Effet and toleranes of RF phase and amplitude errors in the ILC Crab Cavity G. Burt, A. Deter, P. Goudket, Lanaster University, Cokroft Institute, Lanaster, LA 4YR, UK ASTeC, Daresbury, Warrington, WA4 4AD, UK Abstrat A large rossing angle (~ 0 mrad) for eletron and positron bunhes through the beam delivery system of International Linear Collider (ILC) is urrently a favoured option of the Global Design Effort (GDE). For rossing angles greater than about mrad, mathing rab avity systems are required on the eletron and positron beam lines to impart an angular kik to the bunhes so that they beome aligned at the interation point (I.P.). The avoidane of differential transverse defletion of eletron and positron bunhes at the I.P. requires almost perfet mathing of the RF phase of the rab avity system on the positron beam line to the rab avity system on the eletron beam line. In this paper the effets of phase and amplitude errors in the ILC rab avity have been alulated, and these have been used to estimate the toleranes on the rab system. - -
Introdution The most fleible designs of the international linear ollider (ILC) beam delivery system in terms of operating parameters typially have a rossing angle between the eletron and positron beamlines greater than about 4 mrad so that eletron and positron bunhes do not pass through eah other s final fousing quadrupole doublets. At the ILC interation point (IP) the bunh length will be of the order of 600 times the bunh width and 60,000 times the bunh height. If positron and eletron bunhes, aligned with their beamlines, meet with a rossing angle θ, the luminosity is redued by the luminosity redution fator S with respet to a head on ollision given as S = () σzθ + σ where σ is the bunh width and σ z is the bunh length []. Equation () applies for Gaussian bunhes. For the 0 mrad rossing angle nominal parameter set as defined in [], the luminosity redution fator would be 0.6. This loss an be reovered by aligning bunhes prior to ollision. One method of ahieving this alignment is to use a defleting avity in a rabbing phase to rotate bunhes rather than to deflet them. A defleting avity is a RF avity that uses the first dipole mode for its operation instead of the aelerating monopole mode. The dipole mode has zero longitudinal eletri field along its beam ais and a large transverse magneti field (vertial in this ase) that imparts varying transverse momentum (horizontal) along the bunh. If the phase of the RF is timed so that the entre of the bunh passes through the avity when the magneti field is zero then the head and tail of the bunh will eperiene equal and opposite Lorentz fores, ausing the bunh to start rotating. To maimise the luminosity bunhes should be in line at the Interation Point (IP), and hene must have rotated by half the rossing angle when they reah the IP. It is ommon to refer to a defleting avity phased to give a rotational kik rather than a defletional kik as a rab avity. Transverse kik provided by the rab avity. Approimate treatment assuming a pillbo avity The ation of a rab avity is most simply understood with referene to a pillbo avity. For a pillbo avity without beam-pipes, the TM0 dipole mode has no eletri field on ais and a onstant magneti field on ais. The transverse momentum kik for a relativisti partile passing through a rab avity is given as t p(t o ) = ebos( t) dt = eb { t o sin ( t ) sin( t o )} () where B is the Magneti flu density, is the speed of light, e is the harge of an eletron, is the angular frequeny, t o is the entry time of the partile and t is the time the partile leaves the avity. The time t is given as, d t = t o + (3) - -
π where d is the avity length. The maimum defletional kik p ma is obtained when t o = π and t =. For a relativisti bunh the energy of the kik in eletron volts is given as ev = p hene e B evma = pma = (4) V ma is the maimum transverse kik (as transverse kik depends on phase). The rab rotation effet of a defleting avity is also determined from equation () by onsidering the relative transverse momentum kik between the front and the entre of a short bunh. For a short bunh of length σ z = δt the relative momentum kik between front and entre is given as t t+δt Δp = eb dt os o to to +δt ( t) dt os( t) = eb{ os( t ) os( t )} δt The maimum kik ours when t o = 0 and t = π hene the optimum avity length is given by, π d = optimum (6) The maimum relative momentum kik (rab kik) for the front of the bunh with respet to the bunh entre and where the entre of the bunh