Optimisation pour la Source SOLEIL

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25 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

102 εμφανίσεις

A. Nadji

1

Rencontres

LAL / SOLEIL ( 17
Avril

2008)


Optimisation

de l’Optique de l’Anneau de Stockage
pour la Source SOLEIL


Amor NADJI, Synchrotron SOLEIL


A. Nadji

2

Rencontres

LAL / SOLEIL ( 17
Avril

2008)



Extensive

use of Insertion Devices such as
Undulators

and Wigglers (highest


ratio of available straight sections to the circumference ).
Variable Polarisation.


Design Criteria for SOLEIL



Tunability
: the right photon energy for the experiment



Upgrade potential



Long

straight sections



Stability
: intensity (Beam Lifetime), position, size and energy



High
Brilliance

and Coherenc
e



Compactness (
Budget
)




Large beam lifetime and injection rate (
Top
-
Up
)

A. Nadji

3



A parameter of prime importance in experiments with
synchrotron radiation


sources is the spectral
brilliance
(brightness) defined as :

High Brilliance



/
d
dt
d
dA
B
dN
ph


w
b
mrad
mm
ond
per
photons
.
%
1
.
0
sec
2
2


Apart from
diffraction effects
, we have :

z
x
d
dA




HIGH
PHOTON BEAM BRILLIANCE


LOW

ELECTRON BEAM EMITTANCE


Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

4

Rencontres

LAL / SOLEIL ( 17
Avril

2008)


Generation of emittance by radiation
Two e
-
with same E and same trajectory at the
entrance of the bending magnet (they cannot be
distinguished : no e- beam size)
They both radiate

E and exit with E-

E
1 emits

E at the origin, smaller


2 emits


E but at the end
They follow different trajectories
Due to the random location (random amplitude has
the same effect) of the emission, at the exit they are
well separated :
the e
-
beam is heated up by
radiation
introducing a sort of noise or dilution.
An e- beam size and divergence are created
(
plus E spread
)
Concept of e- beam emittance in [m*rad] units

associated with a given E


x
m
rad
dxdx
[
*
]
'


1
Note that the phase space dilution is less for a
short bending magnet

E

a
t

d
i
f
f
e
r
e
n
t

l
o
c
a
t
i
o
n
s
d
i
f
f
e
r
e
n
t


E
e- beam horizontal phase space
Size
D
i
v
e
r
g
e
n
c
e
X
dx
ds
'

X
(m
)
dx
s
s
E
E
dx
s
s
E
E
(
)
(
)
(
)
(
)
'
'







(s) and

’(s)
are respectively the dispersion
function and its derivative

A. Nadji

5

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

The heating in the horizontal X plane produced by radiation in every bending ma
gnet
would lead to a continuous blow up of the emittance, size and divergence
however, there is the RF cavity that restores the energy lost per turn and cool
s down
the transverse horizontal oscillations
e
l
e
c
t
r
o
n

c
i
r
c
u
l
a
r

t
r
a
j
e
c
t
o
r
y
E
E
acc


2
E
E
acc


2
RF
Cavity








E
E
E
acc
keV
GeV
m




88
5
4
.

p
//
cavity exit

p
acc
1
1
1
p
dp
dt
E
ET
t
E
E
X
X
acc
acc











(
)
p
after
one
turn
p
cavity
exit
p
p
p
E
E
E
x
x
acc
acc




/
/
/
/


Damping time is of the order of the
time it would take for the particle to
lose all its energy
Memory of initial conditions is
totally lost when the RF has
restored the total energy of the
particle
The two adverse effects (heating+cooling) lead to an equilibrium emittance

X
and gaussian distributions in X and X’
Fortunately there is a cooling effect
A. Nadji

6

Rencontres

LAL / SOLEIL ( 17
Avril

2008)




x
q
dipole
x
C
H
J

2
J
x
is the horizontal damping partition number.

J
x

~
1 (zero field gradient in bending magnet)

J
x

< 2 (vertical focusing in bending magnet) :

(potentially) emittance reduction of a factor two


Equilibrium Beam Emittance



The natural horizontal emittance for an isomagnetic ring, i.e. all bending
magnets having same bending radius is :


H

is the so called lattice invariant or dispersion’s emittance

or
H
-
function

H
s
s
s
s
s
s
s
s
x
x
x
(
)
(
)
(
)
(
)
(
)
'
(
)
(
)
(
)
'










2
2
2


.
.
.
an average taken only in the part of the circumference where photons are


emitted, that is in the bending magnets (
and Insertion Devices
).







x
dipole
x
nm
rad
E
GeV
H
J
.

1470
2

In practical units,

x

is given by :


x

is completely determined by the energy,
bending field and lattice functions.


