Power dissipation by Higher Order Modes

Transcription

1 slhc-project-note Power dissipatio by Higher Order Modes M. Schuh, W. Weigarte Beams Departmet Keywords: SPL, supercoductig cavities, HOM, power dissipatio, cryo power Summary The additioal power loss due to HOMs i cavities ad HOM couplers is estimated as fuctio of the dampig with aalytic ad beam dyamics simulatio based models. Based o these results, HOM dampig requiremets for the Supercoductig Proto Liac (SPL) ca be defied to limit the additioal heat load to the watt level. Itroductio slhc-project-note-00 /0/0 A cavity ca be see as a power trasformer. The power from the power source e.g. a klystro is trasferred to the beam by the acceleratig mode, which is usually the TM 0,π mode i supercoductig elliptical cavities. This works also i the other directio. A buch passig through a cavity excites modes, ot oly the acceleratig mode but also other so called Higher Order Modes (HOMs), i the cavity. This has cosequeces for the beam stability ad the power dissipatio i the cavity. The power trasferred back to the cavity has to be dissipated either i the cavity walls or has to be extracted via HOM couplers. Some of the power is also trasferred back to the beam, which ca excite beam istabilities. I case of supercoductig cavities the heat load is a importat desig parameter for the cryostat. Depedet o the desig of the HOM couplers the power is trasferred to room temperature or dissipated i the helium bath, which has a impact o the eeded cryo capacity. The additioal heat load should ot exceed a few watts per cavity to keep the overall cryo power eeds withi reasoable limits. First some basic formulae are itroduced i the ext sectio. The the HOM power dissipatio per cavity is calculated, usig a aalytical model, ad bechmarked agaist data extracted from beam dyamics simulatios. Afterwards various supercoductig proto liac (SPL) operatioal coditios ad their ifluece o the HOM power are aalysed. Moopole excitatio ad power dissipatio The formulae derived i this sectio are mostly based o []. There the special case of a relativistic beam withβ = is discussed. All formulae are adopted toβ < accordig to [, ] ad some further equatios are take form [ ]. This is a iteral CERN publicatio ad does ot ecessarily reflect the views of the CERN maagemet.

2 For simplificatio, a complex otatio for the voltage is used. By covetio, the observable physical quatities are real ad the sig of the imagiary phase is positive (e +iωt ). Compared to the cosidered HOM wavelegth, the buch legth is assumed to be very short, meaig, that buches ca be treated as poit-like charges. The coordiate system is chose such, that the beam travels always alog the z-axis (beam axis) i positive directio. The electrical z-axis i the cavity is assumed to be idetical with the geometrical z-axis. A poit charge, travellig alog the beam axis, excites all moopole modes iside a cavity. Sice all modes are orthogoal to each other they ca be discussed separately. The modespecific voltage iduced by a poit-chargeq is V q, = q ω (R/Q) (β), () where ω is the agular frequecy of mode ad β is the relativistic factor v of the particle c velocity. (R/Q) (β) of a particulate mode with a agular frequecy ω takes ito accout the velocity-depedet trasit time factor ad is defied as (R/Q) (β) = E z, (r = 0,z)e iω z βc dz ω U, () wheree z is the field compoet alog the beam axis adu the stored eergy i the mode. The itegratio borders are set to ifiity to cosider also the fields leakig out of the cavity ito the beam pipe. I EM simulatio tools thee-field is usually forced to zero at a certai poit i the beam pipe by the boudary coditios. The mode loss factor ca be defied as ad the eergy deposit i the cavity by a charge q i this mode is k = ω (R/Q) (β) () U q, = k q. () The voltage iduced by the buch starts oscillatig at t = 0 whe the buch passes the cavity mid plae, cosiderig the case, that there is o voltage preset i the cavity before. Due to losses the amplitude decays over time accordig to where the decay time costatt d, is defied as V (t) = V q, e t/t d, e iωt, () T d, = Q L, ω, () where Q L, is the quality factor of the complete system, icludig all couplers, for this mode. It is the iverse sum of all subsystems Q L, =, + Q 0,, () The deviatio to a Gaussia distributed buch with a rms buch legth of cm is exp( ω σ z c ). Throughout this paper the liac defiitio is used. To covert ito the circuit defiitio the value has to be divided by a factor.

