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Start-to-end simulation of x-ray radiation of a next generation light sourceusing the real number of electrons
J. Qiang,* J. Corlett, C. E. Mitchell, C. F. Papadopoulos, G. Penn, M. Placidi, M. Reinsch,R. D. Ryne, F. Sannibale, C. Sun, and M. Venturini
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
P. EmmaSLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
S. ReichePaul Scherrer Institute, 5232 Villigen PSI, Switzerland
(Received 12 December 2013; published 20 March 2014)
In this paper we report on start-to-end simulation of a next generation light source based on a highrepetition rate free electron laser (FEL) driven by a CW superconducting linac. The simulation integratedthe entire system in a seamless start-to-end model, including birth of photoelectrons, transport of electronbeam through 600 m of the accelerator beam delivery system, and generation of coherent x-ray radiation ina two-stage self-seeding undulator beam line. The entire simulation used the real number of electrons(∼2 billion electrons/bunch) to capture the details of the physical shot noise without resorting to artificialfiltering to suppress numerical noise. The simulation results shed light on several issues including theimportance of space-charge effects near the laser heater and the reliability of x-ray radiation powerpredictions when using a smaller number of simulation particles. The results show that the microbunchinginstability in the linac can be controlled with 15 keV uncorrelated energy spread induced by a laser heaterand demonstrate that high brightness and flux 1 nm x-ray radiation (∼1012 photons=pulse) with fullyspatial and temporal coherence is achievable.
DOI: 10.1103/PhysRevSTAB.17.030701 PACS numbers: 29.27.-a, 41.60.Cr
I. INTRODUCTION
A high brightness/flux x-ray free electron laser with fullyspatially and temporally coherent radiation would providean invaluable tool for scientific discovery in condensedmatter physics, material science, chemistry, and biology.Recently, a high repetition rate soft x-ray FEL (a nextgeneration light source) was studied at Lawrence BerkeleyNational Laboratory [1] and is currently being activelypursued at SLAC National Accelerator Laboratory. Thislight source will not only provide spatially coherent x-rayradiation generated by self-amplified spontaneous emission(SASE) [2–5], but it will also provide both spatially andtemporally coherent x-ray radiation through two-stage selfseeding [6], where radiation from the first stage is used as aseed for amplification in the second stage. In both schemes,the quality of the electron beam entering the FEL undulatorwill play a critical role in determining the final x-rayradiation performance. During acceleration and transport
through the accelerator beam delivery system, these elec-trons are subject to various collective effects (e.g., space-charge effects) and the beam quality may be significantlydegraded due to the microbunching instability that growsfrom the electron beam shot noise. An accurate modeling ofthe shot noise in the electron beam together with variouscollective effects, including the electron x-ray radiationinteraction in the undulator, is essential in predicting theproperties of the final x-ray radiation.Start-to-end macroparticle simulation has been used in a
number of previous studies as an important tool forevaluating the design of light sources [7–13]. In this paper,we report on high resolution start-to-end simulation of thex-ray radiation in a high repetition rate, soft x-ray FEL. Arealistic number of electrons are used in this study toaccurately model the initial shot noise inside the beam. Fora given number of macroparticles, Nmp, the shot noise inthe simulation can be artificially magnified by a factor offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN=Nmp
p, where N is the real number of electrons. A low
pass filter was proposed to suppress the numerical noiseassociated with the use of a small number of macroparticlesin comparison with the real number of electrons and tocalculate the microbunching instability gain curve throughthe accelerator [14]. Applying such a low pass filter doesnot completely suppress the artificial noise in the numericalsimulation. This is because the shot noise covers a broad
