# Accurate measurement of uncorrelated energy spread in electron beam

Sergey Tomin, Igor Zagorodnov,
Winfried Decking, Nina Golubeva and Matthias Scholz  Deutsches Elektronen-
Synchrotron, Notkestrasse 85, 22603 Hamburg, Germany

###### Abstract

We present measurements of slice energy spread at the injector section of the
European X-Ray Free Electron Laser for an electron bunch with charge of 250
pC. Two methods considered in the paper are based on measurements at the
dispersive section after a transverse deflecting structure (TDS). The first
approach uses measurements at different beam energies. We show that with a
proper scaling of the TDS voltage with the beam energy the rms error of the
measurement is less than 0.3 keV for the energy spread of 6 keV. In the second
approach we demonstrate that keeping the beam energy constant but adjusting
only the optics we are able to simplify the measurement complexity and to
reduce the rms error below 0.1 keV. The accuracy of the measurement is
confirmed by numerical modelling including beam transport effects and
collective beam dynamics of the electron beam. The slice energy spread
measured at the European XFEL for the beam charge of 250 pC is nearly 3 times
lower as the one reported recently at SwissFEL for the same cathode material
and the beam charge of 200 pC.

##  I Introduction

The small emittance and the low energy spread of the electron beam required at
X-Ray Free Electron Lasers (XFELs) can cause the microbunching instability MB1
; MB2  and destroy the lasing. On the other hand, a large initial energy
spread will hinder a proper compression of the bunch and will lead to
intolerable energy spread after compression. Hence a reliable high resolution
method of the measurements of the uncorrelated (or slice) energy spread is
crucial for a proper operation of the modern facilities.

In order to measure the slice energy spread a standard approach with a
transverse deflector and the dispersive section is used. However, it shows
only a low resolution (of several keV) due to impact of OTR screen resolution,
the betatron beam size and the deflector strength on the measurement. The
energy spread induced by deflector can be excluded with a set of measurements
with different deflector amplitudes. Such experiments have been done at PITZ
MK20 .

Recently in Prat20  it was suggested to carry out the measurements at
different electron beam energies. For the setup used at SwissFEL - deflector
at constant energy and an acceleration section after it - the authors have
written a polynomial equation of the second order and analyzed the accuracy of
the coefficient reconstruction relative to statistical and systematic errors.
On the basis of this analysis they concluded that the accuracy of the
measurements for their setup could be better than 1 keV. In order to exclude
the impact of the deflector the authors in Prat20  suggested to carry out
additional sets of measurements, but it was not done and an analytical
estimation was used instead.

The energy scan method seems to be simple. However at the layout of the
European XFEL this method requires modifications and even with the changes
described in this paper it has not shown the expected performance.

At this paper we present the measurement at the European XFEL with layout
shown in Fig. 1. The situation is different as compared to SwissFEL case. The
deflector is placed after the acceleration section, and the beam energy in it
changes during the experiment. This case was mentioned in Prat20  too and it
was suggested to use a polynomial equation of the third order with additional
sets of measurements in order to exclude the impact of the deflector.

If we use only energy scan then in our setup the accuracy of the
reconstruction of the polynomial of the third order is low and very sensitive
to errors. In order to overcome it we suggest a proper scaling of the
deflector voltage with the beam energy. It allows (1) to reduce the order of
the polynomial equation; (2) to avoid a need in additional sets of measurement
with different deflector amplitudes. And the most important advantage of the
suggested method is (3) a considerable higher accuracy of the reconstruction
of the coefficients and as a consequence a higher accuracy of the energy
spread measurements.

Figure 1: The setup of the experiment at the injector section of the European
XFEL.

The experiments described in our paper are done at low energetic part of the
facility. In order to analyze the accuracy of the measurement we have done
numerical modeling including beam transport effects and collective beam
dynamics of the electron beam. We have found that the radio-frequency (RF)
focusing impacts the beam considerably and a matching of the beam to the
optics before the TDS is necessary for each beam energy. Additionally we see
in the modeling and in the experiment that due to collective effects the
emittance and the energy spread are not constant during the energy scan.

Applying the method based on the energy scan in practice we have found that
(for our setup and a small interval of energy change available) the method is
very time-intensive due to beam matching and the optics scaling. In the
measurements we have failed to obtain reliable data which allow to carry out
an accurate reconstruction. Hence we developed another method and have used
the data from energy scan experiments only to show the consistence with the
results obtained by the second method.

