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Emittance Diagnostics ¾Emittance, what is it ?
¾Emittance, why measuring it ?
¾How to measure it ?
¾What can go wrong ?
CAS Beam diagnostic
Dourdan, 2.6.2008
Hans Braun / CERN
Emittance, what is it ?
particle angle x’
particle position x
ε = Area in x, x’ plane occupied by beam particles
divided by π
Beam envelope along beamline
x
w
z
converging
beam
beam waist
x’
x’
x’
x
x
2w
Along a beamline the orientation and aspect ratio of beam ellipse
in x, x’ plane varies, but area πε remains constant
Beam width along z is described with w( z ) =
β ( z) ε
diverging
beam
x
Beam ellipse and its orientation is described by 4 parameters ε , β ,α , γ
called Courant - Snyder or Twiss parameters
ε = γ x 2 + 2 α x x' + β x' 2
the three ellipse orientation parameters β ,α , γ are connected by the relation
γ =
1+ α 2
β
therefore only of these parameters are independen d
α
βε
is the beam half width
γε
is the beam half divergence
describes how strong x and x' are correlated.
for α > 0 beam is converging ,
for α < 0 beam is diverging.
for α = 0 beam size has minimum (waist) or maximum (anti - waist)
Transport of single particle described with matrix algebra
drift
quadrupole
drift
x
⎛ x1 ⎞
⎜⎜ ⎟⎟
⎝ x1 ' ⎠
1
z
2
LA
⎛ x2 ⎞
⎜⎜ ⎟⎟ = M
⎝ x2 ' ⎠
LC
LB
(
(
⎛ x ⎞ ⎛ 1 LC ⎞ ⎛
⎛x ⎞
cos k LB
⎟⎟ ⋅ ⎜⎜
⋅ ⎜⎜ 1 ⎟⎟ = M C ⋅ M B ⋅ M A ⋅ ⎜⎜ 1 ⎟⎟ = ⎜⎜
⎝ x1 ' ⎠ ⎝ 0 1 ⎠ ⎝ − k sin k LB
⎝ x1 ' ⎠
⎛1 L⎞
⎟⎟
M Drift = ⎜⎜
⎝0 1 ⎠
⎛ x2 ⎞
⎜⎜ ⎟⎟
⎝ x2 ' ⎠
( )
( )
⎛
cos k L
M Quadrupole = ⎜⎜
⎝ − k sin k L
generic names of matrix elements
1
( )
( )
)
)
k sin k L ⎞
⎟
cos k L ⎟⎠
⎛c s⎞
⎟⎟
M = ⎜⎜
c
'
s
'
⎝
⎠
1
(
(
)
)
k sin k LB ⎞⎟ ⎛ 1 L A ⎞ ⎛ x1 ⎞
⎟⋅⎜ ⎟
⋅⎜
cos k LB ⎟⎠ ⎜⎝ 0 1 ⎟⎠ ⎜⎝ x1 ' ⎟⎠
Transport of Twiss parameters β, α, γ of beam ellipse
drift
drift
quadrupole
⎛ β2 ⎞
⎜α ⎟
⎜ 2⎟
⎜γ ⎟
⎝ 2⎠
x
⎛ β1 ⎞
⎜α ⎟
⎜ 1⎟
⎜γ ⎟
⎝ 1⎠
1
z
2
LA
LC
LB
⎛ x 2 ⎞ ⎛ c s ⎞ ⎛ x1 ⎞
⎜⎜ ⎟⎟ = ⋅⎜⎜
⎟⎟ ⋅ ⎜⎜ ⎟⎟
x
'
⎝ 2 ⎠ ⎝ c' s ' ⎠ ⎝ x1 ' ⎠
⇒
β 2 = c 2 β1 − 2 c s α 1 + s 2 γ 1
w22 = c 2 β1 ε − 2 c s α 1 ε + s 2 γ 1 ε
⎛ β2 ⎞
⎜α ⎟
⎜ 2⎟=
⎜γ ⎟
⎝ 2⎠
| ⋅ε
⎛ c2
s 2 ⎞ ⎛ β1 ⎞
− 2c s
⎜
⎟ ⎜ ⎟
⎜ − c c' c s '+c' s − s s ' ⎟ ⋅ ⎜ α 1 ⎟
⎜ c' 2
2 ⎟ ⎜
γ 1 ⎟⎠
c
s
s
2
'
'
'
−
⎝
⎝
⎠
So far we implicitly assumed that beam occupies
x, x’ space area with a sharp boundary
particle density in x, x’ space
transverse beam profile
surface πε
projection on x axis
∞
f ( x) =
∫ ρ ( x, x' ) dx'
−∞
2w
beamwidth
w = βε
In reality beam density in x, x’ space is rarely a constant
on a well defined area but often like this
particle density in x, x’ space
transverse beam profile
surface πε ?