C
q
= 3.83 x 10
-
13
m and


is the Lorentz factor.




is the bending radius.

A. Nadji

7


















(
)
,
(
)
,
(
)
,
(
)
(
cos
),
(
)
.
'
'
'
s
s
s
s
s
s
s
s
s
sin












0
0
0
2
0
0
0
0
0
0
2
1



0
,

0
,

0
,

0
,


0
,
are the values of the lattice functions
at the beginning

of the
BM




We assume a lattice where

0
=


0

=0 (achromatic condition)
.




We get after integration over one
BM
:

H
B
A
C








0
0
0
2






B
A
C







1
3
2
1
5
2
1
4
3
5
18
2
1
20
4
5
14
2
1
1
1






,
,
.


l
b

l
b
the length of the
BM



the full
BM
deflexion angle

For small bending angles:

(
sufficiently accurate for

most storage rings design

application
)

B
ending
M
agnet

betatron function

(s)

dispersion function



(s)

(

0
,

0
)

(

0
=


0
=0)

,


=s/


s

H
-
function in a Bending Magnet

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

8

Rencontres LAL / SOLEIL ( 17 Avril 2008)






x
q
x
b
b
C
J
l
l











2
3
1
3
0
1
4
0
1
20
0



We get for the natural horizontal e
-

beam emittance :


There
is

clearly

a
cubic

dependence

of the
beam

emittance

on the
deflexion

angle




It is a general lattice property,
there is no assumption on the lattice type
.

Should use
many short
BMs

to get

low emittance
.

Need for a magnet focusing (quadrupoles) providing small waist for


the optical functions


N

2


If the ring consists of N identical
BMs

:


x
N

1
3
A. Nadji

9

Rencontres

LAL / SOLEIL ( 17
Avril

2008)



















b
b
x
q
x
l
l
J
C
0
2
0
1
20
1
0
4
1
0
3
1
0
3
2
0








.
0
10
0
0
4
1
3
2
















b
l
x
q
J
C
.
0
2
0
2
0
1
20
3
1
3
2
0




















b
l
x
q
x
J
C
Minimum Emittance

(
with Achromatic Arc Condition)




In order to get the minimum possible emittance we have to vary the initial conditions



0

and

0

until the minimum is found.


and



We can calculate the unknown initial conditions


0

and

0
:

15
5
12
min
,
0
min
,
0




b
l
The minimum possible emittance is determined only by the
BM

length,

b
l


The minimum equilibrium beam emittance in an isomagnetic ring with an
Achromatic


Arc Condition
,


0
=


0

=0,
at the entrance of the
BM

is:

x
q
x
J
C
15
4
3
2
min
,




A. Nadji

10

Rencontres LAL / SOLEIL ( 17 Avril 2008)

Minimum Emittance (without Achromatic Condition)



By breaking the achromatic condition (non
-
zero dispersion in straight sections)


we can obtain the configuration in which the emittance becomes the smallest.

x
q
x
J
C
15
12
3
2
min
,






It is smaller by a factor
3

than in the achromatic arc configuration.



The optimum values of betatron functions at the entrance of the
BM

are:








2
1
6
1
15
15
8
'
min
,
0
2
min
,
0
min
,
0
min
,
0





b
l


It is interesting to note that in this case, the dispersion and the betatron function are


parabola with the symmetry axis at the middle of the bending magnet :



16
2
1
4
15
0
2
0












x
x

Minimum at center


of the
BM
.




(
)
x
x











2
1
2
24
2
2
x



A. Nadji

11



The ideal value


0
15
,
min

causes the betatron function to reach a sharp minimum


inside the
BM

and then to increase from there on to large values in the quadrupoles, leading
to extremely high
chromaticity
(the quadrupoles do not provide the same focussing strength for particle
with energy deviation).


Ideal values (without achromatic condition):

Design values :




0
0
0
15
3
873
2
17
1
8
,
,
,
.
.
.
.
min
min
x
min
m
nm
rad







0
0
0
1
8
1
5
3
7



.
.
.
.
m
nm
rad
x
Emittance Achieved


x,z

= d

x,z

(d

p

p)


x,z
: betatron tunes

Rencontres LAL / SOLEIL ( 17 Avril 2008)

A. Nadji

12

Rencontres

LAL / SOLEIL ( 17
Avril

2008)




This must not be tolerated for two reasons :

Momentum acceptance

:
some variation of energy deviation has to be
accepted by the storage ring for reasons of
beam lifetime
.


Head tail instability
:
collective oscillation of electrons in head and tail of the bunch

leading to very fast beam loss.

Chromaticity :







(
-

K
Q



+
m
s




) ds


0


quadrupole strength

(introduce negative

)




We must operate with zero or positive chromaticity

sextupole strength

(correct the

)


Strong chromaticity correction
sextupoles reduce the
dynamic
aperture

and this negatively
impacts on the

beam lifetime
.

