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 β October 28th, 2017
Outline
β’ Motivation β Brightness, emittance and photocathodes
β’ Simulations GPT in space-charge scenarios β Core brightness computation
β Simulations and analysis
β’ Experimental techniques β Pepper-pot
β TEM grids
β Slit + deflector system
β’ Summary
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Motivation
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Brightness represents the charge density in phase-space (π₯, ππ₯, π¦, ππ¦, π§, ππ§).
π΅6π· =π π
π6π· β
π
πππ₯ πππ¦ πππ§
According to Liouvilleβs theorem, phase space density for a Hamiltonian system is invariant throughout the accelerator. β’ Under linear forces: rms emittance is conserved β’ Under non-linear forces (e.g. space-charge):
- rms emittance is not conserved butβ¦ - β¦βcoreβ emittance is conserved
πππππ
ππππ
Motivation
β’ The simulations presented here aim at finding a transport invariant quantity in presence of strong space-charge forces, rather than the rms emittance.
β’ C. Gulliford et al, APL 106 - 094101 (2015): found core 2D-emittance preservation of 80-90% in DC gun-based photoinjector.
β’ Figure of merit should be found in 6D phase-space core density (difficult to verify experimentally)
β’ An invariant core phase-space density means that it contains information about the beam source: cathode thermal emittance. Experimental possibilities of new cathode physics.
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Ph
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cath
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electrons ππ₯,π¦ β MTE
π΅4π· βπ πΈπ§MTE
GPT simulations and analysis
β’ Simulations were made with General Particle Tracer (GPT) to evaluate the core-brightness as figure of merit for different transport optics:
β Drift + Solenoid + Drift
β Drift +Solenoid + Drift + RF-LINAC + Drift
β’ Initial particle distribution: full 6D gaussian
β Energy 5 MeV
β Total charge 1 pC
β Beam size 100x100x300 um
β Transverse normalized emittance 10 nm rad
β Energy spread 0.01%
β Number of simulated macro-particles: 5000 β 25000
β’ Space-charge is implemented with GPT built-in routines:
β spacecharge3Dmesh: solves Poissonβs equations on a mesh adapted to the beam geometry, with coordinates and fields in rest frame.
β spacecharge3D: point-to-point particle relativistic interaction.
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Core-brightness computation
π β‘ π β π0β€ Ξ£β1 π β π0
Coordinates π =
π₯1π₯2β―π₯6
Covariance matrix Ξ£ = β¨ π β β¨πβ© (π β β¨πβ©)β€β©
π₯1 β‘ π₯ [π]
π₯2 β‘ππ₯π0π
= πΎπ½π₯
π₯3 β‘ π¦ [π]
π₯4 β‘ππ¦
π0π= πΎπ½π¦
π₯5 β‘ π§ [π]
π₯6 β‘ππ§π0π
= πΎπ½π§
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 βcoreβ.
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Core-brightness computation β’ Blue: all particles β’ Red line: contour full distribution β’ Pink: filtered particles (4%) β’ Green line: contour filtered distribution
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Core-brightness computation
Volume is proportional to π 6 and it is used as a fraction factor of the rms emittance of the full distribution:
π6π·(πsubset) β π6π·,πππ ππ’ππ
β πΉπ
Volume is calculated as the rms emittance of the subset of smallest π that contains πsubset particles
π6π· πsubset β π6π·,πππ π π’ππ ππ‘
= det Ξ£π π’ππ ππ‘
The core-brightness π©ππ¨π«π is extrapolated to the center (π β 0) with a linear fit between the number of particles and the volume occupied in 6D phase space.
π΅core βπππ
ππ6π· π6π·β0 πsubset π6π· = π©ππ¨π«π β π6π· + π π6π·
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Method 1: fraction πΉπ Method 2: subset emittance
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Core-brightness computation
The center of the distribution is not clearly defined. We compare the microscopic density at the average position π 6π· = ( π₯ , ππ₯ , π¦ , ππ¦ , π§ , β¨ππ§β©) 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
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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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.
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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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 π 6π· 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
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge MESH routine
Random initial distribution
Hammersley sampling
rms emittance x40
Analysis: The first 1000 macro-particles with respect to average π 6π· of the first frame are tracked to compute the core brightness.
π΅ππππ well preserved
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge POINT-to-POINT routine
β’ (blue) π 6π· of each frame β’ (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
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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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
β’ Hammersley sampling β’ Track of the filtered
1000 particles in the first frame
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge MESH routine
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
π΅ππππ x0.0001
π6π·,πππ
Experimental techniques
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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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
Experimental techniques
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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TEM grid: 4D transverse phase-space
4D transverse phase space density reconstruction
Analysis of the βcoreβ emittance using reconstructed phase space
Projected 2D phase-space
Experimental techniques
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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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.
