Submittedto ’Chinese Physics C’ Bunch evolution study in optimization of MeV ultrafast electron diffraction* LU Xian-Hai1,2 DU Ying-Chao1,2 HUANG Wen-Hui1,2 TANG Chuan-Xiang1,2,1) 1 DepartmentofEngineeringPhysics,TsinghuaUniversity,Beijing100084, China 4 2 KeyLaboratoryofParticleRadiationImaging(TsinghuaUniversity),MinistryofEducation, Beijing100084, China 1 0 2 Abstract: Megaelectronvoltultrafastelectrondiffraction(UED)isapromisingdetectiontoolforultrafastprocesses. The quality of diffraction image is determined by the transverse evolution of the probe bunch. In this paper, we n a studythecontributingtermsoftheemittanceandspacecharge effectstothebunchevolution inMeV UEDscheme, J employingamean-fieldmodelwithanellipsoidaldistributionaswellasparticletrackingsimulation. Smalltransverse 7 dimension of the drive laser is found critical to improve the reciprocal resolution, exploiting both smaller emittance and larger transverse bunch size before the solenoid. The degradation of reciprocal spatial resolution caused by the ] h space charge effects should be carefully controlled. p Key words: ultrafast electron diffraction, reciprocal spatial resolution, space charge effects, bunchevolution - c c PACS: 07.78.+s, 41.75.-i, 41.85.-p a . s c 1 Introduction to-noise ratio (SNR) for resolving diffraction pattern[5]. i One proposed solution is generating a long bunch con- s y taining millions of electrons and then compressed the h With the advancement in research on microcosmic bunch to a sub-picosecond length at the sample [8–10]. p and ultrafast processes, a remarkable demand for new An alternative method circumvents the space charge ef- [ toolsfordirectvisualizationarises. Discoveriesaremade fects by accelerating the electrons to megaelectronvolt using ultrashort X-ray diffraction in the biology and 1 (MeV) energy using a radio frequency (RF) gun [11]. In v solid-state physics field [1, 2]. Compared to large X- anacceleratingfieldashighas60MV/m,the bunchcan 4 ray facilities such as the Linac Coherent Light Source be boosted to the relativistic regime in several centime- 5 (LCLS) [3], ultrafast electron diffraction (UED) is an 2 tersbeforesignificantlongitudinalexpansionoccurs[12– idealtoolforlargercross-sectionandtabletop-scalecon- 1 15]. Benefittingfromthedevelopmentandsuccessofthe venience [4–6]. In the pump-probe experiment, electron . free electronlaseras wellas the synchronizationsystem, 1 bunch of sub picosecond length can be generated and 0 the RF photocathode gun turns a promising candidate sent to the interested sample to record its structural in- 4 for the UED systems. 1 formation with the diffraction pattern. When the sam- The primary concern of the UED system is the ul- : ple is pumped by an intense laser pulse, by adjusting v trashort bunch length, which have been studied by ple- the arrival time of the electron bunch with respect to i narymodelsandsimulations[16–18]. Thesestudiesfocus X thepumpingtime,aseriesofdiffractionpatternsareob- on the longitudinal evolution of bunch, considering the r tained and the dynamical process is retrieved. a transverseevolutionasa partofmodel,ifnotneglecting Currently most UED systems employ high-voltage it at all. In the respect of experiment, the transversedi- static electric field to accelerate the electron bunch to mension of the probe bunch is of equivalent importance. a range of 30 keV to 60 keV. In this scheme, a major The spot size on detector determines the the reciprocal hurdle is the dramatic longitudinal expansion caused by spatialresolution of the diffraction patterns (see the fol- significant space charge effects, and the bunch length lowing text). With poor reciprocal spatial resolution, puts an up limit of temporal resolution of the system crucialdetailsofdiffractionpatternwillbeoverwhelmed [7]. To alleviate the impact of space charge effects, the andlost. Inthe contextofstaticelectric-field-basedkeV electronfluxislimitedto3000–6000electronsperbunch, UEDscheme,simulationstudiesonpatterndisplacement which results in a sub-picosecond resolutionat the price