SOME 　 FUNDAMENTAL 　PROCESSES

IN PULSE-PARTICLE INTERACTION Kaz AKIMOTO

School of Science & EngineeringTEIKYO UNIVERSITY

For US-Japan Workshop on Heavy Ion Fusion and High Energy Density Physics,

Utsunomiya UniversitySeptember 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 fiel ｄ ?

 

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:

 • ES , • EM .   What do they look like?

E(z,t) Ee {(z vgt)/ l}

2 i(koz ot )

Ey(z, t)Eo e {(z v

gt)/ l}2 i(k

oz

ot )

 <WAVEPACKET> ln = lωo / c = 2.0

<IMPULSE> ln=0.2

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-1 -0.5 0 0.5 1

En

zn

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-4 -2 0 2 4

En

zn

＜ 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 

Z Vp

V PULSE

Non-dispersive pulse can accelerate particles via 2 ways.

1. transit-time acceleration （ｖ o≠ ｖ p ）

 

 2. linear reflection （ｖ o ～ｖ p ）

How about dispersive pulse?3. Quasi-Trapping [QT]4. Ponderomotive Reflection [PR] 

vqE cos tmo

e o

2t2 /4

v2 vp vo

Quasi-Trapping

if vp-vtr < vo < vp+vtr （ｖ o ～ｖ p ） , where

vtr=

Linear reflection （ｖ o ～ｖ p ） 　　

Nonlinear (ponderomotive) reflection （ｖ o ～ｖｇ） if vg-vref < vo < vg+vref,

2 q E0mk0

v2 vp vo

vref 1

2m

qEo0 k0 vg

v2 vg vo

v2 vp vo

<WAVEPACKET>

• Hamiltonian Contours in Wave Frame

　 　 　 　 　

-0.01

-0.005

0

0.005

0.01

-4 -2 0 2 4Zn

0.4

0.6

0.8

1

1.2

1.4

1.6

-3 -2 -1 0 1 2 3

V/V

p

Zn'

<MONOCYCLE PULSE>

• Hamiltonian Contours in Wave Frame

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

V/V

p

Zn'

Question:

What happens if the pulse is nonlinear EM,

& Bo is applied?

<theory> Linear Polarization

　　　　

transit-time acceleration

& cyclotron acceleration

Ey(z, t)Eo e {(z v

gt)/ l}2 i(k

oz

ot )

(v)max 2qEtmo

e (o/o )2t2 /4 e (o /o )

2t2 /4

1 v0 / vp

t l / ( vp )

(v0 vg ) / (v0 vp )

ACCELERATIONS DUE TO EM

PULSES

Vperpp

Vz

Ho: Hamiltonian

Voz Vp

NUMERICAL RESULTS

We solve the equation of motion numerically as a function of v0, increasing En=　　　　　　 .

<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.

00 / mceE

NUMERICAL RESULTS

En= 　　　　　　＝ 0.001 .00 / mceE

0

0.01

0.02

0.03

0.04

-0.02 -0.01 0 0.01 0.02

dvzmaxdvzmindvperptheoryno-disp theoryno-disp calc

v0 /c

dispersive

non-dispersive

vz/c

En=0.01

En=0.1

Now the center of resonance has moved to =0.1c.

0

0.02

0.04

0.06

0.08

0.1

-0.04-0.03-0.02-0.01 0 0.01 0.02

dvzmaxdvzmindvperptheorydvpnodisp

v0 /c

LinearTheoryNo Disp.

00 / kv p

ACCELERATIONS DUE TO EM

PULSES

Vperpp

Vz

Ho: Hamiltonian

Voz Vp

Phase-Trapping

IF

then v/ / 2 vp vzo

pzo

o

o vvm

qEv

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

Vperpp

Vz

Ho: Hamiltonian

Voz Vp

trapping ＆ band structure

Owing to trapping, some electrons exit pulse when accelerated, while others do when not.

• The trapping period is given by 　 .

• If this becomes comparable to the transit time= , the trapping becomes important and multi-resonance occurs.

0

3.3

vv

l

g

vE

cvT

n

ptr

/20

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.