Introduction to linear Beam Optics

Transcription

1 Introduction to linear Beam Optic A part of the lecture on Ion Source at Univerit Frankfurt Oliver Keter and Peter Forck, GSI Outline of the lecture on linear beam optic: Motivation: Beam qualit Definition of tem of coordinate Optical propertie and tranfer matri for ome magnet Definition of emittance Tranformation of beam envelope Precaution: Bending b dipole onl in horizontal plane Baed on: F. Hinterberger, Phik der Teilchenbechleuniger und Ionenoptik, Springer 8 K. Wille, The Phic of Particle Accelerator, Oford Univerit Pre J. Robach, P. Schmüer, Baic Coure on Acc. Optic, CAS CERN 94-, 994 L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

2 Motivation: Beam Qualit Beam qualit within the plama region: The ource ha a finite volume: Start poition with ditribution σ Finite temperature of plama Mawell velocit ditribution p m p m Acceleration in direction ielding velocit β and momentum p mc β p p Beam qualit defined b ditribution: Definition of emittanceε kt mkt mc β kt mc β conervation from ource to target preferred. ε σ σ ' σ kt mc β L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

3 Motivation: Beam Qualit During tranportation the beam qualit i influenced b Poition and velocit ditribution internal degree of freedom Internal force due to the interaction of the beam charged particle i.e. beam pace charge ditribution internal force Non-linear force b magnet eternal force e.g. octupole component from a quadrupole magnet Conervation of beam qualit b linear (conervative) force! Definition of brilliance: B πε I beam πε High current, high qualit (low emittanceε) beam high brilliance δ rm Plot: Robach, Schmüer L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 3p-phic at the future GSI facilitielinear Beam Optic

4 Beam Diagnotic for LEBT at GSI UNILAC The meaurement at a LEBT (Low Energ Beam Tranport) comprie of: Current, profile, emittance, charge compoition Eample: LEBT at GSI UNILAC Length:.8 m Emittance Cup L. Dahl et al, LINAC L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

5 UNILAC at GSI: Overview RFQ, IH, IH Alvarez DTL Single Gap Reonator Tranfer to Snchrotron To SIS All ion, HLI: high(ecr,rfq,ih) current, 5 m@5 Hz, 36&8 MHz MEVVA MUCIS PIG Foil Stripper Alvarez DTL RFQ IH IH U4+ U8+ Ga Stripper. kev/u β. kev/u β.6.4 MeV/u β.6 Contructed in the 7th, Upgrade 999, further upgrade in preparation.4 MeV/u β.54 Injector for FAIR ion operation P.Groening, Forck, Lecture on ion ource, Uni Frankfurt, L. Sept. GSI-Palaver, Dec.5th, th,3 3, A dedicated proton.jan accelerator for5p-phic at the future GSI facilitielinear Beam Optic

6 Tranfer Line and Snchrotron The beam i tranported b: Bending b dipole magnet Focue b quadrupole magnet Correction are done b higher order multipole (etupole, octupole.) Eample: Part of nchrotron SIS L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 6p-phic at the future GSI facilitielinear Beam Optic

7 Definition for Stem of Coordinate Beam i tranported from ource to target: A co-moving frame i ued for the equation of motion: Definition compared to reference particle reference orbit : horizontal : vertical : in beam direction : radiu of curvature Lorentz force centrifugal force: F l F z -ev B mv / h() / e/p B (,,) on reference orbit L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 7p-phic at the future GSI facilitielinear Beam Optic

8 Talor Epanion for magnetic Field Definition compared to reference orbit : horizontal : vertical : in beam direction : radiu of curvature e p F l F z h() / e/p B (,,) on reference orbit Talor epanion of magnetic field at location for field in direction: B ( ) e p B () dipole + + e p db d k quadrupole + + etupole octupole Drift ( ), dipole and quadrupole linear beam optic +... L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 8p-phic at the future GSI facilitielinear Beam Optic e p d d m B e p 6 d 3 d o B

9 Definition of Offet and Divergence Horizontal and vertical coordinate at : : Offet from reference orbit in [mm] : Angle of trajector in unit [mrad] d / d Aumption: par-aial beam: l i mall compared to Small angle with p / p << Longitudinal coordinate: Longitudinal orbit difference: l - v ( t t ) in unit [mm] Momentum deviation:δ ( p p ) / p ometime in unit [mrad] [ ] For continuou beam: l ha no meaning et l! Reference particle: no horizontal and vertical offet and l for all L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 9p-phic at the future GSI facilitielinear Beam Optic

