csr induced emittance growth and related design strategies
TRANSCRIPT
CSR induced emittance growth and related design strategies
S. Di Mitri, M. Venturini (1.0 hr.)
19/06/2013 1S. Di Mitri - USPAS CO - Lecture_We_11
CSR longitudinal model
In the following, we will adopt a simplified picture for the CSR transverse effect. Experimental
results suggest that it is accurate enough for describing most of the practical cases.
1. Photons are emitted in the beam direction of motion, at any point along the curved
trajectory in a dipole magnet ⇒ CSR longitudinal effect, pz(s) � pz(s) – δpz(s).
DIPOLE e-
γ
2. At any point of emission, particle transverse position does not change: ∆x = 0 (same for
Remind: particle transv. motion
is the linear superposition
x(s) = xββββ(s) + xηηηη(s)
2. At any point of emission, particle transverse position does not change: ∆x = 0 (same for
angular divergence). Since the emission happens in an energy-dispersive region: ∆xβ = –∆xη.
That is, the particle starts ββββ-oscillating (∆xβ) around a new dispersive trajectory (–∆x η).
Initial reference
(dispersive) trajectory, xη
Initial betatron
oscillation, xβ
Point of emission,
∆x = ∆xβ+ ∆xη = 0New betatron oscillation, with
amplitude ∆xβ,emis = –∆xη,emis
New dispersive trajectory,
xη,emis = xη + ∆xη,emis
19/06/2013 2S. Di Mitri - USPAS CO - Lecture_We_11
Single-kick approximation
3. δpz,CSR is correlated with z along the bunch
⇒ all particles at the same z (slice) feel same CSR kick,
⇒ the slice centroid C-S invariant is increased, albeit
the slice emittance is not.
x
z
4. The projected emittance is increased by the slices misalignment in the phase space
bunch
tail
bunch
head
Picture by
D. Douglas, JLAB-TN-98
4. The projected emittance is increased by the slices misalignment in the phase space
(bending plane only). Use the beam matrix to compute the CSR single-kick effect.
,1'
1'
'
det2
,
0
0
2/1
2
,
2
2
0
2
,0
2
,0
2
,
2
0
CSR
CSRCSR
CSRCSR Hδ
δδ
δδ
σε
εση
β
αεσηηαε
σηηαεσηβε
ε +=
++
+−
+−+
≅
2
βx 22
ηβ xx ∆=∆
( )[ ] βαηβηη22
'++=H takes care of the coupled betatron and dispersive motion;2
2
,
∆=
E
ECSRδ
σ is the CSR-induced energy spread relative to the reference energy and it goes like
where τ ≥ 1.τσ z/1∝
19/06/2013 3S. Di Mitri - USPAS CO - Lecture_We_11
CSR effect with varying bunch duration (e.g., compressor)
� In a small bending angle dipole magnet (θ<<1): θθη ≈≈ sin'
1
2 3
4
=
=
0'
max
η
ηη
≅
≅
θη
ηη
'
max
≅
≅
θη
η
'
0
=
=
0'
0
η
η
� Assume a beam waist (α=0) in the second half of the chicane and typical parameters:
θ≤0.1 rad, l≈0.3 m, η≤0.2 m and β≥3 m.θ≤0.1 rad, l≈0.3 m, η≤0.2 m and β≥3 m.
⇒ at the exit of 3rd dipole,
⇒ at the exit of 4th dipole, .mH 01.0
2 ≈≈ βη
mH 03.0'22 ≈≈≈ βθβη
dipoleth
CSR
CSR
H
40
2
,
2
0
2
,
0
02
11
+≈+=
ε
σθβεσ
εεε δ
δ
(
� Bunch length reaches its shortest duration right at the end of 3rd dipole, and remains
approximately constant along the 4th.
⇒ CSR emission is dominated by 4th dipole:
19/06/2013 4S. Di Mitri - USPAS CO - Lecture_We_11
Warning! CSR propagation in
drifts can be important, but it is
neglected here!
Projected emittance growth in a 4-dipoles compressor
We found that CSR-induced emittance growth is minimized when:
i) H-function is small,
ii) that is, as an example, when β-function has a minimum in the second-half of the chicane,
iii) and the beam energy is high.
LCLS first bucnch
length compressor
Bunch length vs. upstream
linac pahsing
Horizontal emittance vs.
upstream linac pahsing
K.L.Bane et al., PRSTAB
linac pahsing upstream linac pahsing
19/06/2013 5S. Di Mitri - USPAS CO - Lecture_We_11
Slice emittance growth in a 4-dipoles compressorSlice emittance might be affected by CSR if the bunch becomes so short that particle cross
over large portions of the bunch and, at the end of compression, lie in a slice different from
the initial one (“phase mixing”) ⇒ incoherent “sum” of C-S invariants.
This effect is more subtle than projected emittance growthand it is usually investigated with
particle tracking codes, with a large number of particles and control of numerical noise.
S. Di Mitri et al. NIM A 608 (2009) 19
Pictures also courtesy of M. Dohlus
Elegant – 1D CSR, 1D SC
IMPACT – 1D CSR, 3D SC
CSRTrack3D
19/06/2013 6S. Di Mitri - USPAS CO - Lecture_We_11
CSR effect with constant bunch duration (e.g., transfer line)
Problem.As bunch length is now the same all along the line and assuming identical dipoles,
we cannot recognize “dominant” points of CSR emission.
