Empirical Optimization of Undulator Tapering at FLASH2 and Comparison with Numerical Simulations
aa r X i v : . [ phy s i c s . acc - ph ] A ug Empirical Optimization of Undulator Tapering at FLASH2and Comparison with Numerical Simulations
Alan Mak a, ∗ , Francesca Curbis a , Bart Faatz b , Sverker Werin a a MAX IV Laboratory, Lund University, Fotongatan 2, S-22594 Lund, Sweden b Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany
DESY Report 16-168 (Internally reviewed; approved for publication 29 th August, 2018)
Abstract
In a free-electron laser equipped with variable-gap undulator modules, the technique of un-dulator tapering opens up the possibility to increase the radiation power beyond the initialsaturation point, thus enhancing the efficiency of the laser. The effectiveness of the enhance-ment relies on the proper optimization of the taper profile. In this work, a multidimensionaloptimization approach is implemented empirically in the x-ray free-electron laser FLASH2. Theempirical results are compared with numerical simulations.
Keywords: free-electron laser, undulator tapering, experiment, numerical simulation
1. Introduction
FLASH [1] is the free-electron laser (FEL) facility at the Deutsches Elektronen-Synchrotron(DESY) in Hamburg, Germany. It contains two undulator beamlines, FLASH1 and FLASH2,driven by the same linear accelerator. While FLASH1 consists of fixed-gap undulator modules,FLASH2 is equipped with variable-gap undulator modules. The variable-gap feature enablesthe simultaneous operation of FLASH1 and FLASH2 at different wavelengths [2]. It also enablesthe implementation of undulator tapering in FLASH2.Undulator tapering involves the variation of the undulator parameter K as a function ofthe distance z along the undulator line, for the purpose of enhancing the radiation power (andhence the efficiency) of the FEL. This has been demonstrated empirically in x-ray FELs, suchas LCLS [3] and SACLA [4]. In order to maximize the enhancement of radiation power, thetaper profile K ( z ) needs to be properly optimized.Present-day imaging experiments at x-ray FELs call for an increased number of photonswithin a shorter pulse duration [5, 6]. To meet the stringent demand on the radiation power,the theory of taper optimization has been revisited in recent years. In Ref. [7], an importantstep is made towards the formulation of a universal taper law. In Refs. [8, 9], taper optimizationmethods based on the classic Kroll-Morton-Rosenbluth (KMR) model [10] are demonstrated innumerical simulations. In Refs. [11, 12], a multidimensional optimization method is performed innumerical simulations, whereby the optimal taper profile K ( z ) is obtained by scanning througha parameter space comprising the taper order (such as linear and quadratic), the taper startpoint, the taper amplitude etc.The multidimensional optimization approach is relatively straightforward. Guided by thetheoretical studies, this approach is implemented empirically in FLASH2 at a wavelength of44 nm, and the results are presented in this article. The empirical results of the taper optim-ization are then compared with the corresponding numerical simulations. The agreement anddiscrepancies between the empirical and simulation results are analyzed. The article concludesby excluding a number of otherwise possible causes of the discrepancies. ∗ Corresponding author
Email address: [email protected] . Empirical Study FLASH2 contains a total of 12 undulator modules. Between every two adjacent modules,there is a drift section for beam focusing, trajectory correction, phase shifting, diagnostics etc.Table 1 shows the known machine parameters. For machine parameters not listed in Table 1,the nominal design values [1] are assumed.
Parameter Symbol Value
Electron beam energy γm e c
646 MeVBunch charge Q
300 pCRadiation wavelength λ
44 nmRepetition rate R λ w L mod L FODO
Table 1: Machine parameters
Each of the 12 undulator modules ( m = 1 , , ...,
12) is set to an undulator parameter K m .Within each module, the undulator parameter is uniform. The taper profiles considered in thisempirical study are defined by three parameters: the taper order d , the start module n and thetaper amplitude ∆ K/K . These taper profiles are given by the ansatz K m = K for 1 ≤ m < nK " − (cid:18) ∆ KK (cid:19) (cid:18) m − n + 112 − n + 1 (cid:19) d for n ≤ m ≤ . (1)In Eq. (1), K is the initial undulator parameter, in resonance with the initial energy of theelectron beam. The undulator parameter remains K from modules 1 to n −
1, and decreasesin steps from module n onwards. The taper order d equals 1 for linear tapering, and 2 forquadratic tapering. The taper amplitude ∆ K/K is defined such that the undulator parameterof the last module is K = K − ∆ K .Multidimensional optimization is performed by scanning d , n and ∆ K/K empirically for thehighest final radiation energy. The same type of multidimensional optimization in numericalsimulations is presented in Refs. [11, 12].
