Gigahertz repetition rate thermionic electron gun concept
W. F. Toonen, X. F. D. Stragier, P. H. A. Mutsaers, O. J. Luiten
GGigahertz repetition rate thermionic electron gun concept
W.F. Toonen, X.F.D. Stragier, P.H.A. Mutsaers, and O.J. Luiten ∗ Department of Applied Physics, Coherence and Quantum Technology Group,Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands (Dated: October 23, 2019)We present a novel concept for the generation of gigahertz repetition rate high brightness electronbunches. A custom design 100 kV thermionic gun provides a continuous electron beam, with thecurrent determined by the filament size and temperature. A 1 GHz rectangular RF cavity deflectsthe beam across a knife-edge, creating a pulsed beam. Adding a higher harmonic mode to thiscavity results in a flattened magnetic field profile which increases the duty cycle to 30%. Finally, acompression cavity induces a negative longitudinal velocity-time chirp in a bunch, initiating ballisticcompression. Adding a higher harmonic mode to this cavity increases the linearity of this chirp andthus decreases the final bunch length. Charged particle simulations show that with a 0 .
15 mm radiusLaB filament held at 1760 K, this method can create 279 fs, 3 . .
089 mm mrad at a repetition rate of 1 GHz.
I. INTRODUCTION
High-brightness x-ray sources are used in a widerange of fields from chemistry, biology and medicineto material sciences, both in science and industry.Since the construction of the first synchrotron lightsource in 1986, they have provided researchers with anindispensable tool for non-destructive and high spatialresolution inspection of a wide variety of samples. Withthe growing demand for these high-brightness sourcesas well as the large size of the facilities that housethem, spanning hundreds to thousands of meters, thereis a desire for robust, compact and affordable x-raylight sources. One technique proposed to achieve thisis in an Inverse Compton Scattering (ICS) scheme [1].This method requires an electron injector providinghigh-brightness, high charge, pulsed electron bunchesthat are subsequently accelerated in a radio-frequency(RF) accelerator. The bunch properties required at thepoint of electron-light interaction can be traced back tothe properties of the bunches prior to injection into theRF accelerator. While current injectors are capable ofproviding a high peak brightness electron beam, theyare mainly limited in their average brightness.Many avenues of research are being explored forthe next-generation electron injector that will supplyx-ray light sources with high charge, low emittanceand high average current electron bunches. Since 1988,the common approach is to use an RF photoinjector,where a metal or semiconductor cathode is irradiatedby a short, intense laser pulse [2]. While these sourcescan achieve the highest brightness, metal cathodeshave a relatively low quantum efficiency (QE), i.e. thenumber of electrons produced per incident photon.Semiconductor materials have a higher QE but in turn,require pressures below 10 − mbar and have a limited ∗ Electronic mail: [email protected] lifetime of typically 1 to 100 hours, with performanceoften a trade-off between QE and durability. Advancesin cathode development [3, 4] and cavity design [5] keeppushing these boundaries, with the APEX VHF-Guna prime example [6]. Future advances may includesuperconducting RF photoinjectors, capable of achievingeven higher field gradients [7]. Alternatively, Cornell’sDC photogun can deliver electron bunches with a bunchcharge of 19 pC and an emittance of 0 .
33 mm mrad ata 50 MHz repetition rate or 77 pC with an emittance of0 .
