Soft X-Ray pulse length measurement

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