Transcript K x - Agenda INFN

Accelerator Laboratory
OPTICS BASICS
S. Guiducci
Bibliography
1. S. Y. Lee, Accelerator Physics, 2nd Ed., (World Scientific,
2004)
2. CERN Accelerator School, 5th General Accelerator
Physics Course, CERN 94-01, 1994
http://cdsweb.cern.ch/record/235242/files/full_document_V1.pdf
3. K. Wille, The Physics of Particle Accelerators – an
introduction, translated by J. McFall, (Oxford University
Press, 2000)
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Optics Basics
2
Magnetic components of a storage ring
DIPOLE
QUADRUPOLE
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SEXTUPOLE
Optics Basics
3
Components of a storage ring
 Magnets with different characteristics are used to keep confined a beam
of charged particles in a storage ring:
 Dipoles: to guide the beam along a circular trajectory and to correct
deviations from the ideal orbit
 Quadrupoles: to focuse the beam around the reference orbit and achieve
small beam sizes at some positions
 Sextupoles, octupoles, etc: magnets with non linear fields used to correct
unwanted effects (chromaticity, etc…)
 Wigglers and undulators: magnets with many poles with alternating
polarity used to achieve synchrotron light beams with various wavelengths
in the synchrotron light sources storage rings
 Charged particles bent on a circular trajectory in dipoles loose energy
for synchrotron radiation. A Radio Frequency (RF) cavity, with a
longitudinal electromagnetic field varying at high frequency, is used to
restore the particle energy
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Optics Basics
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Components of a storage ring 2
 The beam travels in a vacuum chamber where a very low pressure is




achieved by means of different pumping systems in order to minimize
the interactions with the particles of the residual gas
A cooling system is necessary for the magnets and RF
A series of diagnostic systems is used to monitor the beam
characteristics (current, beam position monitors, beam size monitor,
luminosity monitor,…) and the accelerator performance
To inject the beam special pulsed magnets are used
Collimators and masks are used to intercept the large amplitude
particles and avoid damage of the accelerator sytems and of the
detector’s components and performance
 A control system is managing the operation of the accelerator
 An injector system is used to produce, accelerate and transport
the beam inside the accelerator
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Optics Basics
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Schematic layout of DAFNE accelerator complex
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Optics Basics
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The DAFNE main rings with KLOE-2
ECM = 1020 MeV
Crab-Waist collision scheme
implemented for the first time
Max Luminosity achieved at DAFNE 4 1032 cm-2s-1 is by far the
Susanna Guiduccihighest achieved at this energy Optics Basics
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The DAFNE accumulator ring
e- extraction
Dipoles
Quadrupoles
e-
e-
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e- injection
e+ rotate in the
opposite direction
Optics Basics
8
Lorentz force
F = q éëE + v ´ Bùû
• The magnetic force FB = qvB is perpendicular to the particle
velocity and bends the trajectory with a radius of curvature
r
• The centrifugal force Fc =
mv 2 balances the magnetic force:
r
mv 2
mv p
qvB =
Þr =
=
r
qB qB
For relativistic particles the strength of a 1 Tesla
bending magnet is equivalent to an electric field of
da 3x108 V/m (far beyond technical limits)
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Optics Basics
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Dipole magnets
 Magnets with 2 poles separated by a gap
 The dipole field is uniform and perpendicular to the orbit
plane
 The particle is bent by an angle q with a radius of
curvature r
 Given the length L and the field B the angle is:
q
æ q ö L 1 LB
sin ç ÷ =
=
è 2 ø 2 r 2 ( Br )
 For small q : q=LB/(Br)
 Br = p/e is the magnetic rigidity
of a particle with charge e
B
q
2
[T·m] = 3.3356·p [GeV/c]
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L
2
L
2
Optics Basics
q
2
r
10
10
“C” and “H” Dipole magnets
• Room temperature electromagnetic
dipole have a maximum field of ~2 Tesla
• To increase the field Super Conducting
(SC) dipoles are needed as the LHC
dipoles with B = 8 T
• Future colliders aim at SC dipoles with
very high fields: B = 16 T or more
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Optics Basics
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11
Quadrupole focusing
• The quadrupole is a magnet with 4 poles. The field is zero in the center
and varies linearly both in the horizontal and vertical direction
• The quadrupole has a focusing effect, similar to a lens
• Depending on the field sign it is focusing in the horizontal, called QF, or
in the vertical plane called QD, and defocusing in the other direction
S
Defocusing in
vertical
N
N
S
“QF”
“QD”
Horizontal
oscillation
Focusing in horizontal
Vertical
oscillation
Doublet
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Optics Basics
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Quadrupole magnet
Field
lines
• 4 poles with hiperbolic
contour
• Poles are symmetric with
respect to x and y axes
• The field is zero at the
center and varies linearly
both in the x and y
direction
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Hyperbolic
pole shape
Optics Basics
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13
Quadrupole field
Magnetic
field
Force on the
particles
• On the X (horizontal) axis the field is
vertical:
By = G
x
• On the Y (vertical) axis the field is
horizontal:
Bx = G y
• The gradient G is defined as:
dBy
dx
[Tm-1 ]
• The ‘normalized’ gradient K is:
G
[m-2 ]
( Br )
• The focal length is:
K=
QF: Focusing in x,
defocusing in y
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f=
1
[m]
KLq
Optics Basics
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14
campo
magnetic
o
Quadrupole kick
• The focal length is:
f=
1
[m]
KLq
• The angular “kick” given to
the particles is linear
Dx ' = -Klx
Focusing from quadrupole
Dy' = Kly
Polo di forma
iperbolica
x · y = costante
Δx
x
s
f
x
=
l
=l
qBy
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=l
qB'
x
1 qB'l
=
= kl
Optics Basics
f g mv
15
15
Chromatic
effects
Aberrazione cromatica di un quadrupolo
 A quadrupole acts as a focusing lens with focal length:
• Un quadrupolo focheggia un fascio1 di particelle
Br
p 1 cariche come una lente focheggia
f 


