FEL Simulations: Undulator Modeling

1
FEL Simulations Undulator Modeling

• Sven Reiche
• Start-end Workshop
• DESY-Zeuthen
• 08/20/03

2
FEL Simulations

• Start-end simulation aims to predict the FEL
  performance as realistic as possible
• Modeling the electron beam,
• Modeling the undulator.
• VUV/X-Ray undulator with large array of undulator
  poles (gt1000), grouped in modules and
  combined/interleaved with strong focusing
  elements (quadrupoles).

3
Resolving the Undulator Period

• Non-period averaged codes can use any arbitrary
  field profile but are bound to a sub-period
  length integration step size
• Integration step size of period averaged codes
  can scale with gain length, thus, independent of
  undulator length

Example Integration steps for LCLS
Non-averaged
Averaged
100 x 3000
10 x 20
Periods
Gain lengths
Resolve harmonics
Resolve gain
4
Resolving the Undulator Lattice

• Integration step sizes are limited due to
• Undulator drift sections,
• Field Errors,
• Quadrupole length,
• Phase shifter

Integration step size of period-averaged codes
scales rather with undulator period than with
gain length.
5
Speeding-up the Calculation

• Increasing length of quadrupoles, while reducing
  the field strength (thin lens approximation).
• Insert virtual phase shifter to control phase
  slippage in drift section.
• Replace effect of field errors by a single net
  kick per integration step.

6
Undulator Elements

• Essential
• Undulator field taper,
• Quadrupoles sextupole focusing of undulator
  field,
• Drifts with phase shifter.
• Optional (Exotic)
• Undulator endfield,
• Solenoid field,
• Higher multipoles of undulator field (e.g. for
  apple-type undulators),
• Coupling of motion in x and y.

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Example LCLS Undulator

• Superperiod of 6 quadrupoles undulator modules
• Permanent magnet quadrupoles (107 T/m)

F
D
F
D
F
D
F
Module 3.42 m
Long Gap 42.1 cm
Short Gap 18.7 cm
Quadrupole 5 cm
Super Period 22.11 m
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Effects of Field Errors

• Introduces steering in beam orbit.
• Reduced overlap between beam and radiation field
• Phase shake

Dp
Centroid motion
bz
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Correlated Errors

• Undulator assemble depends on measuring, sorting
  and arranging all permanent magnets with the goal
  of
• Minimizing 1st and 2nd field integral
• Minimizing phase shake

Residual variation in the field strength of
undulator field is highly correlated
10
Correlated vs Uncorrelated

• RMS variation not a good parameter to describe
  FEL performance

11
Quadrupole Misplacement

• Interface with BBA-Procedures.
• Similar effect as field errors.
• 2D codes overestimate effect.

12
Undulator Misplacement

• Possibility for undulator tapering,
• but
• Stronger alignment tolerances,
• Larger K-spread over beam size.

13
Undulator Misplacement (cont)

• Impractical for LCLS as a possibility for a
  controlled field taper.

LCLS alignment tolerance 100 mm
K-spread
tolerance
14
Undulator Taper

• Constant energy loss due to spontaneous
  radiation, compensated by taper.
• Enhancing FEL efficiency after saturation.

15
Wakefields

• Contributions by
• Resistive wall,
• Aperture changes (geometric wakes),
• Surface Roughness,

99 from resistive wall wakefields
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Wakefields (cont)

• FEL degradation by 30-50
• Weak dependence on chamber radius due to large
  transient at bunch head.

Gap due to wakefields
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Conclusion

• Non-averaged codes too CPU-intensive for full
  time-dependent simulations.
• Transverse and longitudinal motion decoupled in
  period averaged codes -gt different solvers.
• All essential features of undulators covered by
  FEL codes.
• Standard support for arbitrary undulator
  lattices.
• 2D codes limited precision for certain effects
  such as transverse centroid motion.