A. Nadji

13

Rencontres

LAL / SOLEIL ( 17
Avril

2008)



2 main limitations :



Touschek

Effect:
scattering within the bunch

(SOLEIL:

acc

= 4 to 6%)



Elastic Scattering

:
scattering with residual gaz of pression p

Beam Lifetime

Chromatic orbit +

E
Chromatic orbit -

E
Ring axis
Dynamic Aperture

Transverse deflexion and subsequent lossin regions

of low aperture, which are usually the narrow vertical
gaps in undulators (g).

(SOLEIL)

























L
acc
x
z
x
x
acc
l
e
T
ds
s
s
s
s
s
s
C
L
cN
r
0
2
'
2
'
3
2
)
(
)
(
)
(
)
(
)
(
)
(
1
.
8
1
2
1













T
acc


2
1
2
2


vacuum
z
ID
f
p
E
g

(
)
(
)
High density
-
Touschek scattering of particles
-

large longitudinal transfers of energy
-

loss
unless large acceptance : RF acceptance,
physical aperture, dynamic aperture for large
E deviations

A. Nadji

14

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Lattice Design Interface



Magnet Design
: technological limits, coil space,
multipolar

errors




Vacuum
: impedance, pressure, physical apertures, space




Radiofrequency
: Energy acceptance, bunch length, space




Diagnostics
: Beam Position Monitors,…, space




Alignment
: Orbit distortions and correction




Mechanical Engineering
: Girders, vibrations




Design Engineering
: Assembling and feasibility



Space requirements
: Magnet, Vacuum, RF, Diagnostics and

Engineering

A. Nadji

15

Linear Optimisation



Reasonable maximum for and


z
<
30m (sensitivity)




Reasonable beta split at the centre of the achromat



Natural chromaticities :


x

<
-
100

and

z

<
-
50



Dispersion

(

x
)
at the centre of the achromat > 0.25m




Low


z (~1m)
in the centre of undulator straight sections


high brilliance and accomodation of low gap IDs



Minimum Beam Stay Clear for efficient injection


minimum ratio of

(

x)
max
/(

x
)
inj


(

xinj
> 10m)




Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

16

Sextupoles Positions



2 purposes :
-

correction of both chromaticities


x

,

z


-

on momentum and off momentum dynamic aperture optimisation




P
hase optimisation to minimise nonlinear effects



Large number of sextupole families



LOW sextupoles strengths



Positions where


x
<<


z

then


x

>>


z




At least 2 such positions where the


x
is large






Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

17

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Optical functions for the APD lattice




0
5
10
15
20
25
30
0
10
20
30
40
50
60
70
80
s (m)

x

(m)

z

(m)
10*

x
(m)

x
= 18.28


z
= 8.38


x
= 3nm.rad at 2.5 GeV

One Superperiod

A. Nadji

18

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Optical functions for the present lattice




0
5
10
15
20
25
30
0
10
20
30
40
50
60
70
80
s (m)

x

(m)

x
(m)

z
(m)

x
= 18.20


z
= 10.30


x
= 3.73nm.rad at 2.75 GeV

One Superperiod

A. Nadji

19







F
L
Circumfere
nce
Energy
SS
n
n
x













10
5
2
0
2
3
/




Source

Energy
(GeV)




C(m)

S
L
SS

ALS

1.9

0.1745 197

81

BESSYII

1.9

0.1963 240

89

DIAMOND

3

0.1309 562

218.2

ESRF

6

0.09817 844

201.6

ELETTRA

2

0.2618 258

74.78

SLS

2.4

0.2440 288

63

SOLEIL

2.75

0.1963 354

159.6


x0

(nm.rad)

5.6

6.4

2.74

4

7

5

3.7

F

0.48

0.68

2.11

1.73

3.05

6.13

10.66

Example of Factor of Merit


Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

20

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Working point : Tune Diagram

Systematic Resonances


2
nd

order

3
rd

order

4
th

order

5
th

order


z


x

Smaller dynamic aperture

Lowest emittance

Good Compromise between

dynamic aperture and emittance

Larger dynamic aperture

Higher emittance

m nx+n nz =p

A. Nadji

21

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Tune vs Emittance

Resonances

Error sensitivity vs Low emittance

(resistive wall)

Beta functions

Emittance, Brilliance, Injection,

Error sensitivity, Chromaticity,

Symmetry

Sextupoles positions


x




z

Large

x

Chromaticity correction

and Dynamic Aperture Optimisation

Resonances

Momentum Acceptance

OPTIMISATION STRATEGY

A. Nadji

22

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Optimization Strategy



Tune shift w/ amplitude


Tune shift w/ energy


Robustness
to errors

multipoles

coupling

IDs

Lattice design

Fine tuning

Improvement

Needed

Tracking

NAFF

NAFF suggestions

Yes


(x
-
z) fmap


injection eff.



