1. Ultra-intense Laser Pulse Propagation in Gaseous and Condensed Media Jerome V Moloney and Miroslav Kolesik Arizona Center for Mathematical Sciences

2. Overview of Talk Why envelope equations don’t work Rigorous bi-directional pulse propagator Collapse regularization in ultrafast nonlinear optics Some real world examples – novel beams ACMS Terawatt femtosecond laser laboratory

3. Maxwell’s Equations Phenomenology Long distance propagation Ultrafast waveforms Electromagnetic shocks Spectral broadening Direct solution of Maxwell’s equation not feasible!

4. Waves with the same frequency propagate with different phase and group velocities Decomposition into two envelope contribution not unique Which envelope at this frequency? Breakdown of SVEA – Third Harmonic Generation in Air Spectrally narrow slowly-varying envelopes at  and 3  Classic two envelope model fails!

5. Full Scalar Bidirectional UPPE Model Exact linear dispersion

6. Unidirectional Pulse Propagating Equation (z-UPPE) Plasma-related current Nonlinear polarization evaluated from real field Accurate chromatic dispersion Second Harmonic component = source of TH Carrier based approach, no envelope approximations used Unidirectional Maxwell - Scalar UPPE Spectral representation natural in optics – Fourier transforms

8. Collapse Regularization in NLO NLSE in 2D (critical) and 3D (supercritical) exhibits blow-up in finite time (distance) Fibich et al study Nonlinear Helmholtz equation – propose combination of nonparaxial and backward wave generation for regularization. However they ignore linear and nonlinear dispersion! All physical collapse regularization mechanisms to date involve either dispersive regularization, plasma limiting or, possibly, nonlinear saturation. Bidirectional UPPE provides a natural platform for rigorously exploring collapse regularization Dispersive regularization – Luther et al. (1994)

9. Scattered field Incident field medium wave Incident optical field is scattered from nonlinear response Effective Three-Wave Mixing: Qualitative Picture

10. Dispersion Maps – X’s, O’s and Fish Qualitative picture from linear dispersion landscape! Water Dispersion Maps Silica Dispersion Map Normal Mixed Anomalous Normal Mixed Anomalous carrier group velocity

11. Induced Nonlinear Dynamical Grating - dynamical 3 wave interaction - dynamical phase matching: Local time Angular Frequency Angle Material response perturbation

12. Filamentation of Airy beams in water  spectra (angularly resolved spectra) Optical frequency – horizontal axis Transverse K-vector (conical angle) – vertical axis Analysis of  spectra reveals details of pulse evolution P. Polynkin, M. Kolesik, J. Moloney, to appear in September 25 issue of Phys. Rev. Lett. (2009)

13. Asymptotic Structure in Spectral Space Experiment UPPE Simulation

14. Analytical Structure in Angularly Resolved Spectra Pump X-wave = Pump scattered off peak p: Stokes X-wave = Stokes scattered off peak p:Mixing two stokes photons with one pump X-wave photon: Mixing two pump photons with one stokes X-wave photon: Angularly-resolved spectrum in water – pump pulse at 1100nm, seed at 527nm

15. Beam shapes commonly used in filamentation studies: Gaussian beams Flat-top beams Beam shaping: Bessel beams Axicon Approximate extent of linear focus cm Plasma density, experiment Observe single, stable filament at pulse energies up to 15mJ Plasma channels cover the entire extent of linear focus zone of BB Optics Express, vol. 16, p. 15733 (2008)

16. X Y Linear properties of Airy beams: Self-healing Resist diffraction Similar to Bessel beams Self-bend or “accelerate” Center of mass propagates along straight line G. Siviloglou, J. Broky, A. Dogairu, D. Christodoulides, Phys. Rev. Lett., vol. 99, 213901 (2007) Beam shaping: 2D Airy beams

17. Filamentation of Airy beams in Air 35fs pulses 800nm wavelength 5-15mJ energy per pulse Meter-long propagation fs pulses Far-Field f f Phase Mask Lens Fourier Plane Plasma Channel P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, D. Christodoulides, Science, vol. 324, p. 229 (2009)

18. Challenge in simulation of Airy-beam ultrashort pulses Large spatial extent Fine-scale structure in the near field Fine-scale structure in the far-field Temporal pulsed dynamics All imply: Large numerical grids, large-scale simulation Near field fluence profile Curved plasma channels Far-field structure These simulation capture the intense filament core. Capturing weak supercontinuum spectra is MUCH more challenging...

