1. 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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. Generalized transfer matrix
With
Dipoles:
Quadrupoles:
Drifts:

17. 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

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

19. 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

20. Optics functions

21. 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)!

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

23. 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

24. 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

25. 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