THE SURFACE ROUGHNESS WAKEFIELD EFFECT

Transcription

1 THE SURFACE ROUGHNESS WAKEFIELD EFFECT A. Novokhtski, M. Timm nd T. Weilnd, TU-Drmstdt, TEMF, Schlossgrtenstr. 8, 6489 Drmstdt, Germny Abstrct In the Liner Colliders FEL projects ultr short bunches re foreseen to be used. In ddition to usul wkefields, coming from irregulrities in the chmber, these bunches excite fields in trnsporting lines nd undultors due to the surfce roughness. This effect cn be lrge for the extruded tubes, usully used in ccelertors. Bsed on computer results it is shown, tht the roughness wkefield effect cn be described by simple model for the monopole nd dipoles wkefields of tube with thin dielectric coting. 1 INTRODUCTION The surfce roughness wkefield is the field, excited by bunch trveling t the speed of light in vcuum chmber with rough wll surfce. These wkefields might ply crucil role for the Liner Colliders nd Free Electron Lsers(FEL). Since it is required, tht the longitudinl nd the trnsverse emittnce is kept smll, every dditionl contribution hs to be studied crefully. The roughness depth of n extruded luminum pipe is in the order of.5 m in verge or even m pek to pek. In FEL opertion the bunchlength is below 5 m. The bunch smples the surfce structure of the tube. Wkefields due to the rough surfce of the vcuum chmber influence the longitudinl s well s the trnsverse bem dynmics. It is shown tht the phse velocity of the fundmentl tube mode is decresed by the disturbnce of mnufcture roughness to the speed of light. Accordingly there is synchronous wve ccompnying the, which is clled the rough tube mode. ANALYTICAL APPROACH To pproch description of the surfce roughness effect cylindricl tube with rdius is ssumed. The boundry of this tube is disturbed by surfce structure with the depth (Fig. 1). For structures s shown in Fig. 1.I) nd Fig. 1.II) the dispersion curve cn be clculted esily..1 Dispersion of the Fundmentl Rough Tube Mode The surfce roughness of the tube decreses the phse velocity of the fundmentl mode. The speed of light curve nd the dispersion curve re nerly prllel for wide rnge of phse dvnces per cell (Fig. ). The first higher mode, with rdil dependency of 1, behves nerly like mode in smooth tube. The phse velocity pproches the speed Work supported in prt by DESY, Hmburg, Germny IV.) III.) r II.) I.) Rectngulr nd Tringulr Structure: Period = Rndom Structure: Averge Period p= Rndom Rectngulr Structure: Averge Period p= Tube Rdius Roughness Depth p= z Figure 1: A cylindricl tube with 4 models of surfce roughness: I.) periodiclly rectngulr,ii.) periodiclly tringulr, III.) rndom with rectngulr shpe, IV rndom in longitudinl nd rdil direction. The tube rdius is, the depth of the roughness is, the period (I., II.) or verge period (III.) respectively is of light curve, but does not cross it. Consequently only the fundmentl mode contributes to the wkefields. The digrm of phse nd group velocity Fig. shows, tht the curve of the rough surfce mode crosses the speed of light line t n rbitrry single, frequency. The group velocity does not rech this line. The reltive difference between group velocity v gr nd speed of light c determines the length of the wke field pulse t = c ; v gr c L (1) where L is the length of the vcuum chmber.. Dielectric Lyer Model To clculte the wkefields inside tube the model of wve guide covered with thin dielectric lyer is used. The pplicbility of this model to tubes with corrugtions ws demonstrted in [1]. It hs been shown tht this pproch is extendble to the trnsverse wkefields creted by rough surfce [].

