Ultra-Compact Ka-band linearizer for the Ultra-Compact X-Ray Free-Electron Laser at UCLA
Bruno Spataro, Mostafa Behtouei, Luigi Faillace, Alessandro Variola, Valery Dolgashev, James Rosenzweig, Giuseppe Torrisi, Mauro Migliorati
UUltra-Compact Ka-band linearizer for the Ultra-Compact X-Ray Free-Electron Laser atUCLA
B. Spataro a , M. Behtouei a , L. Faillace a , d , A. Variola a , V.A. Dolgashev b , J. Rosenzweig c , G. Torrisi f , and M. Migliorati d , e a INFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy; b SLAC- National Accelerator Laboratory, Menlo Park, CA 94025, USA; c Department of Physics and Astronomy, University of California, Los Angeles, California 90095; f INFN, Laboratori Nazionali del Sud, Catania, Italy; d Dipartimento di Scienze di Base e Applicate per l’Ingegneria (SBAI), Sapienza University of Rome, Rome, Italy; e INFN / Roma1, Istituto Nazionale di Fisica Nucleare, Piazzale Aldo Moro, 2, 00185, Rome, Italy
Abstract
There is a strong demand for accelerating structures able to achieve higher gradients and more compact dimensions for the nextgeneration of linear accelerators for research, industrial and medical applications.Notably innovative technologies will permit compact and a ff ordable advanced accelerators as the linear collider and X-ray free-electron lasers (XFELs) with accelerating gradients over twice the value achieved with current technologies. In particular XFELis able to produce coherent X-ray pulses with peak brightness 10 orders of magnitude greater than preceding approaches, whichhas revolutionized numerous fields through imaging of the nanoscopic world at the time and length scale of atom-based systems,that is of femtosecond and Angstrom. There is a strong interest for combining these two fields, to form a proper tool with the goalof producing a very compact XFEL in order to investigate multi-disciplinary researches in chemistry, biology, materials science,medicine and physics.In the framework of the Ultra -Compact XFEL project (UC-XFEL) under study at the University of California of Los Ange-les (UCLA), an ultra high gradient higher harmonic radio-frequency (RF) accelerating structure for the longitudinal space phaselinearization is foreseen. To this aim, a Ka-Band linearizer (34.2 GHz) with an integrated voltage of at least 15 MV working on6th harmonic with respect to the main Linac frequency (5.712 GHz) is required. We here present the electromagnetic design of acold ultra compact Ka-band standing wave (SW) linearizer, 8 cm long, working on π mode with an ultra high accelerating gradient(beyond 100 MV / m) and minimum surface electric field for minimizing the probability of RF breakdown. Moreover, we discussa traveling-wave (TW) option and compare it with the initial SW structure, by means of main RF parameters as well as beam-dynamics considerations. The numerical electromagnetic studies have been performed by using the well known SuperFish, HFSSand CST computing software. Keywords:
Particle Acceleration, Linear Accelerators, Free Electron Laser, Accelerator applications, Accelerator Subsystems andTechnologies
1. Introduction
The development of ever more progressed accelerating struc-tures [1, 2] is one of the driving action of the accelerator com-munity. High gradients, e ffi ciency and unwavering quality ofthe accelerating structures play a fundamental role on linearaccelerators. There is a strong demand of these acceleratingstructures able to achieve higher gradients and more compactdimensions for the next generation of linear accelerators for re-search in chemistry, biology, materials science, medicine andphysics. As an example, a XFELS photoinjector with a highfield level of >
250 MV / m over a factor of 25 increased beambrightness, would have 1 / ff raction and microscopy.The UCLA Particle Beam Physics Laboratory program(PBPL) has established a nearly 30-year track record, as a lead-ing university-based group in beam and accelerator researchin both experiment and theory. Its multi-disciplinary approachpermits study over a wide spectrum of frontier topics, rangingfrom very high brightness electron sources using sophisticatedmicrowave structure (RF photoinjectors) [3, 4, 5, 6], to highgradient wakefield acceleration in plasmas and dielectrics, andextending to fundamental ultra-fast beam physics and diagnos-tics, and beyond, to advanced light sources such as free-electronlasers and Compton scattering sources necessary for wide va-riety of applications [7, 8, 9]. Recent progress in very highgradient radio-frequency acceleration techniques has been car-ried out in recent years by a UCLA-SLAC collaboration [10], Preprint submitted to Nuclear Physics A February 17, 2021 a r X i v : . [ phy s i c s . acc - ph ] F e b igure 1: Schematic layout of the Ultra-Compact X-Ray Free-Electron Laser of UCLA. with notable contributions also arising from INFN-LNF andLANL [11, 12, 13, 14, 15].With this work we considered a new technique, that of cryo-genic cooling of RF cavities, successfully at the frontier ofachievable gradients using microwave accelerators, with main-tenance of surface electric fields up to 500 MV / m [16]. On thelinac application side, SLAC has recently produced a designstudy of a collider based on cryogenic high gradient RF acceler-ation. This study points out that the limits of performance pro-foundly changed in cryogenic operation, and C-band, with itsattendant lower wakefields, may be the optimal RF frequencychoice for a >
