Interstage Optics Design for a PWFA Linear Collider FACETII Workshop - - PowerPoint PPT Presentation

FACETII Workshop 2015 – Oct 14, 2015


Carl A Lindstrøm

PhD Student University of Oslo, Department of Physics Advisor: Erik Adli

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Interstage Optics Design

for a PWFA Linear Collider

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Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Proposed layout of a PWFA Linear Collider*

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* E Adli, JP Delahaye, et al. (presented at IPAC’13)

• Basis for this study + source of parameters.

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Interstage optics

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• Problem at hand:

Switch old with fresh drive beam, keep the main beam focused and preserve its emittance.

Plasma Plasma

Beam dump Kicker system DELAY CHICANE for drive beam

INTERSTAGE
 Yet undefined system of magnetic optics

TRANSFER LINE

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Formal requirements

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• Drive beam injection/extraction
• Collinearity
• Beta function matching
• Dispersion cancellation
• Limit bunch lengthening (R56)
• Emittance preservation


– limit chromaticity
 – limit synchrotron radiation

• Minimize length subject to all the above

βx(L) = βy(L) = βmat αx(L) = αy(L) = 0 Dx(L) = D0

x(L) = 0

R56(L) ⌧ σz δ ⇡ 1 mm ∆✏ ✏ (L) ⌧ 1%

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Injection/extraction of drive beams

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• Using dipoles to create dispersion:


Separate beams spatially by energy.

• Injection and extraction are 


symmetric processes ⇒ mirror symmetric lattice.

• Injection/extraction dipoles 


must be fjrst and last magnets, 
 as main beam quads would destroy the drive beam.

• Important: Dipoles do not scale with main beam energy

(only drive beam energy).

• Defjnes regimes:


– Low energy (Emain ≈ Edrive) ⇒ Dipoles “visible”
 – High energy (Emain >> Edrive) ⇒ Dipoles “invisible”

• Scalings:


– Dipole fjeld & length: Ldipole, B ~ constant 
 – SR power: PSR ~ Emain

2


– Main beam dispersion: Dx ~ Linterstage/Emain ~ 1/√Emain
 (assuming focusing ⇒ Linterstage ~ √Emain)

PLASMA PLASMA MAIN BEAM

−θm

θm

plasma

D R I V E B E A M MAIN BEAM

SYMMETRY LINE

Dipole comes first Defocusing quadrupole

PLASMA PLASMA DRIVE BEAM TRAIN MAIN BEAM DRIVE BEAM TRAIN

Option 2: “S-chicane" 


Stronger bending, more space for beam dump.

Option 1: “C-chicane”


Weaker bending

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 6

Dispersion cancellation

• Both S- and C-chicane has inherent dispersion cancellation. Quadrupoles change this.
• 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.

• Becomes less important with higher energies: Scaling: Dx(s) ~ 1/√Emain

Limiting bunch lengthening (R56)

• Not big problem due to relatively weak dipoles.
• Becomes easier with higher energy.


Scaling: R56(s) ~ Dx(s) ~ 1/√Emain

• If necessary, a method for matching R56 ≈ 0: 


– Low dispersion in dipoles.

R56(L) ⌧ σz δ ⇡ 1 mm

Dx without quads Dx with quads R56 with quads

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Matching beam to plasma

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• Matched beta in a plasma: ⇒ 3.3 cm @ E = 100 GeV, np = 1016 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 fjnal focus system, for every stage.

Plasma cell Plasma cell

QUADRUPOLE STRUCTURE

Beam envelope

βx(L) = βy(L) = βmat αx(L) = αy(L) = 0

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 8

Chromaticity cancellation

• Small beams in plasma lead to large chromaticity.
• PWFALC study assumes a ~1% rms. energy spread.
• Conventionally corrected using sextupoles. 


Seen in fjnal focus systems (SLC, ILC…): very long lattices.

• We will consider a sextupole solution, and a novel solution.

Traditional FF @ 500GeV ~ 1750 m Improved FF @ 500GeV ~ 300 m

*Figures from “Beam Delivery & beam-beam” by Andrei Seryi (SLAC)

Initial phase space (x, x’)

No chromaticity correction: Emittance increases by 
 many orders of magnitude

Example:

Final phase space 


(no energy spread)

Final phase space 


(1% energy spread)

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Plasma density ramp

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• A plasma density ramp mitigates the chromaticity problem.
• Plasma ramp ⇒ larger/less diverging beam 


⇒ smaller beam in quadrupoles ⇒ less chromaticity

• Any adiabatic density profjle will do. Use pressure gradients or partial laser-ionization.
• Scaling (for magnifjcation factor ∏p): Lramp ~ ∏p √Emain

Plasma Plasma ramp

Uniform plasma density Varying
 plasma density Beam envelope * Plasma ramp paper by Xu et al. (2015): 
 http://arxiv.org/pdf/1411.4386v2.pdf

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 10

• Sextupole efgect is stronger with:


– Larger dispersion 
 – Larger beam size

• Long lattices for geometric term cancellation.
• Introduces new problems:



 – Dipoles must ramp with main beam energy 
 ⇒ Dispersion / R56 / SR scales poorly with energy
 – -I transforms require repeated sections 
 ⇒ “unnecessarily” long lattices
 – Tiick sextupoles (imperfect -I transforms)
 ⇒ geometric errors (emittance growth)
 – Sextupoles need large beam sizes
 ⇒ increased energy spread from SR (Oide efgect)

Conventional solution: Sextupoles

y0

y0 y0

0 + ∆y0

sextupole y0

• I transfer

matrix

sextupole

Bx ∼ xy + δDxy

By ∼ 1 2(x2 − y2) + xδDx + 1 2δ2D2

x

Linear chromatic terms CORRECT CHROMATICITY Non-linear chromatic term NEED TO BE CANCELLED Non-linear geometric terms NEEDS TO BE SMALL

Sextupole B-fields: Geometric term cancellation:

0.0 5.0 10.0 15.0 20.0 25.0 0.0 20. 40. 60. 80. 100. 120. 140. 160.

β x β y

• I transform
• I transform
• I transform

“Working” interstage using sextupoles:

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015 11

“Working” interstage using sextupoles

• Tiis solution requires stronger sextupoles than currently manufacturable. Not a conceptual show-stopper.

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Beta functions Dispersion W-functions (chrom.) Second order dispersion R56 Footprint (x-z)

0.0 5.0 10.0 15.0 20.0 25.0 30.0

s (m) inex

0.0 20. 40. 60. 80. 100. 120. 140. 160.

β x β y

0.0 5.0 10.0 15.0 20.0 25.0 30.0

s (m) inex

• 0.0100
• 0.0075
• 0.0050
• 0.0025

0.0 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150

Dx Dy

0.0 5.0 10.0 15.0 20.0 25.0 30.0

s (m) inex

0.0 50. 100. 150. 200. 250. 300. 350. 400. 450. 500.

Wx Wy 0.0 5.0 10.0 15.0 20.0 25.0 30.0

z inex

• 0.08
• 0.07
• 0.06
• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.0

x

x

0.0 5.0 10.0 15.0 20.0 25.0 30.0

s (m) inex

• 0.060
• 0.055
• 0.050
• 0.045
• 0.040
• 0.035
• 0.030
• 0.025
• 0.020
• 0.015
• 0.010
• 0.005

0.0

re56 [*10**( -3)]

re56

0.0 5.0 10.0 15.0 20.0 25.0 30.0

s (m) inex

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.0 0.02 0.04 0.06 0.08 0.10

Dx’ Dy’

100 GeV, ßmat = 0.1m

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Novel solution: Getting rid of sextupoles

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• We are developing a new method for fjnding

quadrupole-only lattices which cancel chromaticity 
 to the required order in energy ofgset.

• Benefjts of using quadrupoles only:


– no geometric terms ⇒ keeps it linear
 – 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 ofgset (δ) 


expansion of β and α to be fmat around δ = 0.

Simpler version used in light optics: “Superachromat” by Carl Zeiss β and α vs. energy offset δ. 
 Flat regions ⇒ no chromaticity

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Examples of chromaticity-free quadrupole lattices

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• 8 quads: cancel chromaticity to 1st order.
• 12 quads: cancel chromaticity to 2nd order.

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Length estimates and scalings

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• Any beta matching imposes:
• High energy regime: retains beta-shape, constant emittance

preservation, good synch. rad. scaling (PSR ~ Emain

2)

• Low energy regime: complex lattices/many quads 


(high chrom. correction order), possibly use of sextupoles.

• Interstage length estimate: ~30 m @ 300 GeV, 


(E-spread: 1% rms, dipole length: 1m, plasma ramp: 10x, 
 quads: 150 T/m, emit. growth: ~1%, plasma density: 1016 cm-3)

• Note: work in progress.

Emittance growth vs. Energy Interstage length vs. Energy

⇒

~E0.5 ~const

Low High Low High Current best solution (9 quads):

Energy spreads:

Approximate scalings for high energy regime:

Interstage length Emittance growth Energy

Emain0.5 const

Quad strength

gmax-0.5 gmax-1.5

Energy spread

(1st order chromaticity correction)

const σE4

Ramp magnification

const ∏p-3

Interstage Optics Design – Carl A Lindstrøm – Oct 14, 2015

Summary

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• Chromaticity is a big challenge facing a PWFA interstage.
• Traditional chromaticity correction designs (using sextupoles) have

unfavourable energy scaling laws.

• A new type of lattice: 


Chromaticity-free quadrupole-only lattices have been developed
 (shorter, less SR, no non-linear terms).

• At high energies, same optics solution applies to any energy 


(length scales, emittance growth is constant)

• PWFA/LWFA interstage and collider lengths will scale as:
• Next up: 

• Integration of plasma density ramps

• Emittance growth studies from SR/misalignment

• Details of injection/extraction optics