Part II Effect of Insertion Devices on the Electron Beam

Transcription

1 Part II Effect of Insertion Devices on the Electron Beam Pascal ELLEAUME European Synchrotron Radiation Facility, Grenoble II, 1/14, P. Elleaume, CAS, Brunnen July -9, 3.

2 Effect of an Insertion Device on the electron beam of a storage ring Perturbation of the lattice functions Affect the synchrotron radiation integrals: Increase energy spread, length the bunch Reduce emittance The effect is in general weak with the eception of the damping wigglers. Deflection, linear and non linear focusing from: Residual field errors Nominal field II, /14, P. Elleaume, CAS, Brunnen July -9, 3.

3 Closed Orbit Distortion from Residual Field Integrals θ Angle induced by field integral Displacement induced By double field integral δs () = θ β () sβ cos( πυ φ ( s) φ ) sin( πυ ) B () s ds, B () s ds < 1 3Gcm II, 3/14, P. Elleaume, CAS, Brunnen July -9, 3.

4 Magnetic Field Errors X θ θ e 1 = B() s ds o( ) γ mc + γ e 1 = B() s ds o( ) γ mc + γ 1 θ e = = F γ mc B () s ds 1 θ e 1 = = B () s ds = F γ mc F 1 θ e = = F γ mc c B () s ds II, 4/14, P. Elleaume, CAS, Brunnen July -9, 3.

5 Focusing Effect An Insertion Device which presents a local focusing and introduces a tune shift And beta beat which are dependent on the field of the Insertion Device δυ = δυ = 1 1 β β Kds 4π 4π F 1 1 β β Kds 4π 4π F δυ = c 1 4π β β F and a Beta Beat : c β πδυ β πδυ =, = β sin( πυ ) β sin( πυ ) Beam sie variation along circumference Stop band around half intereger resonances => Avoid large Beta functions in Insertion Devices II, 5/14, P. Elleaume, CAS, Brunnen July -9, 3.

6 Reduction of Dynamic Aperture induced by Insertion Devices => Reduction of lifetime - The linear focusing of an Insertion Devcie change the betatron function all over the circumference => break the N symmetry => ecites non systematic resonances (normally very weak) which generate beam losses => important to locally correct the focusing and restore the beta functions - The non linear focusing ecites the non systematic resonances and may create additional losses. ν Operating Point - Both the nominal field and field errors can be responsible - The effect can be serious on low energy rings with many Insertion Devices. II, 6/14, P. Elleaume, CAS, Brunnen July -9, 3. 3Ν 3Ν 1 3Ν ν Systematic Resonances : mν +pν = Nq Non Systematic Resonances : mν +pν = q m.n,q : integers

7 with e B () s nd order 1 K = + K + o( ) γmc γ e B () s nd order 1 K = + K + o( ) γmc γ e B () s nd order 1 Kc = + Kc + o( ) γmc γ II, 7/14, P. Elleaume, CAS, Brunnen July -9, 3.

8 Simple Theory of nd Order Undulator Focusing Applied to Planar Undulator A planar undulator presents a D magnetic field which in free space can be derived from a scalar potential satisfying : ϕ(,) s= A solution is : s= B λ s ϕ(, ) sinh( π )cos( π ) π λ λ B B s ϕ(,) s s = = B cosh( π )cos( π ) λ λ ϕ(,) s s = = B sinh( π ) sin( π ) s λ λ The Lorent Force equation in such a field remains to be solved II, 8/14, P. Elleaume, CAS, Brunnen July -9, 3. dv γ m = ev B dt