In the drift section between every two undulator modules, there is a phase shifter for theproper matching of the phase between the electron beam and the optical field. The phaseshifters are characterized in Ref. [13]. The required phase shift in each drift section dependssolely on the undulator parameter of the preceding undulator module. The phase shifts areimplemented automatically by a baseline procedure to ensure constructive interference betweenthe optical fields emitted before and after each drift section. The procedure also accounts forthe phase advance caused by the fringe fields at the two ends of each undulator module.
The lasing of FLASH2 takes place through the process of self-amplified spontaneous emis-sion (SASE). The energy of a radiation pulse is measured with a micro-channel plate (MCP)detector [14], located downstream after the 12 undulator modules. The MCP detector offersa relatively high accuracy over a dynamic range of radiation intensities. To account for theshot-to-shot variability, each energy measurement is averaged over about 100 pulses.2 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 1: Empirical data. The final pulse energy is plotted as a function of the taper amplitude ∆
K/K for (a)linear tapering ( d = 1) and (b) quadratic tapering ( d = 2). The blue solid curve, red dashed curve and yellowdotted curve correspond respectively to start modules n = 6, 7 and 8. Undulator module number P u l s e ene r g y [ µ J ] Pulse Energy Evolution (a)
No taperOptimal linear taperOptimal quadratic taper 1 2 3 4 5 6 7 8 9 10 11 12
Undulator module number N o r m a li z ed K Input Taper Profile (b)
No taperOptimal linear taperOptimal quadratic taper
Figure 2: Empirical data. The evolution of the (a) optical pulse energy and (b) input undulator parameter alongthe undulator line. The undulator parameter is normalized to the initial value. The blue solid curve, red dashedcurve and yellow dotted curve correspond respectively to no taper (∆
K/K = 0), the optimal linear taper ( n = 7,∆ K/K = 4%) and the optimal quadratic taper ( n = 6, ∆ K/K = 6%).
The gas-monitor detector (GMD) [15], which is also located downstream after the 12 un-dulator modules, measures the optical pulse energy in parallel. The GMD reading is used as across check.With all the 12 undulator modules engaged, the MCP and GMD measure the final pulseenergy. To examine the evolution of the pulse energy along the undulator line, it is necessaryto measure the intermediate pulse energy upstream. To measure the pulse energy immediatelyafter an upstream undulator module, the gaps of all subsequent modules are opened, so thatthe optical pulse propagates towards the detectors without further interacting with the electronbeam. During the propagation, the optical pulse undergoes vacuum diffraction, and its trans-verse size can increase. So long as the detectors collect the signal of the entire optical pulse,the pulse energy remains unchanged.
The final optical pulse energy is measured for different taper profiles given by Eq. (1). Themeasurement is done for taper orders d = 1 , n = 6 , ,
8. The resultsare shown in Fig. 1. Each data point in Fig. 1 is obtained with the MCP detector, and isthe average over 140 ±
50 pulses. The error bar indicates the standard deviation of the MCPreadings. Among all the taper profiles considered in Fig. 1, the optimal linear taper occurs at n = 7 and ∆ K/K = 4%, whereas the optimal quadratic taper occurs at n = 6 and ∆ K/K = 6%.For the optimal linear taper, the optimal quadratic taper and no taper, the intermediatepulse energies are measured. The evolution of the pulse energy along the undulator line is shown3 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 3: Simulation results. The final pulse energy is plotted as a function of the taper amplitude ∆
K/K for(a) linear tapering ( d = 1) and (b) quadratic tapering ( d = 2). The blue solid curve, red dashed curve and yellowdotted curve correspond respectively to start modules n = 6, 7 and 8. in Fig. 2(a). The corresponding taper profiles, as input from the control room, are shown inFig. 2(b) for reference. Each data point in Fig. 2(a) is obtained with the MCP detector, and isthe average over 110 ±
30 pulses. Among all the data points in Figs. 1 and 2(a), the absolutedifference between the MCP and GMD values is 19 µ J on average, with a standard deviationof 16 µ J.In the absence of tapering, the saturation of pulse energy is reached in module 8 [see solidcurve in Fig. 2(a)]. In other words, the start modules ( n = 6 , ,
8) considered in Fig. 1 are inthe vicinity of the initial saturation point.
