1. Beam Dynamics in Curved ILC Main Linac (following earth curvature)
K.KUBO
This presentation is a summary of two reports in
ILC-Asia Note:

2. Design and simulation of curved ILC Main Linac
All Quad-magnets are aligned along the earth curvature, and accelerating cavities are aligned along the straight lines between quads. (Coordinate change at the middle of each quad)
Thin (zero-length) steering magnet is inserted at the middle of each quad.
Used SAD for optics calculations, SAD and SLEPT for tracking simulations.

3. Simulated Two Solutions
Strength of steering magnets are set
(a) Eliminating beam offset and dispersion at center of every vertically defocusing quad.
(b) Making the beam goes through the centers of the all quad-magnets. The initial dispersion ( and ’) is adjusted to minimize the final ‘Linear Dispersion Corrected emittance’

4. Optics: Old. Not ILC baseline, but not too different
Optics: Old. Not ILC baseline, but not too different. FODO Beam energy: From 5 GeV to 250 GeV Acc. cavity: 35 MV/m, 10 cavities/module from 5 GeV to 125 GeV: 2 modules/quad from 125GeV to 250 GeV: 3 modules/quad

5. “Design” orbit and dispersion of (a)
Offset from the curvature.

6. “Design” orbit and dispersion of (a), near the beginning of linac.
The curvature and design orbit.
Offset from the curvature.
Dispersion

7. Result of simulation without errors in case (a), using SLEPT.
Projected emittance along the linac.
Initial vertical emittance: 0=2E-8 m,
Initial energy spread: 2.8%,
TESLA-TDR wake.

8. Dispersion along the linac in the case (b)
(Initial dispersion is optimized to minimize the emittance growth.)

9. Projected emittance along the linac.
‘Linear Dispersion corrected’ emittance along the linac.

10. Definition of
Projected emittance and
Linear Dispersion Corrected emittance

11. Simulation including alignment errors and orbit (Kick Minimum) correction
Initial normalized vertical emittance: 2E-8 m
Initial energy spread: %
Random unknown offset of BPM with respect to quad: 10 micron
Random error of BPM(resolution): micron
Every quad has a BPM and a steering at its middle.
Correction: Steering correction.
Minimize additional offset and additional kick” at quads.

12. Distribution of final projected-normalized emittance.
Misalignment of quads and cavities: 300 micron
(Total 100 random seeds in each case.)
Left: Following Earth Curvature, case (a)
Right: Laser Straight
Curved (a)
Laser straight

13. Distribution of Linear Dispersion Corrected emittance
Misalignment of quads and cavities: 200 micron
Left: Following Earth Curvature, case (b)
Right: Laser Straight

14. Vertical position change due to magnet strength error.
Calculated by SAD, 100 random seeds.
Relative magnet strength error 0.01% (sigma).
Relative error of a quad and steering magnet at its middle was set to be the same.
Errors of magnets at different locations were set independently.
Distribution of y/y of the same data.
To make the rms of y/y less than 0.14, the strength error should be less than %.
(Random 0.14  offset of each beam will make average position offset 0.2  between two beams at IP, which will decrease luminosity about 3%, without beam-beam force.)
Required stability: order of 10-5.
Very tight but,
Relevant only for fast jitter, faster than orbit feedback

15. Required stability: order of 10-3.
Emittance increase due to magnet strength error.
Simulated by SLEPT, 100 random seeds.
Relative magnet strength error 0.1% (sigma).
Relative error of a quad and steering magnet at its middle was set to be the same.
Errors of magnets at different locations were set independently.
To make the average of /0 less than 0.063, the strength error should be less than 0.17%.
(Emittance increase by will decrease luminosity about 3%, without beam-beam force.)
Required stability: order of 10-3.

16. Summary-1
Simulations were done for 250 GeV beam energy linac, all Quad-magnets are aligned along the earth curvature and accelerating cavities are aligned along the straight lines between quads.
Two cases:
(a) Zero dispersion at every defocusing quad
(b) Zero orbit every where, tuned initial dispersion
Emittance increase was small without errors.
Emittance increase with misalignments and orbit correction in the curved linac and the straight linac were almost the same.

17. Summary-2
Tolerance of random magnet strength jitters (magnet by magnet independent)
For beam offset less than 0.14  (3% Lunimosity reduction from simple calc.)
0.0025% (case (a)) and %(case (b))
For emittance increase less than 0 (3% Lunimosity reduction from simple calc.)
0.17% (case (a)) and 0.4% (case (b))
The results suggests:
Curved Linac (following earth curvature) will be OK,
if magnet strength can be stable enough.