9 dv dt e = ( vb s vb s ) ds ds γ m dv dt e = ( vb s vb s) ds ds γ m dvs dt e = ( vb vb ) ds ds γ m with B B B s = = B = B s cosh( π ) cos( π ) λ λ s sinh( π )sin( π ) λ λ order in 1/γ v = s = ct [,, c] 1 st order in 1/γ e λ s = π π s = ct v B cosh( )sin( ),, c γm π λ λ dv e e s cosh( π ) cos( π ) ds γmc γm λ λ nd order in 1/γ = vb s = B II, 9/14, P. Elleaume, CAS, Brunnen July -9, 3. λ cosh( π )sinh( π )sin ( π ) dv e eb s = vb s = c ds γ mc γmc π λ λ λ λ cosh ( )cos( )sin( ) dvs e eb s s = vb = c π π π ds γmc γmc π λ λ λ

10 averaging over one period or dv ds = dv c eb λ = sinh(4 π ) ds γ mc 4π λ dvs = ds ( ) ( v ) eb π (...) eb = = if d d d dt d λ ds ds dt ds ds c γmc λ γmc π Planar Undulators are vertically focusing with a focal length : 1 eb 1 1 eb e =, = γmc F = = γmc γ m c K K ds L B ds II, 1/14, P. Elleaume, CAS, Brunnen July -9, 3.

11 General Theory of nd Order Focusing Start from the Lorent Force Equation of motion of an electron in an arbitrary magnetic field epressed in a fied Cartesian frame (Os) dv e = v B dt γ m e = + + B + B + B γ mc '' 1 ' ' ' s (1 ' ) ' ' e = + + B + B + B γ mc '' 1 ' ' ' s (1 ' ) ' ' with dy y' = ds, y'' = d y ds Solve these equations in power series of 1/γ making use of the Mawell Equation : B=, B= II, 11/14, P. Elleaume, CAS, Brunnen July -9, 3.

12 Insertion Device etends from s = to L : Field Errors Nominal Field L 1 Φ 1 ( L) = () + Bds ( ) o( ) γ + γ γ d d e e ds ds mc mc L 1 Φ 1 ( L) = () Bds ( ) o( ) γ + γ γ d d e e ds ds mc mc L s s with Φ (, ) = B( s,, ') ds' + B( s,, ') ds' ds For a periodic field with period λ : s λ s Φ (, ) = N B( s,, ') ds' + B( s,, ') ds' ds The detailed deflection, focusing and non-linear focusing can be predicted From the function Φ(,) computed from the transverse field II, 1/14, P. Elleaume, CAS, Brunnen July -9, 3.

13 Tracking of e- beam in an Insertion Device Split the Undulator into n thin Lenses separated by drift spaces L L n Non linear thin lens 1 e '(, ) = Φ(, ) n γ mc 1 e '(, ) = Φ(, ) n γ mc Drift Space L = + ' n L = + ' n II, 13/14, P. Elleaume, CAS, Brunnen July -9, 3.

14 nd Order Focusing from High Field Wigglers Technology PPM Period [mm] 1 Gap [mm] 1 Length [m]. Peak Field [T] Magnet Width [mm] 6 B Vertical Field under Pole Energy [GeV] 6 Vertical Beta [m] X[ mm] [ T] θ [ µ rad] Horiontal Deflection [micro-rad] vs Horiontal Position Vertical Tune Shift vs Horiontal Position X[ mm] -4-4 X[ mm] In etreme cases (high field Wiggler, Narrow Pole, Low Energy), one may not be able to inject in a Wiggler because of the horiontal non linearity. Eample : SPEAR BL11 Wiggler II, 14/14, P. Elleaume, CAS, Brunnen July -9, 3.

15 Reduction of Dynamic Aperture from Apple II Period = 88 mm Gap = 16 mm Length = 3. m Beta, = 35,.5 m II, 15/14, P. Elleaume, CAS, Brunnen July -9, 3.

16 Conclusion Insertion devices may be the source of perturbations : Closed Orbit distortion Tune shift Lifetime reduction The problem is most severe on low energy rings with many insertion devices. Nowadays the technique of field shimming allows to get rid of most of the perturbations induced by the residual field errors. For high field devices or complicated field geometries (Apple II), one may need local correctors. II, 16/14, P. Elleaume, CAS, Brunnen July -9, 3.