and distortion have been conducted[19]. In this paper, ofhundredsofrepeatingpumping,requiredbythesignal- ReceivedXXXX ∗SupportedbyNationalNaturalScienceFoundationofChina(11127507 and10925523) 1)E-mail:[email protected] (cid:13)c2013 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of ModernPhysicsoftheChineseAcademyofSciences andIOPPublishingLtd 010201-1 Submittedto ’Chinese Physics C’ we study the transverse evolution of the bunch in the ∆Rofthe Debye–Scherrerring[22]. The ringwidth∆R RFgun-basedMeVUEDschemewithanalyticalmodels is determined by the full width at half maximum of the dedicatedtothisissue,aswellasinsimulationsmethod. transversal size of the undiffracted beam at the detec- It should be noted that RF gun has been studied by the tor, which is 2.35 times of the root mean square (rms) accelerator community for decades as an intense elec- spot size σ for a Gaussian distribution [23]. Then, the x tron source for subsequent acceleration, but not yet as reciprocal spatial resolution is: R an independent diagnostic instrument generating bunch R 2θαd of low charge (few pC) and ultrashort length (sub pi- = = , (1) R ∆R 2.35σ cosecond). Here, we present a thorough analysis of fac- x where σ is the rms spot size of the bunch at the de- torsonreciprocalspatialresolutioninthecontextofthe x tector, θ is the diffraction angle, d is the distance from MeVUEDsystembasedonTsinghuaS-bandMeVUED solenoidto detector,and αd is the distance fromsample setup,consideringtheRFgun,thefocusingsolenoid,and todetector. AccordingtoEq.1,thespotsizeondetector the parameters of the drive laser. σ shouldbeminimizedtoobtainoptimalreciprocalres- Therestofthispaperisorganizedasfollows: Section x olution, since the system setup (αd) and characteristics 2 explains the models of the transverse bunch evolution of the sample (θ) are fixed in the experiment. fortheMeVUEDsystem,includingtheunderlyingprin- ciple. Section3presentstheresultsofthesimulationand 2.1 Model without space charge effects corresponding discussion. First,we considerthe situationwithoutspacecharge 2 Models of transverse evolution effects. Using transfer matrix and thin-lens approxima- tionforthe solenoid,wecanderivethe coordinateofthe electron at the detector as: A schematic of the MeV UED setup is shown in Fig. 1. The electron bunch is generated by a 1.6 cell S- x 1 d 1 0 x = 0 , (2) band RF photocathode gun, which is synchronized with " x′ # " 0 1 #" 1/f 1 #" x′ # the laser system[20]. A solenoid coil is attached next to − 0 the RF gun for restricting the bunch transversely and where d, f, x and x′ denote the distance between the foremittance compensation. The solenoidis followedby solenoid to detector, focal length of the solenoid, as well thesampleofinterestdownstream. Adetectorcamerais as position and deviation angle from the axis. The sub- implemented at the end of the beamline. The positions script 0 denotes the position before the solenoid. Then of elements are presented in Table 1. To accelerate the the rms value of x can be derived as: bunch to a kinetic energy of E = 2.8 MeV with a small energyspread,anlaunching phkase of 20degree anda 60 σ = σ2 d σx20+dσx0,x′0 2+ǫ2xd2, (3) MV/m peak field at the cathode are chosen. More pa- x s x0(cid:18)f − σx20 (cid:19) σx20 rameters of operation and probe electrons based on the where σx =√<x2>, σxx′ =<xx′ > and the emittance Tsinghua S-band MeV UED setup can be found in [21]. ǫ = σ2 σ2 σ2 . The <> defines the second cen- x x0 x′0− x0,x′0 solenoid coil Faraday cup detector tral mqoment of the particle distribution [24]. The mini- steering coils mal value of Eq. 3 can be obtained with proper f as: RF gun ǫ d σ = x . (4) x,min σ x0 UV lasercolimator sample ThenwesubstituteEq.(4)intoEq.(1)toobtaintheex- pressionofspatialresolutionwithoutspacechargeeffect: Fig. 1. Schematicof theMeV UED setup. 