Simulation of Positron Production and Capturing.

W. Gai, W. Liu, H. Wang and K. Kim

Working with SLAC & DESY

Start-to-end Model of the Positron Source

e-Undulator

Electron Gamma Shower code: EGSnrc

Particle dynamic code:PARMELA

Analytical + Monte carlo

AMD profile and PreAccelerator Filed Map

AMD field:5T-0.25T in 50cm

Accelerating gradient in preaccelerator: 12MV/m

Photon number spectrum and distribution functions

The spectrum of photon generated by helical undulator is known as

1

222'2

2

220

26

))(][)((4

10]

1[ xJ

x

n

KxJ

K

hc

e

MeVmdE

dNphn

nn

))(][)((4

10 222'2

2

220

26

xJx

n

KxJ

K

hc

e

dE

dNphn

n

nn

1 01 0

)(i i

n

i i

dEdE

dNphdE

dE

dNphnD

Photon number spectrum in terms of harmonics

nE

n

E

nn dE

dE

dNphdE

dE

dNphED

00

)(

(1)

Harmonics distribution function

Energy distribution function

(2)

(3)

(4)

functionsBesselJ

KKx

KK

n

n

n

n

)1(2

0]1)1(

[

21

22

12

1

2

2

11 )1(

4

][*][*934.0

nEE

K

cE

cmTBK

n

u

u

e- Undulator

Photons from Helical Undulator– polarization and radiation angle

223222

22

1

2,0,

yx

yx

yx

yx

II

II

II

II

•The amplitude of radiation from helical undulator in terms of harmonic can be obtained as

From which the Stokes parameters are obtained as:

(1)

(2)

(3)2/122

1 ]1)1(

[1

KK

n

)('4

10),(

)(][4

10),(

220

36

220

36

xJK

c

einEIy

xJx

n

K

K

c

enEIx

n

nn

where describes the linear polarization, 2 describes the linear polarization in coordinate system rotated by 45o with respect to the original one for 1, 3 describes the circular polarization.

After knowing its energy and originating harmonic, the radiation angle of photon is also determined as:

An uniform distribution on azimuthal direction is assumed

e- Undulator

Photons from helical undualtor– originating point and collimating

In addition to distribution functions in previous slides:

• Uniform distribution along the length of undulator is assumed

• Temporal and transverse distribution is inherited from drive e- beam

Collimating:

With the originating point ( x0 , y0 , z0 ) and direction vector (dx, dy, dz) assigned, the state of a photon after collimator is determined by its position on the collimator screen as:

200

200 ]/)([]/)([ dzdyzzydzdxzzxr cc

Where zc is the location of collimator.

If r is greater than the collimator iris, the photon is then ignored and will not allowed to strike the target to produce positrons

(1)

e- Undulator

Undulator Based Positron Source

e-Undulator

Polarized source:

For unpolarized source, the photon collimator and drift between target and undulator can b

e elliminated

Photon Spectrum and Polarization of ILC baseline undulator

Results of photon number spectrum and polarization characteristic of ILC undulator are given here as examples. The parameter of ILC undulator is K=1, u=1cm and the energy of electron beam is 150GeV.

Figure1. Photon Number spectrum and polarization characteristics of ILC undulator up to the 9th harmonic. Only those have energy closed to critical energy of its corresponding harmonics have higher polarization

e- Undulator

Positron polarization distribution after target

without collimator with collimator

The collimator is 2.5mm in radius and is 700m away from the end of a 100m long undulator. Photons are radiated uniformly along the undulator. For the case without collimator, the photon source is treated as a point source.

Photon collimator screened out those photon with lower polarization and thus improved the positron polarization.

Comparison of the captured positron polarization spectrum: with/without collimator.

Without collimator With collimator

The capturing optics are the same for both with and without collimator. The collimator screened out those photons with lower polarization and thus improved the polarization of resulting positron beam. The polarization of captured positron beam is about 30% without collimator and 63% with the collimator setting used here.

Better polarization but lower positron yield due to less photons were used

Yield contribution from different harmonics.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10

Harmonic index

Co

ntr

ibu

tion

(%

)

200m

700m

1100m

1st harmonic contributed <15% to the positron yield 2nd and 3rd harmonic together contributed >45% Contributions from 4th and 5th harmonic are comparable to 1st harmonicPositron yield is mostly contributed by high order harmonics

The cutting angle of collimator is fixed at 3.85rad

e- Undulator

The Effect of Photon Collimator

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

300 500 700 900Length of Drift (m)

Pol

ariz

atio

n

0

0.5

1

1.5

2

2.5

Yie

ld

PolarizationYield

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5Radius of Collimator (mm)

Pol

ariz

atio

n

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Yie

ld

PolarizationYield

Figure 1. Figure 2.

