Download Intense light propagation in air

Intense optical pulses at UV wavelength
Alejandro Aceves
University of New Mexico, Department of Mathematics and Statistics
in collaboration with A, Sukhinin and Olivier Chalus, Jean-Claude Diels,
UNM Department of Physics and Astronomy
6th ICIAM Meeting, Zurich Switzerland, July 2007
Work funded by ARO grant W911NF-06-1-0024
Recent work





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S. Skupin and L. Berge, Physica D 220, pp. 14-30 (2006), (numerical)
Moloney et.al, Phys. Rev E, 72:016618 (numerical, looking at
instabilities of CW solutions, pulse splitting)
Many authors have studied singular collapse phenomena
Fibich, Papanicolau, Developed modulation theory to study perturbed
NLSE at critical values of dimension/nonlinearity
A. Braun et.al, Opt. Lett 20, pp 73-75 (1994) (experimental, short
pulses at 755nm)
J.C. Diels et. al., QELS proceedings (1995) (experimental verification
of UV filamentation for femtosecond pulses)
Background and Motivation
•There have been several experiments which confirm the self-filamentation
of femtosecond laser beams. Almost all in the IR regime.
•It is possible, for example, to control the path of the discharge of
electrical charge by creating a suitable filament. The discharge will
follow the pass of the filament instead of a random pass.
•In air, the laser beam size remains relatively of the same size after
a propagation distance of hundreds of meters up into the normally
cloudy and damp atmospheric conditions.
• Gain a complete understanding of the filamentation of intense UV
picosecond pulses in air. On the theoretical side, the interest is to find
stationary solutions of the modeling equations and determine their stability.
Thus we expect we will give insight to the experimental conditions for
propagation in long distances.
(A)
(A) The discharge is triggered
by the laser filament
(B)
(B) There is no filament which
bring a random pass between the
electrodes.
Applications
Light Detection and Ranging(LIDAR)
Directed Energy
Remote Diagnostics
Laser guide stars
Laser Induced Lightening
Formation of a filament
High power pulses self-focus during their propagation through
air due to the nonlinear index of refraction.
At some critical power this self-focusing can overcome diffraction
and possibly lead to a collapse of the beam.
Short pulses of high peak intensity create their own plasma due
to multi-photon ionization of air. When the laser intensity exceeds
the threshold of multiphoton ionization, the produced plasma will defocus
the beam. If the self-focusing is balanced by multiphoton ionization
defocusing, a stable filament can form.
CCD +
Filters
Aerodynamic
window
Vacuum
UV Beam
Filament array in air
Figure. Setup of the Aerodynamic window, Focus of the
beam into the vacuum then propagation of the filament in
atmospheric pressure
The possible propagation of filament is dependent on input power.
Most of the energy loss occurs in the formation of the filament. The
propagation of the filament once formed, is practically lossless. If we
match the shape of the intensity at the input we can minimize loss of
energy in the filament as it propagates in Aerodynamic window.
Equation for the plasma
The number of electrons N e in the medium is the function of time
and the intensity of the beam [Jens Schwarz and J.C. Diels,2001]
dN e
  ( 3) 6 N 0   ep N e2  N e
dt
where  is the third order multiphoton ionization coefficient,
 ep the electron-positive-ion recombination coefficient
 the electron oxygen attachment coefficient
( 3)
 (3) third order multi-photon ionization coefficient
N 0 atom density at sea level
Wave Equation for the electric field
2

n
2
2
i ( wt  kz )
2
0






e



zz
tt 
0 tt PNL
 tr
2
c


P  PL  PNL 0    0 
(1)
The change of index
In terms of intensity
( 3)
n due to the electron plasma can be expressed
 p2
 p2  ei 
P 2  i
 2 1  i 
 (ei  i )  

 ei

p
the electron-ion collision frequency
the laser frequency
the plasma frequency
N ee2
 
me 0
2
p
   0 P 2
2
 ei  
Reduced equation for the model
The model to be considered is an unidirected beam described by an
envelope approximation that leads to the following equation:
2
2


n2 n0 2
n0e 0

i 1   
  
 2   i 0
   i 2 N e
z
2k  r r r 
377
cm
where the second and third terms on the right-hand side describe
the second and third order nonlinearities of the propagation which
respectively introduce the focusing and defocusing phenomena
dN e
  (3) 6 N 0   ep N e2  N e
dt
Let
 ( z, r , t )   (r )e
 iz
Dimensionless equations
1   
2
 2  C1   C2 N e  
r r r
2
N e
 C3 6  C4 N e2  C5 N e
t
(1)
(2)
where
k0 e 2
2kn02 n2 2
2
C2  2  0 Ne0
C1 
r0  0 ,
c m
3776
( 3)
 N 0 t 0 0
C3 
, C4   ep Ne0t0 , C  t
5
0
Ne0
C1 = 1.155, C2 = 3.5405, C3 = 1.62 × 10−4, C4 = 1.3 × 10−4, C5 = 1.5 × 10−4
Search of the stationary solution
N e
6
2
 C3  C4 N e  C5 N e  0
t

 C5  C  4C3C4
Ne ( ) 
2C4
2
5
6
Equation for

becomes a nonlinear eigenvalue problem
1   2
2
 2  C1   C2 N e  
r r r
 C  C  4C C 
is
an
eigenvalue
and
N
(

)

where 
2C
5
2
5
3
6
4
e
4
Our approach is a continuation method beginning from the
A member of the Townes soliton family of 2D NLSE which is also
the solution of our model if C3  0
boundary conditions:
 12
 (r , t )  AR r e  r AR  3.52
 r (0, t )  0
r  1
Numerical Approach
 C5  C52  4C3C4 6
Ne( ) 
2C4
n
n
n
n
n
1   j 1   j 1   j 1  2 j   j 1
n n 2
n n
n


C



C
N
(

)



1 j
j
2 e
j j
j
2

jh 
2h
h

for j  2...m  1
Near r equal to zero we have
1 
 2

r r
r 2
2 1n  2 0n
n
n 2
n n
n
2




N
(

)



0
0
e
o 0
0
h2
F ( )  A  f ( )  0
where
 4 4

3
 1 2 2
1
3
2
A 2 
4

h



5
4
2 m 3
2 m2
  0n 

 n 

 1 


  






 n 
 2 
  m 1 
  n 2  n  N ( ) n  n 
0
0
e
0 0




n 2 n
n n



N
(

)

1
1
e
1
1


f ( )  





2
  n  n  N ( ) n  n 
e
m1 m1 
 m1 m1
Using the continuation method along with Newton’s method we can find
the solution

( n 1)

  tol
( n)
   
( n)
 F (
( n)
)  F ( ( n ) )
1
Results (relevant to the experimental realization)
C3  1.62   0.023
C3 vs 
C3  10 1   0.8
C3  1
  0.156
Profile at 1.5m propagation
Profile at 2m propagation
Immediate future work
1. Stability analysis.
Helpful is stability with CW case as it will give us some insight of the
full Linear stability analysis.
2. Modulation theory.
3. Full
( z, t , r )
simulation. (see the buildup of the plasma leading towards
steady state)