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Beam Transfer Lines
• Distinctions between transfer lines and circular machines
• Linking machines together
• Trajectory correction
• Emittance and mismatch measurement
• Blow-up from steering errors, optics mismatch and thin screens
• Phase-plane exchange
Brennan Goddard (presented by Malika Meddahi)
CERN
Injection, extraction and transfer
•
An accelerator has limited
dynamic range
•
Chain of stages needed to
reach high energy
•
Periodic re-filling of
storage rings, like LHC
•
External experiments, like
CNGS
Transfer lines transport the
beam between accelerators,
and onto targets, dumps,
instruments etc.
LHC:
SPS:
AD:
ISOLDE:
PSB:
PS:
LINAC:
LEIR:
CNGS:
Large Hadron Collider
Super Proton Synchrotron
Antiproton Decelerator
Isotope Separator Online Device
Proton Synchrotron Booster
Proton Synchrotron
LINear Accelerator
Low Energy Ring
CERN Neutrino to Gran Sasso
CERN Complex
Normalised phase space
• Transform real transverse coordinates x, x’ by
X 
x
1
   N   
S
 x'
 X' 
X
X' 
1
S
1
S
1

 S
x
  S x   S x'
0  x
 

 S   x'
Normalised phase space
Real phase space
x’
x'max   

1

 



1
Normalised phase space
X'
X' max  

x

X
Area = p
Area = p
xmax   
    x 2  2  x  x'   x'2
X max  
  X 2  X' 2
General transport
Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi
y
s
y
s
2
x
1
x
 x2 
 x1   C S   x1 
 x '   M12   x'   C ' S '   x' 
  1
 2
 1 
Twiss
parameterisation
M12
n
M12   Mn

 2 cos    sin  
1

1

 1
1 2 1   2  cos   1  1 2 sin  

i 1



1 cos    sin  
2
2

1 2 sin 
Circular Machine
Circumference = L
One turn
M12  M 0 L
 sin 2pQ
 cos 2pQ   sin 2pQ

  1

2
1


sin
2
p
Q
cos
2
p
Q


sin
2
p
Q
 



• The solution is periodic
• Periodicity condition for one turn (closed ring) imposes 1=2, 1=2, D1=D2
• This condition uniquely determines (s), (s), (s), D(s) around the whole ring
Circular Machine
• Periodicity of the structure leads to regular motion
– Map single particle coordinates on each turn at any location
– Describes an ellipse in phase space, defined by one set of  and 
values  Matched Ellipse (for this location)
x’
x'max  a
a    x 2  2  x  x'   x'2
1  2

x
Area = pa
xmax  a 

1  2

Circular Machine
• For a location with matched ellipse (, , an injected beam of
emittance , characterised by a different ellipse (*, *) generates
(via filamentation) a large ellipse with the original , , but larger 
x’
, 
Turn 1
Turn 2
x
,  , 
After filamentation
Turn 3
  , , 
Turn n>>1
Matched ellipse
determines beam shape
Transfer line
One pass:
 x2 
 x1 
 x '   M12   x ' 
 2
 1
 x2 
 x' 
 2
 x1 
x' 
 1
M12

