Framework for Apochromatic Linear Lattice Design CLIC Beam Physics Meeting – Nov 26, 2015 Carl A Lindstrøm PhD Student University of Oslo, Department of Physics Advisor: Erik Adli 1 Overview • Chromaticity problem, motivation for using linear lattice correction. • Quick literature review. • Conceptual explanation of how linear lattice chromaticity correction works. • How to design apochromatic lattices + demonstration. • Some working examples (PWFA staging / plasma lenses / fnal focus). • Note: Tat this work focuses on linacs/transfer lines (not rings). Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 2 Sequence of events leading up to now • Late 2014: – Study of plasma wakefeld acceleration (PWFA) staging optics. – Chromaticity correction challenging: Sextupoles found unfavorable. • Early 2015: – Searched for and found sextupole-free chromaticity-free lattices. – Literature review: no sources found at the time… • Mid 2015: – Developed theoretical framework / understanding. – Developed computational methodology and sofware. • Late 2015: – LCWS2015 talk to get feedback. – Feedback from R. Tomas: Basic ideas introduced in 1987 (Montague/Ruggiero) • Now: – CLIC Beam Physics meeting to present framework + get feedback Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 3 Problem: Chromaticity @B @x • Problem: Quadrupole focusing is energy dependent: ∆x0 ⇠ E • Chromaticity: Energy dependent focusing. • Small beams (β ≪ drif length) + energy spread = Large chromaticity • Examples: – LWFA/PWFA: Catch small and highly diverging beam exiting plasma. – Final Focus: Demagnify large beam to very small beam at the IP. E + �E E Quadrupole E – �E Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 4 Conventional solution: Sextupoles + Dipoles • Dispersion from dipoles + position dependent focusing from sextupoles ⇒ Controlled energy dependent focusing ⇒ Use to cancel chromaticity (1st order) • Introduces new problems: – Dipoles introduce synchrotron radiation and dispersion (must be cancelled) – Sextupoles introduce non-linear terms (must be cancelled) • Paradox: Linear colliders were born to avoid bending high energy electrons/positrons. ⇒ Question: Can chromaticity be cancelled without dipoles + sextupoles? Sextupole B-fields: B xy + δD y x ⇠ x Non-linear Linear chromatic terms Dipoles geometric terms CORRECT CHROMATICITY 1 2 2 1 2 2 By (x y )+xδDx + δ Dx Quadrupole Sextupole ⇠ 2 − 2 Cartoon of how dipoles and sextupoles can cancel chromaticity (dispersion greatly exaggerated). Non-linear chromatic term Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 5 Some inspiration from light optics • Light optics has the same chromaticity problem. • Teir solution: Image several colors simultaneously using several lenses – 2 colors: Achromat – 3 colors: Apochromat – 4 colors: Superachromat • Chromaticity correction in a straight system Camera optics with superachromatic glass optics. with only linear optics. Image source: ic.pics.livejournal.com Focus shift (arb. units) of different glass optics lattices. Image source: toothwalker.org Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 6 Literature review • Framework for chromaticity correction (W-function). 1977: “Strategie pour la correction de chromaticité” (Zyngier) 1979: “Linear optics for improved chromaticity correction” (Montague) • Linear lattice solution to chromaticity correction. 1987: “Apochromatic focusing for linear colliders” (Montague, Ruggiero) • General apochromatic linear lattice solution. 2010: “Apochromatic beam transport in drif-quadrupole systems” (Balandin et al.) 2011: “Apochromatic Twiss parameters of drif-quadrupole systems with symmetries” (Balandin et al.) • Proof of principle apochromatic solution. 2012: “Tird-order apochromatic drif-quadrupole beamline” (Balandin et al.) Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 7 Chromaticity defnitions • In beam optics, we can defne two types of chromaticity: 1 @µ @ 1 1 – Single particle (phase) chromaticity: ⇠ = = ds δ = ∆p/p 2⇡ @ @ 2⇡ β ✓ Z ◆ @↵ ↵ @ 1 @ – Beam ellipse (Twiss) chromaticity: W = A2 + B2 A = B = @ − β @ β @ p • Rings: Phase advance important (error resonances) ⇒ use ξ (and W) Linacs: Beam size/divergence is important ⇒ use W • ξ requires sextupoles for cancellation. • W does not require sextupoles (although they can be used). Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 8 Confusing nomenclature • Some confusion about what to call chromaticities and their corrected lattices: Variable Symbol Chromatic derivative Symbol Corrected lattice Beam ̻x̼ Dispersion Dx or ηx Achromatic centroid Chromaticity Phase µx > “Phase chromaticity” ξx Achromatic advance “Single particle chromaticity” Chromaticity Twiss “W-function” Wx Apochromatic βx, αx parameters > “Twiss chromaticity” Ax, Bx Achromatic “Beam ellipse chromaticity” • Not consistent with light optics… Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 9 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 10 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 11 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 12 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 13 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 14 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 15 Intuitively: What’s going on in phase space • Question: How can α, β be constant (to 1st order) when the phase advance is not? Phase space ellipse and tracked particles with varying energy offset for a linear lattice with Wx=Wy=0 • Ellipse (α, β) stagnates (W = 0) around nominal energy (� = 0). • Single particle phase advance (µ) varies with energy (ξ ≠ 0) around nominal energy (� = 0) . Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 16 Matching multiple energies PHASE SPACE (ELEGANT tracking) � �=0 �=-1% • So where do we start? x, x’ y, y’ Follow example from light optics. 0% • Can we fnd a lattice which matches two diferent energies? -0.5% • Answer: YES -1% �=0 �=-1% �=-0.5% • What about matching three energies? 0% • Again: YES -0.5% • If matched energies are close: 2 �’s: No 1st order chromaticity. 3 �’s: No 1st + 2nd order chromaticity. … etc. -1% Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 17 Alternative: Chromatic expansions • Expand α, β, µ for energy ofset �: • Goal: Shaping α(�), β(�) around � = 0. • New constraints (1st order chromaticity correction): ∂βx/∂� = ∂αx/∂� = ∂βy/∂� = ∂αy/∂� = 0 ⇒ Need 4 more degrees of freedom (quads/drifs). • Complexity O(order of �): Number of constraints increases linearly with chromatic order. • Chromaticity cancellation to nth order: 4n constraints, 4n degrees of freedom. Example (no chromatic correction) : µ(�)/2π β(�)/β0 α(�) Apochromatic Linear Lattice Design – Carl A Lindstrøm – Nov 26, 2015 18 Methodology (1): Computing chromatic derivatives 1. Defne lattice with variable quads and drifs ({k}, {d}). R({k}, {d}) Compute R-matrix. 2. Rewrite all k → k/(1+�). (Including non-variable k’s). R({k/(1+�)}, {d}) 3. Express α, β in terms of R-elements.