projection on x axis
∞
f ( x) =
∫ ρ ( x, x' ) dx'
−∞
beamwidth ?
Definition of wRMS
+∞
wRMS =
∫ (x − x
COG
) 2 f ( x) dx
−∞
+∞
∫ f ( x) dx
Commonly used definitions of emittance
−∞
+∞
εrms , 1 sigma r.m.s. emittance
ε2rms , 2 sigma r.m.s. emittance
⇔
wRMS = βε
⇔ wRMS =
xCOG =
βε
∫x
f ( x) dx
−∞
+∞
∫ f ( x) dx
−∞
2
Definition of w90
ε90% ,
90% emittance
⇔ w90 =
βε
+W90
∫ f ( x) dx
−W90
+∞
∫ f ( x) dx
−∞
= 90%
Relations between different emittance definitions
Always
For Gauss
beam distribution only
εrms , 68.3% of beam
ε2rms = 4 εrms
ε90% = 2.7055 εrms
wrms = σX
w90% =1.6449 σX
ε90% , 90% of beam
ε2rms , 95.5% of beam
Slice emittance
important for free electron lasers !
x
slice A
slice B
z
x’
x’
εΑ
x
εΒ
x
Emittance is only constant in beamlines without acceleration
x’
Px
ε1
x1’
x
P1
acceleration
ε 2 = ε1
P1
P2
P = mβ rel γ rel c
with
x’
ε2
Px
x2’
x
P2
normalised emittance
ε N = β rel γ rel ε
To distinguish from normalised emittance εN
,ε
preserved with acceleration !
is quoted as “geometric emittance” !
Commonly used units for emittance
mm·mrad, m·rad, μm, m, nm
1 mm·mrad=10-6 m·rad=1μm=10-6 m=103nm
Often a π is added to the unit to indicate that the numerical value describes
a surface in x, x’ space divided by π, i.e. 1 π·mm·mrad
The units for normalised emittance are the same as for geometric emittance
Due to the various definitions it is recommended to always mention
the emittance definition used when reporting measurement values !
Emittance, why measuring it ?
The emittance tells if a beam fits in the vacuum chamber or not
w( z) = β ( z) ε
< a(z )
LHC IR region
aperture a(z)
10 × β y (z ) ε y
Emittance is one of key parameters for overall performance of an accelerator
o Luminosity of colliders for particle physics
o Brightness of synchrotron radiation sources
o Wavelength range of free electron lasers
o Resolution of fixed target experiments
Therefore emittance measurement is essential to guide tune-up of accelerator !
Measurement of emittance is closely linked
with measurement of Twiss parameter β, α, γ .
o Provide initial beam conditions for a beamline, thus allowing to
compute optimum setting of focusing magnets in a beamline.
Important to minimise beam losses in transfer line !
o Measurement of Twiss parameters at injection in synchrotron or
storage ring allows to “match” Twiss parameters to circulating beam.