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bandwidth in the frequency domain. In the particle-in-cellsimulation, the numerical grid size sets the bandwidth in thefrequency domain. A smaller number of macroparticlescauses larger fluctuation not just in the high frequencyregion but within the entire bandwidth including the lowfrequency region as shown in Fig. 1. Here, we calculatedthe amplitude of the Fourier coefficients difference betweenan analytical Gaussian function and a randomly sampledGaussian function using 10 thousand, 100 thousand, onemillion, and 10 million macroparticles as a function of thenormalized wave number. A low pass filter helps tosuppress the high frequency numerical noise associatedwith sampling using a smaller macroparticle number, butthe low frequency numerical noise still exists and increasesthe final modulation level after being magnified by themicrobunching instability through the linac. In our previousstudy, high resolution beam dynamics simulations werecarried out in the study of an old design of the FEL linac[15]. In this paper, we will report on the start-to-endsimulation of a new design of the next generation lightsource, starting from the photocathode and ending with thefinal FEL x-ray radiation.The computational tool used in this study is the IMPACT
parallel beam dynamics code framework. This frameworkincludes the time-dependent IMPACT-T code [16] and theposition-dependent IMPACT-Z code [17], which are based ona particle-in-cell method. The FEL simulation code,GENESIS [18], is imbedded into the framework to simulatethe x-ray radiation within an undulator. This integratedframework produces a single executable code whichfacilitates seamless start-to-end simulation in a singlerun. The macroparticle electrons pass from one moduleof the code to another module of the code directly throughthe internal memory of the computer. The IMPACT-T moduleis used to simulate the photoelectron production andacceleration inside the injector. The IMPACT-Z module isused to simulate the electron beam acceleration, compres-sion and transport through the linac and the spreader. The
GENESIS module is used to simulate the electron beaminteraction with x-ray radiation inside the undulators. Theself-consistent 3D space-charge effects, the acceleratingcavity structure wakefields, the longitudinal coherentsynchrotron radiation (CSR) wakefields, and the time-dependent x-ray/electron interaction inside the undulatorare included in the simulation.The remainder of the paper is organized as follows: after
the Introduction, the light source machine layout is pre-sented in Sec. II; the beam dynamics study in injector ispresented in Sec. III; the transverse space-charge effects arediscussed in Sec. IV; the study of microbunching instabilityin the whole beam delivery system is discussed in Sec. V;the start-to-end simulations of the x-ray SASE FELradiation and self-seeding FEL radiation are presented inSec. VI; and the summary is given in Sec. VII.
II. THE LIGHT SOURCE MACHINE LAYOUT
A schematic diagram of the next generation light sourcelayout is shown in Fig. 2. It consists of a high brightnesshigh repetition rate injector, a high stability CW super-conducting linac, a beam spreader, and an array of FELundulator beam lines. The injector includes a low rffrequency (187 MHz) high repetition rate (1 MHz orhigher) normal conducting gun to generate an electronbeam with 750 keV kinetic energy [19,20], a 1.3 GHzbuncher cavity to provide initial velocity bunching, andseven 1.3 GHz Tesla superconducting cavities to acceleratethe 300 pC electron beam to an energy of 96 MeV. It is thenfollowed by a laser heater to induce uncorrelated energyspread in the beam to control the effects of the micro-bunching instability. After the laser heater, two 1.3 GHzTesla superconducting cavity cryomodules (each withseven cavities) are used to accelerate the electron beamto an energy of 215 MeV before entering the first bunchcompressor. The first bunch compressor has a momentumcompaction factor of R56 ¼ −94.0 mm, and provides afactor of two longitudinal compression. After the firstbunch compressor, the electron beam is further acceleratedto 720 MeV in six cryomodules before entering the secondbunch compressor. This bunch compressor has a momen-tum compaction factor of R56 ¼ −76.0 mm and providesanother factor of five compression so that the final beampeak current is about 500 A. After the second bunchcompressor, the electron beam is further accelerated in 18
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FIG. 1. Amplitude of the Fourier coefficient difference from theFFT of an analytical Gaussian function and a sampled Gaussianfunction with 10 k, 100 k, 1 M, and 10 M macroparticles.
FIG. 2. A schematic diagram of the layout of the next gen-eration high repetition rate FEL.
J. QIANG et al. Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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superconducting cryomodules to a final energy of 2.4 GeVbefore entering the spreader section, where the 1 MHzelectron beam is distributed into different FEL undulatorbeam lines to generate coherent x-ray radiation. Morediscussions of the injector, linac, spreader, and FELundulator of this light source can be found in Refs. [21–24].