In the second approach we keep the beam energy constant and avoid time-
consuming matching of the beam before TDS. By adjusting the strength of
quadrupoles after TDS we are able to carry out independent scans in
dispersion, in TDS strength and in beta-function on OTR screen. The method is
fast and allows to obtain accurate measurements of slice energy spread with
resolution better than 0.1 keV.

The paper is organized as follows. The methods of the measurement and their
analysis are described in Section II. The beam dynamics modeling of the
approaches with collective effects is considered in Section III. Then, in
Section IV the results of the measurements at the European XFEL injector and
their analysis are presented.

##  II Methods of the measurement and their analysis

For the setup of Fig. 1 the measured beam size $\sigma_{M}$ on the screen can
be written as

$\displaystyle\sigma_{M}^{2}=\sigma_{R}^{2}+\frac{E_{0}}{E}\sigma_{B}^{2}+\frac{D^{2}}{E^{2}}\sigma_{E}^{2}+\frac{(DekV)^{2}E_{0}}{E^{3}}\sigma_{I}^{2},$
(1)
$\displaystyle\sigma_{B}^{2}=\frac{\beta_{x}\epsilon_{n}}{\gamma_{0}},\quad\sigma_{I}^{2}=\frac{\epsilon_{n}(\beta_{y}^{0}+0.25L^{2}\gamma_{y}^{0}-L\alpha_{y}^{0})}{\gamma_{0}},$

where $E$ is the beam energy, $\sigma_{R}$ is the screen resolution,
$\beta_{x}$ is the optical function at the position of the screen,
$\epsilon_{n}$ is the normalized beam emittance, $\gamma_{0}$ is the relative
beam energy, $D$ is the horizontal dispersion at the screen position;
$\beta_{y}^{0}$, $\gamma_{y}^{0}$ and $\alpha_{y}^{0}$ are the twiss
parameters at the beginning of TDS; $k$, $V$, and $L$ are the wave number,
voltage and length of the TDS and $e$ is the electron charge.

Table 1: Simulation parameters. parameter | Units | Value
---|---|---
OTR resolution, $\sigma_{R}$ |  $\mu$m | 28
Normalized emittance, $\epsilon_{n}$ |  $\mu$m | 0.4
Reference optical $\beta$-function at OTR , $\beta_{x}^{0}$ | m | 0.6
Reference dispersion, $D_{0}$ | m | 1.2
Optical $\beta$-function at TDS, $\beta_{y}^{0}$ | m | 4.3
Optical $\alpha$-function at TDS, $\alpha_{y}^{0}$ |  | 1.9
Wave number of TDS, $k$ | 1/m | 58.7
Length of TDS, $L$ | m | 0.7
Reference voltage of TDS, $V_{0}$ | MV | 0.61
Reference energy, $E_{0}$ | MeV | 130

In this section we propose and validate two methods to measure the slice
energy spread. The parameters used in the simulations are listed in Table 1.
These parameters are close to estimations obtained in the experiment. The
resolution of OTR screen is 28 $\mu$m and it agrees with other publications:
10-20 $\mu$m in OTR , 30 $\mu$m in Prat20 .

###  II.1 Method based on energy scan

If the second and the third terms in the right hand side of Eq.(1) are not
known then the energy spread can be estimated as

$\displaystyle\sigma_{E}\approx\frac{E}{D}\sigma_{M}.$ (2)

The error of this estimation is defined by the resolution

$\displaystyle
R_{\sigma_{E}}=\frac{E}{D}\sqrt{\sigma_{R}^{2}+\frac{E_{0}}{E}\sigma_{B}^{2}+\frac{(DekV)^{2}E_{0}}{E^{3}}\sigma_{I}^{2}}.$
(3)

In order to increase the resolution it was suggested in Prat20  to ”perform
beam size measurements for different energies and deflector voltages and to
fit the data” with Eq.(1). Note that Eq.(1) uses a more accurate approximation
of the last term compared to Prat20  where the authors used
$\sigma_{I}^{2}=\epsilon_{n}\beta_{y}^{0}/\gamma_{0}$.

If we keep the voltage of the deflector constant and change only the beam
energies than we can fit the measurements to Eq.(1) in hope to reconstruct all
coefficients of this polynomial. We simulated with Eq.(1) a measurement of the
beam size $\sigma_{M}$ with constant TDS voltage $V_{0}$ and the beam energy
changing between 90 and 190 MeV with step of 10 MeV. At each beam energy we
simulate 30 measurements of the beam size $\sigma_{M}$ with random error of
$2\%$. We consider the slice energy spread between 0.5 and 7 keV. In the fit
we used the simplex search method of Lagarias et al Fit .