 2 cos    sin  
1

1

 1
1 2 1   2  cos   1  1 2 sin  




1 cos    sin  
2
2

1 2 sin 
• No periodic condition exists
• The Twiss parameters are simply propagated from beginning to end of line
• At any point in line, (s) (s) are functions of 1 1
Transfer line
• On a single pass there is no regular motion
– Map single particle coordinates at entrance and exit.
– Infinite number of equally valid possible starting ellipses for single particle
……transported to infinite number of final ellipses…
x’
1, 1
 x1 
x' 
 1
1, 1
x
Entry
M0L
x’
 x2 
 x' 
 2
2, 2
x
Transfer Line
Exit
2, 2
Transfer Line
• Initial ,  defined for transfer line by beam shape at entrance
, 
x’
x’
, 
x
x
Gaussian beam
Non-Gaussian beam
(e.g. slow extracted)
• Propagation of this beam ellipse depends on line elements
• A transfer line optics is different for different input beams
Transfer Line
• The optics functions in the line depend on the initial values
Horizontal optics
350
300
- Design x functions in a transfer line
 x functions with different initial conditions
BetaX [m]
250
200
150
100
50
0
1500
2000
2500
S [m]
• Same considerations are true for Dispersion function:
– Dispersion in ring defined by periodic solution  ring elements
– Dispersion in line defined by initial D and D’ and line elements
3000
Transfer Line
• Another difference….unlike a circular ring, a change of an element
in a line affects only the downstream Twiss values (including
dispersion)
Horizontal optics
350
300
- Unperturbed x functions in a transfer line
 x functions with modification of one quadrupole strength
BetaX [m]
250
10% change in
this QF strength
200
150
100
50
0
1500
2000
2500
S [m]
3000
Linking Machines
• Beams have to be transported from extraction of one machine to
injection of next machine
– Trajectories must be matched, ideally in all 6 geometric degrees of freedom
(x,y,z,q,f,y)
• Other important constraints can include
– Minimum bend radius, maximum quadrupole gradient, magnet aperture,
cost, geology
Linking Machines
Extraction
Transfer
1x, 1x , 1y, 1y
s
x(s), x(s) , y(s), y(s)
2x, 2x , 2y, 2y
Injection
The Twiss parameters can be propagated
when the transfer matrix M is known
 x2 
 x1   C S   x1 
 x '   M12   x '   C ' S '   x ' 
  1
 2
 1 
2
 2CS
S 2   1 
 2   C
    CC ' CS ' SC '  SS '   
  1
 2 
  2   C '2
 2C ' S '
S '2    1 
Linking Machines
• Linking the optics is a complicated process
– Parameters at start of line have to be propagated to matched parameters
at the end of the line
– Need to “match” 8 variables (x x Dx D’x and y y Dy D’y)
– Maximum  and D values are imposed by magnet apertures
– Other constraints can exist
• phase conditions for collimators,
• insertions for special equipment like stripping foils
– Need to use a number of independently powered (“matching”)
quadrupoles
– Matching with computer codes and relying on mixture of theory,
experience, intuition, trial and error, …
Linking Machines
• For long transfer lines we can simplify the problem by designing the
line in separate sections
– Regular central section – e.g. FODO or doublet, with quads at regular
spacing, (+ bending dipoles), with magnets powered in series
 [m]
– Initial and final matching sections – independently powered quadrupoles,
with sometimes irregular spacing.
300
BETX
BETY
250
200
150
100
50
0
0
250
SPS
500
Initial
matching
section
750
1000
1250
1500
1750
2000
2250
2500
Regular lattice (FODO)
(elements all powered in series
with same strengths)
2750
3000
Final
matching
section
3250
LHC
SPS to LHC Transfer Line (3 km)
Extraction
point
Injection
point
3500
3750
4000
S [m ]
Trajectory correction
• Magnet misalignments, field and powering errors cause the
trajectory to deviate from the design
• Use small independently powered dipole magnets (correctors) to
steer the beam
• Measure the response using monitors (pick-ups) downstream of the
corrector (p/2, 3p/2, …)
Pickup
Corrector dipole
QF
Trajectory
QF
QD
QD
p/2
• Horizontal and vertical elements are separated
• H-correctors and pick-ups located at F-quadrupoles (large x )
• V-correctors and pick-ups located at D-quadrupoles (large y)
Trajectory correction
• Global correction can be used which attempts to minimise the RMS
offsets at the BPMs, using all or some of the available corrector
magnets.
• Steering in matching sections, extraction and injection region
requires particular care
– D and  functions can be large  bigger beam size
– Often very limited in aperture
– Injection offsets can be detrimental for performance
Trajectory correction
Uncorrected trajectory.
y growing as a result
of random errors in the
line.
The RMS at the BPMs
is 3.4 mm, and ymax is
12.0mm
Corrected trajectory.
The RMS at the BPMs
is 0.3mm and ymax is
1mm
Trajectory correction
• Sensitivity to BPM errors is an important issue
– If the BPM phase sampling is poor, the loss of a few key BPMs can
allow a very bad trajectory, while all the monitor readings are ~zero
Correction with some
monitors disabled
With poor BPM phase
sampling the correction
algorithm produces a
trajectory with 185mm
ymax
Note the change of vertical scale
Steering (dipole) errors
• Precise delivery of the beam is important.
– To avoid injection oscillations and emittance growth in rings
– For stability on secondary particle production targets
• Convenient to express injection error in s (includes x and x’ errors)
a [s] = ((X2+X’2)/)  x2  2xx’+ x’2/
a
X'
Septum
X
Bumper
magnets
kicker
Mis-steered
injected beam
Steering (dipole) errors
• Static effects (e.g. from errors in alignment, field, calibration, …) are
dealt with by trajectory correction (steering).
• But there are also dynamic effects, from:
– Power supply ripples
– Temperature variations
– Non-trapezoidal kicker waveforms
• These dynamic effects produce a variable injection offset which can
vary from batch to batch, or even within a batch.
• An injection damper system is used to minimise effect on emittance
Blow-up from steering error
• Consider a collection of particles with amplitudes A
• The beam can be injected with a error in angle and position.
• For an injection error ay (in units of sigma = ) the mis-injected
beam is offset in normalised phase space by L = ay
Matched
particles
X'
Misinjected
beam
A
X
L
Blow-up from steering error
• The new particle coordinates in normalised phase space are
X new  X 0  Lcosq
'
X new
 X '0  Lsin q
Matched
particles
• For a general particle distribution,
where A denotes amplitude in
normalised phase space
A 2  X 2  X '2
X'
Misinjected
beam
A
q
L
  A2 / 2
X
Blow-up from steering error
• So if we plug in the new coordinates….

