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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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