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
Separation of Scales in Wave - Particle
Interactions
Laser Wake Field Accelerator
LWFA
Basic Parameters:
Plasma Wake
=
Laser pulse
Vacuum Electronic Device (VED)
Beam Drive Radiation Source
Cavity
Electron
beam
Radiation
8  10-5 cm
(100 fsec) =
3  10-3 cm
Propagation length
= 5 cm
Gyromonotron
Radiation period
(170GHz)
Transit time
Cavity Time
Voltage rise time
=
6  10-12 sec
=
=
>
5  10-10 sec
5  10-9 sec
5  10-5 sec
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
LASER-PLASMA INTERACTION
APPROACHES / APPROXIMATIONS
• Laser
Full EM - Laser Envelope
• Plasma
Particles - Fluid
Full Lorenz force - Ponderomotive
Dynamic response -
Quasi-static
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
Laser Envelope Approximation
• Laser + Wake field:
• Vector Potential:
E = E laser + E wake
A laser = A 0(x ,x ^,t) exp ik 0x
• Laser Frame Coordinate:
+ c.c.
x = ct – z
• Envelope equation:
2  
 ˆ
2 ˆ
4p ˆ
2
j
ik 0  A  2 2 A  ^ Aˆ  
c t 
x 
c t
c
Necessary for:
Raman Forward
Self phase modulation
vg< c
Drop
(eliminates Raman back-scatter)
AXIAL GROUP VELOCITY
v g  c(1
True dispersion :
Extended Paraxial
Requires :
:
k ^2 c 2  w 2p
w
2
)1/2
k ^2 c 2  w 2p
v g  c / (1 
)
2
2w
k^2 c 2,w 2p  w 2
Extended Para - Axial approximation
- Correct treatment of forward and near forward scattered radiation
- Does not treat backscattered radiation
Full Lorenz Force vs. Ponderomotive Description
• Full Lorenz:
d pi
v B
= q E+ i c
dt
• Separation of time scales
E = E laser + E wake
dx i
= vi
dt

g =
p2
1+ 2 2
mc
• Requires small excursion
x(t)   E laser < < Elaser
x(t) = x(t) + x(t)
• Ponderomotive Equations
2
q A laser
Fp = – mc 
mc 2
2g
v  Bwake
dp
= q E wake +
+ Fp
c
dt
g =
q A laser
p2
1+ 2 2 +
mc
mc 2
2
2
PLASMA WAKE
• Maxwell’s Equations for
Wake Fields in Laser
Frame
4p q n
B
p
c g
E

x
part icles
B
E 
x

 E  4pq n
particles
Laser frame coordinate: x = ct -z collapses t and z
Time t is a parameter
Solved using potentials F, A
 n0

Quasi - Static vs.
Laser Pulse
Dynamic Wake
PlasmaWake
Electron transit time:
c
Plasma electron
c - vz
t pulse
1 – vz / c
Electron transit time << Pulse modification time
Trapped electron
Advantages:
te =
d =  + c –v  +v 
z
^
^
dt
t
x
fewer particles, less noise (particles marched in x= ct-z)
Disadvantages: particles are not trapped
CODE STRUCTURE
Laser
Fp
2
w


2

p
2
ˆ
ˆ
ˆ
ik

a


a


 0

^
2 a
c t 
x 
c
Particles and Wake
dp^
1  
v  B 


qE 

F

p

dx
c  v z 
c 
^
dr^
v
 ^
dx c  v z
w 2p
B
4p q n
p
c g
Note: t is a parameter

part icles
E
x
Plasma Particle Motion and Wake Become 2D
Particles marched in x
Motion in r^ - x plane
dx
 c  vz
dt
r
ri +1
dp^
1  
v  B 


q
E


F
p 
dx
c  v z 
c 
 
^
ri
dr^
v
 ^
dx c  v z
xj
xj + 1
x = ct –z
Density
n(r^,x ) 
c
n(r ,0)
c  vz ^ 0
PARTICLES CONTINUED
:
• Hamiltonian:
H  H (x  ct  z,r^, p)  mc 2g  q
• Weak dependence on “t” in the laser frame
Pz  pz 
• Introduce potentials
• Algebraic equation:
B  A
H  cPz  const.
q
A
c z
E   
A
x
2
pz  pz (p ^,    Az, a )
2
g  g (p ^,    Az , a )
WAKE FIELDS
• Maxwell’s Equations for Potentials
2^ A 


4p
j     A
c
x

  E  2 

  A  4p
x
• Iteration required for EM wake
2
 ^ A^ 
4p
j
c ^
dp^
A
 E ^  ^   ^
dx
x
GAUGE
Lorentz QUICKPIC
 ^  A^  