enters the avity at t o = 0 is given as Δ pma = eσz B (7) The maimum momentum kik relative to the entre an be written in terms of the maimum energy kik using (4) so that evma σ z Δ p ma = (8) If we define the rab kik voltage with respet to the momentum kik between the entre and the end of the bunh we have that Δp Δ V rab = (9) e hene applying (8) σz ma{ Δ Vrab} = Vma (0). Generalisation to a real avities The treatment in the previous setion is for a perfet pillbo avity with no beam-pipes. If the avity shape is hanged or beam-pipes are added the dipole mode develops a transverse eletri field on ais that also ontributes to the kik. It is possible to alulate the ombined effet of both the eletri and magneti field using Panofsky Wenzel theorem [3] that states V i ( E z ) dz () where = V is a transverse kik (not neessarily the maimum kik V ma. (5) - 3 -
As E z = 0 for TE modes their transverse kik vanishes. Effetively the transverse kik from the eletri and magneti fields anel for a TE mode but add for a TM mode. In real avities there are no true TE and TM modes just hybrids, however TE-like modes deliver muh smaller transverse kiks than TM-like modes. Applying equation () to a avity eited in a dipole mode suh that the longitudinal eletri field is zero on ais one obtains, z Ez ( a, 0, z) os = V i. dz a where a is a point shifted in the diretion from the ais and the entre of the avity is at z = 0. The osine term omes from the transit time, see (). For a rab avity the transverse defleting voltage ΔV Crab seen by the front of the bunh when the avity is phased orretly is therefore given by z σz E z ( a,0,z) sin + Δ = V Crab i. dz a For a symmetri avity this beomes z σz E z ( a,0,z) os sin σz ΔVCrab = i.dz = sin V a and for small bunhes σz ΔVCrab = V () It is apparent therefore that () and hene (0) are general results. 3 Bunh rotation at the interation point (IP) For a bunh whose entre is at the middle of the rab avity when its magneti field is zero, then partiles either side of the entre move away from the ais in opposite diretions as the bunh travels on towards the final fous quadrupole doublets. As these off ais partiles pass through the final fous quadrupoles, they get a momentum kik that is in opposition to the initial rab kik. If the rab avity is very lose to the final fous then the momentum kik from the final doublet is very small and the bunh keeps rotating in the same diretion up to and beyond the IP. If the rab avity is set bak from the final fous by its nominal foal length, then the kik imparted by the final doublet eatly anels the kik from the rab avity hene the bunh ontinues to and beyond the final doublet at the angle with whih it arrived. When the rab avity is set even further bak, the diretion of rotation is reversed in the final doublet. As the divergene of the beam arriving at the rab avity is nonzero, the angle that a bunh takes at the IP depends on the position of the rab avity with respet to the final fous. The effet of the final fous an be epressed in terms of the matri element R that determines a transverse displaement ip of a partile at the IP that is on ais at the rab avity and has angular diretion on leaving the rab avity, i.e. ip = R (3) The angular defletion imparted by the rab avity for a relativisti partile is given simply by - 4 -
Δp = (4) m hene from (3), (4) and (8) we an write e Vmaσz Vmaσz ip = R = R (5) m Eo where E o is the energy of the beam in ev. Defining the rab angle θ r as half the rossing angle θ then ip Vma θ r = = R (6) σz Eo Assuming a rab angle of 0 mrad, a beam energy of 500 GeV, a avity frequeny of 3.9 GHz and R = 6.3 m rad - (horizontal) then V ma = 3.74 MV. The value of R is typially determined omputationally hene its dependene on the position of the rab avity is not immediately apparent. If however we write the general transfer matri M f in the horizontal plane of a non-dispersive fousing lens system of foal length f as + δ δ M f = + δ (7) + δ + δ3 f where δ, δ and δ 3 beome zero for a thin lens, then it an be shown by onsidering a drift L from the rab avity to front of the lens system followed by a drift L p from the end of the lens system to the IP that R is given by LpL ip = R = ( + δ) Lp + L + ( δ + Lpδ3 ) f (8) Equation (8) also gives the displaement either side of the IP for partiles that enter the rab avity on ais. This formula shows that when L p > f then the displaement ip and hene the rab rotation dereases as the rab avity is moved further bak from the final fous doublet. When fousing is at the IP from a transfer matri analysis one an show that Lp LpL ( + δ ) ( Lu L ) ( ) Lp L + ( δ + Lpδ3 ) = 0 f + + δ + f where L u is the distane from the β funtion minimum to the front of the lens system. It follows that (8) an be written L p ip = ( + δ ) ( L L u ) f (9) Care should be used when interpreting this equation as when L p = f the value of L u goes to infinity and ip stays finite. By onsideration of how the β funtion transforms equation (9) is equivalent to = β β β (0) ip ip rab o The usefulness of this epression is that R an be determined from knowledge of the β funtion. - 5 -