From numerical experiment we have found that the rms error of the
reconstruction of energy spread is larger than 2 keV. Under the rms error of
reconstruction in the paper we mean the value defined as

$\displaystyle\Delta_{\sigma_{E}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\sigma_{E}-\sigma_{E}^{0})^{2}},$
(4)

where N is the number of shots (reconstructions), $\sigma_{E}$ is the energy
spread obtained from the reconstruction (of the polynomial coefficient from
the simulated measurements) and $\sigma_{E}^{0}$ is true energy spread used in
the simulation of the reconstruction procedure. In order to estimate this
error we used 100 shots at each energy spread point.

In order to reduce the error we can do an additional scan with different
deflector voltages to estimate the last term in Eq.(1). With this estimation
we reduce the error of the reconstruction. However, we will not analyze this
approach here and suggest below another technique to reduce the order of the
polynomial and to increase the accuracy of the reconstruction of the
polynomial coefficients.

It can be achieved if we will keep constant not the voltage $V$ but the
streaking parameter of the deflector:

$\displaystyle
S_{0}=\sqrt{\beta_{y}\beta_{y}^{0}}\sin(\Delta\mu_{y})K_{0},\quad
K_{0}=\frac{eV_{0}k}{E_{0}},$ (5)

where $\Delta\mu_{y}$ is the phase advance between the middle of TDS and the
OTR screen, $\beta_{y}^{0}$ is the optical function at the TDS, $\beta_{y}$ is
the optical function at the position of OTR and the voltage $V_{0}$ is a fixed
value which produces the desired streak $S_{0}$ at the fixed beam energy
$E_{0}$.

In the following we adjust the voltage of TDS proportionally to the beam
energy:

$\displaystyle V(E)=\frac{V_{0}}{E_{0}}E.$ (6)

Figure 2: Performance of the method based on energy scan. Circles shows
reconstructed mean values. Error bars give the standard deviation error. The
red dashed curve presents the rms error of the reconstruction. Dotted black
curves show true values used for the simulation.

If we put Eq.(6) in Eq.(1) then we reduce the order of the polynomial from the
third to the second one:

$\displaystyle\sigma_{M}^{2}=\sigma_{R}^{2}+\frac{E_{0}}{E}\sigma_{BI}^{2}+\frac{D^{2}}{E^{2}}\sigma_{E}^{2},\quad\sigma_{BI}^{2}=\sigma_{B}^{2}+(DK_{0}\sigma_{I})^{2}.$
(7)

We simulated with Eq.(7) a measurement of the beam size $\sigma_{M}$ with the
beam energy changing between 90 and 190 MeV with step of 10 MeV. We used the
same errors and the reconstruction algorithm as in the previous example. The
results of the reconstruction are shown in Fig. 2 and the error of the
reconstruction of energy spread is nearly 0.3 keV at the energy spread of 6
keV.

In experiment and in the beam dynamics simulations we have not been able to
show this accuracy. We show with beam dynamics simulations that in order to
use this method we have to match the beam to the optics at each beam energy.
It requires considerable efforts and we failed to make it with high accuracy
at the experiment. Additionally, in the modelling and the experiment we have
seen that the slice emittance is not constant.

###  II.2 Method based on dispersion scan

In this section we present another method which use constant beam energy
$E_{0}$ and avoids above described difficulties. The method shows much better
resolution theoretically and it is easy to use experimentally.

We have developed a special optics described in the next section. Using only
few quadrupoles between TDS and the OTR screen we are able to change the
dispersion $D$ at the OTR position keeping $\beta_{x}$ -function constant with
only moderate changes in $\beta_{y}$-function and in the streaking $S$.

We start with changing of TDS voltage $V$ and fit the measured slice size
$\sigma_{M}$ to the quadratic polynomial:

$\displaystyle\sigma_{M}^{2}=A_{V}+B_{V}V^{2}.$ (8)

During the scan we keep the dispersion at constant value $D_{0}$.

Figure 3: Performance of the method based on dispersion scan. Circles shows
reconstructed mean values. Error bars give the standart deviation error. The
red dashed curve presents the rms error of the reconstruction. Dotted black
curves show true values used for the simulation.