10 Definition of Coordinate The baic vector i 6 dimenional: The tranformation from a location to ( i given b the Tranfer matri R ( ) ) R() ' ' l δ ( ) hori. patial deviation horizontal divergence vert.patial deviation vertical divergence longitudinal deviation momentum deviation R R R3 R4 R5 R R R3 R4 R5 R3 R3 R33 R34 R35 R4 R4 R43 R44 R45 R5 R5 R53 R54 R55 R6 R6 R63 R64 R65 [mm] [mrad] [mm] [mrad] [mm] [% o] R6 R6 ' R 36 R46 ' R 56 l R 66 δ L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

11 Some Propertie of the Tranfer Matri The tranformation can be done ucceive: with with R R( ),, R n R( n- n ) It i R R n R n- R The element decribe the coupling between the component R ( ), R ( ), R 3 ( ), R 4 ( ), R 5 ( l), R 6 ( δ) R ( ), R ( ).. If all force are mmetric along the reference orbit than the horizontal and vertical plane are decoupled: ub-matri i ufficient ( ) ( ' ) ( l ) It i det(r) (Liouville Theorem) ( ') ( ' ') ( l ') For un-bunched beam: delete row 5 and column 5 R ( ) ( ' ) ( ') ( ' ') ( δ ) ( ' δ ) ( l l) L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

12 L. Groening, Sept. 5th, 3 GSI-Palaver, Dec. th, 3, A dedicated proton accelerator for p-phic at the future GSI facilitie P. Forck, Lecture on ion ource, Uni Frankfurt,.Jan Linear Beam Optic Tranfer Matri for a Drift Tranformation from of ditance L / R ) ( ) ( γ L L L R R L / R γ L R L Within the hori., vert. and long. ubpace γ: Lorentz factor, for ion ource γ L L

13 Movement within a Sector Dipole Magnet Homogeneou field region B-field component: Plot: Wille B B L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 3p-phic at the future GSI facilitielinear Beam Optic

14 Trajector within a Sector Dipole Magnet Horizontal ector field dipole magnet: Edge rectangular to reference orbit R R co in in co Care: Other tpe of dipole have a different tranfer matri Plot: Wille L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

15 Trajector within a Sector Dipole Magnet Horizontal ector field dipole magnet: Edge rectangular to reference orbit Horizontal effect:. Focuing effect due to R, R. Fat particle le bending Slower particle more bending Coupling and δ due to R 6 Can be ued a pectrometer L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 5p-phic at the future GSI facilitielinear Beam Optic

16 L. Groening, Sept. 5th, 3 GSI-Palaver, Dec. th, 3, A dedicated proton accelerator for p-phic at the future GSI facilitie P. Forck, Lecture on ion ource, Uni Frankfurt,.Jan Linear Beam Optic 6 Eample: 3 parallel trajectorie,δ, 45 o focal point at L Weak Focuing of a Sector Dipole Magnet R R drift R dipole + / co in in co R L L L L Final poition independent on tart poition if R ( ) L ( parallel-to-point image) L

17 Quadrupole Magnet Uage: For focuing (here horizontal) Field gradient: g B g ( B) B Focuing trength: k > Plot: Wille L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 7p-phic at the future GSI facilitielinear Beam Optic

18 Tranfer Matri of a horizontal focuing Quadrupole k g ( B) > Plot: Wille L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 8p-phic at the future GSI facilitielinear Beam Optic

19 Tranfer Matri of a horizontal de-focuing Quadrupole Role of horizontal and vertical plane are interchanged! Horizontal focuing trength i negative horizontal de-focuing k g ( B) > & vertical focuing L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 9p-phic at the future GSI facilitielinear Beam Optic

20 Imaging uing Quadropule: Here Duplett Focuing Horizontal focuing Horizontal de-focuing trajectorie a L l L b Tranfer matri: R channel R drift (b) R de-focu (kl) R drift (l) R focu (kl) R drift (a) Plot: Robach, Schmüer L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

21 L. Groening, Sept. 5th, 3 GSI-Palaver, Dec. th, 3, A dedicated proton accelerator for p-phic at the future GSI facilitie P. Forck, Lecture on ion ource, Uni Frankfurt,.Jan Linear Beam Optic Thin Len Approimation for Quadrupole Thin len approimation with the condition: focal length >> len thickne The action can be approimated b a hort kick at the center of the quadrupole / / R focu f f / / R focu - de f f Thin len: no etenion in direction! Finite length L of quadrupole: R R drift (L/) R focu R drift (L/) Horizontal de-focuing: Kick: - /f Horizontal focuing: f

