1. Phase space moment equation model of highly relativistic electron beams in plasma wakefield accelerators
Robert Robson1, Timon Mehrling2, Jan-Hendrik Erbe2 and Jens Osterhoff2
School of Natural Sciences, Griffith University, Brisbane, Australia
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

2. What this talk is about
A new theoretical technique for rapid, accurate calculation of average phase space properties of an electron beam injected into & extracted from a laser wakefield 
2. At first plasma properties are assumed uniform (benchmark model)
3. Generalize the procedure to realistic profiles  quick, accurate alternative to PIC

3. Electron “bunch” vs “swarm”
Fx ~ -kx x
Fig 1
Axial symmetry 
Fz  const
z = ct Fx
Low energy electron “swarm” in a gas (ANU, Heidelberg, t < 2011)
Highly relativistic bunch of electrons in a laser wakefield plasma (DESY)
What is in common? Kinetic theory, fluid modelling, meV  MeV
Quantities of interest are phase space averages <x2 >, <x px>, <px2>, emittance, …
PIC is accurate and comprehensive but computationally expensive

4. Phase space moment method
Fig 2
( t + v. + F. p ) f = (t f )collisions  0 (Vlasov equation)
“Short cut" method  set of phase space averages <i(r,p) > directly
Form a set of moment equations
 d (phase space) i (r,p)  Vlasov Eqn (i=1,2,3, …)
Generally more unknowns than equations, therefore require a closure Ansatz
Like “fluid models” of plasma physics  significant computational economy
Accuracy critically dependent upon Ansatz

5. Basic moment equations

6. Benchmark model – analytic solution

7. Semi-analytical numerical analysis (SANA)

8. Comparison of SANA with analytic solution for constant k
Fig 3 Evolution of emittance and <x2> after injection into a blowout plasma wakefield of density cm-3 as calculated from the analytic expression (solid line) and SANA numerical model (dashed lined), for  = 0.1, <uz> = 2000, k = 0.94 mm-1 , with initial conditions <x2>0 = (5 m)2 , <xx>0 = 0 and <x2>0 =(0.5 mrad)2. Small oscillatory differences are due to an approximation in analytic model

9. PIC simulation (HIPACE) for constant k
Fig 4 Same model as last page, including PIC simulation (HIPACE) results. See T. Mehrling, C. Benedetti, C. B. Schroeder, and J. Osterhoff., Plasma Physics & Controlled Fusion 56, (2014) .
Direct comparison with PIC is possible only if the effects of the longitudinal force term Fz are included in the moments equations. Then <uz> and  vary with s, no analytic solution available, SANA still OK.

10. Further moments for constant k
To this point all details are for constant kx and can be found in R.E. Robson, T.J. Mehrling and J. Osterhoff, Ann. Phys. 356, (2015)
Now extend the moment equations and generalize SANA solution

11. Application of SANA to more general plasma profiles
Now include contributions from terms arising from Fz in moment equations, generalize SANA procedure
Work with phase space moments <x2 >, <x px >, <px2> instead of trace space
Fig. 5 Two plasma profiles in capillaries of the same length. The standard profile changes at the plasma-vacuum interfaces much more rapidly than the tailored profile.

12. Fig. 6 Phase space emittance evolution of beams externally injected into plasma-wakefields in the blowout regime for standard and tailored plasma profiles, as calculated by PIC simulation and SANA. The tailored profile mitigates growth during injection and in the vacuum drift.

13. SANA - compares well with PIC simulations + computationally economical
SUMMARY
Vlasov kinetic equation  equivalent set of phase-space moment equations, closure achieved through an Ansatz
An analytic solution has been obtained for constant betatron wavenumber, for use as a benchmark
An analytic/numerical procedure (SANA) has been developed for arbitrary plasma density profiles and betatron wavenumbers
SANA - compares well with PIC simulations + computationally economical
- quickly calculates emittance growth during injection/ extraction phases
- enables rapid optimization of plasma target density profiles
- this is the key to generation of high quality beams in plasma-based
acceleration

14. THANK YOU FOR YOUR ATTENTION!
References
R.E. Robson, T.J. Mehrling and J. Osterhoff, Ann. Phys. 356, (2015) [Methodology & details]
T. Mehrling , C. Benedetti, C. B. Schroeder, and J. Osterhoff., Plasma Physics & Controlled Fusion 56, (2014) [HIPACE PIC simulation details]
R.E. Robson, et al, Rev. Mod. Phys 77, (2005) [Fluid analysis in configuration space]
K. Kumar, J. Phys. D14, 2199 (1981) [ Phase space moment method for low energy electrons]
THANK YOU FOR YOUR ATTENTION!
Isola d’Elba
Isola di Coochiemudlo
1.6  104 km