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Separation of Scales in Wave - Particle Interactions

QUASI-STATIC MODELING of PARTICLE –FIELD INTERACTIONS Thomas M. Antonsen Jr, IREAP, University of Maryland College Park MD, 20742 Multiscale Processes in Fusion Plasmas IPAM 2005 Work supported by NSF, ONR, and DOE - HEP. Laser Wake Field Accelerator LWFA. Plasma Wake. Laser pulse.

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Separation of Scales in Wave - Particle Interactions

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  1. QUASI-STATIC MODELING of PARTICLE –FIELD INTERACTIONSThomas M. Antonsen Jr, IREAP, University of MarylandCollege Park MD, 20742Multiscale Processes in Fusion PlasmasIPAM 2005Work supported by NSF, ONR, and DOE - HEP

  2. Laser Wake Field Accelerator LWFA Plasma Wake Laser pulse Vacuum Electronic Device (VED) Beam Drive Radiation Source Cavity Electron beam Radiation Separation of Scales in Wave - ParticleInteractions Basic Parameters:   = 8  10-5 cm  (100 fsec) = 3  10-3 cm Propagation length = 5 cm Gyromonotron Radiation period = 6  10-12 sec (170GHz) Transit time = 5  10-10 sec Cavity Time = 5  10-9 sec Voltage rise time > 5  10-5 sec

  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 INTERACTIONAPPROACHES / APPROXIMATIONS • Laser 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: wlasertpulse >> 1, rspot >> l wp /wlaser <<1 • Advantages of envelope model: -Larger time steps Full EM stability: Dt < Dx/c Envelope accuracy: Dt < 2pDx2/lc -Further approximations • Advantages of full EM: Includes Stimulated Raman back-scattering

  6. • Laser + Wake field: E = E + E laser wake • Vector Potential: A = A ( , x , t ) exp i k + c . c . x x laser 0 0 ^ • Laser Frame Coordinate: = ct – z x • Envelope equation: Necessary for: Raman Forward Self phase modulation vg< c Drop (eliminates Raman back-scatter) Laser Envelope Approximation

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

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

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

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

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

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

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

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

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

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

  18. UCLA QUICKPIC 3DUCLA/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. UCLA Laser Pulse x Axial Electric Field 1.8 nC electron bunch 25 MeV injection energy Reduced amplitude due to effects of beam loading

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

  21. VED ModelingInteraction 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 Doppler curve kz vz Dispersion curve w (kz)

  23. time t+dt t z 0 L Time Domain SimulationStandard PIC Trajectories Positions interpolated to a grid in z Signal and particles injected Fields advanced in time domain Carrier and its harmonics must be resolved System state specified

  24. time 0 L z z+dz Frequency & Time Domain Hybrid SimulationRF phase sampled Trajectories t+dt Signal Period T=2p/w Ensemble of particles samples RF phase

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

  26. Code Verification: Comparison with MAGIC Operating Frequency 3.23 GHz Output Power (Pout) MAGIC 214.2 kW TESLA 214.0 kW Input Power (Pin) 49.06 kW B0=1kG rwall=1.4 cm rbeam=1.0 cm zgap = 1 cm Q= 115 115 R/Q= 85.685.6 (on axis) fres3.225 GHz3.225 GHz

  27. TESLA : Sub-Cycling to Improve Performance MAGIC: Pout=214.2 kW TESLA: Pout=214.0 kW 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 Particles with small z Resonators: 1, 2, 3, 4

  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. t trajectories Followed in t Sum over time steps of duration dt Followed in z z zj zj+dz Particle Characteristicsand Current Assignment

  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 Modelsbased 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?

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