F. Antoniou, E. Gazis (NTUA, CERN) and Y. Papaphilippou (CERN) Analytical considerations for Theoretical Minimum Emittance Cell Optics F. Antoniou, E. Gazis (NTUA, CERN) and Y. Papaphilippou (CERN) April 17th-19th, 2007

Outline CLIC damping rings design and open issues Summary M. Korostelev (PhD thesis, EPFL 2006) CLIC damping rings design and open issues Design goals and challenges Input parameters - Pre-damping rings Lattice choice and optics optimisation Circumference (realistic drift space and magnets) Wiggler design and parameter scan (prototypes) Synchrotron radiation absorption Chromaticity correction and dynamic aperture Low emittance tuning in the presence of coupling (tolerances) e-cloud and other collective effects Summary 19/4/2008 F. Antoniou, HEP'08

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 19/4/2008 F. Antoniou, HEP'08

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

CLIC Pre-damping rings PDR Parameters CLIC PDR Energy [GeV] 2.424 1.98 Bunch population [109] 4.5 7.5 Bunch length [mm] 10 5.1 Energy Spread [%] 0.5 0.09 Long. emittance [eV.m] 121000 9000 Hor. Norm. emittance [nm] 63000 46000 Ver. Norm. emittance [nm] 1500 4600 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 Injected Parameters e- e+ Bunch population [109] 4.7 6.4 Bunch length [mm] 1 5 Energy Spread [%] 0.07 1.5 Long. emittance [eV.m] 1700 240000 Hor.,Ver Norm. emittance [nm] 100 x 103 9.7 x 106 19/4/2008 F. Antoniou, HEP'08 L. Rinolfi

Beam guidance Consider uniform magnetic field B in the direction perpendicular to particle motion. From the ideal trajectory and after considering that the transverse velocities vx<< vs,vy<<vs, the radius of curvature is The cyclotron or Larmor frequency We define the magnetic rigidity In more practical units For ions with charge multiplicity Z and atomic number A, the energy per nucleon is

Magnet definitions 2n-pole: dipole quadrupole sextupole octupole … Normal: gap appears at the horizontal plane Skew: rotate around beam axis by p/2n angle Symmetry: rotating around beam axis by p/n angle, the field is reversed (polarity flipped) N S N S N S N S

Dipoles Consider an accelerator ring for particles with energy E with N dipoles of length L Bending angle Bending radius Integrated dipole strength SNS ring dipole B θ ρ L Comments: By choosing a dipole field, the dipole length is imposed and vice versa The higher the field, shorter or smaller number of dipoles can be used Ring circumference (cost) is influenced by the field choice

Equations of motion – Linear fields Consider s-dependent fields from dipoles and normal quadrupoles The total momentum can be written With magnetic rigidity and normalized gradient the equations of motion are Inhomogeneous equations with s-dependent coefficients The term 1/ρ2 corresponds to the dipole week focusing The term ΔP/(Pρ) represents off-momentum particles

Hill’s equations George Hill Solutions are combination of the ones from the homogeneous and inhomogeneous equations Consider particles with the design momentum. The equations of motion become with Hill’s equations of linear transverse particle motion Linear equations with s-dependent coefficients (harmonic oscillator with time dependent frequency) In a ring (or in transport line with symmetries), coefficients are periodic Not straightforward to derive analytical solutions for whole accelerator

Betatron motion The on-momentum linear betatron motion of a particle is described by with the twiss functions the betatron phase and the beta function is defined by the envelope equation By differentiation, we have that the angle is

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 and the phase advance

Periodic transfer matrix Consider a periodic cell of length C The optics functions are and the phase advance The transfer matrix is The cell matrix can be also written as with and the Twiss matrix

Effect of dipole on off-momentum particles Up to now all particles had the same momentum P0 What happens for off-momentum particles, i.e. particles with momentum P0+ΔP? Consider a dipole with field B and bending radius ρ Recall that the magnetic rigidity is and for off-momentum particles Considering the effective length of the dipole unchanged Off-momentum particles get different deflection (different orbit) θ P0+ΔP P0 ρ ρ+δρ

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

Generalized transfer matrix With Dipoles: Quadrupoles: Drifts:

Betatron motion reminder The linear betatron motion of a particle is described by and with the twiss functions the betatron phase The beta function defines the envelope (machine aperture) Twiss parameters evolve as

Lattice section transfer matrix Generalized transfer matrix Periodic cell Mirror symmetric cell

Equilibrium emittance reminder with the dispersion emittance defined as For isomagnetic ring with separated function magnets the equilibrium emittance is written Smaller bending angle and lower energy reduce emittance

Optics functions

Deviation from the minimum emittance Introduce the dimensionless quantities and Introduce them into the expression of the mean dispersion emittance to get The curves of equal relative emittance are ellipses The phase advance for a mirror symmetric cell is The optimum phase advance for reaching the absolute minimum emittance (F=1) is unique (284.5o)!

Low emittance lattices dispersion Double Bend Achromat (DBA) Triple Bend Achromat (TBA) Quadruple Bend Achromat (QBA) Minimum Emittance Lattice (MEL)

Constraints for general double bend cells Consider a general double bend with the ideal effective emittance (drifts are parameters) In the straight section between the ID and the dipole entrance, there are three constraints, thus at least three quadrupoles are needed In the “achromat”, there are two constraints, thus at least two quadrupoles are needed (one and a half for a symmetric cell) Note that there is no control in the vertical plane The vertical phase advance is also fixed!!!! Expressions for the quadrupole gradients can be obtained, parameterized with the drift lengths, the initial optics functions and the beta on the IDs All the optics functions are thus uniquely determined for both planes and can be minimized (the gradients as well) by varying the drifts The chromaticities are also uniquely defined

TME arc cell TME cell chosen for compactness and efficient emittance minimisation over Multiple Bend Structures (or achromats) used in light sources Large phase advance necessary to achieve optimum equilibrium emittance Very low dispersion Strong sextupoles needed to correct chromaticity Impact in dynamic aperture 19/4/2008

Concluding remarks Detailed and robust design of the CLIC damping rings, delivering target emittance with the help of super-conducting wigglers Prototype to be built and tested at ANKA synchrotron Radiation absorption and quench protection Areas needing further optimisation and/or detailed studies Pre-damping ring optics design Realistic damping ring cell length and magnet parameters Sextupole optimisation and non-linear dynamics including wiggler field errors Linear and non-linear correction schemes Low emittance tuning and alignment tolerances IBS theory, numerical tools and experimental demonstration of low emittance Collective effects including electron cloud and fast ion instability Detailed vacuum chamber design – impedance budget Injection and extraction elements Design of HOM free high frequency RF cavities Diagnostics and feedback 19/4/2008 F. Antoniou, HEP'08