22 Focuing b a Solenoid beam Plot: M. Maier, GSI Focuing effect due to: Inner part: Rotation of particle on cclotron radii coupling of horizontal and vertical plane Edge: Strong ammetric field line reulting in kick Focuing with the ame trength in hor. and vert. plane ueful for round beam L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for p-phic at the future GSI facilitielinear Beam Optic

23 L. Groening, Sept. 5th, 3 GSI-Palaver, Dec. th, 3, A dedicated proton accelerator for p-phic at the future GSI facilitie P. Forck, Lecture on ion ource, Uni Frankfurt,.Jan Linear Beam Optic 3 Focuing b a Solenoid: Tranfer Matri Inner part: Rotation of particle b t v BL m q mv r qb / co in / in ) / co ( in co ) / co ( / in inner R γ L L L L L Edge: Focuing due to in-homogeneou field with B / (B) and B / (B) ± ) /( ) /( edge R B B B B m Total matri: R R edge R inner R edge Focuing in & with ame trength

24 Precaution for Differential Equation for Particle Trajector Generall it i: d dt p qv B Onl linear part of magnetic field: d q r v B dt γm e B ( ) B () + p Velocit along reference orbit in dipole of curvature : v ω e p + k Within the Carteian frame (,,z): v z ( + ) ω Subtituting time derivative into derivative with rep. to uing d/dt ω e p db d Par-aial beam v, v << v Field of a dipole magnet: B & B << B v v z ( + + ) / v z Small longitudinal momentum deviation: p << p Detail ee tet book: K. Wille etc. L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

25 L. Groening, Sept. 5th, 3 GSI-Palaver, Dec. th, 3, A dedicated proton accelerator for p-phic at the future GSI facilitie P. Forck, Lecture on ion ource, Uni Frankfurt,.Jan Linear Beam Optic 5 Uing the aumption for previou lide a differential equation can be calculated: ) ( ) ( ) ''( ) ( ) ( ) ( ) ''( + + k p p k Hill Tpe Equation Solution within a dipole magnet: k() and for reference particle of p/p in ' co ) ( Anatz : ) ( ) ''( + + Equation like for ocillator but with location dependent retoring force! co ' in ) '( then : it i + co in in co R R and with / i.e. drift ' ) '( and ' ) ( ) '( ' + Differential Hill Equation for Particle Trajector in Dipole

26 Differential Hill Equation for Particle Trajector in Quadupole Uing the aumption for previou lide a differential equation can be calculated: ''( ) + ( ) ''( ) k( ) ( ) + k( ) ( ) L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 6p-phic at the future GSI facilitielinear Beam Optic p p Hill Tpe Equation Equation like for ocillator but with location dependent retoring force! Solution within a quadrupole magnet: k() for reference particle of p/p ' ' '( ) - k ( ) Anatz : ( ) co k + in k k it i then : '( ) k in k + ' co k ' '( ) ' + k ( ) Anatz : ( ) coh k + inh k k in k inh k co k R k and coh k R k k in k co k k inh k coh k

27 Concluion Tranfer Matri and Motivation of Emittance Concluion for tranfer matri: Stem of coordinate: 3 poition and divergence within a co-moving frame [,,] () (,,,, l, p/p) T Linear differential equation for drift, dipole, quadrupole and olenoid decription b matri tranformation R Baic propertie of a ingle particle trajector However: In regular cae ingle particle are not meaured The whole beam hould be decribed The tart poition of the particle have a tatitical ditribution phae pace Thi enemble i tranferred from ource to target L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 7p-phic at the future GSI facilitielinear Beam Optic

28 Differential Equation for Particle Trajector Single particle trajectorie are forming a beam The have a ditribution of tart poition and angle Characteritic quantit i the beam envelope Goal: Tranformation of envelope Behavior of whole enemble Focu. quad. drift Focu. quad. drift Plot: Wille L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 8p-phic at the future GSI facilitielinear Beam Optic

29 Anatz: Beam matri at one location: σ It decribe a -dim probabilit ditr. The value of emittance i: ε Definition of tranvere Emittance The emittance characterize the whole beam qualit: detσ σ σ σ σ σ σ σ ε β ε π γ dd' A with ' For the profile and angular meaurement: σ ' σ σ εβ εγ and σ Geometrical interpretation: All point fulfilling t σ - are located on a ellipe σ σ + σ detσ ε L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 9p-phic at the future GSI facilitielinear Beam Optic

30 The Emittance for Gauian Beam The denit function for a -dim Gauian ditribution i: It decribe an ellipe with the characteritic profile and angle Gauian ditribution of width σ ' σ σ σ and and the correlation or covariance cov σ L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 3p-phic at the future GSI facilitielinear Beam Optic