72 mm mrad at 1 . − to 10 − mbar and having thousands of hours of lifetime,common thermionic cathode materials such as tungstenor tantalum do not provide sufficient current densityat a low enough emittance [10]. However, with today’sdispenser cathodes, as well as CeB and LaB crystals,ever lower work functions are reached. Researchersat SACLA have shown that a DC electron gun witha thermionic cathode can be successfully used as aninjector for an x-ray free electron laser [11]. Thermioniccathodes in principle provide continuous electron beams,while acceleration to energies >
10 MeV generallyrequires RF accelerators and thus a pulsed beam. AtSACLA the method used is a beam chopper that canpick out a 2 ns pulse at a 60 Hz repetition rate. Anothermethod often employed is through a gridded cathode,where a small grid in front of the cathode set at apotential at or higher than at the cathode. This way theamount of current reaching the anode can be regulated.The major downside of this method is the significantincrease in beam emittance due to the presence of thegrid. Thermionic cathodes can also be used as the a r X i v : . [ phy s i c s . acc - ph ] O c t Emitter chop compresssolenoid
FIG. 1. Schematic view of the complete thermionic electron gun. The blue and red arrows depict the electromagnetic forceon the green electron beam due to the time-dependent magnetic and electric field respectively. emitter in RF guns. Here, the gating is provided by theoscillating electric field, but this also induces increasedenergy spreads, as well as back-bombardment whichresults in cathode deterioration.Finally, the injector at CEBAF uses a 100 keV beamchopper somewhat similar to the one proposed in thispaper [12]. There, a square cavity with two orthogonalTM modes operating at the third subharmonic to theirdesired frequency sweeps an electron beam radiallyonto a circular aperture with three holes. This chopsthe beam into three beamlets, after which an identicalsecond square cavity cancels the effects of the first one.Lens aberrations however lead to a minimum of 20%emittance growth.In this paper, we present a novel approach to high-brightness, high repetition rate injectors based onthermionic emission and beam manipulation using RFcavities, as is schematically illustrated in Fig. 1. Thethermionic electron gun, consisting of three separatestages, will first generate a continuous electron beam.Next, an RF deflection cavity containing a fundamentaland higher harmonic mode will chop the beam intoseparate bunches. Finally, an RF bunching cavity, alsorunning on two modes, will longitudinally compress thebunch to a sub-ps pulse length. The compressed bunchcan then, for example, be injected into a booster linaccapable of increasing the beam energy to >
10 MeV. Thisinjection occurs slightly before the point of maximumcompression, so the beam will become relativistic atits minimal length. Hence, the point of maximumcompression is the Point of Interest (PoI).The first stage in the electron gun is a thermionicemitter, either a LaB or CeB crystal housed in acustom designed 100 kV DC accelerator, generating ahigh average current continuous electron beam whichis subsequently accelerated in the positive z -direction.With current technology these emitter crystals can bemanufactured with an increasingly lower work function,resulting in a higher quality beam. Furthermore,these crystals can operate in a background pressure ashigh as 10 − mbar while being capable of operatingfor thousands of hours, creating both a simple androbust system. Operating at 100 kV instead of themore common 500 kV reduces the requirement of highvoltage insulation, leading to a smaller HV source and a smaller gun. The design of the gun is shown in Fig. 2.Construction of the DC accelerator is finished and initialtesting has begun.In order to inject this beam into a booster linac,it has to satisfy certain entry conditions. First andforemost the beam should be pulsed with a maximumroot-mean-square (rms) pulse duration of typically 0 . π to 0 . π radians of the booster linac RF phase [13]. Inthe second part of the thermionic gun, the continuousbeam is chopped into bunches by deflecting the beamonto a knife-edge using the on-axis transverse magneticfield of an RF deflection cavity. It has been shown thatthis chopping can be done without loss of beam quality[14]. However, the low duty cycle of this process defeatsthe purpose of using a continuous, high current densityemitter. We will show that the addition of a secondharmonic mode into the chopping cavity can increasethe duty cycle to up to 0 . π radians phase angle of thefundamental mode, i.e. a 30% duty cycle.This increased duty cycle means the bunch is now toolong to be injected into a booster linac. It is thereforerequired to compress the bunches by a factor of at least6 f boost f chop , with f boost the booster linac frequency and f chop the chopping cavity fundamental mode frequency.The third part of the thermionic gun will use velocitybunching to achieve this compression. By inducing alinear correlation between the longitudinal velocity andtime, i.e. a negative v z - t chirp, the front of the bunch willovertake the rear, ballistically compressing the bunchover a drift space. In order to achieve this, we will usethe on-axis longitudinal electric field of an RF bunchingcavity. This method assumes that the change in electricfield over time will be approximately linear. Near thezero-crossing of the field, this is certainly the case. Yetdue to the increased duty cycle of the chopping process,the bunch will encounter distinctly non-linear partsof the field, resulting in insufficient acceleration anddeceleration of respectively the front and rear end of thebunch as compared to the bunch center. This aberrationdecreases the compression ratio such that the buncheswill be too long to enter the booster linac. The additionof a second harmonic mode into the cavity can accountfor these non-linearities, achieving sub-ps pulse lengthsat the PoI. With this, the thermionic-based electrongun can deliver ultra-short, high quality, picocoulombcharge bunches at GHz repetition rates. An electrostatic FIG. 2. CAD model of the custom design thermionic gun DC accelerator. The 100 kV insulator is a Claymount R10 HVreceptacle. deflector could then select a train of pulses for injectioninto the booster linac, allowing the electron gun tooperate in either continuous wave or burst mode.The second part of this paper (Sec. II) will describe theprocess of thermionic emission, leading to the designof the cathode and anode. In order to achieve tens ofpC per bunch, the emission current is in the order of100 mA. At 100 kV, this will require an HV generatorcapable of delivering a few tens of kW, combined witha beam block capable of dissipating it. In the firstexperiments, the gun will be operated at a power of1 kW where a 0 .