III. BEAM DYNAMICS IN THE INJECTOR
The injector was designed from a multiobjective opti-mization with 13 control parameters [21]. The projectedrms emittance and the rms bunch length at the exit of theinjector are defined as two objective functions in theoptimization. A lower emittance can be obtained with alonger bunch length that corresponds to a lower peakcurrent, while a shorter bunch length with higher peakcurrent leads to a higher emittance. In this study, weadopted a solution around the middle of the optimalPareto front that has a reasonable peak current (∼50 A)and rms emittance (< 1 mmmrad). Here, the optimalPareto front is a set of solutions in the objective functionspace so that any solution in this front is no worse than theother solution in the feasible solution space. The initiallaser pulse is assumed to have a longitudinal 44 ps flattemporal profile with 2 ps rising time on both ends. Thetransverse spatial distribution is assumed to be a rounduniform cylinder with 0.6 mm radius. The initial thermalemittance of the beam is 0.3 mm mrad, following aconservative assumption of 1 mm mrad thermal emittanceper 1 mm rms radius beam. A three-step model is used togenerate the initial particle distribution in momentum space[25]. The detailed description of the model of particlemomentum distribution is given in Appendix A. In order toimprove the computational efficiency, we have developed asecond-order electron emission model to simulate theprocess of photoelectrons coming out of the cathode duringthe laser interaction with the photocathode. A detaileddiscussion of the second-order emission model is givenin Appendix B. Using the real number of electron
macroparticles for 300 pC charge, we simulated the photo-electron emission from the photocathode and the transportthrough the injector. The evolution of the transverseprojected rms emittance is given in Fig. 3. The finalprojected emittance at the exit of the injector is about0.6 mm mrad. Figure 4 shows the final current profile, sliceemittances and longitudinal phase space at the exit of theinjector. The peak current at the exit of the injector is about50 A. The initial current out of the photocathode (∼6 A) isamplified by a factor of about 9 at the exit of the injectorthrough the velocity and ballistic bunching by the bunchercavity and the boosting cavities. The slice emittances arebetween 0.5 and 0.6 mm mrad for most parts of the beam.The longitudinal phase space at the exit of the injector is not
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FIG. 3. Evolution of projected transverse rms emittance (100%)through the injector.
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FIG. 4. The current profile (top), slice emittances (middle), andlongitudinal phase distribution (bottom) at the exit of the injector.
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completely flat, with a small relative rms energy spread ofabout 0.02%.
IV. EFFECTS OF TRANSVERSE SPACE-CHARGENEAR LASER HEATER
A laser heater is used after the injector to increase theelectron beam uncorrelated energy spread. The transverserms size of the beam is designed to match the laser spot size(with the same horizontal and vertical sizes) at the center ofthe undulator in order to optimize the laser electron beaminteraction. A diagnostic section right after the laser heateralso requires the same sizes in both horizontal and verticaldirections. Figure 5 shows the evolution of rms sizes andemittances without and with transverse space charge effectsthrough the matching section, the laser heater, the diag-nostic section, and two accelerating modules. The electronbeam enters the matching section with a kinetic energy of95 MeV and leaves the two accelerating modules with215 MeV. Transverse space charge is seen to cause both amismatch in the rms envelopes (particularly evident in thevertical plane) and more than 20% emittance growth in thevertical plane, with a smaller growth in the horizontal
plane. The importance of the transverse space-chargeeffects can be estimated from an envelope equation model,where the contribution from the space-charge effects can becalculated in comparison with that from the emittance.Using the rms size and emittance of the beam at the exit ofthe injector, we found that the space-charge term and theemittance term are comparable. There are two mechanismsrelated to the space-charge induced emittance growth. Thefirst mechanism involves a misalignment of the beamellipses in the transverse phase space corresponding todifferent longitudinal beam slices. The second mechanisminvolves the nonlinear space-charge fields presented whenthe beam charge density is transversally nonuniform. Thefirst mechanism might be compensated by appropriatelytuning the machine lattice to match the distribution withspace-charge forces. The second mechanism affects indi-vidual slices and is generally irreversible. Figure 6 showsthe rms size and emittance evolution of the solution afterrematching the beam line lattice with the space-chargeeffects. Here, we have reoptimized the six quadrupoles inthe matching section and the two quadrupoles in thediagnostic section after including the transverse space-charge effects. It is seen that the envelopes are rematchedwith correct sizes at the location of the laser heater and
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FIG. 5. Evolution of transverse rms sizes (top) and transverserms emittances (bottom) without (red) and with (green) space-charge effects in the simulation.
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FIG. 6. Evolution of transverse rms sizes (top) and transverserms emittances (bottom) with rematched space-charge effects inthe simulation.
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inside the diagnostic section. The rms vertical emittancegrowth is also significantly reduced after rematching thetransverse space-charge effects. The remaining emittancegrowth seen in Fig. 6 is mostly due to the secondmechanism of space-charge induced emittance growth.