At the second step we keep constant the TDS voltage at $V_{0}$ and change the
dispersion $D$. We fit the measured slice size $\sigma_{M}$ to the quadratic
polynomial:

$\displaystyle\sigma_{M}^{2}=A_{D}+B_{D}D^{2}.$ (9)

After these two fits we are able to find out all terms of Eq.(1):

$\displaystyle\sigma_{E}=\frac{E_{0}}{D_{0}}\sqrt{A_{D}-A_{V}},\quad\sigma_{I}=\frac{E_{0}}{D_{0}ek}\sqrt{B_{V}},$
(10)
$\displaystyle\sigma_{B}=\sqrt{B_{\beta}\beta_{x}^{0}},\quad\sigma_{R}=\sqrt{A_{D}-\sigma_{B}^{2}},$

where

$\displaystyle
B_{\beta}=\sigma_{I}^{2}(\beta_{y}^{0}+0.25L^{2}\gamma_{y}^{0}-L\alpha_{y}^{0})^{-1}.$
(11)

Eq. (11) calculates the coefficient $B_{\beta}$ from the results of the TDS
voltage scan, Eq. (8) . Otherwise, if we had measured the slice emittance
$\epsilon_{n}$ independently, then we can use more accurate estimation of
$B_{\beta}$ through the relation $B_{\beta}=\epsilon_{n}/\gamma_{0}$ . For
example, we can estimate $B_{\beta}$ (or emittance $\epsilon_{n}$) changing
only $\beta_{x}$ function at the OTR screen position and keeping the
dispersion $D$ constant and fitting the measured slice size $\sigma_{M}$ to
the linear polynomial:

$\displaystyle\sigma_{M}^{2}=A_{\beta}+B_{\beta}\beta_{x}.$ (12)

We simulated with Eq. (1) the measurement of the beam size $\sigma_{M}$ for
two scans as given by Eqs.(8)-(11). For the dispersion scan we used the values
of 0.6, 0.8, 1.0 and 1.2 meters. For the TDS voltage scan we used values 0.38,
0.47, 0.56, 0.65 and 0.75 MV. We used the same errors and the reconstruction
algorithm as in the previous examples. The results of the reconstruction are
shown in Fig. 3 and the error of the reconstruction of energy spread is
smaller than 0.1 keV at the energy spread of 6 keV.

###  II.3 Impact of systematic and random instrumental errors

Finally, let us consider instrumental errors in the setup of TDS voltage $V$
and the dispersion $D$ during the scans used in the dispersion scan method.

Figure 4: Impact of instrumental errors in setup of voltage and dispersion on
the reconstruction error from dispersion scan method.

If the errors are systematic with the same sign then the reconstruction of
energy spread only weakly affected by them. Indeed, we calculate energy spread
by Eq. (10) and use only the constant terms $A_{D}$ and $A_{V}$. If we suggest
that during the TDS voltage scan we set the voltage with the same negative
error, for example it is 10 %, then it has only impact on coefficient $B_{V}$
which in this case will be increased by factor $0.9^{-2}$, but the constant
term $A_{V}$ is not changed. The same is true for the impact of the systematic
error in the dispersion $D$ during the dispersion scan.

Hence we analyze only random errors in the setup of voltage or dispersion. The
results of the analysis are shown in Fig. 4. In the suggestion of rms error of
5% the rms reconstruction error remains below 0.3 keV for the energy spread of
6 keV.

##  III Modelling of the experiment with collective effects and the beam
transport

The electron beam dynamics at the European XFEL accelerator has been recently
discussed in Zag19  and Zag20 . In the last work, an experimental validation
of the collective effects modeling at the European XFEL injector was
presented. Here we use the same approach from Zag20  to simulate the beam
dynamics, namely (1) the dynamics of the electron beam in the gun was
simulated using ASTRA code ASTRA , (2) the beam tracking starting from 3.2 m
from the gun cathode, was performed using Ocelot code OCELOT  with the space
charge and the wakefield effects included, (3) the coherent synchrotron
radiation was omitted as negligible for this section.

Figure 5:  Electron beam distribution after the gun used in the modeling.