15 mm radius emitter will delivera 10 mA average emission current to create a lowemittance continuous beam. Appendix A shows howhigher average currents could be achieved.Sec. III details the process of beam chopping, showinghow a second harmonic theoretically improves the chop-ping efficiency. Charged particle tracking simulationsare used for further optimization, demonstrating theincreased effectiveness of the cavity operating at boththe fundamental and second harmonic mode.Sec. IV describes the compression of the chopped beam,including a similar derivation for the higher harmonicmode. Simulation results will compare the bunchparameters with and without the addition of the secondharmonic as well as the final parameters at the exit ofthe thermionic gun.
II. THERMIONIC GUNA. Thermionic emission
In thermionic emission the current density J achievedby the emitter is described by the modified Richardsonequation [15] J = A g T exp( − ( W − ∆ W ) k b T ) , (1)with A g a material constant called the Richardson con-stant, T the crystal temperature, k b the Boltzmann con-stant, W the work function of the material and ∆ W theeffective lowering of the work function due to the Schot-tky effect, given by [16]∆ W = (cid:115) e E πε . (2)Here, e is the electron charge, E the electric field at thecathode and ε the vacuum permittivity. For LaB theRichardson constant is 29 A cm − K − , while work func-tions are reported between 2 . . ε n,rms = γβ (cid:113) (cid:104) x (cid:105) (cid:104) x (cid:48) (cid:105) − (cid:104) xx (cid:48) (cid:105) , (3)with γ the Lorentz factor, β = v/c the velocity normal-ized to the speed of light c , the transverse position x andthe divergence in the paraxial approximation given by x (cid:48) ≈ v x /v z . As in x-ray generation schemes, the inter-action happens mostly with the core of the phase spaceregion of the beam [20]. A better figure of merit thereforeis the 90% core emittance ε core , which entails removingthe outermost 10% of the 4D transverse phase space priorto the calculation of the emittance.At the cathode, the electrons are still being acceleratedand this approximation is not yet valid. Instead, theinitial emittance for a circular and uniform thermionicemitter can be calculated as the thermal emittance ε thn,rms = r (cid:114) k b Tm e c , (4)with r the crystal radius and m e the electron rest mass.From Eqs. (1) and (4) follows that, for a given current,a smaller radius with a higher temperature results in thehighest beam quality. The lifetime of the emitter, how-ever, is adversely affected by higher temperatures. ForLaB , limiting the temperature to below 1800 K ensuresa few thousands of operation hours; higher temperaturesresult in quickly decreasing lifetimes. Even if higher tem-peratures were possible, the current density cannot be ar-bitrarily increased. The limit is described by the Child-Langmuir law which states that as the current densityincreases, the electric field on the cathode caused by thespace charge of the electrons will counteract the electricfield of the DC accelerating structure. Current densitiesclose to this limit are thus called space-charge limited.When operating in this regime, the electrons that escapethe cathode are only accelerated slowly due to the smallernet electric field. This gives space-charge effects moretime to influence the beam which may cause a signifi-cant growth in transverse emittance. This regime shouldtherefore be avoided. For an infinite area cathode emit-ting electrons with zero velocity towards a parallel infi-nite area anode, the Child-Langmuir law is determinedas [21] J max = 4 ε (cid:114) em e V / d , (5)with V and d the electric potential and the distance be-tween the cathode and anode respectively. In real elec-tron sources, the cathode surface is small compared to thedistance between cathode and anode, requiring the Child-Langmuir law to be multiplied by a factor F dependingon the ratio r/d [11]. Limiting the cathode temperatureto 1760 K and taking a crystal radius of 0 .
15 mm resultsin a thermal emittance of only 0 .
04 mm mrad. With acurrent of 10 mA and a chopping duty cycle of 30%, acharge per bunch of Q = 3 . / mis required to reach this current. At an electric poten-tial of 100 kV, a gap of 10 mm and the correction factor F ≈ .
7, the adjusted Child-Langmuir limit is approxi-mately 245 mA. This is well above the intended operating current of 10 mA. Therefore, the electron source is notoperating in the space-charge limited region.