V. MICROBUNCHING INSTABILITY IN THEBEAM DELIVERY SYSTEM
The microbunching instability in the beam deliverysystem causes degradation of electron beam quality andlowers the performance of the x-ray FEL. Here, we studiedthe microbunching instability seeded by shot noise as wellas by small current perturbations at the cathode (caused, forexample, by nonuniformities in the photo-gun laser pulsetemporal profile).We modeled the effect of irregularities in the photo-gun
laser profile by a placing few-percent sinusoidal modu-lations of a given period on the nominal beam currentgenerated at the cathode. The evolution of the modulationamplitude of the excited mode along the injector for a givenperturbation period is shown in Fig. 7 from the simulationand from a linear analytical model discussed in this section.The simulation shows some magnification in the amplitudeof the modulation, while the beam undergoes compressionby velocity bunching.We attempted to model the evolution of these modu-
lations with a simple, linear 1D model aiming at capturingthe qualitative behavior of the longitudinal beam dynamicsthrough the first few meters of the injector, including thebuncher, the downstream drift, and the first cavity of thebooster. We describe the particle dynamics in terms of(Δz;Δ ~pz), deviations from the reference orbit, where Δz isthe longitudinal coordinate and Δ ~pz ¼ Δpz=mc is thescaled longitudinal momentum. Time is the independentvariable. We denote M as the 2 × 2 matrix for the linearunperturbed motion (no space-charge). Through a driftspace (including the gun gap, where all beam particlesexperience about the same accelerating field) the nontrivialentry of the matrix readsM12ðt0 → tÞ ¼ c
Rtt0 dt
0=γ3ðt0Þ. Wemodel the buncher as a thin cavity operated at zero-field
crossing, with transfer matrix M11 ¼ M22 ¼ 1, M12 ¼ 0and M21 ¼ −αB, and assume that the beam stays on crestthrough the first cavity of the buncher (implying that all theRF compression is induced by the buncher, which is closeto a typical setup for the injector). The compressionfactor at time t is CðtÞ ¼ 1 for t < tB and CðtÞ ¼ j1 −αBM12ðtB → tÞj−1 for t > tB, where tB is the time thereference particle reaches the buncher. The parameter αB, ischosen empirically so as to yield a compression factor CðtÞalong the injector that is comparable to that resulting fromthe IMPACT simulations.The bunching function relative to the nominal wave
number k0 for the sinusoidal perturbation is found as thesolution of the integral equation [26]
b ¼ b0 þZ
t
0
dt0Kðt0; tÞbðt0Þ; (1)
where k ¼ CðtÞk0=βðtÞ and k0 ¼ Cðt0Þk0=βðt0Þ, and thekernel reads
K ¼ 4πiCðt0ÞIðt0Þ
IAM12ðt0 → tÞβðt0Þk Zðk
0; sÞZ0
: (2)
Space charge is described by a 1D model using theimpedance ZðkÞ ¼ Z0½1 − 2K1ðxÞI1ðxÞ�=kπr2b where x ¼krb=γ and rb is an effective transverse beam radius [15].The evolution of the relative amplitude of the sinusoidalcharge density perturbation from the model is shown inFig. 7 as the solid line. The model roughly overlaps withthe simulation data (dots) provided that we chooserbðtÞ ¼ 2.1σ⊥ðtÞ, where σ⊥ðtÞ is the rms transverse beamsize as determined by the IMPACT simulations. The modeamplitude is in units of the amplitude observed in the beamcore at the time the beam tail leaves the cathode. Thebehavior of the mode as observed in the figure has thesignature of a plasma oscillation with increasing period asthe beam undergoes acceleration.The uncorrelated energy spread of the beam helps to
damp the microbunching instability. Figure 8 shows thefinal electron beam current profile, slice energy, anduncorrelated energy spread at the entrance of the firstundulator radiator when different levels of uncorrelatedenergy spread are induced by the laser heater. It is seen thatthe introduction of 15 keV uncorrelated energy spread isneeded in order to suppress the microbunching instability.In addition to the initial current modulation from the
electron beam shot noise, the laser temporal profile ripplingat the photocathode also leads to initial current modulation.This temporal rippling might come from the stacking of anumber of short laser pulses. Figure 9 shows the finalelectron beam current profile, slice energy profile, anduncorrelated energy spread with an initial 5% temporallaser modulation and different modulation periods throughthe beam delivery system with an initial 15 keV energy
FIG. 7. Initial sinusoidal perturbation gain evolution insidethe injector from the IMPACT simulations (dots) and the linearmodel (line).
START-TO-END SIMULATION OF X-RAY RADIATION OF … Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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spread from the laser heater. The longer temporal period(3.4 ps) modulation causes significant fluctuation in thefinal beam current and energy due to the microbunchinginstability. Such a modulation in slice energy might causean increase in the x-ray FEL radiation bandwidth. Theinitial shorter period (0.8 ps) modulation is significantlydamped by the uncorrelated energy spread and results inonly slight final beam fluctuation.