3. Comparison with Numerical Simulation
The empirical results are compared with numerical simulation, after the experiment hasbeen completed. The simulation is performed using the three-dimensional and time-dependentsimulation code GENESIS [16], with parameter values as close as possible to the empirical ones(see Table 1). Parameters not specified in Table 1 are assumed to have the nominal valuesshown in Table 2.
Parameter Symbol Value
Peak current I σ z µ mRMS energy spread σ γ m e c ε x,y β x,y Table 2: Nominal FLASH2 parameter values used in the simulation
In the simulation, the initial values of the optical functions and the quadrupole strengthsare chosen self-consistently to give the desired average beta value, independent of the valuesused in the experiment.
The same multidimensional optimization is performed in simulation. Using Eq. (1) as theansatz, the parameters d , n and ∆ K/K are scanned for the highest final radiation energy. Theresults are shown in Fig. 3. Among all the taper profiles considered in Fig. 3, the optimal lineartaper occurs at n = 7 and ∆ K/K = 4%, whereas the optimal quadratic taper occurs at n = 6and ∆ K/K = 6%. 4
Undulator module number P u l s e ene r g y [ µ J ] Pulse Energy Evolution (a)(a)(a)
No taperOptimal linear taperOptimal quadratic taper 1 2 3 4 5 6 7 8 9 10 11 12
Undulator module number N o r m a li z ed K Taper Profile (b)(b)(b)
No taperOptimal linear taperOptimal quadratic taper
Figure 4: Simulation results. The evolution of the (a) optical pulse energy and (b) undulator parameter alongthe undulator line. The undulator parameter is normalized to the initial value. The blue solid curve, red dashedcurve and yellow dotted curve correspond respectively to no taper (∆
K/K = 0), the optimal linear taper ( n = 7,∆ K/K = 4%) and the optimal quadratic taper ( n = 6, ∆ K/K = 6%).
For the optimal linear taper, the optimal quadratic taper and no taper, the simulated pulseenergy evolutions along the undulator line are shown in Fig. 4(a). The corresponding taperprofiles are shown in Fig. 4(b) for reference.
Comparing Fig. 1 (empirical) and Fig. 3 (simulation), the optimal taper profiles are consist-ent. In both cases, the optimal linear taper occurs at n = 7 and ∆ K/K = 4%, and the optimalquadratic taper occurs at n = 6 and ∆ K/K = 6%. Figs. 1 and 3 also show good agreement inthe overall trend for the final optical pulse energy E . In both cases, the overall trend for lineartapering ( d = 1) is E ( n = 7 , ∆ K/K ) > E ( n = 6 , ∆ K/K ) > E ( n = 8 , ∆ K/K ) , whereas the overall trend for quadratic tapering ( d = 2) is E ( n = 6 , ∆ K/K ) > E ( n = 7 , ∆ K/K ) > E ( n = 8 , ∆ K/K ) . However, Figs. 1 and 3 show disagreement in terms of the absolute pulse energies. The rangeof pulse energies is generally higher in the simulation than in the experiment.Next, the pulse energy evolution along the undulator line is compared between simulation[see Fig. 4(a)] and experiment [see Fig. 2(a)]. In both cases, the pulse energy remains in theorder of 1 µ J before module 5, and exceeds the 10- µ J threshold in module 5. In the absenceof tapering, the initial saturation point is situated around module 8 in both cases (see solidcurves). With the optimal linear and quadratic tapers, final saturation is reached within the12 undulator modules in both simulation and experiment, but occurs earlier in the experimentthan in the simulation (see dashed and dotted curves).In the experiment, the optimal linear taper and the optimal quadratic taper yield almostidentical final pulse energy. But in the simulation, the final pulse energy for the optimal quad-ratic taper is 1.2 times higher than that for the optimal linear taper.In the experiment, the enhancement factor is E (optimal taper) E (no taper) = 1 . . But in the simulation, the enhancement factor is E (optimal taper) E (no taper) = 3 . , which is 2.6 times higher than that in the experiment.5 Undulator module number E [ G V / m ] Field Amplitude Evolution (a)
No taperOptimal linear taperOptimal quadratic taper 1 2 3 4 5 6 7 8 9 10 11 12
Undulator module number -1001020304050 ψ R [ deg r ee s ] Resonant Phase Evolution (b)
No taperOptimal linear taperOptimal quadratic taper
Figure 5: Simulation results. The evolution of the (a) optical field amplitude E and (b) resonant phase ψ R alongthe undulator line. The blue solid curve, red dashed curve and yellow dotted curve correspond respectively tono taper (∆ K/K = 0), the optimal linear taper ( n = 7, ∆ K/K = 4%) and the optimal quadratic taper ( n = 6,∆ K/K = 6%).
