Abstract The dependence of emittance growth due to Coherent Synchrotron Radiation (CSR) in bunch compressor chicanes on optics has been noticed and empirically studied in the past. We revisit the subject, suggesting a model to explain slice emittance growth dependence on chicane optics. A simplified model to calculate projected emittance growth when it is mainly caused by transverse slice centroid offsets is presented. It is then used to find optimal compensation of centroid kicks in the single chicanes of a two-stage compression system by adjusting the phase advance of the transport in between and the ration of the compression factors

DESY, Hamburg, Germany T Limberg

M Dohlus

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IMPACT OF OPTICS ON CSR-RELATED EMITTANCE GROWTH IN 
BUNCH COMPRESSOR CHICANES 
M. Dohlus, T. Limberg, DESY, Hamburg, Germany
Abstract 
The dependence of emittance growth due to Coherent 
Synchrotron Radiation (CSR) in bunch compressor 
chicanes on optics has been noticed and empirically 
studied in the past. We revisit the subject, suggesting a 
model to explain slice emittance growth dependence on 
chicane optics. 
A simplified model to calculate projected emittance 
growth when it is mainly caused by transverse slice 
centroid offsets is presented. It is then used to find 
optimal compensation of centroid kicks in the single 
chicanes of a two-stage compression system by adjusting 
the phase advance of the transport in between and the 
ration of the compression factors.  
INTRODUCTION 
In a previous paper [1], we published a 2D scan of the 
incoming Twiss parameters into the compressor chicane 
of the TTF, calculating the resulting horizontal slice 
emittance. According to this scan, a beam waist close to 
the 4th bending magnet was necessary to keep slice 
emittance growth small. 
Since this is a severe restriction for the optics design 
around bunch compressors, we repeated the calculation 
with the code CSRtrack, using a better modelling of the 
transverse phase space as previously possible for 3D 
calculations of CSR fields. In addition, we did a scan 
where the waist position is kept constant, but its β-
function is varied.  
It has been pointed out [2,3] that in the case of a bunch 
compression system using two magnet chicanes, the 
phase advance between the chicanes can be optimized to 
reduce projected emittance growth.  
We developed a fast method to estimate and optimize 
projected emittance. The method uses the projected or 1D 
model for the particle dynamics in bunch compressors in 
combination with perturbation theory. The longitudinal 
charge profile for any position in the bunch compressor is 
calculated without self forces and used to compute the 
‘renormalized’ longitudinal field [4,5]. This perturbation 
of the equation of motion is integrated in a second step.  
A further simplification estimates the charge profile by 
a linear compression of the initial shape. It is used here to 
determine the centroid positions of individual slices at the 
exit of the bunch compressor. In cases like the European 
XFEL, where space charge effects are negligible, the 
growth of projected emittance is essentially caused by the 
individual offset of slices. 
 SLICE EMITTANCE 
For the calculations in this chapter, we use the ‘Zeuthen 
chicane’ a benchmark case available on the web where a 
bunch of 1 nC charge compressed to reach a peak current 
of 6 kA at a beam energy of 500 MeV [6]. 
Fig. 1 shows a few of the different optics used for the 
calculations, the dotted lines mark the bending magnets. 
The earlier result was confirmed. An optics with a 
divergent beam like the one shown with initial Twiss 
parameters α = −0.8 and β = 15 m leads to slice emittance 
growth by a factor of 10, while a convergent beam with a 
narrow waist increases it by just a few per cent. Fig. 2 
shows the results for different β−functions at the waist; 
the waist position is kept at the 4th bending magnet. 
 
Fig. 1: β−functions through the chicane. The dotted lines mark 
the bending magnets. βω and sw are β−function and position at 
the waist. 
0 5 10 15
1
1.2
1.4
1.6
1.8
2
β - function at waist [m]
H
or
. n
or
m
al
iz
ed
 E
m
itt
an
ce
 [ 
µm
]
 