2θασ = x0. (5) Rmax 2.35ǫ Table 1. Positions of elements along thebeamline. x 2.2 Model with space charge effects Elements Positions Units Cathode 0 cm When the space charge effects is significant, the Solenoid 22 cm model without Coulomb force underestimates the spot Sample 73 cm size and the transverse evolution depends on the charge Detector 373 cm density. In turn, the charge density depends on the di- mension of the bunch, resulting in the coupling of the For diffraction patterns, sharper patterns indicate expansionrates in both dimensions. This close loop fea- better reciprocal spatial resolution, which can be quan- ture suggests differential equations are proper for mod- tified by the ratio of the diameter R and the width eling the space charge effects. It should be noted that R 010201-2 Submittedto ’Chinese Physics C’ in accelerator community where RF gun is regarded as For the uniform ellipsoidal charge distribution, the re- source followed by acceleration and focusing, attention lation between the semi-principal axis and rms value is areusuallyfocusedonemittanceorpeakcurrent,instead σ =R/√5. Then,we addthe spacechargetermofEqs. x of the spot size. Here we perform a dedicated analysis (9)to(11)andobtainthe radialdifferentialequationfor for the issue of spot size optimization. the ellipsoidal bunch (we omit “ ’ ” here for the labora- First, we assume the bunch of a uniform three- tory frame): dimensionalellipsoidaldistribution,whichmaintainsthe shapeofbunchaswellasthe uniformityofchargeinlin- ear field[25]. The ellipsoidal bunch can be generated by intense short laser with “half-circle” radial profile [25]. d2R 25ǫ2 4Qe2M =k2R x+ x . (13) The blow-out regime[26] is not a widely used regime for dz2 0 − R3 3πRLε mβ2γ3c2 0 conventional photo injector due to limited current, but the feature of uniform ellipsoidal distribution is favor- able forUEDsystem. We denote the semi-principalaxis R inthe transversedirectionandthe semi-principalaxis Combining Eqs. (10) and (13), we can investigate L in the longitudinal direction. In the rest frame, the the evolution of the transverse size of the bunch quanti- transversalelectric field in the ellipse is [25]: tativelyandevaluatethespacechargeeffectsunderspec- ρ ified charge. The result is discussed in Section 3. E = 0M x, (6) x ε x 0 1 1+Γ2 2.3 Simulation code M = 1 (Γ arctanΓ) , (7) x 2 − Γ3 − (cid:20) (cid:21) where ρ is the charge density, ε is the permittivity of We use the particle tracking code ASTRA [28] for 0 0 vacuum, and Γ = R2/L2 1 is the eccentricity. We simulationpurpose. ASTRAisaparticle-in-cellcodefor − derive the equations of motion for the particles at the photoinjector simulation, where different initial distri- p transverseand longitudinaledges of the ellipse [with co- butions of the bunch on the photocathode can be spec- ordinates of (R, 0, 0) and (0, 0, L)]. Then, we obtain ified. In the simulation, the space charge effects can be the evolution of the transverse semi-principal axis: switched off for comparisonpurpose. Using ASTRA, we can obtain bunch parameters such as σ , L, and expan- d2R 4QeM x = x , (8) sion rates at the exit of the RF gun. These parame- dt2 3πRLε0m ters can be used as initial values for the transfer ma- where Q is the charge of the bunch, e is the charge of trixmodelwithoutspacechargeeffectsorthedifferential electron, and m is the mass of the electron. Then, we model with space charge effects. We can also track the transformthecoordinate(x, y, z, t)fromtherestframe bunchdimensionsfromstarttoendtocheckthe validity ofthebunchtothelaboratoryframewithR′=R,t′=γt, of the models above. L′=L/γ and d2z′/dt′2=β2c2. The result is: d2R′ 4Qe2M′ 3 Results and discussion = x , (9) dz′2 3πR′L′ε mβ2γ3c2 0 where γ is the Lorentz factor, c is the velocity of light 3.1 Model without space charge effects in vacuum, β is the velocity of the bunch relative to the speed of light c. Similarly, we obtain the equation of motion for the longitudinal direction: First we generate bunch distributions of different thermal emittance at the cathode and switch off the d2L′ 4Qe2M′ = z . (10) space charge effects in ASTRA. Given the absence of dz′2 3πR′2ε mβ2γ3c2 0 the space charge effects and the weak nonlinear field of the RF gun for a sub-picosecond bunch, the emittance Whenexcludingtheeffectofspacecharge,weusethe is preserved from the cathode [29]. For each specified equation of transverse motion with magnetic field B of z emittance, we conduct solenoid scanning and obtain the the solenoid and emittance ǫ [27] as: x minimized σ . In the transfer matrix model, we choose x d2σx+k2σ ǫ2x =0 (11) the middle of the solenoid as the thin-lens plane and dz2 0 x−σ3 use the spot size σ from simulation as input values. x x0 eB Theresultsofthemodelandsimulationarecomparedin k = z . (12) 0 2γmβc Fig. 2. 