In both the simulation and empirical results, the optimal linear taper starts from module7, while the optimal quadratic taper starts from module 6. The reason for this difference inthe optimal start point is that the undulator parameter decreases much more slowly at thebeginning of the quadratic taper. This is seen in Figs. 2(b) and 4(b). In module 6 from whichthe quadratic taper starts, the undulator parameter K is effectively identical to the initial value K , as K = 99 . × K ≈ K . It is in module 7 where the undulator parameter starts toshow a significant difference from the initial value. In other words, the optimal quadratic taperstarts effectively from module 7, the same module from which the optimal linear taper starts.Refs. [7, 17] suggest that the optimal taper start point is two gain lengths before the initialsaturation point. In one-dimensional theory, the gain length is given by L g = λ w √ πρ , (2)where ρ = 14 (cid:18) II A ε x,y ¯ β x,y (cid:19) / (cid:18) λ w Kf B πγ (cid:19) / (3)is the dimensionless Pierce parameter, I A = m e c /e = 17 .
045 kA is the Alfv´en current, σ x isthe rms radius of the electron beam, and f B = J ( ξ ) − J ( ξ ) is the Bessel factor for planarundulators, with ξ = K / [2( K + 2)].With the parameters in Tables 1 and 2, the Pierce parameter is ρ = 3 . × − , and thegain length is L g = 0 .
41 m. Thus, the optimal taper start point is predicted to be 2 L g = 0 .
82 mbefore the initial saturation point, excluding the length of the drift section between undulatormodules. If we assume that the precise initial saturation point is at the beginning of module8, then the optimal taper start point should lie within module 7. This rough prediction agreeswith the simulation and empirical results.
The Kroll-Morton-Rosenbluth (KMR) model [10] is a theoretical analysis of undulator taper-ing in FELs based on a one-dimensional relativistic Hamiltonian formulation. In Refs. [8, 9], theKMR model is used as a method to optimize FEL taper profiles in numerical simulations. Afterchoosing the resonant phase ψ R ( z ), the taper profile K ( z ) is computed from the differentialequation dKdz = − em e c λλ w f B ( z ) E ( z ) sin[ ψ R ( z )] , (4)6here E is the on-axis field amplitude and z is the position along the undulator line. Witha constant ψ R , the optimization is known as the ordinary KMR method. With a variable ψ R which increases gradually from zero, the optimization is known as the modified KMR method.With the simulation results at hand, the evolution of the resonant phase ψ R along theundulator line can be back-calculated from Eq. (4). This back-calculation requires the taperprofile K ( z ) [see Fig. 4(b)] and the field amplitude evolution E ( z ) [see Fig. 5(a)] as inputs.Carrying out this back-calculation for the optimal linear taper, the optimal quadratic taper andno taper, the resulting ψ R ( z ) functions are shown in Fig. 5(b).The optimal linear and quadratic tapers start from module 7 and module 6, respectively.Before the taper starts, ψ R = 0 [see dashed and dotted curves in Fig. 5(b)]. This is expected,as dK/dz = 0 implies ψ R = 0 according to Eq. (4). For the same reason, in the absence of anytapering, ψ R remains zero at all times [see solid line in Fig. 5(b)].When the optimal linear taper starts in module 7, ψ R increases abruptly from 0 to 12 ◦ ,and remains almost constant afterwards [see dashed curve in Fig. 5(b)]. When the optimalquadratic taper starts in module 6, ψ R increases gradually and monotonically from 0, until itreaches a value of 36 ◦ in the final module [see dotted curve in Fig. 5(b)]. The ψ R ( z ) function forthe optimal linear taper resembles one used in the ordinary KMR method, whereas the ψ R ( z )function for the optimal quadratic taper resembles one used in the modified KMR method.