In the measurements described in Section IV we have found that the
uncorrelated energy spread is equal to approximately 5.9 keV. As it is
discussed in Section V one of the possible reasons of such large energy spread
could be intrabeam scattering (IBS). The codes ASTRA and Ocelot do not model
IBS. In the simulation of RF gun with ASTRA for charge of 250 pC we obtain the
energy spread of 0.6 keV. Hence we apply random generator at distance of 3.2 m
from the cathode to increase the energy spread artificially to 5.9 keV. The
properties of the electron bunch after this procedure at position $z=3.2$ m
are shown in Fig. 5. Let us note here that the projected emittance at this
distance from the cathode is relatively large. The emittance will reduce in
the booster considerably according with the emittance compensation process
Carlsten .

###  III.1 Magnetic lattice and its properties

A special optics (shown in Fig. 6 and Table 1) was developed to fulfil
requirements of the experiments described above. The important properties of
the optics are: (1) high dispersion at the screen position to maximize energy
spread contribution to the beam size on the screen; (2) the high value of
$R_{34}$ element of the transport matrix between the TDS and the screen to
minimize voltage of the TDS which in turn minimizes the induced energy spread;
(3) low sensitivity of $\beta_{x}$ in the screen position to the beam mismatch
at the matching point before TDS.

Figure 6: Special optics for the slice energy spread measurement. Optics is
shown from the matching point. Note, $\beta$-functions are calculated in
linear approximation without any collective effects.

Another feature of this optics is the ability to vary the horizontal
dispersion over a wide range without change in the horizontal $\beta$-function
on the OTR screen. The main contribution for the dispersion change comes from
the quadrupole QI.63.I1D while minor changes in the $\beta_{x}$-function are
compensated by quadrupoles QI.60.I1 and QI.61.I1, Fig 7. The described feature
is used to performed the dispersion scan.

Figure 7: Changes of the twiss parameters in the dump section during
dispersion scan.

###  III.2 Numerical modelling of the experiment based on the energy scan

Following the measurement procedure described in the Section II.1, we
performed beam dynamics simulations changing the voltage of module A1 (see
Fig. 1) from 80 to 180 MV with step 10 MV. The beam comes from the gun with
initial energy of 6.5 MeV.

The transport matrix from TDS to the screen in the dump section has non-zero
$R_{51}$, $R_{52}$ elements (contribution form the dump magnet). These
elements create couplings between the horizontal and longitudinal planes,
which leads to the expansion of the beam in the longitudinal direction in case
of non zero horizontal emittance. In the presence of the correlated energy
spread it will cause of the slice widening on the OTR screen. This effect in
simulations can be see in Fig. 8. On the left are two plots of the LPS beam
distribution (a) and the slice energy spread (b) of the beam in front of the
dump magnet, and on the right two plots are the image (c) and the horizontal
slice size (d) on the OTR screen of the same beam that was tracked through the
dump section. To avoid the influence of this effect on the measurement, the
beam slice of interest must have zero energy chirp. For completeness, it is
worth noting that the element $R_{56}$ is small and its effect can be
neglected.

Figure 8: Some details of longitudinal beam dynamics for the beam energy 136
MeV. a) The LPS beam distribution in front of the dump magnet, b) slice energy
spread of the beam in front of dump magnet, c) the beam image on the OTR
screen and d) horizontal slice beam size on the OTR screen without effect of
the screen resolution.

Taking into account the effect described above, the third harmonic cavity AH1
was turned off. measurement of the slice energy spread was carried out at the
extremum of the mean slice energy. The horizontal twiss parameters of the
slice have been matched to the magnetic lattice before TDS. Since the RF-
focusing effect is strong, the beam should be matched for all beam energies.
At the Fig. 9 are shown $\beta$-functions of the central slice for highest and
lowest energies. Twiss parameters were calculated from the beam transported in
Ocelot with collective effects included.

In the simulations and in the experiment we have seen increase of the slice
emittance by 30% at the highest voltage of RF module A1. It is due to very
strong RF focusing and very small $\beta$-functions in module A1 which, in
turn, enhance the SC effect. Additionally we think that IBS would change the
energy spread during the energy scan as well. But this effect was omitted in
the simulation.

The true values used in the simulation are listed in the first row of Table 2.

Figure 9:  $\beta$-functions for lowest and highest energy with taking into
account the SC effect and RF focusing.

The results of the simulations are shown on Fig. 10 and in the second row of
Table 2. The left plot shows the results for the beam matched at each energy.
The right plot shows the results for the beam matched only at the reference
energy $E_{0}=130$ MeV. The black circles show the slice width $\sigma_{M}$
from the simulations. The black dotted line at the left plot shows the results
of the reconstruction with method of Section II.1 using Eq. (7). The other
lines at this plot show contribution of different terms of Eq. (7) as found
from the reconstruction.