3 whereq 0, is the value for the uloaded cavity ad, is the total quality factor of the exteral loads, which are the fudametal power coupler, HOM coupler ad dampers like bellows. The aturalq 0 of a supercoductig cavity is i the order of. SoQ L is domiated by ad i this discussio the simplificatioq L is valid., deped o the couplig efficiecy to the particular mode ad ca vary over several orders of magitude from mode to mode. Modes with a very high are called trapped modes. I this paper the is used as a simulatio parameter ad is ot calculated from EM simulatios for the particular modes. The trasit time of the buch through cavity is several orders of magitude smaller tha the decay timet d,. Hece the voltage does ot decay durig the buch passage ad the cavitybuch iteractio ca be assumed to be istataeous. The voltage iduced i the same mode by the ext buch arrivig after the timet b is added as vector sum to the voltage preset i the cavity. { } V, = V q, +e iωt b e T b T d, () is the voltage preset i the cavity directly after the secod buch, if o voltage was preset before the first buch passed the cavity. Uder the assumptio, that the time betwee buches is costat ad there is o charge scatter the equilibrium HOM voltage i cotiuous wave (CW) operatio ca be derived by evaluatig the sum of the iduced voltage directly V cw, = V q, e kt b/t d, +ikω T b = k=0 V q, e T b/t d, +iω T b. (9) From this formula oe ca easily see, that a HOM with f = k/t b,k N will be resoatly excited by the fudametal machie lies. I this case, whe additioally T b << T d,, (9) ca be simplified usig a Tailor expasio to V res, = V q, T d, T b, = ω (R/Q) (β)i b T d, = (R/Q) (β)i b Q L,, () where I b = q /T b is the average beam curret. I the ati-resoace case (ω T b = (k + )π,k N) (9) evaluates, also assumigt b << T d,, to V ati, = V q, = π (R/Q) (β)i b. () I pulsed operatio, the voltage after oe pulse with N buches ad o HOM voltage preset before the first buch arrives, is the give by V p, = V q, N Aftermpulses the voltage k= e kt b/t d, +ikω T b = V q, e NT b/t d, +inω T b e T b/t d, +iω T b. () Vp, m = V e NT b/t d, +inω T b e mtp/td,+imωtp q, e T b/t d, +iω T b e Tp/T d,+iω T p () is preset i the cavity, wheret p is the pulse period. If there are sub-structures i the pulse trai the formula ca be exteded to V p, = V q, M j=0 e k jt j /T d, +ik j ω T j e T j/t d, +iω T j, ()

4 wherek j is the umber of repetitios,t j the period legth of the sub-structure adm the umber of structure elemets. M = 0 correspods to CW operatio, M = to pulsed operatio, ad M > to pulsed operatio with further sub-structures. Due to the choppig of the regular buch sequece, additioal weaker machie lies appear at f = k/t j, k N, () which ca also excite resoatly a HOM. These resoaces, created by the pulse sub-structure, are called choppig machie lies i the further discussio. The smaller T j, the stroger is the resoace, which ca be explaied by lookig at the Fourier compoets of the excitatio. So the most dagerous resoaces are the fudametal machie lies. This is illustrated i Figure, where the HOM voltage spectrum with differet pulse sub-structures is show. [h!] [V] V HOM HOM voltage spectrum Choppig: o / 0/0 00/ f HOM [MHz] Figure : HOM voltage preset i a cavity after 0 pulses for differet pulse substructures (choppig patters) aroud the th fudametal machie lie, where f is varied cotiuously. Further calculatio settigs: R/Q =, I b = 0mA, =. The actual power dissipated by modei the cavity walls ca be directly calculated from the voltage preset i the cavity at timet P c, (t) = V (t) R a, (β) = V (t) (R/Q) (β)q 0. () I CW operatio the power dissipatio ca be assumed as costat after the system has reached a equilibrium state ad i pulsed operatio the average power dissipatio ca be calculated by itegratig over oe pulse periodt p P c, = t+tp P c, dt. () T p t