100 MeV / m linear accelerator. Such acceleratorsmay be employed for compact FELs, Compton sources, wake-field accelerators, and many other applications. Indeed, due tothe higher frequency and lower thermal load, C-band is alsofavored for many photoinjector applications [3, 4, 17], particu-larly in FEL due to the desire for very low emittance.This project would represent the first use of such an accel-erator structure operating at cryogenic temperatures. It is alsothe initial development of a cryogenic, ultra-high brightness RFphotoinjector. Together, these structures permit testing of thesystem with a high level of beam loading, by exciting a bunchtrain from the photocathode. The low loss of the system impliesa high RF-to-beam power e ffi ciency, in excess of 60% [8]. TheUC-XFEL machine to be constructed at UCLA [9], includes a cryogenic short C band linac able of providing a beam energyof 300 MeV with an emittance 0.2 mm-mrad. A conceptual lay-out is reported in Figure 1 of [9], and it shows the necessity of aKa-Band RF structure for correcting longitudinal beam phase-space non-linearity as arises in the C band linac in an ultra com-pact X-ray FEL is required. There are no dedicated dampers ofthe parasitic higher order modes for the linearizer structure be-cause the beam dynamics is not influenced by the long-rangewake-fields due to the foreseeing of the single bunch operation.Several laboratories worldwide, such as the LaboratoriNazionali di Frascati (INFN-LNF) and UCLA, are involvedin the modeling, development and tests of RF structures de-voted to particles acceleration with higher gradient electricfield through metal device, minimizing the breakdown and thedark current. In particular, new manufacturing techniques forhard-copper structures are being investigated in order to de-termine the maximum sustainable gradients well above 100MV / m and extremely low probability of RF breakdown. Thepreferences of utilizing high frequency accelerating structuresare well known: smaller size, higher shunt impedance, higherbreakdown threshold level and short filling time. In refs.[18, 19, 20, 21, 22, 23, 24, 25], it is shown that there are rea-sonable candidates for microwave tube sources, which, togetherwith RF pulse compressor (SLED), are capable of supplying therequired RF power. The technologies in the Ka-Band acceler-2ting structures, high power sources and modulators have alsobeen developed by getting very promising results in order toreach a RF power output of (40-50) MW by using the SLEDsystem.RF stability during operation and tuning tolerances are im-portant points for the RF structure design in the high frequencyrange. The complexity of machining, tight mechanical toler-ances and alignments are therefore important aspects whichhave to be taken into account in the design activity. In orderto obtain a longitudinal phase space linearization we have de-signed a compact sixth harmonic standing wave (SW) accelerat-ing structure operating at a frequency of F = π mode at about 100 MV / m accelerating gradient.This report discusses the fundamental RF parameters depen-dence of the structure as function of the iris’s aperture and thecavity’s geometry in order to get the optimized RF design. Thefigures of merit, like the longitudinal shunt impedance, qual-ity factor, coupling coe ffi cient, transit time factor and so on,are calculated with the well known Superfish, HFSS and CSTnumerical computing software [26, 27, 28]. Moreover, in or-der to estimate the breakdown thresholds, investigations on themodified Poynting vector and pulse heating are discussed. Acomparison with a linearizer traveling wave structure operatingon the 2 π /
2. Motivation
One of the most important requirements, for any acceleratingstructure design, is related to the stability of the particles inthe beam. Like in any RF accelerating structure, the build-upof the wakefields and corresponding wake potentials generatedfrom the beam-cavity interaction and acting back on the bunchitself, can be destructive, leading to the beam break-up (BBU).Our concern regards here the transverse wakefield, since thebeam has very high energy ( ≥
