1. High quality plasma wakefield acceleration experiment in linear regime at SPARC_LAB
Stefano Romeo on behalf of SPARC_LAB collaboration
European Advanced Accelerator Concepts, La Biodola 26/09/2017

2. Blow-out regime 𝜶≫𝟏 𝐸 𝑟 = 𝑛 0 2 𝜀 0 𝑟
There is not a complete analytical solution of the fields inside plasma
Ion column model
1 𝑟 𝜕 𝜕𝑟 𝑟( 𝐸 𝑟 −𝑐 𝐵 𝜃 ) = 𝜌 𝜀 0 − 𝜇 0 𝑐 𝐽 𝑧
Since inside bubble there is no current, the transverse field can be derived from electrostatic component only (cylinder with a constant charge density 𝑛 0 )
𝐸 𝑟 = 𝑛 0 2 𝜀 0 𝑟
Blow out field
In the position of the bubble closing there is a spike in the accelerating field
Linear dependency of the focusing
Accelerating field doesn’t depend on transverse position
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

3. Linear regime 𝒏 𝒃 𝒏 𝟎 =𝜶≪𝟏 𝑊 𝑧,𝑟 =𝑒 𝐸 + 𝑣 × 𝐵 𝑧,𝑟
𝒏 𝒃 𝒏 𝟎 =𝜶≪𝟏
Linear field
In linear approximation it is possible to derive an analytical solution for the fields
It is possible to inject a beam in the crest region (no focusing)
Non linear dependency of the focusing
Accelerating field depends on transverse position
Lower field
𝑊 𝑧,𝑟 =𝑒 𝐸 + 𝑣 × 𝐵 𝑧,𝑟
𝐸 𝑧 𝑟,𝜉 =𝑅 𝑟 𝑍 ′ 𝜉
𝐸 𝑟 𝑟,𝜉 = 𝑅 ′ 𝑟 𝑍 𝜉
𝑍 𝜉 = 2𝜋 𝑚 𝑒 𝑐 2 𝑒 𝛼 𝑘 𝑝 𝜎 𝑧 𝑒 − 𝑘 𝑝 2 𝜎 𝑧 2 /2 𝑠𝑖𝑛 𝑘 𝑝 𝜉
𝑍 ′ (𝜉)= 2𝜋 𝑚 𝑒 𝑐 2 𝑒 𝛼 𝑘 𝑝 2 𝜎 𝑧 𝑒 − 𝑘 𝑝 2 𝜎 𝑧 2 /2 𝑐𝑜𝑠 𝑘 𝑝 𝜉
𝑅 0 = 𝑘 𝑝 2 𝜎 𝑟 𝑒 𝑘 𝑝 2 𝜎 𝑧 𝛤 0, 𝑘 𝑝 2 𝜎 𝑟 2 2
𝑅 𝑟 =𝑅 0 − 𝑘 𝑝 𝑟 2 + 𝑘 𝑝 𝑟 4 +…
𝑅 ′ 𝑟 =− 𝑘 𝑝 2 2 𝑟+ 𝑘 𝑝 𝑟 3 +…
𝑘 𝑝 = 2𝜋 𝜆 𝑝 = 𝑒 2 𝑛 0 𝜀 0 𝑚 𝑒 𝑐 2
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

4. High quality acceleration in linear regime
Transverse contribution (sliced)
𝐶 𝑆 =− 𝑒 𝑘 𝑝 2 𝑍 𝜉 𝐿 8 𝜎 𝑟,𝐷 2 𝛾 𝑚 𝑒 𝑐 2
Spherical aberration coefficient
𝜎 𝐸,𝑓 = 𝜎 𝐸,𝑖 − 𝛾 0 𝛾 tan 2 𝜙 0 𝑘 𝑝 2 𝜎 𝑧,𝑤 𝑘 𝑝 4 𝜎 𝑧,𝑤 𝑘 𝑝 4 𝜎 𝑟,𝑤 𝑅 2 (0) − 𝑘 𝑝 4 𝜎 𝑟,𝑤 2 𝜎 𝑧,𝑤 2 4𝑅 (0)
Bunch length
LT correlation
Chirp
Energy spread growth
High quality requires:
𝜎 𝑟,𝑤 ≪ 𝜎 𝑟,𝐷
𝑘 𝑝 𝜎 𝑧,𝑤 ≪1
In many realistic cases, beam loading is not negligible
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