This helps to minimise injection losses and to avoid emittance growth
due to filamentation of unmatched beam.
o Measurement of emittance as function of time in synchrotrons and storage
rings allows to identify, understand and possibly mitigate mechanism of
emittance growth.
o Measurement of emittance at different locations in a beam line or in a linear
accelerator allows to verify and/or improve beam optics model of this line.
Moreover, it allows to identify, understand and possibly mitigate mechanism
of emittance growth.
Emittance, how to measure it ?
¾
Methods based on transverse beam profile measurements
¾
Slit and pepperpot methods
¾
Methods based on Schottky signal analysis
⇒ F. Caspers Lecture, June 3rd
¾
Direct methods to measure transverse profiles and beam divergence
⇒ see literature list
Emittance measurement in transfer line or linac
Twiss parameter β,α,γ are a priori not known, they have to be
determined together with ε.
Method A
wC
x
wA
wB
La
Reference point where
β, α, γ will be determined
⇒
2
Lc
profile monitors
see Enrico Bravin’s lecture
w = c β ε − 2 c s α ε + s γ ε , for drift
2
z
Lb
2
w 2 = β ε − 2 L α ε + L2 γ ε
⎛c
⎜⎜
⎝ c'
s ⎞ ⎛1 L⎞
⎟=⎜
⎟
s' ⎟⎠ ⎜⎝ 0 1 ⎟⎠
Derivation of emittance and Twiss parameters from beam width measurements
w A2 = β ε − 2 L A α ε + L2A γ ε
wB2 = β ε − 2 LB α ε + L2B γ ε
wC2 = β ε − 2 LC α ε + L2C γ ε
⇓ can be rewritten in Matrix notation
⎛ w A2 ⎞ ⎛1 − 2 L A
⎜ 2⎟ ⎜
⎜ w B ⎟ = ⎜1 − 2 L B
⎜ w 2 ⎟ ⎜1 − 2 L
C
⎝ C⎠ ⎝
⎛1 − 2 L A
L2A ⎞ ⎛ βε ⎞
⎟
⎜
⎜
⎟
2
LB ⎟ ⋅ ⎜ αε ⎟ ⇒ ⎜1 − 2 LB
⎜1 − 2 L
L2C ⎟⎠ ⎜⎝ γε ⎟⎠
C
⎝
βε ⋅ γε − (αε ) 2 = ε 2 (β ⋅ γ − α 2 ) = ε 2 ⇒
−1
L2A ⎞ ⎛ w A2 ⎞ ⎛ βε ⎞
⎟ ⎜ 2⎟ ⎜ ⎟
2
LB ⎟ ⋅ ⎜ wB ⎟ = ⎜ αε ⎟
L2C ⎟⎠ ⎜⎝ wC2 ⎟⎠ ⎜⎝ γε ⎟⎠
βε ⋅ γε − (αε ) 2 = ε , β =
βε
αε
, α=
ε
ε
Emittance measurements often require inversion of 3x3 Matrices.
Many Math and spreadsheet programs and programming languages
have functions for this.