un fascio di luce
KLq GLq e GLq
• Analogamente alle aberrazioni cromatiche delle lenti ottiche, anche qui abbiamo
lo The
focal
length
depends
on the particle
momentum
stesso
effetto:
l’effetto
focheggiante
o defocheggiante
di un quadrupolo dipende
dall’energia
Since thedella
beam
has
particella
 an energy spread, the high energy particles will
• Inbe
un under-focused
quadrupolo focheggiante
più energetica
è piu’
rigida p=q (Br),
and the una
low particella
energy particles
will be
over-focused
è deviata
meno
(chromaticity)
E > Eo
energia
superiore
quadrupolo
energia
inferiore
E = Eo
f(E-DE)
f(E+DE)
E < Eo
sestupole
magnets
• Questo effetto siSolution:
cura con i introduce
sestupoli magnetici,
che
introducono campi
magnetici non-lineari (forza quadratica~x2).
• Esistono poi ottupoli
magnetici (forza cubica ~ x3) che, insieme
a magnetici di 1616
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Optics Basics
Sextupole magnets
 6 poles with hyperbolic shape
 The field is zero at the center and varies quadratically with the
transverse coordinate:
Bx = m xy
1
By = m ( x 2 - y 2 )
2
 Normalized gradient:
 Kick:
2
1 d By
-3
m=
[m
]
2
Br dx
Dx ' = -mx 2, Dy' = 2mxy
 A sextupole is like a quadrupole with a gradient proportional to the
transverse diplacement
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Optics Basics
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17
Magnetic Field Multipole Expansion
Magnetic elements with 2-dimensional fields of the form
B  Bx  x, y  xˆ  By  x, y  yˆ
can be expanded in a complex multipole expansion:

By ( x, y )  iBx ( x, y )  B0   bn  ian  x  iy 
n
n 0
n

By
1
with bn 
n ! B0 x n
 x , y   0,0 
1  n Bx
and an 
n ! B0 x n
 x , y   0,0 
In this form, we can normalize to the main guide field strength, Bŷ, by setting b0=1 to yield:
1
e
By  iBx    By  iBx  

Br
p0
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1

 bn  ian  x  iy  for  q
n
r n 0
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18
Multipole Moments
Upright Fields
Skew Fields
Dipole (q  90°):
Dipole:
e
Bx  0
p0
e
1
By  
p0
r
e
1
By 
p0
r
Quadrupole (q  45°):
Quadrupole:
e
Bx  ky
p0
e
By  kx
p0

e
Bx  kskew x
p0
e
By  kskew y
p0
Sextupole (q  30°):
Sextupole:
e
Bx  mxy
p0
e
By  0
p0
e
1
By  m  x 2  y 2 
p0
2
Octupole:
e
1
Bx  r  3x 2 y  y 3 
p0
6
e
1
By  r  x3  3xy 2 
p0
6
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e
1
Bx   mskew  x 2  y 2 
p0
2
e
By  mskew xy
p0
Octupole (q  22.5°):
e
1
Bx   rskew  x 3  3xy 2 
p0
6
e
1
By  rskew  3x 2 y  y 3 
p0
6
Optics Basics
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19
Coordinate system
y
x
• For circular machines, it is convenient to convert
z
to a curvilinear coordinate system (Frenet-Serret)
Curverlinear
coordina
and change the independent variable from
time
s
to the longitudinal abscissa “s”, which is the
•magnets
Coordinateand
system to describe particle m
rr
reference orbit given by the bending
• Moves with the particle
is moving with the beam
yˆ
0
xˆ