 
2
2
'2
A new
 X new
 X new
 X 0  Lcosq  X '0  Lsin q
2


 X 02  X '02  2L X 0cosq  X '0sin q  L2


2
A new
 X 02  X '02  2L X 0cosq  X '0sin q  L2

0
0

 2 0  2L cosq X 0  sin q X '0  L2
 2 0  L2
• Giving for the emittance increase
 new  A new 2 / 2   0  L2 / 2

  0 1  a 2 / 2


2
Blow-up from steering error
A numerical example….
Consider an offset a of 0.5 sigma for
injected beam
X'
Misinjected beam
 new   0 1  a 2 / 2
 1.125 0

0.5
X
Matched
Beam
Blow-up from betatron mismatch
• Optical errors occur in transfer line and ring, such that the beam can be
injected with a mismatch.
• Filamentation will produce an
emittance increase.
• In normalised phase space, consider
the matched beam as a circle, and the
mismatched beam as an ellipse.
X'
Mismatched
beam
X
Matched
beam
Blow-up from betatron mismatch
General betatron motion
x 2   2  2 sin(f  f o ),
x' 2   2  2 cos(f  f o )   2 sin(f  f o )
applying the normalising transformation for the matched beam
X2 


X
'
 2
1

 1  1
1
0   x2 

 1   x' 2 
an ellipse is obtained in normalised phase space
A 
2


X 22  1
2


 2
1
2

1  

 1   2
   X' 22 2  2X 2 X' 2
2  
1


2

  1

 
 1   2 1 
 2 

characterised by new, new and new, where
 new
 2 
1 

,

1  2

1 
2 
 new

 2,
1
 new
1  2 
1 




1  2

 2 1 
2 
2
Blow-up from betatron mismatch
From the general ellipse properties
a
A
2


