Wakefields and Impedances I. Wakefields and Impedances I

Transcription

1 Wakefields and Impedances I Wakefields and Impedances I An Introduction R. Wanzenberg M. Dohlus CAS Nov. 2, 2015 R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 1

2 Introduction Wake Field = the track left by a moving body (as a ship) in a fluid (as water); broadly : a track or path left Impedance = Fourier Transform (Wake Field) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 2

3 Electric Field of a Bunch Point charge in a beam pipe Gaussian Bunch, Step out transition Opening angle φ = MeV E E = 10 MeV, => φ = 50 mrad = 2.89 R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 3

4 Effects on a test charge Change of energy E(s) = q 2 dz Ez (r,z,t = (s + z)/c) Bunch charge q 1 Test charge q 2 Equation of motion q 1 : z = c t q 2 s q 1 q 2 : z = c t s E z (r, z, t) depends on the total bunch charge R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 4

5 Wakepotential W (s) = 1 q 1 Z dz E z (r, z, t(s, z)) t s t(s, z) = 1 c (s + z) E(s) = q 2 q 1 W(s) Bunch charge q 1 Test charge q 2 Equation of motion: q 1 : z = c t q 2 : z = c t s 1 Longitudinal Wakepotential Wake Bunch 0.5 Wake / V/pC s / σ z R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 5 z

6 Wake: point charge versus bunch Longitudinal Wakepotential r Point charge W (s) = 1 q 1 Z Bunch W (s) = 1 q 1 Z dz E z,p.c. (r, z, t(s, z)) dz E z (r, z, t(s, z)) ϕ r 2 s q 2 q 1 r 1 v = c u z W (r 2, s) = Z 0 ds λ(s s ) W (r 2, s ) The fields E and B are generated by the charge distribution ρ and the current density j. They are solutions of the Maxwell equations and have to obey several boundary conditions. B = µ 0 j + 1 c 2 t E B = 0 j = c u z ρ E = t B E = 1 ǫ 0 ρ ρ = q 1 λ λ λ(z) = «1 σ 2 π exp z2 2σ 2 R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 6

7 Approximations Wake Field Rigid beam approximation The wake field does not affect the motion of the beam The wake field does not affect the motion of the test charge (only the energy or momentum change is calculated) The interaction of the beam with the environment is not self consistent. Nevertheless, one can use results from wake field calculations for a turn by turn tracking code - sometimes cutting a beam into slices. R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 7

8 Catch-up distance b z c d Catch-up distance z c z c + d c = t = zc2 + b 2 c z c = b2 d 2 2 d = b2 2 d d 2 Example: z c 75 mm b = 35 mm, d 2 σ z = 8 mm R = 60 mm, L = 50 mm lim z c = LARGE d small R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 8

9 Numerical Calculations There exist several numerical codes to calculate wakefields. Examples are: Non commercial codes (2D, r-z geometry) ABCI, Yong Ho Chin, KEK Echo 2D, Igor Zagorodnov, DESY zagor/wakefieldcode_echoz/ Commercial codes (3D) GdfidL CST (Particle Studio, Microwave Studio) The Maxwell equations are solved on a grid R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 9

10 Example: wakepotential of a cavity Numerical calculation of the wakepotential with CST Particle Studio: R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 10

11 Loss Parameter Wakepotential W(s) = 1 q 1 Z dz E z (r, z, (s + z)/c) 1 Longitudinal Wakepotential Wake Bunch Line charge density λ(s) = 1 σ z 2 π exp 1 2 s 2 σ z 2 «Wake / V/pC Total loss parameter k tot = ds λ(s) W(s) Energy loss s / σ z k tot = 0.46 V pc, σ z = 4 mm, Cavity: L = 5 cm, R = 6 cm, pipe radius 3.5 cm E = P = q bunch 2 k tot E = (16 nc) V pc = mv C Bunch population N = , PETRA C = 2304m, T = 7.68µs P = 1 E = 15.2 W T R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 11

12 Transverse Wake Fields Lorentz Force F(r 2, t) = q 2 (E(r 2, t) +v B(r 2, t)) Longitudinal Wakepotential W (r 2, s) = 1 q 1 Z dz E z (r 2, z,(s + z)/c) q 2 Wakepotential W(r 2, s) = 1 Z q 1 dz[e(r 2, z, t) + ce z B(r 2, z, t)] t=(s+z)/c q 1 Change of momentum of a test charge q 2 p(r 2, s) = 1 c q 2 q 1 W(r 2, s) kick on an electron (q 2 ) due to the bunch charge q 1 θ(r 2, s) = e E q 1W (r 2, s) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 12