VI. START-END-SIMULATION OF THEX-RAY RADIATION
We carried out full start-to-end simulations of thegeneration of FEL x-ray radiation starting from the electronemission at the photocathode to the end of the undulatorusing about two billion (the real number) electrons for a300 pC charged beam. Besides the effects from external
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FIG. 9. Final electron beam current profiles with initial 5%temporal laser modulation and different modulation periodsthrough the nominal two bunch compressor linac.
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focusing and acceleration, the simulation also includes self-consistent three-dimensional space-charge effects, longi-tudinal and transverse structure wakefield effects inaccelerating cavities, longitudinal CSR effects and inco-herent synchrotron radiation (ISR) effects in bendingmagnets, and three-dimensional self-consistent interactionsbetween the coherent x-ray radiation and the electron beamin undulators. The one-dimensional CSRmodel used in thisstudy includes only the radiative force [27] and is imple-mented inside the simulation using an integrated Greenfunction method [28]. The space-charge effects are addedto the CSR effects inside the chicane to account for theCoulomb force. The simulations were carried out on aCray-XC30 parallel computer, Edison, at the NationalEnergy Research Scientific Computing Center using2048 processors (cores). In the simulation, we have used64 × 64 × 1024 numerical grid points for the space-chargecalculation, 1024 grid points for the wakefield calculation,and 6000 (each slice represents 20 wavelengths) slices forthe three-dimensional time-dependent FEL calculation inmost simulations. The choice of the numerical grid param-eters is based on some numerical convergence tests and theconsideration of minimizing computing time needed for thesimulation. The grid also should resolve the minimumphysical length scale of interest in the simulation, whichcan be obtained from the analytical microbunching gaincalculation. Each computing processor has a theoreticalpeak performance of about 19 GFLOP/sec and about2.7 GB of memory. The computing node is connectedthrough Cray Aires high-speed interconnect with aDragonfly topology to ensure high performance, low-latency communication for message passing interface jobs[29]. The computing time for the start-to-end simulation isabout six hours. The total memory usage of the simulationis about 900 GB. The rms information of the beam iscalculated internally during the process of simulation. Theslice information such as slice emittance or energy spreadand current profile are calculated internally at a givenlocation using the macroparticle distribution. SNAPSHOT isused to output a fraction of randomly sampled macro-particles at a given location. The phase-space plots aremade after projecting onto a two-dimensional grid andplotted using software such as MATLAB.Figure 10 shows the projected rms emittance evolution
through the whole accelerator beam delivery system. It isseen that the normalized rms emittances are reasonablywell preserved and the final emittances are below 1 mmmrad. The major projected emittance growth is after thesecond bunch compressor and it is caused by the coherentsynchrotron radiation effects. Figure 11 shows the finalelectron beam slice emittance and longitudinal phase-spacedistribution coming out of the accelerator beam deliverysystem at the entrance of the first undulator. There is arelatively flat region of the beam with a length of about120 um. The slice emittance in this region is 0.5–0.6 mm
mrad with a current between 400 and 600 A. Thelongitudinal phase space still shows small long-wavelength(∼10 μm) modulation from the microbunching instability.(The modulation wavelength is consistent with the
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FIG. 10. Evolution of transverse rms emittances (red and green)through the whole accelerator beam delivery system.