Idea.Take into account all CSR kicks along the line. Use optics arrangement that make
successive CSR kicks canceling each other (similarly to “emittance bumps” for
transverse wakes). A note by D.Douglas (JLAB-TN-98) proposes a perfect optics
symmetry and π phase advance between identical CSR emission “points”.
Real case.
19/06/2013 7S. Di Mitri - USPAS CO - Lecture_We_11
Real case.In real life, spatial and other optics constraints make hard to achieve such a
symmetry. Hence, try to cancel CSR even in the presence of optics asymmetry. This
would allow to satisfy other design constraints and suppres CSR induced ε-groth.
Double bend achromatic lines are often used to
bring the e-beam from the linac end to the
undulator. Optics design may include:
achromaticity, isochronicity, matching section,
time-compression, diagnostics, collimation....
A. Write down the particle coordinates (x,x’) with C-S formalism. Initial invariant is zero.
B. While traversing a dipole, add the CSR induced η-terms to the initial β-coordinates. This
corresponds to an increase of the particle C-S invariant:
2
1
2'
11
'
111
2
11122 CSRHxxxxJ δβαγ =++=
( )
−=∆+∆−≡=
=∆≡=
=∆
=∆
1
1
101
1
1'
1
110111
2sincos
2'
2cos2
βαµµα
βδη
βµβηδ
µ
µ
JJx
JJx
C. At the second dipole, after π phase advance and in the presence of the second CSR kick we
have:
=+−=+= 022 ββηδ JJxx Coordinates along the line are
Linear optics, single-kick model
19/06/2013 8S. Di Mitri - USPAS CO - Lecture_We_11
( )
+=+=−=
=+−=+=
21
1
1
1
1
1
2
1
2
'
2
'
3
112123
222'
022
ααββ
αβ
αδη
ββηδ
JJJxx
JJxx Coordinates along the line are
function of the initial invariant J1.
Special choice here: same β, η at
the dipoles of the same achromat.
D. Repeat until the end of the line. Each new invariant will be locally defined in terms of J1.and
the local C-S paramaters. After the last dipole we obtain:
−−=
−=
,22
22
1
1
1
4
115
7
'
7
1141157
βα
βα
ββ
JJXx
JJXx
( ) ( )
+−+++
−+==
4
1
5
1
4
1315315
2
4
1
5
1
4
1
4
1
1
5
1522
β
βα
β
βααααααα
β
βα
β
βα
β
β
J
JX
+= ββ
β
αββ xxw
x
xxxx ''
( )', ηη ++
πµ =∆3,2
Representation in the phase space
Normalize to local C-S
params: Floquet phase
space, ellipse � circle.
171
2
4
1
7
1
4
1
2
4
1
1517222 XJXJJ ≡
++
−=
β
βα
β
βα
β
β
−+=∆ 11
17
2
,1X
H CSR
ε
σγεγε δ
� Final particle invariant and emittance growth is:
x
x
xw
β
β=
πµ =∆2,1
#1
( )', ηη ++
#3
( )', ηη −+
#5( )', ηη −−
πµ =∆4,3
( )', ηη +−
#7
πµ =∆2,1 πµ =∆
3,2
πµ =∆4,3
� Final CSR induced C-S invariant is
zero (cancellation).
19/06/2013 9S. Di Mitri - USPAS CO - Lecture_We_11
Projected emittance growth in a multi-bend transfer line
Double-bend achromat (ηf = 0, η’f = 0) with 4
quadrupoles in between.
ηηηηx
Many quadrupoles ensure π-phase
advance between dipoles and proper
values of β, α to cancel the final
emittance growth.
This quadrupole’s strength is varied to
change the phase advance between the two
achromats (“optics tuning”).
ββββx,y
S. Di Mitri et al., PRL
109 (2013)
Final emittance is measured as
function of the quadrupole strength,
that is of the phase advance between
the achromats.
19/06/2013 10S. Di Mitri - USPAS CO - Lecture_We_11
Applies to two-stage magnetic compression
� Assume your linac includes two (or more) bunch compressors (chicanes, BCs).
⇒ define Twiss functions and phase advance that allow cancellation of CSR kicks in the two
(consecutive) BCs.
⇒ since bunch length is different in the two BCs, cancellation will be partial (unless optics
symmetry is broken to take into account a stronger effect in the 2nd BC than in the 1st).
( )( )xx
xx
',
,
ηη
αβ ( )( )xx
xx
',
,
ηη
αβ
…
BC1 BC2
( )xx ',ηη ( )xx ',ηη…
LINAC
?2,1
=∆µ
� All planned and existing linac-driven x-ray FELs includes at least 2 compressors. While
optics symmetry is usually adopted in transfer lines, no one has so far demonstrated CSR
cancellation in the two-stage compression scheme. Major efforts tend to reduce CSR induced
emittance growth in the individual chicanes.
19/06/2013 11S. Di Mitri - USPAS CO - Lecture_We_11
General formalism to find CSRGeneral formalism to find CSR--induced induced emittanceemittance growth growth
On-momentum,
zero-emittance
On-momentum,
finite-emittance
Particle energy Particle energy
changeschanges
Add up all contributions to energy change:
Transverse cood. at
exit of chicane
• To find emittance calculate central moments:
12
Vanishes
(Centered unperturbed beam)
Relative rms
energy spread
per unit length
induced by CSR 19/06/2013 12M. Venturini - USPAS CO - Lecture_We_11
– CSR-induced relative energy spreadenergy spread:
Similarly:
• Finally, we can determine the emittance at the exit of chicane
131319/06/2013 13M. Venturini - USPAS CO - Lecture_We_11