5 Oly a small fractio of the power is dissipated i cavity ad most of the power goes ito the load, e.g. the HOM coupler. The power dissipated i a load with a couplig stregth β = Q 0 is the P ex, (t) = βp V (t) c, (t) =. () (R/Q) (β), A worst case power dissipatio i the load ca be estimated usig () ad for the cavity P ex, = (R/Q) (β)i b Q L,, (R/Q) (β)i b, (9) P c, = (R/Q) (β)ibq L, Q (R/Q) (β)ib ex,. (0) Q 0, Q 0, The same ca be doe for a best case estimatio usig (). The total power dissipated i HOMs is the sum over all modes sice the modes are orthogoal P tot = P c,. () The voltage as well as the power are proportioal to R/Q. So the voltage ad power ca be ormalised with R/Q, which is used i geeral for a R/Q idepedet aalysis. For large values the decay time gets loger tha the gap betwee two pulses i case of pulsed operatio. The the voltage icreases from pulse to pulse util a steady state is reached. I order to calculate average HOM power dissipatio correctly, the limit of ifiite pulses has to be used. All the formulae above are oly valid uder the assumptio of a perfect beam without ay variatios i buch charge ad arrival time. Applyig beam oise, the aalytic formulae ca o loger be used reliably to calculate the HOM voltage. The iduced voltage has to be summed up explicitly, cosiderig phase, eergy ad charge variatios, eve i case of a Gaussia distributio because of the o liearity of the voltage decay betwee the buches ad the phase deviatio, which itroduces also a o liearity. Boudary coditios of the aalytic model Usig the aalytic expressios to calculate the HOM power dissipated i the system is much more coveiet tha performig beam dyamics simulatios ad extract the power dissipatio data. I this sectio, it is evaluated uder which coditios the aalytic formulae ca still be used without big systematic errors. All simulated ad calculated values for the dissipated power are the average values over oe pulse period. I case of beam dyamic simulatios the average value of all simulatios results is used, where a sigle result is the average value over oe pulse period. At the fudametal machie lies a perfect beam without phase jitter is the best driver. The effect of charge jitter ca be eglected because o average the same voltage is iduced by each buch ad the phase of the HOM voltage is always the same moduloπ at the buch arrival time. The differece i the voltage decay betwee the buches is zero i the first order at the ivestigated dampig regime. Off resoace, especially at ati resoaces (f = k+ f b,k Æ) the voltage iduced by a perfect beam cacels itself out via destructive iterferece. This coditio is ot fulfilled ay more, whe phase ad charge jitter are preset.

6 . Test sceario The HOM power dissipatio is calculated usig the aalytical formula derived i the last sectio. Additioally a semi aalytic approach is used, where a Gaussia distributed charge scatter is applied. These results are compared with data obtaied from ed to ed beam dyamics simulatios of the complete SPL, where HOM voltage developmet durig a pulse of oe cavity i the liac is recorded. Afterwards the average HOM power dissipatio i this moitored cavity is calculated. For this task the beam dyamics code Simulate higher order Mode Dyamics (SMD), itroduced i [, ], is used to geerate the voltage versus time developmet over oe pulse i the moitorig cavity. SMD performs ed to ed buch trackig simulatios of the complete liac, specialised o the beam-hom iteractio ad time of flight variatios from cavity to cavity due to eergy deviatios ad particle velocities of β <. It takes also ito accout beam oise i terms of charge scatter, ijectio oise ad RF errors. Oe HOM is preset per cavity i all simulatios ad the geeral simulatio parameters are listed i Table. I the moitorig cavity differet HOM settigs are applied to study the HOM power dissipatio for a HOM close to the th fudametal machie lie (. MHz) with a R/Q value of 0Ω. Table : Simulatio iput parameters Parameter Uit Value RMS-Error E Iput [MeV] φ syc [deg] - 0. I b [ma] 0 % Cavities (β g =0./.0) /9 Desig gradiet E 0 T(β g ) [MV/m] 9./ R/Q(β g ) [Ω] 90/0 φ rf [deg] - 0. V rf [MV] E 0 T(β)L 0.% Buch frequecy f b [MHz]. Pulse legth T p [ms].00 Pulse repetitio rate [Hz] 0 HOM frequecy f [MHz] /.0 R/Q max [Ω] /0 Q 0 Average pulse curret; peak curret of ma i case of choppig to keep the charge per pulse costat liac defiitio. Aalysis Simulatios are executed, where the dampig ( ) ad the HOM frequecy is varied. A system check is doe usig o beam oise ad moitorig the first cavity, which should give the same result as the aalytic calculatio. No HOM voltage is preset i the cavity at simulatio start. Oe pulse is simulated ad the resultig P c, (crosses) perfectly agree with the aalytic values (solid lies) as show i Figure.