19. Challenge in simulation of Airy-beam ultrashort pulses Large spatial extent Fine-scale structure in the near field Fine-scale structure in the far-field Temporal pulsed dynamics All imply: Large numerical grids, large-scale simulation Simulations: Large, 3D domain Fine grid resolution (1536 – 4096)^2 x (128 – 256)‏ Simplified model: ●diffraction ●gvd + 3-order dispersion ●instantaneous Kerr ●plasma MPI generation ●plasma induced defocusing

20. Short Pulse Equation (1D)

21. Novel self-compression mechanism for ultrashort pulses Theoretically studied in glass- membrane fibers with anti-guiding thickness profile Experiments are under way at Max Planck Institute for Physics of Light Simulations predict very large self- compression at high efficiency. Better control than normal self-compression in femto-second filaments. Applicable to different media - such as preformed plasma channel, and gas slab wave-guides (next slide). Significant self-compression

22. Novel self-compression mechanism for ultrashort pulses Picture: simulated anti-guiding driven selfcompression from 50fs to 5fs duration in a planar gas- slab wave-guide. Simulations are being used to study different scenarios and optimize the process. Rich system, many potentially interesting regimes! glass argon, air,... Recent interest in slab-geometry gas-filled waveguides (Midorikawa,Mysyrowicz)‏ Advantages: potential for energy scaling, dispersion tuning, off-axis phase matching,...

23. Hollow-core photonic crystal fibers‏ Controlled nonlinear optics in gas-filled hollow core fibers Dispersion management through fabrication

24. Multiple filaments in Atmospheric propagation Propagation up to 30km vertically in atmosphere!

25. Assembled in 2007-2008 under support from AFOSR DURIP Supports on-going computational program at ACMS 35 femtosecond pulsewidth 35 mJ pulse energy 10 Hz PRF Integrated pulse shaper (temporal) Pulse diagnostics (FROG, correlator) Beam shaping via static phase masks (high pulse energy) Beam shaping with programmable 2D LC matrix (<3mJ) High energy OPA: Tunable multi-mJ, <100fs pulses, wavelength coverage from 470nm to 2.6  m Our TW laser facility Pavel Polynkin (OSC)

26. Filamentation Laser filaments in air: Self-focusing are dynamically balanced by plasma de-focusing

27. Useful properties and applications of filaments in air: Extended propagation (up to hundreds of meters) Relative immunity to obscurants and turbulence Forward-emission of broad supercontinuum Electrical conductivity

28. Filamentation of Airy beams in Air Beam displacement proportional to z 2, ~10mm per m 2 Generated plasma channels are curved, follow parabolic beam trajectory P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, D. Christodoulides, Science, vol. 324, p. 229 (2009)

29. 1O1O 0O0O -1 O -2 O 0O0O 1O1O 2O2O -1 O -2 O 0O0O 1O1O 2O2O -1 O -2 O 0O0O 1O1O 2O2O -1 O -2 O Direct emission patterns, 800nm light blocked Full pattern Beginning of filamentEnd of filament Filamentation of Airy beams in Water: Forward emission from different parts of filament is angularly resolved P. Polynkin, M. Kolesik, J. Moloney, to appear in September 25 issue of Phys. Rev. Lett. (2009)

30.  spectra, Airy beams in water Full pattern Beginning of filamentEnd of filament 5.0 O 2.5 O 0.0 O -2.5 O -5.0 O 800nm 633nm 532nm800nm 633nm 532nm800nm 633nm 532nm P. Polynkin, M. Kolesik, J. Moloney, to appear in September 25 issue of Phys. Rev. Lett. (2009)