Fig. 2: Horizontal slice emittance at the chicane exit for different  
β−functions at the waist 
Transverse slice emittance growth can be caused by 
two effects: non-linearly varying CSR forces for different 
transverse positions and directions of particles in the same 
slice on one hand and different longitudinal positions of 
particles that are initially (and approximately finally) in 
the same slice on the other.  
The longitudinal spread is shown in Fig. 3. We plot the 
length of the projection on the bunch center trajectory for 
an ensemble of particles which start in an infinitesimally 
short slice at the chicane entry. In linear optics, it is given 
by: 
2
51
2
51
2
52 rrsr
w
w
wproj +⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
β
εβσ  
βw is the β−function at the beam waist, r51  and r51 are the 
elements of the linear transport matrix from the entry of 
the chicane to the point of interest, coupling initial 
horizontal offsets and angles to longitudinal deviations at 
this position. 
For a 4-dipole chicane, the term ( 51
2
52 rsr w− )/βw is 
almost zero at the waist if it is placed close to the entry of 
the 4th magnet. Further reduction of projected slice length 
and subsequent slice emittance growth is possible by 
minimizing βw. The slice emittance growth at the chicane 
exit scales indeed roughly quadratically with projσ  in the 
vicinity of the end of the 3rd and beginning of the 4th 
magnet, where the CSR forces have maximum strength. 
 
 
Fig. 3: Length of slice projection through the magnet chicane. 
The dotted lines mark the bending magnets. 
PROJECTED EMITTANCE 
Even if the slice emittance is preserved, the projected 
emittance can change significantly due to variation of the 
optics along the bunch or slice centroid shifts caused for 
example by longitudinally varying fields. Fig. 4 shows the 
horizontal phase space distribution of a bunch at the exit 
of a test chicane. The parameters for that calculation are 
listed in Table 1.  
 
Magnet Length 0.3 m 
Drift Length 8.9 m 
Beam Energy 0.5 – 2.5 GeV 
Compression 100µm -> 20µm 
α,βinitial 1.8, 48 m 
Normalized beam emittance 1 mm-mrad 
Table 1: Parameters for Test Chicane calculations 
 
 
Fig. 4: 1-σ ellipses and centroid at the exit of a bunch 
compressor (2.5 GeV). 
 For this example the size and shape is in agreement 
with linear optics, but the shifted centroids increase the 
projected emittance. The slice emittance is increased by 
only about 1%, but the projected emittance calculated 
with CSRtrack (using its Green’s functions method and 
projected method) is nearly doubled [5,7]. Both 
calculation methods are approximately in agreement for 
energies above 1GeV, but for lower energies the 
approximations of the 1D method (neglect space charge 
effects, transverse forces and transverse dependence of 
forces) are insufficient (see Fig. 5). 
 
Fig. 5: Emittance growth in test chicane. 
 
Simplified Calculation of Projected Emittance 
Supposed the shift of the slice centroids is the main 
reason for an increased projected emittance, this effect 
can be estimated by the computation of the centroid curve 
cx , cx′ . The first and second momenta of x and x′ are 
∫ xddsdxssxxsxxxx cc ′′−′−= )())(),(( λψ  
∫ xddsdxssxxsxxxx cc ′′−′−′=′ )())(),(( λψ  
( )( )∫ xddsdxsxxxxxx ′−−= )()( λψ L  
( )( )∫ xddsdxsxxxxxx ′′−′−=′ )()( λψ L  
( )( )∫ xddsdxsxxxxxx ′′−′′−′=′′ )()( λψ L  
with ),( xx ′ψ  calculated by linear optics and )(sλ  the 
longitudinal profile. These momenta can be expressed as 
superposition 
εβ+= cc xxxx , εα−′=′ cc xxxx , εγ+′′=′′ cc xxxx  
of second momenta of the centroids, e.g.,: 
( )∫ dssxsxxx cccc )()( 2 λ−=  
and second momenta of the original distribution that are 
determind by the Twiss parameters α, β, γ, ε of the 
unperturbed optic. The projected emittance is 
2
xxxxxxp ′−′′⋅=ε . 
To solve the equation of motion 
E
E∆
=+′′
KxKx cc
2  
for the centroids, the longitudinal electrical field CSRE  has 
to be calculated for all positions S  along the trajectory. 
( )(SK  is the curvature radius of the trajectory, E  the 
beam energy and ∫ dSEq CSR=∆E .)  This can be done 
with low numerical effort for the unperturbed and linearly 
compressed bunch profile ))(()(),( SCsSCSs ⋅= λλ  
with )(SC  the compression ratio. The comparison of this 
estimation with CSRtrack calculations can be seen in Fig. 
5. 
Fig. 6 shows the projected emittance for different initial 
parameters α and β. With an initial β-function of 100 m 
and stronger focussing (α = 3.5), the projected emittance 
is reduced to a value below 1.6 µm. The beam waist is 
positioned is a few meters after the compressor end. 
 
