Report

CLIC Analytical considerations for Theoretical Minimum Emittance Cell Optics F. Antoniou, E. Gazis (NTUA, CERN) and Y. Papaphilippou (CERN) 17 April 2008 Outline CLIC CLIC pre-damping rings design Design goals and challenges Theoretical background Lattice choice and optics optimisation Analytical solutions Open issues 17/4/2008 F. Antoniou/NTUA 2 CLIC The CLIC Project Compact Linear Collider : multi-TeV electronpositron 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 17/4/2008 F. Antoniou/NTUA 3 CLIC Injector complex CLIC 100 m e- Main Linac 12 GHz, 100 MV/m, 21 km 12 GHz, 100 MV/m, 21 km RTM L 9 GeV Booster Linac 6.6 GeV e+ 12 GHz BC2 2.4 GV e+ Main Linac 5m DR 2.424 GeV 3 GHz 88 MV Thermionic gun Unpolarized e- 17/4/2008 Positron Drive beam Linac 2 GeV 1.5 GHz 200 m Base line configuration e-/e+ Target (L. Rinolfi) 5m 3 GHz 88 MV e- DR ePDR e+ PDR 230 m Pre-injector Linac for e+ 200 MeV 1.5 GHz 5m 15 m 12 GHz 2.4 GV 3 TeV 3 GHz 500 m 30 m 30 m RTM L 48 km e- BC1 Injector Linac 2.2 GeV 2.424 GeV 365 m e+ e+ BC1 e- BC2 L ~ 1100 m 100 m 1.5 GHz 220 m 2.424 GeV 30 m Pre-injector Linac for e200 MeV 1.5 GHz F. Antoniou/NTUA 2.424 GeV 365 m Laser DC gun Polarized e- 4 CLIC Pre-Damping Rings (PDR) CLIC PDR Parameters Pre-damping rings needed in order Energy [GeV] to achieve injected beam size Bunch population [109] tolerances at the entrance of the Bunch length [mm] damping rings Energy Spread [%] Most critical the positron damping Long. emittance [eV.m] ring Hor. Norm. emittance [nm] Injected emittances ~ 3 orders of Ver. Norm. emittance [nm] magnitude larger than for electrons Injected Parameters CLIC PDR parameters very close to those of NLC Bunch population [109] (I. Raichel and A. Wolski, EPAC04) Bunch length [mm] Similar design may be adapted to Energy Spread [%] CLIC Long. emittance [eV.m] Lower vertical emittance Hor.,Ver Norm. emittance [nm] Higher energy spread 17/4/2008 F. Antoniou/NTUA L. Rinolfi CLIC NLC 2.424 1.98 4.5 7.5 10 5.1 0.5 0.09 121000 9000 63000 46000 1500 4600 e- e+ 4.7 6.4 1 5 0.07 1.5 1700 240000 100 x 103 9.7 x 106 5 CLIC Equations of motion Accelerator main beam elements • Dipoles (constant magnetic field) • Quadrupoles (linear magnetic fields) guidance 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 17/4/2008 F. Antoniou/NTUA 6 CLIC 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 two parts The dispersion function can be defined as The dispersion equation is 17/4/2008 F. Antoniou/NTUA 7 CLIC Generalized transfer matrix The particle trajectory can be then written in the general form: Xi+1 = M Xi Where X= X px y py Δp/p M= Dipoles: Quadrupoles: Drifts: 17/4/2008 Using the above generalized transfer matrix, the equations can be solved piecewise F. Antoniou/NTUA 8 CLIC 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 17/4/2008 F. Antoniou/NTUA 9 CLIC 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 s with: For a periodic cell of length C we have: Where μ is the phase advance per cell: 17/4/2008 F. Antoniou/NTUA 10 CLIC Equilibrium emittance The horizontal emittance of an electron beam is defined as: For isomagnetic ring : the dispersion emittance 3 One can prove that H ~ ρθ and the normalized emittance can be written as: 3 εn= γ εx= FlatticeCq (γθ) 17/4/2008 Where the scaling factor F lattice depends on the design of the storage ring lattices F. Antoniou/NTUA 11 CLIC 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 CLIC 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 bend = 2π/Ν 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. Optics functions for minimum emittance CLIC Constraints for general MEL CLIC 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 CLIC 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) l1=l2=l3 l1>l2,l3 l2>l1,l3 l3>l1,l2 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 17/4/2008 F. Antoniou/NTUA 16 …Quad strengths CLIC Emittance parameterization (for fixed drift lengths) F=1 F=1.2 F=1.4 F=1.6 F=1.8 F=2 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. The horizontal plane is uniquely defined 17/4/2008 F. Antoniou/NTUA 17 CLIC 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 17/4/2008 F. Antoniou/NTUA 18 CLIC 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 17/4/2008 F. Antoniou/NTUA 19