7 [W] Average HOM power dissipatio i coupler f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: (a) Coupler f f res [Hz] E+00 E+00 E+00 E+00 E+00 [W] 9 Average HOM power dissipatio i cavity f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: (b) Cavity (Q 0 = ) E+00 E+00 E+00 E+00 E+00 Figure : HOM power dissipatio i the HOM coupler ad cavity wall for differet distaces of f to the th fudametal machie lie ad differet dampig. Simulated (crosses) ad calculated (solid lies) values perfectly agree i the absece of beam oise. I the ext step beam oise is added ad cavity umber 0 i the β = sectio is chose as moitorig cavity. 0 simulatios with differet oise patters are executed for each dampig ad HOM frequecy value. The power dissipatio i the cavity (Q 0 = ) ad the HOM coupler is show i Figure, where the simulated values (cross) are directly compared with the aalytic values (solid lies). Additioally a semi aalytic approach, where oly a Gaussia charge jitter is used is also show (circles). [W] Average HOM power dissipatio i coupler f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (a) Coupler E+00 E+00 E+00 E+00 E+00 [W] 9 Average HOM power dissipatio i cavity f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (b) Cavity (Q 0 = ) E+00 E+00 E+00 E+00 E+00 Figure : HOM power dissipatio i the HOM coupler ad cavity walls as fuctio of the dampig ad differet distaces of f to the th machie lie, calculated with differet methods: simulatio (crosses), semi aalytic (circles) ad aalytic (lie). Oly at low dampig ( > ) ad away from the machie lie ( f f res MHz) a deviatio betwee the simulated ad aalytic values is observed. The simulatio values are averaged over 0 differet pulses. At resoace the dissipated power icreases with ad seems to saturate at =, which is a effect of the pulse structure. Icreasig further would lead to a pulse to pulse couplig ad the HOM voltage does ot decay completely betwee two pulses ay more. I that regime the used simulatio cofiguratio would lead to a systematic error, because the HOM voltage icreases from pulse to pulse ad the assumptio, that at the begiig of the pulse

8 o HOM voltage is preset is ot valid ay more. The power dissipated i the HOM coupler exceeds kw beyod = ad is more tha W i the cavity wall. Off resoace there is a peak i the power dissipatio of the HOM coupler aroud =. The values i the cavities are several orders of magitude lower. They are almost costatly below = i case the HOM frequecy deviates more tha 0 khz from the machie lie. Simulatio ad aalytic values agree very well o resoace ad close to the resoace. At low dampig ad away from the machie lie the aalytic values decrease further, while the simulated values seem to saturate at a level of W. The semi aalytic value follows directly the aalytic value with mior deviatio. Oly oe charge patter is used for the calculatio. Repeatig the calculatio with 0 differet charge patters at = ad MHz distace to the th fudametal machie lie leads to a average value of 9±µW, which is slightly higher tha the aalytic value (.µw) but still almost oe order of magitude lower tha the simulatio value ( ± µw). The distributios for the semi aalytic ad simulated values are show i Figure. Couts Semi aalytic Etries 0 Mea 9.0 RMS. Couts Simulatio Etries 0 Mea. RMS [µw] (a) Semi aalytic model [µw] (b) Beam dyamics simulatio Figure : Histogram: Average HOM power dissipatio i the HOM coupler with the followig settigs: =, f f res = MHz, R/Q = 0, I b = 0±0.mA. The semi aalytic values are closer to the aalytic values tha to the simulated oes. From this result oe ca coclude, that ot oly the charge jitter is resposible for the saturatio of the simulatio value. Also the phase jitter has to be icluded. Addig also a Gaussia phase jitter to the semi aalytic model leads to slightly higher values but they are still sigificatly below the simulatio values. Repeatig the beam dyamics simulatio oly with a phase jitter ad lookig at the first cavity the values of the semi aalytic model ca be reproduced. This leads to the coclusio, that the complete buch to buch phase modulatio icludig all effects like eergy error, arrival time errors ad impact of HOMs has to be cosidered for the off-resoace coditio with a frequecy distace of more tha MHz to a machie lie. Further simulatios i the middle of two fudametal machie lies (f =. f b ) are carried out to check the mismatch betwee simulatio ad aalytic approach. The results are show i Figure. Oly at = the aalytic ad the simulated values are the same. The values decrease with icreasig, but the simulatio values oly getly declie while the aalytic values drop by several orders of magitude. However, these discrepacies are oly of academic iterest sice they occur oly for very low power while for high power, where it is importat, good agreemet exits.