31 The Emittance for Gauian and non-gauian Beam The beam ditribution can be non-gauian, e.g. at: beam behind ion ource pace charged dominated beam at LINAC & nchrotron Variance Covariance i.e. correlation General decription of emittance uing term of -dim ditribution: ε rm ' ' It decribe the value for tand. derivation L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 3p-phic at the future GSI facilitielinear Beam Optic

32 Conervation of Emittance Liouville Theorem: The phae pace denit can not change with conervative e.g. linear force. The beam ditribution at one location i decribed b the beam matri σ( ) Thi beam matri i tranported from location to via the tranfer matri σ( ) R σ( ) R T Prove via the relation of ellipe equation: L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 3p-phic at the future GSI facilitielinear Beam Optic

33 Conervation of Emittance Liouville Theorem: The phae pace denit can not change with conervative e.g. linear force. The beam ditribution at one location i decribed b the beam matri σ( ) Thi beam matri i tranported from location to via the tranfer matri σ( ) R σ( ) R T 6-dim beam matri with decoupled horizontal and vertical plane: The beam width concerning the three coordinate are: l rm rm rm σ σ σ L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 33p-phic at the future GSI facilitielinear Beam Optic

34 Eample for Emittance Evolution Radiu () Envelope Trajector Divergence (') Simulation done b Peter Gerhard, GSI Beam envelope (here code MIRKO): Beam Matriσ( ) R σ( ) R T Plot of beam width σ εβ L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 34p-phic at the future GSI facilitielinear Beam Optic

35 Eample for Emittance Evolution Dipla of Horizontal envelope Vertical envelope Uage of eparated (, ) and (, ) ub-pace No coupling & p/p Envelope in and Quadrupole: Hor. focuing vert. de-focuing Simulation done b Peter Gerhard, GSI Beam envelope (here code MIRKO): Beam Matriσ( ) R σ( ) R T Plot of beam width σ and σ 33 L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 35p-phic at the future GSI facilitielinear Beam Optic

36 Emittance Evolution: Divergence Beam in and ' ' Simulation done b Peter Gerhard, GSI L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 36p-phic at the future GSI facilitielinear Beam Optic

37 Emittance: Ellipe (Beam Envelope) and Part. Ditribution ' ' L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 37p-phic at the future GSI facilitielinear Beam Optic

38 Emittance Evolution: Drift Sheering in Phae Space Simulation done b Peter Gerhard, GSI - [mrad] ' ' - [mm] - [mm] L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 38p-phic at the future GSI facilitielinear Beam Optic

39 Emittance Evolution: At Wait upright Ellipe i.e conv Simulation done b Peter Gerhard, GSI - [mrad] ' ' - [mm] - [mm] L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 39p-phic at the future GSI facilitielinear Beam Optic

40 Emittance Evolution: Behind Wait divergent in and 3 Simulation done b Peter Gerhard, GSI - [mrad] ' ' - [mm] - [mm] L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

41 Emittance Evolution: Important General Finding 3 ' ' [mm] - [mm] Liouville Theorem: The area within phae pace i preerved Within a drift: the ellipe i tranformed via heering Within a quadrupole: the ellipe i rotated Simulation done b Peter Gerhard, GSI L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

42 ' ' Simulation done b Peter Gerhard, GSI L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 4p-phic at the future GSI facilitielinear Beam Optic

43 Concluion for linear Beam Optic (without Acceleration) Stem of coordinate for a ingle particle () (,,,, l, p/p) T Second order linear differential equation of Hill Tpe with focuing k() and bending () Decription of linear part via Tranfer Matri R with ( ) R ( ) Enemble of particle characterized b phae pace ditribution In man cae approimated b Gauian ditribution with parameter < >, < > and < > Conervation of emittance (for - ub pace): Definition of Beam Matri σ() :, rm ' ' uing the three parameterσ, σ and σ i.e variance & co-variance for eparated phae pace Tranformation of Beam Matri via σ( ) R σ( ) R T Beam envelope i repreented b σ and σ 33 auming a Gauian ditribution Deign of an tranport line uing matri calculu (Code: MADX, WinAgile, MIRKO ) Non-linear force can not be decribed with thi method particle trajector required L. GSI-Palaver, P. Groening, Forck, Lecture Sept. Dec. 5th, on ion, 3, ource, A dedicated Uni Frankfurt, proton.jan accelerator for 43p-phic at the future GSI facilitielinear Beam Optic ε σ σ σ