1. Beam Optics of the TTF2 Nina Golubeva DESY

2. Beam optics from the BC2 up to the undulators General introduction to linear optics: – constraints for different sections – matching between sections (flexibility, chromatic behaviour) Chromatic properties Effect of the transverse space charge (after BC3) Transverse beam dynamics

3. Layout & properties & constraints BC2 bunch compressor FODO cells (μ x,y =45°) ACC2 & ACC3 BC3 bunch compressor ACC4, ACC5, TB_ACC6, TB_ACC7 COLLIMATOR (with the dogleg) FODO in TB_SEED UNDULATOR · Beam waist and small β between two last dipoles (CSR effects) · L = 1.9 m => βmax ~ 3.5 m · Matching (BC2, BC3) is ƒ (E, Eacc, …) · Beam waist and small β between two last dipoles (CSR effects) · Matching (BC3, COLL) is f (E, Eacc, …) · β (dogleg) ~ 5 m, 2 sextupoles, 4 Collimators: β(CT1&2) > 20m · μ x,y = 45º => βmax ~ 17 m · ~ 4.5 m

4. Linear optics BC2BC3COL FODO βx=6m ACC2&3ACC4&5 β~40m MAD code no RF focusing UND β x ~2m

5. Linear optics: transverse focusing of RF cavities Example: E 0 (BC2) = 130 MeV E (BC3) = 380 MeV E f (COL) = 445 MeV ACC2 & ACC3: Φ = 80º, E acc = 15 MV/m ACC4: Φ = 90º, E acc = 8 MV/m ACC5: Switch off Elegant code, M.Borland Twiss parameters at the entrance of BC3 and COLL sections are fixed: Matching (BC2, BC3) = ƒ (E, E acc ) Matching(BC3, COLL) = f (E, E acc )

6. BC2 section BC2FODO ACC1 2 quads of ACC1 3 quads in the front of BC2 5 quads between BC2 and FODO lattice FODO cells => => BC2 area as a matching section to FODO lattice is flexible to beam parameters from injector

7. Collimation system Protection of undulators (transv. & energy collimation) Protection against off-energy and mis-steered beams 4 copper collimators : Recommended apertures radii: R(CT1) = 2 mm R(CT2) = 2 mm R(CE1) = 6 mm R(CE2) = 2 mm |ΔE/E 0 | > 3% will be stopped L col = 50 cm TESLA Report 2003-17 CT1CT2CE Collimator section L = 50 cm

8. Chromatic properties TrackFMN code, V.Balandin 1.All matching sections (esp. from small/large to large/small β-functions): are optimized taking into account the chromatic effects tuned to let the particles with the energy offset less ±3% pass through with small distortion of beam parameters 2. Collimator section with the dogleg: second order dispersion is suppressed by 2 sextupoles particles with the energy offset less ±1.5% pass through with small distortion of beam parameters The total optics has an energy acceptance of about ±1.5%.

9. Chromatic properties Section between BC3 and COLLCollimator section (final) Sext. off Sext. on Initial Final 3 ellipses: -3%, 0%, +3%3 ellipses: -1.5%, 0%, +1.5% -1.5% +1.5% E = 380 MeVE = 445 MeV

10. Effect of transverse space charge TrackFMN code, V.Balandin ¤ Beam line after last bunch compressor BC3: after ACC4 module to undulators ¤ Beam parameters: E 0 = 450 MeV Î = (0.5 – 3) kA ε n = 2 mm mrad TrackFMN : 2D Poisson equation solver 4D particle tracking (x, Px, y, Py)

11. Transverse space charge after BC3 Î=0 Î = 3 kA Î = 1 kA Î = 2 kA 0 kA 1 kA 2 kA 3 kA hor ver Simulations: N p = 10^5 - 10^6 Grid: 128 – 512 N sp-kick / elem = 2 – 20 mismatch at und. entrance = 0.5 ( γ 0 β i – 2 α 0 α i + β 0 β i ) Results for the total beam statistical horizontal β-functions

12. Transverse space charge after BC3 Vertical plane Statistical β-functions Current: 0, 1, 2, 3 kA

13. Transverse space charge after BC3 1 σ 2 σ final: undulator entrance Î=0 Î=2kA initial: ACC4 exit Red: 10 000 particles, Gaussian distr. Green (I=0) & Blue (I=2kA): test particles: – do not contribute in space charge forces – are tracked in space charge field of the main beam CPU time = 90 sec (grid: 256x256, Nsp/elem=5)

14. Summary & next steps Linear optics: * BC2 area is working as matching section between injector and FODO lattice * large β-functions in sections between BC2&BC3 and BC3&COLLIMATOR * optics of COLLIMATOR section is assumed to be fixed Energy acceptance: |ΔE/E 0 | is about ±1.5% Particles with |ΔE/E 0 | > 3% are stopped by collimators Error and misalignment analysis Detailed studies of the space charge effects, ƒ(E, Î, ε n ), (optics less sensitive to the space charge effect ?) …