4 Transverse offset As well as giving bunhes a distributed momentum kik the rab avity also gives the bunh a horizontal displaement. As the entre of the bunh enters the avity it first sees a defleting fore in the horizontal plane that auses it to move off-ais. After the bunh rosses the entre of the avity it eperienes a defleting fore equal and opposite to the fore seen in the first half of the avity. This fore stops the transverse motion but does not restore the bunh position bak on-ais. The displaement of the partiles ourring in the rab avity is omputed as t p = T (t') avity dt' () m t0 For the TM0 mode, the displaement for optimum avity length (5) as the bunh leaves the rab avity is given as eb Vma avity = ( πsin( t o ) os( t o )) = ( πsin( t o ) os( t o )) () m Eo For a beam energy of 500 GeV, a avity frequeny of 3.9 GHz and V ma = 3.74 MV then this displaement ~ 0.09 μm Speifying the fousing doublet by the matri M f of epression (6) the displaement avity is Lp redued at the IP by the fator ( ) + δ. As L p is lose to f the resulting f displaement at the IP is an order of magnitude smaller than the displaement at the rab avity and hene an be negleted. 5 Timing Errors Zero displaement of the bunh entres at the IP only ours when the bunhes are perfetly synhronised with the RF field in the rab avities suh that the bunh entre is at the middle of the rab avity when the magneti field is zero. The antiipated jitter on the arrival time of bunhes at the rab avities from the linas will be about 0.4 ps. For a bunh that enters a rab avity of optimum length at time t o and leaves at time t then from (3) and (6) t = t o + π (3) (Note that perfet synhronism for a rab avity with field time dependene speified as in () ours when t o = 0.) For a avity of optimum length () gives eb eb Δ popt ( t o ) = { sin( t o + π) sin( t o )} = sin ( t o ) (4) and using (4), (3) and (4) this may be written Vma Δ ip = R sin( t o ) (5) Eo using (6) this may also be written as ( t ) sin o Δ ip = θr θr t o (6) Hene for a timing error of t o = 0.4 ps and when half the rossing angle θ r = 0.0 rads then the bunh offset is. μm. This is unaeptably large unless the eletron rab avity has - 6 -
nearly the same kik as the positron rab avity so that eletron and positron bunhes have the nearly the same offsets at the i.p. To estimate timing errors there are three harateristi ases:-. Beams and one avity are in synhronism but other avity is out of time,. Crab avities and one beam are in synhronism but other beam is out of time, 3. Crab avities are in synhronism but beams are in differing synhronism. We shall refer to ase as a avity timing error and ase as a beam timing error. Case 3 will be regarded as a sub ase of rather than. 5. Bunh displaement due to avity timing errors Displaement for a avity timing error is given below by equation (7). To obtain the result replae os( t) in () with os ( t + φ) where φ is the avity phase error and set t o = 0. The steps that gave (6) now give ( φ) sin θr φ Δ ip = θr (7) If one onsiders the atual bunh to be part of a muh longer virtual bunh there will be some part of the virtual bunh that enters the avity at time t o suh that t o φ = 0 onsequently this part travels without defletion along the beam-line. The atual bunh an therefore be onsidered as having been rotated about the un-defleted part. This onept only works for small timing errors as the sine term in (7) bends the virtual bunh, see figure. Path of bunh with no rab kik Atual virtual bunh Ideal virtual bunh Figure. Shape of very long bunh rotated by a rab avity 5. Geometrial luminosity loss due to bunh displaement If at the IP the positron bunh has a horizontal displaement of 0.5Δ and the eletron bunh has a displaement of 0.5Δ and both bunhes have Gaussian profiles then the integral that determines the geometri luminosity ontains the term ( ) ( ) + 0.5Δ ( 0.5Δ) f = ep ep (8) πσ σ σ The luminosity redution fator therefore given as Δ S = ep (9) 4σ Putting numbers in (9) for the 500 GeV entre of mass nominal parameter set one sees that a luminosity redution of % ( S = 0.98) when the horizontal beam size is 665 nm - 7 -
