core brightness in presence of space-charge...effects in bright charged particle beams. 3 brightness...

Core brightness in presence of space-charge

Jorge Giner-Navarro Pietro Musumeci

Workshop on Methods in Collective and Nonlinear Effects in

Bright Charged Particle Beams University of Chicago – October 28th, 2017

Outline

• Motivation – Brightness, emittance and photocathodes

• Simulations GPT in space-charge scenarios – Core brightness computation

– Simulations and analysis

• Experimental techniques – Pepper-pot

– TEM grids

– Slit + deflector system

• Summary

Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.

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Motivation


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Brightness represents the charge density in phase-space (𝑥, 𝑝𝑥, 𝑦, 𝑝𝑦, 𝑧, 𝑝𝑧).

𝐵6𝐷 =𝑁 𝑒

𝑉6𝐷 ∝

𝑄

𝜀𝑛𝑥 𝜀𝑛𝑦 𝜀𝑛𝑧

According to Liouville’s theorem, phase space density for a Hamiltonian system is invariant throughout the accelerator. • Under linear forces: rms emittance is conserved • Under non-linear forces (e.g. space-charge):

- rms emittance is not conserved but… - …“core” emittance is conserved

𝜀𝑐𝑜𝑟𝑒

𝜀𝑟𝑚𝑠

Motivation

• The simulations presented here aim at finding a transport invariant quantity in presence of strong space-charge forces, rather than the rms emittance.

• C. Gulliford et al, APL 106 - 094101 (2015): found core 2D-emittance preservation of 80-90% in DC gun-based photoinjector.

• Figure of merit should be found in 6D phase-space core density (difficult to verify experimentally)

• An invariant core phase-space density means that it contains information about the beam source: cathode thermal emittance. Experimental possibilities of new cathode physics.


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x

px

Ph

oto

cath

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-Ez

electrons 𝜀𝑥,𝑦 ∝ MTE

𝐵4𝐷 ∝𝑄 𝐸𝑧MTE

GPT simulations and analysis

• Simulations were made with General Particle Tracer (GPT) to evaluate the core-brightness as figure of merit for different transport optics:

– Drift + Solenoid + Drift

– Drift +Solenoid + Drift + RF-LINAC + Drift

• Initial particle distribution: full 6D gaussian

– Energy 5 MeV

– Total charge 1 pC

– Beam size 100x100x300 um

– Transverse normalized emittance 10 nm rad

– Energy spread 0.01%

– Number of simulated macro-particles: 5000 – 25000

• Space-charge is implemented with GPT built-in routines:

– spacecharge3Dmesh: solves Poisson’s equations on a mesh adapted to the beam geometry, with coordinates and fields in rest frame.

– spacecharge3D: point-to-point particle relativistic interaction.


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Simulations and analysis


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Core-brightness computation

𝑅 ≡ 𝑟 − 𝑟0⊤ Σ−1 𝑟 − 𝑟0

Coordinates 𝑟 =

𝑥1𝑥2⋯𝑥6

Covariance matrix Σ = ⟨ 𝑟 − ⟨𝑟⟩ (𝑟 − ⟨𝑟⟩)⊤⟩

𝑥1 ≡ 𝑥 [𝑚]

𝑥2 ≡𝑝𝑥𝑚0𝑐

= 𝛾𝛽𝑥

𝑥3 ≡ 𝑦 [𝑚]

𝑥4 ≡𝑝𝑦

𝑚0𝑐= 𝛾𝛽𝑦

𝑥5 ≡ 𝑧 [𝑚]

𝑥6 ≡𝑝𝑧𝑚0𝑐

= 𝛾𝛽𝑧

Normalized distance: defines a distance with respect to the center or reference (r0) taking into account the geometry and orientation of the distribution in the 6D phase space. This distance is used to sort and filter the particles of the “core”.



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Core-brightness computation • Blue: all particles • Red line: contour full distribution • Pink: filtered particles (4%) • Green line: contour filtered distribution



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Volume is proportional to 𝑅6 and it is used as a fraction factor of the rms emittance of the full distribution:

𝑉6𝐷(𝑛subset) ∝ 𝜀6𝐷,𝑟𝑚𝑠𝑓𝑢𝑙𝑙

⋅ 𝑹𝟔

Volume is calculated as the rms emittance of the subset of smallest 𝑅 that contains 𝑛subset particles

𝑉6𝐷 𝑛subset ∝ 𝜀6𝐷,𝑟𝑚𝑠𝑠𝑢𝑏𝑠𝑒𝑡

= det Σ𝑠𝑢𝑏𝑠𝑒𝑡

The core-brightness 𝑩𝐜𝐨𝐫𝐞 is extrapolated to the center (𝑅 → 0) with a linear fit between the number of particles and the volume occupied in 6D phase space.

