Measurement of Incoherent Radiation Fluctuations
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Transcript Measurement of Incoherent Radiation Fluctuations
Measurement of Incoherent Radiation
Fluctuations and Bunch Profile Recovery
Vadim Sajaev
Advanced Photon Source
Argonne National Laboratory
07/27/2004
XFEL 2004
Theory
• Each particle in the bunch radiates an electromagnetic
pulse e(t)
N
• Total radiated field is
E ( t ) e( t t k )
k 1
N
Eˆ ( ) eˆ( ) eitk
• Fourier transform of the field is
k 1
• Power spectrum of the radiation is
N
N
k 1
m1
P( ) eˆ( )eˆ* ( ) eitk eitm eˆ( )
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2
N
i ( tk tm )
e
k ,m1
Average power
Average power is
2
2 ˆ
P( ) eˆ( ) N N f ( )
2
incoherent
radiation
coherent
radiation
The coherent radiation term carries information about the
distribution of the beam at low frequencies of the order of
-1
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Difference between coherent and incoherent
power is huge
R( ) N fˆ ( )
• Coherent to incoherent ratio
2
• consider a Gaussian beam with t=1 ps and total charge of
1 nC (approximately 1010 electrons)
f (t )
1
e
2 t
t2
2 t2
At 1 THz:
At 10 THz:
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and
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fˆ ( ) e
2 t2
2
High frequencies still contain information
P( ) eˆ( )
• Power spectrum before averaging:
N
2
i ( tk tm )
e
k ,m 1
• Each separate term of the summation oscillates with the
period =2/(tk-tm)~2/t
FITTED_SINC( Data_to_fitNcutoff
Gues s_am
• Because of the random distribution of particles in the
bunch, the summation fluctuates randomly as a function of
frequency .
377.306
400
300
Sm N mix21
200
S1 m
100
0
1.874 100
515
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520
516.593
525
530
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Radiated s pectrum
535
2 3
m
540
5
10
545
550
555
554.787
Example: incoherent radiation in a wiggler
Single electron
2
Electron bunch
Spectrometer
N
N
E ( t ) e( t t k )
2
e(t)
E ( ) e( ) e itk
k 1
1
483.063
FITTED_SINC(
k 1
500
551.617
600
400
F ( x)
0
0
Sm N mix21
S1m
200
Rad_Field_Gaussi
0
500
1
Spike width is inversely
proportional to the bunch
length
2.621 200
515
2
2
5
5
~10 0fs
x
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520
525
0
05
2000
~1 ps
4000
6000
i
5
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8000
530
535
2 3
516.593
541.862 1000
1 10Radiated spectrum
Measured
spectrum
4
110
4
m
540
5
10
545
550
555
554.787
Bunch profile measurements using
fluctuations of incoherent radiation
• The method was proposed by M. Zolotorev and G.
Stupakov and also by E. Saldin, E. Schneidmiller and M.
Yurkov
• Emission can be produced by any kind of incoherent
radiation: synchrotron radiation in a bend or wiggler,
transition, Cerenkov, etc.
• The method does not set any conditions on the bandwidth
of the radiation
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Quantitative analysis
We can calculate autocorrelation of the spectrum:
P( ) P( ) eˆ( ) eˆ( )
2
N
2
e
i ( tk tm ) i ( t p t q )
k , m , p , q 1
2
ˆ
P( ) P( ) N eˆ( ) eˆ( ) 1 f ( )
2
2
2
2
1
ˆ
g () 1 f ()
2
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Variance of the Fourier transform of the
spectrum
Fourier transform of the spectrum:
Its variance:
G( ) P( )ei d
D( ) G ( ) G ( )
2
It can be shown, that the variance is related to the
convolution function of the particle distribution:
D( ) A f (t ) f (t )dt
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Some of the limitations
• Bandwidth of the radiation has to be larger than the spike
width
• In order to neglect quantum fluctuations, number of
photons has to be large
1
n ph N e
2
• Transverse bunch size – radiation has to be fully coherent
to observe 100% intensity fluctuations
rad
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2
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LEUTL at APS
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Spectrometer
Grooves/mm
600
Curv. radius [mm]
1000
Blaze wavelength[nm]
482
Grating
CCD
camera
Number of pixels
Pixel size [m]
Concave mirror curv. radius [mm]
24
4000
Spectral resolution [Å]
0.4
Bandpass [nm]
44
Resolving power at 530 nm
Wavelength range [nm]
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10000
250 – 1100
Single shot spectrum
Typical single-shot spectrum
Average spectrum
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Spectrum for different bunch length
Long 2-ps rms bunch
Short 0.4-ps rms bunch
Note: Total spectrum width (defined by the number of
poles in the wiggler) is barely enough for the short bunch.
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Spectrum correlation
Cn
P(i ) P(in ) /
i
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2
P
(
)
i
i
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Bunch length
From the plot the correlation width is 2 pixels.
Frequency step corresponding to one pixel is 2.4·1011 rad/s.
Assuming the beam to be Gaussian, from equation
2
1
ˆ
g () 1 f ()
2
we get
1
b
2 ps
n
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Convolution of the bunch profile
N ch
Gk ,n Pm,n e
Np
Dk Gk ,m
2imk / N ch
m 1
m 1
Bunch profile
1
0
2
0
2
Step function
Gaus sian
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1
Np
2
Np
G
n 1
k ,n
Convolution recovered from the
measurements
Convolution of the Gaussian
is also a Gaussian with
2 t
The Gaussian fit gives us
b 1.8 ps
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Phase retrieval
The amplitude and the phase information of the radiation
source can be recovered by applying a Kramers-Kronig
relation to the convolution function in combination with the
minimal phase approach.
ln ( x) / ( )
m ( )
P dx
0
x2 2
2
1
z
S ( z ) d ( ) cos m ( )
c 0
c
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Phase retrieval example
Calculation of longitudinal distribution for different bunches
Calculated shape
Exact shape
Time (arb. units)
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Bunch profile
Two different measurements (two sets of 100 single-shot spectra)
FWHM4ps
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Conclusions
• Measurements of incoherent radiation spectrum showing
intensity fluctuations were done.
• A technique for recovering a longitudinal bunch profile
from spectral fluctuations of incoherent radiation has been
implemented. Although we used synchrotron radiation, the
nature of the radiation is not important.
• Typically, analysis of many single shots is required,
however one can perform statistical analysis over wide
spectral intervals in a single pulse
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Conclusions
• An important feature of the method is that it can be used
for bunches with lengths varying from a centimeter to tens
of microns (30 ps – 30 fs)
• There are several important conditions for this technique.
In order to be able to measure a bunch of length t, the
spectral resolution of the spectrometer should be
comparable with 1/t. Also, the spectral width of the
radiation and the spectrometer must be larger than the
inverse bunch length
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