4. Post-Experimental Analysis of the Discrepancies
The empirical and simulation results are in good agreement in terms of: • the ( n, ∆ K/K ) values for the optimal linear and quadratic tapers; • the overall trend in the plots of the final energy E versus ∆ K/K (see Figs. 1 and 3); and • the module in which the exponential gain crosses the 10- µ J threshold (see Figs. 2 and 4).However, there are three main discrepancies between the empirical and simulation results: • In the parameter space ( d, n, ∆ K/K ) considered, the E range is generally lower in theexperiment than in the simulation (see Figs. 1 and 3). • The enhancement factor E (optimal taper) / E (no taper) is 3.9 in the simulation, but only1.5 in the experiment. • With the optimal linear and quadratic tapers, final saturation occurs earlier in the exper-iment than in the simulation (see Figs. 2 and 4).The exact causes of these discrepancies are not known. Yet, it is possible to exclude anumber of otherwise possible causes, such as the shot-to-shot variability, drift of the machineand wakefield effects. These are addressed in the upcoming subsections.The discrepancies in question can also be caused by incorrect assumptions of parametervalues. For the simulation, the nominal FLASH2 parameter values in Table 2 are assumed. Theassumed nominal values in the simulation can be different from the unknown actual values inthe experiment.As illustrated in the sensitivity study in Ref. [18], a slight change in the emittance, energyspread or peak current can have a huge impact on the optimized radiation power of a taperedFEL. In other words, if the actual emittance, energy spread or peak current is worse thanassumed, then the optimized radiation energy will be lower than expected. This will, in turn,influence the enhancement factor. This can possibly explain the discrepancies in question.However, the proposition that the emittance, energy spread or peak current is worse thanassumed will be disproved in the following subsections.7 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 6: Simulation results with normalized emittance increased from 1.4 mm mrad to 1.6 mm mrad. The finalpulse energy is plotted as a function of the taper amplitude ∆
K/K for (a) linear tapering and (b) quadratictapering.
In the empirical results (Figs. 1 and 2), the shot-to-shot fluctuations are accounted for bythe error bar, which indicates the standard deviation of many shots. All the error bars arewithin ± µ J, which is too small to account for the discrepancies between the simulation andempirical results.
Consider two scenarios in particular, the optimal linear taper and no taper. Since theoptimal linear taper only starts from undulator module 7, the two scenarios are identical beforemodule 7. In principle, the two scenarios should yield the same pulse energy evolution beforemodule 7. This is precisely the case in the simulation [see solid and dashed curves in Fig. 4(a)],which is the ideal case free of any drift. But in the empirical results [see solid and dashed curvesin Fig. 2(a)], the two scenarios yield slightly different energies in modules 5 and 6. The energydifferences can be partly attributed to the drift of the machine. But despite the drift, the energydifferences are still within 24 µ J, which is too small to account for the discrepancies betweenthe simulation and empirical results.
In order to disprove that the emittance is underestimated, the simulation is repeated withthe normalized emittance slightly increased, from 1.4 mm mrad to 1.6 mm mrad. All otherparameters in Tables 1 and 2 are kept unchanged. With the average beta function ¯ β x,y keptunchanged, this requires increasing the RMS beam radius σ x,y from 82 µ m to 87 µ m. The newsimulation results are shown in Fig. 6.If the emittance were indeed underestimated in the original simulation, then the new sim-ulation (with an increased emittance) would show an improved agreement with the empiricalresults. But in the new simulation results, the overall trends of the final pulse energy E change.As seen in Fig. 6, the overall trend for linear tapering ( d = 1) becomes E ( n = 7 , ∆ K/K ) > E ( n = 8 , ∆ K/K ) > E ( n = 6 , ∆ K/K ) , whereas the overall trend for quadratic tapering ( d = 2) becomes E ( n = 6 , ∆ K/K ) ≈ E ( n = 7 , ∆ K/K ) > E ( n = 8 , ∆ K/K ) . The overall trends actually become further off from those in the empirical results (see Fig. 1).Meanwhile, there is no improved agreement in the E range and in the enhancement factor. Thisdisproves that the emittance is underestimated in the original simulation.8 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 7: Simulation results with peak current decreased from 1.5 kA to 1.2 kA. The final pulse energy is plottedas a function of the taper amplitude ∆
K/K for (a) linear tapering and (b) quadratic tapering.
