Wake Field Calculations at DESY

Wake Field Calculations at DESY Impedance Budget for the European XFEL Igor Zagorodnov Collaboration Meeting at PAL Pohang, Korea 2-6. September 213

Overview Numerical Methods for Wake Filed Calculations low-dispersive schemes indirect integration algorithm modelling of conductive walls optical approximation slowly tapered transitions The European XFEL Impedance Budget cavity and couplers wakes collimator wakes high-frequency impedances longitudinal impedance budget Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 2

Low-dispersive schemes Wake field calculation estimation of the effect of the geometry variations on the bunch Ω r r j = cρ + First codes in time domain ~ 198 A. Novokhatski (BINP), T. Weiland (CERN) Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 3

Low-dispersive schemes TESLA cavity Gaussian bunch with RMS length σ = 5µ m New projects new needs - short bunches; - long structures; 3 cryomodules to reach steady state about 36 m - tapered collimators New methods are required - without dispersion error accumulation; - staircase free; - fast 3D calculations on PC Solutions - zero dispersion in longitudinal direction; - conformal meshing; - moving mesh and explicit or split methods Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 4

Low-dispersive schemes j = cρ Long smooth structures in 3D MAFIA, ABCI ECHO-3D E/M * scheme grid dispersion staircase geometry approximation moving mesh demands interpolation TE/TM ** scheme zero dispersion in longitudinal direction. staircase free (second order convergent) travelling mesh easily (mesh step is equal to time step) * E/M- electric - magnetic splitting of the field components in time = Yee s FDTD scheme ** TE/TM- transversal electric - transversal magnetic splitting of the field components in time Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 5

Low-dispersive schemes E/M and TE/TM splitting e ) x e ) y h ) z e ) h ) x z h ) y Subdue the updating procedure to the bunch motion τ τ +τ / 2 τ +τ E/M splitting TE/TM splitting h n n hx n = hy n hz n+.5 ex n+.5 n+.5 e = e y n+.5 ez MAFIA v n n ex n = e y u n hz ECHO-3D h = n+.5 x n+.5 n+.5 hy n+.5 ez Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 6

Low-dispersive schemes 3D simulation. Test examples Moving mesh 2 TESLA cells structure 3 geometry elements The geometric elements are loaded at the instant when the moving mesh reach them. During the calculation only 2 geometric elements are in memory. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 7

Low-dispersive schemes W V / pc -1 E / M 2.5D ( z = σ /5) ( z = σ /1) MAFIA W V / pc -1 ECHO-3D POT 2.5D -2-2 -3-4 POT 2.5D -.5.5 1 1.5 s / cm ( z = x/3 = y /3 = σ / 2.5) -4 -.5.5 1 1.5 Comparison of the wake potentials obtained by different methods for structure consisting of 2 TESLA cells excited by Gaussian bunch σ = 1mm E/M splitting -3 TE / TM 3D Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 8 s / cm 3 σ z ~ TE/TM splitting z ~ σ L

Low-dispersive schemes 3D simulation. Collimator Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 9

Low-dispersive schemes ABCI ECHO-3D W V / pc 4 2 E / M 2.5D ( r = z = σ /5) ( r = z = σ /1) W V / pc 15 1 5 TE / TM 3D ( z =.5 y = =.5 x = σ /5) POT 2.5D -5 POT 2.5D -2-1 -4 -.5.5 s / cm -.5.5 Comparison of the wake potentials obtained by different methods for round collimator excited by Gaussian bunch σ = 1mm TE/TM method fast, stable and accurate with coarse mesh Zagorodnov I., Weiland T., TE/TM Scheme for Computation of Electromagnetic Fields in Accelerators // Journal of Computational Physics, 24. -15 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 1 s / cm

Low-dispersive schemes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 11

Low-dispersive schemes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 12

Indirect Integration Algorithm Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 13

Indirect Integration Algorithm Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 14

Indirect Integration Algorithm Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 15

Modelling of Conductive Walls Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 16

Modelling of Conductive Walls Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 17

Modelling of Conductive Walls Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 18

Optical Approximation, L << a 2 / σ 2 Z ( r 1, r ε 2) = ϕb( 1, ) ϕb( 2, ) ds ϕa( 1, ) ϕb ( 2, ) ds c r r r r r r r r SB Sap 1 A ri r r ri ϕ (, ) = ε δ ( ) r S A ϕ r r = A (, ) i r S A 1 B ri r r ri ϕ (, ) = ε δ ( ) r S B ϕ ( r, r) = B i r S A i = 1, 2 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 19