𝐵core ∝𝑑𝑁𝑝

𝑑𝑉6𝐷 𝑉6𝐷→0 𝑛subset 𝑉6𝐷 = 𝑩𝐜𝐨𝐫𝐞 ⋅ 𝑉6𝐷 + 𝜗 𝑉6𝐷

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Method 1: fraction 𝑹𝟔 Method 2: subset emittance



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The center of the distribution is not clearly defined. We compare the microscopic density at the average position 𝑟 6𝐷 = ( 𝑥 , 𝑝𝑥 , 𝑦 , 𝑝𝑦 , 𝑧 , ⟨𝑝𝑧⟩) and at the neighboring

particles. We can take either the maximum density or an average.

Particle distribution with respect to neighboring particles

Maximum density at R=0



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Initial distribution

No sampling With Hammersley sampling

From GPT User Manual

One remark, one may consider relevant for the space-charge models and density analysis that the finite number of simulated particles (<0.1%) requires a suitable sampling to minimize statistical errors. GPT includes Hammersley sequences that generates a quasi-random initial 6D distribution.



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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge MESH routine

rms emittance x40

~70% core brightness preserved

Analysis: The average 𝑟 6𝐷 is considered here as the center of the distribution in order to calculate normalized distances. The maximum local core brightness is taken among a small subset in the center.

Random initial distribution

Hammersley sampling



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Random initial distribution

Hammersley sampling

rms emittance x40

Analysis: The first 1000 macro-particles with respect to average 𝑟 6𝐷 of the first frame are tracked to compute the core brightness.

𝐵𝑐𝑜𝑟𝑒 well preserved



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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge POINT-to-POINT routine

• (blue) 𝑟 6𝐷 of each frame • (red) track same 1000

particles of the first frame

Jumps in rms emittance at the waist of the beam (~10um) as evidence of very strong space-charge forces. Core brightness calculations follow the same jumps: no preservation.

Loss core brightness to 4%

(Jump x4)

rms emittance x130



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Drift + Solenoid + Drift + Linac + Drift

- Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm - Linac: PEGASUS linac field map (S-band, SW, 10-cell) Space-Charge MESH routine rms emittance x70

RF Linac does not perturb core emittance.

Solenoid Linac

• Hammersley sampling • Track of the filtered

1000 particles in the first frame



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Higher charge (x100) Stronger space charge forces!

Core brightness drops dramatically even at drift sections. Bad discretization (more charge means more particles)

rms emittance x2000

𝐵𝑐𝑜𝑟𝑒 x0.0001

𝜀6𝐷,𝑟𝑚𝑠

Experimental techniques


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Simulation

In order to measure the core density we need an experimental technique to reconstruct the phase space density. - Pepper-pot and TEM grid techniques allow the reconstruction of the transverse

phase space density (4D)

R.K. Li et al, PRSTAB 15, 090702 (2012)

Pepper-pot TEM grid



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TEM grid: 4D transverse phase-space

4D transverse phase space density reconstruction

Analysis of the “core” emittance using reconstructed phase space

Projected 2D phase-space



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Simulation

TEM grid: 4D transverse phase-space We developed analysis algorithms to reconstruct 4D emittance (including correlations) from TEM grid images.

𝑥2 = 𝐼𝑖𝑗𝑥𝑖𝑗

2𝑛𝑦𝑗=1

𝑛𝑥𝑖=1

𝐼𝑖𝑗𝑛𝑦𝑗=1

𝑛𝑥𝑖=1

𝑥𝑥′ = 𝐼𝑖𝑗𝑥𝑖𝑗

𝑛𝑦𝑗=1

𝑥′𝑖𝑗𝑛𝑥𝑖=1


𝑛𝑥𝑖=1

𝑥′2 =

𝐼𝑖𝑗 𝑥𝑖𝑗′ 2 + 𝜎𝑥𝑖𝑗

′2𝑛𝑦

𝑗=1𝑛𝑥𝑖=1


𝑛𝑥𝑖=1

erf𝑋 − 𝑥𝑏 −𝑀𝑥𝑎/2

2𝐿𝜎𝑥′−erf

𝑋 − 𝑥𝑏 +𝑀𝑥𝑎/2

2𝐿𝜎𝑥′

𝑥′𝑦′ =

𝐼𝑖𝑗 𝑥′𝑖𝑗 𝑦′𝑖𝑗 + 𝜎𝑥′𝑦′𝑖𝑗2𝑛𝑦



𝑛𝑥𝑖=1

Correlation terms:

J. Giner-Navarro, D. Marx, P. Musumeci

𝑥′𝑖𝑗 =𝑥𝑖𝑗𝑠𝑐𝑟𝑒𝑒𝑛

𝐿𝑑𝑟𝑖𝑓𝑡



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Simulation

Slit + Deflector: Longitudinal phase-space

XTCAV

TEM Grid

Slit (10 𝜇𝑚)

J.Maxson, D. Cesar, P. Musumeci (2016)

~4.5 ps

High charge transmission and single-shot 4D reconstruction allows the extension to 6D phase-space using the slit+deflector system. A slice of the beam is striked to measure the temporal distribution.

