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 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?