Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Interstage Optics Design for a PWFA Linear Collider FACETII Workshop 2015 – Oct 14, 2015 Carl A Lindstrøm PhD Student University of Oslo, Department of Physics Advisor: Erik Adli 1
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Proposed layout of a PWFA Linear Collider * • Basis for this study + source of parameters. * E Adli, JP Delahaye, et al. (presented at IPAC’13) 2
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Interstage optics • Problem at hand: Switch old with fresh drive beam, keep the main beam focused and preserve its emittance. Kicker system DELAY CHICANE for drive beam Beam dump TRANSFER LINE INTERSTAGE Plasma Plasma Yet undefined system of magnetic optics 3
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Formal requirements • Drive beam injection/extraction • Collinearity β x ( L ) = β y ( L ) = β mat • Beta function matching α x ( L ) = α y ( L ) = 0 • Dispersion cancellation D x ( L ) = D 0 x ( L ) = 0 • Limit bunch lengthening (R 56 ) R 56 ( L ) ⌧ σ z δ ⇡ 1 mm • Emittance preservation – limit chromaticity ∆ ✏ ✏ ( L ) ⌧ 1% – limit synchrotron radiation • Minimize length subject to all the above 4
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Injection/extraction of drive beams Option 1: “C-chicane” Weaker bending • Using dipoles to create dispersion: Separate beams spatially by energy. DRIVE BEAM TRAIN MAIN BEAM • Injection and extraction are PLASMA PLASMA symmetric processes ⇒ mirror symmetric lattice. • Injection/extraction dipoles Option 2: “S-chicane" must be fj rst and last magnets , Stronger bending, more space for beam dump. as main beam quads would destroy the drive beam. DRIVE BEAM TRAIN • Important: Dipoles do not scale with main beam energy MAIN BEAM PLASMA PLASMA (only drive beam energy). • De fj nes regimes: – Low energy (E main ≈ E drive ) ⇒ Dipoles “ visible ” – High energy (E main >> E drive ) ⇒ Dipoles “ invisible ” Defocusing quadrupole E V M − θ m Dipole I R A D E • Scalings: B comes first – Dipole fj eld & length: L dipole , B ~ constant MAIN 2 BEAM – SR power: P SR ~ E main plasma – Main beam dispersion: D x ~ L interstage /E main ~ 1/ √ E main SYMMETRY LINE (assuming focusing ⇒ L interstage ~ √ E main ) θ m 5
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Dispersion cancellation D x • Both S- and C-chicane has inherent dispersion cancellation. Quadrupoles change this. without quads • Cancel by: – Matching quadrupoles (not independent of beam matching) – Inserting extra dipoles (independent of beam matching) • Low energy regime (large main beam dispersion) may require second order dispersion cancellation. D x with • Becomes less important with higher energies: Scaling: D x (s) ~ 1/ √ E main quads Limiting bunch lengthening (R 56 ) • Not big problem due to relatively weak dipoles. R 56 • Becomes easier with higher energy. with Scaling: R 56 (s) ~ D x (s) ~ 1/ √ E main quads • If necessary, a method for matching R 56 ≈ 0: – Low dispersion in dipoles . R 56 ( L ) ⌧ σ z δ ⇡ 1 mm 6
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Matching beam to plasma α x ( L ) = α y ( L ) = 0 β x ( L ) = β y ( L ) = β mat • Matched beta in a plasma: ⇒ 3.3 cm @ E = 100 GeV, n p = 10 16 cm -3 • Naive matching is simple: place quadrupoles, match strengths/separations. • Strong plasma channel focusing ⇒ small plasma betas ⇒ strongly diverging/large beams ⇒ long/strong quadrupoles ⇒ large chromaticity (big challenge) • Requirements similar to those of a fj nal focus system, for every stage. Beam envelope QUADRUPOLE Plasma cell Plasma cell STRUCTURE 7
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Chromaticity cancellation Example: • Small beams in plasma lead to large chromaticity. Initial phase space (x, x’) • PWFALC study assumes a ~1% rms. energy spread. • Conventionally corrected using sextupoles . Seen in fj nal focus systems (SLC, ILC…): very long lattices. • We will consider a sextupole solution, and a novel solution. Final phase space Final phase space ( no energy spread ) ( 1% energy spread ) No chromaticity correction: Emittance increases by many orders of magnitude Traditional FF @ 500GeV ~ 1750 m Improved FF @ 500GeV ~ 300 m *Figures from “Beam Delivery & beam-beam” by Andrei Seryi (SLAC) 8