5. BLAST scheme 𝛽 𝑥 = 2 𝛾 𝑘 𝑝 𝜎 𝑥 = 4 𝛾Λ𝒵 𝜀 𝑛 𝜎 𝑥 = 4 1 𝛾 2 𝜀 𝑛 𝑘 𝑝
The driver generates a linear field
Witness can be injected on crest
The field generated by the witness is non linear
Focusing is guaranteed by a beam loading effect
𝜎 𝑥 = 4 𝛾Λ𝒵 𝜀 𝑛
𝜎 𝑥 = 4 1 𝛾 𝜀 𝑛 𝑘 𝑝
𝛽 𝑥 = 2 𝛾 𝑘 𝑝
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

6. Energy spread growth 𝜕 𝐸 𝑧 𝜕𝜉 =0 Beam loading compensation
The longitudinal field generated by a low charge high density bunch can be described by linear equations
Barov, Nick, et al. "Energy loss of a high-charge bunched electron beam in plasma: Analysis." Physical Review Special Topics-Accelerators and Beams 7.6 (2004):
The beam loading compensation occurs when the first derivative of the accelerating field in the center of the witness is 0
Chiou, T. C., and T. Katsouleas. "High beam quality and efficiency in plasma-based accelerators." Physical review letters 81.16 (1998): 3411.
From the analytical model of field it is possible to evaluate the energy spread
LCLS, CDR, ch07. SLAC
𝜕 𝐸 𝑧 𝜕𝜉 =0
sin 𝜙 0 =− 𝑘 𝑝 𝑍 𝑤 0 𝑅 𝑤 𝐸 0
Beam loading compensation
For any witness in a great range of parameters it’s possible to find a bunch separation that guarantees beam loading compensation
𝜎 𝐸,𝑓 = 𝜎 𝐸,𝑖 − 𝛾 0 𝛾 𝑘 𝑝 4 𝜎 𝑧,𝑤 4 4
Energy spread growth depends only on witness length
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

7. BLAST scheme transverse matching
A correlated focusing on longitudinal dimension causes emittance growth
𝜀 𝑛,𝑐 = 𝑘 2 𝜎 𝑥 2 𝐿 𝑐 𝑓 2 𝜉
Hypotesis of energy spread compensation using beam loading
(NO BUNCH SHAPING)
𝜕 𝐸 𝑧 𝜕𝜉 =0
𝜕( 𝐸 𝑟 −𝑐 𝐵 𝜃 ) 𝜕𝜉 =0
Hypotesis of quasi-linearity
𝐽 𝑧 =0
Lu, W., et al. "A nonlinear theory for multidimensional relativistic plasma wave wakefields a." Physics of Plasmas 13.5 (2006):
𝜕𝜌 𝜕𝑟,𝜉 =0
𝜕( 𝐸 𝑟 −𝑐 𝐵 𝜃 ) 𝜕𝜉 =0
Consequences
𝐽 𝑟 =0
𝜎 𝑚 = 4 1 𝛾 𝜀 𝑛 𝑘 𝑝
𝑛= 𝑛 0 2
Ion column model can be applied
Stupakov, G., et al. "Wake excited in plasma by an ultrarelativistic pointlike bunch." Physical Review Accelerators and Beams (2016):
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

8. Working point parameters
SPARC_LAB Working Point
Driver
Injection
Witness
200
10.3
1.26
37.2 3
0.1 17
0.3 10
0.68
0.42
1.2
0.06
2⋅ 10 16 2 5 𝛾
𝜎 𝑟 [μm]
𝜎 𝑧 [μm]
𝜎 𝐸 [%]
𝜀 𝑛 [μm]
𝑄[pC]
𝐼[kA] 𝑄
𝑛 0 [ cm −3 ]
𝐸 𝑧 [GV/m]
Δ𝑠[cm]
𝝀 𝒑 ≈𝟐𝟑𝟓 𝛍𝐦
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

9. Hybrid Tool: Architect
Bunch(es) are treated kinetically
background plasma as a fluid
cylindrical symmetry assumed
run time figure of merit: 1cm/h
no-Quasi Static Approximation
Code reliability respect to full PIC code showed both for linear and quasi-linear regimes
Massimo, F., S. Atzeni, and A. Marocchino. "Comparisons of time explicit hybrid kinetic-fluid code Architect for Plasma Wakefield Acceleration with a full PIC code. " Journal of Computational Physics 327 (2016):
Marocchino, A., et al. "Efficient modeling of plasma wakefield acceleration in quasi-non-linear-regimes with the hybrid code Architect.“ Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 829 (2016):
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