B. Gun Design
In order to achieve an electric field of 10 MV / m at theemitter surface in a DC setup without breakdown, thecustom cathode-anode assembly shown in Fig. 2 has beendesigned. The geometry is optimized so that the electricfield strength near the crystal is maximized, while simul-taneously minimizing the increase everywhere else, as isshown in Fig. 3. The principal way this is done is byraising the cathode towards the anode hole while taper-ing the anode. This ensures minimal distance betweenthe emitter and anode hole, while not decreasing the dis-tance between cathode and anode, thus keeping the peakelectric field strength low. Electrostatic simulations us-ing cst studio suite [22] show that when 10 MV / m atthe emitter is reached, the peak field strength is approx-imately 15 MV / m, near the edge of the emitter. Thisedge is a graphite ring with rounded edges that ensuresno emission occurs on the side of the crystal. By makingthis graphite ring as large as 3 mm across, the differencein the electric field strength between the crystal edge andcenter will be minimal. The size of the graphite ring,however, is currently limited by manufacturing restric-tions and is set at 2 mm across. With a field strengthof 15 MV / m, electron emission from the graphite will beover four orders of magnitude smaller than that of thecrystal. Moreover, if an electrical breakdown should oc-cur here, it shall be part of the thermionic current. Thiswill only result in a temporary emitter current fluctua-tion and will not damage the accelerator structure.After the electrons have been liberated from the crys-tal surface they first have to pick up speed and thereforespend a relatively large amount of time near the cathode. E - f i e l d ( V / m ) AnodeCathodeEmitter zr E - f i e l d ( V / m ) AnodeCathodeEmitter zr FIG. 3. Electric field map between the anode and cathode.The field is rotationally symmetric with respect to the dashedred line, while the hatched area depicts the LaB crystal. chop(a) chop(b) FIG. 4. (a) The principle of RF chopping. (b) The proposed method of chopping to increase the charge per pulse, using afundamental mode, a higher harmonic mode and a constant magnetic field. The electron beam is shown in green and the forcedue to the oscillating magnetic field in blue.
As the space charge forces rapidly expand the beam, theouter electrons will sample the off-axis non-linear fields,causing significant emittance growth. In order to coun-teract this, a magnetic solenoid lens is placed directlybehind the anode to control the transverse beam size.However, the solenoid has a residual longitudinal mag-netic field B res at the cathode, causing emittance growthin the beam due to an asymmetry between the azimuthalentrance and exit kick. This normalized rms magneticemittance growth is given by [23] ε mag = er | B res | m e c . (6)Since this emittance growth scales with r B res , combinedwith the small size of the emitter as well as careful place-ment and tuning of the solenoid, the contribution of thiseffect should remain small. This will be investigated inSection III B. III. BEAM CHOPPINGA. RF cavity theory
The second part of the proposed setup is to chop thecontinuous beam of electrons into bunches with minimalloss of beam current, while maintaining beam quality.Chopping an electron beam can be done using RF cavitieswith an on-axis transverse magnetic field, as illustratedin Fig. 4(a). The time-dependent transverse magneticfield deflects the beam periodically, after which an aper-ture blocks parts of the continuous beam, creating a trainof ultra-short electron bunches [24]. While this methoddoes maintain beam quality, the downside is the limitedduty cycle. Losing typically ∼
99% of the beam currenton the aperture, a far greater initial current is requiredto achieve a decent charge per bunch.In order to avoid this, the duty cycle can be increased byadding a higher harmonic mode to the cavity and choos-ing a different RF phase range in which electrons will passthe aperture. In the initial method that phase range isnear the zero-crossing of the field. By adding a constantmagnetic field, the electrons that pass the aperture arethose that experienced the peak of the sinusoidal field. A higher harmonic mode can then be used to flatten thatpeak, as shown in Fig. 4(b). Since chopping now onlyoccurs on the top part of the beam, the aperture is re-placed by a knife-edge.Because the standing wave in an ideal cylindrical cavityis dependent on a Bessel function in the radial direction,a higher order mode is not an integer multiple of the fun-damental frequency. This means a cylindrical cavity isinherently unfit for higher harmonic operation, increas-ing the difficulty of design. However, the magnetic