Transcript ipf

INSTITUTE OF APPLIED PHYSICS RAS
RUSSIA
NIZHNY NOVGOROD
ULTRASHORT RELATIVISTICALLY STRONG
SOLITONS IN PLASMA
Authors:
Fadeev D. A. ([email protected])
Mironov V. A.
introduction
Initial 3d laser pulse
y
x
V ~c
z
rare plasma
nonlinear response factors:
relativistic electron weighting
the push out of electrons from area occupied by laser
pulse with ponderomotive force (transverse striction)
excitation of nonlinear oscillations of electrons behind
laser pulse (wake field)
ions supposed to be motionless
nonlinear wave equation
dimensionless wave equation:
2 A
A
2
  A 
0
2
z
1 A
A
- complex vector potential
Re A  Ax
    p1
[ x, y , z ]   p1  c
A  A0  mc 2 / e
Im A  Ay
the model consists of a single equation for complex value

 tz
1 
2
2
symmetry:  
r
,
where
r

x

y



r r  r 
- co-moving time:
cylindrical
initial conditions:
Ax ~ cos 
Ay ~ sin  
circular polarization
no multiple spectrum generation
problem evolution
1. NSE research (smooth envelopes).
I
A.B. Borisov, A.V. Borovskii, V.V. Korobkin et al., Sov. Phys. JETP 74(4), 604 (1992)
II
“OPTICAL SOLITONS from fibers to photonic crystals" Y.S.Kivshar, G.P. Agrawal (2003)
III
“SOLITONS nonlinear pulses and beams” N.N. Akhmediev and A. Ankiewicz (1997)
existence and steadiness of 3d soliton structures was determined
2. Switching to investigation of real field dynamics.
IV
A.A. Balakin and V.A. Mironov, JETP Lett. 75 (12), 617 (2002)
The common features of electromagnetic pulses dynamics were
investigated in the case of Kerr type nonlinearity.
limitations of the model
quasi-flat model:
3
L||
L
L||
 A
  
L
 A0 
1
here A0  c me
is the
typical
relativistic
vector potential
2
thus
the
transverse
push
out
electrons from the area occupied
electromagnetic laser pulse could
neglected
of
by
be
non-refractive approach:
 p  
no wake field:
L||

p
c
1
wake field excitation is suppressed if the duration
of laser pulse is much greater than plasma wave
period
hamiltonian formalism
integrals:
energy integral
2
A
2
quanta number integral
2


A
H   
 2 1  A  2 r dr d
I  
r dr d
  r
 

impulse integral
 A * A* 
P  i   A 
A r dr d (could not be obtained by
 
standard routine for
  
momenta evolution:
d c
H

dz
I
hamiltonian systems)
centre of mass description
centre of mass
conditions only
velocity
defined
2
2


2

A

2
1

A
d r
8  2
r dr d

A


r
2

dz 2 I  
1 A

2 2
by
initial
Laser pulse
width
description
allow to investigate initial evolution of some types
solutions according to the set of it’s initial parameters
of
solitonsI
soliton general form:
A( , z, r )  f ( , r )ei ( )

), (  V –
where:
f ,  
  Vz
  Vz
V
, 
V
is the free parameter
solution:
 ( )  a
solution for phase
)1,1(  a
1 d  2 df  2
f


a
f

0


2
2
 d  d 
1 f
   r
2
2
equation for amplitude
(to be solved numerically)
Assumption of central symmetry for all 3
coordinates (two transverse spatial coordinates
and co-moving time) allows to obtain quite
simple differential equation for amplitude of
soliton
solitonsII
two parameters:
70

Are
2
1.4
0.28
0
0
60
0
50
40
a=0.95
30
-2
a=0.97
-1.4
a=0.999
-0.28
r
0
10 20 30 40 0
10 20 30 40
0
10 20 30 40
- selects soliton amplitude, transverse width and spectral
characteristics (number of optical periods in wave packet)
a
a1
V1
V-
V1  V2
a1
V2
scales whole soliton field along co-moving time coordinate
steadiness
Lapunov’s method for partial derivative equations:
F [ A]  H  I  P
F [ A]
A*
1   f   2 f
f
2