13 Transverse Wakepotential W (r 2, s) = 1 q 1 Z Bunch charge q 1 Test charge q 2 Equation of motion: q 1 : z = c t q 2 : z = c t s Transverse Wakepotential dz[e (r 2, z, t) + ce z B (r 2, z, t)] t=(s+z)/c t s Wake Bunch 15 Wake / V/(pC m) s / σ z z R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 13

14 Kick Parameter Wakepotential W(r 2,s) = 1 q 1 dz[e(r 2,z,t) + c e z B(r 2,z,t)] t=(s+z)/c r Line charge density λ(s) = 1 σ z 2 π exp 1 2 Kick parameter ( V/(pCm) ) k = s 2 σ z 2 «ds λ(s) 1 r 2 W (r 2,s) ϕ 15 s r 2 q 2 Transverse Wakepotential Wake Bunch r 1 v = c u z q 1 z W radial, vertical or horizontal component of the wake potential Wake / V/(pC m) 10 5 Betatron Tune shift in a storage ring Q β = I bunch T 0 4π E/e β k, (β function, optics) s / σ z k = 3.6 V pc m R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 14

15 From Wakes to Impedance Wake Field = the track left by a moving ship in water Impedance = Fourier Transform (Wake Field) Warning: Wakefields have a bad reputation and the generation of wakes may be even illegal in some communities: NO WAKE please on Chicago river (PAC 2001) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 15

16 Impedance Longitudinal impedance r Z (r 2, ω) = 1 c Z ds W (r 2, s) exp( i ω c s) ϕ s r2 q2 r 1 q 1 v = c u z z Transverse impedance 1 Longitudinal Wakepotential Wake Bunch Z (r 2, ω) = i c Z phase factor i = exp( i π/2) Wakepotential of a Gaussian bunch W (r 2, s) = Z Fourier Transform of the Wake 0 dsw (r 2, s) exp( i ω c s) ds λ(s s ) W (r 2, s ) Wake / V/pC s / σ z eλ(ω) Z (r 2, ω) = 1 c Z ds W (r 2, s) exp( i ω c s) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 16

17 You get the Impedance if you click here! R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 17

18 Equivalent Circuit Model L C R Impedance Z (ω) = 1 + i Q R ( ) ωr ω ω ω r 2 k = ω r R Q = 1 C ω r = 1 L C Q = ω r R C Wake Potential s > 0 (point charge) long range wakefield of one mode W (s) 2 k cos(ω r s c ) exp( ω r 2 Q Loss parameter (Gaussian bunch) k tot k exp ( ω 2 r ( σ ) z c )2 s c ) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 18

19 Panofsky-Wenzel-Theorem r s W (r 2, s) = 2 W (r 2, s) s W (r 2, s) = 1 q 1 Z» 1 dz c q r 2 2 t E (r 2, z, t) + ce z 1 c s v = c u z q r 1 1 t B(r 2, z, t) t=(s+z)/c z Maxwell Equation: E = t B e z t B = z E E z d dz E (r 2, z, z + s ) = c z + 1 «E (r 2, z, z + s ) c t c s W (r 2, s) = 1 q 1 Z dz ( d dz E )(r 2, z + s c ) E z (r 2, z + s «) c s W (r 2, s) = 2 W (r 2, s) + 1 q 1 Z dz( d dz E )(r 2, z + s ) c R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 19

20 Panofsky-Wenzel-Theorem (cont.) s W (r 2, s) = 2 W (r 2, s) r Frequency domain version of the Panofsky-Wenzel-Theorem: ω c Z (r 2, ω) = Z (r 2, ω) q r 2 2 s v = c u z q r 1 1 z Integration of the transverse gradient of the longitudinal wake potential provides the transverse wake potential: W (r 2, s) = Z s ds W (r 2, s ). W.K.H. Panofsky, W.A. Wenzel, Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields, Rev. Sci. Instrum. 27 (1956), 947 R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 20