2 f/[ghz] 15 1 f/[ghz] φ Phse nd Group Velocity speed of light v ph rough surfce mode v ph 1. higher tube mode v gr rough surfce mode v gr 1. higher tube mode 5 Speed of light Roughness mode 1. mode with rdil dependency φ Figure : Dispersion digrm of the rough tube mode (structure s in Fig. 1.I). The rdius of the tube is =5 mm. The period of the roughness is 1m, the roughness depth is =5m. The frequency is plotted ginst the phse dvnce per cell. The roughness mode curve crosses the speed of light curve t GHz...1 Monopole Cse In the monopole cse (m =)thewve number of wve guide covered with thin dielectric lyer is given by k = " () (" ; 1) where " is the reltive permittivity, the tube rdius nd the thickness of the dielectric lyer. The longitudinl wkefunction is W k (s) =Z c cos(k s) () p using the impednce of free spce Z = =". c denotes the speed of light... Dipole Cse In the dipole cse the wve number is the sme, k 1 = " (4) (" ; 1) s in the monopole cse. The longitudinl wkefunction is given by: W k 1 (s) = r r1 Z c cos(k 1s) (5) where r nd r 1 re the offset of the driving chrge nd the witness respectively. Note tht for r = r 1 = the mplitude of the longitudinl dipole wkefield is twice s lrge s in the monopole cse. Finlly the trnsverse dipole wkefunction reds: W? r Z 1 (s) = c sin(k 1 s): (6) k Frequency [GHz] Figure : Dispersion digrm of the rough tube mode (structure s in Fig. 1.I). The rdius of the tube is =5 mm. The period of the roughness is 1m, the roughness depth is =5m. The phse velocity v ph =!= nd the group velocity v gr re plotted ginst the frequency. The roughness mode curve crosses the speed of light curve t GHz. To describe the consequences of the trnsverse wkefields on the bunch in the trnsfer line the grdient G(s) of the wkefunction s the quotient of the trnsverse dipole wkefunction nd the offset to the xis is introduced: G(s) = W? 1 (s) (7) r Assuming tht bunch of n energy E enters the tube t n offset r it will double its offset fter certin distnce, to which in this pper is referred to s the instbility length: z inst = s E G( z ).. Appliction of the Dielectric Lyer Model to Surfce Roughness The pplicbility of the dielectric lyer model to the longitudinl nd trnsverse wkefields of vcuum chmber with rough surfce modeled s shown in Fig. 1 hs been demonstrted in [1, ]. It is importnt to note, tht neither the ppering of the rf-pulse nor its frequency depend on strict periodicity of the structure []...4 Vlidity in Three Dimensions There re mny uncertinties in the trnsformtion of this model to three dimensionl problems, but the effective roughness depth in D is expected to be times less thn in D.. Normlized Description The surfce roughness wkefield, the loss prmeter nd the energy spred re given s functions of k z. The fre- (8)

3 quency k cn be derived from the model bove nd is essentilly property of the tube nd z of course bunch property. The normliztion is chosen in this wy, tht the mximum loss fctor equls 1. The wkefunction is: w k (z) =cos(k z z): (9) The wkepotentil derived from this wke is Normlized Wke Field. 1.. k =.5 k =1 k =1.5 W k (z) = 1 p Z z ;1 Thus the normlized lossfctor is: e ;y : cos(k z (z ; y))dy: (1) H(k z )=e ;(k z ) : (11) It gives the energy trnsported by the rf-wve trveling long the tube. The normlized energy spred is: (k z )= 1 p Z ;1 W k (z)e;z dz ; e ;(k z ) k Lossfctor Energy Spred k (1) Figure 4: Loss fctor H nd Energyspred due to the rough tube mode in normlized description. Depending on the vlue of k z the wkefields ct cpcitive k z or inductive k z >> 1. Fig. 4 shows flt top of the Energyspred in the region :75 <k z < 1:5. The energy spred stys the sme while the loss fctor, nd thus the energy of the rf-pulse is decresing. The mplitude of the wkefield decreses in this region of k z, It is the trnsition from cpcitive to inductive wke chrcteristic. The til of the bunch is now ccelerted. Therefore the energy spred is not chnging..4 Dielectric Lyer Model nd Normlized Description To clculte the lossfctor nd the energyspred from the normlized description the frequency k hs to be determined ccording to eq. Assuming rdius =mm nd Normlized Distnce z Figure 5: The wkefields in the flt top region of the normlized energy spred (compre Fig. 4). roughness depth = 1m, the permittivity of the dielectric lyer is found to be round " 1:9 in cses Fig. 1.I.) nd Fig. 1.III.) nd " 1:4 in cse Fig. 1.II.). Note tht the equivlent permittivity depends on different prmeters s e.g. the roughness shpe. Subsequently the vlue of the lossfctor nd the energy spred cn be clculted by nd respectively. k loss = E = Z c H(k z ) (1) Z c (k z ) (14) NUMERICAL RESULTS As n exmple of the ppliction of the dielectric lyer model tube with rdius of = mm nd roughness depth of = m, modeled s Fig.1, is tken. The bunch length z =5mis more thn 1 times lrger thn the gps of the surfce roughness. The longitudinl wkefield resulting from the numericl clcultion, is compred to the wkefield, s derived by convolution from the nlyticl solution 6. The curves show good greement. The reltive permittivity is " r = 1:515. The longer the clculted tube is, the more the mplitude of the wkefield pproches the nlytic curve. The wkefield creted by z =5mbunch in side tube with surfce modeled rndomly s well in longitudinl nd rdil direction is used s nother exmple. The tube rdius is = 5mm, the men vlue of the rndom distribution is 5m in rdil nd longitudinl direction. The field lines of the electric field 7 derived by the numericl simultion in the time domin. show hrmonic oscillting field. The wve length of the field is much higher thn the period of the surfce roughness. The size of single roughness cvity does not correspond directly to the wve length. This is in greement to the fct, tht the strict periodicity is not necessry. The rndom distribution

4 W z (s)/[v/pc/m] Numericl clcultion Anlyticl Approch s/[mm] Figure 6: Comprison of the nlyticl pproch to numericl clcultions. Wkefield of bunch with the length of z =5m, crrying chrge of 1pC is pssing tube with rdius =mm. The structure of the surfce is modeled s Fig. 1period of the roughness is p = 4m, the roughness depth is =m. does not led to ny decoherence of the rf-pulse following the bunch. 4 APPLICATION OF THE MODEL OF SURFACE ROUGHNESS The derived theory is now pplied to some components of Liner Colliders nd Free Electron Lser. As n exmple the numbers of three tubes re given. One with comprtively lrge dimeter nd very smooth surfce. with two different roughness depth, nd of very nrrow undultor chmber, where wkefields re pprehended in prticulr, re given. Tube 1 Tube Undultor Tube Rdius [mm] Rough. Depth [m] z [m] 1 1 Q [nc] frep [Hz] 1 5 Spcing [ns] Pulse Length [s] Tble 1: Electron bem nd geometric prmeters of the exmples for surfce roughness wke fields. Numbers re given for tube of 1 meter length. Severl ssumptions re mde to derive the number. The roughness depth is hs rectngulr shpe. For the equivlent permittivity " the vlue is tken. To clculte the verge power different repetition rtes of re ssumed. This number s well s the bunch spcing nd the pulse length, ffects the verge power only. The ppliction of the surfce roughness model on some Figure 7: Field lines of the electric wkefield of bunch with the length of z = 5m, crrying chrge of 1pC.The rdius of the tube is 5 mm. The verge period of the roughness is =5m, the verge roughness depth is =5m. elements which might pper in future Liner Colliders nd FEL's shows, tht enormous pek powers re chieved even with surfces usully regrded s smooth (Tb. 4). The pek power stys constnt s the structure length increses, becuse the rf-pulse length depends on the structure length too. Thus the power stored in the rf-pulse grow proportionl to the length. Note tht the numbers given in Tb. 4 expected to be smller in three dimensionl structure with n rbitrry shped surfce. Nevertheless the effect on bem dynmics cnnot be neglected. Furthermore the utiliztion of this wkefields in the bem dynmics clcultion in dmping rings shows, tht Tube 1 Tube Undultor Energyloss [kv] Energyspred [MV] k Frequency [GHz] Pulselength [fs] Pek Power [MW] Averge Power [W] Tble : Wkefields due to the vcuum chmber roughness in severl Liner Collider components. Prmeters re listed in 4.

