separation of scales in wave - particle interactions
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
QUASI-STATIC MODELING of PARTICLE –FIELD
INTERACTIONS
Thomas M. Antonsen Jr,
IREAP, University of Maryland
College Park MD, 20742
Multiscale Processes in Fusion PlasmasIPAM 2005
Work supported by NSF, ONR, and DOE - HEP
Separation of Scales in Wave - ParticleInteractions
Laser Wake Field AcceleratorLWFA
Laser pulse
Plasma Wake
Vacuum Electronic Device (VED)Beam Drive Radiation Source
Electronbeam
Cavity
Radiation
Basic Parameters:
= 8 10-5 cm (100 fsec) = 3 10-3 cmPropagation length = 5 cm
Gyromonotron
Radiation period = 6 10-12 sec (170GHz)Transit time = 5 10-10 secCavity Time = 5 10-9 secVoltage rise time > 5 10-5 sec
Hierarchy of Time Scales
• Radiation period << Transit time
Simplified equations of motionLaser - Plasma: - Ponderomotive ForceRadiation Source: - Period averaged equations
• Transit time << Radiation Evolution Time
Quasi Static ApproximationPulse Shape/Field Envelope constant during transit time
• Limited interaction time for some electrons -Transit timeLaser - Plasma: - pulse durationRadiation Source: - electron transit time
• Radiation period << Radiation Evolution Time
Simplified equations for radiationEnvelope Equations
LASER-PLASMA INTERACTIONAPPROACHES / APPROXIMATIONS
• LaserFull EM - Laser Envelope
• Plasma
Particles - Fluid
Full Lorenz force - Ponderomotive
Dynamic response - Quasi-static
Full EM vs. Laser Envelope
• Required Approximation for Laser envelope:
laser pulse >> 1, rspot >>
p /laser <<1
• Advantages of envelope model:
-Larger time steps
Full EM stability: t < x/c
Envelope accuracy: t < 2 x2/c
-Further approximations
• Advantages of full EM: Includes Stimulated Raman back-scattering
Laser Envelope Approximation
• Laser Frame Coordinate: = ct – z
• Laser + Wake field: E = E laser + E wake
• Vector Potential: Alaser = A 0(,x,t) exp ik 0 + c.c.
• Envelope equation:
2c
t
ik0
A 2
c2t2ˆ A
2 ˆ A 4c
ˆ j
Necessary for: Raman ForwardSelf phase modulationvg< c
Drop(eliminates Raman back-scatter)
AXIAL GROUP VELOCITY
Extended Para - Axial approximation
- Correct treatment of forward and near forward scattered radiation
- Does not treat backscattered radiation
True dispersion :
vg c(1k
2c2 p2
2 )1/2
Extended Paraxial :
vg c / (1 k
2c2 p2
22 )
Requires :
k2c 2, p
2 2
Full Lorenz Force vs. Ponderomotive Description
• Ponderomotive Equations
dpdt
= q Ewake +v Bwake
c + Fp
Fp = – mc 2
2
q A laser
mc 2
2
= 1 +
p 2
m2c2 +q A laser
mc 2
2
dpi
dt= q E +
vi Bc
dxi
dt= vi
= 1 +
p 2
m2c2• Full Lorenz:
E = Elaser + Ewake
x(t) = x(t) + x(t)
• Separation of time scales
x(t) Elaser < < Elaser
• Requires small excursion
PLASMA WAKE
B 4qc
n
pparticles
E
E B
E 4q n particles n0
• Maxwell’s Equations for Wake Fields in Laser Frame
Laser frame coordinate: = ct -z collapses t and zTime t is a parameterSolved using potentials , A
Quasi - Static vs. Dynamic Wake
ddt
= t
+ c – vz
+ v
Laser Pulse PlasmaWake
Plasma electron
c
Trapped electron
c - vz
Electron transit time:
e = pulse
1 – vz / c
Electron transit time << Pulse modification time
Advantages: fewer particles, less noise (particles marched in = ct-z)
Disadvantages: particles are not trapped
CODE STRUCTURE
Laser
2c
t
ik0
a 2 ˆ a
p2
c 2ˆ a
Fp
p2
Particles and Wake
dp
d
1c vz
q E v B
c
Fp
drd
v
c vz
B 4qc
n
pparticles
E
Note: t is a parameter
Plasma Particle Motion and Wake Become 2D
r
= ct –z
j j +1
ri
ri +1
Motion in r - plane
dp
d
1c vz
q E v B
c
Fp
drd
v
c vz
n(r,) c
c vz
n(r 0,0)
Density
ddt
c vz
Particles marched in
PARTICLES CONTINUED
• Hamiltonian::
H H ( ct z,r, p) mc 2 q
• Algebraic equation:
pz pz (p, Az, a2)
(p, Az , a2)
H cPz const.• Weak dependence on “t” in the laser frame
Pz pz qc
Az
• Introduce potentials
B A
E A
WAKE FIELDS
• Maxwell’s Equations for Potentials
E 2
A 4
2 A
4c
j
A
• Iteration required for EM wake
dp
d E
A
2 A
4c
j
GAUGE
Lorentz QUICKPIC
2 A
4c
j
2 4
Pro:
Simple structure
Compatible with 2D PIC
Con:
A carries “electrostatic” field
Transverse Coulomb WAKE
A 0
2 A
4c
j Az
Pro:
A = 0 in electrostatic limit
Con:
non-standard field equations
A
Az
Numerical Simulation of Plasma Wave
Viewed in laser frame
-4
-3
-2
-1