fieldsin a rectangular vacuum cavity operating in the TM klm mode are described by sinusoidal functions of position B y ( x, y, z, t ) = B cos (cid:18) kπa x (cid:19) sin (cid:18) lπb y (cid:19) cos (cid:16) mπd z (cid:17) sin ( ωt ) , (7)with k , l and m the number of anti-nodes and a , b and d the cavity dimensions in respectively the x -, y -, and z -direction and B the magnetic field amplitude. Thepropagation of the electrons is in the positive z -direction.This sinusoidal behavior means that with the correct cav-ity dimensions higher harmonics are possible. In orderto determine which mode has to be added, we will lookat the resulting time-dependent magnetic field along the z -axis. To achieve the greatest duty cycle without lossof beam quality, the magnetic field experienced by theelectrons should remain constant for as long as possible.The ideal would be a rectangular wave. Approximat-ing a rectangular wave with its Fourier series, however,requires many modes to reach suitable flatness. For ex-perimental feasibility, we will only include two modes inthe cavity.With k even, l odd and m = 0, the on-axis magnetic fieldof Eq. (7) for the first and the η -th order mode simplifiesto B ,ηy ( t ) = B (cid:18) sin ( ω t + φ ) + 1 ζ sin ( ω η t + φ η ) (cid:19) , (8)with ω η = ηω , ζ = B /B η the amplitude scaling be-tween the two modes and φ η the phase of the η -th har-monic. To achieve a flat-top magnetic field profile aroundtime t = 0, all derivatives d n B y /dt n = 0 for n ≤ p , with p as large as possible, so that B ,ηy ( t ) ≈ C + O (cid:0) t p +1 (cid:1) , (9)with C some constant. The higher p is, the larger t hasto be in order for the field to deviate significantly from itsvalue at t = 0. Since φ , φ η and ζ can be controlled freely,the first three derivatives can be set to zero, resulting in B ,ηy ( t ) = B (cid:18) cos ( ω t ) − η cos ( ηω t ) (cid:19) . (10)In order to determine the optimal η , the fourth derivativeat t = 0 should be close to zero: d B ,ηy dt (cid:12)(cid:12)(cid:12) t =0 = B ω (cid:0) − η (cid:1) . (11)Since integer η >
1, the smallest absolute value is ob-tained for η = 2, which means a second harmonic shouldbe added to the cavity. Note that this is distinctly dif-ferent from a square wave where the added term wouldbe a third harmonic.A suitable combination of field modes for chopping isthe TM and TM mode, of which the on-axis time-dependent magnetic field is shown in Fig. 5. In order forthe latter mode to be the second harmonic of the first,the cavity dimensions must satisfy the relation f = 2 f , (12)with the resonance frequency given by f klm = c (cid:114) ( ka ) + ( lb ) + ( md ) , (13)resulting in a ratio of a = b (cid:114) . (14) tB – Fundamental TM210 mode – Second order TM230 mode – Sum of modes
FIG. 5. The time-dependent magnetic field in the centerof the chopping cavity. The dashed green line represents thefield probed by an electron bunch.
For a fundamental frequency of 1 GHz the dimensionsthen are: a = 379 mm, b = 245 mm. The length ofthe cavity in the z -direction can still be chosen freely, aslong as it does not cause other modes to be excited at orclose to the fundamental or second harmonic frequency.The fundamental frequency of 1 GHz has been chosen forseveral reasons. With the higher order modes, the usedfrequencies are f and 2 f . In the 1 to 3 GHz range theRF equipment is readily available. With a higher fun-damental frequency, such as 3 GHz, the required equip-ment is both more expensive and difficult to obtain. Alower frequency implies a lower repetition rate of the elec-tron bunches, but also a proportional increase in chargeper bunch. The downside of going to frequencies below1 GHz is the size of the cavities, which will exceed 0 . . ∼
96 ps rms,which will be shown in the following section.
B. Beam chopping simulations
The entire beamline has been simulated using GeneralParticle Tracer ( gpt ) [25]. Since gpt can only simulatecharged particle bunches, the continuous beam was sim-ulated by using a 2 ns long macro-bunch. This ensuresthat a large enough fraction in the center of this macro-bunch is not influenced by space charge effects due tothe presence of a front and back end. This fraction, ef-fectively coming from a continuous beam, is the part thatwill eventually pass the chopping knife-edge.Using the electric field distribution shown in Fig. 3 as theaccelerator field, the 100 keV electron beam is generatedat t = 0 and z = 0. As was demonstrated at SACLA,creating a thermionic source whose actual beam param-eters agree with their theoretical value is experimentallyachievable [11]. With a filament radius of 0 .
15 mm, atemperature of 1760 K and a field strength of 10 MV/m,the 10 mA beam is then generated with an emittanceof 0 .
