Computational Nonlinear and Relativistic Optics

In the mid-1990's, first experiments on the meter-range propagation of femtosecond (fs) laser pulsed beams were performed. In these experiments, infrared laser pulses with a duration of about 100 fs produced narrow filaments of several meters. More than 10 % of the energy was observed to be localized in the near-axis area. Similar results, but on smaller length scales, are known for propagation in dense media like silica or water. This so called filamentation is attributed to the initial self-focusing of laser radiation, which originates from the Kerr response of air and leads to an increase of the light intensity. This growth is then saturated by the defocusing action of the electron plasma created by photoionization of the medium. The understanding of the complex dynamics of these filaments is crucial for potential applications such as supercontinuum generation, pulse compression, generation of very high harmonics, remote sensing, material processing, etc.
Pulse compression in pressurized cells filled with noble gases by femtosecond filaments is one of the hot and challenging topics in nonlinear optics. The impressively simple setup and the high compression rates achieved so far attracts much interest at recent international conferences. Fully space-time resolved simulations can give insight into the details of the compression mechanism which are not accessible in experiments.
One of the most spectacular features of femtosecond filamentation is the huge spectral broadening during the filamentation process. Therefore, this process is often termed as the ''white light laser''. The underlying mechanism of this supercontinuum generation is basically self-phase modulation by the optical Kerr effect, but the complicated spatio-temporal dynamics, especially the formation of shocks can introduce a strong asymmetry in the spectrum.

• L. Bergé, C.-L. Soulez, C. Köhler, and S. Skupin. Role of the carrier-envelope phase in laser filamentation. Appl. Phys. B: Lasers & Optics, 103:563, 2011.
• C. Köhler, L. Bergé, and S. Skupin. Effect of nonlinear dispersion on pulse self-compression in a defocusing noble gas. Physica D, 240:963, 2011.

The propagation and dynamics of localized nonlinear waves is a subject of great interest in a range of physical settings stretching from nonlinear optics to plasmas and ultracold atomic gases. The structure and stability of nonlinear optical modes is determined by the interplay of the radiation field with the functional form of the material nonlinearity. In the case of optical beams the nonlinear response can be described in terms of the induced change in the refractive index n which is often approximated as a local function of the wave intensity. However, in many real physical systems the nonlinear response is spatially nonlocal which means that the refractive index depends on the beam intensity in the neighborhood of each spatial point. This occurs, for instance, when the nonlinearity is associated with some sort of transport processes such as heat conduction in media with thermal response, diffusion of charge carriers or atoms or molecules in atomic vapors. It is also the case in systems exhibiting a long-range interaction of constituent molecules or particles such as in nematic liquid crystals or dipolar Bose-Einstein condensates.
Nonlocality is thus a feature of a large number of nonlinear systems leading to novel phenomena of a generic nature. For instance, it may promote modulational instability in self-defocusing media, as well as suppress wave collapse of multidimensional beams in self-focusing media. Nonlocal nonlinearity may even represent parametric wave mixing, both in spatial and spatio-temporal domain where it describes formation of the so called X-waves. Furthermore, nonlocality significantly affects soliton interaction leading to formation of bound state of otherwise repelling bright or dark solitons. It has been also shown that nonlocal media may support formation of stable complex localized structures. They include multihump and vortex ring solitons.

• F. Maucher, S. Skupin, and W. Królikowski. Collapse in the nonlocal nonlinear Schrödinger equation. Nonlinearity, 24:1987, 2011.
• F. Maucher, N. Henkel, M. Saffman, W. Królikowski, S. Skupin, and T. Pohl. Rydberg-Induced Solitons: Three-Dimensional Self-Trapping of Matter Waves Phys. Rev. Lett., 106:170401, 2011 

