Embed Size (px)
DESCRIPTION
Soft X-Ray pulse length measurement. Alberto Lutman Jacek Krzywinski Juhao Wu Zhirong Huang Marc Messerschmidt …. 17.February.2011. Soft X-Ray Pulse length Determination. Electron Beam. Spectrometer. SASE FEL Amplifier. t. T. Goal: recover T from the spectra - PowerPoint PPT Presentation
Citation preview
Soft X-Ray pulse length measurement
17.February.2011
Alberto LutmanJacek Krzywinski
Juhao WuZhirong Huang
Marc Messerschmidt…
Electron Beam
SASE FEL Amplifier
T t
Goal: recover T from the spectra(correlation technique proposed by Jacek K.)
Soft X-Ray Pulse length Determination
10 5 0 5 10
Spectrometer
Electron Beam
1
( ) ( ) ( )N
kk
I t e t t
t
tk is a random variable having probability density f(t)
Current profile
0
( ) ( ) k
Ni t
k
I e e
Current Fourier Transform
2( ) *( ') ( ')I I e NF 2 2 24 2( ) ( ') (1 ( ') )I I e N F
( ) ( ) i tF f t e dt
10 5 0 5 10
( )f t
( , )aH z( , ) ( , ) ( )aE z H z I
FEL transfer function in Exponential growth
The spectrometer transfer function ( )mH ( , ) ( ) ( , ) ( )m aE z H H z I
For Saturation, we run numerical simulations
Correlation of the radiation intensity at the exit of the spectrometer
2 2
', ''2 2 2
' ''
( ', ) ( '', ) ' ''
( , ) 1( ', ) ' ( '', ) ''
a b
a b
a b
E E d d
GE d E d
0
0
/ 2
/ 2a
b
Frequencies that we correlate
0
Single shot spectrum
Single shot correlation,to be:averaged on many shots normalized
Calculation of the G2 function for different profiles
2 2 2 20 0 0 02 2 2 2
2 2 2 20 0 0 02 2 2 2
( ' /2) ( '' /2) ( ' ) ( '' )22 2 2 2
', ''2 0 ( ' /2) ( ' ) ( '' /2) ( '' )
2 2 2 2
' ''
( ' '', ) ' ''
( , )
' ''
m m a a
m a m a
e e e e F T d d
G
e e d e e d
202
( )2 2( ) a
aH e
2
2
( )2 2( )
m
mmH e
2
02 2
2
2
,( )
2
x x
Se F x TG dx
S
2 2
022 ( ) ( , ) ( , )
Z SiZx
G e f t T f Z t T dZ
2
0 02 2a
m a
x
02 22 a m
m a
S
To find an analytical expression for G2 we need just to plug in the f(t) function
Gaussian Electrons Profile2
21 2
2 2( )
1 2
g
g
x
S
g
eG
S
m
2 2
2 2 2 2 2 20
1(0)
(1 4 )
a m
m a T m
GMS
M: “number of modes”
With our spectrometer resolution, the rms of the bell shape is due to the spectrometer bandwidth
0 .00006 0 .00004 0 .00002 0 0 .00002 0 .00004 0 .00006 0 .0001 0 .0002 0 .0003 0 .0004
0 .02
0 .04
0 .06
0 .08
bunchprofile
G2
2
22
( , )2
T
t
T
T
ef t
0g Tx x
g TS S
* *
° °# #
Flat top vs Gaussian
m
G2
0 .0001 0 .0002 0 .0003 0 .0004
0 .01
0 .02
0 .03
0 .04
0 .00005 0 0 .00005
We cannot distinguish between:
- Gaussian profile with rms length
- Flat Top profile with full length T
12 TT
2
22
( , )2
T
t
T
T
ef t
1
2( , )0 2
TtTf t TTt
( )f t
2
21 2
2 2( )
1 2
g
g
x
S
g
eG
S
2 21
22
0
( ) 2 (1 )cos( )fS
fG e x d
Some adressed Issues
Statistical Gain
Scales the measured function by
2
2
gain
gain
Central Frequency Jitter
2
02 2
2
2
,( )
2
x x
Se F x TG K dx
S
20 020
0
2
0( )2
ep
2 20
2 2 2 2 20 0
0
2 2 24
2 2 2 2 2( )
2
m m m mm m
m m m m
K e
Numerical Simulations
2) Recover bunch length and spectrometer bandwith
1) Verified that relations hold well enough in saturation
40mExponential growth
60mAt saturation
100mDeep saturation
m 2.99 x 10-5 3.00 x 10-5 3.06 x 10-5
BL 9.70 m 9.79 m 9.81 m
