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

Intense optical pulses at UV wavelength

Work funded by ARO grant W911NF-06-1-0024

6th ICIAM Meeting, Zurich Switzerland, July 2007

Recent work

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-filamentationof femtosecond laser beams. Almost all in the IR regime.

•It is possible, for example, to control the path of the discharge ofelectrical charge by creating a suitable filament. The discharge willfollow the pass of the filament instead of a random pass.

•In air, the laser beam size remains relatively of the same size aftera propagation distance of hundreds of meters up into the normallycloudy 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 forpropagation in long distances.

(A) (B)

(A) The discharge is triggeredby the laser filament

(B) There is no filament whichbring a random pass between the

electrodes.

Applications

Laser Induced Lightening

Light Detection and Ranging(LIDAR)

Directed Energy

Remote Diagnostics

Laser guide stars

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 diffractionand 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. Thepropagation of the filament once formed, is practically lossless. If wematch the shape of the intensity at the input we can minimize loss ofenergy in the filament as it propagates in Aerodynamic window.

Equation for the plasma

eeepe NNN

dt

dN 20

6)3(

and the intensity of the beam

where )3( is the third order multiphoton ionization coefficient,

ep the electron-positive-ion recombination coefficient the electron oxygen attachment coefficient

eNThe number of electrons in the medium is the function of time

)3( third order multi-photon ionization coefficient

0N atom density at sea level

[Jens Schwarz and J.C. Diels,2001]

NLttkzwti

ttzztr Pec

n 20

)(22

202

eip

ei

p ii

iP 1)( 2

22

2

The change of index n due to the electron plasma can be expressed

In terms of intensity

0

22

e

ep m

eN

ei the electron-ion collision frequency

the laser frequency

p the plasma frequency

ei

Wave Equation for the electric field

20

2)3(0

)1(0 PPPP NLL

Reduced equation for the modelThe model to be considered is an unidirected beam described by anenvelope approximation that leads to the following equation:

where the second and third terms on the right-hand side describethe second and third order nonlinearities of the propagation whichrespectively introduce the focusing and defocusing phenomena

eeepe NNN

dt

dN 20

6)3(

Letziertrz )(),,(

eNmc

eni

nni

rrrk

i

z 20

20202

02

2

377

1

2

Dimensionless equations

eee NCNCCt

N5

24

63

,377

2 2

02

02

20

1

r

nnkC

where

002

20

2 Nemc

ekC

,0

6000

)3(

3 Ne

tNC

05 tC ,004 tNeC ep

(1)

(2)

C1 = 1.155, C2 = 3.5405, C3 = 1.62 × 10−4, C4 = 1.3 × 10−4, C5 = 1.5 × 10−4

eNCCrrr 2

2

12

21

Search of the stationary solution

052

46

3

eee NCNCCt

N

4

643

255

2

4)(

C

CCCCNe

eNCCrrr 2

2

12

21

Equation for becomes a nonlinear eigenvalue problem

where is an eigenvalue and

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 03 C

4

643

255

2

4)(

C

CCCCN e

boundary conditions:

0),0( tr

52.3RArR erAtr 2

1

),( 1r

Numerical Approach

for 1...2 mj

2

21

rrr

Near r equal to zero we have

4

643

255

2

4)(

C

CCCCNe

nj

nj

nje

nj

nj

nj

nj

nj

nj

nj NCC

hhjh

)(2

2

12

2

12

1111

nnnoe

nnnn

Nh 00

2

00201 )(

222

0)()( fAF

where

Using the continuation method along with Newton’s method we can find the solution

)()( )(1)()()()1( nnnnn FF

tol

2

2

21

44

1

2232

45

43

23

2

mm

hA

nm

n

n

1

1

0

nm

nme

nm

nm

nne

nn

nne

nn

N

N

N

f

111

2

1

111

2

1

000

2

0

)(

)(

)(

)(

62.13 C 023.0

Results (relevant to the experimental realization)

3C vs

13 10C 8.0 13 C 156.0

Profile at 1.5m propagation

Profile at 2m propagation

Immediate future work

1. Stability analysis.

3. Full

simulation. (see the buildup of the plasma leading towardssteady state)

),,( rtz

Helpful is stability with CW case as it will give us some insight of the

full Linear stability analysis.

2. Modulation theory.