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START-TO-END SIMULATION OF X-RAY RADIATION OF … Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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analytical calculation of the microbunching instability gaincurve, which has a maximum gain ∼100 μm before the firstbunch compressor [30]). Such a long wavelength energymodulation does not have significant impact to the perfor-mance of the SASE x-ray FEL radiation but limits thesmallest bandwidth that can be achieved in the seeded FELradiation scheme. The final uncorrelated energy spread(i.e., slice energy spread) of the beam is about 150 keV atthe entrance of the undulator beam line.The x-ray FEL beam line in this study consists of two
sections of undulators. The first section of the undulatorbeam line is used for producing SASE FEL radiation, whilethe second section is used for producing narrow bandwidthFEL radiation through a self-seeding scheme [24]. We firststudied the 1 nm x-ray FEL radiation in the SASEundulator beam line section using start-to-end simulationwith the real number of electrons. This section consists ofabout 12 superconducting undulator cryomodules, each3.3 m long. The magnetic break between the undulatorcryomodules is about 1.1 m. The average undulatorparameter is about 1.1 with an undulator period of 2 cm.In the simulation, the electron beam coming out of thebeam delivery system is directly sent into the FELundulator beam line through the internal memory of theprogram to produce coherent x-ray radiation. Figure 12shows the average radiation power evolution along theundulator beam line for the fundamental 1 nm x-rayradiation with different levels of uncorrelated energy spreadcoming out of the laser heater. The 5 keV uncorrelatedenergy spread induced by the laser heater is not sufficient tosuppress the microbunching instability. This results in alower x-ray FEL radiation power inside the undulator (withthe fixed length) and a longer radiation gain length due tothe degradation of the electron beam quality in the accel-erator beam delivery system. Using 10 and 15 keV uncor-related energy spread from the laser heater significantly
suppresses the microbunching instability and the x-rayradiation saturates around 50 m with about 250 MWaverage radiation power. Further increasing the uncorre-lated energy spread from the laser heater causes even largerelectron beam uncorrelated energy spread and lowers thex-ray radiation power.The above start-to-end simulation used the real number
of electrons (1.9 billion) from the cathode to the end of thebeam delivery system. A section of 120 μm long beam(1.2 billion electrons) around the core of the bunch withpeak current beyond 500 Ampere was directly used in theFEL undulator simulation. As a comparison, we also ranthe simulation using one million macroparticles and64 × 64 × 64 grid points. The number of numerical gridpoints is reduced in order to reduce artificial numericalnoise from the use of the small number of macroparticles.The macroparticle distribution (120 μm long around thecore) coming out of the accelerator was resampled in theGenesis simulation using 16,384 macroparticles per sliceand total 6000 slices. Figure 13 shows the average x-rayradiation power evolution along the undulator beam line forthe fundamental 1 nm and the third harmonic 0.33 nm
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MW
)
distance (m)
0
50
100
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250
30020 keV 15 keV 10 keV
5 keV
FIG. 12. Evolution of the 1 nm x-ray average radiation powerinside the first section of the SASE undulator beam line usingelectron beams with 5, 10, 15 and 20 keV uncorrelated energyspread from the laser heater.
0
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1.8
2
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)
distance (m)
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FIG. 13. Evolution of the average x-ray radiation power at 1 nm(top) and 0.33 nm (bottom) wavelength inside the first undulatorsection of the SASE FEL scheme with the real number ofelectrons (red) and the one million macroparticles (green).
J. QIANG et al. Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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radiation using the real number of electrons and the onemillion macroparticles. It is seen that both simulationsagree reasonably well in their predictions of the averagepower for the fundamental 1 nm x-ray radiation. However,for the third harmonic radiation, the one million macro-particle simulation significantly overpredicts the radiationpower in comparison with that from the simulation usingthe real number of electrons. This could be due to the largerthird harmonic component in the shot noise of the initialresampled beam distribution from the one million macro-particle simulation.The x-ray radiation from the SASE FEL has a good
spatial coherence but lacks longitudinal temporal coher-ence. Figure 14 shows an example of the final x-rayradiation temporal and spectral profiles at the end of theSASE undulator beam line with 15 keV uncorrelatedenergy spread from the laser heater in the electron beam.There are multiple spikes along the radiation pulse lengthand the full-width half maximum bandwidth is about0.5 × 10−3. The bandwidth and the temporal coherencefrom the SASE FEL radiation can be improved by usingthe self-seeding scheme in the following start-to-endsimulation.
In the start-to-end self-seeding x-ray radiation simula-tion, the first undulator section is shortened to 34 m so thatthe SASE x-ray radiation in this section is not saturated.The x-ray radiation from this stage is transported through amonochromator with 10−5 bandwidth to provide a narrowbandwidth seeding signal for the second stage amplifica-tion. In this simulation, we assumed an initial seedingsignal of 1 nm radiation with 8.6 kW power coming out ofthe monochromator. The electron beam coming out of thefirst stage of the undulator is transported through anidealized chicane with R56 ¼ 600 μm to account for thedelay between the electron beam and the x-ray pulse and tosmear out the microbunching inside the electron beaminduced in the first section of the undulator. The secondundulator section has the same parameter settings as thefirst section of the undulator except with longer length(∼53 m). Figure 15 shows the evolution of the average1 nm x-ray radiation power inside the second stage of theundulator. As a comparison, we also show the simulationresults from using one million macroparticles following aresampling procedure for the FEL simulation describedabove. The radiation powers from both simulations are inreasonable agreement before saturation and show signifi-cant differences after saturation. The slower decrease of theradiation power from the one million macroparticle sim-ulation is due to the artificial shot-noise amplificationpresent in the simulation with the use of smaller numberof macroparticles. Figure 16 shows the temporal andspectral profiles of the 1 nm x-ray radiation at 41 mlocation from the beginning of the second undulatorsection. Here, we see that in the temporal domain, theone-million macroparticle simulation case shows largerfluctuation of the radiation power along the pulse length. Inthe spectral domain, both simulations show a narrowbandwidth with a relative FWHM of 1–2 × 10−5. Theslightly larger bandwidth than the transform limit is due to
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1
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wave length deviation (nm)
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400
600
800
1000
1200
1400
1600
1800
0
0.05
0.1
0.15
0.2
0.25
0.3
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FIG. 14. Temporal (top) and spectral (bottom) profiles of the1 nm x-ray radiation at the end of the first undulator section in theSASE FEL scheme.