H 1  H 1 ,
b
A
2

H 1  H 1

Mismatched
beam
X'
where
1
H   new   new 
2
2



1  1  2
1
 2 
 

  1   2

2  2 1 
 2  1 


a
X
A
giving
1

2


H 1  H 1 ,
X new    A sin( f  f1 ),
1
1


2

X' new 
b
H  1  H 1
1


A cos(f  f1 )
generally
aA 
b  A 
Matched
Beam
Blow-up from betatron mismatch
We can evaluate the square of the distance of a particle from the origin as
2
2
A 2new  X new
 X' new
 2  A 02 sin2 (f  f1 ) 
1

2
A 02 cos 2 (f  f1 )
The new emittance is the average over all phases
 new 
1
1
1
2
A new
   2 A 02 sin 2 (f  f1 )  2 A 02 cos 2 (f  f1 )
2
2

0.5
0.5
1
 2
  sin 2 (f  f1 )  2 cos 2 (f  f1 )



1
A 02
2

1  2 1
 0   2
2 










If we’re feeling diligent, we can substitute back for  to give
2

1  2 1
1  1  2 
1   2 
 new   0    2   H 0   0
 1   2  
2 
 
2   2 1 
 2  1 


where subscript 1 refers to matched ellipse, 2 to mismatched ellipse.
Blow-up from betatron mismatch
A numerical example….consider b = 3a for the mismatched ellipse
  b/a  3
X'
Mismatched
beam
Then
 new   0 2  1 2 
1
2
 1.67 0
a
b=3a
o
X
Matched
Beam
Emittance and mismatch measurement
• A profile monitor is needed to measure the beam size
– e.g. beam screen (luminescent) provides 2D density profile of the beam
• Profile fit gives transverse beam sizes s.
• In a ring,  is ‘known’ so  can be calculated from a single screen
Emittance and mismatch measurement
• Emittance measurement in a line needs 3 profile measurements in a
dispersion-free region
• Measurements of s0,s1,s2, plus the two transfer matrices M01 and
M12 allows determination of ,  and 
s0
s1
M12
s2
M23
s 02 s 12 s 22



 0 1  2
s0
s1
s2
Emittance and mismatch measurement
 2C1 S 1
S 12    0 
  1   C12
  
  



C
C
'
C
S
'

S
C
'

S
S
'

   0 
1
1
1
1
1
1
1
1
1
 
  1   C1 ' 2
 2C1 ' S 1 '
S 1 ' 2    0 


We have
where
so that
Using
we find
where
C1
C '
 1

1 cos    sin  
0
S1  
0

S1'   1
11  0  1  cos   1   01 sin  

1   ,
1  C  0  2C1S1 0 
S12
s 02
0 
,


 0,


2
1
s
 1   1
s0
0
2
2
0



 0 cos    sin  
1
1

 0 1 sin 
 2  C  0  2C2 S 2 0 
s
 2   2
s0
2
2
2

 0


1
2
 0  0W

s 2 / s 0 2 / S 22  s 1 / s 0 2 / S12  C2 / S 2 2  C1 / S1 2
W
C1 / S1   C2 / S 2 
S 22
0
1   
2
0
Emittance and mismatch measurement
Some algebra with above equations gives
0  1
s 2 / s 0 2 / S22  C2 / S2 2  WC2 / S2 2  W 2 / 4
And finally we are in a position to evaluate  and 0

 s 02  0
1
2
 0  0W
Comparing measured o, 0 with expected values gives
measurement of mismatch
Blow-up from thin scatterer
• Scattering elements are sometimes required in the beam
– Thin beam screens (Al2O3,Ti) used to generate profiles.
– Metal windows also used to separate vacuum of transfer lines from
vacuum in circular machines.
– Foils are used to strip electrons to change charge state
• The emittance of the beam increases when it passes through, due
to multiple Coulomb scattering.
qs
rms angle increase:
14.1
L 
L 


q [mrad ] 
Z inc
1  0.11 log 10

 c p[ MeV / c]
Lrad 
Lrad 
2
s
c = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length
Blow-up from thin scatterer
Each particles gets a random angle change
qs but there is no effect on the positions at
the scatterer
X'
Ellipse after
scattering
X new  X 0
'
X new
 X '0   q s
X
After filamentation the particles have
different amplitudes and the beam has
a larger emittance
2
  A new
/2
Matched
ellipse
Blow-up from thin scatterer
2
2
'2
A new
 X new
 X new

