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Optics design considerations for the minimum emittance lattices Application to CLIC pre-damping rings. F.Antoniou, CERN, NTUAthens, Y. Papaphilippou, CERN CLIC meeting 5 Sept. 2008. Outline. CLIC PDR PDR Parameters TME cells Analytical Solution Constraints Stability criteria - PowerPoint PPT Presentation
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F.Antoniou, CERN, NTUAthens,Y. Papaphilippou, CERN
CLIC meeting 5 Sept. 2008
Optics design considerations
for the minimum emittance lattices
Application to CLIC pre-damping rings
OUTLINE
CLIC PDR PDR Parameters
TME cells
Analytical Solution Constraints Stability criteria Parameterization
2
CLIC INJECTOR COMPLEX
Thermionic gunUnpolarized e-
3 TeV
Base line configuration
(L. Rinolfi)
LaserDC gunPolarized e-
Pre-injector Linac for e-
200 MeV
e-/e+ TargetPositron Drive
beam Linac 2 GeV
Inje
ctor
Lin
ac
2.2
GeV
e+ DR
e+ PDR
2.424 GeV365 m
Boo
ster
Lin
ac
6.6
GeV 3
GHz
e+ BC1 e- BC1
e+ BC2 e- BC2e+ Main Linac e- Main Linac
12 GHz, 100 MV/m, 21 km 12 GHz, 100 MV/m, 21 km
1.5 GHz
e- DR
e- PDR
1.5 GHz 1.5 GHz 1.5 GHz
3 GHz88 MV
3 GHz88 MV
12 GHz2.4 GV
12 GHz2.4 GV
9 GeV 48
km
5 m 5 m
500 m
220 m 30 m
15 m 200 m
2.424 GeV365 m
2.424 GeV 2.424 GeV
100 m 100 m
Pre-injector Linac for e+
200 MeV
RTML
RTML
30 m 30 m
L ~
110
0 m
R ~ 130 m
5 m
230 m
3
CLIC PRE-DAMPING RINGS
The pre-damping rings (PDR) are an essential part of the CLIC injector complex.
Most critical the e+ PDR Injected e+ emittance ~ 2
orders of magnitude larger than for e-, i.e. aperture limited if injected directly into DR
PDR for e- beam necessary as well A “zero current” linac e-
beam (no IBS) would need ~ 17ms to reach equilibrium in DR, (very close to repetition time of 20ms)
CLIC PDR parameters at extraction close to those of NLC PDR(I. Raichel and A. Wolski, EPAC04)
PDR Extracted Parameters
CLIC
NLC
Energy [GeV]2.42
41.98
Bunch population [109] 4.5 7.5
Bunch length [mm] 10 5.1
Energy Spread [%] 0.5 0.09
Long. emittance [eV.m]1210
009000
Hor. Norm. emittance [nm]
63000
46000
Ver. Norm. emittance [nm]
1500 4600
Injected Parameters
e- e+
Bunch population [109] 4.7 6.4
Bunch length [mm] 1 5
Energy Spread [%] 0.07 1
Long. emittance [eV.m] 170024000
0
Hor.,Ver Norm. emittance [nm]
100 x 103
9.7 x 106
PDR DESIGN CHALLENGES Momentum compaction factor
for minimum number (18) of TME cells needed for PDR is quite large
For increasing momentum acceptance Reduce αc with more bends or TMEs with
higher emittance and negative dispersion in the bends
Increase RF voltage or use harmonic cavities
Damping times quite long Incompatible with polarized positron
stacking of 12ms Damping times should be twice faster than
in the DR and cannot be achieved with damping wigglers
Staggered trains injection scheme High beam power handling (radiation
absorption, HOM free cavities, other collective effects)
The TME cell may be a good structure for the design of the PDR lattice.
The beta function and the dispersion function has a minimum in the center of the bending magnet.
They get the values:
The Normalized TME is then: where , Cq = 3.84∙10-3 m and Jx ≈ 1 in the case
of isomagnetic, separated function ring
6
THEORETICAL MINIMUM EMITTANCE CELL
6
1
1
THEORETICAL MINIMUM EMITTANCE CELL
The TME cells provide 3 times lower horizontal emittances than the DBA cells.
Only one pair of (eta,beta) values can achieve the TME.
Higher emittance values can be achieved by several pairs of (eta, beta)
7Andrea Streunhttp://slsbd.psi.ch/pub/cas/cas/node41.html
For achieving a certain emittance in a standard TME cell, there are two independent horizontal optical parameters to be fixed at the entrance (and exit) of the cell
Two quadrupole families are needed to achieve this The vertical plane is not controlled and for this additional
quadrupoles may be needed
The problem can be solved analytically using thin lens approximation
Parameterization of the solutions with respect to the drift lengths (for constant emittance) and with the emittance (for constant drift lengths)
ANALYTICAL SOLUTION
8
The general solutions for the focusing strengths are:
ηs is the dispersion function in the center of the cell and
it is a complicated function of the initial optic functions at the center of the dipole and the drift lengths. There are 2 solutions for ηs
One of them results in horizontal focusing for both f1 and f2
This solution is unstable in the vertical plane and is rejected
Solutions and Constraints
9
Stability criteria
Trace(M)=2 cosμ where M is the transfer matrix of all the cell
and μ the phase advance per cell.