For the data shown in the right plot of Fig. 10 the reconstruction was
impossible. The black dotted line shows the expected values calculated by Eq.
(7) using the true data listed in the first row of Table 2. The other lines at
this plot show contribution of different terms of Eq. (7) as found from the
true values of Table 2.

Figure 10: The left plot shows the results for the beam matched at each energy. The right plot shows the results for the beam matched only at the reference energy $E_{0}=130$ MeV. The black circles show the slice width $\sigma_{M}$ from the simulations. The black dotted line at the left plot shows the results of the reconstruction with Eq. (7). The other lines at this plot show contribution of different terms. For the data shown in the right plot the reconstruction is impossible. The lines show the expected values calculated using the true data of Table 2. Table 2: The true and the reconstructed data from the beam dynamics simulations at the reference energy $E_{0}=130$ MeV. Parameter | $\sigma_{E}$ | $\sigma_{I}$ | $\sigma_{B}$ | $\sigma_{R}$ | $\epsilon_{n}$
---|---|---|---|---|---
Units | keV |  $\mu$m |  $\mu$m |  $\mu$m |  $\mu$m
True values | $5.90$ | $80.3$ | $35.4$ | $28$ | $0.53$
Energy scan method | $5.89$ |  |  | 41 |
Dispersion scan method | $5.97$ | $81.8$ | $36.0$ | $26.4$ | $0.55$

Thus, in the setup of the European XFEL the slice matching procedure should be
applied on each step. However even with the matching the reconstruction could
be non-accurate due to changes in the slice emittance and the energy spread at
different energies.

###  III.3 Numerical modeling of the experiment based on dispersion scan

Another method for the slice energy spread measurement proposed in the Section
II.2 uses the variation of the dispersion instead of the energy. To test the
viability of the method, we carried out numerical experiments similar to that
described above with two scans: dispersion scan and TDS voltage scan.

The simulations have been done at the reference beam energy of 130 MeV for two
different uncorrelated energy spreads: 5.9 keV and 2 keV. The first scan was
performed with dispersion on the OTR screen of 0.6, 0.8, 1.0 and 1.2 m. The
TDS voltage was kept constant at 0.61 MV. The second scan was performed with
TDS voltages of 0.47, 0.56, 0.65, 0.75, 0.84 MV. The dispersion was kept
constant at 1.2 m.

The results of the modelling are shown in Fig. 11. The black circles show the
central slice width $\sigma_{M}$ obtained from beam dynamics simulations. The
blue dotted lines presents the curves reconstructed by method of Section II.2.

Using reconstruction procedure described in Section II.2 for the true energy
spread of 5.9 keV we got the reconstructed values listed in the last row of
Table 2. We see that all values are reconstructed with high accuracy. For the
true energy spread of 2 keV the reconstructed energy spread is 2.13 keV.

Figure 11: Beam size $\sigma_{M}$ (black circles) from the dispersion scan and
the TDS voltage scans as obtained from beam dynamics simulations. The blue
dotted lines presents the curves of the numerical fit with Eq.(9) and Eq.(8)..

Contrary to the first method based on energy scan in the second method the
slice emittance does not change during the dispersion and the TDS voltage
scans.

##  IV Measurements

In the analysis of the images obtained in the experiment we have followed the
same procedure as in the simulations. At each point we took 30 images and for
each of them we calculated the mean slice energy and the slice size. The slice
length was taken about 0.2 ps and the slice width was found by fitting to the
Gaussian shape. Then the slice width at the extremum of the mean slice energy
curve has been taken as $\sigma_{M}$.

From the measurements we estimate that the standard deviation error in
$\sigma_{M}$ is below 1.5 %. Hence the error in the mean value from the 30
measurement is below 0.3%.

Due to substantial difficulties with the energy scan method in the experiment
we change the order of consideration and consider the energy scan method at
the beginning.

###  IV.1 Results obtained with the dispersion scan

The measurements presented in this section have been conducted at constant
electron beam energy $E_{0}=130$ MeV. The parameters of the gun has been
optimized to have small normalized slice emittance $\epsilon_{n}\approx 0.4$
$\mu$m.

Figure 12: Measured curves with the dispersion scan method. (a) Mean vertical
position of the slices on the screen along the bunch for different dispersion
values. (b) Mean vertical position of the slices on the screen along the bunch
for different TDS voltages. (c) Vertical size of the slices on the screen
along the bunch for different dispersion values. (b) Vertical size of the
slices on the screen along the bunch for different TDS voltages. The gray
dotted lines present the position of the reference slice.