(x
-
d
) fmap


Lifetime


Touschek computation


Resonance identification



4D tracking



6D tracking


Knobs :

quadrupoles

sextupoles

Dynamics analysis

Good Working Point

No

(NAFF: J. Laskar)

A. Nadji

23

Minimizing the
Strengths

of Sextupolar Resonances

2
2
2
2
2
1





L
H
G
F
D
f














i
j
xi
i
x
e
H
xi
D
M
R





2
3
sin
1
2
1









i
j
xi
i
x
e
H
xi
F
M
R





3
2
3
3
sin
1
2
1









i
j
xi
i
x
e
H
xi
G
M
R





2
1
sin
1
2
1











i
j
zi
xi
i
z
x
e
H
zi
xi
H
M
R
)
(
)
2
(
sin
1
2
1
2
2
1



















i
j
zi
xi
i
z
x
e
H
zi
xi
L
M
R
)
(
)
2
(
sin
1
2
1
2
2
1








Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

24

Adjusting the Linear Terms of Tune Shift with Amplitude

2
2
2
2













































z
z
z
x
x
x
f





x
m
n
x
n
x
m
x
n
F
m
D


















3
3
4
3
2
2






z
m
m
r
n
x
m
m
z
x
r
q
z
x
q
g
d
n
G
D
r
v
L
q
H






















)
cos(
2
2
2
1
2
2







x
m
m
r
n
x
m
m
z
x
r
q
z
x
q
z
g
d
n
G
D
r
v
L
q
H























)
cos(
2
2
2
1
2
2





z
q
r
z
x
r
z
x
q
p
x
p
r
v
L
q
v
H
p
G





















2
2
4
1
2
2
2
Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

25

2
5
2
4
2
3
2
2
2
1





L
H
G
F
D
p
p
p
p
p
f





2
8
2
7
2
6













































z
z
z
x
x
x
p
p
p


2
10
2
9
z
x
p
p


Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Simultaneous Minimization

A. Nadji

26

Nonlinear Optimisation


Tune shift with amplitude



Phase space plots



Tune shift with energy



Dynamic aperture (on and off momentum)



Maximum sextupole strengths



Frequency Map Analysis




THE QUALITY FACTORS :



TOOLS :



BETA
-
SOLEIL code



TRACYII code



NAFF

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

27

Horizontal tune shift with amplitude

5

x
= 92

4

x
= 73

3

x
= 55

X(m)


x

(z=0.)

RESONANCES

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

28

On
-
Momentum dynamic aperture

500 turns

X(m)

Z(m)

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

29

Phase Space diagram

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

30

Working point1

Working point 2

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Frequency Map Analysis (NAFF)

Launched particles over a fine X
-
Z grid plotting

Numerical tunes

Highlighting, nonlinearity (diffusion rate
)

Without IDs, Physical aperture included and 1% coupling

A. Nadji

31

Frequency Map Analysis

Off
-
momentum

Working point1

Working point 2

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Without IDs, Physical aperture included and 1% coupling

A. Nadji

32

With 3 x U20 (full gap=5mm) and 1% coupling

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Frequency Map Analysis

Working point1

Working point 2

A. Nadji

33

Frequency Map Analysis

With 3 x U20 (full gap=5mm) and 1% coupling

Off
-
momentum

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

A. Nadji

34

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

High Brilliance


B


I



x


z

Constant Intensity

Impedance

Heat load, vacuum…

High current I

Ring size


Max


Max

Magnet gaps

Budget

Long beam lifetime


Touschek
-

scattering

dominates

Large Energy

Acceptance

Low Emittance



Bend angle









3


Small

,



in dipole centres

Strong focusing

Short cell length

A. Nadji

35

Rencontres

LAL / SOLEIL ( 17
Avril

2008)

Strong focusing

Large Quadrupole Strength K
Q


Chromaticity


= d

(d

p

p)




(
-

K
Q


+ m
s



) ds


0


Large Sextupole Strength m
s


Chaos B
Z



x
2



z
2



B
X



x z

Closed Orbit Distorsion

from Magnet displacements

AC & DC

Steerers

Magnet groups

on girders

Dynamic Aperture

(Acceptance)



Physical Aperture

Lattice Energy

Acceptance



RF Energy

Acceptance

Beam Lifetime

Bump

Injection

RF

A. Nadji

36


SOLEIL Machine Physics group


P. Brunelle

A.
Loulergue

A. Nadji

L. Nadolski

M. A. Tordeux


Rencontres

LAL / SOLEIL ( 17
Avril

2008)