3 Design of the TDS longitudinal diagnostic sections for the European XFEL
3.4 Simulations with S2E bunch
3.4.3 Longitudinal phase space measurement
3.4 Simulations with S2E bunch
−800 −400 0 400 800
0.5 1 1.5
t(fs) εy(µm)
−800 −400 0 400 800
1 2 3 4 5
t(fs) My
ref. (@S1, TDS off, kicker off) simulation:V0=15 MV simulation:V0=4 MV
Figure 3.33:Slice emittanceεyand mismatch parameterMydetermined from simulations with the TDS op-erated at the effective voltages of (blue)V0=15 MV and (green) 4 MV. The grey lines represent the scaled current profile (in arbitrary unit) obtained at the screen S1 with the TDS effective voltage ofV0 =15 MV in the simulations.
parameters due to the variation of the twiss parameters in the slices along the bunch. It is common to match the central slice of the bunch, the slice containing the peak current9or the projection. For example, when the central slice is matched to the design twiss parameters (see, e.g., Fig.3.31), which is in particular for the S2E bunch at the same time the case of matching the slice containing the peak current, the mismatch parameter increases up toM =4 towards the ends of the bunch. Figure3.35 compares the simulation results using input distributions with matched central slice and matched projection. No obvious difference is observed in the two cases. Since the twiss parameters of the central slice are similar to these of the projected bunch, the two different matching procedures do not differ much from each other. The evolutions of the slice mismatch parameters in the reference distribution of the case with matched central slice (red solid) and matched projection (red dashed)are comparable.
3 Design of the TDS longitudinal diagnostic sections for the European XFEL
−800 −600 −400 −200 0 200 400 600 800 50
100 150
equivalent slice width ∆xslice
@S1,V0=4 MV
t(fs) slicebeamsizeσx0(µm)
Figure 3.34:Initial horizontal slice beam sizeσx0 determined from the reference particle distribution. The equivalent slice width ∆xsl iceat S1 in the case ofV0=4 MV is indicated for comparison.
−800 −400 0 400 800
0.5 1 1.5 2
t(fs) εy(µm)
−800 −400 0 400 800
1 2 3 4 5
t(fs) My
ref.: central slice matched ref.: projection matched simu.: central slice matched simu.: projection matched
Figure 3.35:Slice emittanceεyand mismatch parameterMydetermined from simulations using input particle distribution with (blue) matched central slice and (green) matched projection. The slice parameters of the reference particle distribution with (red solid) matched central slice and (red dashed) matched projection are shown. The TDS is assumed to be operated at an effective voltage ofV0=15 MV, and the slice width is chosen as ∆tsl ice =60 fs. The grey lines represent the scaled current profile (in arbitrary unit) obtained at the screen S1 using a bunch with matched central slice in the simulations.
Calibration
Calibration of the streak parameterS is simulated by determining the horizontal position of the bunch centre on the screen in dependence of the change of the TDS RF phase around the zero-crossing (see Eq.2.36). Calibration of the dispersionDyat the screen location is simulated by record-ing the vertical position of the bunch centre in dependence of the change in the current of the dipole magnet (see Eq.2.43). The simulation results of the calibrations are shown in Fig.3.36. The obtained calibration constants are comparable to the design values as listed in Table3.4. Using the streak
pa-3.4 Simulations with S2E bunch
rameter obtained from the simulation, the longitudinal resolution measurement can be simulated (see Eq.2.21) and has been determined to beRt=12 fs.
−0.4 −0.2 0 0.2 0.4
−4
−2 0 2 4
Rt=12 fs
2π fc ∆ϕ(mm)
∆⟨x⟩(mm)
simulation fit:S= −15.5
−1 −0.5 0 0.5 1
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
δI· 103
∆⟨y⟩(mm)
simulation fit: Dy =0.54 m
Figure 3.36:Simulation of the calibration measurement for (left) the streak parameterSand (right) the verti-cal dispersionDyin the longitudinal phase space measurement. In the calibration of the streak parameter, the horizontal position change of the bunch ∆⟨x⟩at the screen is determined for different TDS RF phases.
The TDS phases are changed to an offset ∆ϕfrom the zero-crossing phase, and scaled to 2π fc ∆ϕwithc be-ing the speed of light and fthe frequency of the TDS. The linear fit to the simulation data yields the streak parameter. The longitudinal resolutionRt at the screen is determined to be 12 fs from the simulation. In the calibration of the dispersion, the current of the dipole magnet is changed byδIand the corresponding vertical position change of the bunch ∆⟨y⟩is recorded. The linear fit yields the additive inverse value of the dispersion.
Longitudinal phase space
The longitudinal phase space can be then reconstructed by calibrating the horizontal and vertical axis of the simulated image on the screen with the calibration constants. Figure3.37compares the recon-struction at the screen with the reference longitudinal phase space of the input particle distribution at the entrance of the TDS. The colour code of the simulated image scales with the electron density.
Very good agreement has been achieved, except in the leading part (att< −600 fs) and trailing part (att >550 fs) of the bunch, which contain only a fraction of 2% and 3% of the total charges in the bunch, respectively.
Projection of the image onto the two axis yields the longitudinal and energy profile (see Fig.3.38).
Both profiles are in excellent agreement with these of the reference particle distribution. The rms bunch length ofσt =304 fs and rms energy spread ofσδ =10.5 · 10−3determined from the profiles are within a deviation of<1% compared to the reference values.
3 Design of the TDS longitudinal diagnostic sections for the European XFEL
−800−600−400−200 0 200 400 600 800
−0.02
−0.01 0 0.01 0.02
t(fs)
δ
reference (@in TDS)
0 0.2 0.4 0.6 0.8 1
Figure 3.37:Calibrated image from the simulation representing the longitudinal phase space. The colour code of the image scales with the electron density. The longitudinal phase space of the reference particle distri-bution at the entrance of the TDS is shown in red.