9 [W] 9 Average HOM power dissipatio i coupler f =.90 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 Figure : Normalised average HOM power dissipatio i the HOM coupler i the middle betwee the th ad th fudametal machie lie as fuctio of for the omial beam (0mA) without choppig calculated aalytically (solid lie) ad via simulatio (cross). Not oly fudametal machie lies are evaluated, but also resoat lies itroduced by choppig are ivestigated. There the same cofiguratio as before is used, but the resoace frequecy is adapted. The used choppig patters ad the distace betwee two resoaces are listed i Table. Table : Choppig patters used i HOM power simulatio Patter (m/n) f c [MHz] /.0 0/0.0 00/ Sice the choppig resoace frequecies are closer together oly frequecy detuig steps are doe util the same order of magitude, as the distace betwee two resoaces, is reached. This meas for the 00/00 choppig patter the simulatios are carried out util the detuig is 0 khz. The power dissipatio i the cavity ad the HOM coupler for the differet choppig patters is show i Figure -. The same geeral behaviour is observed as for the fudametal machie lies but the power dissipatio is much less i case of resoat lies itroduced by choppig. Here also a deviatio of aalytic ad simulated values i case of high ad off resoace is observed. 9

10 [W] Average HOM power dissipatio i coupler / choppig f res =.09 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (a) Coupler E+00 E+00 E+00 E+00 E+00 [W] 9 Average HOM power dissipatio i cavity / choppig f res =.09 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (b) Cavity (Q 0 = ) E+00 E+00 E+00 E+00 E+00 Figure : Average HOM power dissipatio i the HOM coupler ad cavity walls as fuctio of the dampig ad differet distaces to a / choppig machie lie is calculated with differet methods: simulatio (crosses) aalytic (lie). Oly at low dampig ( > ) ad away from the machie lie a deviatio betwee the simulated ad aalytic values is observed. [W] Average HOM power dissipatio i coupler 0/0 choppig f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (a) Coupler E+00 E+00 E+00 [W] 9 Average HOM power dissipatio i cavity 0/0 choppig f res =.0 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (b) Cavity (Q 0 = ) E+00 E+00 E+00 Figure : Average HOM power dissipatio i the HOM coupler ad cavity walls as fuctio of the dampig ad differet distaces to a 0/0 choppig machie lie is calculated with differet methods: simulatio (crosses) aalytic (lie). Oly at low dampig ( > ) ad away from the machie lie a deviatio betwee the simulated ad aalytic values is observed.. Results Both power calculatio methods itroduced i this ote agree very well, as log as the distaces to a resoace lie is ot larger tha MHz. Off-resoace ad at high the values obtaied with the differet methods deviate. I that regime the power dissipatio is less tha mw ad so far below ay critical value. Form this compariso oe ca coclude, that the aalytic model ca be used to estimate the maximum power dissipatio due to HOM without cosiderig beam oise. Off resoace, a threshold value of mw ca be used.

