F.Antoniou, CERN, NTUAthens, Y. Papaphilippou, CERN CLIC meeting 5 Sept. 2008

1. 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

2. Outline • CLIC PDR • PDR Parameters • TME cells • Analytical Solution • Constraints • Stability criteria • Parameterization

3. R ~ 130 m e- Main Linac e+ Main Linac e- BC2 e+ BC2 12 GHz 2.4 GV 12 GHz 2.4 GV 12 GHz, 100 MV/m, 21 km 12 GHz, 100 MV/m, 21 km RTML RTML L ~ 1100 m 9 GeV 48 km 3 TeV Base line configuration (L. Rinolfi) Booster Linac 6.6 GeV 3 GHz  500 m  100 m  100 m e+ BC1 e- BC1  5 m  5 m 30 m 30 m 3 GHz 88 MV 3 GHz 88 MV 2.424 GeV 365 m 2.424 GeV 365 m e+ DR e- DR e- PDR e+ PDR 2.424 GeV 2.424 GeV Injector Linac 2.2 GeV 1.5 GHz  220 m  230 m  30 m e-/e+ Target Pre-injector Linac for e+ 200 MeV Laser Thermionic gun Unpolarized e- Pre-injector Linac for e- 200 MeV Positron Drive beam Linac 2 GeV DC gun Polarized e- 1.5 GHz 1.5 GHz 1.5 GHz  5 m  15 m  200 m CLIC Injector complex

4. 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)

5. 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)

6. Theoretical Minimum Emittance Cell • 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 1 1 6

7. 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) Andrea Streun http://slsbd.psi.ch/pub/cas/cas/node41.html

8. 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

9. 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

10. 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

11. 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. • l1is 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)

12. …Drift length parameterization • 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

13. Emittance parameterization • A possible configuration for the l1, l2, l3for 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 TME 1.2 TME 1.4TME 1.6 TME 1.8 TME 2 TME

14. Solution dependence on each drift length. For higher values of l1greater values of 1/f1 and 1/f2 are needed in order to achieve the TME For higher values of l2or l3 smaller values of 1/f1 and 1/f2 are needed in order to achieve the TME l1=0.66 l3=2 l2=0.5 l3=2 l1=0.66 l1=1 l1=1.5 l2=0.5 l2=1 l2=1.5 l1=0.66 l2=0.5 l3=1 l3=1.5 l3=2

15. STABILITY REGIONS • Phase advance curves in the horizontal and vertical plane for the solutions • There is always stability in the horizontal plane (by construction) but only very few solutions are stable in the vertical plane • The horizontal phase advance • for the TME is 284.5 ◦ and is • independent of the cell details • The vertical phase advance • depends on the drift lengths • and for this cell is 277.16 ◦ TME 1.2 TME 1.4TME 1.6 TME 1.8 TME 2 TME

16. Stability Regions • 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.

17. Stability Regions • 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

18. STABILITY REGIONS WITH EACH DRIFT LENGTH • The stability region dependence on each drift length Interesting to study mathematically the stability regions behavior!!! (under study) Varying l1 Varying l2 Varying l3

19. MADX EXAMPLES FOR TME CELLS

20. Chromaticity studies • 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.

21. Momentum Compaction factor And Chromaticity Minimization • 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!!!!

22. Application to the DR • The previous analysis was applied in the case of the damping rings (DR) of CLIC.