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