1. QUASI-STATIC MODELING of PARTICLE –FIELD INTERACTIONS Thomas M
QUASI-STATIC MODELING of PARTICLE –FIELD INTERACTIONS Thomas M. Antonsen Jr, IREAP, University of Maryland College Park MD, Multiscale Processes in Fusion Plasmas IPAM 2005 Work supported by NSF, ONR, and DOE - HEP

2. Separation of Scales in Wave - Particle Interactions
Laser Wake Field Accelerator
LWFA
Laser pulse
Plasma Wake
Basic Parameters:

 = 8  10-5 cm
 (100 fsec) = 3  10-3 cm
Propagation length = 5 cm
Gyromonotron
Radiation period =  sec
(170GHz)
Transit time =  sec
Cavity Time =  10-9 sec
Voltage rise time >  10-5 sec
Vacuum Electronic Device (VED)
Beam Drive Radiation Source
Electron
beam
Cavity
Radiation

3. Hierarchy of Time Scales
• Limited interaction time for some electrons -Transit time
Laser - Plasma: - pulse duration
Radiation Source: - electron transit time
• Radiation period << Transit time
Simplified equations of motion
Laser - Plasma: - Ponderomotive Force
Radiation Source: - Period averaged equations
• Transit time << Radiation Evolution Time
Quasi Static Approximation
Pulse Shape/Field Envelope constant during transit time
• Radiation period << Radiation Evolution Time
Simplified equations for radiation
Envelope Equations

4. LASER-PLASMA INTERACTION APPROACHES / APPROXIMATIONS
Full EM - Laser Envelope
• Plasma
Particles - Fluid
Full Lorenz force - Ponderomotive
Dynamic response Quasi-static

5. Full EM vs. Laser Envelope
• Required Approximation for Laser envelope:
wlaser tpulse >> 1, rspot >> l
wp /wlaser <<1
• Advantages of envelope model:
-Larger time steps
Full EM stability: Dt < Dx/c
Envelope accuracy: Dt < 2p Dx2/lc
-Further approximations
• Advantages of full EM: Includes Stimulated Raman back-scattering

6. Laser Envelope Approximation
• Laser Frame Coordinate: x = ct – z
• Laser + Wake field: E
laser +
wake
• Vector Potential: A ( , ^ t )
exp i k c .
• Envelope equation:
Necessary for:
Raman Forward
Self phase modulation
vg< c
Drop
(eliminates Raman back-scatter)

7. AXIAL GROUP VELOCITY True dispersion : Extended Paraxial : Requires :
Extended Para - Axial approximation
- Correct treatment of forward and near forward scattered radiation
- Does not treat backscattered radiation

8. Full Lorenz Force vs. Ponderomotive Descriptiondt = q E + v B c x
• Full Lorenz: E =
laser +
wake x ( t )
• Separation of time scales × Ñ
<
• Requires small excursion
• Ponderomotive Equations d p dt = q E
wake + v B c F – m 2 g Ñ A
laser 1

9. Laser frame coordinate: x = ct -z collapses t and z
PLASMA WAKE
• Maxwell’s Equations for Wake Fields in Laser Frame
Laser frame coordinate: x = ct -z collapses t and z
Time t is a parameter
Solved using potentials F, A

10. Quasi - Static vs. Dynamic Wake
Laser Pulse
PlasmaWake
Electron transit time: t e =
pulse 1 – v z / c
Plasma electron c
Electron transit time << Pulse modification time
Trapped electron
c - vz d dt = t + c – v z x ^ × Ñ
Advantages: fewer particles, less noise (particles marched in x= ct-z)
Disadvantages: particles are not trapped

11. CODE STRUCTURE
Laser
Particles and Wake
Note: t is a parameter

12. Plasma Particle Motion and Wake Become 2D
Particles marched in x
Motion in r - x plane r x = ct – z j + 1 i
Density

13. PARTICLES CONTINUED • Hamiltonian:
• Weak dependence on “t” in the laser frame
• Introduce potentials
• Algebraic equation:

14. WAKE FIELDS • Maxwell’s Equations for Potentials
• Iteration required for EM wake

15. GAUGE Lorentz QUICKPIC Transverse Coulomb WAKE Pro: Simple structure
Compatible with 2D PIC
Con:
A carries “electrostatic” field
Pro:
A = 0 in electrostatic limit
Con:
non-standard field equations