π₯2 = πΌπππ₯ππ
2ππ¦π=1
ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
π₯π₯β² = πΌπππ₯ππ
ππ¦π=1
π₯β²ππππ₯π=1
πΌππππ¦π=1
ππ₯π=1
π₯β²2 =
πΌππ π₯ππβ² 2 + ππ₯ππ
β²2ππ¦
π=1ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
erfπ β π₯π βππ₯π/2
2πΏππ₯β²βerf
π β π₯π +ππ₯π/2
2πΏππ₯β²
π₯β²π¦β² =
πΌππ π₯β²ππ π¦β²ππ + ππ₯β²π¦β²ππ2ππ¦
π=1ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
Correlation terms:
J. Giner-Navarro, D. Marx, P. Musumeci
π₯β²ππ =π₯πππ πππππ
πΏπππππ‘
Experimental techniques
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Simulation
Slit + Deflector: Longitudinal phase-space
XTCAV
TEM Grid
Slit (10 ππ)
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
β’ Numerical simulations have been performed using GPT to analyse the evolution of the βcoreβ 6D-brightness in beam transport systems in presence of non-linear space-charge forces.
β’ 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.
β’ 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.
β’ 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.
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Thank you for your attention!
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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BACK UP Slides:
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Drift + Solenoid + Drift
First check: no space-charge forces! β’ Rms emittance is doubled as it enters
inside the solenoid but is back to initial value at the output.
β’ Core brightness keeps constant as expected.
Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm
Simulations and analysis
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Drift + Solenoid + Drift Solenoid: Main body field 0.8T, Length 50cm, Aperture 5cm Space-Charge MESH routine
Reference: π 6π· of each frame First 1000 particles of the first frame
To consider statistical density in the vicinities of the βcore centerβ, we can take an average of the fitted brightness (first 20 particles here).
Daniel Marx | Center for Bright Beams Meeting | July 26, 2017 | Page 25
Technique
π₯β²2 =
πΌππ π₯ππβ² 2 + ππ₯ππ
β²2ππ¦
π=1ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
π π =π
2πΏππ₯β²
π
22 + erf
π β π₯π βππ₯π/2
2πΏππ₯β²βerf
π β π₯π +ππ₯π/2
2πΏππ₯β²
Daniel Marx | Center for Bright Beams Meeting | July 26, 2017 | Page 26
Technique
π°ππ« πΏ,π = π πβ
π/π
π π π¨ β ππ±π© βπ
π π β ππβ²πβ²π
πΏβπ΄ππ βπ΄ππππ
ππβ²π³π
+πβπ΄πππ βπ΄ππ
π
ππβ²π³π β
πππβ²πβ²
ππβ²ππβ²π³ππΏβπ΄ππβπ΄πππ π βπ΄πππ βπ΄ππ
β
π/π
ππ₯β²π¦β² = 0
ππ₯ = 25 ππ¦ = 15 ππ₯π¦ = 0 ππ¦π₯ = 0
ππ₯β²π¦β² =ππ₯β²π¦β²
ππ₯β²ππ¦β²
ππ₯β²π¦β² = 0
ππ₯ = 25 ππ¦ = 15 ππ₯π¦ = π ππ¦π₯ = π
π₯β²π¦β² =
πΌππ π₯β²ππ π¦β²ππ + ππ₯β²π¦β²ππ2ππ¦
π=1ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
ππ₯β²π¦β² = π. π
ππ₯ = 25 ππ¦ = 15 ππ₯π¦ = π ππ¦π₯ = π
Daniel Marx | Center for Bright Beams Meeting | July 26, 2017 | Page 27
Technique
π₯β²π¦β² =
πΌππ π₯β²ππ π¦β²ππ + ππ₯β²π¦β²ππ2ππ¦
π=1ππ₯π=1
πΌππππ¦π=1
ππ₯π=1
We are fitting a range of parameters for π and looking for π2 minimum.
Problem is π2 has very shallow minimum.
Experimental techniques
Oct 28th, 2017 J.Giner-Navarro. Workshop: Methods in Collective and Nonlinear Effects in Bright Charged Particle Beams.
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Simulation
TEM grid: 4D transverse phase-space
ASTRA Analysis
routine
ππ₯norm 4.49e-8 3.99e-8
ππ¦norm 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
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