at the IP, ours for a displaement of 86 nm, whih for a 0mrad rossing angle orresponds to a 0.09 o jitter at.3ghz and 0.087 o at 3.9 GHz. This phase error orresponds to a timing error of 0.06 ps. For the TeV nominal parameter set the horizontal beam size at the IP is 554 nm hene % luminosity loss ours for a displaement of 39 nm, whih for a 0mrad rossing angle orresponds to a 0.065 o jitter at 3.9 GHz. This phase error orresponds to a timing error of 0.047 ps. The RMS luminosity loss from phase jitter is found to be approimately equal to the single bunh luminosity loss. 5.3 Luminosity loss from bunh timing errors If the avities are synhronised with eah other, one bunh arrives on time and the other is late (or early) then onsidering figure, the bunh that arrives on time an be regarded as a small segment of the virtual bunh that lies on the beam-line and the late (or early) bunh an be regarded as a small segment of the virtual bunh that lies off the beamline. Effetively figure and equation (6) tells us that late bunhes get more kik and if the are not too late they get just enough etra kik to be in line with where they ought to be at any instant. This means that they still ollide head on, but at a distane 0.5Δ t for the intended IP and with a horizontal offset of 0.5θ r Δt. The late (or early) bunh is only in perfet alignment with the orretly timed bunh when the latter is at the IP. Beause the atual interation ours at a small distane from the intended IP, there will be a small amount of over/under rotation, see figure. Δt error Referene position on eletron beam quadrupole t t 3 t 3 Δ error t Referene position on positron beam quadrupole t θ t d ip d p Position of positron bunh entre when eletron bunh entre is at IP Position where bunhes pass IP Figure. Graphial analysis of bunh timing errors Epliitly the horizontal offset for bunh that is late by Δ t with respet to a bunh that arrives on time is given as θr offset = [ sin( Δt) Δt] (30) The luminosity redution fator an then be found using equation (9). As before to ensure that the luminosity loss is less than % the offset must be less than 4 nm hene for a 0 mrad rossing angle and a rab avity frequeny of 3.9 GHz equation (30) indiates that Δ t should be less than 7.7 ps. This tolerane of 7.7 ps does not inlude effets of bunh over rotation, inorret rotation assoiated with inorret kik, hange in bunh size due to fousing (the hourglass effet) and beam to beam interation effets. Fousing and beam to beam interations effets do not depend on the rab avity however it is of interest to - 8 -
onsider the loss in luminosity resulting from moving the IP with onsequential loss of fous as it is far more restritive than beam timing issues assoiated with the rab avity. Considering the fousing effet first, size variation Δ σ at a distane Δ z from the ε fous an be estimated as Δ σ σ + Δz σ * where for the ILC nominal TeV β 6 parameter set the horizontal emittaneε =.0 0 and size at the IP σ = 0.554 0 m and the beta funtion at the ip β * = 0.03m. A similar equation applies to vertial fousing 4 9 where the vertial emittaneε y = 4.09 0 and size at the IP σ y = 3.5 0 m and the beta funtion at the ip * β = 3.0 0 4 m. For a bunh that is 7.7 ps late then the shift in the IP is given by Δ z =. 6mm hene Δσ = 0. 4nm and Δσ y = 0. 4nm The relative variation in the horizontal is tiny however the relative variation in the vertial size is huge. It is of interest therefore to estimate the luminosity redution assoiated with a hange in the bunhes lateral dimensions. Epliitly we onsider the horizontal variation in size however the resulting formula generalises to the vertial dimension. If at the IP the positron bunh has an inreased size of + Δσ and the eletron bunh has an inreased size of + Δσ and both bunhes have Gaussian profiles then the integral that determines the geometri luminosity ontains the term ( ) ( )( ) ( ) ( ) f = ep ep (3) π σ + Δσ σ + Δσ σ + Δσ σ + Δσ Performing the integration one an show that the luminosity redution fator with respet to a ollision at the fous is given as S = = (3) Δσ Δσ ε 0.5 0.5 + ( Δz + Δz ) * + + + β σ σ σ For the urrent ILC beam delivery system the beam waist foi for eah beam are planned to be at Δ f = 30μm upstream of the IP hene the luminosity redution fator beomes ε