Calculation of Coherent Synchrotron Radiation in General Particle Tracer

Transcription

1 Calculaton of Coherent Synchrotron Raaton n General Partcle Tracer T. Myajma, Ivan V. Bazarov KEK-PF, Cornell Unversty 9 July, 008

2 CSR n GPT D CSR wake calculaton n GPT usng D. Sagan s formula. General Partcle Tracer (GPT) s a partcle trackng coe, whch nclues 3D space charge effect base on a nonequstant multgr Posson solver or a pont-to-pont metho. The routne can calculate D-wake functons for arbtrary beam trajectores as well as CSR shelng effect. In partcular, the CSR routne oes not assume ultrarelatvstc electron beam an s therefore applcable at low beam energes n the njector. I. V. Bazarov an T. Myajma, Calculaton of Coherent Synchrotron Raaton n General Partcle Tracer, Proc of EPAC 008, MOPC04 D. Sagan, AN EFFICIENT FORMALISM FOR SIMULATING THE LONGITUDINAL KICK FROM COHERENT SYNCHROTRON RADIATION, Proc of EPAC 006, THPCH04

3 Sagan s formula

4 Sagan s formula Two partcle nteracton The source partcle at pont P. An electrc fel E(P) at the poston of the kcke partcle at pont P an tme. The Lenar-Wechert formula e E( P) The CSR term E E L Lβ' e ( L L β' ) c CSR E SC [( L Lβ') a' ] L ( L L β') 3 Here, the space charge term s E SC e ( P) [ zsˆ xxˆ yyˆ ] ( z x y ) 3/ The rate of energy change s gven by K eβˆ E eβˆ E E CSR CSR ( ) CSR β ˆ β / β β β β'

5 The space charge term e [ zsˆ xxˆ yyˆ ] E SC ( P) Sagan s formula Space charge term ( z x y ) 3/ The longtunal stance s requre to calculate the space charge term. The change of the longtunal poston of the source partcle s L s ( t t' ) βl z β c The longtunal stance between P an P s z Ls βl

6 Sagan s formula 3 Calculaton of z on arbtrary orbt The orbt s ve nto N elements from O. The path length: v an w components of the vector L: ν L s N ν ) cos ( sn w v w v L L L R w L R v L φ φ 3 ω ν ν w v N N g g g v 3 6 ψ ω ψ ψ The stance, z: L L z s β ( ) g g z ν ω ν ν

7 Sagan s formula 4 Calculaton of CSR kck on arbtrary orbt CSR kck: ( )( ) ( ) ( ) z e g e K CSR α τ τακ α τ α τ τκ α α τ τ ( ) ( ) g g g θ κ ν ω α ν τ ( ) g g z ν ω ν ν

8 Proceure of CSR calculaton. Calculate, whch satsfy z z 0. Calculate ν, ω, g an wth 3. Calculate K CSR wth 4. Shft z0 an repeat to 3. z 0 P s P

9 CSR calculaton n GPT

10 Commans of GPT/CSR Comman name csrdwakexz(); Assumpton It s assume that the partcles move on x-z plane. Namely, the vertcal component of the average velocty s zero. Optons The GPT/CSR has 6 optons.

11 Optons of GPT/CSR. CSRTmestep (ouble) (s). CSRCalcTstep (ouble) (s) 3. CSRMeshNbn (long) 4. CSRBGTolerance (ouble) 5. CSRMeshBoxSze (ouble) 6. CSRMeshNbfac (ouble) 7. CSRMeshStep (ouble) (m) 8. CSRTrangleWth (ouble) (m) 9. CSRSgn (ouble) 0. CSRHshel (ouble) (m). CSRNmage (nt). CSRDrftLength (ouble) (m) 3. CSRCalcArea (ouble) (m) 4. CSRArcRaus (ouble) (m) 5. CSRArcAngle (ouble) (ra) 6. CSROutputWake (ouble) (m) # # example of CSR calculaton # csr_t 0.0e-; csr_tstep 0.0; csr_nb 0; csr_bgtol.0e-; csr_nst 0.0; csr_mnbfac 0.; csr_ml 0.06e-3; csr_tr 0.6e-3; csr_sgn -.0; csr_h.0; csr_nh 0; csr_ns 0.0; csr_xn -0.0; csr_xout 0.0; csr_zn -0.0; csr_zout 0.0; csr_arcr 0.0; csr_arcang 0.0; csr_wfrom 0.0; csr_wto 0.0; csr_wstep 0.0; # # please comment out the followng lne # for calculaton wthout CSR # csrdwakexz("csrtmestep", csr_t, "CSRCalcTstep", csr_tstep, "CSRMeshNbn", csr_nb, "CSRBGTolerance", csr_bgtol, "CSRMeshBoxSze", csr_nst, "CSRMeshNbfac", csr_mnbfac, "CSRMeshStep", csr_ml, "CSRTrangleWth", csr_tr, "CSRSgn", csr_sgn, "CSRHshel", csr_h, "CSRNmage", csr_nh, "CSRDrftLength", csr_ns, "CSRCalcArea", csr_xn, csr_xout, csr_zn, csr_zout, "CSRArcRaus", csr_arcr, "CSRArcAngle", csr_arcang, "CSROutputWake", csr_wfrom, csr_wto, csr_wstep);