Optical Approximation Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 2

Optical Approximation Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 21

Slowly Tapered Transitions L λl Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 22

Slowly Tapered Transitions Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 23

Cavity and Coupler Wakes Cryomodule 1 Cryomodule 2 Cryomodule 3 1 8 1 8 1 8 8 cavities + 9 belows =12m 12m 12m 3 cryomodules = 36 meters Transverse wakes for short bunches up to σ = 5µ m have been studied. To reach steady state solution the structure from 3 cryomodules is considered. For longitudinal case the wakes were studied earlier by Novokhatski et al *. The transverse results are calculated with ECHO **. * Novokhatski A, Timm M, Weiland T. Single Bunch Energy Spread in the TESLA Cryomodule, DESY, TESLA-1999-16, 1999 ** Weiland T., Zagorodnov I, The Short-Range Transverse Wake Function for TESLA Accelerating Structure, DESY, TESLA-23-19, 23 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 24

Cavity and Coupler Wakes The wake functions at short distance are approximately related by 1 2 s w ( s) w 2 ( z) dz a 2 () () a w 1 w s 2 (1) (2) K.L.F.Bane, SLAC-PUB- 9663, LCC-116, 23 Different behavior! One-cell structure Z c g w s O s 2π a s.5 ( ) = ~ ( ) 2 2 2Z c w s gs O s a π a.5 ( ) = ~ ( ) 2 2 Periodic structure w ( s) = A Z c exp( s s 2 ) ~ O(1) π a s s1 ( ( ) ) 2 Z c w ( s) = A 2s 2 2 1 1 1 + s s1 e ~ O( s) a π a A, s, s 1 - fit parameters s s a iris rtadius, g cavity gap = s 1 s 1 relations (1), (2) hold exactly only relation (2) holds exactly Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 25

Cavity and Coupler Wakes 15 1 8 1 8 1 8 V / p C V / pc / m W 1 2 1 1 W 5 ( W ) 1 2 2 1-2 1-1 / cm O(1), σ σ 1 1 1 ( W ) 1 2 2 1-2 σ / cm 1-1 O( σ ), σ Comparison of numerical (points) and analytical (lines) integral parameters for the third cryomodule Like periodic structure! w ( s) = 344exp( s s )[V/pC/module] w s s V = + pc m module 3 ( s) 1 1 1 exp s1 s1 s = 1.74 1 3 3 s 1 =.92 1 A = 1.46 a = a = 35.57mm O(1), s O( s), s Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 26

Cavity and Coupler Wakes W V pc / / W V pc / / -1 σ = 5 µ m -1 σ = 7 µ m -2-2 σ = 5 µ m wake function -3-3 -5 5.1.2.3 s / σ s / cm Comparison of numerical (grays) and analytical (dashes) longitudinal wakes for the third cryomodule W -6 1 /V / pc / m 6 W 1 /V / pc / m -4 σ = 7 µ m 4-2 -5 5 2 wake function.1.2.3 s / σ s / cm Comparison of numerical (grays) and analytical (dashes) transverse wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 27

Cavity and Coupler Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 28

Cavity and Coupler Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 29

Collimator Wakes δ b r ϕ z The bunch moves very close to the aperture wall! b 1 α d b 2 ~ b σ 2 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 3

Collimator Wakes I.Zagorodnov et al, DESY, TESLA-23-19, 23 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 31

Collimator Wakes L ( δ ) V / pc 6 5 4 3 2 1 Near-wall wakefields of the optimized geometry (σ=5µm) L ( n ) 36 L (7, δ ) V / pc L (7, δ ) L (1, δ ) 34 32 3 28 L (, δ ).5 1 1.5 2 δ b / mm The kick factor for the aperture b=2 mm 26.5 1 1.5 2 δ b / mm The loss factor for the aperture b=2 mm The loss factor increases only by ~2% up to the aperture wall. The kick factor shows fast grows near the wall - ln(1 δ 2 ), δ > 1.. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 32

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 33

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 34

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 35

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 36

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 37

Collimator Wakes b 1 d α b 2 Figure 1: Top half of a symmetric collimator. Regimes diffractive inductive intermediate k ρ round 1 1 αb2 σ << 1 2α 1 1 Zc = πσ b2 b1 4π (1) Inductive k ρ 1 << 1, rectangular παh 1 1 = σ b 2 1 1 2 h b2 << 1 ρ α σ 2 2 2 b1 Zc 4π (2) Intermediate 1 ~1,.5 1.5 1 2 k 2.7 Aσ b α Z c(4 π) (3) = ρ 2 ρ2 π ρ 1 >> 1 Diffractive ρ 1 >> 1 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 38

