Uniform Beam Distributions of Charged Particle Beams

Uniform Beam Distributions of Charged Particle Beams

Workshop on Non-linear beam expander systems in high-power accelerator facilities

ISA, Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

26th and 27th March 2012

N. Tsoupas Brookhaven National Laboratory

f(x,y)

x

y

Workshop on Non Linear Expander

Aarhus University March 26 27 2012

http://fuau.dk/main.aspx?pagetype=5&id=7

Conclusion: In a linear beam transport line (Dipole, Quadrupoles) ;

Target

Entrance Gaussian Distribution

Quads Quads Quads

Beam Line

By introducing two Octupoles;

Oct1 Oct2

We can transform the Gaussian distribution at the Target;

Into a a beam with Uniform distribution;




Booster

AGS

Aerial view of the NSRL facility where Uniform beam distributions are generated




R-Line: From Booster to NSRL Facility where Uniform beam distribution are generated

Uniform Beam at NSRL Target

Gaussian Beam at Start of R-Line

Octupoles

Photos Courtesy Adam Rusek Workshop on Non Linear Expander



Measured 2Dimentional Beam profile at the location of the NSRL target

4”

Measured beam profiles: Vertical (Green line) and Horizontal (Red Line) at the location of the NSRL target

Experimental Results of Uniform beam Distributions at the target location of the NSRL facility at BNL

Courtesy Adam Rusek BNL

8”




Measured Horizontal and Vertical Beam Profiles

at the location of the NSRL target NSRL Facility at BNL

“Bad” Harp Wire

with Octupoles ON

with Octupole OFF




Objective of the presentation

• Present an example of the method.

x

y

f(x,y)

• Present a method to transform a beam with a Gaussian distribution into beam with a Uniform distribution.




An Isometric view of a theoretical Gaussian beam distribution in the transverse (x,y) plane.

The vertical-axis corresponds to the density distribution function.

To a good approximation the particle distribution of a beam generated by an accelerator, is Gaussian in any of its six dimensions (x,x ,́y,y ,́𝓁, p)

x

y

f(x,y)

Beam direction

Projection of the six dimensional distribution on a plane will generate a 2D Normal distribution




In Linear Beam Transport (Dipole Quads) The distribution of the transported beam

remains Normal (next four slides)




A 2-D Gaussian (“Normal”) Distribution function expressed in terms of the “s-matrix”.

)~

(2

1

2

2

))det()det(

2

)det((

2

1

2

2

1

21112

222

)det()2(

1

)det()2(

1),(

XXxxxx

eexxf

s

s

s

s

s

s

s

ss

2211

1212TRANSPORTin used as )(n Correlatio

12

ss

s rr

12

22

)x(x,in n correlatio

xin rms)deviation( standard

in x rms)deviation( standard

s

s

s

xxX ~

x

xX

2212

1211

ss

sss

x

11s

11

12

s

sslope

(x,x´)

22s

x´

-




)~

(2

1

2

2

))det()det(

2

)det((

2

1

2

2

1

21112

222

)det()2(

1

)det()2(

1),(

XXxxxx

eexxf

s

s

s

s

s

s

s

ss

A 2-D Gaussian (“Normal”) Distribution function expressed in terms of the “s-matrix”.

x

11s

x´

Particles inside ellipse 39.4%

Particles inside ellipse 39.4%

Particles projected on x-axis 68.2%



22s

11

12

s

sslope

http://upload.wikimedia.org/wikipedia/commons/8/8c/Standard_deviation_diagram.svg