04 mm mrad in both the x - and y -direction. The100 keV continuous electron beam passes through thefirst magnetic solenoid, which has a residual field on thecathode of approximately 8 . ε tot = (cid:113) ε + ε ≈ . . This result is similar to the x - and y -emittances of 0 . . z = 0 .
23 m. 3D electromagnetic field maps of both the .
75 4 .
80 4 .
85 4 .
90 4 .
95 5 .
00 5 .
05 5 . − . − . − . . . . . t (ns) x ( mm ) N particles ≈ . × (a) .
75 4 .
80 4 .
85 4 .
90 4 .
95 5 .
00 5 .
05 5 . − . − . − . . . . . t (ns) y ( mm ) N particles ≈ . × (b) FIG. 6. A single bunch after passing the knife-edge viewed (a) perpendicular and (b) parallel to the transverse chopping field.Total charge Q = 3 . chopping and compression cavity were generated using cst studio , with the magnetic field of the chopping cav-ity oriented in the transverse y -direction. Details can befound in Appendix B. The fundamental and second har-monic chopping modes operate at a 1 and 2 GHz fre-quency respectively, with separate control over the RFphases and amplitudes. Also added is a constant mag-netic field in the y -direction. The strength of this field isset in such a way that the direction of propagation for theelectrons of interest is not changed during the choppingof the beam, as is shown in Fig. 4(b).The amplitude of the fundamental mode is set at B =1 . B = B / . /η in Eq. (10). This is be-cause the electrons are influenced by the integral of thefield over their transit time through the cavity. This in-tegration smooths out the field so that a higher relativestrength can be used to increase the charge per pulse.After passing the knife-edge at z = 0 .
57 m, the electronbeam now consists of 3 . y -direction,the bunch oscillates and is chopped in the x -direction, asshown in Fig. 6(a). The grey particles indicate the elec-trons that hit the knife-edge, choosing the cut-off pointso that there are no protrusions at the front and back ofthe bunch. The top of the bunch shows a small indent,which is caused by the aforementioned increased relativeamplitude. Fig. 6(b) displays the same bunch, but sinceno chopping occurs in the y -direction the bunch shapewill remain rectangular. IV. PULSED BEAM COMPRESSION
Like the cavity used for chopping, a rectangular RFcavity can be used for compression. With an on-axislongitudinal electric field, electrons passing through thiscavity will experience a force solely in the propagationdirection, causing them to be accelerated or decelerated.The amount of acceleration is determined by the inte-gral of the electric field experienced and therefore de-pends upon the RF phase. Due to the time-dependentfield, different parts of an electron bunch will thus expe-rience different amounts of acceleration. By setting theRF phase correctly, the center of the electron bunch willtravel through the center of the cavity when the electricfield goes through its zero-crossing. The electrons at thefront of the bunch will then be decelerated while those atthe rear will be accelerated, initiating a ballistic compres-sion. This method of compression uses the fact that theelectric field around the zero-crossing is approximatelylinear. For electron bunches at 30% the length of oneRF cycle this approximation is no longer valid. To allowlonger bunch lengths to be compressed a higher harmonicmode can be used in order to shape the waveform.To achieve optimal compression, the resulting velocitydistribution of the electrons should be linearly increasingfrom front to rear, i.e. a negative velocity-chirp. Assum-ing all electrons have the same longitudinal velocity, thismeans the change in momentum and thus the integratedelectric field should be linear in time. This integratedfield takes the shape of the moving average of the elec-tric field, where the cavity length determines the inter-val of the integration. Therefore, if the electric field issufficiently linear, the change in momentum and thus ve-locity will be as well. Taking a similar approach as withthe chopping cavity, the Taylor series of the electric field tE – Fundamental TM110 mode – Second order TM310 mode – Sum of modes
FIG. 7. The time-dependent electric field in the center ofthe compression cavity. The dashed green line represents thefield probed by an electron bunch. around t = 0 should then be of the form E ,ηz ( t ) ≈ Ct + O (cid:0) t p +1 (cid:1) , (15)with C <
0. This means setting the zeroth, secondand third derivative to zero. Doing so will also set thefourth derivative to zero, so η is used to minimize thefifth derivative, leading to E , z ( t ) = E (cid:18) − sin ( ω t ) + 18 sin (2 ω t ) (cid:19) , (16)which is schematically illustrated in Fig. 7.In general, the modes suitable for compression are theTM κ +1,2 λ +1,0 modes with κ and λ non-negative integers.The fundamental mode at frequency f is TM , wherethe ratio of dimensions for the double frequency 2 f modecan be determined similar to the chopping cavity usingEqs. (12) and (13), resulting in a = b (cid:114)
53 (17)for the TM and TM mode. For a 1 GHz cavity, a = 245 mm and b = 190 mm. A. Bunch compression simulations
After the chopping cavity and knife-edge, the bunchesdrift towards a magnetic solenoid that controls the beamradius, after which it finally enters the RF compressioncavity at z = 0 .
