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

1. 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

2. 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)

3. 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.

4. The discharge is triggered • by the laser filament (B) There is no filament which bring a random pass between the electrodes.

5. Applications Light Detection and Ranging(LIDAR) Directed Energy Remote Diagnostics Laser guide stars Laser Induced Lightening

6. 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.

7. Aerodynamic window CCD + Filters Vacuum UV Beam

8. 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.

9. Equation for the plasma 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

10. 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

11. 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

12. 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

13. Search of the stationary solution

14. 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:

15. Numerical Approach for Near r equal to zero we have

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

17. Results (relevant to the experimental realization)

20. vs

24. Profile at 1.5m propagation

25. Profile at 2m propagation

26. 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 simulation. (see the buildup of the plasma leading towards steady state)