16. Numerical Simulation of Plasma Wave
2D WAKE Mora and Antonsen, Phys Plasmas 4, 217 (1997)
Viewed in laser frame
Particle trajectories
Density maxima

17. Cavitation and Wave Breaking
WAKE - Particle Mode
Intensity
Density
Trajectories
Cavitation and Wave Breaking

18. QUICKPIC 3D UCLA/UMD/USC Collaboration
UCLA: Chengkun Huang, V. K. Decyk, C. Ren, M. Zhou, W. Lu, W. B. Mori
UMD: J. Cooley, T. M. Antonsen Jr.
USC: T. Katsouleas
Beam particles equations: 3D
3D simulation
- Laser pulse evolution
Plasma particles
- Beam particles
Numerics
Parallel
Object Oriented
Beam charge and current

19. Reduced amplitude due to effects of beam loading
UCLA
Axial Electric Field x
Laser Pulse
1.8 nC electron bunch
25 MeV injection energy
Reduced amplitude due to effects of beam loading

20. Electron Distribution and Axial Field
UCLA
Electron Distribution and Axial Field
Laser Pulse
Electron Bunch Distribution
~1.7x108 electrons
Axial Electric Field

21. VED Modeling Interaction Circuit Types
Interaction requires:
Beam
Structure
Synchronism
Wiggler FEL
S. H. Gold and G. S. Nusinovich, Rev. Sci. Instrum. 68 (11), 3945 (1997)

22. Synchronism in a Linear Beam Device
Dispersion curve
w (kz)
Doppler curve
kz vz

23. Time Domain Simulation Standard PICL z
Trajectories
Positions interpolated to a grid in z
Signal and
particles injected t
t+dt
Fields advanced in time domain
Carrier and its harmonics must be resolved
System state specified

24. Frequency & Time Domain Hybrid Simulation RF phase sampledz
time L
z+dz
Trajectories
t+dt
Signal
Period T=2p/w
Ensemble of particles samples RF phase

25. Separate Beam Region from Structure Region
Cavities coupled
through slots
Electrodynamic
structures
Cavity fields
Penetrate to
Beam tunnel
trough gaps e-
Beam region
simulation
boundary
Cavity fields ,
jth cavity:
Beam tunnel fields

26. Code Verification: Comparison with MAGIC
Operating Frequency 3.23 GHz
Output Power (Pout)
MAGIC kW
TESLA kW
Input Power (Pin) kW
B0=1kG
rwall=1.4 cm
rbeam=1.0 cm
zgap = 1 cm Q=
R/Q= (on axis)
fres GHz GHz

27. TESLA : Sub-Cycling to Improve Performance
MAGIC: Pout=214.2 kW
TESLA: Pout=214.0 kW
Frequency of trajectory update with respect to field update (each nth step)
CPU Time [sec]
Pentium IV 2.2 GHz 10
7.4 5
11.7 2
24.6 1
46.0
MAGIC 2D ~ 2 hours
MAGIC 3D ~72 hours

28. Parallelization - Multiple Beam Klystrons (MBK)
Output Power
Input Power
Beams surrounded
by individual beam
tunnel
Beam Tunnels
Resonators
(Common)

29. Code development for multiple beam case
Code is being developed to exploit multiple processors
Each beam tunnel assigned to a processor
Communication through cavity fields
Each processor evolves independently cavity equations

30. Technical Challenge: Simulations of Saturated Regimes
Phase Space
NRL 4 cavity MBK
Saturated regime of operation:
Particles may stop
Analogous situation in LWFA: plasma electrons accelerated
Resonators: 1, , , 4
Particles with small z

31. Reflected Particles
Equations of particles motion (EQM):
d/dz representation
d/dt representation
If vz 0 right hand side of EQM  
Switch to d/dt equations for selected particles with small vz,i
Numerical solution of EQM lost accuracy currently these particles are removed

32. Particle Characteristics and Current Assignmentz
trajectories
zj zj+dz
Followed in t
Sum over time steps of duration dt
Followed in z

33. Sample Trajectories in MBK
Direction reverses

34. Accelerated Plasma Particles
Plasma particles with E > 500 keV promoted to status of passive test particles

35. Conclusions
• Reduced Models based on separation of time scales yield efficient programs
• Simplifications take various forms
- Envelope equations
- Ponderomotive force
- Resonant phase
- Quasi-static fields
• Breakdown of assumptions can cause models to fail
- Reflected particles
- Accelerated electrons
- Spurious modes (VEDs)
• Ad hoc fixes are being considered. Is there are more general approach?