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Alejandro Aceves University of New Mexico, Department of Mathematics and Statistics

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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

- 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)

- 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.

- The discharge is triggered
- by the laser filament

(B) There is no filament which

bring a random pass between the electrodes.

Light Detection and Ranging(LIDAR)

Directed Energy

Remote Diagnostics

Laser guide stars

Laser Induced Lightening

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.

Aerodynamic window

CCD + Filters

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.

The number of electrons

in the medium is the function of time

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

and the intensity of the beam

where

is the third order multiphoton ionization coefficient,

the electron-positive-ion recombination coefficient

the electron oxygen attachment coefficient

third order multi-photon ionization coefficient

atom density at sea level

Wave Equation for the electric field

The change of index

due to the electron plasma can be expressed

In terms of intensity

the electron-ion collision frequency

the laser frequency

the plasma frequency

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:

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

Let

Dimensionless equations

(1)

(2)

where

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

becomes a nonlinear eigenvalue problem

Equation for

is an eigenvalue and

where

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

boundary conditions:

for

Near r equal to zero we have

where

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

the solution

Results (relevant to the experimental realization)

vs

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

simulation. (see the buildup of the plasma leading towards

steady state)