Modeling Circularly-Polarizing ID Effects at APS

Transcription

1 Modeling Circularly-Polarizing ID Effects at APS L. Emery and A. Xiao Accelerator Systems Division Argonne National Laboratory Mini-workshop on Dynamic Aperture Issues of USR 11 th November 2010

2 Outline ID types at APS Nonlinearities, multipole and dynamic Fitting of calculated fields to reduce noise Kick maps for tracking 2

3 Type of Insertion Devices at the APS Planar undulators Regular permanent magnet-iron pole hybrid Superconducting undulator under development Circular polarizing undulators CPU: fast-switched magnet (10 ms transition) IEX (Intermediate-Energy X-ray) beamline: no fast switching Possible APPLE (Advanced Planar Polarized Light Emitter) devices: no fast switching Undulators can be installed in-line or canted 3

4 ID Perturbations First and second integrals are corrected with dipole correctors Quadrupole (normal and skew) can be compensated as well Multipoles can be compensated up to a point by external multipole magnets Dynamic multipoles from field roll-off don't have the same field shapes of multipole correctors correctors. Detailed simulation studies required. Field strength and roll-off may need to be restricted Some references on Dynamic multipoles: P. Elleaume, EPAC 1992, p.661 J. Safranek, PRST-AB 2002, vol. 5,

5 Non-linear Fields from Insertion Device On-axis By s Undulating trajectory of one period from exact integration x0 ~ 5mm x mm Trajectory change due to By(x) By weaker away from x=0 Trajectory if pole is infinitely wide (and no multipole) s 5

6 Non-linear Fields from Insertion Device (integrating from a different point) On-axis By s Zoomed-in scale of Undulating trajectory x0 x mm Trajectory if pole is infinitely wide (and no multipole) ~ 5mm Trajectory change due to By(x) By is not constant along this direction s 6

7 Non-linear Fields from Insertion Device In a simple case (a sinusoidal horizontally-deflecting wiggler), the nonlinear kick in x-plane: Roll-off of magnetic field. High energy of APS non-linear effects are reduced. Normal planar-deflecting undulator (including SCU) effects are negligible. short period length w (typical length cm) wider pole face smaller df 2 (x)/dx 7

8 Non-linear Fields from Insertion Device Circular polarizing undulator (CPU, IEX,...) longer period length w (12.8 cm for CPU, 12.5 cm for IEX) narrow pole face larger df 2 (x)/dx 8

9 Elements for Tracking Wigglers 3D magnet calculation (B fields or potential)) Fitting to analytical model Track analytical model as CWIGGLER element in elegant Make a kick map table from CWIGGLER tracking result Track kick map for DA and MA calculations 9

10 Model of Insertion Device An analytical wiggler model (CWIGGLER) is used in elegant for simulating the non-linear effect with a canonical integrator (Y. Wu, Duke Univ.) Based on pole configuration, the field is expressed with 10

11 Fitting Designed/Measured Magnetic Field to The Model Doing FFT on calculated B y (z) and B x (z) from a 3D magnet code. Real part goes to horizontal-deflecting mode, imaginary part goes to vertical-deflecting mode. Real Imaginary Use harmonic series for x-dependence, k x,mn = mk x,1n, to fit parameters in the red box. Simultaneously fitting of k x,1n and A mn is very hard (multidimensional non-linear problem). 11

12 Fitting Designed/Measured Magnetic Field to The Model An improvement is to fix k x,1n and do linear fit of A mn. 1D scan of k x,1n over possible parameter space, a best fit is reached. The fitting procedure also removes unphysical random noise error from field calculation. 12

13 Fitting Result of IEX Field On-axis field of circular polarizing mode Overall fitting errors: ±17mm(x) x ±4mm(y) (on-axis: < 1%, off-axis: 10% but most of it is random noise) 13

14 Tracking with Kick-map Fitted field model is used as input of CWIGGLER in elegant for tracking particles. More than 200 harmonic terms are used for precisely expressing the magnetic field A kick map replaces canonical integration to speed up tracking Kick map is created from a (x,y) grid of runs with CWIGGLER singleelement beamline of one-period wiggler Element UKICKMAP is defined with a (x 0, y 0, x' 1, y' 1 ) table for the kick that is sandwiched between two drift space. Changes of particles position is ignored since they are very small. The total CWIGGLER is effectively replaced by n (total number of periods) UKICKMAP elements Comparison of tracking results (DA and MA) between using CWIGGLER and UKICKMAP was made, and results show no noticeable difference. 14

15 Kick-and-Drift Derived from Exact Integration x mm (x 1,x 1 ') CWIGGLER elegant element x 0 ~ 5mm x By is not constant along this direction (x 1,x 1 ') s UKICKMAP elegant element x 0 Kick of x' drift drift Output angle is made the same as integration s 15

16 Simulation Result for IEX There are 3 operation modes of IEX pure H/V polarizing modes + circular polarizing mode. All results are for C-mode except those with a special notation. Two major design versions IEX1: narrow vertical poles which resulted in large roll-off of magnetic field df 2 (x)/dx we learned our lesson there! IEX2: wider vertical poles, value of df 2 (x)/dx is reduced. The CPU (an existing device) is also simulated for comparison (we used double length of actual magnet to get L=4.8 m) It's already known from APS operation that the non-linear effects from CPU is negligible. 16

17 Kick-map of Three Wigglers CPU: Existing device with no effect on DA IEX1: High gradient at origin and bad for DA IEX2: Bump moved to large x but OK for DA One period C mode Entire length C mode 17

18 Dynamic Aperture CPU: Existing device with no effect on DA IEX1: High gradient at origin and bad for DA IEX2: Bump moved to large x but OK for DA 8 random error seeds Quad+Sext: Field error of 2E-4 Quad+Sext: Tilt error of 5E-4 rms beta beating 1% 18

19 Momentum Aperture CPU: Existing device with no effect on DA IEX1: High gradient at origin and bad for DA IEX2: Bump moved to large x but OK for DA No decrease of MA is observed! 19

20 Correction of Non-Linear Fields of Good IEX2 with Multipole V-mode Note: the reduction in slope at x=0 20

21 Kick-map for entire IEX On-axis field: H: B y,max = T V: B x,max =0.371 T C: B x,max = T, B y,max =

22 Conclusion Method for rapid tracking of complex wiggler has been worked out Particular design of a wiggler was modified to ensure good DA Comment from Bengsston of BNL: The noise of the 3d magnetic field calculation can be suppressed by dealing directly with the vector potential for kick maps. 22