Considerations on Short Range Wakefield for CLIC Main Beam Accelerating Structures

Transcription

1 Conidertion on Short Rne Wkefield for CLIC Min Be Accelertin Structure R. Zennro 1, Deceber 16, 8 Abtrct The coprehenion of hort rne wkefield i eentil for the dein of CLIC. Ueful tool re the Krl Bne forul which predict the hort rne wke for periodic D yetry tructure. The coprion of D coputtion with predicted reult nd the tudy of the rne of vlidity of thee forul re the ubject of thi note. A odel for rounded iri tructure i lo propoed. 1 CERN, Genev, Switzerlnd - 1 -

2 1 Introduction One of the in liittion to be tken into ccount in the dein of ccelertin tructure for the in be of CLIC i the hort rne wkefield. Since no dpin i poible the hort rne wke deterine the xiu chre per bunch nd the iniu perture of the tructure. Short rne wke h thu tron ipct on the luinoity of the collider nd on the efficiency of the ccelertor. The in tool for the tudy of the hort rne wke hve been provided by Krl Bne in ter of nlyticl forul tht correlte the wke to the eoetricl preter of periodic rry of cell. K. Bne work h been done for the NLC ccelertin tructure. The vlidtion of thee forul for different eoetricl rne CLIC rne i the firt tret of the preent work. A tudy of the effect of rounded irie inted of rectnulr one i lo prt of thi report. 1.1 Hih frequency lonitudinl ipednce nd hort rne wkefield For periodic rry of cvitie ccordin to Yokoy nd Bne [1,] the hih frequency ipedence (o for lre wve nuber k) i pproxitely iven by: p iz ( ) 1 ( 1 ) p p π Z L k = + + i with Z = 377Ω, p the πk k periodicity, the iri perture nd the cvity lenth (ee Fi. 1). p 1 Fiure 1: Bic hpe of D periodic tructure. The hort-rne wke i obtined by Invere Fourier trnforin: Z () c π π W L = exp erfc π with = p p p for > nd W L ( ) = for <. For hort (1) cn be rewritten in the followin ipler wy: Z c () W L exp () π The reltion between the lonitudinl wke W L nd trnvere wke W x for ll i the followin [,3]: Wx () = W ( ' ) d' l (3) (1) - -

3 Cobinin () nd (3) we et: Z () = c W x exp () π The () i the K. Bne forul for trnvere hort-rne wkefield; the vlue of S nd it dependence on,,p h been odified by K. Bne by fittin with coputtionl reult nd it i rppreented by the followin expreion []: S =. 169 (5) 1.17 L Nuericl coputtion The hort rne wkefield hve been copute with the D code [] which olve the Mxwell eqution in the tie doin for xi-yetric tructure. ke ue of ovin eh which drticlly reduce the nuber of eh point which hould be tored. The ovin eh option llow thu the coputtion of very hort bunche. In order to evlute the liit of the coputtion, everl Guin be hve been coputed with the e eoetry. The rtio between the i nd the eh denity h been kept contnt for the different coputtion. The reult how very ood reeent for i lrer thn 7 icron but lo not tolerble dicrepncie for hort be bunche (i of the order of 3-5 icron); ee fi.. In the ret of thi note ll the coputtion hve been done for i of 1 icron to hve reonbly correct reult. The CLIC bunch lenth, which i round -5 icron, i on the border line; for thi reon i very iportnt to hve lo non-coputtionl pproch to tudy the hort rne wkefield epecilly to decribe non Guin ditribution. K.B. forul re obviouly independent fro the bunch lenth nd ditribution nd re the bet tool to decribe hort rne wkefield. Lonitudinl wke wkepotentil poition (i) Fiure : (red line) nd convolution of K. Bne (blue line) lonitudinl wke for different i (1=.15, =.13, 3=.11, =.9, 5=.7, 6=.5, 7=.3; for Guin ditribution). K. Bne ive for trnvere wkefield () (5) the followin eoetricl rne of vlidity:.3 /p nd.5 /p.89 []

4 The preent CLIC tudy cover uch lrer re in the /p, /p plne, for thi reion check of vlidity of the () nd (5) h been done; ee fi CLIC reion /p CLIC tructure checked point K.B. rne of vlidity Fiure 3: The rne of vlidity of () nd (5) doe not cover the full CLIC rne. In red re hown the firt nd lt cell of the preent CLIC reference tructure. /p The coputtionl reult how tht the K. Bne forul rne of vlidity i uch lrer for the lonitudinl wkefield nd it cover the full CLIC reion in the pce /p, /p. For the trnvere wkefield, coputtionl reult how ood reeent with K. Bne in the full /p rne but lo not neliible dicrepncie for low vlue of /p; ee fi.. /p=. /p=.3 wkepotrntil wkepotrntil poition (i) /p=. poition (i) /p=.16 wkepotrntil wkepotrntil poition (i) poition (i) Fiure : The rne of vlidity of () nd (5) doe not cover the full CLIC rne. For ll perture (/p<.3) the predicted vlue (blue) re inificntly different of the coputed one (red). 3 Vlidity of the K.B. for tpered tructure K. Bne forul re vlid for contnt eoetry, i.e. contnt ipednce tructure; for thi reon the utilition of thee forul for tpered tructure hd to be verified. - -