The control of laser beam coherence properties is crucial to optimize the coupling between laser beams and target in Inertial Confinement Fusion (ICF). This control is currently achieved by optical smoothing techniques. However, recent works have underlined the capability of the plasma to modify these coherence properties.
At sufficiently high intensities, the interplay between filamentation and Forward Stimulated Brillouin Scattering (FSBS) is responsible for the observed coherence loss. This regime shows a strong reduction of the temporal coherence, but filamentation is associated with a strong spreading of the transmitted light and enhanced back-scattering instabilities.
We are interested in a lower intensity regime below the filamentation threshold where only collective effects of an ensemble of speckles can explain the observed laser beam smoothing. Here laser beam multiple scattering (MS) on the self-induced density fluctuations reduces spatial and temporal coherence of the transmitted light. Also, MS serves as a strong seed for FSBS. It turns out that it is possible to obtain induced smoothing in a low density plasma without strong angular spreading. This smoothing could be enhanced by introducing a low density foam layer surrounding the target. The additional MS on the naturally inhomogeneous plasma created from the laser interaction with the foam should further reduce the coherence of the transmitted light.

• L. Lancia, M. Grech, S. Weber, J.-R. Marquès, L. Romagnani, M. Nakatsutsumi, P. Antici, A. Bellue, N. Bourgeois, J.-L. Feugeas, T. Grismayer, T. Lin, Ph. Nicolaï, B. Nkonga, P. Audebert, R. Kodama, V. T. Tikhonchuk, and J. Fuchs. Anomalous Self-Generated Electrostatic Fields in Nanosecond Laser-Plasma Interaction. Phys. Plasmas, 18:030705, 2011 
• S. Depierreux, C. Labaune D. T. Michel, C. Stenz, P. Nicolaï, M. Grech, G. Riazuelo, S. Weber, C. Riconda, V. T. Tikhonchuk, P. Loiseau, N. G. Borisenko, W. Nazarov, S. Hüller, D. Pesme, M. Casanova, J. Limpouch, C. Meyer, P. Di-Nicola, R. Wrobel, E. Alozy, P. Romary, G. Thiell, G. Soullié, C. Reverdin, and B. Villette. Laser Smoothing and Imprint Reduction with a Foam Layer in the Multikilojoule Regime. Phys. Rev. Lett., 102:195005, 2009. 

The possibility of using high power lasers to generate multi MeV proton beams is extremely interesting for applications in physics, engineering, or medicine. In physics, such beams can be used to initiate the thermonuclear reaction in the so called fast ignitor scheme for ICF if one can improve the efficiency of energy conversion and control the energy dispersion. Because these proton beams are laminar, well collimated, and due to their short duration (of the order of a ps), they can also be used for time resolved radiography in plasma experiments. Laser created proton beams with energy up to a few MeV are already used in current experiments. However, probing the high density compressed hot spot in ICF experiments requires proton energies of up to 60 MeV. For medical applications in cancer therapy, even higher energies (up to 200 MeV) and a sufficiently small dispersion in energy are required.
For all applications the proton beam energy distribution, angular divergence, duration and the laser to proton energy conversion efficiency have to be controlled. The laser pulse duration, energy, peak intensity, polarization and the geometry of the focal spot strongly influence the proton beam characteristics. There is so far no clear understanding of all these dependencies. Their investigation by means of numerical simulations and theoretical modeling should provide practical propositions to optimize the particle acceleration and beam quality.

• M. Grech, S. Skupin, A. Diaw, T. Schlegel, and V. T. Tikhonchuk. Energy dispersion in radiation pressure accelerated ion beams. New J. Phys., 13:1230033, 2011         
• M. Grech, R. Nuter, A. Mikaberidze, P. Di Cintio, L. Gremillet, E. Lefebvre, U. Saalmann J. M. Rost, and S. Skupin. Coulomb explosion of uniformly charged spheroids. Phys. Rev. E, 84:056404, 2011. 

In recent years, the range of wavelengths where coherent radiation can be generated has grown dramatically into both high and low frequency domain. Remarkably, most of the methods to obtain radiation at extreme frequencies use, in one or the other way, nonlinear processes in laser-induced plasma. One prominent example is high harmonic generation (HHG) where frequencies thousand times larger than the frequency of the pump pulse are excited, exploiting the recollision dynamics of electrons ionized by the intense light pulses. More recently it was demonstrated that a two-color fs beam allows generation of new frequencies just in the opposite part of the spectrum, namely in the THz range, hundreds times smaller than the optical pump frequency. To this end, a short two-color pulse of fundamental frequency and second harmonic is strongly focused into a plasma spot. The observed THz emission generated in this scheme has been attributed to the laser-induced plasma current in the asymmetric twocolor field. Using such scheme generation of strong THz radiation was reported, with a spectrum which can be as broad as 70 THz. Such broad-band coherent radiation can allow new applications providing the possibility to probe complex molecules or as an analytical and imaging tool in a broad range of fields.