11 [W] Average HOM power dissipatio i coupler 00/00 choppig f res =.00 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (a) Coupler E+00 E+00 [W] 9 Average HOM power dissipatio i cavity 00/00 choppig f res =.00 GHz I b = 0 ma R/Q = 0 Ω Rus: 0 (b) Cavity (Q 0 = ) E+00 E+00 Figure : Average HOM power dissipatio i the HOM coupler ad cavity walls as fuctio of the dampig ad differet distaces to a 00/00 choppig machie lie is calculated with differet methods: simulatio (crosses) aalytic (lie). Oly at low dampig ( > ) ad away from the machie lie a deviatio betwee the simulated ad aalytic values is observed. Geeral aalysis The aalysis is divided i two subsectios. First the HOM power dissipatio i the cavity wall is calculated ad the the power extracted by the HOM coupler is aalysed.. Power dissipatio i the cavity walls A fractio of the HOM power is dissipated i the cavity walls. From the results obtaied i the last sectio, oe ca see directly, that oly i case of the fudametal machie lie resoace a sigificat amout of power is dissipated i the cavity. I that regime the aalytic model ca be used for a further aalysis. AQ 0 = is assumed for all HOMs. Figure 9 shows the ormalised average equilibrium HOM power dissipatio i the cavity at the th fudametal machie lie as fuctio of for the omial beam (0 ma) without choppig. There is a liear icrease i the log-log scale above =. The dip aroud = is due to pulsed operatio. If the power dissipatio should be less tha W, a dampig i the order of = is eeded for HOMs withr/q values of 0Ω. Icludig additioally a pulse sub-structure by choppig, the plot i Figure 9 would oly chage very little below = ad above = ad does ot chage the geeral result. Differet fudametal machie lies have a mior effect o the dip area aroud =, ad do ot ifluece the defied limit.. Power dissipatio i the HOM coupler Oly resoaces are critical for the HOM power dissipatio. Far from resoace the dissipated HOM power i the HOM coupler is ot a cocer as already illustrated i Figure. Hece, the area aroud the fudametal machie lies is ivestigated i more detail. Additioally the effect of the the choppig machie lies betwee the th ad th fudametal machie lie is aalysed. The power dissipatio for differet values is show i Figure -.

12 /(R/Q) [W/Ω] Average HOM power dissipatio i cavity 9 Figure 9: Normalised average HOM power dissipatio i the cavity at the th fudametal machie lie as fuctio of for the omial beam (0mA) without choppig ad a Q 0 = of the cavity. /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 (a) Aroud the th fudametal machie lie. (b) Choppig resoaces betwee the th ad th fudametal machie lie. Figure : Normalised average HOM power dissipatio i the HOM coupler at = for differet choppig patters ad omial beam curret aroud the th fudametal machie lie (a) ad at resoaces betwee th ad th fudametal machie lie (b). Without choppig oly a very arrow bad (less tha MHz) aroud the fudametal machie lie has to be take ito accout at =. The peak value is more tha 0 W/Ω but decreases fast dow to less tha 0 mw/ω. Decreasig leads to a broadeig of the resoace but also to a decrease of the peak value dow to 0. W/Ω i case of =. At the distace of MHz to the machie lie the level of mw/ω is reached. Itroducig choppig causes a icrease of the resoaces. The peak values for the /

13 /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 (a) Aroud the th fudametal machie lie. (b) Choppig resoaces betwee the th ad th fudametal machie lie. Figure : Normalised HOM power dissipatio i the HOM coupler at = for differet choppig patters ad omial beam curret aroud the th fudametal machie lie (a) ad at resoaces betwee th ad th fudametal machie lie (b). /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 /(R/Q) [W/Ω] Resoace Spectrum f HOM [MHz] Choppig: o / 0/0 00/00 (a) Aroud the th fudametal machie lie. (b) Choppig resoaces betwee the th ad th fudametal machie lie. Figure : Normalised HOM power dissipatio i the HOM coupler at = for differet choppig patters ad omial beam curret aroud the th fudametal machie lie (a) ad at resoaces betwee th ad th fudametal machie lie (b). patter are all above W/Ω at = ad some reach almost the same value as the fudametal machie lies. The other patters have more resoaces but sigificat lower peak values. I case of the00/00 patter the peak values drop fast below 0. W/Ω with icreasig distace to the fudametal machie lie. The choppig resoaces are very sharp. A detuig of a few khz is sufficiet to drop the power by several orders of magitude. Decreasig the to leads to a sigificat decrease i the dissipated HOM power. For the 0/0 ad 00/00 patter dissipated power is below mw/ω, oly the / patter shows higher peak values. The drawback of this is the broadeig of the resoace.

