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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 2-9, 2003. 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 exception of the damping wigglers. • Deflection, linear and non linear focusing from: – Residual field errors – Nominal field II, 2/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Closed Orbit Distortion from Residual Field Integrals θ Angle induced by field integral Displacement induced By double field integral cos(πυ x − φx ( s) − φxID ) δ x( s ) = θ β x ( s ) β xID 2sin(πυ x ) ⇒ ∫ B (s)ds , ∫ B (s)ds ID x ID z < 10 − 30 Gcm II, 3/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Magnetic Field Errors X 1 ∂θ x e ∂ e 1 = = ∫ B (s)ds θx = ∫ Bz ( s )ds + o( ) ∂x γ mc ∂x z Fx γ mc ID γ ID 1 ∂θ z e ∂ 1 −e 1 = ∂z =− γ mc ∂z ∫ Bx (s)ds = − ∫ Fz Fx θz = Bx ( s )ds + o( ) ID γ mc ID γ 1 ∂θ x e ∂ Fc = = ∂z γ mc ∂z ∫ B (s)ds ID z II, 4/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. 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 1 1 β xID δυ x = 4π ∫β ID x K x ds 4π Fx 1 1 β z ID δυ z = 4π ∫β ID z K z ds 4π Fz 1 β xID β xID δυc = 4π Fc and a Beta Beat : ∆β x 2πδυ x ∆β z 2πδυ z = , = βx sin(2πυ x ) β z sin(2πυ z ) ⇒ Beam size variation along circumference ⇒ Stop band around half intereger resonances => Avoid large Beta functions in Insertion Devices II, 5/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Reduction of Dynamic Aperture induced by Insertion Devices => Reduction of lifetimeOperating Point - The linear focusing of an Insertion Devcie change the betatron function νz all over the circumference => break the N symmetry => excites 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 excites the non systematic resonances and may create additional losses. - Both the nominal field and field errors can be responsible 3Ν−2 3Ν−1 3Ν - The effect can be serious on low νx energy rings with many Insertion Systematic Resonances : mνx+pνz= Nq Devices. Non Systematic Resonances : mνx+pνz= q II, 6/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. m.n,q : integers with e ∂Bz ( s ) 1 Kx = + K x 2 nd order + o( 2 ) γ mc ∂x γ e ∂Bz ( s ) 1 Kz = − + Kz 2 nd order + o( 2 ) γ mc ∂x γ e ∂Bx ( s ) 1 Kc = + K c 2 nd order + o( 2 ) γ mc ∂x γ II, 7/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Simple Theory of 2nd Order Undulator Focusing Applied to Planar Undulator • A planar undulator presents a 2D magnetic field which in free space can be derived from a scalar potential satisfying : ∆ϕ ( z , s ) = 0 B0 λ0 z s • A solution is : ϕ ( z, s) = sinh(2π ) cos(2π ) 2π λ0 λ0 ∂ϕ ( z , s ) z s Bz = = B0 cosh(2π ) cos(2π ) ∂z λ0 λ0 ∂ϕ ( z , s ) z s Bs = = − B0 sinh(2π ) sin(2π ) ∂s λ0 λ0 • The Lorentz Force equation in such a field remains to be solved dv γm = ev × B dt II, 8/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. dvx dt e = (vz Bs − vs Bz ) Bx = 0 ds ds γ m z s dvz dt e with Bz = B0 cosh(2π ) cos(2π ) = (vs Bx − vx Bs ) λ0 λ0 ds ds γ m z s dvs dt e Bs = − B0 sinh(2π ) sin(2π ) = (vx Bz − vz Bx ) λ0 λ0 ds ds γ m v = [ 0, 0, c ] • 0 order in 1/γ s = ct ⎡ −e λ z s ⎤ v=⎢ B0 0 cosh(2π ) sin(2π ), 0, c ⎥ • 1st order in 1/γ ⎣ γ m 2π λ0 λ0 ⎦ s = ct dvx e −e z s • 2nd order in 1/γ =− vs Bz = B0 cosh(2π ) cos(2π ) ds γ mc γm