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 39

Collimator Wakes k [V/pC/mm] 5 long 4.5 4 in +out 3.5 σ =.5mm σ =.3mm 3 σ=.1mm 2.5 short.1.1 1 1 d[cm] 5 4 3 2 k [V/pC/mm] long short.1.1 1 1 d[cm] Table1: Loss and kick factors as estimated by 2D electrostatic calculation. The bunch length σ =.3 mm. ``Short'' means using Eq. 6, ``long'' Eq. 5 Type k [V/pC] k tr [V/pC/mm] short long short long round 78 78 2.5 5.1 rect. 56 72 2.43 6.11 square 74 78 1.99 4.25 Figure 2:Kick factor vs. collimator length. A round collimator (left), a square or rectangular collimator (σ =.3 mm, right). The good agreement we have found between direct time-domain calculation [1] and the approximations (5, 6), suggests that the latter method can be used to approximate short-bunch wakes for a large class of 3D collimators. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 4

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 41

Collimator Wakes Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 42

High-Frequency Impedances TRANSVERSE IMPEDANCE OF LASER MIRROR OF RF GUN Z ( R a ) ad 2 ( m) 1 2 2 R y ( ω) = 2 α + 1 ln 2 2 + 2 2 2ε π ωar a + d β = cot ( ( )) ( d) Z ( ) 1 4 4 3 3 4 2 2 2 3 4 2 y ω = A ar d a d + R d α a d Q + R β + R d Q 4 d( α + β ) ( q ) 1 Z y ( ) = ad R 4 ( d 2 a 2 ) + ( a 2 + d 2 )( R 2 + 6d 2 ) aq + B a 2 R 4 8d 4 AB + R 4 8a 4 d 2 ( ) ( ( ) ( ) ) ω β α σ =.5 mm k y (,), k y (d), k y (q), V/pC V/pC/ V/pC/ m m Analytical.124 13.1 12.1 Numerical.12 13.1 11.6 σ = 2 mm k y (,), k y (d), α = -1 tan Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 43 d a d a tan Q d -1-1 Q = R d 2 2 2 2 B = a + d A = 4π ε ω a R d k y (q), V/pC V/pC/ V/pC/ m m Analytical.12 13 12 Numerical.8 24 7.5 2 2 4 2

High-Frequency Impedances TRANSVERSE IMPEDANCE OF OTR SCREENS d h a R S ap ( m) 1 Z y ( ω) = F( a, h) F( h, a) 4π ε ωar h 2 2 [ ] ( ) F( x, y) = R 2x y cot + tan + ay R x + d ln d + y 2 2 R x x 2 2 1 x 1 d 2 2 2 2 ( ) ( ) 2 1.5 1.5 kv k y nc d = 4 mm d = 6 mm d = 12.5 mm 2 4 6 8 1 a [mm] 6 5 4 3 2 1 ε x ε x [%] ε x 2 4 6 8 1 a [mm] ε y ε y 2 2 = 1+ S β 1 S β ε 3ε 6ε y y y eqk S = β y 2 z E Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 44

High-Frequency Impedances LONGITUDINAL IMPEDANCE OF ELLIPTICAL TO ROUND TRANSITIONS IN UNDULATOR SECTION g 1 =.4cm w 1 =.45cm S ap ϕ S A R S B g w w =.75cm g =.44 cm 15 kv k nc 25 2 kv k nc 1 5 E2R Total R2E.4.5.6.7 R[cm] 15 1 5 A2R Total R2E.4.5.6.7 R [cm] Dependence of the loss factor from the radius of the round pipe. The left graph presents the results without the absorber, the right graph presents the results with the absorber included. The black dots show the numerical results from CST Particle Studio. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 45

High-Frequency Impedances LONGITUDINAL IMPEDANCE OF ROUND TO RECTANGULAR TRANSITIONS IN BUNCH COMPRESSORS R g w S ap ϕ ( R2 S ) 9 Z = Ω ( S 2 R) Z = 8 7 Z [ Ω ] 16 14 Z [ Ω ] 6 5 4 3 2 1 Total w = 1 cm g = 2 cm rct2r R2rct 2 2.5 3 3.5 4 4.5 5 R[cm] 12 1 8 6 4 2 w = 1 cm R = 5cm R2rct Total rct2r 1 2 3 4 5 g [cm] Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 46