300 MeV) and it is ultra-short(350 fs rms). In our project, operation is foreseen in single-bunch regime, therefore the only concern is due to the e ff ect ofthe so-called single-bunch BBU. We will discuss the e ff ect ofmulti-bunch BBU in a forthcoming paper.Here, we give an analytical estimation of the wake potentialinside the Ka-Band cavity by using the di ff raction theory [29].The transverse wakefield of a pillbox cavity of radius b , gap g and iris aperture radius a under the hypothesis that2 g σ ( b − a ) (cid:28) σ the rms bunch length, can be written as w ⊥ ( z ) = / Z c π a √ gz (2)where Z is the vacuum impedance and c the speed of light. Inour case, the ratio of Eq. (1) is about 0.13 in the worst case ofstanding wave structure (having the largest gap of 4.374 mm)with an iris radius of 1.33 mm and the cavity radius of 3.8776mm. This wakefield can be convoluted with a Gaussian or rectan-gular bunch to obtain the corresponding wake potentials. Forexample, the wake potential of a rectangular bunch of length l can be written as W ⊥ ( z ) = / Z c π a (cid:112) gl (cid:32) + z − l / l (cid:33) / (3)The wake potentials for the two bunch shapes are shown inFig. 2. For the rectangular case we have used a total bunchlength equal to l = √ σ . W a k e ( V / p C / m ) Wake pot. Gaussian bunchWake pot. Rect. bunch
Figure 2: Transverse wake potential of a Gaussian and rectangular bunch shapefor the wakefield obtained with the di ff raction theory. We can see from the figure that the wake is similar in bothcases and it can be considered linear inside the bunch. As aconsequence, for the analytical approach here described, we usethe linear approximation for the wake potential and a rectangu-lar bunch distribution.To evaluate the strength of the BBU, we consider two ex-treme cases. Under the further assumptions of mono-energeticbeam and constant beta function β inside the structure of totallength L , we can define the parameter η = β QW ⊥ ( l / LgE / e (4)where Q is the bunch charge, E is the bunch energy, e is theelectron charge, and the wake potential is evaluated at the bunchtail. If η (cid:29)
1, that is in case of very strong BBU, we can usethe following asymptotic expression [30] to evaluate the finaltransverse displacement of the tail ( x f ) with respect to the initialone ( x i ) x f x i = (cid:114) π η ( L ) − / e √ η ( L ) / (5)On the other hand, if η (cid:28)
1, that is in the weak BBU regime,we can use a perturbation method to obtain the first order solu-tion of the BBU equation, that, in case of constant energy, givessimply x f x i = η x f / x i perturbation methodasymptotic expression Figure 3: bbu
We observe that in the perturbation method case, since η islinear with the transverse wake potential and with the machinelength, we can easily deduce the BBU dependence by the di ff er-ent parameters. In particular the BBU increases inversely to thethird power of the iris radius and to the square root of the cavitygap. Therefore, for a travelling wave structure, we expect anincrease of the BBU by a factor of about 20 % with respect tothe standing wave one. More complicated is the behaviour incase of strong BBU. However, around an iris aperture of about1 mm, we obtain similar results. In particular, for a total cavitylength of L =
16 cm, we observe a ratio of about 2 which meansthat for an initial o ff set of the injection bunch of 100 um, the tailis displaced at the exit by about 200 um, which is still a factorof 4 lower than the iris radius.As for the longitudinal plane, in case of a rectangular shape,the wake potential can be written as: w || ( z ) = / Z c π a (cid:114) gl (cid:114) − z − l / l (7)In both SW and TW cases, one obtains a loss factor per unitlength equal to nearly 800 keV / m, which is a negligible valuewith respect to the beam operation energy.By choosing a length of 16 cm and an iris radius of 1 mm,we then provide the evaluation for the best performing type ofaccelerating scheme, for both standing-wave (SW) or traveling-wave (TW) cases. As published in [39], we demonstrated thatthe optimal maximum length for the SW structure is around 8cm, in order to avoid resonant mode overlapping. In SW op-eration, our proposal is then to utilize two SW structures, each8 cm long, fed by a 3dB hybrid coupler. This setup is alsobeneficial for protecting the high-power source and avoiding ahigh-power circulator, which is not trivial to fabricate in Ka-Band. From simulations, it turns out that the required input RF power for an 8 cm long SW cavity is around 8 MW. On theother hand, for the TW case, the required input RF power isshown in Figure 2 for an 8 cm long structure as well as 16 cm,that is possible only in TW operation. The group velocity is as-sumed to be an average value in the estimation of the requiredpower for both constant-gradient and constant-impedance TWstructures. In any case, the needed RF power for the same 8 cmlong structure is a factor of 3 less for the SW option. Figure 4: Required input RF power vs. accelerating gradient for the TW struc-ture in the case of two lengths, 8 cm and 16 cm. E acc corresponds to an inte-grated voltage of 20 MV. Eventually, we give the final comparison of the main RF pa-rameters between a SW and TW structure with a length L =