λ0 λ0 2 dvz e ⎛ eB ⎞ λ z z s =− vx Bs = − c ⎜ 0 ⎟ 0 cosh(2π ) sinh(2π ) sin 2 (2π ) ds γ mc ⎝ γ mc ⎠ 2π λ0 λ0 λ0 2 dvs e ⎛ eB ⎞ λ z s s = vx Bz = −c ⎜ 0 ⎟ 0 cosh 2 (2π ) cos(2π ) sin(2π ) ds γ mc ⎝ γ mc ⎠ 2π λ0 λ0 λ0 II, 9/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. • averaging over one period dvx =0 ds 2 dvz c ⎛ eB ⎞ λ z = − ⎜ 0 ⎟ 0 sinh(4π ) ds 2 ⎝ γ mc ⎠ 4π λ0 dvs =0 ds • or 2 2 d 2 z d dz dt d v 1 ⎛ eB0 ⎞ 16π 2 3 1 ⎛ eB ⎞ λ0 = ( )= ( z) − ⎜ ⎟ (z + z + ...) ≅ − ⎜ 0 ⎟ z if z ds 2 ds dt ds ds c 2 ⎝ γ mc ⎠ 6λ02 2 ⎝ γ mc ⎠ 4π Planar Undulators are vertically focusing with a focal length : 2 2 1 ⎛ eB0 ⎞ 1 1 ⎛ eB0 ⎞ e2 Kz = ⎜ ⎟ , = ∫ K z ds = ⎜ ⎟ L = 2 2 2 ∫ Bz2 ds 2 ⎝ γ mc ⎠ Fz ID 2 ⎝ γ mc ⎠ γ mc II, 10/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. General Theory of 2nd Order Focusing Start from the Lorentz Force Equation of motion of an electron in an arbitrary magnetic field expressed in a fixed Cartesian frame (Oxzs) e x '' = − 1 + x '2 + z '2 ⎡ z ' Bs − (1 + x '2 ) Bz + x ' z ' Bx ⎤ ⎣ ⎦ γ mc dv e = v×B z '' = e 1 + x '2 + z '2 ⎡ x ' Bs − (1 + z '2 ) Bx + x ' z ' Bz ⎤ dt γ m γ mc ⎣ ⎦ with dy d2y y' = , y '' = 2 ds ds Solve these equations in power series of 1/γ making use of the Maxwell Equation : ∇B = 0 , ∇ × B = 0 II, 11/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Insertion Device extends from s = 0 to L : Field Errors Nominal Field 1 e 2 ∂Φ L dx dx e 1 γ mc ∫ ( L) = (0) + Bz ds − ( ) + o( 2 ) ds ds 0 2 γ mc ∂x γ 1 e 2 ∂Φ L dz dz e 1 γ mc ∫ ( L) = (0) − Bx ds − ( ) + o( 2 ) ds ds 0 2 γ mc ∂z γ 2 2 ⎛s L ⎞ ⎛s ⎞ with Φ ( x, z ) = ∫ ⎜ ∫ Bx ( x, z , s ')ds ' ⎟ + ⎜ ∫ Bz ( x, z , s ')ds ' ⎟ ds 0⎝0 ⎠ ⎝0 ⎠ For a periodic field with period λ0 : λ0 2 2 ⎛s ⎞ ⎛s ⎞ Φ ( x, z ) = N ∫ ⎜ ∫ Bx ( x, z , s ')ds ' ⎟ + ⎜ ∫ Bz ( x, z , s ')ds ' ⎟ ds 0 ⎝0 ⎠ ⎝0 ⎠ The detailed deflection, focusing and non-linear focusing can be predicted From the function Φ(x,z) computed from the transverse field II, 12/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. 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 Drift Space 2 1 ⎛ e ⎞ ∂ L ∆x '( x, z ) = − ⎜ ⎟ Φ ( x, z ) ∆x = x + x ' 2n ⎝ γ mc ⎠ ∂x n 2 1 ⎛ e ⎞ ∂ L ∆z '( x, z ) = − ⎜ ⎟ Φ ( x, z ) ∆z = z + z ' 2n ⎝ γ mc ⎠ ∂z n II, 13/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. 2nd Order Focusing from High Field Wigglers Vertical Field under Pole Technology PPM B z [T ] Period [mm] 100 Gap [mm] 10 1.5 Length [m] 2.2 Peak Field [T] 2 1 Magnet W idth [mm] 60 0.5 Energy [GeV] 6 0 Vertical Beta [m] 4 -60 -40 -20 0 20 40 60 X [ mm ] Horizontal Deflection [micro-rad] vs Horizontal Position 100 Vertical Tune Shift vs Horizontal Position 0.01 θ x [ µ rad ] 50 0.008 0 0.006 0.004 -50 0.002 -100 0 -40 -20 0 20 40 -40 -20 0 20 40 X [ mm] X [ mm] In extreme cases (high field Wiggler, Narrow Pole, Low Energy), one may not be able to inject in a Wiggler because of the horizontal non linearity . Example : SPEAR BL11 Wiggler II, 14/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. Reduction of Dynamic Aperture from Apple II Period = 88 mm Gap = 16 mm Length = 3.2 m Betax,z = 35, 2.5 m II, 15/14 , P. Elleaume, CAS, Brunnen July 2-9, 2003. 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 2-9, 2003.

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