Analytical considerations for Theoretical Minimum Emittance Cell Optics

C L I CC L I C

Analytical considerations for

Theoretical Minimum Emittance

Cell Optics

17 April 2008

F. Antoniou, E. Gazis (NTUA, CERN)and Y. Papaphilippou (CERN)

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F. Antoniou/NTUA 217/4/2008

Outline

CLIC pre-damping rings design Design goals and challenges Theoretical background Lattice choice and optics optimisation Analytical solutions Open issues

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F. Antoniou/NTUA3

17/4/2008

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

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CLIC Injector complex

4

Thermionic gunUnpolarized e-

3 TeV

Base line configuration

(L. Rinolfi)

LaserDC gunPolarized e-

Pre-injector Linac for e-

200 MeV

e-/e+ TargetPositron Drive

beam Linac 2 GeV

Inje

ctor

Lin

ac

2.2

GeV

e+ DR

e+ PDR

2.424 GeV365 m

Boo

ster

Lin

ac

6.6

GeV 3

GHz

e+ BC1 e- BC1

e+ BC2 e- BC2e+ Main Linac e- Main Linac

12 GHz, 100 MV/m, 21 km 12 GHz, 100 MV/m, 21 km

1.5 GHz

e- DR

e- PDR

1.5 GHz 1.5 GHz 1.5 GHz

3 GHz88 MV

3 GHz88 MV

12 GHz2.4 GV

12 GHz2.4 GV

9 GeV 48

km

5 m 5 m

500 m

220 m 30 m

15 m 200 m

2.424 GeV365 m

2.424 GeV 2.424 GeV

100 m 100 m

Pre-injector Linac for e+

200 MeV

RTML

RTML

30 m 30 m

L ~

110

0 m

R ~ 130 m

5 m

230 m

17/4/2008 F. Antoniou/NTUA

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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

PDR ParametersCLIC

NLC

Energy [GeV]2.42

41.98

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

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] 170024000

0

Hor.,Ver Norm. emittance [nm]

100 x 103

9.7 x 106

L. Rinolfi17/4/2008 5F. Antoniou/NTUA

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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

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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


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Generalized transfer matrix

Dipoles:

Quadrupoles:

Drifts:

8

M =

The particle trajectory can be then written in the general form:

Xpxy

pyΔp/p

Xi+1 = M Xi Where X=

Using the above generalized transfer matrix, the equations can be solved piecewise


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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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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 10F. Antoniou/NTUA

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For isomagnetic ring :


Equilibrium emittance

The horizontal emittance of an electron beam is defined as:

the dispersion emittance

One can prove that H ~ ρθ and the normalized emittance can be

written as:

3

ε = γ ε = F C (γθ)n x lattice q3

Where the scaling factor F depends on the design of the storage ring lattices

lattice

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Low emittance lattices

Double Bend Achromat (DBA)

Triple Bend Achromat (TBA)

Quadruple Bend Achromat (QBA)

Theoretical Minimum Emittance cell (TME)

dispersion

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:

Only dipoles are shown but there are also quadrupoles in between for providing focusing

C L I CC L I CUsing 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 .

bendΘ = 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.

Cell choice

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Optics functions for minimum emittance

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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

C L I CC L I CQuad 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=l3l1>l2,l3l2>l1,l3l3>l1,l2

C L I CC L I C…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.

The horizontal plane is uniquely defined

F=1F=1.2F=1.4F=1.6F=1.8F=2

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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

C L I CC L I COpen 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

Documents

Analytical considerations for Theoretical Minimum Emittance Cell Optics