m 1.47 x 10-5 1.48 x 10-5 1.51 x 10-5
BL 9.53 m 9.71 m 10.02 m
53 10m
51.5 10m
Bunch Length = 10 m
A Matalab GUI to process the data
Calculate G2
FunctionRe-alignSpectra
Plot spectraand
Shot-to-shot recorded quantities
Select a Subset of the
collected spectra
Datasets
Show Bunch Length Result
Shot by shot quantities
Bunch Length vs # of Undulators (2 November 2010 Data)
FWHMGaussian
fs
Undulators282522191613
Spectrometer relative bandwidth (2 November 2010 Data)Spectrometer
Relativebandwidth
282522191613Undulators
Bunch Length vs different peak current (26 January 2011 Data)
FWHMGaussian
fs
Peak current
kA
Bunch Length using slotted foil Measured Photon Bunch length
10 fs 13.5 fs
18 fs 27 fs
Next Steps
Include cases with non monoenergetic electron bunch
- Two gaussian with different energies case- Including a linear energy chirp
Analyze in detail data collected January 26 2011
Finish to write the paper
Adapt the matlab GUI to be used in Control Room
THE END
Different pulses have different Gain
( , )f t T Electron arrivals density probability
T as a random variable with probability density p(t)
( )gain T Gain is function of T (e.g. smaller T, gives higher peak current and higher gain)
( ) ( ) ( ) ( ) ( )m aE H gain T H I
2 2 2 20 0 0 0
2 2 2
2 2 2 2 220 0 0 0
', ''2 0 2( ' /2) ( ' ) ( '' /2) ( '' )
2 2 2 2
'
( ', ) ( '', ) ( ', ) ( '', ) ( ) ( ) (1 ( ' '', ) ) ' ''
( , )
( ) ( ) 'm a m a
a a m mH H H H p T gain T F T dTd d
G
e e p T gain T dT d e e
2
2
''
( ) ( ) ''p T gain T dT d
Statistical gain and FEL gain depending on profile length
We are using indeed a different average profile
The correlation function is affected by the statistical gain
In case the gain is independent of T, the relation between G2 with and without the gain is the following:
2
2, 0 2
2 0 2
2
( , )
( , )
1
gain
gainG
gainG
gain
gain
Statistical gain and FEL gain depending on profile length
We can observe that
Both approaches are not easy to apply when analyzing the real noisy spectral data
2
2 0 2( , )gain
Ggain
normalizing shot by shot each spectrum with its energy
And get rid of this effect
Using the offset of G2
Double Gaussian Electrons Profile 2 2
0 02 21 22 2
1 2
( , ) ( 1)2 2
T T
t t t t
T
T T
e ef t H H
1 0 1g Tx x 1 1g TS S
0 0 0gx x t2 2g TS S
2 2 2 2 21 2 0 1 22 2 2 21 2 1 2
4
1 2 1 2 2(1 )0
2 2 22 2 2 21 21 2 1 2
2( 1) 2 ( 1)( ) cos
11 2 1 2 1
g g g g g
g g g g
x x S x x
S S S Sg
g gg g g g
xHe H e H H eG
S SS S S S
2 0 2g Tx x
0 0gS St
2
0 02 2a
m a
x
02 22 a m
m a
S
Gaussian Electrons Profile2
22
( , )2
T
t
T
T
ef t
2
21 2
2 2( )
1 2
g
g
x
S
g
eG
S
0g Tx x
g TS S
2
0 02 2a
m a
x
02 2
2 a m
m a
S
GASSIAN WITH SIGMA T
Flat Top Electrons Profile1
2( , )0 2
TtTf t TTt
2 21
22
0
( ) 2 (1 )cos( )fS
fG e x d
0fx x T
fS ST
2 22 2 22
2 2 23
Dawson2cos(2 ) 1 2 2
( ) erf2 22 2 2
f
f f
f xfS S
ff f f f f
f ff f f
xx
Se x S ix S ixeG
S SS S S
2
0 02 2a
m a
x
02 2
2 a m
m a
S
Statistical Gain
Considering the model of incoherent radiation, the intensity at a certain frequency can be written
( )S CK
We let the charge C fluctuate, and correlate intensities at two different frequencies ’ and ’’