At the first step of the energy scan method we conducted the measurements at
constant dispersion $D_{0}=1.181$ m with different voltages $V$ of TDS. The
voltages are listed in the first row of the Table 3. The measured values of
the slice width $\sigma_{M}$ together with the errors are listed in the second
row of the Table.

At the second step we conducted the measurements at constant TDS voltage
$V_{0}=0.61$ MV with different dispersion values $D$ as listed in the third
row of the Table 3. The measured values of the slice width $\sigma_{M}$
together with the errors are listed in the last row of the Table.

Table 3: Two first rows show the beam sizes measured at different TDS voltages. The last two rows present the beam sizes measured at different dispersion values. $V$ | MV | 0.375 | 0.469 | 0.563 | 0.657 | 0.751
---|---|---|---|---|---|---
$\sigma_{M}$ |  $\mu$m | $69.87\pm 0.12$ | $70.64\pm 0.10$ | $71.86\pm 0.13$ | $72.85\pm 0.17$ | $74.12\pm 0.14$
$D$ | m | 0.578 | 0.789 | 1.006 | 1.181 |
$\sigma_{M}$ |  $\mu$m | $50.62\pm 0.08$ | $57.49\pm 0.09$ | $65.43\pm 0.1$ | $72.05\pm 0.1$ |

Fig. 12 shows the curves measured in the core of the electron bunch. At the
position of the OTR screen the electron bunch has rms length of 4 ps. Figures
(a) and (b) show vertical position of the slice on the screen. Figures (c) and
(d) show the slice width on the screen. The gray dotted line defines the
position of the reference slice. It is the extremum of the curves shown in
plots (a) and (b). The same as in the simulations the reference slice do not
have the minimal width but its position well defined by the extremum of the
mean slice position curve.

The measured values from Table 3 correspond to the coordinate $t=0$ ps at Fig.
12. They are plotted in Fig. 13 by black circles with error bars. The blue
dotted lines are obtained by the numerical fit to Eq.(9) (left plot) and
Eq.(8) (right plot).

The coefficients of Eq.(8), Eq.(9) obtained from the scans are listed in Table
4. In the same Table are presented the physical values of interest with the
estimated errors. They are obtained with the help of Eqs.(10)-(11). The errors
are estimated by the numerical experiment described in Section II.1.

If we take into account that the estimated instrumental errors in the setup of
the TDS voltage and dispersion are smaller than 2 % then we can state that the
uncorrelated energy spread in the core of the beam is equal to $5.9\pm 0.1$ k
eV.

Figure 13: The black circles with error bars at the left plot show the measured slice width $\sigma_{M}$ for different dispersion values $D$. The black circles with error bars at the right plot show the measured slice width $\sigma_{M}$ for different values of TDS voltage $V$. The blue dotted lines are obtained by the numerical fit to Eq.(9) and Eq.(8). Table 4: The reconstructed data from the measurements with the dispersion scan method. $A_{V}$ | $B_{V}$ | $A_{D}$ | $B_{D}$ | $\sigma_{E}$ | $\sigma_{I}$ | $\sigma_{B}$ | $\sigma_{R}$ | $\epsilon_{n}$
---|---|---|---|---|---|---|---|---
$m^{2}$ | $m^{2}/MV^{2}$ | $m^{2}$ |  | keV |  $\mu$m |  $\mu$m |  $\mu$m |  $\mu$m
4.68e-9 | 1.45e-9 | 1.75e-9 | 2.48e-9 | $5.948\pm 0.06$ | $71.4\pm 3$ | $31.4\pm 1.3$ | $27.6\pm 1.5$ | $0.42\pm 0.02$

Note that the emittance estimation agrees well with the independent method of
the mesurement of the beam emittance (see Fig. 15). Finally the estimated
screen resolution $\sigma_{R}$ agrees with the numbers published in OTR ;
Prat20 .

###  IV.2 Results obtained with the energy scan

We have done the energy scans with dispersion values of 0.6 and 1.2 meters.
The results are shown in Fig. 14. We were not able to do the reconstruction
from the data measured and simply compare the measurements with the expected
values calculated from the results of the previous method presented in Table
4.

Taking into account the issues with the beam matching and non-constant slice
emittance (see Section III.2) we think that there is no contradiction between
the data.