21 Multipole Expansion of the Wake Longitudinal Wakepotential W (r 1, r 2, ϕ 1, ϕ 2, s) = X r m 1 r m 2 W (m) (s) cos (m(ϕ 2 ϕ 1 )) m=0 Panofsky Wenzel theorem s W (x 2, y 2, x 1, y 1, s) = 2 W (x 2, y 2, x 1, y 1, s) Multipole expansion in Cartesian coordinates W (x 1, y 1, x 2, y 2, s) W (0) (s) r ϕ + (x 2 x 1 + y 2 y 1 ) W (1) (s) r 2 s q 2 q 1 r 1 v = c u z + `(x 2 2 y 2 2 )(x 2 1 y 2 1 ) + 2 x 2 y 2 2 x 1 y 1 W(2) (s) W (x 1, y 1, x 2, y 2, s) (x 1 u x + y 1 u y ) W (1) (s) +(x 2 u x y 2 u y ) 2 (x 2 1 y 2 1 ) W (2) (s) +(y 2 u x + x 2 u y )2 (2 x 1 y 1 ) W (2) (s). Z s W (m) (s) = ds W (m) (s ) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 21

22 Monopole and Dipole Wake Longitudinal Wakepotential W (x 1, y 1, x 2, y 2, s) W (0) (s) + (x 2 x 1 + y 2 y 1 ) W (1) (s) Transverse Wakepotential W (x 1, y 1, x 2, y 2, s) (x 1 u x + y 1 u y ) W (1) (s) r ϕ s q 2 x 2, y 2 q 1 x 1, y 1 v = c u z Monopole Wakepotential W (0) (s) Longitudinal Dipole Wakepotential W (1) (s) Z s Transverse Dipole Wakepotential W (1) (s) = ds W (1) (s ) There is no transverse monopole wake potential. There is no a priori relation between the wake potentials of different azimuthal order. But for resistive wall impedance (wake) in a pipe with radius b there is a relation: Z (1) (ω) = c 2 ω b 2 Z(0) (ω) R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 22

23 W (r, s) as a harmonic function W (r, s) is a harmonic function of the transverse coordinates: 2 W (r,s) = 0 Line charge density λ(z) Charge density ρ(r, t) = q 1 λ (x, y) λ(z ct) Current density j(r, t) = cu z ρ(r, t) Maxwell Equation + ( E) = ( E) 2 E 2 E 1 2 c 2 2 t E = 1 ρ + 1 «ε 0 c 2 t j ρ and j z = c ρ are functions of z ct 1 2 E 2 «z = c 2 2 t 2 2 E z z 2 W (x, y, s) = 1 q 1 Z t j z = c 2 z ρ dz 2 E z(x, y, z, (s + z)/c) = 0 2 W 1 γ 2 0, for γ R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 23

24 Indirect test beams 2 W (x, y, s) = 0 It is possible to find W (x, y, s) for all possible beam positions (x, y) by a numerical solution of Poissons equation if one knows the wake potential on the boundary. Beam and Test Beams on the boundary. Lines of constant longitudinal wake potential. Gradient of the longitudinal wake potential. An integration provides the transverse wake potential according to Testbeam Beam the Panofsky Wenzel theorem. R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 24

25 PETRA III Application of Wake Field Calculations: The Light Source PETRA III R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 25

26 PETRA III undulator PETRA III undulators: Length: 10 m / 5 m / 2 m Mag. Field: 0.9 T Period: 30 mm Synchrotron radiation from an undulator R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 26

27 Undulator Chamber Wakefields Model of the undulator chamber and the absorber Wakepotential 90 mm x 38 mm y Taper Bellow BPM Undulator L = 115 mm 56 mm x 9 mm 66 mm x 11 mm k = 62.8 V/(pC m) 10 mm 57 mm x 7 mm Longt. Wake / V/pC z Trans. Wake V/(pC mm) L. Wake, on axis L. Wake, off axis Bunch Trans. Wake s / σ z E. Gjonaj, et al. Wake Computations for Undulator Vacuum Chambers of PETRA III, PAC 07 R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 27

28 Prediction versus Measurement wiggler West Vertical frequency shift Betatron oszillations: y = q ǫ y β(s) cos(φ(s)) f / I / khz / ma wiggler North SW W NW N NE E SE S (ORM Data: J. Keil, DESY) W NW Damping Wigglers 10 wigglers (10 x 4 m) 54.0 m Arc 14 FODO cells m 32.4 m N Damping Wigglers 10 wigglers (10 x 4 m) Insertion Devices 8 DBA cells NE 8 FB Cavities Injection E s / m new octant rf section φ(c) = 2π Q y Measurement: phase φ(s) Orbit Responce Matrix (ORM) with diffenrent single bunch intensities 240 x ma = 20 ma 10 x 2 ma = 20 ma f I = 1 2 E/e New Octant: Intensity dependend phase shift khz/ma Prediction khz/ma Measurement ( 20 % larger) n β n k n SW RF Section 12 cavities S SE R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 28

29 Acknowledgment Thank you for your attention! R. Wanzenberg Wakefields and Impedances I CAS, Nov Page 29