Fig. 6: Projected Emittance (mormalized, ×106) for different 
Twiss parameters at the entrance, E = 2.5 GeV. 
Two Stage Bunch Compression System 
In the following, we will demonstrate that a 
compensation of CSR effects in a two stage bunch 
compression can be utilized to reduce the projected 
emittance without increasing the β function. 
A two stage bunch compression system is proposed for 
the European XFEL to decrease the initial rms length of 
about 2 mm down to 20 µm [8]. Both chicanes have the 
same geometry as the test chicane. BC1 is operated at 500 
MeV and the second compression takes place at energies 
between 2 GeV and 2.5 GeV. The absolute energy chirp 
of 10 MeV/rms-length is the same in both compressors as 
well as the optics at the entrance (β = 48m,  α = 1.8) that 
is chosen to maintain the slice emittance.  
There are only two degrees of freedom in the 
characterization of the system: the compression per stage 
(with C1C2 = 100) and the phase advance µ of the 
transport between the chicanes. The simplified calculation 
of the projected emittance described above was used to 
explore this two dimensional parameter space. Fig. 7 
shows the centroid curves after BC2 for the optimal 
setting C1 = 20, C2 = 5 and µ = 1.4. For comparison the 
centroid positions are calculated with CSR forces in one 
of the two chicanes as well as the 1-σ ellipse according to 
linear optics. It can be seen that the CSR effects of both 
compressors compensate each other so that the estimated 
projected emittance is only slightly increased. 
This result was verified with CSRtrack calculations 
(ideal particle distribution at entrance of BC1, transport 
exit BC1 to entrance BC2 by transport matrix). It is found 
that the cancellation of the centroid shifts is less perfect 
than predicted by the simplified model but significant 
(normalized slice emittance increased by few %, 
normalized projected emittance 1.4 µm) and that the 
working point is nearly optimal (projected emittance 
increases for other compression ratios and phase 
advance). 
 
Fig. 7: Centroid curves after two stage bunch compression. 
SHIELDING, RESISTIVE WALLS 
All CSR fields have been calculated for free space 
conditions. The assumption that shielding effects are 
negligible has to be reviewed: a bunch (with length σ) on 
a circular trajectory (with radius R) between conducting 
planes radiates the same power as in free space if the gap 
g  between the planes is large compared to  3 2σR . For 
g  of the order of centimeters this condition is violated in 
BC1 but fulfilled in BC2. Usually shielding helps to 
reduce CSR related emittance growth in a single BC, but 
it counteracts to the compensation of effects in a two 
stage system as described above. Furthermore, it has to be 
mentioned that the criterion for the gap width holds for 
circular motion but not for chicanes with long drifts. The 
conducting planes build a waveguide system with cut-off 
effects, reduced group velocity and wall losses. Therefore 
the bunch escapes from CSR fields if the drift is long 
compared to ( )σπ 222g , but it builds up a resistive wall 
field. Shielding and resistive wall effects have to be 
investigated. 
 
REFERENCES 
[1] M. Dohlus, A. Kabel, T. Limberg: “Optimal Beam Optics in 
the TTF-FEL Bunch Compression System”, PAC 99. 
[2] P.Emma: private communication  
[3] A. Loulergue, A. Mosnier: private communication 
[4] E.Saldin, E.Schneidmiller, M.Yurkov: “Radiative Interaction 
of Electrons in a Bunch …”,  NIM A417 (1998) 158 – 168.  
[5] M. Dohlus: “Two Methods for the Calculation of CSR 
Fields”, TESLA-FEL-2003-05. 
[6] http://www.desy.de/csr/ 
[7] M. Dohlus, T. Limberg, “CSRtrack: Faster  Calculation of 
3D CSR effects”, FEL 2004, 2004 
[8] J.V.Balandin et al: “Optimized Bunch Compression System 
for the European XFEL”. PAC 2005. 
 
 


Impact of Optics on CSR-Related Emittance Growth in Bunch Compressor Chicanes

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Impact of Optics on CSR-Related Emittance Growth in Bunch Compressor Chicanes

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