Summary

• Numerical simulations have been performed using GPT to analyse the evolution of the “core” 6D-brightness in beam transport systems in presence of non-linear space-charge forces.

• Mesh routines of space-charge forces show fair agreement of core brightness preservation, rather than point-to-point interaction routines which demand larger number of particles and computation time.

• Ultimate goal is the use of this invariant for the characterization of new photocathodes from the analysis of the produced beam properties in the diagnostics section.

• The use of TEM grids combined with a slit-deflector system is presented to be a good candidate for phase space density reconstruction and analysis of the core 6D brightness.


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Thank you for your attention!


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BACK UP Slides:


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Drift + Solenoid + Drift

First check: no space-charge forces! • Rms emittance is doubled as it enters

inside the solenoid but is back to initial value at the output.

• Core brightness keeps constant as expected.

Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm



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Reference: 𝑟 6𝐷 of each frame First 1000 particles of the first frame

To consider statistical density in the vicinities of the “core center”, we can take an average of the fitted brightness (first 20 particles here).

Daniel Marx | Center for Bright Beams Meeting | July 26, 2017 |

Technique

𝑥′2 =

𝐼𝑖𝑗 𝑥𝑖𝑗′ 2 + 𝜎𝑥𝑖𝑗

′2𝑛𝑦



𝑛𝑥𝑖=1

𝑓 𝑋 =𝜌

2𝐿𝜎𝑥′

𝜋

22 + erf

𝑋 − 𝑥𝑏 −𝑀𝑥𝑎/2

2𝐿𝜎𝑥′−erf

𝑋 − 𝑥𝑏 +𝑀𝑥𝑎/2

2𝐿𝜎𝑥′


Technique

𝑰𝟐𝑫 𝑿,𝒀 = 𝒅𝒙∞

𝒂/𝟐

𝒅𝒚 𝑨 ⋅ 𝐞𝐱𝐩 −𝟏

𝟐 𝟏 − 𝝆𝒙′𝒚′𝟐

𝑿−𝑴𝒙𝒙 −𝑴𝒙𝒚𝒚𝟐

𝝈𝒙′𝑳𝟐

+𝒀−𝑴𝒚𝒙𝒙 −𝑴𝒚𝒚

𝟐

𝝈𝒚′𝑳𝟐 −

𝟐𝝆𝒙′𝒚′

𝝈𝒙′𝝈𝒚′𝑳𝟐𝑿−𝑴𝒙𝒙−𝑴𝒙𝒚𝒚 𝒀 −𝑴𝒚𝒙𝒙 −𝑴𝒚𝒚

∞

𝒂/𝟐

𝜌𝑥′𝑦′ = 0

𝑀𝑥 = 25 𝑀𝑦 = 15 𝑀𝑥𝑦 = 0 𝑀𝑦𝑥 = 0

𝜌𝑥′𝑦′ =𝜎𝑥′𝑦′

𝜎𝑥′𝜎𝑦′

𝜌𝑥′𝑦′ = 0

𝑀𝑥 = 25 𝑀𝑦 = 15 𝑀𝑥𝑦 = 𝟐 𝑀𝑦𝑥 = 𝟐

𝑥′𝑦′ =




𝑛𝑥𝑖=1

𝜌𝑥′𝑦′ = 𝟎. 𝟓

𝑀𝑥 = 25 𝑀𝑦 = 15 𝑀𝑥𝑦 = 𝟐 𝑀𝑦𝑥 = 𝟐


Technique

𝑥′𝑦′ =




𝑛𝑥𝑖=1

We are fitting a range of parameters for 𝜌 and looking for 𝜒2 minimum.

Problem is 𝜒2 has very shallow minimum.



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Simulation

TEM grid: 4D transverse phase-space

ASTRA Analysis

routine

𝜖𝑥norm 4.49e-8 3.99e-8

𝜖𝑦norm 4.49e-8 3.97e-8

Reconstruction in ASTRA simulations:

Energy 3.05 MeV, Bunch charge 1 pC, TEM grid: 83um pitch/ 25um bar width

D. Marx

Transverse beam emittance measurements at Pegasus beamline (Aug 2017) Oblique incidence (elliptical) TEM grid (300): 54um pitch/ 31um bar width

core brightness in presence of space-charge...effects in bright charged particle beams. 3 brightness...

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