ε + Δf + Δf * * β σ β σ S = = (33) ε ε + {( Δf Δz) + ( Δf + Δz) } + ( Δf + Δz ) * * β σ β σ The luminosity redution fator arising as a onsequene of fousing effets when one bunh is late as determined by (33) is plotted in figure 3. Parameters orrespond to TeV entre of mass. The variation of the plot with Δf is very small. It is a apparent from figure 3 that a timing error that gives a signifiant luminosity redution from vertial fousing effets is very small ompared a timing error that gives luminosity loss assoiated with redued kik at the rab avity. For the ILC to ahieve its luminosity target beam timing will need to be better than 0.5 ps with respet to vertial fousing and additional effets from the rab avity are very small when this beam timing target is ahieved. - 9 -
.00 luminsoity redution fator 0.99 fous effet (nominal) y fous effet (nominal) 0.98 fous effet (LowQ) y fous effet (LowQ) 0.97 0.96 0.0.0.0 3.0 4.0 5.0 6.0 7.0 8.0 time delay of one bunh (ps) Figure 3 Luminosity redution fator as a onsequene of a late bunh shifting the interation point onsidering fousing effets only. Beam timing errors also lead to inorret rotation. Most of the ontribution to inorret rotation omes from inorret kik rather than additional rotation between the intended IP and the atual IP. If the rab avity is right net to the final fousing doublet then the rab angle is set up over a distane of the order of 5 metres. In this ase the bunh passes through the first fousing quadrupole with little hange to its initial rate of rotation. On passing through the defousing quadrupole the rate of rotation is signifiantly redued hene the rate of rotation near the IP should be estimated on the basis that the angle of rotation was ahieved over a distane of 6.3 metres for the TeV nominal ase. For the beam timing error of 7.7 ps giving % luminosity loss from displaement errors, then the IP will be shift by.6 mm hene additional rotation will be less than half the rossing angle times this shift divided by 5 metres whih equals 0.0007 mrads. From () it is apparent that this etra rotation will give no signifiant luminosity loss and indeed a timing error of 7.7 ps was intolerable from fousing effets anyway. Looking now at the under rotation due to an inorret kik, the rotation angle between the two bunh at ollision is determined as ( Δt + σ ) ( Δt σ ) offset z offset z sin( θ ) = σz (34) whih using (30) gives σz ( ) =θ sin os( Δt) θ ( 0.5 Δt ) sin θ r r (35) σz The luminosity redution fator an then be alulated using equation () where replaed by θ r θ i.e. θ is S = (36) σz ( θr θ) z r t σ θ Δ + + σ 4σ For a beam timing error of 7.7 ps the loss in luminosity predited by (36) is frations of a perent hene this orretion is also not an issue. - 0 -
6 Amplitude error From (6) one sees that the rab rotation angle of the bunh at the IP is linearly proportional to the avity voltage, hene any variation in the voltage Δ V produes a proportional variation in the rotation of the bunh. The luminosity redution fator S due to the voltage variation is then determined by () where θ is replaed by Δθ whih is determined from (6) as ΔV ΔV Δ θ = R = θr (37) Eo Vma Hene the aeptable amplitude variation is given as ΔV σ = Vma θr σz S Taking the luminosity redution fator as 0.98, 6 θ r = 0.0 rads then ΔV = V 4.4% (38) σ = 0.665 0 m, σ = 0.3 0 m and ma S= 0.98 whih at a glane is a omfortably large. However this tolerane may atually be diffiult to ahieve if the bunhes arriving at the rab avity arrive off ais. It has been estimated that bunhes ould arrive at the rab avity with a horizontal offset as a muh as.5mm. Arriving off ais will result is a slightly differing transverse kik but more worryingly the bunh will deposit or etrat energy from the avity. This will alter the amplitude and indeed the phase of the field in the avity for the net bunh. Detailed alulations to determine whether the toleranes established an be maintained with off-ais bunhes are required. If the toleranes annot be met, one solution might be to onsider a larger avity operating at a lower frequeny. 