12 Energy Loss an Sprea () The steay-state energy loss an sprea for varous beam energes are compare as calculate by GPT/CSR, elegant, an analytcal expresson for a crcular orbt. TheCSRroutnenelegantncluesthe assumpton of ultrarelatvstc beam. GPT/CSR reprouces the analytcal result accurately. Benng raus: ρ.0 m Bunch length: σ s 0.6 mm Intal strbuton: Gaussan Bunch charge: Q 80 pc. Analytcal expresson erve by C. Mayes 4 4 ε ( re mec ) cβ N( ( N ) T ( a)) t 3 ρ a 3/ 3σ s /( βρ) 9 a 8 T( a) e π K 5/6 πa 3 3π a 8a K 5/6 (x) : the mofe Bessel functon N : the number of electon n the bunch r e : the classcal electron raus

13 Energy loss an sprea () The results of GPT/CSR an elegant both reprouce well the analytcal result for hgher beam energy, E 0 > 40MeV. The results of elegant an the theory verge to nfnty for E 0 0. The result of GPT/CSR approaches zero as expecte. Analytcal expresson wth the assumpton of >> (r/σ s ) /3 [,] σ δ r 0. e Ncβ /3 4/3 t β ( / ρ σ c : the spee of lght : the Lorentz energy factor ) / [] P. Emma an R. Brnkmann, Proceengs of PAC97, Vancouver, B.C., Canaa, 997, pp [] Ya. S. Derbenev. et.al., TESLA FEL-Report s These results show that the GPT/CSR s effectve for we range of beam energes, an can be use to nvestgate beam ynamcs n ERL an FEL photonjectors.

14 CSR shelng effect Image charge layer Chamber heght, h h h Benng raus: ρ 0.0 m Bunch length: σ s.0 mm Intal strbuton: Gaussan Bunch charge: Q 80 pc. Number of mage charge layers: 3 The effect of CSR shelng s calculate by GPT/CSR for a crcular orbt. As the shelng heght ncreases, the energy loss approaches to the analytcal value.

15 CSR n transent state wthout shelng As an example of CSR effect n a transent state, the CSR wake form s calculate by GPT/CSR after the ext of a benng magnet. Beam energy: 8 MeV Benng raus: ρ 0.0 m Bunch length: σ s 0.3 mm Intal strbuton: Gaussan Bunch charge: Q 80 pc Shelng chamber heght: h Number of mage charge layers: 3

16 Beam energy: 8 MeV Benng raus: ρ 0.0 m Bunch length: σ s 0.3 mm Intal strbuton: Gaussan Bunch charge: Q 80 pc Shelng chamber heght: h cm Number of mage charge layers: 3 CSR n transent state wth shelng The fgures show that the CSR wake reuces as the stance from the ext of the benng magnet ncreases as expecte.

17 CSR n merger secton Asanexample,thetransverseemttancena3-polemergerof ERL project at Cornell Unversty s calculate by GPT/CSR an elegant for two fferent contons: (a) p 0 0MeV/can (b) p MeV/c. Bunch length: σ s 0.3 mm Intal strbuton: Gaussan Bunch charge: Q 80 pc Intal emttance : ε nx 0 - m ra Intal betatron functon : β x β y 9 m Wthout shelng an space charge

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19 For (a) p 0 0MeV/c, the GPT/CSR an elegant results sagree. For (b) p MeV/c, the agreement s goo emonstratng that GPT/CSR reprouces elegant CSR calculatons at hgher beam energes as expecte.

20 Enhance 3D Space Charge Routne n GPT

21 Enhance 3D Space Charge Routne To calculate the space charge fel n the 3D mesh-base routne n GPT, the partcle coornates are transforme from the laboratory frame to the rest frame accorng to r' r, r' r relatve to the recton of moton. When the bunch oes not move along the z-axs, the bounng box ens up mproperly orente. n GPT

22 In ths case, for example, the transverse emttance ncorrectly epens on the angle relatve to the z-axs n a straght trajectory. To fx ths problem, we have ae a transformaton of rotaton n the rest frame n the space charge routne. Orgnal routne Enhance routne

23 Summary We have evelope a new CSR routne for GPT n orer to nvestgate beam ynamcs n ERL an FEL njectors. To check GPT/CSR, energy loss an energy sprea are calculate by GPT/CSR, elegant an analytcal expresson. The results show GPT/CSR to be effectve n a we range of beam energes. We have correcte 3D spacecharge routne n GPT so that t s mae applcable to calculatng the space charge effect n benng magnets.