 2af  0
r   2  a f 
r r  r  
1 f 2
A  f ( r, )eia

f
f
 a  0    a


Lapunov’s functional is combined of integrals of the equation
under investigation thus it stays constant during the evolution
minimization under the conditions of
2
constant pulse P
P
W
H I 8 2
W
1 W
here
W   A r dr d
2

numerical investigation of steadiness:
16
duration
100
transverse width
12
80
60
8
40
4
20
evolution step/30
0
0
400
800
1200
1600
evolution step/30
0
0
400
800
1200
1600
solitons interactionI
by
varying
initial
delay Td
between solitons
the relative phase before
collision could be set
Td  52.9
-50
0
0
50
1
-50
0
Td  58.650
1
0
50
2
 100 A
3
2V
100
A
V
2.4
1.6
2.4
50 r
2
2
4
0.8
3
1.6
5
1.2
4
5
0
6
0.8
0.4
0
7
solitons interactionII
results of two solitary waves interaction for different parameters
of initial solitons.
surfaces of amplitude after interaction on axis of the system

A( r  0, )
  0..2
longitudinal instability
of a wave packet
986
1
6
10
23
A
2.4
2.1
1.8
1.5
1.2
0.9
0.6
0.3
0
121
r
0 .6
156

initial gaussian pulse breaks apart on
two solitary waves
soltons have different velocities
transverse instability of a
flat wave (2D simulations)I
The break-up of non perturbed super gaussian wave packet
x
9
0
17
31

72
120
165
47
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0
A
200
2 A 2 A
A
 2
0
4
z x
1 A
nonliarity of 4th
order allows to
assume same processes
in 3D case
transverse instability
of a flat wave (2D simulations)II
x
The break-up of initial super gaussian pulse on a quantity of
solitons in 2D model for nonsymmetrical perturbation
nonliarity of 4th order allows to
2 A 2 A
A


0
assume same processes in 3D case
4
z x 2
1 A
11
0
18
24
49
78
140

x
in case of 2nd order of nonliarity  2 A  2 A
A


0
the break-up is slower
2
z x 2
1 A
11
0

18
24
49
78
140
A
4.8
4.2
3.6
3
2.4
1.8
1.2
0.6
0
A
4.2
3.6
3
2.4
1.8
1.2
0.6
0
solitons in the full wave
equation
For dense plasma the assumption of non refractive propagation is
incorrect. In that case the full wave equation is appropriate:


2
2
2

A

A

A
A
2
1 v
 2v
 2   A 
0
2
2
z z

1 A
The last equation is obtained in new coordinates,
moving with the velocity v :
  t  v  z, z  z
common form for solitary wave solution:

1  v2  ~
b
a
v
A( , z, r )  f ( , r )e
i  a~ bz 
the parameter b could be obtained with substitution of
the solitary wave form to the wave equation
1   2 f  1  v 2 ~ 2
f
 2


a
f

0


2
2
     v
1 f
  1  v2   2  r 2
1
thus the soltions of full wave equation could be obtained from
ones
demonstrated
previously
(with
V=1)
with
simple
transformations:
v
1  v2
->
frequency
multiplication
1
1  v2
-> longitudinal scaling for
envelope
results
•
The mathematical approach for investigation of cold plasma and
electromagnetic
pulse
interaction
was
developed.
The
generalization for the case of relativistic motions of plasma
particles was performed.
•
The
numerical
code
for
the
model
describing
nonlinear
interaction of ultrashort laser pulse with plasma was developed
(the code was optimized for distributed computation systems)
•
The soltions for the equation under investigation was obtained
analytically as a generalization of NSE (Nonlinear Shroedinger
equation)
•
The steadiness of found solitons was checked both analytically
and numerically
•
The detailed
performed.
•
The processes of transverse and longitudinal instability of wave
packets accompanied by formation of soliton type structures was
demonstrated in numerical simulations
numerical
analysis
of
solitons
interaction
was
thanks
mail to: [email protected]
fadeev daniil
find detailed information in article:
V.A. Mironov, D.A. Fadeev, JETP, 106, 974-982 (2008)