Transcript SOME FUNDAMENTAL PROCESSES IN PULSE

SOME FUNDAMENTAL PROCESSES
IN PULSE-PARTICLE INTERACTION
Kaz AKIMOTO
School of Science & Engineering
TEIKYO UNIVERSITY
For US-Japan Workshop on Heavy Ion Fusion and
High Energy Density Physics,
Utsunomiya University
September 28-30, 2005
＜METHOD＞
• Velocity shifts of particles are calculated
after interaction with an ES or EM pulse that
is dispersive and propagating.
•
•
•
•
＜APPLICATIONS＞
particle acceleration
（cosmic rays/accelerators）
particle heating （laser fusion etc.）
plasma instabilities and turbulence
plasma processing etc.
What you will learn out of this talk.
1.
What kind of waves have more acceleration
mechanisms?
=> By breaking the symmetry of a wave
acceleration mechanisms can be pair-produced.
2.
What happens to cyclotron resonance if instead of
a sinusoidal wave a pulse is used?
3.
What happens to cyclotorn resonance if wave
ampitude becomes greater than the external
magnetic fielｄ?
METHOD ２：
• Equation of motion for a particle with charge q, mass
m is solved analytically and numerically in the
presence of a generalized wavepacket:
{(z vg t)/ l}2 i(k o z o t  )
• ES ,
E(z,t)  E e
• EM .
{(zv t)/l}2 i(k z  t  )
g
o
o
Ey (z,t)  Eo e
What do they look like?
<WAVEPACKET> ln = lωo / c = 2.0
0.03
0.02
En
0.01
0
-0.01
-0.02
-0.03
-4
-2
0
zn
2
4
<IMPULSE> ln=0.2
0.03
0.02
En
0.01
0
-0.01
-0.02
-0.03
-1
-0.5
0
zn
0.5
1
＜background＞
●Acceleration of particles by a standing-wave pulse
had been studied (e.g. Morales and Lee, 1974)
extreme dispersion：ｖｇ ＝０，ｖｐ＝ ωo/ ko ＝∞
(ko=0)
●Non-dispersive pulse was also studied.
(Akimoto, 1997)
ｖｇ＝ｖｐ ≠０
●Then results were extended to dispersive pulse:
arbitrary dispersion：－∞＜ｖｇ，ｖｐ＜ ∞
(ES （EM） cases solved. Akimoto 2002(2003))
■ sinusoidal wave ( l → ∞)
highly symmetric
⇒ no net acceleration
■ nondispersive pulse
• transit-time acceleration
• reflection
SINUSOIDAL WAVE VS. PULSE
V
Vp
PULSE
Z
Non-dispersive pulse can
accelerate particles via 2 ways.
1. transit-time acceleration（ｖo≠ｖp）
 qEcos t   o2 t 2 /4
v 
e
m o 
2. linear reflection（ｖo～ｖp）

v  2 v p  vo

How about dispersive pulse?
3. Quasi-Trapping [QT]
4. Ponderomotive Reflection [PR]
Quasi-Trapping
if vp-vtr < vo < vp+vtr （ｖo～ｖp）, where
2 q E0
vtr=

mk 0
v  2 v p  vo

Linear reflection （ｖo～ｖp）

v  2 v p  vo

Nonlinear (ponderomotive) reflection （ｖo～ｖｇ）
1
qEo
if vg-vref < vo < vg+vref,
vref 

v  2 v g  vo

2m  0  k0 vg
<WAVEPACKET>
• Hamiltonian Contours in Wave Frame
1.6
1.4
V/Vp
1.2
1
0.8
0.6
0.4
-3
-2
-1
0
Zn'
1
2
3
<MONOCYCLE PULSE>
• Hamiltonian
Contours
in
Wave
Frame
2
V/Vp
1.5
1
0.5
0
-1
-0.5
0
Zn'
0.5
1
Question:
What happens if the pulse is nonlinear EM,
& Bo is applied?
<theory> Linear Polarization
{(zv t)/l}2 i(k z t  )
g
o
o
Ey (z,t)  Eo e
transit-time acceleration
& cyclotron acceleration
( v ) max 
 qEt

2 m o
2 t 2 / 4 
 (  o /  o )2 t 2 / 4
(


/

)
o
o
e
e



  1 v0 / v p
  (v 0  v g ) / (v0  v p )
 t  l / ( v p )
ACCELERATIONS DUE TO EM PULSES
V pe r p
H o : H a m ilt o n ian
p
Vo z
Vp
Vz
NUMERICAL RESULTS
We solve the equation of motion
numerically as a function of v0,
increasing En= eE0 / mc0 .
<Parameters>
1. Phase Velocity: Vp=0.1c
2. Group Velocity: Vg=0.1c &0.05c
3. Field Strength: Ωe＝ωo
4. Pulse Length: ln=2.0 etc.
NUMERICAL RESULTS
En= eE0 / mc0 ＝0.001 .
En=0.01
En=0.1
Now the center of resonance has moved
to v p   0 / k0 =0.1c.
ACCELERATIONS DUE TO EM PULSES
V pe r p
H o : H a m ilt o n ian
p
Vo z
Vp
Vz
Phase-Trapping
IF
qEo
v 
 vzo  v p
m o
then

v/ /  2 v p  vzo

What is the mechanism
for multi-peaking?
• The band structure becomes more
significant as the pulse is elongated.
• En=0.01, ln=5 or 10(ｖｇ＝ｖｐ)
• En=0.001, ln=20
ANALYSIS OF PARTICLE VELOCITIES
• En=0.01, ln=2
TEMPORAL EVOLUTION OF
PARTICLE VELOCITIES
ACCELERATIONS DUE TO EM PULSES
V pe r p
H o : H a m ilt o n ian
p
Vo z
Vp
Vz
trapping ＆ band structure
Owing to trapping, some electrons exit
pulse when accelerated, while others do
when not.
vp /c
• The trapping period is given by 0Ttr  2 En v.
• If this becomes comparable to the transit
3.3l
time= v g  v0 , the trapping becomes
important and multi-resonance occurs.
CONCLUSIONS
• AS WAVE IS MADE LESS SYMMETRIC, MORE
ACCELERATION MECHANISMS EMERGE.
• AS PULSE AMPLITUDE AND/OR PULSE WIDTH
ARE ENHANCED, LINEAR CYCLOTRON
ACCELERATION BY A PULSE BECOMES
NONLINEAR, AND TENDS TO SHOW BAND
STRUCTURE. IT IS DUE TO PARTICLE TRAPPING
AND THE FINITE SIZE OF PULSE.
• AS THE NONLINEARITY IS FURTHER ENHANCED,
THE INTERACTION TRANSFORMS INTO PHASE
TRAPPING.