# Measurement of Bunch Length Using Spectral Analysis of Incoherent

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```					 Measurement of Incoherent Radiation
Fluctuations and Bunch Profile Recovery

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
• Fourier transform of the field is                 E (ω ) = e(ω )∑ eiωtk
ˆ        ˆ
k =1

• Power spectrum of the radiation is
N          N                           N
P (ω ) = e(ω )e* (ω )∑ eiωtk ∑ e −iωtm = e(ω )           ∑ eiω ( tk −tm )
2
ˆ    ˆ                          ˆ
k =1     m =1                        k ,m =1

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Average power
⎛ N + N 2 f (ω ) 2 ⎞
P(ω ) = e(ω ) ⎜
2
ˆ
Average power is            ˆ                        ⎟
⎝                  ⎠

incoherent       coherent

distribution of the beam at low frequencies of the order of
ω≈σ -1

07/27/2004                  XFEL 2004
Difference between coherent and incoherent
power is huge
2
• Coherent to incoherent ratio                      R(ω ) = N f (ω )
ˆ

• consider a Gaussian beam with σt=1 ps and total charge of
1 nC (approximately 1010 electrons)
t2                                 ω 2σ t2
−
f (t ) =
1
e       2σ t2
and          ˆ (ω ) = e −
f                2
2π σ t

At 1 THz:               R≈1010
At 10 THz:              R≈10-34

07/27/2004                              XFEL 2004
High frequencies still contain information
N
P (ω ) = e(ω )        ∑ eiω ( tk −tm )
2
• Power spectrum before averaging:             ˆ
k ,m =1

• Each separate term of the summation oscillates with the
period ∆ω=2π/(tk-tm)~2π/σt

• Because of the random distribution of particles in the
bunch, the summation fluctuates randomly as a function of
frequency ω.

07/27/2004                XFEL 2004
Example: incoherent radiation in a wiggler
Single electron    Electron bunch                Spectrometer

N                                   N
e(t)         E ( t ) = ∑ e( t − t k )    E (ω ) = e(ω ) ⋅ ∑ e iωtk
k =1                                k =1

Spike width is inversely
proportional to the bunch
~10 fs                ~1 ps
length

07/27/2004                 XFEL 2004
Bunch profile measurements using

• 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
transition, Cerenkov, etc.
• The method does not set any conditions on the bandwidth

07/27/2004                  XFEL 2004
Quantitative analysis

We can calculate autocorrelation of the spectrum:
N

∑e
iω ( t k − t m ) + iω ′( t p − t q )
P (ω ) P (ω ′) = e(ω ) e(ω ′)
2         2
ˆ     ˆ
k , m , p , q =1

⎛1 + f (ω − ω ′) 2 ⎞
P (ω ) P (ω ′) = N e(ω ) e(ω ′) ⎜
2
ˆ2
⎟
2
ˆ     ˆ
⎝                  ⎠

1⎛    ˆ (Ω ) ⎞
2
g (Ω) = ⎜1 + f       ⎟
2 ⎝           ⎠

07/27/2004                      XFEL 2004
Variance of the Fourier transform of the
spectrum

Fourier transform of the spectrum:
∞
G (τ ) =   ∫
−∞
P (ω )e iωτ dω
Its variance:
2
D(τ ) = G (τ ) − G (τ )
It can be shown, that the variance is related to the
convolution function of the particle distribution:
∞
D (τ ) = A ∫ f (t ) f (t − τ )dt
−∞

07/27/2004                     XFEL 2004
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     ∆ω
n ph ≈ αN e
2      ω
• Transverse bunch size – radiation has to be fully coherent
to observe 100% intensity fluctuations

2πσ θ

07/27/2004                XFEL 2004
LEUTL at APS

07/27/2004      XFEL 2004
Spectrometer
Grooves/mm                600
Grating

Blaze wavelength[nm]      482

CCD              Number of pixels       1100×330
camera
Pixel size [µm]           24

Concave mirror curv. radius [mm]          4000

Spectral resolution [Å]                    0.4

Bandpass [nm]                              44

Resolving power at 530 nm                 10000

Wavelength range [nm]                   250 – 1100

07/27/2004      XFEL 2004
Single shot spectrum
Typical single-shot spectrum

Average spectrum

07/27/2004                     XFEL 2004
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.

07/27/2004                 XFEL 2004
Spectrum correlation

Cn =   ∑ P(ω i ) P(ω i+n ) /
i
∑i
P (ω i ) 2

07/27/2004                     XFEL 2004
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

1⎛    ˆ (Ω ) ⎞
2
g (Ω) = ⎜1 + f       ⎟
2 ⎝           ⎠
we get
1
τb ≈        ≈ 2 ps
n ⋅ δω

07/27/2004                  XFEL 2004
Convolution of the bunch profile
2
N ch                                       Np             Np
1
Gk ,n = ∑ Pm ,n e       2πimk / N ch
Dk = ∑ Gk ,m −      ∑G     k ,n
m =1                                       m =1      Np   n =1

Bunch profile

1

0
2           0        2
Step function
Gaussian

07/27/2004                                  XFEL 2004
Convolution recovered from the
measurements

Convolution of the Gaussian
is also a Gaussian with
σ = 2 ⋅σ t

The Gaussian fit gives us
τ b = 1.8 ps

07/27/2004               XFEL 2004
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[ρ ( x) / ρ (ω )]
∞
2ω
ψ m (ω ) = −  P ∫ dx
π 0         x2 −ω 2

∞
1                    ⎛          ωz ⎞
S ( z) =    ∫ dω ⋅ ρ (ω ) ⋅ cos⎜ψ m (ω ) − c ⎟
πc 0                  ⎝             ⎠

07/27/2004                       XFEL 2004
Phase retrieval example

Calculation of longitudinal distribution for different bunches

Calculated shape
Exact shape

Time (arb. units)

07/27/2004                           XFEL 2004
Bunch profile
Two different measurements (two sets of 100 single-shot spectra)

FWHM≈4ps

07/27/2004                 XFEL 2004
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

07/27/2004                 XFEL 2004
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 σt, the
spectral resolution of the spectrometer should be
comparable with 1/σt. Also, the spectral width of the
radiation and the spectrometer must be larger than the
inverse bunch length

07/27/2004                XFEL 2004

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