A charge distribution in 6D can be expressed in terms of a 6-D s-matrix

δplyyxxX ~

X => Phase space vector of a particle in beam

p

l

y

y

x

x

X

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

ssssss

ssssss

ssssss

ssssss

ssssss

ssssss

s

)~

(2

1

2

6

1

)det()2(

1),,,,,(

XX

eplyyxxf

s

s

jiij ss

654321 :Notation theuse Also xxxxxxdpdlyyxx




Linear Transformation of a beam with Gaussian distribution

Out

Out

Out

Out

Out

Out

Out

x

x

x

x

x

x

X

6

5

4

3

2

1

)(

In

In

In

In

In

In

In

x

x

x

x

x

x

X

6

5

4

3

2

1

)()()( InOut RXX

The R Matrix transforms the coordinates X(In) of a particle of the distribution in the CS at Pin

to the coordinates X(Out) in the CS in Pout

)()( In

jij

Out

i xRx

SSRRT

010000

101000

010100

001010

000101

000010

ST

inout RRss insouts

SRS , RR T-- 111

1. Under Linear Transformation R the beam distribution remains Gaussian

2. Transformation of a Gaussian is equivalent to Transformation of the smatrix

Quadrupoles Transport the particle distribution from

CoSys at Pin to CoSys at Pout Beam

Pin matrixin s

)(

1

Inx

)(

2

InxPout

)(

1

Outx

)(

2

Outx

matrixout s




Linear elements cannot change the Gaussian beam distribution

How can we change a Gaussian beam distribution To a Non-Gaussian one?




Effect of an octupole on a Gaussian beam

1. Beam is distributed Normally in angle q as it emanates from point O.

l

B 3. Inserting an octupole at a point A, the x

and θ are related by x ≈ ℓθ+kθ3 The distribution along the x, (at z=l) is

modified to that of the yellow curve.

2. At z= ℓ , x ≈ ℓθ the beam is distributed Normally along the x direction at a distance z=l , (red curve).

2

2

2

2

1)( qs

q

qsq

eP

x

x ́

Important constraint: At the location of the Octupole

there is perfect (x,q) correlation.

qq

0

01

0 lx




Effect of an octupole on a Gaussian beam

l

An octupole can alter the distribution of a one dimensional Gaussian beam to a beam with more Uniform distribution.

Can this simple example be modified and applied to a six dimensional realistic beam which has non-zero emittance?

2

2

2

2

1)( qs

q

qsq

eP

Important constraints: a) At the location of the Octupole

there is perfect (x,q) correlation. b) Beam is “flat”

x

θ

x

y

x




Questions to be answered for generating a uniform rectangular beam in two-dimensions

using “realistic” beam and octupoles.

3

3

032

3

03

3

0 )3()sin(),( yR

Byyx

R

Bnr

R

BrBx qq

3

3

023

3

03

3

0 )3()cos(),( xR

Bxyx

R

Bnr

R

BrBy qq For σy σx

x

x ́

x

y

x

y

y

y ́

Will the finite emittance of the beam provide the high correlation in phase space (x,x )́ and (y,y )́ required at the location of the octupoles, and also keep the beam within the available aperture?

Will the finite dimensions of the “flat beam” at the location of the octupoles be affected by the aberrations introduced by the octupoles, thus affecting the beam uniformity at the target?

For σx σy

0

0

A”flat” beam minimizes coupling.




How do the equations of motion of a charged particle moving in the magnetic field of dipoles, quadrupoles, and octupoles, look like?

Do I need a computer code to generate a uniform beam distribution starting from a beam with normal distribution?

Is there a closed form solution of the exact (or approximate) equations of motion?