0 10 20 30 40 50 60
aver
age
pow
er (
MW
)
distance (m)
real number of electrons1 million macroparticles
0
50
100
150
200
250
300
FIG. 15. Evolution of the average 1 nm x-ray radiation powerinside the second undulator section of the self-seeded FEL beamline with the real number of electrons (red) and the one millionmacroparticle simulations (green).
START-TO-END SIMULATION OF X-RAY RADIATION OF … Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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slice energy variation along the beam at the undulatorentrance and the energy modulation caused by the resistivewall wakefield inside the undulator with a 6 mm aperturesize. The simulation using onemillionmacroparticles showsslightly larger bandwidth with larger background powerfluctuation among different wavelengths due to the shotnoise amplification. The growth of those shot noise modes(observed in the spectral profile of the radiation at the end ofthe simulation) accounts for the slower decrease of theaverage radiation power after saturation in Fig. 15.
VII. CONCLUSIONS
In this paper, we have carried out self-consistent multi-particle start-to-end simulations of the SASE and the self-seeded x-ray radiation in a next generation FEL-based lightsource design with a multiphysics model to account forvarious collective effects, using a real number of electrons(300 pC). The simulation integrated the electron beamdynamics modeling (inside the accelerator beam deliverysystem) and the x-ray FEL radiation modeling (inside thetwo stage undulator) into a seamless start-to-end modeling
of the entire system. The simulation started from thephotoelectron emission process at the photocathode in alow frequency rf gun, and electrons were transportedthrough a 600 m accelerator beam delivery system toattain a final 2.4 GeV energy before entering a two stageundulator for generating 1 nm SASE and self-seeded FELx-ray radiation. The simulation showed that the emittancesof the electron beam are well preserved through theaccelerator with final total projected rms emittances ofabout 0.8 mm mrad in the horizontal plane and 0.7 mmmrad in the vertical plane and 0.5–0.6 mm mrad sliceemittances in the core of the beam. Final current inside thecore of the beam is between 400 and 600 A. The micro-bunching instability driven by space-charge and wakefieldeffects can be controlled by the use of a laser heater toinduce 15 keV uncorrelated energy spread in the beam, andby controlling the initial laser pulse intensity fluctuationsbelow 5% and with less than 1 ps period. The finaluncorrelated energy spread of the beam is about150 keV. Such an electron beam produces about 100 μJ1 nm spatially coherent x-ray radiation through about 50 mSASE undulator beam line, and about 100 μJ fully spatiallyand temporally coherent radiation through the self-seedingundulator beam line.In this study, we also observed strong transverse space-
charge effects at about 100 MeV electron energy near thelaser heater that cause significant beam envelope mismatchand emittance growth. This mismatch can be mitigated byretuning the matching lattice beam line elements afterincluding space-charge effects.In this study, we also compared the final x-ray radiation
from the simulation using the real number of electrons andthe simulation using one million macroparticles. Bothsimulations show reasonable agreement in the predictionof 1 nm SASE FEL radiation power while the one millionmacroparticle simulation significantly overpredicts theradiation power for the third harmonic 0.33 nm x-rayradiation. In the self-seeding x-ray radiation, the onemillion macroparticle simulation case also shows largertemporal radiation power fluctuation in comparison to theuse of the real number of electrons. Those discrepanciescould be due to the lack of resolution in the use of fewermacroparticles in the one million microparticle simulation.
ACKNOWLEDGMENTS
This research used computer resources at the NationalEnergy Research Scientific Computing Center and wassupported by the U.S. Department of Energy underContract No. DE-AC02-05CH11231.
APPENDIX A: THREE-STEP MODEL OF THEINITIAL DISTRIBUTION
In the start-to-end simulation, the first step is to generatean initial ensemble of macroparticles in six-dimensional
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real number of electrons1 million macroparticles
0
20000
40000
60000
80000
100000
FIG. 16. Temporal (top) and spectral (bottom) profiles of the1 nm x-ray radiation at 41 m undulator location in the self-seededFEL scheme with the real number of electrons (red) and the onemillion macroparticle simulations (green).