5 the sw tooth instbility reported in mny cses might be consequence of the surfce roughness[]. 5 RESISTIVE AND SURFACE ROUGHNESS WAKEFIELDS The collective effect of the surfce roughness nd the resistive wll wkefield effect is studied. Therefore periodic rectngulr structure is chosen (Fig. 1 I.). The rdius = mm nd the roughness depth = 1m sty unchnged for the comprison. For the conductivity severl different vlues where ssumed. The lowest conductivity = 1 1/m corresponds to skin depth of skin =1m, the sme vlue s the roughness depth t the bunch frequency. In steps of times the preceding conductivity, is incresed, i.e. the skin depth is decresed by p. Finlly perfect conducting mteril is used, s comprison to the usul surfce roughness wkefields. W z (s) [kv/pc/m] perfect conducting s [mm] Figure 8: Wkefield of rough resistive Tube. The tube rdius is mm the roughness depth is 1m. The conductivity =11/m corresponds to skin depth of t the bunch frequency. Fig. 8 shows the wkefields in the bunch region. The lower the conductivity is, the higher is the mplitude of the wkefield. The resistive wll nd the rough surfce effect fortify ech other. Regrding the wkefield in longer rnge Fig. 9, the lower conductivity dmps the wkefield strongly. The rfpulse does not rech the length of the perfectly conducting cse. The frequency of the pulse is lower. Compred with the description of the resistive wll wkefield in [4], the rough surfce wke field is the dominting effect if > 4 p c (15) where, the conductivity, is 5: s 1, for copper tube, : 1 17 s 1 for luminum nd 1: 116 s 1 for stinless steel. In cse of the luminum undultor pipe (Tb. 4) the W Z (s)[kv/pc/m] s [mm] Figure 9: Wkefield of rough resistive tube. The tube rdius is mm, the roughness depth is 1m. The conductivity =11/m corresponds to skin depth of t the bunch frequency. roughness depth >64nm nd in cse of the trnsfer line >18nm. In the exmple bove Fig. 8, 9 the trnsition between the regimes is :5. 6 CONCLUSION This pproch tkes into ccount the ccelertors vcuum chmber disturbed by rough surfce. There is rough tube mode with the phse velocity equl to the speed of light ccompnying the. The bunch does not experience every single detil of the surfce corrugtion, but verges over the fults. The wkefields due to this mode re lrge. Estimtions of the influences re given. 7 ACKNOWLEDGMENTS The uthors wnt to thnk Reinhrd Brinkmnn, Jörg Rossbch nd Holger Schlrb for their interest in this work. 8 REFERENCES [1] A. Novokhtski nd A. Mosnier, `Wkefields of Short es in the Cnl Covered with thin dielectric lyer', PAC'97,p. 1444, Vncouver, Cnd, My [] A. Novokhtski, M. Timm nd T. Weilnd, `The Trnsverse Wke Fields in the TESLA-FEL Trnsfer Line', EPAC-98, p. 515, Stockholm, Sweden, June [] A. Novokhtski nd T. Weilnd, `Self-Consistent Model for the Bems in Accelertors ', ICAP'98, Monterey C., USA, September [4] K.L.F. Bne nd M. Snds, `The Short-Rnge Resistive Wll Wkefields', SLAC-PUB-95774, USA, December [5] The MAFIA collbortion User Guide, CST GmbH, Drmstdt, Germny.