10. Bunch separation scan Optimal injection distance 𝟎.𝟓𝟎𝟓 𝝀 𝒑
The model developed for BLAST scheme doesn’t forsee the driver head erosion effects
A simulation scan is required in order to evaluate 𝑬 𝟎 and 𝝓 𝟎
sin 𝜙 0 =− 𝑘 𝑝 𝑍 𝑤 0 𝑅 𝑤 𝐸 0
Optimal injection distance 𝟎.𝟓𝟎𝟓 𝝀 𝒑
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

11. Witness longitudinal phase space
Unoptimized injection phase
Optimized injection phase
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

12. Energy spread evolution and accelerating gradient
𝜎 𝐸,𝑓 = 𝜎 𝐸,𝑖 − 𝛾 0 𝛾 𝑘 𝑝 4 𝜎 𝑧,𝑤 4 4
Energy spread growth depends only on witness length
Final accelerating gradient 1.07 GV/m
𝑬 𝟎 is not constant for the first cm
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

13. Witness envelope
Envelope doesn’t follow the prevision for the first cm
After the first cm the simulated background approaches the theoretical one
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

14. SPARC_LAB layout for beam driven PWFA experiments
Incoming witness
Outcoming witness
𝛾=200
𝛾=300
𝜀 𝑛 =0.3 𝑚𝑚 𝑚𝑟𝑎𝑑
𝜀 𝑛 =0.308 𝑚𝑚 𝑚𝑟𝑎𝑑
𝜎 𝐸 =0.1%
𝜎 𝐸 =0.22%
Acceleration parameters
Δ𝑠=5𝑐𝑚
𝐸 𝑧 =1.07𝐺𝑉/𝑚
𝑬≅𝟏𝟎𝟎−𝟏𝟓𝟎 𝐌𝐞𝐕
𝜺≅𝟎.𝟑−𝟒 𝐦𝐦 𝐦𝐫𝐚𝐝
𝝈 𝑬 <𝟎.𝟏%
𝝈 𝒛 =𝟒−𝟕𝟓 𝛍𝐦
High quality PWFA
Beam diagnostic
Photoinjector that allows to create trains of high quality electron bunches
2 S band accelerating cavities
(velocity bunching)
C band accelerating cavity
FEL
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

15. Stability analysis 𝜎 𝐸 =0.4±0.2% 𝛾=306±14
A scan of 30 simulation was performed using the latin hypercube sample in order to analyze the working point robustness
The chosen parameter range of the scan is consistent with measurements performed at SPARC_LAB injector (worst results)
𝜎 𝑟 ±15%
𝜎 𝑧 ±8%
𝑄±10%
Δ𝑧±8%
𝜎 𝐸 =0.4±0.2%
𝜀 𝑛 =0.307±0.005 mm mrad
𝛾=306±14
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

16. Conclusions Results Theoretical basis for working point design
Low energy working point fullfilled
First working point stability test performed with promising results
Future perspectives
Start to end simulation of a BLAST working point at SPARC_LAB
Stability analysis in the context of a S2E simulation
Stefano Romeo EAAC 2017, La Biodola – 26/09/2017

17. Thanks for the attention
Acknowledgements:
SPARC_LAB GROUP
EAAC organizing committee
Old and new friends here
Stefano Romeo PhD defense La Sapienza, University of Rome 2017/09/20

18. Half density ion column
𝜕 𝐸 𝑧 𝜕𝜉 =0
𝜕 𝐸 𝑧 𝜕𝑟 = 𝜇 0 𝑐 𝐽 𝑟
𝜕 𝐽 𝑟 𝜕𝜉 =0
𝐽 𝑟 =0
𝐽 z =0
𝜌= 𝑛 0 2
𝜕 (𝐸 𝑟 −𝑐 𝐵 𝜃 ) 𝜕𝜉 = 𝜇 0 𝑐 𝐽 𝑟 =0
𝜕𝜌 𝜕𝜉 =0
1 𝑟 𝜕 𝜕𝑟 𝑟( 𝐸 𝑟 −𝑐 𝐵 𝜃 ) = 𝜌 𝜖 0 − 𝜇 0 𝑐 𝐽 𝑧
Stefano Romeo PhD defense La Sapienza, University of Rome 2017/09/20