J. QIANG et al. Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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phase space. The spatial coordinates of the macroparticlescan be sampled by following the transverse spatial profileand the longitudinal temporal profile of the input laserpulse. The momentum coordinates of these particles can begenerated following the three-step model [25].In the three-step model, electrons are first excited inside
the cathode material by absorption of photons with energyhν. Then, those electrons migrate to the surface and mayexperience e-e scattering or e-phonon scattering. In thethird step, the electrons with kinetic energy above thebarrier potential will escape into the vacuum. To includethis model in our simulations, we first assumed an excitedelectron energy distribution inside the cathode materialgiven by the following:
fðEÞ ¼ ½1 − fFDðEÞ�fFDðE − hνÞ; (A1)
where hν is the single photon energy of the laser, fFD is theFermi-Dirac distribution function representing the initialdensity of the state:
fFDðEÞ ¼1
1þ eðE−EFÞ=kBT ; (A2)
where kBT is the electron gas thermal energy, and EF is theFermi energy. If an electron succeeds in moving to the innersurface of the cathode, a three-dimensional momentum isgenerated by assuming an angular distribution
fðθ;ϕÞ ¼ sinðθÞ: (A3)
After the angular distribution is sampled, the electrontransverse and longitudinal momentum can be calculated.Only an electron with normal momentum satisfying thefollowing condition will be emitted:
pinz ¼
ffiffiffiffiffiffiffiffiffiffi2mE
pcosðθÞ ≥
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mðEF þ ϕeffÞ
p; (A4)
where ϕeff is the effective work function of the photo-cathode material (including both the material work functionand the Schottky work function). The transverse andlongitudinal momenta for an electron outside the cathodesurface will be
px ¼ffiffiffiffiffiffiffiffiffiffi2mE
psinðθÞ cosðϕÞ; (A5)
py ¼ffiffiffiffiffiffiffiffiffiffi2mE
psinðθÞ sinðϕÞ; (A6)
pz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mðE − EF − ϕeffÞ − p2
x − p2y
q: (A7)
The above process is repeated many times until a specifiedamount of electron charge is generated.
APPENDIX B: A SECOND-ORDERPHOTOELECTRON EMISSION MODEL
A fast and accurate model is important to simulate thephotoelectron emission process for start-to-end light sourcemodeling. In this study, we have developed a second-ordercomputational model to simulate the production of photo-electrons from a photocathode driven by an external laser.For a given laser temporal pulse distribution and spatialdistribution, a number of electrons carrying the totalemitted charge are generated behind the cathode with thesame transverse distribution as the laser’s and the samelongitudinal distribution as the laser’s temporal profiletimes a reference longitudinal velocity v0. Those electronsare moved outside the photocathode during N time steps. Aschematic plot of the electron emission process at thephotocathode is shown in Fig. 17. Here, the time step size isΔt ¼ tlaser=N, where tlaser is the total laser pulse length. Inthe second order photoelectron emission model, the posi-tions and the velocities of an electron after emission aregiven by
x ¼ x0 þ vx0δti þ1
2axðδtiÞ2; (B1)
vx ¼ vx0 þ axδti; (B2)
z ¼ vz0δti þ1
2azðδtiÞ2; (B3)
vz ¼ vz0 þ azδti; (B4)
where δti ¼ zi=v0, zi is the electron longitudinal coordinateout of the photocathode right after emission during the timestep Δt, and a is the acceleration that can be calculatedusing the field at the photocathode surface. The y positionand velocity can be obtained by replacing x with y in aboveequations. As a comparison, we generate 300 pC photo-electrons from a photo-injector gun using the second-orderemission model and the first-order emission model (withoutincluding acceleration). The current profile of the beamshortly after emission is shown in Fig. 18 using differentnumbers of emission steps in the above emission models. It
FIG. 17. A schematic plot of the electron emission process atthe photocathode.
START-TO-END SIMULATION OF X-RAY RADIATION OF … Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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is seen that the crude first-order emission model canintroduce artificial modulation of the beam. A much largernumber of emission steps (a factor of 4) i.e., smalleremission step size, are needed in order to achieve the samelevel smoothness of the current profile.
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2nd order - 7501st order - 750
1st order - 15001st order - 3000
0.88
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FIG. 18. A section of the current profile of the beam after theemission using the second-order emission model and the first-order emission model with different numbers of emission steps.
J. QIANG et al. Phys. Rev. ST Accel. Beams 17, 030701 (2014)
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