 X 02  X '0   q s

X'

2

 X 02  X '02  2  X '0q s  q s2
Ellipse after
filamentation
uncorrelated
2
A new
 X 02  X '02  2  X '0q s   q s2
X
0
 2 0  2  X '0 q s   q s2
 2 0   q s2
 new   0 

2
Matched
ellipse
q s2
Need to keep  small to minimise blow-up (small  means large spread in
angles in beam distribution, so additional angle has small effect on distn.)
Blow-up from charge stripping foil
• For LHC heavy ions, Pb53+ is stripped to Pb82+ at 4.25GeV/u using a
0.8mm thick Al foil, in the PS to SPS line
 is minimised with low- insertion (xy ~5 m) in the transfer line
•
• Emittance increase expected is about 8%
TT10 optics
120
beta X
Stripping foil
beta Y
100
Beta [m]
80
60
40
20
0
0
50
100
150
S [m]
200
250
300
Emittance exchange insertion
• Acceptances of circular accelerators tend to be larger in horizontal
plane (bending dipole gap height small as possible)
• Several multiturn extraction process produce beams which have
emittances which are larger in the vertical plane  larger losses
• We can overcome this by exchanging the H and V phase planes
(emittance exchange)
Low energy machine
After multi-turn
extraction
After emittance
exchange
High energy machine
y
x
In the following, remember that the matrix is our friend…
Emittance exchange
Phase-plane exchange requires a transformation of the form:
 x1   0

 
 x'1   0
 y   m
 1   31
 y'   m
 1   41
0
m13
0
m 23
m 32
0
m42
0
m14  x 0 


m 24  x' 0 
0  y 0 




0  y' 0 
A skew quadrupole is a normal quadrupole rotated by an angle q.
The transfer matrix S obtained by a rotation of the normal transfer matrix Mq:
S = R-1MqR
where R is the rotation matrix
 cos q

 0
  sinq

 0

0
cos q
sinq
0
0
cos q
 sinq
0
0 

sinq 
0 

cos q 
(you can convince yourself of what R does by checking that x0 is transformed to
x1 = x0cosq y0sinq, y0 into -x0sinq y0cosq, etc.)
Emittance exchange
1


M q  
0

0

For a thin-lens approximation
So that
 cos q

0
S  R 1M q R  
sinq

 0

0
 sinq
cos q
0
0
cos q
sinq
0
0
1
0
0
0
1
0

0
0

1 
0 
 1

 sinq  
0  0

cos q  0
1


  cos 2q

0

  sin 2q

0
(where  = kl = 1/f is the
quadrupole strength)
0
0
1
0
0
1
0 
0
0
1
 sin 2q
0
1
0   cos 2q
0  cos q

0  0
0   sinq

1  0
0
sinq
cos q
0
0
cos q
 sinq
0
0

0
0

1 
45º skew quad
For the case of q = 45º,
this reduces to
1

0
S
0



0 0 0

1  0
0 1 0

0 0 1 
0 

sinq 
0 

cos q 
Normal quad
Emittance exchange
The transformation required can be achieved with 3 such skew quads in a lattice,
of strengths 1, 2, 3, with transfer matrices S1, S2, S3
A
B
Skew quad
Skew quad
1
Skew quad
2
3
The transfer matrix without the skew quads is C = B A .