' ( )( ' ')S C C K K '' ( )( '' '')S C C K K
with
0C ' 0K '' 0K
Spectra Central Frequency jitter
Considering the model of incoherent radiation, the intensity at a certain frequency can be written
( )S CK
We let the charge C fluctuate, and correlate intensities at two different frequencies ’ and ’’
' ( )( ' ')S C C K K '' ( )( '' '')S C C K K
with
0C ' 0K '' 0K
Different pulses have different Gain
2
' ' '' ''
' ' '' ''s
C C K K C C K KG
C C K K C C K K
2
2 2
' ''1 1
' ''s
CK KG
K K C
The G2 function is multiplied by
2
21
C
C
We can observe that for large
Both approaches are not easy to apply when analyzing the real noisy spectral data
normalize shot by shot each spectrum with its integral
To get rid of the multiplicative effect we can:
Use the offset of G2s
' ''
0' ''
K K
K K
2
2 21s
CG
C
Numerical Simulations
Simulations have been also done for Gaussian electrons profile
Flat top electrons profile
Full Length = 10 umRadiation Wavelength = 1.5 ARho = 4.5 x 10-4
Gain Length = 2.98 mUndulator Length = 100 mNumber of Shots = 2000Slippage Length = 0.5 um
Analytical theory has been dereived in the linear regime.
Simulation have been carried to determine if the theory is still applicable in saturation.
Power vs Undulator Distance
Gaussian electrons profileFlat top electrons profile
12 2
( ) *( ')( , ') ( ')
( ) ( ')
E Eg F
E E
Flat top electrons profile
Agreement:
Z=30mLinear regime
Z=60msaturation
Z=100mDeep saturation
Flat top electrons profile
Agreement:2 2
2
2 12 2
( ) ( ')( , ') 1 ( , ')
( ) ( ')
E Eg g
E E
2
2
2
Z=30mLinear regime
Z=60msaturation
Z=100mDeep saturation
Flat top electrons profileG2 function
Simulations with Peak Current Jitter
30 m 60 m
Shot by Shot quantities
x-ray pulse energyElectron bunch chargeElectron bunch energyX and Y positionX and Y anglePeak current
For each shot, beside the spectra, other quantities have been recorded:
Filtering the datasets
Theory assumes that different shots differ only by the arrival time of the electrons.
This is not the case for the real data.
We use the data collected to filter and keep only a subset of the spectra
Filter:- bunch charge- bunch energy- peak current
Correlation between Energy and First moment
The Gui allows to:show each spectrum profileplot recorded quantities
Spectra first moment
Electron bunch energy
Spectra Realignment
We can realign the spectra, using the strong linear correlation between electron bunch energy and first moment
We can either:
Realign the spectra Saving around 50% shots
Use a smaller electron bunch energy window
Saving around 10% shots
Both approaches lead to the same bunch length result
Evaluation of the correlation function
We calculate the correlation function.
Interface leaves some freedom to:
- Deal with backgroud noise issue
- Deal with gain issue
Bunch length is calculated with flat top and gaussian models
Results from data collected November 2nd, 2010
Only 6 sets of data collected with:
200 lines/mm monochromator 3 kA peak currentDifferent number of undulators
Can be used to calculate the bunch length.Other sets have:
Too low spectrometer resolution
Too low signal to noise ratio