3. Vacuum RF breakdown
It is well-known that the vacuum RF breakdown, pulsed sur-face heating and field emission are among the major issues lim-iting the highest achievable accelerating gradient inside an RFcavity.In particular, the RF breakdown instantaneously a ff ects theRF power fed into the accelerating structure, thus loweringits performance [11]. Being probabilistic, many experimentshave been carried out in order to study and then measure thebreakdown-rate probability of such phenomenon. As of today,most of these studies have been performed at X-Band frequen-cies [14, 15, 31], showing gradients well beyond 100 MV / m.This is the demonstration of the accelerating gradient scalingwith frequency.Material technology and handling represents another crucialfactor for the determination of the ultimate performance of anaccelerating structure. In order to study the contribution of this4 able 1: Comparison between SW and TW. Main RF parameters SW TWFrequency [GHz] 34.272 34.272Accelerating Gradient [MV / m] 125 125Input RF power [MW] 8 23.9 Input RF power @77K [MW] 3.63 10.8
Attenuation [ m − ] - 2.51Shunt impedance [M Ω/ m] 154 158Unloaded quality factor, Q 5728 4110Group velocity [%] - 3.65Coupling coe ffi cient, K% 0.83 -Structure length [cm] 8 8Build-up [ns] 12.5 -Filling Time, T f [ns] - 7.3RF Pulse length flat top [ns] 50 50Repetition rate [Hz] 100 100Average RF power / m [kW / m] 0.52 0.52Average RF power / m @ 77K [kW / m] 0.23 0.23 factor to the RF beakdown-rate statistics, various technologi-cal techniques have been investigated and used here as refer-ence [32, 33, 34, 35, 36]. From these experiments, we havelearned that hard-copper alloys, at warm temperature, show thebest performance in terms of accelerating gradients. In particu-lar the best performance was obtained with CuAg that reachedan accelerating gradient of 200 MV / m at 10 − / pulse / m break-down probability using a shaped pulse with a 150 ns flat part.On the other hand, under cryogenics operation (cold tempera-ture below 77K), accelerating gradients up to 250 MV / m withsurface fields of 500 MV / m can be reached [10, 16].Therefore, the combination of high RF frequency and lowtemperature operation will bring the proposed Ka-Band lin-earizing cavity to the achievement of unprecedented values,both in terms of accelerating gradient and vacuum breakdownrate probability. In the next section, we discuss the details atcryogienic operation of the Ka-Band RF linearizer.
4. Cryogenic Operation of a Ka-band RF Linearizer
The use of cryogenic structures to both diminish the RF dis-sipation and to mitigate breakdown is by now well established,particularly through testing of X-band and S-band devices. Inthese experiments, the scaling of RF dissipation according tothe theory of the anomalous skin e ff ect (ASE) [37] has beenverified, and surface fields over 500 MV / m have been achievedbefore breakdown is observed [10]. The advantage in dissipa-tion e ff ects diminishes somewhat at high frequency, but is stillnotable up to Ka-band, the operating frequency that has beenproposed for a high gradient, compact linearizer. This compo-nent is critically important for applications such as the MaRIEXFEL [38], the CompactLight FEL, the Ultra-Compact XFELat UCLA [9]. Paired with the CompactLight sponsored initia-tive to develop a 15 MW-class klystron at 36 GHz, a compact,high gradient cryogenic linearizer in this frequency range nowseems within reach [39]. Here we review the scaling laws that allow approximate pre-diction of the performance of such a linearizer, based on a der-ated 5 MW input. To orient the expected performance, we notethat the shunt impedance calculated for an optimized 36 GHzstructure at room temperature is 158 M Ω/ m. We can scale theexpected behavior of this shunt impedance from detailed calcu-lations of ASE enhancement at low temperature in S-band by afactor of 5. To extend this to Ka-band, we note that the ohmicmodel scaling of surface resistivity is R s , Ω ∝ ω / , while forASE, the scaling in the low temperature limit is R s , AS E ∝ ω / .This means that the expected enhancement in the quality factorhas a scaling Q enh ∝ ω − / , and for low temperature (below 40K), one may expect in Ka-band Q enh (cid:39) .