14 Discussio Detuig the HOM frequecy from the resoace showed a peak i the power dissipatio aroud =, which is the recommeded dampig value from the beam dyamics studies [, ] doe for the SPL. Icreasig the leads to HOM power drop i the off resoat case, but a icrease o resoace. Detuig of HOMs away from machie lies is possible i pricipal [9], but ot suitable for the large umber of resoaces to deal with i case of the SPL. At moderate dampig = the dissipated HOM power at a choppig resoace lie ca easily exceed W/Ω. This is a eormous load for the cryogeic system, if the power is dissipated at k. A of is expected whe usig slightly lossy material betwee cavities as the oly HOM dampig mechaism. Istead, a dedicated HOM coupler ca reduce the further ad decreases the dissipated HOM power o resoace. O the other had it icreases the width of the resoace ad the power dissipatio close to a resoace. Nevertheless, this situatio is preferable, because it eables the operatio with HOMs at choppig machie lies ad does ot limit the operatio to low frequecy choppig patters. I the situatio without ay choppig, a higher is preferable as already show i [] to reduce the overall power dissipatio ad have o complicated HOM coupler desig, which ca also cause problems. There HOMs at fudametal machie lies have to be expelled from machie lies i the cavity desig stage. The pulse sub-structure used at SNS [] correspods to a 0/0 choppig patter, which does ot create ay potetially dagerous choppig machie lies. Oly the fudametal machie lies ca drive istabilities sigificatly. Sice all moopole modes i the SNS cavities [] are far away from the resoace lies, o HOM caused istabilities are observed durig operatio []. That is why the dampig requiremets are differet compared to the discussed sceario here. Coclusio It is show, that aalytic calculatios without beam oise ca be used to estimate the HOM power dissipatio close to resoaces, which are the areas of iterest. A strog dampig below = is recommeded to limit the HOM power dissipatio to the watt level ad to eable operatio with arbitrary choppig patters. Ackowledgemets We would like to thak J. Tückmatel for fruitful discussios ad carefully readig the formulae i this ote ad also F. Gerigk for his commets. Refereces [] H. Padamsee, J. Kobloch ad T. Hays, RF Supercoductivity for Accelerators (WILEY- VCH, 00), d ed. [] S. S. Kureoy, Depedece of buch eergy loss i cavities o beam velocity, Phys. Rev. ST Accel. Beams (999)() 000 [] D. Jeo, Measuremet of beam loadig at the SNS supercoductig liac for a beam with β <, Nuclear Istrumets ad Methods i Physics Research A 9 (00)(-)

15 [] S.-H. Kim et al., Higher-order-mode (HOM) power i elliptical supercoductig cavities for itese pulsed proto accelerators, Nuclear Istrumets ad Methods i Physics Research A 9 (00) [] T. P. Wagler, Priciples of RF Liear Acceerators (Wiley-VCH, 00), d ed. [] J. Tückmatel, Beam simulatios with iitial buch oise i supercoductig rf proto liacs, Phys. Rev. ST Accel. Beams (0)() 00 [] M. Schuh et al., Ifluece of Higher Order Modes o the Beam Stability i a High Power Supercoductig Proto Liac, submitted to PRST-AB, 0 [] M. Schuh et al., Higher Order Mode Aalysis of the SPL Cavities, Proceedigs of IPAC, Kyoto, Japa (0) [9] N. Solyak et al., Logitudial ad Trasverse Effects of HOMs i the Project X Liac, Proceedigs of IPAC, Kyoto, Japa, TUPEA00 (0) [] J.-L. Biarrotte, A Statistical Aalysis of the Dager Iduced by HOM Excitatio i a Supercoductig Liacs, Proceedigs of the 00 Particle Accelerator Coferece, Chicago, MPPH (00) [] S.-H. Kim, M. Doleas ad R. Sudeli, HOM fidigs ad HOM iduced power i the supercoductig liac of the itese pulsed proto accelerator, Techical ote, Oak Ridge Natioal Laboratory, 00 [] A. Lombardi, S.-H. Kim ad W. Weigarte, Summary of the workshop o HOM i SPL, slhc-project-note 000, CERN, Geeva, 009