73 m. Here, the 1 GHz and 2 GHzmode initiate the bunch compression. In the absenceof space charge effects, a perfectly linear relationship be-tween t and β z would result in a minimal compressed -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.200.480.500.520.540.560.580.60 st + 2 nd harmonic 1 st harmonic only b z t-t (ns) FIG. 8. gpt simulated longitudinal phase space ( t, β z ) of abunch directly behind the compression cavity. pulse length, directly related to the energy spread of thebunch prior to compression. Fig. 8 shows that addinga second harmonic to the cavity significantly increasesthe linearity of this relationship. As with the chop-ping cavity, the optimal mode parameters are differentfrom their theoretical value, as they can somewhat ac-count for non-linearities in the bunch, such as the ini-tial energy spread and space-charge effects. Using gpt for optimization, the mode parameters are { φ , φ η , ζ } = { . π, . π, . } instead of the theoretical { π, , } ,with E = 7 .
06 MV/m. The large difference for ζ ismainly because the cavity length is not taken into ac-count in the theoretical value. As a longer cavity resultsin more averaging of the field, greater deviations from alinear electric field can still result in a linear integratedfield, which is ultimately what leads to ballistic compres-sion.As shown in Fig. 9, after drifting for about 0 .
34 m andthrough another solenoid at z = 0 .
97 m, the electronbunches will pass through the PoI. Comparing this re-sult with that of the optimal compression using only thefirst harmonic shows a significant decrease in rms pulselength, creating 3 . t = 0 in thelongitudinal phase space is slightly lower for the higherharmonic compression.The emittance from Eq. (3) can be used to comparethe bunch quality during compression. However, sincethe chopped bunch was part of the top of a modifiedsine wave, it is asymmetrically shaped. Added to thisis that the range of RF phases that will allow electrons st + 2 nd harmonic 1 st harmonic only s t ( p s ) Position (m) t,min =0.28 ps
FIG. 9. RMS temporal bunch length during compression.Initial pulse length is 95 .
57 ps.
PoI
Cathode emittance: 0.04 mm mrad e corex e x e corey e corer e n , r m s ( mm m r ad ) Position (m)
FIG. 10. Emittances of the bunch at the end of the beamline.The bunches pass through the PoI at z = 1 .
07 m. to pass the knife-edge is changed, depending on how theknife-edge is positioned. If a greater range is allowed, thebunch in Fig. 6(a) will be more ”banana-shaped”, withthe front and rear of the bunch extending further in thenegative x -direction, as depicted by the grey particles.These parts have a significant effect on the emittance buta lesser effect on x-ray generation, which the 90% coreemittance takes into account. Furthermore, beamline el-ements such as a magnetic solenoid introduce large cor-relations between the transverse positions and velocitiesof the electrons, causing a large but temporary emittancegrowth. By looking at the radial emittance ε r, n,rms = γβ (cid:113) ε x, rms ε y, rms − |(cid:104) xy (cid:105) (cid:104) x (cid:48) y (cid:48) (cid:105) − (cid:104) xy (cid:48) (cid:105) (cid:104) x (cid:48) y (cid:105)| the contributions to this growth can be reduced. Fig. 10shows the core x -, y - and r -emittances after goingthrough the compression cavity at z = 0 .
73 m. At po-sition z = 0 .
81 m the core emittance has increased to0 .
041 mm mrad. Comparing this to the emittance of0 .
049 mm mrad shows that the majority of emittancegrowth is due to the outer 10% of the phase space.Comparing Figs. 9 and 10 shows a significant increasein transverse emittance around the PoI. This is due tospace-charge effects that start to dominate the short elec-tron bunch, but this increase can be reduced by de-creasing the amount of compression or decreasing thecharge per pulse. In short, at the PoI the thermionicelectron gun provides 279 fs electron bunches with a0 .
089 mm mrad radial core emittance at a 1 GHz rep-etition rate.