It is shown in Section III.2 that the energy scan method could be used but
requires stringent control of the shape of the longitudinal phase space and
very accurate matching of the beam to the optics before TDS. Unfortunately, we
have not managed in two very time-consuming experiments to show it.

Figure 14: Comparison of the measurements of the energy scan (dots with error
bars) with the values calculated from the data of the dispersion scan

###  IV.3 Validation of the experimental results

In this section we consider several arguments to confirm the accuracy of the
obtained data.

The energy spread estimation based on Eq.(10) uses only coefficient $A_{V}$
and $A_{D}$. But there is another equation

$\displaystyle\sigma_{E}=\frac{E_{0}}{D_{0}}\sqrt{D_{0}^{2}B_{D}-V_{0}^{2}B_{V}},$
(13)

based on two other coefficients, $B_{D}$ and $B_{V}$, from the numerical fits.
From Eq.(13) we obtain that the energy spread is equal to 5.946 keV that
agrees with the previous estimation (see Table 4) with accuracy 0.03%.

Figure 15: The slice emittance along the bunch measured by standard method
Bolko . Black line on the right plot is BMAG parameter Sands .

In order to check the estimation of the emittance $\epsilon_{n}$ we have done
an independent measurement of the slice emittance with the standard tools
Bolko  used by operators of the facility. The results of independent
measurement of the slice emittance are shown in Fig. 15 and the emittance of
the central slice (slice index 0) agrees with the value listed in Table 4.

We had additional possibility to do the measurement of the slice energy spread
with the laser heater tuned for maximal SASE radiation energy. We have found
that the energy spread in the electron bunch was $7.5\pm 0.1$ keV.

In theoretical studies of microbunching carried out by our colleague M. Dohlus
(see, for example, Dohlus ) the optimal energy spread after laser heater for
microbunching suppression is nearly 8 keV. This number agrees reasonable with
the measured one.

##  V Discussion

The theoretical calculations with different numerical models predict the
uncorrelated energy spread below 1 keV. The discrepancy between the
theoretical estimations and the measurements could be caused by neglecting of
full physics in the simplified numerical models. For example, it could be that
the emission process from the cathode should be simulated differently.
Additionally we do not take into account the intrabeam scattering and
wakefields in the RF gun cavity. The number of macroparticles used in the
simulations does not allow to take into account the microbunching during the
transport from the gun to the OTR screen.

It was shown in DiMitri ; Huang  that the intrabeam scattering in the injector
section increases the energy spread considerably and has to be taken into
account. For example, a simple estimation of the induced energy spread due to
IBS from Huang  reads

$\displaystyle\sigma_{E}^{IBS}=\frac{2r_{e}^{2}N_{b}}{\epsilon_{n}}\int{\frac{ds}{\sigma_{x}\sigma_{z}}},$
(14)

where $r_{e}$ is the electron radius, $N_{b}$ is the number of electrons,
$\sigma_{z}$ is the rms length of the bunch, $\sigma_{x}$ is the transverse
rms size and integration is done along the bunch path $s$. If we use this
equation with the parameters used in the paper we obtain that the energy
spread introduced during the beam transport from the gun to OTR is about 2
keV. Hence it is considerable effect and should be taken into account in the
simulations. We are considering now different models of IBS to include IBS in
the beam dynamics codes

The energy spread from the RF gun measured at the European XFEL for charge of
250 pC is $5.9\pm 0.1$ keV. This number is approximately 3 times lower then
the energy spread of $14.8\pm 0.6$ keV reported recently by SwissFEL for the
bunch charge of 200 pC Prat20  . The both guns use cesium telluride cathodes
and the larger difference between these results requires additional efforts to
understand.

##  VI Summary

We have described two methods for measurement of the slice energy spread of
electron bunch. With the beam dynamics simulations we have identified
substantial difficulties of the first method based on energy scan: we need
match the beam and the slice emittance changes. The difficulties are confirmed
in the real experiment.

We have shown with the beam dynamics simulations and the measurements that the
second method based on dispersion scan at the constant beam energy shows high
accuracy and easy to conduct.

At the same time the measured slice energy spread of $5.9\pm 0.1$ keV is
several times higher than theoretically estimated and it requires additional
theoretical research to clarify.

##  VII Acknowledgement

The authors thank M. Dohlus and M. Krasilnikov for helpful discussions. We
thank members of the European XFEL team for providing help and conditions to
carry out the measurements.

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