3. For less ambitiouscooling designs, operating with liquid nitrogen at 77 K, we mayforesee an enhancement of 2.2.To give an idea of what is possible with this approach, weassume, as stated above, a 5 MW matched input into a 10cm long structure, operating at 77 K, with estimated shuntimpedance of 349 M Ω / m . In this case, the accelerating fieldis 130 MV / m, which is well below the breakdown limit of 250MV / m. The corresponding surface field of 260 MV / m is alsobelow the threshold of dark-current emission of ∼
300 MV / mthat is strong enough to beam load the structure [16]. Further,at this frequency, the normalized vector potential is a factor ofthree below that needed to capture and accelerate dark current,further mitigating potential issues with spurious field emissione ff ects.
5. Cavity choice criteria
Exhaustive discussions on the design cavity criteria in orderto choose the cavity RF parameters have already described else-where [39, 40, 41] in case of Ka-band structure to be used forthe Compact Light XLS project.The ultra compact Ka-band linearizer has to fulfill many re-quirements achievable as a compromise among the availablepower source, maximum e ff ective accelerating gradient, sim-ple geometry (for easy fabrication with reasonable toleranceswithin 2 µ m for getting small sensitivity to construction er-rors), narrow spread of electron energy, beam dynamics ef-fects in terms of beam loading, beam break-up etc. To mini-mize the input power required for a given accelerating gradient,the geometry should be designed with the aim of maximizingthe e ff ective shunt impedance per unit length. This one occurswith an about (cavity gap) / (cavity radius) ratio of 1 as it is alsoshown by using an analytical check of a pill-box cavity. On theother hand, because of the interaction between the beam and thesurroundings, the accelerating section performances are limitedby e ff ects such as the beam loading, instabilities, beam break-up etc. [30, 42]. Since we work with a low charge and singlebunch, we expect no specific problems on the beam dynam-ics quality. The third harmonic frequency of the main Linacimplies small physical dimensions and thereby the dissipatedpower constitutes one of the main constraints. A reasonable up-per limit on the average power dissipation has been estimatedto be the in the (4-5) kW / m range [43]. To meet the full require-ments by keeping a flexibility margin, a structure with simple5a) (b) (c) Figure 5: a) Shunt impedance b) unloaded quality factor and c) coupling coe ffi cient K (%) as a function of the a / λ ratio, where a is aperture radius. geometry and of reliable construction with satisfactory mechan-ical tolerances has to be chosen. In this paper, the main concernis to design an accelerating structure operating on the π modewith the requirements referring to the Table 1. Table 2: Parameters list for the cavity design
Design parametersFrequency [GHz] 34.272E ff ective Accelerating Electric Field [MV / m] 125Axial length[cm] 8Iris Aperture radius (a / λ )* 0.114Iris thickness (h / λ ) 0.076Ratio of phase to light velocity ( ν φ / c ) 1Pulse charge [pC] 75Rms bunch length ( σ τ ( f s )) 350Pulse repetition rate frequency [Hz] 100 * a: iris radius of the structure; λ = free space wavelengthIt should be emphasized that in normal conducting operationa 125 MV / m e ff ective accelerating electric field can be con-sidered as upper limit. Basically only experimental tests canconfirm reliability and stable operation to this level of field.For sake of completeness, high power tests on a 100 GHz π mode structure, a 10 ns long pulses and a / λ = / m with a peak surface of more than 520 MV / m [44].Technological advancements in terms of precision and assem-bly of the accelerating structure will allow to reach the designvalue of 300 MV / m accelerating gradient for 600 kW of dissi-pated power. As a result, it is reasonable to predict to work inthe (100-150) MV / m range in the Ka-band regime. It is alsoreasonable that by working at a cryogenic temperature of 77 Kdegree, the field can reach a level up to 150 MV / m.