• The local radius of curvature is denoted by r
The unit vectors sˆ, xˆ, yˆ
are the basis for the
coordinate system

dr0 (s)
ŝ(s) =
ds
dŝ(s)
x̂(s) = - r
ds
ŷ(s) = x̂(s) ´ ŝ(s)
r (s)
r0 (s)
sˆ
20
Motion in a circular accelerator
• x and y are the betatron coordinates representing small
amplitude motion around the reference orbit
• In each plane (x,s) and (y,s) the motion of a particle in a
transverse magnetic field is described by two variables:
– Position x(s), displacement perpendicular to the reference orbit
– Angle x’(s)= dx/ds with respect to the reference orbit
x’
x
x
s
ds
dx
• The motion is similar to an harmonic oscillator
21
Equations of motion
• Particle motion in electromagnetic fields is governed by the
Lorentz force:
dp
=e E+v´B
dt
(
r
r
E  F  A t
)
r
r
B  A
F = scalar potential
A = vector potential
• with the corresponding
Hamiltonian in Cartesian coordinates:


2ù
é 2 2
H = c êm c + P - eA ú + eF
ë
û
(
x=
Susanna Guiducci
)
1/2
¶H
¶H
, Px = ,...
¶Px
¶x
Optics Basics
22
Hamiltonian in the curvilinear coordinate system
• Using a canonical transformation we get a new Hamiltonian in the
reference orbit coordinate system (x, s, y)
12

2
 x  H  eF2
2
2 2
  eAs
H˜  1 

m
c

p

eA

x
x   py  eAy
2
 r 
c




• Because the reference orbit is a closed curve the new Hamiltonian on s is
periodic
• Using the relations:
E = H - eF ,
E2
p = 2 - m2c 2
c
• and expanding to 2nd order in px and py yields:
æ x ö 1+ x r é
2ù
2
H » - p ç1+ ÷ +
px - eAx - p y - eAy ú - eAs
ê
û
2p ë
è rø
(
Susanna Guiducci
) (
)
Optics Basics
23
Equations of motion (2)
• In the absence of synchrotron motion, we can
generate the equations of motion with:
x¢ =
¶H
,
¶px
p¢x = -
Bx  
¶H
,
¶x
1 As
1 x r y
y¢ =
¶H
,
¶p y
By 
p¢y = -
¶H
¶y
1 As
1 x r x
• Which yields (top/bottom sign for +/- charge):

By p0 æ x ö
r+x
x¢¢ - 2 = ±
ç1+ ÷
Br p è r ø
r
2
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y 
Bx p0 
x
1



Br p  r 
Optics Basics
2
24
Equations of motion (Hill’s Equation)
• We next want to consider the equations of motion for a ring with
only guide (dipole) and focusing (quadrupole) elements:
1
x  K x  s  x  0,
Kx  s 
y  K y  s  y  0,
K y  s   k  s 
r s
2
k s
also commonly
denoted as k1
• Kx and Ky are periodic functions of s
• The period length Lp is the circumference or a fraction of it, in case
the lattice, i.e. the layout of dipole and quadrupoles, has a periodic
structure
• A horizontal bending dipole has Kx = 1/r2 and Ky =0
• In a quadrupole 1/r = 0 and Kx = - Ky is the focusing strength
• A horizontally focusing quadrupole is vertically defocusing
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Optics Basics
25
General Solution to Hill’s Equation
• The general solution to Hill’s equation can now be written as:
x  s   A b x  s  cos  x  s   0  where  x  s   
s
0
with bx(s) a periodic function of s:
ds
bx  s 
bx(s+LP) = bx(s)
• The linear betatron motion is like an harmonic oscillation with
amplitude and phase varying along the ring as a function of s
• We can now define the betatron tune for a ring as:
Fturn
1 s C ds
Qx   x 

where C  ring circumference

2
2 s b x  s 
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Optics Basics
26
Betatron
oscillations
Betatron oscillation
• The periodic function b(s) describes the envelope of the
betatron oscillations that the particles perform with respect to
•the
Beta
function orbit
:
b x (s)given
reference
by guide field of the dipoles
– Describes the envelope of the betatron oscillation in an accelerator
s 1
Phase
advance: are in both
• •The
oscillations
x and y
y (s) = ò 0 planes,
ds
b x (s)
• The number of betatron oscillations
per turn, “betatron tune”,
•orBetatron
tune: number
of betatron
oscillations
in one orbital turn
“phase advance”,
is an
important
ring parameter
y (0 | C)
Qx =
=
2p
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ò
ds
R With C ring circumference
/2p =
R ring radius
b x (s)
áb x ñ
Optics Basics
27
Comments about the solutions to Hill’s equations
– The solutions to Hill’s equation describe the particle motion around a
reference orbit, the closed orbit. This motion is known as betatron
motion. We are generally interested in small amplitude motions around
the closed orbit
– Accelerators are generally designed with discrete components which have
locally uniform magnetic fields and the focusing functions, K(s), can be
represented in a piecewise constant manner
– This allows us to locally solve for the characteristics of the motion and
implement the solution in terms of a transfer matrix M
– For each segment for which we have a solution, we can then take a
particle’s initial conditions at the entrance to the segment and transform it
to the final conditions at the exit
 x   m11 m12   x0 
     m m   x 
 x   21
22  0 
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Optics Basics
28
Transfer Matrices
We now write the solutions of the Hill’s equations in transfer
matrix form:

 cos k


  k sin k

 1 
M  s s0   

0
1




 cosh k


 k sinh k

 
 

 s  s0 .
k
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1 B
Br x
Focusing
Quadrupole
Drift
Region


where
 k  

cos  k  

1
sin
k
1
sinh
k

cosh


k
k

 




Defocusing
Quadrupole
All Matrices have Det = 1
Optics Basics
29
Transfer Matrices
All Matrices have Det = 1
Examples:
– Quadrupole in thin lens approximation:
 0,
M focusing
 1

 1 f
0

1
1
f = lim
0 K
Mdefocusing
 1

1 f
0

1
– Sector dipole (entrance and exit faces ┴ to closed orbit):
c.o.
M sector
 cos q
 1
  sin q
 r

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r sin q   1

cos q    2
  r


1 

where q 
Optics Basics
r
30
Twiss Parameters
The generalized one turn matrix can be written as:
b sin F
 cos F  a sin F

M 
  I cos F  J sin F
cos F  a sin F 
 g sin F
Identity matrix
This is the most general form of the matrix. a, b, and g are known as either
the Courant-Snyder or Twiss parameters (note: they have nothing to do with
the familiar relativistic parameters) and F is the betatron phase advance.
The matrix J has the properties:
a
J 
 g
b 
,
a 
J 2  I  bg  1  a 2
The n-turn matrix can be expressed as:
Mn  I cos  nF   J sin  nF 
which leads to the stability requirement for betatron motion:
Trace  M   2 cos F  2
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Optics Basics
31
Lattice
Exampleofofmagnets:
a lattice:
the
FODO
• Arrangement
structure
of beam
line
cell
– Bending
dipoles,
Quadrupoles,
Steering
dipoles, Driftand
space
• Lattice
is the
sequence
of dipole,
quadrupoles
and Other insertion elements
other magnets which constitutes the accelerator
• Example:
• The FODO cell is a series of focusing and defocusing
– FODO cell: alternating arrangement between focusing and
quadrupoles
defocusing quadrupoles
f
-f
L
L
One FODO cell
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Optics Basics
32
Thin Lens FODO Cell
In thin lens approximation the matrix of a sequence QF-Drift-QDDrift-QF (FODO cell) in the horizontal plane can be written as
(we start from the center of QF, then its focal length is half and
the sign is opposite with respect to QD):
2

L
L 
2L(1 
)
 1 2
 1 01 L 1 01 L 1 0
2f
2f



1


1


1

M 





2
10 1 
10 1 
1



L
L
L
 2f

f

 2f

 2 (1  2 f ) 1  2 
 2f
2f

For the vertical plane change f in –f
The total effect is focusing
in both planes

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Optics Basics
33
Exercise on FODO cell
Comparing the FODO cell matrix with the matrix of a periodic structure evaluate the
betatron phase advance  and the Twiss functions b and a at the center of QF and
QD
Since the cell is periodic we have a1 = a2 = aF and
b1 = b2 = bF and the transport matrix is
 cos
 sin 

 bF
bF sin  

cos 

1
L2
cos   Trace(M) 1 2
2
2f

 L
2
cos =1- 2sin ; sin 
2
2 2f
2f 2f L
2L(1 sin(  2)
bF 
bF 
2f L
sin 


2L(1 sin(  2)
bD 
sin 
Susanna Guiducci
or
aF = aD = 0

bD 
2f 2f L
2f L
Optics Basics
34
bF and bD vs phase advance for FODO cell
30.00
For =180º
25.00
betM
betm
20.00
15.00
L=1.5 m
bF
bD0
10.00
5.00
The motion
is unstable
bD
0.00
0.000
bF
20.000 40.000 60.000 80.000 100.000 120.000 140.000 160.000 180.000
The minimum
f = L/2 = 0.75m
Susanna Guiducci
Optics Basics
35
An example of ring
based on FODO cell
12 FODO cells
Qx = 3.15, Qz = 3.15
Single FODO cell
x= y= 94.5º
Susanna Guiducci
Optics Basics
36
Thank you for the attention