  Az 
x
4p
 A 
j
c
Transverse Coulomb WAKE
 ^  A^  0
2
^
2^  4p
Pro:
Simple structure
Compatible with 2D PIC
Con:
A carries “electrostatic” field
2^ A 
  Az 
4p
j  

c

x




Pro:
A^ = 0 in electrostatic limit
Con:
non-standard field equations
Numerical Simulation of Plasma Wave
2D WAKE Mora and Antonsen, Phys Plasmas 4, 217 (1997)
Viewed in laser frame
Particle trajectories
Density maxima
WAKE - Particle Mode
Intensity
Density
Trajectories
5
4
3
r
2
1
0
Cavitation and Wave Breaking
0
10
20
x
30
40
QUICKPIC 3D
UCLA/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
Beam particles equations: 3D
dPb ^
 qb ^ 
dt
dPb ||

 qb
dt
x
dxb^
P
 b^
dt
gmc
    A||
dx b
P
 1 b||
dt
gmc
Beam charge and current
b 
1
q
Volume i b i
jb 
1
q V
Volume i bi bi
Axial Electric Field
UCLA
x  ct  z
Laser Pulse
x
1.8 nC electron
bunch
25 MeV injection
energy
Reduced amplitude
due to effects of
beam loading
UCLA
x  ct  z
Electron Distribution and Axial
Field
Laser Pulse
Electron Bunch
Distribution
~1.7x108 electrons
Axial Electric Field
VED Modeling
Interaction Circuit Types
Interaction requires:
Wiggler FEL
Beam
Structure
Synchronism
S. H. Gold and G. S. Nusinovich, Rev. Sci. Instrum. 68 (11), 3945 (1997)
Synchronism in a Linear
Beam Device

E(x,t)
= Re {E exp [ik zz – wt]}
w
w = kz vz
Dispersion curve
w(kz)
Doppler curve
kz v z
BWO
TWT
-p/d
0
p/d
2p/d
kz
Time Domain Simulation
Standard PIC
time
dz
 vz
dt
Trajectories
Positions interpolated to a
grid in z
Signal and
particles injected
t+dt
Carrier and
its harmonics
must be
resolved
Fields advanced in time
domain
t
System state specified
0
z
L
Frequency & Time Domain Hybrid Simulation
RF phase sampled
time
Trajectories
Vs (t)exp[iwt]
t+dt
dp q 
v  B
 E 
dz vz
c 
dt
1

dz vz
Signal
Period T=2p/w
Ensemble of particles
samples RF phase
0
z
z+dz
L
Separate Beam Region from Structure Region
Cavities coupled
through slots
Cavity fields
Penetrate to
Beam tunnel
trough gaps
Electrodynamic
structures
e-
Beam region
simulation
boundary
Cavity fields ,
jth cavity:
E(x,t) 

Vs (t)e s (x)exp[iw t]  c.c.
j
smodes
B(x,t) 

s modes
Beam tunnel fields
E T  Vk z,t e k rT ,z exp[iw n t]
k ,n
I sj (t)bs (x)exp[iw t]  c.c.
B T   I k z,tbk rT ,zexp[iw nt]
k ,n
Code Verification:
Comparison with MAGIC
Operating Frequency 3.23 GHz
Input Power (Pin)
49.06 kW
B0=1kG
rwall=1.4 cm
rbeam=1.0 cm
Q=
R/Q=
fres
Output Power (Pout)
MAGIC 214.2 kW
TESLA 214.0 kW
Dzgap = 1 cm
115
85.6
3.225 GHz
115
85.6 (on axis)
3.225 GHz
TESLA : Sub-Cycling to Improve Performance
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
MAGIC 3D
~ 2 hours
~72 hours
Output Power [W]
MAGIC: Pout=214.2 kW
TESLA: Pout=214.0 kW
2.5 10
5
2 10
5
1.5 10
5
every time step
1 10
nd
each 2
5
time step
th
each 5 time step
5 10
th
each 10 time step
4
0
0
1 10
-8
2 10
-8
3 10
-8
Time [sec]
4 10
-8
5 10
-8
6 10
-8
Parallelization - Multiple Beam Klystrons
(MBK)
Input Power
Output Power
Beams surrounded
by individual beam
tunnel
Beam Tunnels
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
NRL 4 cavity MBK
Saturated regime of operation:
Particles may stop
Analogous situation in
LWFA: plasma electrons
accelerated
Resonators: 1,
2,
3, 4
Particles with small z
Reflected Particles
Equations of particles motion (EQM):
d/dz representation
d pi
q

dz
v z ,i
d ri v i

dz vz ,i
vi  B 

E  c 
If vz 0 right hand side of EQM  
Numerical solution of EQM lost accuracy
currently these particles are removed
d/dt representation
d pi
vi  B 

 q E 
dt
c 

d ri
 vi
dt
Switch to d/dt equations for
selected particles with
small vz,i
Particle Characteristics
and Current Assignment
t
Followed in t
trajectories
j(x ^ ,z j )    Iiv i (x ^  x^i )D(z j  zi )e
dt i
Sum over time steps of duration dt
Followed in z
j(x ^ ,z j )   Ii
i
zj zj+dz
z
vi
 (x ^  x ^i )e iwt i
v zi
iwti
Sample Trajectories in MBK
3 10 -9
0.6
1 10
-10
8 10
-11
0.1
t-integr
2.5 10
-9
0.08
0.5
t, sec
0.06

-9
0.04
0.02
0.3
1 10 -9
0.2

z
1.5 10
-9

t [sec]
4 10-11
2 10-11
0
19.35
0
19.36
19.37
19.38
19.39
-0.02
19.4
z [cm]
z
5 10 -10
0.1
0
0
0
5
10
15
z [cm]
20
Direction reverses
z
0.4
z

2 10
t [sec]
6 10-11
Accelerated Plasma Particles
Plasma particles with E > 500 keV
promoted to status of passive test
particles
1.5
1
p^/mc
0.5
0
-0.5
-1
-1.5
-10
0
10
20
30
pz/mc
40
50
60
70
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?