6. Accelerating structure design
The Ka-band structure linearizing the longitudinal phasespace has to fulfill many requirements dominated mainly by thelack of high power klystron amplifier at 36 GHz to be adopted for feeding the linearizer. The key requirement of the ultra-compact linearizer Ka-band is to achieve an ultra high e ff ec-tive accelerating gradient using an available amount of powersource with negligible wakefield e ff ects on the beam dynamicsquality. Therefore special care should be taken in order to ful-fill the structure aperture for achieving a high shunt impedanceand use the available power source. Currently, high gradientstructure in the Ka-band regime can be operated with the gyro-klystrons devices which are able to provide an output power upto 3 MW, and with the SLED system which could provide 12MW by assuming a compression factor of 4. It is well knownthat the performances of high gradient structures are limited bybreakdown due to the pulse heating and field emission e ff ects.On the other hand, the RF properties depend strongly on thea / λ ratio where a is the iris aperture and λ the free space wave-length. A detailed analysis on the SW structure operating in theKa-band regime for the Compact Light XLS project has beendiscussed in a recent paper[39]. Much care was used in choos-ing the cavity shape in order to reduce or avoid breakdown ef-fects at a high accelerating gradient operation. As a result, anextensive electromagnetic analysis of the main RF parametersas function of the cavity geometry has been performed for get-ting a RF design with a minimum surface electric field. By as-suming an iris thickness h = π mode, we have quantified the influence of the iris radius andof its geometry on the main RF parameters. As a result, the RFstructure optimization has been reached with a iris radius in the(0.11- 0.14) a / λ range in case of elliptical geometry iris, with a5 / / λ = π mode inorder to get a satisfactory longitudinal shunt impedance and anacceptable iris aperture for practical beam dynamics considera-tions.Fig. 5 shows the main RF parameters estimations of the ef-fective shunt impedance Rsh / m, unloaded quality factor Q andcell to cell coupling coe ffi cient K% as function of the iris radiusa / λ obtained with the simulation code HFSS. In Fig. 3 we showthe peak electric and magnetic fields E p / E a and H p / E a , with E a Figure 6: a) E p / E a Cavity radius as function of the a /λ ratio. b) H p / E a Cavityradius as function of the a /λ ratio. the e ff ective accelerating as function of the iris radius a / λ with5 / / λ range forgetting a satisfactory compromise with the beam dynamics re-quirements. In this case, we observe a variation of about 22%of the R sh / m , 2% of Q, 187% of K%, 22% of E p / E a and 13% of H p / E a . Since the di ffi cult problems limiting the linearizer per-formance are the ultra-high gradient operation achievable andthe 36 GHz power source, we choose the geometry with a / λ = E p / E a with H p / E a and R sh / m satisfactory values if theyare compared with the best estimations. This choice is also agood compromise with the beam dynamics requirements sincewe suppose the beam emittance is estimated to be 0.2 µ rad at300 MeV [49]. As a result, the optimized cavity geometry ofthe linearizer has a rounded profile, an elliptical iris geometryof 5 / = Table 3: RF parameters of the designed cavity
Design parametersE ff ective Shunt impedamce [M Ω/ m] 189Unloaded quality factor, Q 5778Coupling coe ffi cient, K% 0.83 E p / E a H p / E a [mT / MV / m] 2.69Iris thickness [mm] 0.667Cavity radius [mm] 3.374Cavity gap [mm] 4.374Iris radius [mm] 1 These RF parameters are to be chosen to achieve an ultraaccelerating gradient by assuming the necessary RF power isavailable. However, we also note that the cavity RF parametersare conservative if compared to the usual accelerators.Additional checks of the chosen cavity geometry have beencarried out by estimating the quantities referring to the modi-fied Poynting vector (MPV) and the pulse heating (PH) e ff ectsin order to estimate the allowed levels of the breakdown safetythresholds and confirm the above discussed cavity optimiza-tion. Figs.7 (a) and (b) illustrate the MPV and PH as functionof the e ff ective accelerating field by assuming a flat top 50 nspulse length for di ff erent irises geometries, as it was alreadydiscussed for the Compact light XLS project linearizer [39] .We observe, the MPV and the PH are in agreement with theanalysis developed so far by comparing the E p / E a and H p / E a behavior. At a given e ff ective accelerating gradient, the worsecase occurs for the circular iris for both MPV and PH quan-tities. In all cases, we are well below the safety threshold ofMPV and PH which are about 5 MW / mm and 50 Celsius de-gree, respectively. The other geometries provide comparablevalues each other and they are smaller with respect to the circu-lar iris by factor of about 2.5 for the MPV and of the same orderof magnitude for the PH, by confirming the above analysis.For the sake of completeness, the estimated MPV for the TWcase is around 8 MW / mm and thus exceeds the threshold limitof 5 MW / mm .
