Analytical considerations for Theoretical Minimum Emittance Cell Optics

1. Analytical considerations for Theoretical Minimum Emittance Cell Optics F. Antoniou, E. Gazis (NTUA, CERN) and Y. Papaphilippou (CERN) 17 April 2008

2. Outline • CLIC pre-damping rings design • Design goals and challenges • Theoretical background • Lattice choice and optics optimisation • Analytical solutions • Open issues F. Antoniou/NTUA

3. The CLIC Project • Compact Linear Collider : multi-TeV electron-positron collider for high energy physics beyond today's particle accelerators • Center-of-mass energy from 0.5 to 3 TeV • RF gradient and frequencies are very high • 100 MV/m in room temperature accelerating structures at 12 GHz • Two-beam-acceleration concept • High current “drive” beam, decelerated in special power extraction structures (PETS) , generates RF power for main beam. • Challenges: • Efficient generation of drive beam • PETS generating the required power • 12 GHz RF structures for the required gradient • Generation/preservation of small emittance beam • Focusing to nanometer beam size • Precise alignment of the different components F. Antoniou/NTUA

4. 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 F. Antoniou/NTUA

5. CLIC Pre-Damping Rings (PDR) • Pre-damping rings needed in order to achieve injected beam size tolerances at the entrance of the damping rings • Most critical the positron damping ring • Injected emittances ~ 3 orders of magnitude larger than for electrons • CLIC PDR parameters very close to those of NLC (I. Raichel and A. Wolski, EPAC04) • Similar design may be adapted to CLIC • Lower vertical emittance • Higher energy spread F. Antoniou/NTUA L. Rinolfi

6. Equations of motion • Accelerator main beam elements • Dipoles (constant magnetic field) guidance • Quadrupoles (linear magnetic fields) beam focusing • Consider particles with the design momentum. The Lorentz equations of motion become with • Hill’s equations of linear transverse particle motion • Linear equations with s-dependent coefficients (harmonic oscillator) • In a ring (or in transport line with symmetries), coefficients are periodic • Not straightforward to derive analytical solutions for whole accelerator F. Antoniou/NTUA

7. Dispersion equation Consider the equations of motion for off-momentum particles The solution is a sum of the homogeneous equation (on-momentum) and the inhomogeneous (off-momentum) In that way, the equations of motion are split in twoparts The dispersion function can be defined as The dispersion equation is F. Antoniou/NTUA 7

8. Generalized transfer matrix The particle trajectory can be then written in the general form: X px y py Δp/p Xi+1 = M Xi Where X= M= Dipoles: Using the above generalized transfer matrix, the equations can be solved piecewise Quadrupoles: Drifts: F. Antoniou/NTUA 8

9. Betatron motion • The linear betatron motion of a particle is described by: and α, β, γ the twiss functions: Ψ the betatron phase: • The beta function defines the envelope (machine aperture): • Twiss parameters evolve as F. Antoniou/NTUA

10. General transfer matrix From equation for position and angle we have Expand the trigonometric formulas and set ψ(0)=0 to get the transfer matrix from location 0 to swith: For a periodic cell of length C we have: Where μ is the phase advance per cell: F. Antoniou/NTUA

11. Equilibrium emittance • The horizontal emittance of an electron beam is defined as: • For isomagnetic ring : the dispersion emittance • One can prove that H ~ ρθ and the normalized emittance can be • written as: 3 Where the scaling factor F depends on the design of the storage ring lattices lattice 3 ε = γ ε = F C (γθ) n x lattice q F. Antoniou/NTUA

12. Low emittance lattices • FODO cell: the most common and simple structure that is made of a pair of focusing and defocusing quadrupoles with or without dipoles in between • There are also other structures more complex but giving lower emittance: dispersion • Double Bend Achromat (DBA) • Triple Bend Achromat (TBA) • Quadruple Bend Achromat (QBA) • Theoretical Minimum Emittance cell (TME) Only dipoles are shown but there are also quadrupoles in between for providing focusing

13. Cell choice Using the values for the F factor and the relation between the bending angle and the number of dipoles, we can calculate the minimum number of dipoles needed to achieve a required normalized minimum emittance of 50 μm for the FODO, the DBA and the TME cells . Θ = 2π/Ν bend • FFODO = 1.3 NFODO > 67 NCELL > 33 • FDBA = 1/(4√15Jx) NDBA > 24 NCELL > 24 • FTME = 1/(12√15Jx) NTME > 17 NCELL > 17 • Straightforward solutions for FODO cells but do not achieve very low emittances • TME cell chosen for compactness and efficient emittance minimisation over Multiple • Bend Structures (or achromats) used in light sources • TME more complex to tune over other cell types • We want to parameterize the solutions for the three types of cells • We start from the TME that is the more difficult one and there is nothing been • done for this yet.

14. Optics functions for minimum emittance

15. Constraints for general MEL • Consider a general MEL with the theoretical minimum emittance (drifts are parameters) • In the straight section, there are two independent constraints, thus at least two quadrupoles are needed • Note that there is no control in the vertical plane!! • Expressions for the quadrupole gradients can be obtained, parameterized with the drift lengths and the initial optics functions • All the optics functions are thus uniquely determined for both planes and can be minimized (the gradients as well) by varying the drifts • The vertical phase advance is also fixed!!!! • The chromaticities are also uniquely defined There are tools like the MADX program that can provide a numerical solution, but an analytical solution is preferable in order to completely parameterize the problem

16. Quad strengths • The quad strengths were derived analytically and parameterized with the drift lengths and the emittance • Drift lengths parameterization (for the minimum emittance optics) • 2 solutions: • The first solution is not • acceptable as it gives negative • values for both quadrupole • strengths (focusing quads) instability • in the vertical plane • The second solution gives all possible values for the quads to achieve the minimum emittance l1=l2=l3 l1>l2,l3 l2>l1,l3 l3>l1,l2 F. Antoniou/NTUA

17. …Quad strengths • Emittance parameterization (for fixed drift lengths) • F = (achieved emittance)/(TME emittance) • All quad strength values • for emittance values from the • theoretical minimum emittance • to 2 times the TME. • The point (F=1) represents the • values of the quand strengths • for the TME. F=1 F=1.2 F=1.4 F=1.6 F=1.8 F=2 • The horizontal plane is uniquely defined F. Antoniou/NTUA

18. The vertical plane is also uniquely defined by these solutions (opposite signs in the quad strengths) • Certain values should be excluded because they do not provide • stability to both the planes • The drift strengths should be constrained to provide stability • The stability criterion is: • Trace(M) = 2 cos μ Abs[Trace(M)] < 2 • The criterion has to be valid in both the planes F. Antoniou/NTUA

19. Open issues • Find all the restrictions and all the regions of stability • Parameterize the problem with other parameters, like phase • advance and chromaticity • Lattice design with MADX • Follow the same strategy for other lattice options • Non-linear dynamics optimization and lattice comparison for • CLIC pre-damping rings F. Antoniou/NTUA