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Plasma density ramp * Plasma ramp paper by Xu et al. (2015): http://arxiv.org/pdf/1411.4386v2.pdf • A plasma density ramp mitigates the chromaticity problem . • Plasma ramp ⇒ larger/less diverging beam ⇒ smaller beam in quadrupoles ⇒ less chromaticity • Any adiabatic density pro fj le will do. Use pressure gradients or partial laser-ionization. • Scaling (for magni fj cation factor ∏ p ) : L ramp ~ ∏ p √ E main Beam envelope Varying Uniform plasma density plasma density Plasma ramp Plasma 9
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Conventional solution: Sextupoles Sextupole B-fields: • Sextupole e fg ect is stronger with: Non-linear geometric terms Non-linear chromatic term – Larger dispersion NEEDS TO BE SMALL NEED TO BE CANCELLED – Larger beam size B y ∼ 1 2( x 2 − y 2 ) + x δ D x + 1 2 δ 2 D 2 B x ∼ xy + δ D x y x • Long lattices for geometric term cancellation. Linear chromatic terms CORRECT CHROMATICITY • Introduces new problems: Geometric term cancellation: – Dipoles must ramp with main beam energy y 0 sextupole y 0 0 + ∆ y 0 0 0 ⇒ Dispersion / R 56 / SR scales poorly with energy - I transfer – -I transforms require repeated sections matrix ⇒ “ unnecessarily ” long lattices sextupole y 0 1 = − ( y 0 0 + ∆ y 0 0 ) y 0 1 + ∆ y 0 1 = − y 0 0 – Ti ick sextupoles (imperfect -I transforms) ⇒ geometric errors (emittance growth) “Working” interstage using sextupoles: – Sextupoles need large beam sizes - I transform - I transform - I transform ⇒ increased energy spread from SR (Oide e fg ect) 160. β x β y 140. 120. 100. 80. 60. 40. 20. 0.0 0.0 5.0 10.0 15.0 20.0 25.0 10
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 “ Working ” interstage using sextupoles 100 GeV, ß mat = 0.1m Beta functions Dispersion W-functions (chrom.) inex inex inex 160. 0.0150 500. D x D y β x β y W x W y 0.0125 450. 140. 0.0100 400. 120. 0.0075 350. 100. 0.0050 300. 80. 0.0025 250. 0.0 200. 60. -0.0025 150. 40. -0.0050 100. 20. -0.0075 50. 0.0 -0.0100 0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 s (m) s (m) s (m) Second order dispersion R 56 Footprint (x-z) inex 0.0 x x inex inex 0.10 0.0 -0.01 D x ’ D y ’ re56 [*10**( -3)] re56 -0.005 0.08 -0.02 -0.010 0.06 -0.015 0.04 -0.03 -0.020 0.02 -0.025 -0.04 0.0 -0.030 -0.035 -0.02 -0.05 -0.040 -0.04 -0.045 -0.06 -0.06 -0.050 -0.08 -0.07 -0.055 -0.10 -0.060 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 -0.08 0.0 5.0 10.0 15.0 20.0 25.0 30.0 s (m) s (m) z • Ti is solution requires stronger sextupoles than currently manufacturable. Not a conceptual show-stopper. 11 11
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Novel solution: Getting rid of sextupoles • We are developing a new method for fj nding quadrupole-only lattices which cancel chromaticity to the required order in energy o fg set. • Bene fj ts of using quadrupoles only: Simpler version used in light optics: – no geometric terms ⇒ keeps it linear “Superachromat” by Carl Zeiss – no -I transforms ⇒ much shorter – no ramping dipoles ⇒ better SR scaling • Similar solutions in light optics: Superachromats (however, beam optics is x/y-asymmetric). • Achieved by tailoring the energy o fg set ( δ ) expansion of β and α to be fm at around δ = 0. β and α vs. energy offset δ . Flat regions ⇒ no chromaticity 12
Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Examples of chromaticity-free quadrupole lattices • 8 quads: cancel chromaticity to 1st order. • 12 quads: cancel chromaticity to 2nd order. 13
⇒ Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 Length estimates and scalings • Any beta matching imposes: Current best solution (9 quads): • High energy regime: retains beta-shape , constant emittance 2 ) preservation, good synch. rad. scaling (P SR ~ E main • Low energy regime: complex lattices/many quads (high chrom. correction order), possibly use of sextupoles. • Interstage length estimate: ~30 m @ 300 GeV, Interstage length vs. Energy (E-spread: 1% rms , dipole length: 1m , plasma ramp: 10x , ~E 0.5 quads: 150 T/m , emit. growth: ~1%, plasma density: 10 16 cm -3 ) • Note: work in progress. Approximate scalings for high energy regime: Low High Interstage length Emittance growth Emittance growth vs. Energy E main0.5 const Energy Energy spreads: g max-0.5 g max-1.5 Quad strength ~const Energy spread σ E4 const (1st order chromaticity correction) ∏ p-3 const Ramp magnification Low High 14
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