EXPERIMENTAL FACILITIES OVERVIEW

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

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Design goals and challenges
Theoretical background
Lattice choice and optics optimisation
Analytical solutions

Open issues
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CLIC


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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
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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-
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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

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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
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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
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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
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CLIC
Dispersion equation
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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)
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In that way, the equations of motion are split in two parts

The dispersion function can be defined as
The dispersion equation is
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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:
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Using the above
generalized transfer
matrix, the equations can
be solved piecewise
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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
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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:
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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)
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Where the scaling factor F lattice
depends on the design of the
storage ring lattices
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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
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…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
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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
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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
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