7. Power source
As we already mentioned, so far the technologies in the Ka-Band regime for accelerating structures, high power sourcesand modulators have been developed by getting promising re-sults in order to reach a RF power output of (40-50) MW byusing the SLED system. However, a lot of work has yet to bedone in order to achieve high power Ka-band devices with per-formances reproducible and reliable. Our main interest is to getat least un integrated voltage of 15 MV [49]. Therefore, we alsoare planning to finalize the linearizer design as well as engineer-ing of the RF power source that will be able to produce up to a10 MW input power by using a SLED system. Analytical esti-mations on the Gyroklystrons devices provide an RF output ofabout 3 MW and with the SLED system we are able to get anRF output power of 12 MW. In addition, to create shorter RF7b) (a)
Figure 7: a) Modified Poynting Vector b) Pulsed Heating as function of E acc pulses, a laser-based RF switch developed in the framework ofthe MIT-SLAC collaboration for the W band structures can beused to select a proper RF pulse length [50]. As a result, it isreasonable to assume an available RF power of about 11 MWin order to feed the Ka-band linearizer. Moreover, for sake ofcompleteness, additional investigations and optimization on the36 GHz klystron design are in progress at INFN-LNF in collab-oration with Tor Vergata University of Rome [51, 52] . So far,numerical studies provided a 42% e ffi ciency with a 20 MW RFoutput even if a confirmation of this estimation has to be carriedout, too.
8. Practical achievable integrated voltage
The linearizer has to be designed following criteria similarto those used to build SLAC X-band high gradient acceleratingstructures. It is our concern that this structure will operate witha surface electric field up to of 250MV / m to keep the break-down rate probability below a reasonable value. Due to thesmall dimensions of the linearizer, one of major performancelimitations are losses.Table 3 reports the RF parameter list of the normal con-ducting linearizers structures SW and SW with a 1 mm and1.333 mm iris radius, respectively, for a comparison in case it isneeded a larger iris for beam dynamics specific requirements.By inspecting the table 3, the amount of RF parameters varia-tion physically sounds and it is reasonable when increasing theiris radius. By inspecting the table 3, in order to sustain thesame e ff ective accelerating gradient, the SW structure has tobe fed with a 21% power more than the SW structure, as it isexpected to be.In normal conducting operation, we can consider an e ff ectiveaccelerating electric field of 125 MV / m as reasonable practi-cal estimation, since E p / E a and H p / E a are enough conserva-tive parameters by providing maximum values no higher than222 MV / m and 379 mT respectively. On the other hand, at 125 Table 4: RF parameters list of the normal conducting linearizer structure
Design parameters SW SW Frequency [GHz] 34.272 34.272E ff ective accelerating Electric field [MV / m] 125 125Input power [MW] 6.6 8Shunt impedance [M Ω/ m] 189 155Unloaded quality factor, Q 5778 5888Coupling coe ffi cient, K% 0.83 2.38 E p / E a H p / E a [mT / MV / m] 2.69 3.03Iris thickness [mm] 0.667 0.667Cavity radius [mm] 3.794 4.878Cavity gap [mm] 4.374 4.374Iris radius [mm] 1 1.333Structure length [cm] 8 8Build-up [ns] 13.4 13.7RF Pulse length flat top [ns] 50 50Repetition rate [Hz] 100 100Average RF power per meter [W / m] 412 500 MV / m the MPV and PH are well below the safety threshold bya factor 4 and 6 respectively. We do not foresee problems forlosses per meter since the duty cycle is expected to be 5 × − .In both structures, the power supply can be provided by a gyro-klystron plus a SLED system. Fig. 8 shows the integrated volt-age as function of the input power for both 1 mm and 1.333 mmradius. At a matched input power of 8 MW, we are able to getabout an integrated voltage of 12 MV and 10 MV, respectively.As a result, two separated structures are needed for providingan integrated voltage higher than 15 MV. Since the enhance-ment factor at 77 K degree is 2.2, only one structure is neededfor getting about an integrated voltage of about 15 MV.8a) (b) Figure 8: Integrated voltage as function of the input power for a) 1 mm and b) 1.333 mm iris radius
9. Conclusions
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