Mathematics to Calculate the beam distribution at the target




Expression of the equation of motion in the curvilinear coordinate system

))1(()]})(1([)1({2 ys BhxByT

p

exhxhhxyyxx

T

xhxhx

The x-component

)()]})(1([)1(

)2{(2 xy ByBxT

p

exhxhhxyyxx

T

hxxhxh

The s-component

2/1222222222 ])()(21[)1()()( )1()()()( yxhxhxyxThxyxT

])1[()]})(1([{2 sx BxBhxT

p

exhxhhxyyxx

T

yy

The y-component

These equations are exact

19

)(sr

)( Bvp

edt

d

1h

K. L. Brown SLAC-PUP 3381




311

02

3

3

0

2120100

))(

)()(),0,0(

(

))()(

()),0,0()(

(),0,0(

ys

sKhsK

e

ph

s

sK

s

sB

e

p

yxs

sKh

s

sK

e

pxy

s

sBh

s

sK

e

py

s

sB

e

pB

y

yy

s

!2)()(

),0,0(),0,0(2

)(

2)(

),0,0(

!2!3)(

22)()(),0,0(

2

1

2

22

2

2

1

2

0

2

12

2

0

23

30

22

20

10

xysKhshK

s

sBh

s

sBh

s

sK

e

p

yshK

s

sB

e

p

xyxsK

e

pyxsK

e

pxsK

e

psBB

yy

y

yy

Fields expansion to 3nd order in coordinates: Bx, By, Bs

!3

])(),0,0(

2),0,0(

[!3

)()(

!32)()()(

3

2

1

2

2

2

0

3

21

20

32

30

20

10

y

s

sK

s

sBh

s

sBh

e

pyshKsKh

e

p

yyxsK

e

pxysK

e

pysK

e

pB

yy

x

Dipole Quadrupole Sextupole Octupole

20

The MAD Program CERN/SL/92-?? (AP) B2 expression Eq. 2.3 page 3




If we assume that the Gaussian beam starts at the exit of the last dipole of the line, we may eliminate the dipole field in the equations of motion and

use quadrupoles and octupoles only.

In an attempt to simplify the equations of motion we start eliminating multipoles.




!2

)()

)((

!2!3)()(}{

2

2

1

2

1

23

31

2 xy

s

sKyxy

s

sKxyxsKxsKyyxxxx

Even if we drop the terms of the fringe field…..

The equations of motion are Simplified !!!! to:

The x-component

!2!3)()(}{

23

31

2 xyxsKxsKyyxxxx

To third order with Quadrupoles and Octupoles only

!2!3)()(

23

31

xyxsKxsKx This approximation Not correct!!!

K3(s) cannot be treated as a perturbation

If we assume 0




xxys

sKy

s

sKyyxsKyyxysKysKyyyxyxy

)

)((

!3

)(

!32)())((

2

1)( 1

3

2

1

232

3

22

11

Even if we drop the terms of the fringe field…..

!32)())((

2

1)(

32

3

22

11

yyxsKyyxysKysKyyyxyxy


The y-component

!32)())((

2

1)(

32

3

22

11

yyxsKyyxysKysKy

assume 0

This approximation is Not correct!!! K3(s) cannot be treated as a perturbation Workshop on Non Linear Expander



)()]})(1([)1(

)2{(2 xy ByBxT

p

exhxhhxyyxx

T

hxxhxh

The s-component

1/=h=0

xxyysKyyxx )(][ 1





How can one solve these equations?

!32)())((

2

1)(

32

3

22

11

yyxsKyyxysKysKyyyxyxy

!2!3)()(}{

23

31

2 xyxsKxsKyyxxxx

xxyysKyyxx )(][ 1

x-comp

y-comp

s-comp

Quadrupoles and Octupoles only; No fringe field; Correct to 3rd order.

Can I solve these approximate equations in a closed form to calculate the beam distribution at the target?



Answer: NO

I solve the “exact equations”; But Numerically using a computer

Computer codes used: TRANSPORT , or RAYTRACE


Procedure to generate a uniform beam distribution



I use the TRANSPORT or RAYTRACE computer code to calculate the beam distribution at the target


Procedure to Generate uniform beam distribution at target. 1st Order Optics

• First Order Optics:

– Should satisfy the required beam constraints at the location of the Octupoles. • High correlation in (x,x’) and “flat beam” in y-direction at the location of one Octupole

• High correlation in (y,y’) and “flat beam” in x-direction at the location of other Octupole

– Beam size at the target should be adjusted by the first order optics.

Beam distribution at the entrance of beam line is assumed to be Gaussian

x

x ́

x

y

x

y

y

y ́

Target Entrance O2 O1 Q3 Q2 Q1 Q6 Q5 Q4

Beam Line Workshop on Non Linear Expander



Procedure to Generate uniform beam distribution at target. 3rd Order Optics

• Third order Optics:

– With Octupoles ON, Calculate the first order (Rij ) , second order (Tijk ) , and third order (Wijkl ) , aberrations coefficients of the beam line.

