Matching recipe and trackingfor the final focus

T. Asaka†, J. Resta López‡

and F. Zimmermann†CERN, Geneve / SPring-8, Japan

‡CERN, Geneve / University of Valencia

CERN, Geneve

CLIC BDS Day, CERN, November 22 2005

One of the most important issues in the final focus design is to take care about the beam size growth () due to the energy spread (.

* ≈ 0* : chromaticity, 0*: linear optics beam size)

The beam size easily becomes about ten times larger.(For example,

In order to correct the chromaticity of the final focus, the CLIC final focus uses sextupoles in a dispersive region next to the final quadrupoles, thereby providing a local correction as proposed by P. Raimondi and A. Seryi.

The nonlinear effects of the sextupoles, namely geometric aberrations, are cancelled by upstream sextupoles with proper phase advance and beta functions.

Introduction and Motivation

Introduction and MotivationWe study the performance of the CLIC final focus by using SAD and MAD.

The optics matching of the CLIC final focus system is obtained by consideration of correction of the 2nd and 3rd order geometric aberration terms (reference to ATF final focus design at KEK by S. Kuroda).

Outline

» Overview of the main features of the SAD code

» Schematic of CLIC final focus

» Calculation setup and results

» Particle tracking

» Energy bandwidth

» Matching recipe of the final focus

» Summary

Computer program SAD6D full-symplectic tracking. Also envelope formalism.

For calculation of the synchrotron radiation, each magnet is split into several slices. In the tracking, the radiation occurs at borders between the slices.

K. Ohmi, et al., Phys. Rev. E49, 751 (1994).K. Oide, et al., Phys. Rev. Lett. 61, 1713 (1988).

http://acc-physics.kek.jp/SAD/sad.html

Schematic of CLIC final focusSF1 SD1 SF2 SD2 QDQF IP

Matching section

The CLIC final focus is based on a compact final focus system, Raimondi-like compact design.

Two sextupole magnets are placed just next to the final quadrupole magnets, which are major chromaticity sources because of large beta function there.

The geometric aberrations can be reduced by using pairs of sextupole magnets compensating each other.

Calculation set up and results

@entrance @IP

x = 0y = 0x = 65 my = 18 mx = 0.23 e-12 my = 3.4e-15 m = 1%

x = 0y = 0x = 7 mmy = 90 mx = 0 m

CLIC final focus system has 40 quadrupoles, 10 sextupoles and one octupole.

This lattice has been matched to the twiss parameters at the entrance and at the IP.

0

100

200

300

400

500

600

0 100 200 300 400 500 600

(x

/y

/[

/]

[]

(x)/

(y)/

-60

-50

-40

-30

-20

-10

0

10

0 100 200 300 400 500 600

x, η

y [mm]

s [m]

ηx

ηy

Calculation results

DP -0.6% -0.4% -0.2% 0% 0.2% 0.4% 0.6%

x -0.34 -0.22 -0.11 0.00 0.11 0.23 0.37

x [mm]

7.3 7.0 6.9 7.0 7.3 7.8 8.7

y 1.74 0.94 0.37 0.00 -0.19 -0.24 -0.17

y [ m]

376 173 103 90 92 93 89

The value of the twiss parameters at the interaction point is shown in the table for several energy offset ( = E/E0).

Vertical beta function increased with less than -0.2% energy offset.

Moreover, since the higher order (3rd order) geometric aberrations remain in vertical space, it is necessary to optimize the optics.

Particle tracking

A distribution of 40000 particles and 1% full width energy spread for a flat square energy distribution has been tracked.

This plots shows an example of the beam transversal profile at IP.

The beam sizes have been calculated taking the size of the beam core by means of gaussian fit.

Particle trackingw/o SR with SR

x[nm] 45.7 53.2

xrms [nm] 47.4 56.5

y[nm] 0.63 0.80

yrms [nm] 0.78 1.52

w/o SR with SR

Without the SR, the result for the horizontal beam size is very similar in gaussian fit and standard deviation. We find a discrepancy for the vertical beam size with the gaussian fit and the standard deviation.With the SR, the horizontal beam size is up to 19% higher than the value without SR, and the vertical beam size increase about 2 times for the standard deviation.

Energy bandwidth

The tracking was made with 10000 particles for the different energy spread.

These plots shows the beam sizes as function of the given by the beam energy spread for the horizontal and vertical beam size.

The results have been normalized to the value of the given by MAD 0% energy spread without SR.

Energy bandwidth

Both case without SR and with SR have been considered.

Without synchrotron radiation, there is good agreement between the two codes for energy spread on.

But, if the synchrotron radiation is included, the value of the vertical sizes obtain from SAD are up to about 25% higher than those from simulations with MAD.

Matching recipe of the final focus

P M Q N

SF1 SD1 SF2 SD2 QDQF IP

Matching section

The optimized optics is calculated by consideration of correction of the geometric aberration terms.

P, M, Q and N represent the transfer matrices here.

The chromatic effect is corrected by sextupole magnets.

Two sextupole magnets are placed just next to the final quadrupole magnets, which are major chromaticity sources because of large beta function there.

Matching recipe of the final focusThe 2nd order geometric aberration is cancelled by the other sextupole magnets with the transfer matrix as follows,

MP =

F

F21

0

0

0

1/F

0

0

0

0

F

F43

0

0

0

1/F

QM =

D

D21

0

0

0

1/D

0

0

0

0

D

D43

0

0

0

1/D

Strength of the sextupole magnets :

K2, SF1 = -F3K2, SF2 and K2, SD1 = -D3K2, SD2

Matching recipe of the final focusIn order to cancel remains of 3rd order geometric aberration, the equations by the coefficients when the nonlinear map including the sextupole magnets action is used.

U3444 ~ N342 Q12 (N33 Q34 + N34 Q44)2

U1244 = U3224

~ Q12 N342 (NQ)12

2 + Q12 N122 (NQ)34

2 - 4 Q34 N12 N34 (NQ)12 (NQ)34

where the indices 1, 2, 3 and 4 represent for x, px, y and py, respectively.

The 3rd order geometric aberration can be expressed with only the transfer matrices Q and N.

With adequate choice of the final quad strength, U1244 = U3224 = 0. And U3444 also can be made small at the same time.

Matching recipe of the final focus» The final drift space length L*.» The distance between QF and SD2.» The distance between quadrupoles and their correction

sextupoles.

======================================================

» The equation for the 3rd order geometric aberration U1244 = 0 is solved and one of the strengths of the final quadrupole magnets is expressed, and the value of U3444 is minimize.

» Strength of QD and QF in consideration chromaticity.» The quadrupole magnets between SF1 and SF2 are adjusted.» Coefficients F and D about the sextupole magnets.» Optimization of quadrupole magnets in the matching section.

Summary» The optics of the CLIC final focus have been calculated by

using SAD.

» The beam size at the IP has been calculated from particle tracking with the SAD.

» Energy bandwidth in the final focus have been obtaind with and without SR.

» We are planning to optimize the optics of final focus by using SAD with the matching recipe.