If you insist to code it yourself you can use :
The inverse of a 3x3 matrix:
| a11 a12 a13 |-1
| a21 a22 a23 |
| a31 a32 a33 |
with DET
=
=
| a33a22-a32a23
1/DET * | a31a23-a33a21
| a32a21-a31a22
a32a13-a33a12
a33a11-a31a13
a32a11-a31a12
a23a12-a22a13 |
a21a13-a23a11 |
a22a11-a21a12 |
a11(a33a22-a32a23)-a21(a33a12-a32a13)+a31(a23a12-a22a13)
Emittance measurement in transfer line or linac, Method B
Adjustable magnetic lens with settings A,B,C
(quadrupole magnet, solenoid, system of quadrupole magnets…)
wA, wB, wC
x
z
Reference point where
β, α, γ will be determined
w = c β ε − 2 c s α ε + s γ ε,
2
2
2
profile monitor
⎛c
⎜⎜
⎝ c'
s ⎞ ⎛ 1 L ⎞ ⎛ m11 ( I mag ) m12 ( I mag ) ⎞
⎟
⎟⎟ = ⎜⎜
⎟⎟ ⋅ ⎜⎜
⎟
(
)
(
)
m
I
m
I
0
1
s' ⎠ ⎝
22
mag ⎠
⎠ ⎝ 21 mag
Derivation of emittance and Twiss parameters from beam width measurements
w A2 = c A2 β ε − 2 c A s A α ε + s A2 γ ε
wB2 = c B2 β ε − 2 c B s B α ε + s B2 γ ε
wC2 = cC2 β ε − 2 cC sC α ε + sC2 γ ε
⇓ can be rewritten in Matrix notation
⎛ w A2 ⎞ ⎛ c A2
⎜ 2⎟ ⎜ 2
⎜ wB ⎟ = ⎜ c B
⎜ w2 ⎟ ⎜ c 2
⎝ C⎠ ⎝ C
− 2c A s A
− 2c B s B
− 2c C s C
⎛ c A2
s A2 ⎞ ⎛ βε ⎞
⎟ ⎜ ⎟
⎜ 2
2
s B ⎟ ⋅ ⎜ αε ⎟ ⇒ ⎜ c B
⎜c2
sC2 ⎟⎠ ⎜⎝ γε ⎟⎠
⎝ C
βε ⋅ γε − (αε ) 2 = ε 2 (β ⋅ γ − α 2 ) = ε 2 ⇒
− 2c A s A
− 2c B s B
− 2cC sC
−1
s A2 ⎞ ⎛ w A2 ⎞ ⎛ βε ⎞
⎟ ⎜ 2⎟ ⎜ ⎟
2
s B ⎟ ⋅ ⎜ wB ⎟ = ⎜ αε ⎟
sC2 ⎟⎠ ⎜⎝ wC2 ⎟⎠ ⎜⎝ γε ⎟⎠
βε ⋅ γε − (αε ) 2 = ε , β =
βε
αε
, α=
ε
ε
To determine ε,
β, α
at a reference point in a beamline one needs
at least three w measurements with different transfer matrices between the
reference point and the w measurements location.
Different transfer matrices can be achieved with different profile monitor locations,
different focusing magnet settings or combinations of both.
Once β, α at one reference point is determined the values of β,
in the beamline can be calculated.
α at every point
Three w measurements are in principle enough to determine ε, β, α
In practice better results are obtained with more measurements.
However, with more than three measurements the problem is over-determined.
χ2 formalism gives the best estimate of ε,
for a set of n measurements wi ,
i=1-n
β, α
with transfer matrix elements ci, si .
χ2 formalism to determine ε,
β, α
Measured half beam width wi
Predicted half beam width
These conditions can be rewritten in matrix notation
ci2 βε − 2ci siαε + si2 γε ,
⎛ n 4
⎜ ∑ ci
⎜ i =1
⎜ n 3
⎜ ∑ ci s i
⎜ in=1
⎜ c2s2
⎜∑ i i
⎝ i =1
but βε , αε , γε a priori not known
