:∆Erms  0.22

φζ

eQLb
4/3

4π 0 ρ2/3 σs,(7)Figure 2: Two-dimensional phase ellipse in the (x, x )plane, and displacement of beam slices due to the CSRkick. (α, β, γ) are Courant-Snyder parameters.

x’x’xxwhere Q is bunch charge, σs is rms bunch length. Thisenergy spread results in the deviation of bunch slices in the(x, x ) phase space after a bending path:(D, D ) = (∆κrms ζx , ∆κrms ζx ) .

(8)∆κrms = ∆Erms /E0 /Lb .

(9)

Figure 3: The CSR-induced emittance dilution becomesminimum, when the CSR kick coincides with the phase ellipse orientation.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

B1

QD QF

QD QF QDB2

QD QF QD

B3

QF QD

Figure 4: A triple bend acromatic cell: ρ = 25m, θ =3 + 6 + 3 = 12degree.
We can use the matrix method to search an optimum configuration by scanning cell parameters. In the design, we
keep isochronous condition, R56 = 0. Figure 5 shows betatron and dispersion functions for a TBA cell, which gives
minimum emittance dilution. It shows that βx and ζx havea similar envelope after the last bending magnet, which corresponds to the coincidence between the CSR kick and thebeam ellipse orientation in the (x, x ) phase space.
30βx
20 ηx20 ζxβx, ηx (m), ζx (m2)25

mum design for the emittance compensation even when thetransient CSR is included.
normalized RMS emittance (mm-mrad)

We consider CSR-induced emittance growth through a
triple-bend achromatic cell (TBA) as a part of an ERL-loop(Fig. 4).

1.2
1
0.8
0.6
0.4
0.2
0-11.5-11-10.5-10-9.5αx (at the 3rd bend exit)-9Figure 6: Emittance growth as a function of αx at the 3rdbend exit. 1st-order matrix (solid line), particle tracking
without transient CSR (◦), particle tracking with transient
CSR (•).

20CONCLUSION

151050-5-100510s (m)1520

Figure 5: Betatron function βx , momentum dispersion
ηx and CSR wake dispersion ζx of an optimized triplebend achromat. The envelope of the CSR wake dispersion
matches the betatron envelope after the last bending magnet.
The emittance compensation in the optimized TBA cellhas been confirmed by particle tracking code ELEGANT including CSR calculation.[7] We assume the initial electron bunch parameters as follows: central energy 3.07GeV,
bunch charge Q = 770pC, normalized rms emittanceεn,0 = 0.1mm-mrad, longitudinal size σs = 30µm (100fs),
uncorrelated energy spread σE /E0 = 0.02%, and Gaussian distribution of electrons in the 6D phase space. Projection emittance at the cell exit is calculated with scanning
Courant-Snyder parameters at the cell entrance: −5.0 <αx < 3.0, γx = 0.29m−1 .
Figure 6 shows the calculated projection emittance as afunction of αx at the third bend exit. It has been confirmedthat the projection emittance is a function of the beam envelope inside the TBA cell, and the CSR-induced emittance
dilution can be estimated precisely by matrix method. We
also plotted simulation results including the transient CSR
effect. It is found that the matrix approach gives an opti-We have shown an analysis of emittance dilution due
to the CSR effect by using a first-order transform matrix
method. Since the matrix approach enables us to scan anumber of parameters quickly, it is widely applicable tothe design optimization of achromatic cells for ERL lightsources. We have seen that the CSR-induced emittance
growth in an achromatic cell of a 3GeV ERL loop can be
minimized by choosing an appropriate beam envelope inside the cell. This optimum cell design is easily obtainedby matrix calculation.

REFERENCES[1] Ya.S. Derbenev, J. Rossbach, E.L. Saldin, V.D. Shiltsev, DESY, TESLA-FEL 95-05 (1995); B.E. Carlsten andT.O. Raubenheimer, Phys. Rev. E51 (1995) 1453; B.E. Carlsten, Phys. Rev. E54 (1996) 838.
[2] R. Hajima et al., Nucl. Instr. Meth. A429 (1999) 264;
H. Braun et al., Phys. Rev. Lett. 84 (2000) 658; S. Okuda
et al., Nucl. Instr. Meth. A445 (2000) 351.
[3] E.L. Saldin et al., Nucl. Instr. Meth. A398 (1997) 373.
[4] R. Hajima, E. Ikeno, Nucl. Instr. Meth. A429 (1999) 315; A.
Kabel et al., Nucl. Instr. Meth. A455 (2000) 185; M. Borland: Phys. Rev. ST Accel. Beams 4 (2001) 070701.
[5] R. Hajima, Jpn. J. Appl. Phys. 42 (2003) L974–L976.
[6] H. Wiedemann: Particle Accelerator Physics (SpringerVerlag, Berlin, 1993) p.107.
[7] ELEGANT version 14.8, Nov. 26, 2002; M. Borland: Argonne
National Laboratory Advanced Photon Source Report No.
LS-287 (2000).