Mes activités de recherche actuelles se situent dans le cadre de la morphogenèse optique et plus généralement de la dynamique non-linéaire. Les systèmes étudiés sont les milieux Kerr (cristaux liquides et fibres optiques) en cavité ou avec feedback optique. J'y étudie plus particulièrement les phénomènes d'instabilités temporelles et spatio-temporelles tels que : - la formation de structures transverses et les instabilités modulationnelles - les solitons dissipatifs et les structures localisées - les systèmes convectifs et leurs instabilités convectives et absolues - les effets du bruit sur ces instabilités, tels que les structures entretenues par le bruit.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

This thesis reports on ultrafast linear and nonlinear optical probing of nanometer thick films. Exfoliated graphene and few-layer graphite are probed through optical second harmonic generation (SHG) with 800 nm, 150 fs pulses. Samples of varying thickness from 1 carbon layer to bulk graphite are deposited onto an oxidized silicon substrate. SHG measurements are taken as a function of azimuthal rotation angle of the films. It is found that the SHG from graphene is much weaker than that from bilayer graphene, and has a qualitatively different azimuthal pattern. As the sample thickness increases from bilayer graphene to bulk graphite, the SHG yield generally decreases. Both of these effects are explained in terms of the symmetry of graphene and graphite, and modeled using multilayer optical transfer matrices, and an identical set of nonlinear susceptibility tensor elements for the front and back surfaces. These tensors are independent of sample thickness. MnAs films of 150 and 190 nm thickness on (001)GaAs are optically excited with 775 nm, 200 fs pump pulses. Specular SHG at 388 nm and first order optical diffraction at ∼ 400 nm are used to probe the samples on timescales up to 2 μs. It is found that the SHG probes the temperature-dependent, spatially averaged, surface strain. This strain reaches a maximum deviation in ∼ 6–100 ps after optical excitation depending on the pump fluence and initial temperature. The strain then recovers in hundreds of picoseconds, a timescale consistent with heat diffusion. The optical diffraction probes the first Fourier component of the paramagnetic–ferromagnetic stripes inherent to MnAs films in the 10–40◦C temperature range. After optical excitation, the diffraction data show highly nonthermal behaviour in the MnAs films. If a sample is excited from the coexistence phase, the diffraction signal shows decaying oscillations with a period of ∼ 335±4 (408±4) ps for the 150 (190) nm films; this is consistent with the release of a standing acoustic wave. Decay occurs on a timescale of ∼ 2 ns consistent with local diffusion through the films. The stripes are restored on a timescale of hundreds of nanoseconds, with a temporal behavior consistent with a diffusion process, possibly thermal in origin.

In this thesis, the spectral behavior of the fundamental and sum-frequency waves, generated from the surface of a thin metal film in the Kretschmann configuration, is theoretically studied with coherent ultrashort pulses. As a first exploration of considering spectral response in nonlinear plasmonics, it is shown that the spectra of reflected sum-frequency waves exhibit pronounced shifts for the incident fundamental waves close to the plasmon coupling angle, whereas meanwhile those of reflected fundamental waves display energy holes. We also demonstrate that the scale of discovered plasmon-enhanced spectral changes is strongly influenced by the magnitude of the incidentce angle and the source pulse duration, and at a certain angle a spectral switch is observed. The appearance of large sum-frequency wave shifts can serve as an unambiguous plasmon signatur in nonlinear surface spectroscopy. Also, the discovered spectral switch can trigger extremely surface-sensitive nonlinear plasmonic sensors.

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