010201-3 Submittedto ’Chinese Physics C’ 0.5 systemcomparedwithseveralcminkeVsystem)results significant degradation of reciprocal spatial resolution. m 0.45 m The significant influence of transverse evolution should σ, x 0.4 be taken into consideration of model construction. r In the ellipsoidal-distribution-based model intro- o ct 0.35 duced above, firstly we choose initial values of (R, e et dR/dz) and (L, dL/dz) from the simulation result of d at 0.3 ASTRA at z = 0.5 m, corresponding to the exit of the h solenoid. In this case, we examine the performance of c 0.25 n the model in free space. Furthermore, we select another u b of 0.2 start point upstream at z = 0.1 m corresponding to the e exit of the gun, including the solenoid scanning in the Siz 0.15 simulation model. The results of bothstarting points arepresented model in Fig. 3, compared with ASTRA simulation. The con- 0.1 0.05 0.1 0.15 0.2 sistency between the model and simulation is excellent Thermal emittance ε ,mm mrad x for region either with or without the magnetic field. Fig. 2. Optimized transverse size of the bunch at thedetectordeterminedbytransfermatrixmodel 0.7 simulation and ASTRA result without space charge effect. ellipsoidal model from z=0.1 m m 0.6 m ellipsoidal model from z=0.5 m The results have similar trends and the discrepancy ,z σ 0.5 between both results is within 10%. As indicated by d n wmEqiot.dh(e4el)ma,nittdhteatnhtcereasniǫmsxv.uelraTstehioesnizsmemoaaynllbtdheiesactdrteerptiebacunttcoeyrdσbtxeotdwtheeceerneeffatesheces- σch ax0.4 n 0.3 tive distance d and the thin-lens approximation. Using u b this model, we can also estimate the influence of energy of 0.2 spread on σ . Firstly, the focal length of the solenoid s x e can be derived as: Siz 0.1 1 f= . (14) (eB /2βγmc)2dz 0 z 0 1 2 3 4 Position z,m Then the smearingRof ring caused by the energy spread Fig.3. Transverse(top)andlongitudinal(bottom) can be calculated as: evolutionsofthebunchforthreecases: simulation ∆x ∆f 2γ∆γ (startingfromz=0m),modelfromtheexitofRF = = . (15) gun(startingfromz=0.1m),andmodelfromthe x − f −γ2 1 − exit of solenoid (starting from z=0.5 m). Foratypicalrelativeenergyspread(∆γ/γ)smallerthan 1% for the MeV UED system, the effect of the energy With the help of differential model, we evaluate the spread on σ can be neglected. space charge effects by examining the extent of trans- x verse expansion for different charge. We choose the exit 3.2 Model with space charge effects ofthe RF gunas the startpoint andobtainthe minimal The mean-fieldmodel has been proposedand proved spot size under different charges. The result (shown in effective and efficient for modeling the evolution of the Fig. 4) indicates that the expansioncausedby the space bunch length and energy spread, in keV UED regime charge effect is almost linear with respect to charge. In [16–18]. In the mean-field model, the bunch is modeled the case of Q = 1 pC, the optimized spot size is four asacylinderwithevolvingaspectratiosmallerthan1(a times larger than that in the case of Q=0 due to space thindisk)intheregionbeforethesample. Thesestudies chargeeffects, under identicalinitialdistribution. While focus on the dramatic longitudinal expansion before the limiting Q is required to improve temporal resolution sample as the firstconcern. However,for the MeV UED in keV UED system [16], here we draw the conclusion regime, the longitudinal expansion is mitigated by the that to improve the performance of MeV UED system, relativistic energy thus not as severe as in the keV UED especially in the respect of spatial reciprocal resolution, regime. On the other hand, the transverse expansion in transverse expansion due to space charge effect must to themuchlongerdriftingdistance(severalmetersinMeV carefully handled. Otherwise, this issue will hinder the 010201-4 Submittedto ’Chinese Physics C’ hard-earned advantage of favorable SNR feature of the Fig. 5. Optimized transverse size with respect to MeV UED system. spot size of laser on cathode. 