Find set of βε , αε , γε values, which minimises
n
(
χ 2 = ∑ ci2 βε − 2ci siαε + si2 γε − wi 2
i =1
)
2
Conditions for χ 2 minimum
n
∂χ 2
= 0 ⇒ βε ∑ ci4
∂βε
i =1
n
n
n
i =1
i =1
i =1
− 2αε ∑ ci3 si + γε ∑ ci2 si2 = ∑ ci2 wi
2
n
n
n
n
∂χ 2
2
= 0 ⇒ βε ∑ ci3 si − 2αε ∑ ci2 si2 + γε ∑ ci si3 = ∑ ci si wi
∂αε
i =1
i =1
i =1
i =1
n
n
n
∂χ 2
2 2
3
= 0 ⇒ βε ∑ ci si − 2αε ∑ ci si + γε ∑ si4
∂γε
i =1
i =1
i =1
n
= ∑ c wi
i =1
2
i
n
− 2 ∑ ci3 s i
i =1
n
− 2 ∑ ci2 s i2
i =1
n
− 2 ∑ ci s i3
i =1
⎛ n 4
⎛ βε ⎞
⎜ ∑ ci
⎜ ⎟
⎜ i =1
,⎜ ⎟
⎜ n
⎜ αε ⎟ = ⎜ ∑ ci3 si
⎜ ⎟
⎜ in=1
⎜⎜ ⎟⎟
⎜ c2s2
⎜∑ i i
⎝ γε ⎠
⎝ i =1
⎛ n 2 2 ⎞
2 2 ⎞
βε
⎛
⎞
c
s
⎟
⎜ ∑ ci wi ⎟
∑
i i
i =1
⎟ ⎜ ⎟
⎜ i =1
⎟
n
n
⎜
⎟
⎟
⎜
2⎟
ci s i3 ⎟ ⋅ ⎜ αε ⎟ = ⎜ ∑ ci s i wi ⎟
∑
i =1
⎟ ⎜ ⎟
⎜ i =n1
⎟
n
⎜
⎟
2
4
2
⎟
⎜
⎟
s i ⎟ ⎜⎝ γε ⎟⎠
∑
⎜ ∑ ci wi ⎟
i =1
⎠
⎝ i =1
⎠
n
n
− 2 ∑c s
i =1
n
3
i i
− 2 ∑ ci2 s i2
i =1
n
− 2 ∑ ci s i3
i =1
−1
⎞ ⎛ n 2 2 ⎞
c s ⎟ ⎜ ∑ ci wi ⎟
∑
i =1
⎟ ⎜ i =1
⎟
n
n
⎟
⎜
2⎟
ci s i3 ⎟ ⋅ ⎜ ∑ ci s i wi ⎟
∑
i =1
⎟ ⎜ i =n1
⎟
n
2 ⎟
4
2
⎟
⎜
s i ⎟ ⎜ ∑ ci wi ⎟
∑
i =1
⎠ ⎝ i =1
⎠
n
2 2
i i
2
βε ⋅ γε − (αε ) 2 = ε , β =
βε
αε
, α=
ε
ε
Emittance measurement insertion in CTF3 linac
4.5 m
quadrupole
triplet
steering
magnet
OTR screen
beam direction
Girder 10
4 April 2007
Example of ε measurement in CTF3 linac
g
Emittance measurement with moveable slit
x’
v/L
Intensity at u,
x’ in x, x’-space is
mapped to position u+x’·L on screen
u
x
x
u
v
g
z
L
Moveable slit
profile monitor
From width and position of slit image mean beam angle and divergence of slice
at position u is readily computed.
By moving slit across the beam complete distribution in x,
Conditions for good resolution:
v >> s
x’ space is reconstructed.
Phase space reconstruction at SPARC
courtesy of Massimo Ferrario / LNF X’
X
Single shot emittance measurement
for 50 MeV proton beam at LINAC II / CERN
SEM grid
profile monitor
quadrupoles for point
to parallel imaging
Slit
fast deflector magnets
position on profile monitor
Single shot emittance measurement
for 50 MeV proton beam at LINAC II / CERN
time
time resolved
profile monitor
quadrupoles for
point to parallel
imaging
fast sweeping
magnet
slit
(fixed position)
beam
Pepperpot
similar to moving slit method
+ allows measurement in both planes simultaneously
- pepperpot dimensions have to be well adapted to beam conditions
hole spacing << beam widths
hole image >> hole diameter
hole image < hole spacing
Where to use what ?