Comparing the two sets of simulation results in Fig. 3 and 6, the increased emittance makesit more favourable to start the taper at a later point down the undulator line. The optimalquadratic taper in Fig. 3 starts from module 6, whereas that in Fig. 6 starts from module 7.As for linear taper, module 7 remains the most favourable start module. Yet, while module 8is the least favourable of the three start modules considered in Fig. 3, it becomes the secondmost favourable in Fig. 6.The shift in the optimal taper start point can be explained as follows. Refs. [7, 17] suggestthat the optimal taper start point z is two gain lengths before the initial saturation point. Inone-dimensional theory, this is given by z = L sat − L g = λ w ρ − (cid:18) λ w √ πρ (cid:19) = (cid:18) − √ π (cid:19) λ w ρ ∝ ρ . (5)With the definition of the Pierce parameter in Eq. (3), one can deduce that z ∝ ρ ∝ ( ε x,y ) / . (6)The proportionality implies that an increased emittance moves the optimal taper start pointdownstream. It also implies that a further increase in emittance would move the optimal taperstart point further downstream, thus making the overall trends of E even further off from thosein the empirical results. In order to disprove that the peak current is overestimated, the simulation is repeated withthe peak current slightly decreased, from 1.5 kA to 1.2 kA. In order to keep the known bunchcharge in Table 1 unchanged, this requires increasing the RMS bunch length from 24 µ m to 30 µ m. All other parameters in Tables 1 and 2 are kept unchanged. The new simulation resultsare shown in Fig. 7.Again, by decreasing the peak current in the simulation, the overall trends in the final pulseenergy E become further off from those in the empirical results (see Fig. 1). This disproves thatthe peak current is overestimated.Comparing the two sets of simulation results in Fig. 3 and 7, the decreased peak currentalso makes it more favourable to start the taper in a later undulator module. This agrees withthe one-dimensional theoretical prediction from Eqs. (3) and (5) that z ∝ ρ ∝ I / . (7)9 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 8: Simulation results with energy spread increased from 0.5 MeV to 0.7 MeV. The final pulse energy isplotted as a function of the taper amplitude ∆
K/K for (a) linear tapering and (b) quadratic tapering.
In order to disprove that the energy spread is underestimated, the simulation is repeatedwith the energy spread slightly increased, from 0.5 MeV to 0.7 MeV. All other parameters inTables 1 and 2 are kept unchanged. The new simulation results are shown in Fig. 8.Again, by increasing the energy spread in the simulation, the overall trends in the final pulseenergy E become further off from those in the empirical results (see Fig. 1). This disproves thatthe energy spread is underestimated.Comparing the two sets of simulation results in Fig. 3 and 8, the increased energy spreadalso makes it more favourable to start the taper in a later undulator module. However, it isimpossible to use the one-dimensional formulation to explain the shift in the optimal taperstart point caused by the increased energy spread, as it is done for the emittance and the peakcurrent. Nonetheless, the energy spread effects can be explained by similar arguments using thegeneralized formulation of Ming Xie [19]. In Ref. [16], a simulation study on the effects of wakefields is performed on a case of the TTF-FEL, which is the predecessor of the FLASH1 and FLASH2 facilities. The machine parametersused in the simulation study are in the same orders of magnitude as those in Tables 1 and 2. Thestudy identifies three major sources of wakefields, namely, the conductivity, surface roughnessand geometrical changes of the beam pipe along the undulator. The simulation on the TTF-FEL case shows that wakefields can reduce the saturation power of the FEL by three ordersof magnitude, while keeping the saturation length almost unchanged. In principle, wakefieldeffects can be a possible explanation for the discrepancies between our empirical and simulationresults for FLASH2. However, this can be disproved as follows.In the empirical optimization of undulator tapering, the optimal taper profile which max-imizes the final radiation energy is also that which best compensates the energy loss due towakefields [4]. Meanwhile, in the simulation which results in Fig. 3, wakefields are not con-sidered. If wakefield effects were significant, then the optimal taper profile should occur at verydifferent ( n, ∆ K/K ) values in the empirical and simulation results. But as seen in Figs. 1 and 3,this is not the case. In fact, the experiment and simulation yield the exact same ( n, ∆ K/K )values for the optimal linear taper, and for the optimal quadratic taper. This leads us to theconclusion that wakefield effects are not significant in the experiment, and therefore do notaccount for the discrepancies in question.