High-Frequency Impedances LONGITUDINAL AND TRANSVERSE IMPEDACES OF ROUND MISALIGNED PIPES y R 2 Rs Rs 2 R x 2g z 2g R R S ap y R ϕ g 15 1 5 x Z ()[ Ω] Total Rs2R R2Rs.1.2.3.4.5 g / R 2 Z g Ω M. Dohlus, I. Zagorodnov, O. Zagorodnova, High Frequency Impedances in European XFEL, DESY 1-63, 21 15 1 5 ( )[ ] Total Rs2R=R2Rs.2.4.6.8 1 g / R Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 47

Longitudinal Impedance Budget Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 48

Longitudinal Impedance Budget s W ( s) = w ( s s ) λ( s ) ds wake (Green) function Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 49

Longitudinal Impedance Budget wake potential s W ( s) = w ( s s ) λ( s ) ds wake (Green) function If we know wake function then we can calculate wake potential for any bunch shape. For beam dynamics simulations we need the wake function. The numerical codes can calculate only wake potentials (usually the Gaussian bunch shape for relatively large rms width is used). But the real bunch shape is far not Gaussian one. How to obtain the wake function? Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 5

Longitudinal Impedance Budget wake potential s W ( s) = w ( s s ) λ( s ) ds How to obtain the wake function? The deconvolution is bad poised operation and does not help. wake (Green) function Well developed analytical estimation for short-range wake functions of different geometries are available. Hence we can fit our numerical results to an analytical model and define free parameters of the model. Such approach is used, for example, in A. Novokhatsky, M. Timm, and T. Weiland, Single bunch energy spread in. the TESLA cryomodule, Tech. Rep. DESY-TESLA-99-16 T. Weiland, I. Zagorodnov, The Short-Range Transverse Wake Function for TESLA Accelerating Structure, DESY-TESLA-3-23 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 51

Longitudinal Impedance Budget There are hundreds of wakefield sources in XFEL beam line. The bunch shape changes along the beam line. Hence, a database with wake functions for all element is required. The wake functions are not functions but distributions (generalized functions). How to keep information about such functions? We need a model. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 52

Longitudinal Impedance Budget Wake function model () 1 ( 1) w( s) = w ( s) + + Rcδ ( s) c Lcδ ( s) + w ( s) C s regular part singular part (cannot be tabulated directly) () 1 ( 1) Z( ω) = Z ( ω) + R + iω L + Z ( ω) iωc s resistive capacitive inductive W ~ λ W ~ λ ( s) ( s) ds W ~ λ ( s) ( 1) 1 w ( s) = o( s ), s. it describes singularities s -α, α<1 Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 53

Longitudinal Impedance Budget () 1 ( 1) w( s) = w ( s) + + Rcδ ( s) c Lcδ ( s) + w ( s) C s Pillbox Cavity Zc g w( s) = ( 2 2π a s 1) Z w ( s) = 2 Step-out transition Z b w( s) = c ln ( s) π a δ Z b R = ln π a w( s) = crδ ( s) Tapered collimator 2π a sg 2 Z w( s) = c r dr ( s) 4π c s δ 2 w( s) = c L ( s) s δ L Z = 4π c r dr Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 54

Longitudinal Impedance Budget () 1 W ( s) = w ( s s ) λ( s ) ds λ( s ) ds Rcλ( s) C 2 Wake potential for arbitrary bunch shape s s ( 1) c Lλ ( s) c w ( s s ) λ ( s) ds derivative of the bunch shape s Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 55

Longitudinal Impedance Budget The main form of data base application contains a list of element types, parameters R, L, C and links to tables w and w_1. Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 56

Longitudinal Impedance Budget Undulator wake Q=1nC W [kv/m] 2 15 1 5 bunch -5-1 resistive wake -15-2 Total wake -6-4 -2 2 4 6 s [µm] Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 57

Longitudinal Impedance Budget Undulator wake Q=1nC Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 58

Longitudinal Impedance Budget Accelerator wakes. Q=1nC warm pipe 14% 2% 4% 19% collimators 1% 1% 1% 1% 2% 4% cavities 42% COL CAV TDS BPMA OTRA BPMR TORAO KICK PIP2 PUMCL FLANG Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 59

Longitudinal Impedance Budget Accelerator wakes. Q=25 pc warm pipe 23% 2% 4% collimators 33% 11% 1% 4% 22% cavities COL CAV TDS BPMA KICK PIP2 PUMCL FLANG Igor Zagorodnov Collaboration Meeting at PAL 2-6. August 213 Seite 6