Figure 3 Yield and polarization strongly correlated to collimator configuration. For 150GeV drive beam with K=1 and lu=1cm, 60% polarization required at least 400m

drift

Optimum yield for 150GeV undulator with u=1cm and different K

Optimum yield dropping from about 1.1 down to

<0.9 when K change from 1 to 0.8

Optimum yield for 150GeV undulator with and different u

Optimum yield decrease from ~1.1 down to ~0.4 when u inc

rease from 1cm up to 1.6cm

Yield and Polarization for undulator with 250GeV drive beam

Yield for polarized undulator based source using 250GeV drive e- beam with K=1.5 and B=1.76T (u~0.91cm) is about 2.5 per 100m und

ulator

The collimator has a fixed cut angle at 3.85rad/This angle is enough for 150GeV option but still a little bit bigger for 250GeV and results in a smaller polarization

Photon collimating

Since the polarization of photon from hellical undulator has strong correlation between it’s radiation angle and harmonics index. For undulator based positron source, a collimator with proper configuration is required in order to achieve the required polarization. For 150GeV option, the shortest drift between target and undulator is 400m where the hardedge radius of the collimator iris is about 1.5mm.

For 250GeV option, since the yield is higher and thus the undulator is shorter, the drift could also become shorter. But for 250GeV drive beam energy, collimator iris will be smaller and this could makes it more difficult to align the collimator with the beam

Unpolarized e+ source

For unpolarized e+ source, we don’t need collimator and drift of several hundreds meters. (you still get 30% polarization)

Since we are using all the available photons, the e+ yield will be higher. For 150GeV drive beam, the yield is about 2.3 per 100m undulator. For 250GeV drive beam with the same undulator parameters where K=1, u=1cm, the yield is about 6.5 per 100m undulator.

For K ~ 1.5 and B=1.76T, for 250GeV drive beam, the yield will be about 12 per 100m undulator

Yield for unpolarized source at different undulator parameters, 250GeV drive beam

For yield of 1.5, the required length of undulator using K=1, lu=1cm and drive be

am of 250GeV will be only 23m

Yield for unpolarized source at different undulator parameters, 150GeV drive beam

For required yield of 1.5, using 150GeV with K=1,u=1cm, the unpolarized source require

d a length of about 65m

Error Analysis Sensitivity of Yield to beam jitter

Yield and polarization as a function of drive beam transverse position-collimator: location 700m, iris radius 2.5mm

Yield and polarization as a function of drive beam transverse position-collimator: location is 300m, iris radius= 1.155mm

e- Undulator

Summary

We studied the e+ yield in the immersed target case. Including both polarized (60%) and unoplarized cases.

Many undulator and drive beam parameters investigated.

Also studied the drive beam jitter and found it is not a severe problem.

A lot more cases to be done.

Back-up

Summary on numerical modeling of helical undulator photon source

A complete numerical model of helical undualtor photon source is established with correlations between drive beam profile, undulator length, collimator location and iris included.

Using this model, we find that the majority of captured positrons are produced by photons from high order harmonics. 1st harmonic contributed only less than 15% of the yield.

By comparing the yield and polarization of resulted positron beam, we noticed that the optimum yield decreasing more quickly with undulator period u than with K.

For the effect of transverse position jittering of drive electron beam, it depends on the location of collimator. The simulation results shows that for collimator with 5mm iris aperture and located at 700m, the yield change is not significant until the offset of the drive beam reaches 0.6mm. But for collimator with aperture of 2.31mm and located at 300m, the yield is decreasing immediately after the offset of drive beam transverse position is over 0.1mm and the polarization of positron beam goes below 0.6 when the offset is over 0.3mm. But in either case, transverse jittering of drive beam is not a problem as long as the collimator is not too close to the undulator

A possible separation optics

e+ toSC Linac

Momentum selection chicane with collimator

Transverse emittance collimator

Bunch compressionchicane with

collimator

Longitudinal phase space rotation Linac

e+ fromPPA

28m

To be inserted between preaccelator and SC linac. Only positrons within 150 RF phase capturing win

dow will pass through. Having insertion lost

At exit of pre-accelerator At exit of momentum selection chicane

At exit of bunch compressing chicaneAt exit of phase space rotation linacs

PARMELA simulation shows after beam pass through the positron separation optics, there are still some particles staying in subsequent RF buckets. So RF deflector should be use to kick off these undesired particles.

Possible ways to improvement

Recently, Yuri has come up with a scheme to enhance the yield by doing phase space shaping of e+ beam while doing spin rotation before inject beam into damping ring. From his results, the capture efficiency enhanced by 50%. If we combine Yuri’s scheme with the positron separation optics presented here, we can overcome the insertion lost and solve the problems associated with dumping 5GeV e+ beam in his original scheme.

Positron separation optics

Since the energy spread e+ beam is too big at beginning it is very difficult to do separation at the beginning.

We also don’t want to separate e+/e- too late because of the problems associated with dumping high energy e+/e- beam

Open questions

Immersed or non-immersed target: Yield changes ~ 35% Magnetic field distortion and cancellation in immersed target, it is difficult

to solve, but will try to work on it.

Multivariate Optimization

Use Start-to-End Model to maximize positron yield delivered to the Damping Ring– Drive beam energy (ILC Baseline:150 GeV)

– Collimator Angle (Drift Length and Collimator Radius)

– Target Material and Thickness (ILC Baseline: 0.4 rl, Ti)

– AMD profile ( 5T-0.25T in 50cm)

– Accelerating gradient: • 12MV/m in pre-accelerator• 25MV/m in super conducting linac

– Damping ring acceptance:

• x+y<=0.09m.rad.

• E/E < 1%