1.6 Table 2. Simulation parameters of thelaser on cathode. 1.4 Parameter Value Transversaldistribution flattop m m 1.2 Laserspotsizeσx,laser (rms) 0.1mm ,x Longitudinaldistribution Gaussian σ ch 1 Pulselengthσz,laser (rms) 300fs un BeamchargeQ 1pC of b 0.8 Thermalemittance γǫx 1µm/mm e z 0.6 Thecurveofspotsizeatscreen(reddashline)shows Si a convex shape with a peak at moderate laser spot size. 0.4 To illustrate this result, we scrutinize the components of emittance and focusing effect (blue and red solid line 0.2 respectively in Fig. 5). The extremely small spot size at 0 0.2 0.4 0.6 0.8 1 Adjusted charge,pC screen in bottom left corner can be explained by 1) the large transverse size at solenoid caused by high charge Fig.4. Optimizedtransversesizeandthelinearfit- ting of different charges calculated by differential density in the gun and 2) the small emittance propor- model. tionaltolaserspotsize. ThisisinconsistentwithEq.4, which has emittance ǫ in the numerator and size σ in 3.3 Optimization of laser spot size x x0 thedenominator. Whenconsideringspacechargeeffects, Besides the optimization of drifting region discussed the effect is favorable due to lower charge density (indi- above, the parameters of the driving laser is also criti- cated by large spot size at solenoid) during drifting. As thelaserspotsizeincreases,theemittancetermincreases cal. One consideration of the probe bunch is the emit- tance, consisting the thermal emittance and the emit- monotonously anda turning point of drifting bunch size occurs. The superposition of both effects explains the tancegrowth[29]. Thethermalemittanceisproportional shape of spot size at screen. Notably, the maximum of tothermslaserspotsizeσ onthecathode,andtakes laser up a significant part in the low charge regime like UED spot size at screen is lagged behind the minimal of spot size at solenoid. The lag also confirms the influence of system. Meanwhile the intensity of laser influences the chargedensity inthe drifting region,thusthe bunchsize the increasing emittance. evolution, which will be discussed in the following part. Based on the analysis above, we conduct a thor- ough simulation for different laser spot size, including Since the dynamicsdue tononlinearityofRFfieldisbe- yond the capability of the models, we study this issue the diffraction process of polycrystalline aluminum film using the kinematic method. To optimize the resolu- relying on the particle tracking code, from the cathode tion,lowchargeQ=0.1pCtoavoidseverespacecharge to the detector. The result of optimized spot size with respect to the laser spot size is shown in Fig. 5. Other degradation. The result is shown in Fig. 6. parameters of laser are listed in Table 2. 120 σ =0.075 mm 0.5 1 x,laser emittance σ =0.050 mm 100 x,laser m spot size at solenoid m u. σ =0.030 mm µ 0.4 spot size at screen 0.8m a. x,laser ε x h, y, 80 γalized emittance 00..23 00..46verse size of bunc cattering intensit 4600 m s S or 0.1 0.2an 20 N Tr 0 0 0 0.4 0.5 0.6 0.7 0.8 0.9 00 00..11 00..22 00..33 00..44 Laser spot size σ , mm s, angstrom−1 x,laser 010201-5 Submittedto ’Chinese Physics C’ Fig. 6. Scattering intensity of the diffraction pat- This work was supported by the National Natural Sci- tern of polycrystalline aluminum with respect to ence Foundation of China (Grant Nos. 11127507 and laser spot size. The momentum transfer s is ex- 10925523). pressed as s=2sin(θ/2)/λ, where θ is the scat- References teredangle,andλisthedeBrogliewavelengthof theelectron. 1 Schotte F, Lim M, Jackson T A, et al. Science, 2003, 300: 1944—1947. 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