The ideal trajectory of the electron beam is the central axis along the undulator line. Butif the electron beam undergoes betatron oscillations as a whole, it deviates from the ideal10rajectory and is subject to trajectory errors. These errors can be caused by a combination ofmany factors, which include • the imperfect alignment of the undulator modules; • the imperfect alignment of the quadrupole magnets; and • the inclined injection of the electron beam to the undulator modules.Trajectory errors can degrade the FEL performance through a number of mechanisms [20].A complete analysis of all these mechanisms is not trivial. But in taper optimization studies,it is the undulator parameter K which characterizes a taper profile. The following discussionsshall focus on the implication of trajectory errors to K .The undulator parameter K is associated with the magnetic field strength B on the centralaxis of the undulator by the definition K = eλ w πm e c B . (8)In the presence of trajectory errors, the electron beam deviates from the central axis. Even ifthe on-axis field strength B were perfectly accurate, the electron beam would still experience afield strength different from the desired value B , hence an undulator parameter different fromthe desired value K . As derived in the Appendix, the effective undulator parameter is K eff = K cosh( k w y ) ≥ K, (9)where y is the deviation of the electron beam from the central axis, and k w = 2 π/λ w is theundulator wavenumber. The magnetic field strength experienced by an electron beam with atrajectory error y in an undulator with parameter K is equivalent to that experienced by anon-axis electron beam in an undulator of parameter K eff .The effective undulator parameter K eff also leads to a phase shift error. As mentionedin Section 2.3, the required phase shift in the drift section depends solely on the K value ofthe preceding undulator module. Given an input value K , the phase shifter is automaticallyadjusted to ensure proper phase matching at the end of the drift section. But if the effectivevalue is K eff = K , then a phase mismatch will occur. As derived in the Appendix, this phasemismatch is given by δφ = − k w L D K (cid:18) K (cid:19) − [cosh( k w y ) − , (10)Here L D is the drift section length, which is 800 mm in FLASH2.The simulation is now repeated with the K eff and δφ associated with a trajectory error of y = 250 µ m, calculated from Eqs. (9) and (10). With a trajectory error of y = 250 µ m, thedifference between K eff and K becomes comparable to the Pierce parameter ρ , and is thereforesignificant. The new simulation results are shown in Fig. 9. Again, the overall trends in the finalpulse energy E become further off from those in the empirical results (see Fig. 1). Thus, the K eff and δφ associated of a trajectory error of y = 250 µ m cannot account for the discrepanciesbetween the empirical and simulation results. In the preceding discussions, the different possible causes of the discrepancies in questionare considered separately. In the following, combinations of these factors will be discussed.Shot-to-shot fluctuations and the drift of the machine can each affect the measured opticalpulse energy by about 20 µ J. The combined effect is then 40 µ J, which is still too small toaccount for the discrepancies in question.The emittance, the peak current and the energy spread have been considered individually.As discussed in Sections 4.4–4.6, if any of these three parameters is worse than assumed, then11 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Linear Taper (a)(a)(a)(a)
Taper starts from module 6Taper starts from module 7Taper starts from module 8 0 2 4 6 8 10 ∆ K / K [%] P u l s e ene r g y a ft e r m odu l e [ µ J ] Quadratic Taper (b)(b)(b)(b)
Taper starts from module 6Taper starts from module 7Taper starts from module 8
Figure 9: Simulation results for K eff and δφ associated with a trajectory error of y = 250 µ m. The final pulseenergy is plotted as a function of the taper amplitude ∆ K/K for (a) linear tapering and (b) quadratic tapering. the optimal taper start point z will be shifted downstream [see e.g. Eqs. (6) and (7)]. Fromthis one can deduce that if all three (or at least two of the three) parameters are worse thanassumed, then the optical taper start point z will be shifted even further downstream. Thiswill, in turn, make the overall trends of the final pulse energy E even further off from thosein the empirical results. Thus, the discrepancies between the simulation and empirical resultscannot be explained by the combination of an underestimated emittance, an overestimated peakcurrent and an underestimated energy spread.There are no indications that the three parameters are much different from their designvalues. But in principle, one could consider different scenarios where one parameter is worsethan assumed while another parameter is better than assumed. One example examined innumerical simulation is the scenario where the normalized emittance is halved while the energyspread is doubled (results not shown). The resulting range of optical pulse energies becomescloser to that in the experiment. Yet, the overall trends of the final pulse energy E , as well asthe ( n, ∆ K/K ) values of the optimal tapers, become further off from those in the experiment.Even though there are possible explanations for some of the discrepancies between the em-pirical and simulation results, there is no simple explanation that would explain all differences.