7 Beam-beam disruption effets Using the ILC TeV nominal parameter set, reent results by Churh [4] using the e + /e - ollision simulation ode Monte Carlo that inorporates beam to beam interation effets and beam energy distributions, show that beam timing errors an be up to 0.67 ps before % luminosity is lost. This figure is in agreement with our preditions in figure 3 derived from defousing effets alone, geometrial effets being negligible. Simulations also performed in [4] predit a % luminosity loss for an amplitude jitter of 4.4% in agreement with our value derived from (35). They also predit a % luminosity loss for unorrelated avity phase timing errors of 0.043 ps, for a bunh with a horizontal width of 655nm. The unorrelated avity phase timing error is related to the phase errors alulated in this paper by fator of /, as we alulated the phase differene between two avities and Churh alulates the error of the avities from a perfet referene. We alulate an unorrelated avity phase timing error of 0.044 ps using the method detailed above. The differene ould be due to beam-beam disruption effets. Churh [4] also shows that the energy tolerane on the beam needs to better than 0.3% for luminosity losses to be less than %. This final result is related to movement of the beam waist fous with energy. 8 Toleranes to Transverse Wakefields The presene of wakefields in the rab avity an have many effets on the beam, however we will onern ourselves only with the defleting kik given to the entre of the bunh. If there is a transverse kik given to the bunh in a rab avity it will produe an offset at the IP that will redue the luminosity as given in (9). For the ILC TeV nominal z 3 - -
parameter set for a luminosity loss of less than % this offset must be less than 39 nm in the horizontal plane and 0.875 nm in the vertial plane. These offset toleranes an be used in (5) to alulate the maimum defleting voltage tolerane at the rab avity. Assuming a 0.5 TeV beam with R = 6.3 m/rad horizontal and R =.4 m/rad vertial, we alulate the maimum voltage in the horizontal plane to be 4.5 kv, and 0.8 kv in the vertial plane. 9 Conlusion Phase and amplitude toleranes have been alulated for the ILC rab avity system. It is shown that the system has high tolerane with respet to luminosity loss for bunh arrival time and amplitude errors, however there are tight limits on the avity-to-avity phase errors. To stay within the luminosity budget the positron rab avity must have on average an idential phase to the eletron rab avity to within 0.066 degrees. This orresponds to a timing auray of 0.047 ps or a signal path length auray of 0 μm. For the 0 mrad rossing angle parameter set the rab avities are 34 metres apart. Given that the beams will have differing offsets and hene will be delivering or etrating differing amounts of power from the avities, the favoured option is to drive the two sets of avities with differing power soures. Sine most of the luminosity loss assoiated with avity timing errors omes from assoiated beam defletion one might onsider measuring horizontal beam defletion some distane after the rab avity and ontrolling this defletion to a set point. This option is not possible unless the rab avities are moved bak from the final doublet. nominal 500GeV Low Q 500GeV Large Y 500GeV Low P 500GeV High Lum 500GeV Jitter (degees) 0.0766 0.0579 0.0579 0.059 0.059 Amplitude % 0.0443 0.0670 0.00 0.0459 0.06 nominal TeV Low Q TeV Large Y TeV Low P TeV High Lum TeV Jitter (degees) 0.0648 0.0459 0.049 0.049 0.0374 Amplitude % 0.0375 0.0398 0.04 0.04 0.035 Table Phase jitter and amplitude toleranes for the suggested ILC parameters Currently it is planned that the avities be synhronised with a LLRF signals. The phase tolerane is on the limit of what is likely to be ahievable. One might antiipate that initially the phase error will be slightly outside the tolerane dropping the luminosity by % or more. Use of beam position monitors or indeed the luminosity signal within the ontrol algorithm should allow us to reover this loss if the loss annot be reovered in other ways. Aknowledgement The authors would like to aknowledge Andrei Seryei and Chris Adolphsen at SLAC, and Mike Churh and Leo Bellantoni at FNAL, for their help and useful disussions. This work is supported by the Commission of the European Communities under the 6 th Framework Programme Struturing the European Researh Area, ontrat number RIDS-0899, and by PPARC. Referenes [] B. Muratori, Luminosity in the presenes of offsets and a rossing angle, CERN-AB- Note-003-06 (ABP) [] T. Raubenhiemer, Suggest ILC beam parameter range, http://wwwprojet.sla.stanford.edu/il/aeldev/beampar/suggested%0ilc%0beam%0para meter%0spae.pdf - -
[3] W. K. H. Panofsky and W. A. Wenzel, Some Considerations Conerning the Transverse Defletion of Charged Partile Radio-Frequeny Fields, Rev. Si. Instr. 7, 967 (956). [4] M. Churh, Timing and amplitude toleranes for Crab Cavity, FNAL ILC dodb-95-3 -