Beam Line

- Use the transformation:

(1)

of entrance beam coordinates x(I) to the exit beam coordinates x(O)

)()()()()()()( I

l

I

k

I

jijkl

I

k

I

jijk

I

jij

O

i xxxWxxTxRx

– Perform Monte Carlo calculations using eq. (1) to calculate the particle distribution at the target.

Quadrupoles Octupoles Transport particles from CoSy at Pi to CoSy at Po X

I

i

)(

x

x ́

Beam

Pi

XO

i

)(

x

x ́

Po

Reminder: The “sampling” beam distribution at the entrance is Gaussian:

x

x ́

x

y

y

y ́

x

y

Entrance

Target

1st Order Beam Optics of an “Example Beam Line” to create Uniform beam distribution




Horizontal (Black/Red circles) and Vertical (Blue/Green squares) Beam Profiles with Octupoles OFF( Black/Blue) and Octupoles ON (Red/Green) for the “Example Beam Line”




Same method of generating uniform beams but Alternative Mathematical approach

Calculations of nonlinear envelopes in beam expanders

F. Meot Physical Review Special Topics 3 10351 (2000)

Comments

Circulating beam

Extracted beam

Location of foil at the entrance of extraction septum.

It is very important that the beam distribution at the beginning of the line is “Gaussian”. How do we achieve a beam with Gaussian distribution from a slowly extracted beam at the entrance of NSRL?




The beam emittance has an effect on the “Sharpness” of the beam distribution

Beam profile at the location of the target x

x ́

x

y

Beam with Low emittance

If (x,x) highly correlated x is small, then the kick from the octupole will not spread the beam at the target. “Sharp fall”

Beam at the location of the octupole

Comments

x

x ́

x

y

Beam with High emittance

(x,x) less correlated x is large. The kick from the octupole will spread the beam at the target. “Not Sharp fall”




Instrumentation for beam diagnostics. Beam profiles in front of the “Horizontal Octupole”

x

x ́

x

y

Horizontal profile Vertical profile




Study the Effect of the vacuum windows, on the beam. How does it affect the beam’s uniformity and the “sharp fall off” of the beam?



Monte Carlo simulation of particle beam flattening Using a dual-scattering-foil technique

E.Detsi deqsolver.com/download/montecarloSim.pdf


Effect of linear coupling of the beam on the beam uniformity

x

x ́

x

y

x

x ́

x

y

Beam with No coupling Linearly Coupled beam




Effect of Higher order coupling of the beam on the beam uniformity

2D beam at target with strong octupoles It is Possible to identify the octupole aberrations from the distribution at the target

Under Normal operating conditions the octupoles introduce “minimal coupling” to affect the beam uniformity and sharpness of the “beam’s fall off”

2D beam profile at target with the required strength of octupoles

Courtesy Adam Rusek BNL




The D6 Septum Start of slides

Where is the D6 Septum Located?

What is its function ?

Isometric view of D6 Septum magnet

Isometric Views of D6-Septum

Magnet

Cross section of the Septum Magnet

at Beam Entrance (looking

downstream). 2D Modeling

Cross section of the Septum Magnet at Beam

Entrance (looking downstream). Picture by Ed Hoey

Cross section of the Septum Magnet at Beam Exit

(looking upstream). Picture by Ed Hoey

The D6 Septum End of slides

Isometric 2D beam distribution at the NSRL Target

Courtesy of Adam Rusek BNL

)3()cos(),( 23

3

03

3

0 xyxR

Bnr

R

BrBy qq )3()sin(),( 32

3

03

3

0 yyxR

Bnr

R

BrBx qq

2D distribution at the NSRL Target with High strength Octupoles


Measured Projected Vertical (Green line) and Horizontal (Red Line) Beam Profiles at the location of the NSRL target

Measured 2Dimentional Beam profile at the location of the NSRL target

4”

8”

Experimental Results of Uniform beam Distributions at the target location of the NSRL facility at BNL


12” beam pipe

8” beam pipe

R-Line: From Booster to NSRL

Do I need a computer code to generate a uniform distribution starting from a beam with normal distribution?

Is there a solution in closed form? even approximate?

Uniform Beam Distributions of Charged Particle Beams

Documents

Transcript of Uniform Beam Distributions of Charged Particle Beams