00 4 8 12 16 20 24 28 32
Laser Intensity p t – z / c
pr / c Particle trajectories
-4
-3
-2
-1
00 4 8 12 16 20 24 28 32
pr / c
p t – z / cElectron Density
Density maxima
2D WAKE Mora and Antonsen, Phys Plasmas 4, 217 (1997)
WAKE - Particle Mode
0
1
2
3
4
5
0 10 20 30 40
r
Intensity Density Trajectories
Cavitation and Wave Breaking
QUICKPIC 3DUCLA/UMD/USC Collaboration
UCLA
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
3D simulation
- Laser pulse evolution
- Plasma particles
- Beam particles
Numerics
Parallel
Object Oriented
dPb
dt qb
dPb ||
dt qb
dxb
dt
Pb
mc
db
dt1
Pb||
mc
A||
Beam particles equations: 3D
b 1
Volumeqbi
i
jb 1
VolumeqbiVbi
i
Beam charge and current
Axial Electric Field
ct z
x
Laser Pulse
1.8 nC electron bunch25 MeV injection energy
Reduced amplitude due to effects of beam loading
UCLA
Electron Distribution and Axial Field
ct z Laser Pulse
Electron Bunch Distribution
Axial Electric Field
~1.7x108 electrons
UCLA
VED ModelingInteraction Circuit Types
Wiggler FEL
S. H. Gold and G. S. Nusinovich, Rev. Sci. Instrum. 68 (11), 3945 (1997)
Interaction requires:
BeamStructureSynchronism
Synchronism in a Linear Beam Device
/d-/d 2/d0 kz
= kzvz
TWT
BWO
E(x,t) = Re {E exp [ik zz – t]}
Dispersion curve(kz)
Doppler curve
kz vz
Time Domain SimulationStandard PIC
t
t+dt
time
0 Lz
System state specified
Signal and particles injected
Trajectories
dzdt
vz
Fields advanced in time domain
Positions interpolated to a grid in z
Carrier and its harmonics must be resolved
SignalPeriod T=2/
Trajectories
z
time
0 Lz+dz
dtdz
1v
z
dpdz
qvz
E v B
c
Frequency & Time Domain Hybrid SimulationRF phase sampled
Vs(t)exp[ it]
t+dt
Ensemble of particles samples RF phase
Separate Beam Region from Structure Region
e-Beam region
simulation boundary
Electrodynamic structures
Cavities coupledthrough slots
Cavity fieldsPenetrate toBeam tunneltrough gaps
Cavity fields , jth cavity:
E(x, t) Vs
j(t)essmodes (x)exp[ it] c.c.
B(x, t) Is
j (t)bssmodes (x)exp[ it] c.c.
BT I k z, t bk rT ,z k ,n exp[ int]
ET Vk z, t ek rT ,z k ,n exp[ int]
Beam tunnel fields
Code Verification: Comparison with MAGIC
Input Power (Pin) 49.06 kW
Output Power (Pout)MAGIC 214.2 kWTESLA 214.0 kW
B0=1kG
Q= 115 115R/Q= 85.6 85.6 (on axis)fres 3.225 GHz 3.225 GHz
rwall=1.4 cmrbeam=1.0 cm
zgap = 1 cm
Operating Frequency 3.23 GHz
TESLA : Sub-Cycling to Improve Performance
MAGIC 2D ~ 2 hoursMAGIC 3D ~72 hours
0
5 104
1 105
1.5 105
2 105
2.5 105
0 1 10-8 2 10-8 3 10-8 4 10-8 5 10-8 6 10-8
every time step
each 2nd time step
each 5th time step
each 10th time step
Ou
tpu
t P
ow
er
[W]
Time [sec]
MAGIC: Pout=214.2 kWTESLA: 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
Parallelization - Multiple Beam Klystrons (MBK)
Input Power Output Power
Beam Tunnels
Beams surroundedby individual beam tunnel
Resonators(Common)
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
Technical Challenge: Simulations of Saturated Regimes
Phase Space
Resonators: 1, 2, 3, 4 Particles with small z
Saturated regime of operation: Particles may stop
NRL 4 cavity MBK
Analogous situation in LWFA: plasma electrons accelerated
Reflected Particles
Equations of particles motion (EQM):
iz
ii
iz
i
vdz
dcv
q
dz
d
,
,
vr
BvE
p i
If vz 0 right hand side of EQM Numerical solution of EQM lost accuracy currently these particles are removed
ii
i
dt
dc
qdt
d
vr
BvE
p i
d/dz representation d/dt representation
Switch to d/dt equations for selected particles with small vz,i
Particle Characteristicsand Current Assignment
t
z
trajectories
zj zj+dz
Followed in t
j(x ,z j ) Iiv ii (x xi)(z j zi)e iti
dt
Sum over time steps of duration dt
Followed in z
j(x ,z j ) Iiv i
vzii (x xi)e it i
0
5 10-10
1 10-9
1.5 10-9
2 10-9
2.5 10-9
3 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20
t, sec
z
t [se
c] z
z [cm]
Sample Trajectories in MBK
0
2 10-11
4 10-11
6 10-11
8 10-11
1 10-10
-0.02
0
0.02
0.04
0.06
0.08
0.1
19.35 19.36 19.37 19.38 19.39 19.4
t-integr
z
t [se
c] z
z [cm]
Direction reverses
Accelerated Plasma Particles
Plasma particles with E > 500 keV promoted to status of passive test particles
-1.5
-1
-0.5
0
0.5
1
1.5
-10 0 10 20 30 40 50 60 70
p /m
c
pz/mc
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?