5. Conclusion
A multidimensional optimization method has been implemented empirically in FLASH2, tooptimize the taper profile for the maximum radiation energy. The empirical results have beencorrelated to simulations.In the empirical study, the taper profile is characterized by the taper order d , the startmodule n and the taper amplitude ∆ K/K . For the optimal linear ( d = 1) and quadratic( d = 2) tapers, the evolution of the optical pulse energy along the undulator line was examined.The empirical results were compared with the corresponding results of numerical simulation.The two sets of results show good agreement in terms of the overall trend in the variation ofthe final pulse energy E with ∆ K/K . They also show good agreement for the optimal linearand quadratic tapers regarding the start module ( n ), the taper amplitude (∆ K/K ) and theexponential gain profile. However, there are discrepancies in terms of the general range of pulseenergies, the enhancement factor from tapering, as well as the final saturation points for theoptimal tapers.Possible causes of the discrepancies have been examined, and a number of them excluded,such as emittance, energy spread and peak current deviations. Also, shot-to-shot variation,the drift of the machine, wakefield effects, as well as the systematic K and phase shift errorsassociated with a beam trajectory error have been excluded.Remaining factors are mainly (i) a poor overlapping between the electron beam and theoptical mode, caused by the misalignment and mismatch of the electron optics; and (ii) phase12ismatch caused by random errors in the phase shifters. These remaining factors need to be in-vestigated in more detail. Further studies in numerical simulations and empirical measurementsare planned for the future. Acknowledgment
The authors would like to thank Katja Honkavaara and Siegfried Schreiber for their crucialroles in facilitating this international collaboration. The authors would also like to thank Ev-geny Schneidmiller, Markus Tischer and Mikhail Yurkov for their participation in the planningmeeting for the experimental work.
Appendix: Derivation of K eff and δφ In Section 4.8, the effective undulator parameter K eff associated with a trajectory error isdiscussed. This Appendix gives a derivation for K eff and the subsequent phase mismatch δφ .Consider a pair of magnetic poles in the undulator, directly opposite to each other. Definethe y -axis as the straight line passing through the middle points of the two pole tips. As usual,the z -axis is in the direction of beam propagation, perpendicular to the y -axis. A trajectoryerror in the y -direction changes the distance between the electron beam and the magnetic pole,which has a strong impact on the magnetic field strength experienced by the beam. Meanwhile,an trajectory error purely in the x -direction imposes no change on the beam-pole distance, andis therefore not treated here.Following the derivation in Ref. [21], the variation of the magnetic field strength B y alongthe y -axis is examined using a two-dimensional model in the yz -plane. Along the z -axis, themagnetic field strength is periodic, with a period of λ w = 2 π/k w . Assuming that the periodicvariation is perfectly sinusoidal, the following ansatz can be written for the magnetic scalarpotential: ϕ ( y, z ) = f ( y ) cos( k w z ) . (11)Here f ( y ) is an unknown function which depends only on y . The scalar potential ϕ has tosatisfy the Laplace equation ∇ ϕ ( y, z ) = 0 . (12)Substituting Eq. (11) into Eq. (12) results in the second-order ordinary differential equation d f ( y ) dy − k w f ( y ) = 0 , (13)to which the general solution is f ( y ) = A sinh( k w y ) + A cosh( k w y ) (14)with arbitrary constants A and A . Inserting this into Eq. (11), the scalar potential may berewritten as ϕ ( y, z ) = A sinh( k w y ) cos( k w z ) + A cosh( k w y ) cos( k w z ) . (15)The y -component of the magnetic field is then B y ( y, z ) = − ∂ϕ∂y = − k w A cosh( k w y ) cos( k w z ) − k w A sinh( k w y ) cos( k w z ) . (16)Recalling that the peak field on the z -axis is B y (0 ,
0) = B , we have A = − B /k w . Giventhe symmetry of the system about the plane y = 0, we have B y (+ y, z ) = B y ( − y, z ) and hence A = 0. With these results, Eq. (16) can be rewritten as B y ( y, z ) = B cosh( k w y ) cos( k w z ) . (17)13
800 -600 -400 -200 0 200 400 600 800 y [ µ m] K e ff Effective Undulator Parameter (a) -800 -600 -400 -200 0 200 400 600 800 y [ µ m] | δ φ | [ ° ] Phase Shift Error (b) K Figure 10: The following quantities are plotted as functions of the trajectory error y : (a) the effective undulatorparameter K eff of an undulator module and (b) the resulting error | δφ | in the phase shift immediately after theundulator module. These plots are made for a desired K value of 2.638, which is in resonance with the initialenergy of the electron beam. To examine the variation of B y along the y -axis, we set z = 0 and obtain B y ( y,
0) = B cosh( k w y ) . (18)In other words, if the electron beam has a trajectory error of y , then it experiences a field B y ( y,
0) as given by Eq. (18). Analogous to Eq. (8), the effective undulator parameter can bedefined as K eff ( y ) ≡ eλ w πm e c B y ( y,