• I. Babushkin, S. Skupin, A. Husakou, C. Köhler, E. Cabrera-Granado, L. Bergé, and J. Herrmann. Tailoring THz radiation by controlling tunnel photoionization events in gases. New J. Phys., 13:123029, 2011 
• C. Köhler, E. Cabrera-Granado, I. Babushkin, L. Bergé, J. Herrmann, and S. Skupin. Directionality of THz emission from photoinduced gas plasmas. Opt. Lett., 36:3166, 2011 

The stimulated scattering of light is one of the major topics in nonlinear optics. Although several types of stimulated scattering (Raman, Brillouin and thermal Rayleigh) were already discovered in the 1960s, related topics are still highly active. Stimulated scattering always involves a pump laser beam and a frequency-shifted scattered wave, coupled by either molecular vibrational transitions (Raman) or acoustic waves (Brillouin). The generation of intense, frequency-shifted radiation through stimulated Brillouin scattering (SBS) is currently used in tunable laser sources, coherent optical communication systems, Brillouin amplifiers and sensors. SBS occurs in a large variety of transparent media and plasmas, from single-mode fibers to all-optical silica devices employed, e.g., in large laser facilities devoted to inertial confinement fusion. In materials with no optical absorption, SBS is mainly driven by the electrostriction strain produced by an intense laser pulse with long enough (nanosecond) durations. This strain excites acoustic waves on which a Stokes wave scatters a significant amount of energy, preferably in the direction opposite to the pump one. For powerful pump beams, the counterpropagating Stokes wave can convey high enough fluence to cause severe damage and SBS appears as a harmful process that limits the pulse energy of high-power laser sources. The standard approach for reducing SBS is the use of broadband lasers, which works quite well in one-dimensional (1D) geometries and discarded Kerr optical response. It turns out that in full three-dimensional (3D) configurations, however, the coupling between SBS and Kerr nonlinearities becomes a crucial issue. Let us indeed recall that Kerr nonlinearities cause modulational instabilities and can even lead to catastrophic wave collapse at high dimensions once the pump peak power exceeds the self- focusing threshold. For understanding the initiation of material damage by powerful lasers, it is necessary to model the interplay between SBS and the self-focusing (SF) process in 3D geometries.

• S. Mauger, L. Bergé, and S. Skupin. Controlling the stimulated Brillouin scattering of self-focusing nanosecond laser pulses in silica glasses. Phys. Rev. A, 83:063829, 2011. 
• S. Mauger, L. Bergé, and S. Skupin. Self-focusing versus stimulated Brillouin scattering of laser pulses in fused silica. New J. Phys., 12:103049, 2010. 

The detailed understanding of nonlinear effects in optical systems has been the goal of many research activities in recent years. With the rapid development of both powerful and controllable light sources, many challenging effects were discovered just by increasing the intensity. For example, by enhancing the intensity of the optical field in a waveguide, self-focusing due to the optical Kerr effect can change the guiding properties dramatically. This self-focusing process is basically described by the nonlinear Schrödinger (NLS) equation, which governs the evolution of the slowly varying envelope of the electric field, and it can partly be tamed by coupling the beam with an appropriate potential. In particular, we can expect stable spatial solitons in such systems. A weakly nonlinear optical multi-mode waveguide is an ideal simple system to investigate various properties of rotating and nonrotating higher order solitons.

• Y. Zhang, S. Skupin, F. Maucher, A. Galestian Pour, K. Lu, and W. Królikowski. Azimuthons in weakly nonlinear waveguides of different symmetries. Opt. Express, 18:27846, 2010.     
• S. Skupin, U. Peschel, L. Bergé, and F. Lederer. Stability of weakly nonlinear localized states in attractive potentials. Phys. Rev. E, 70:016614, 2004.