The beta functions at the quads have values smaller than 30m
All the optic functions are thus uniquely determined
for both planes and can be optimized by varying the drifts
10
Drift lengths parameterization Minimum number of dipoles (18) to achieve target PDR emittance
with a TME cell (with a 30% margin) A constant emittance is first considered (the TME one) Focusing strengths and beta functions on the quads, for l1, l2, l3
(between 0.5 and 2m).
• No restrictions for l3.
• l1 is bounded to values smaller than 1.7 m and l2 to values smaller than 1.6 m.
• The limits are set by the constraint in the beta functions (<30m)
11
…Drift length parameterization
O Region of solutions for the beta functions at the center of the quads and the chromaticities for all possible drifts
• The blue dots represent the case where only the phase advance stability criterion is applied
• The red dots represent the case where also the beta functions are restricted
The beta functions and chromaticities can be controlled in low enough values for certain drift length considerations
Horizontal beta function at f1 versus vertical at f2 and horizontal chromaticity versus vertical
12
Emittance parameterizationA possible configuration for the l1, l2, l3 for minimum chromaticities (and small focusing strengths) at both the planes is chosen for the TME:
l1 =0.66, l2=0.5, l3 =2
The choice of the drift lengths can be done according to what we want to minimize or succeed.
The focal strengths can be now parameterized with respect to the achievable emittance
These solutions do not necessarily provide stability in the vertical plane
TME1.2 TME1.4TME1.6 TME1.8 TME2 TME
13
Solution dependence on each drift length.
o For higher values of l1 greater values of 1/f1 and 1/f2 are needed in order to achieve the TME
o For higher values of l2 or l3 smaller values of 1/f1 and 1/f2 are needed in order to achieve the TME
l2=0.5l3=2
l1=0.66l1=1l1=1.5
l1=0.66l3=2
l2=0.5l2=1l2=1.5
l3=1l3=1.5l3=2
l1=0.66l2=0.5
14
STABILITY REGIONS
Phase advance curves in the horizontal and vertical plane for the solutions
o There is always stability in the horizontal plane (by construction) but only very few solutions are stable in the vertical plane
o The horizontal phase advance for the TME is 284.5 ◦ and is independent of the cell details
o The vertical phase advance depends on the drift lengths and for this cell is 277.16 ◦
15
TME1.2 TME1.4TME1.6 TME1.8 TME2 TME
STABILITY REGIONS
16
Applying also the stability criterion for the vertical plane in all the solutions, we get the stability solutions for both the planes.
Two bands of solutions
The stability region for f2 almost stable for all the values of emittances.
The stability regions are studied for different values of TME (increasing the number of the dipoles, the TME is reduced)
The stability region for f2 is the same in all the cases.
Nd=18
Nd=100 17
STABILITY REGIONS
Varying l1Varying l2
Varying l3
STABILITY REGIONS WITH EACH DRIFT LENGTH The stability region dependence on each drift length
18
Interesting to study mathematically the stability
regions behavior!!! (under study)
MADX EXAMPLES FOR TME CELLS
19
CHROMATICITY STUDIES
20
The blue dots represent the case of stability solutions, for all the possible lengths and for several values of emittance.
The red dots represent the case where ξx, ξy >-1.
The TME cells can not provide small values of chromaticities.
Restriction in the cell length.
MOMENTUM COMPACTION FACTOR AND CHROMATICITY MINIMIZATION
21
Same color convention with the previous plots.
The αc factor is decreased as the number of cells is decreased.
For smaller values of the αc
, higher values of horizontal chromaticities!!!!
APPLICATION TO THE DR
The previous analysis was applied in the case of the damping rings (DR) of CLIC.
22
Parameters CLIC DR V04 Alternative
Energy [GeV]Number of arc cellsCell length [m]Dipole bending angle [grad]Dipole Length [m]Bend field [T]Quad coefficient K1[1/m2]Beta-function in dipole center x,z [m]Dispersion in dipole center [m]Normalized Emittance [nm rad]Natural chromaticity x/zMom. Compaction factor, ac Phase advance x/z
2.424100
1.7293.6
0.5449440.93
27.21/-16.170.112/0.6143.363E-03
398.5-0.85/-1.178.0213E-050.581/0.248
2.424100
2.7293.60.4
1.2713.33/-
8.240.087/1.54
0.0029512.31
-1.59/-0.924.56E-05
0.524/0.183
2.424100
2.74623.6
0.5449440.93
12.609/-7.005
0.1523/0.961
0.0035423.38-0.945/-1.141
10.94E-050.561/0.251