quadrupole scan for beamlines where linear beam optics is valid,
i.e. where space charge forces are small and energy spread is not too large
Slit / pepper-pot for low energy beams,
where beam can be stopped in slit / pepperpot mask
high energy
high space charge forces
large energy spread
slit / pepperpot
‐
+
+
quadrupole scan
+
‐
‐
drift with several monitors
+
‐
+
The "ultimate" emittance measurement set-up
online data acquisition
and analysis system
profile monitor with
variable Z-position
(CCD camera & screen)
beam
switchable lens
f=25-100 mm
pepperpot
beam
generation
Slice emittance
measurement
ATF at BNL
time slice
off crest acceleration
Slice emittance set-up at FLASH / DESY
deflecting
RF cavity
Emittance measurement in storage ring or synchrotron
Beam profile measurement at one location of ring is sufficient to determine ε
⎛c s⎞
⎟⎟
M ( z , z + C ) = ⎜⎜
⎝ c' s' ⎠
is single particle transfer matrix for one ring revolution starting at profile monitor positon z.
M ( z , z + C ) can be calculated from magnet parameters and position.
Twiss functions must be periodic w ith Circumfere nce
⇒
⎛ β ( z) ⎞
⎜α ( z) ⎟
⎜
⎟
⎜ γ ( z) ⎟
⎝
⎠
⎛ c2
s 2 ⎞ ⎛ β ( z) ⎞
− 2c s
⎛ β (z + C)⎞
⎟ ⎜
⎜
⎟
⎜
⎟
= ⎜ α ( z + C ) ⎟ = ⎜ − c c' c s '+ c' s − s s ' ⎟ ⋅ ⎜ α ( z ) ⎟
⎜
⎟
⎜γ (z + C) ⎟
⎜ c' 2
s ' 2 ⎟⎠ ⎝ γ ( z ) ⎠
− 2c ' s '
⎝
⎠
⎝
⎛ c2
s2 ⎞
− 2c s
⎛ β ( z) ⎞
⎟
⎜
⎜
⎟
Therefore ⎜ α ( z ) ⎟ is Eigenvecto r of matrix ⎜ − c c' c s '+ c' s − s s ' ⎟ .
⎜ γ ( z) ⎟
⎟
⎜ c' 2
s' 2 ⎠
− 2c ' s '
⎝
⎠
⎝
2s
Solving for the Eigenvecto r gives β ( z ) =
( 2 − c − s ' )(2 + c + s ' )
Measured beam halfwidth
w = β ( z) ε
w2
⇒ ε=
β ( z)
Complication in storage ring
w( z ) = β ( z ) ε + D( z )
ΔP
P
2
2
Emittance determination requires profile monitors at location with
β (z ) ε
>> D( z )
2
ΔP
P
2
For measurement of β(z) and D(z) see Jörg Wenniger’s lecture.
For measurement of
ΔP
P
see Mario Ferianis’s lecture
Longitudinal emittance
εL =
z
ΔP
P
2
2
ΔP
− z⋅
P
In synchrotron or storage rings with Qs << 1
εL =
ΔP
P
z2
2
If intensity dependend synchrotron tune shift is negligible it is sufficient
z
to measure either
using
I
z
2
2
or
ΔP
P
2
⎛
1 ⎞
c ⎜⎜ α − 2 ⎟⎟
γ ⎠
= ⎝
ωS
and the synchrotron oscillation frequency ωs
ΔP
P
2
Longitudinal emittance measurement for Linacs I
Spectrometer magnet
Y-plane deflection
beam
For good accuracy
( A L ω wt ) 2 >> βε X
RF deflector cavity
X-plane deflection
( D w ΔP ) 2 >> βε Y
P
required
X-Y image monitor
x’
y
A
wY
bunch centered on
zero cresting deflection
t
x
w ΔP
P
wX
x = A L sin (ωt )
wt ≈
1
wx
A Lω
ΔP
P
1
=
wY
DY
y = DY
Longitudinal emittance measurement for Linacs II
Spectrometer magnet
Y-plane deflection
beam
Profile monitor with preservation
of time information like
OTR or Cerenkov radiator
+ observation with streak camera
y
x
w ΔP
P
wt
streak direction
ΔP
P
1
wY
=
DY
y = DY
wY
Pitfalls with emittance measurements
0 Beam width determination
0 Space charge
0 Chromatic effects
0 Calibration errors