 x
C  
0 0

0 0
C
Cx
0 0

0 0

y 
C

 x2

cos f x   x1 sin f x 
 x1


     cos f  1     sin f
x1
x
x1 x 2
x
 x2

 x1  x 2







cos f x   x 2 sin f x 
 x2


 x1  x 2 sin f x
 x1
and similar for Cy
Emittance exchange
With the skew quads the overall matrix is M = S3B S2A S1
c11  b12 a 34 1 2
c12



 c 21  b22 a 34 1 2
 c 22  b34 a12 2  3


   3 c 34 1  b34 a11 2 
M

c 34 1  b34 a11 2
a12 b34 2


  c  b a    
  3 11 12 34 1 2  c   b a 
12 3
44 12 2
 c  b a  
22 34 1 2 
  44 1
c12 1  b12 a 33 2
b12 a 34 2



c 22 1  b22 a 33 2
 b22 a 34 2  c 34 3 







c

b
a



34 12 1 2 
 3 33


c 33  b34 a12 1 2
c 34



 3 c12 1  b12 a 33 2 

 c 44  b12 a 34 2  3 
 c 43  b44 a12 1 2 

Equating the terms with our target matrix form
 0

 0
m
 31
m
 41
0
m13
0
m 23
m 32
0
m 42
0
m14 

m 24 
0 

0 
a list of conditions result which must be met for phase-plane exchange.
Emittance exchange
0  c12
0  c 34
0  c11  b12 a 34 1 2
0  c 22  b34 a12 2  3
0  c 33  b34 a12 1 2
0  c 44  b12 a 34 2  3
0  c 21  b22 a 34 1 2   3 c 34 1  b34 a11 2 
0  c 43  b44 a12 1 2   3 c12 1  b12 a 33 2 
The simplest conditions are c12 = c34 = 0.
Looking back at the matrix C, this means that fx and fy need to be integer
multiples of p (i.e. the phase advance from first to last skew quad should be 180º,
360º, …)
We also have for the strength of the skew quads
 1 2  
c
c11
  33
b12 a 34
b34 a12
 2 3  
c 22
c
  44
b34 a12
b12 a 34
Emittance exchange
Several solutions exist which give M the target form.
One of the simplest is obtained by setting all the skew quadrupole strengths the
same, and putting the skew quads at symmetric locations in a 90º FODO
lattice
A
B (=A)



From symmetry A = B, and the values of  and  at all skew quads are identical.
Therefore A x  B x
 cos f x   x sin f x 


1   x2 sin f x

 
x



 x sin f x




cos f x   x sin f x 

The matrix C is similar, but with phase advances of 2f
with the same form for y
Emittance exchange
Since we have chose a 90º FODO phase advance, fx = fy = p/2, and
2fx = 2fy = p which means we can now write down A,B and C:
 x


 1   x2

x

A B

0




0




x
0
 x
0
0
y
0
1   

2
y
y
0 



0 


y 



 y 


0
0
1 0


0
 0 1 0
C
0
0 1 0 


 0

0
0

1


we can then write down the skew lens strength as
For the 90º FODO with half-cell length L,
 F   D 
i.e. 180º across the
insertion in both planes
1   2   3   s 
2
,
L
s 
1
L 2
1
x y
Summary
• Transfer lines present interesting challenges and differences from
circular machines
– No periodic condition mean optics is defined by transfer line element
strengths and by initial beam ellipse
– Matching at the extremes is subject to many constraints
– Trajectory correction is rather simple compared to circular machine
– Emittance blow-up is an important consideration, and arises from
several sources
– Phase-plane rotation is sometimes required - skew quads
Keywords for related topics
• Transfer lines
– Achromat bends
– Algorithms for optics matching
– The effect of alignment and gradient errors on the trajectory and optics
– Trajectory correction algorithms
– SVD trajectory analysis
– Kick-response optics measurement techniques in transfer lines
– Optics measurements including dispersion and p/p with >3 screens
– Different phase-plane exchange insertion solutions