Post on 23-Feb-2022
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
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;
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
Booster
AGS
Aerial view of the NSRL facility where Uniform beam distributions are generated
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Aarhus University March 26 27 2012
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”
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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.
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
In Linear Beam Transport (Dipole Quads) The distribution of the transported beam
remains Normal (next four slides)
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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´
-
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
)~
(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%
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
22s
11
12
s
sslope
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
Linear elements cannot change the Gaussian beam distribution
How can we change a Gaussian beam distribution To a Non-Gaussian one?
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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.
0
01
0 lx
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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.
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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.
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
!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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
To third order with Quadrupoles and Octupoles only
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
Aarhus University March 26 27 2012
)()]})(1([)1(
)2{(2 xy ByBxT
p
exhxhhxyyxx
T
hxxhxh
The s-component
1/=h=0
xxyysKyyxx )(][ 1
To third order with Quadrupoles and Octupoles only
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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?
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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”
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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?
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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”
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
Instrumentation for beam diagnostics. Beam profiles in front of the “Horizontal Octupole”
x
x ́
x
y
Horizontal profile Vertical profile
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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?
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Workshop on Non Linear Expander
Aarhus University March 26 27 2012
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
Courtesy of Adam Rusek BNL
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
Courtesy of Adam Rusek 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?