−500 0 500
0 500 1000 1500
t(fs)
I(A)
simulation:σt=302 fs reference:σt=304 fs
0 1 · 10−8 2 · 10−8 3 · 10−8 4 · 10−8
−0.02
−0.01 0 0.01 0.02
Q/δ
δ
simulation:σδ=10.4 · 10−3 reference:σδ=10.5 · 10−3
Figure 3.38:Longitudinal current profile (left) and energy profile (right) obtained from the simulated image as shown in Fig.3.37.
Slice energy spread
When the simulated image of the longitudinal phase space is divided into longitudinal slices, the slice energy spread can be derived from the measurement as well. Firstly, a slice width of ∆tslice =60 fs= 5Rtis chosen. As shown in Fig.3.39(left), the rms slice energy spread of the simulated longitudinal phase space (blue dotted) is compared with that of the reference particle distribution at the entrance of
3.4 Simulations with S2E bunch
the TDS (red solid, noted as @in TDS). All slices of the simulation display significantly larger energy spreads than those of the reference. The same behaviour is observed in the results with smaller slice widths of ∆tslice =30 fs≈2.5Rtand ∆tslice =10 fs≈Rtas well (see Fig.3.39middle and right). In all three cases with different definitions of the slice width, the evolution of the slice energy spread along the slices from the simulation has similar shape to that of the reference. With smaller slice width, the discrepancy of the simulation from the reference becomes larger. It is worth to note that the slice energy spreads decrease with reduced slice width in both the simulation and reference. It can be explained by the fact that the large energy chirp (correlation in(t,δ)) of the bunch contributes a lot to the rms value of the energy distribution in the slices.
−600−300 0 300 600 0
0.5 1
t(fs) σδ·103
∆tslice=60 fs
−600−300 0 300 600 t(fs)
∆tslice =30 fs
−600−300 0 300 600 t(fs)
∆tslice =12 fs ref. (@in TDS) ref. (@out TDS) simulation
Figure 3.39:Slice energy spread determined from: (red solid) reference particle distribution at the entrance of the TDS, (red dashed) reference particle distribution at the exit of TDS and (blue dot) simulated image as shown in Fig.3.37. From left to the right, the different definitions of the slice width ∆tsl icerelate to the longitudinal resolution on the screen as 5Rt, 2.5RtandRt, respectively.
In order to understand the discrepancy, the particle distribution directly at the exit of the TDS (noted as @out TDS) is taken as a second reference and its slice energy spread is plotted in Fig.3.39as well (red dashed). The bunch at the exit of the TDS has slightly increased slice energy spread resulting from the induced energy gain from the TDS as expected from Eq.2.44. However, the induced energy spread from the TDSσIESis in the order ofσIES∼10−4and still cannot explain the large discrepancy between the slice energy spreads of simulations and reference.
Another speculation on the reason for the discrepancy is the initial vertical beam sizeσy0of the slices in the dispersion direction. The measurable beam size from the simulated image is given as σy=√
(σ0y)2+D2yσδ2+D2yσIES2 , from which the energy spread is then determined as σy/Dy. As a result, the derived energy spread is actually larger than the real one. In a real longitudinal phase space measurement, the initial vertical beam sizeσ0y in the slices is not accessible and cannot be
3 Design of the TDS longitudinal diagnostic sections for the European XFEL
corrected for the calculation of the slice energy spread10. With the help of elegant simulations, such virtual beamline has been designed. The correction to the slice energy spread due to the initial slice beam size is estimated to be approximatelyσy0/Dy=0.05 · 10−3, which does not account for the large discrepancy as well.
−50 0 50
−2 0 2
t(fs)
δ·103
reference (@in TDS)
−0.4
−0.2 0
0.2
−2 0.4
−1 0 1 2
x(mm)
y(mm)
particle distribution @ screen
Figure 3.40:(Left) Longitudinal phase space of the central part of the reference particle distribution at the entrance of the TDS. Electrons in three different slices are marked in blue, green and yellow. Slices with a separation of 2 · ∆tsl icefrom each other are highlighted for clarification. (Right) Tracked particle distribu-tion at the screen in the transformed longitudinal phase space(x,y). The horizontal and vertical axis can be then calibrated totandδ, respectively. The vertical black lines represent the subdivision of the slices, which is applied for the determination of the slice energy spread presented in Fig.3.39.
Figure3.40(left) shows electrons in three slices (marked as blue, green and yellow) in the longi-tudinal phase space of the reference particle distribution. The tracked distribution in the transverse plane(x,y)(the transformed longitudinal phase space) at the screen is shown in Figure3.40(right), with the electrons in the three slices being marked accordingly. It can be seen that the original slices are distorted in the transformed coordinates(x,y). The initial finite slice beam size in the streak directionxcauses the shearing of the slices in that direction. Slicing of the bunch in thexdirection, which is represented by the black lines in Fig.3.40(right), is not appropriate.
Special slicing procedure
A special slicing procedure has been introduced to take into account the effect of the finite slice beam size in the streak direction. As illustrated in Fig.3.41(left), the mean position of each row on the en-ergy axis (blue line) is determined from the simulated image. The boundary of the slices are parallel to the time axis, in contrast to the common slicing procedures where the slices are perpendicular to
10In order to measure the initial vertical beam size in the slices, it requires to switch off the dipole magnet and measure the slice beam size at a screen downstream of the dipole magnet in a straight beamline that has the same beamline layout as the dispersive beamline.