5 The ipleented function in plcet [5] i ctully the followin: = 1 1 D W ( ( + ) + ( )) + ( ) D tpered W D W D W W W (8) where i the vere perture for linerly tpered tructure where D i totl excurion (D= x - in ). The expreion (8) i iplified pproxition of: N Z () = c Sn Wx exp (9) Nπ n= 1 n Sn Sn Where N i the totl nuber of irie in the tructure, S n i defined by the (5) with the correpondin vlue of n nd n. A iple ce of tpered tructure copoed of 8 different irie (.335 /p.63) h been coputed nd the reult hve been copred with the cobintion of the 8 convolution of the uin be with 8 expreion of the () (5) for the different irie. The reult how ood reeent with the coputtion nd confir the vlidity of the odel ued in plcet [5]; ee fi. 5. non-periodic tructure (8 cell) lonitudinl wke Non-periodic tructure (8 cell), trnvere wke K.B. (8 convolution) K. B. (8 convolution) wke (.u.) K.B. vere rdiu wke (.u.) K.B.; vere rdiu 1/[1/(w(+D)+W(-D)+W()+W(+D/)+W(-D/)] 1/[1/(w(+D)+W(-D)+W()+W(+D/)+W(-D/)] poition (i) poition (i) Fiure 5: Lonitudinl wke (to the left) nd dipole wke (to the riht). The cobintion of the 8 convolution i in ood reeent with the coputtionl reult. Extenion of the K.B. forul to rounded iri tructure One of the in liittion of the K. Bne forul concern the hpe of the irie; the reference eoetry i iple D periodic rry of qured irie. Rel tructure hve obviouly rounded or ellipticl irie for evident reon of pek electric field nd chinin. Both lonitudinl wke nd trnvere wke depend lrely on the perture 3 ( w ; W ); the roundin of the iri introduce thu inificnt pproxition l x epecilly for the trnvere wke. The odifiction of the odel to include the rounded irie or iple cobintion of different,, tructure re two poible option to iprove the etition of the wke. The econd pproch i propoed. A iple eoetry (/p=.33, /p=.83) h been coputed; in thi ce i llet perture (=) nd the rdiu of curvture i (p-)/. The coputed reult of the curved irie hve been copred with the copoition of the convolution of K. B. forul of even different, tructure with the Guin be. In the preent ce the eoetricl vlue (,) of the even tructure hve been defined by iply coniderin in-like vrition of nd co-like vrition of for even different nle t the reulr tep of 15 fro to 9. The propoed odel to repreent rounded irie i thu the followin: - 5 -

6 Wx () M Z S = c Mπ where S p S exp p S = ( = + ) 1 (1) = ( in( α )); co( α ) = nd o α = 9 1 nd M i poitive inteer lre enouh (M=7 in the preent ce) to M 1 provide ood repreenttion of the proble. The reult re in ood reeent with the coputtion epecilly for the trnvere wke; ee fi. 6. Lonitudinl wke Trnvere wke wke potentil (.u.) cobintion of K.B. K.B = cobintion of K.B. K.B.; = poition (i) poition (i) Fiure 6: Lonitudinl wke (to the left) nd dipole wke (to the riht).in both ce the reeent of the odel expreed by the (1)(blue line) nd the coputtionl reult(red line) i very ood. The reult how lo how the conventionl K.B. forul (reen line) overetite the wke epecilly the dipole one. The reult how how the trnvere wke i lrely overetited in the preent ce (=, /p=.33, /p=.8333 nd σ=.1 ). A better coprehenion of the effect of rounded irie nd the vlidtion of the (1) over lre rne of /p nd /p could provide ueful tool for optiiin the dein of RF tructure. 5 Concluion The rne of vlidity of K. Bne forul cover lot ll CLIC requireent with the exception of the dipole wke for low vlue of /p where the wke ee to be overetited. A further pproxition of K.B. forul i due the roundin of the irie; lo in thi ce nlyticl forul tend to overetite the wke. The (1) ee to be proiin tool to decribe the effect of the rounded iri. A new fittin of the (6) for the CLIC eoetricl rne i under wy. Acknowledeent The upport of Dniel Schulte nd the dvice of Krl Bne hve been extreely helpful for thee tudie. Thi work i upported by the Coiion of the Europen Counitie under the 6 th Frework Prore Structurin the Europen Reerch Are, contrct nuber RIDS

7 Reference [1] k. Yokoy nd K. bne, The lonitudinl hih-frequency ipednce of periodic ccelertin tructure, Proceedin of the 1999 IEEE Prticle Accelertor Conference Vol. 3 p. 175, New York, Mrch 1999 [] K. Bne, Short-rne dipole wkefield in ccelertin tructure for the NLC SLAC- PUB-9663 LCC-116, Mrch 3 [3] A.V. Fedotov, R.L. Glucktern, M. Venturini, Trnvere ipednce of periodic rry of cvitie, Phyicl Review Specil Topic-Accelertor nd Be Vol., 61, June 1999 [] Yon Ho Chin, Uer Guide for Verion 9., KEK Report 5-6 [5]