B. Cavity Feasibility
As mentioned in Section III A, since the chopping andthe compression cavity both operate at the fundamentaland the second harmonic mode, a rectangular cavity ispreferred. Research into power efficient RF cavity de-sign, however, has mainly been done for pillbox cavities[26]. This is accomplished either through modifying thegeometry of the cavity or through the introduction ofdielectric material into the cavity, where the cylindricalsymmetry somewhat simplifies the design process. Fora rectangular cavity, such options have not yet been ex-plored, as far as we know. cst studio simulations show that for a regular rectangu-lar copper compression cavity, the total power required todrive both the fundamental and second harmonic modeto the field amplitudes used in this paper is approxi-mately 539 W. For the chopper cavity, 425 W is requiredto reach a peak field of B = 1 mT. No optimizationhas been performed on these geometries, leading to thesemoderately high powers. Still, CW solid-state amplifiersare commercially available in this power and frequencyrange and should pose no problem. V. CONCLUSIONS
With the need for a next-generation high-repetition-rate, high-brightness electron injector, we have presenteda design concept for a 100 keV pulsed electron gun basedon the chopping and compression of a continuous beamfrom a LaB cathode.The use of an RF cavity for beam chopping allows theelectron gun to reach GHz repetition rates without se-vere degradation of beam quality. Adding a higher har-monic mode, the duty cycle of this process is greatlyincreased, reaching a high charge per bunch relative tothe initial current, resulting in 279 fs, 0 .
089 mm mrad,3 . ACKNOWLEDGMENTS
The authors would like to thank Eddy Rietman andHarry van Doorn for their invaluable technical supportin the mechanical and electrical design of the thermionicgun. This research is part of the High Tech Systems andMaterials programme of the Netherlands Organisationfor Scientific Research (NWO-AES) and is supported byASML.
Appendix A: High current operation
As Section III has shown, a 10 mA continuous electronbeam sent through the 1 GHz chopping cavity will re-sult in a charge of Q = 3 . A g and W are determinedby the emitter material, ∆ W is limited by the peak elec-tric field in the gun and T cannot be increased withoutseverely affecting the lifetime of the filament. As such,the only method of increasing current is to increase theemitter radius. Several things have to be taken into ac-count for this greater radius.First, the charge per bunch will increase quadraticallywith the emitter size, while the thermal emittance willincrease linearly. For example, going from r = 0 .
15 mmto r = 0 .
75 mm will increase Q to 75 pC and the ther-mal emittance to 0 .
20 mm mrad. The new radius also changes the multiplication factor for Eq. (5) from F ≈ . F ≈ .
8. Recalculating the currents from Eqs. (1) and(5) comes to an emitted current of 250 mA. Meaning theemission current is still over ten times smaller than thespace-charge limited current of 3 . Appendix B: RF cavity design
In order to use the RF cavities in gpt , a descriptionof the 3D electromagnetic fields is required. To generatethe required fields, a preliminary EM design was madefor the rectangular chopping and compression cavities,shown in Figs. 11 and 12 respectively. The cavity dimen-sions agree to within 1 mm with those calculated withEqs. (14) and (17) and have a length of 36 . . [1] W. S. Graves, J. Bessuille, P. Brown, S. Carbajo, V. Dol-gashev, K.-H. Hong, E. Ihloff, B. Khaykovich, H. Lin,K. Murari, E. A. Nanni, G. Resta, S. Tantawi, L. E.Zapata, F. X. K¨artner, and D. E. Moncton, Compact x-ray source based on burst-mode inverse compton scatter-ing at 100 khz, Phys. Rev. ST Accel. Beams , 120701 (2014).[2] K. Batchelor, H. Kirk, J. Sheehan, M. Woodle, andK. McDonald, Development of a High Brightness Elec-tron Gun for the Accelerator Test Facility at BrookhavenNational Laboratory, Particle accelerator. Proceedings,1st EPAC Conference, Rome, Italy, June 7-11, 1988. (a) (b) xy z FIG. 11. Cross section of the (a) fundamental and (b) higher harmonic chopping mode cavity. Colors represent the relativefield strength of the magnetic field in the y -direction with red positive and blue negative, normalized to the maximum fieldstrength. Cavity dimensions in mm are 379 . × . × . (a) (b) xy z FIG. 12. Cross section of the (a) fundamental and (b) higher harmonic compression mode cavity. Colors represent the relativefield strength of the electric field in the z -direction with red positive and blue negative, normalized to the maximum fieldstrength. Cavity dimensions in mm are 189 . × . × . Vol. 1, 2 , Conf. Proc.
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