My work can be organized around 2 axes, each one concerning mathematical models and their numerical approximations in a particular domain of application. These mathematical models are written using Partial Differential Equations (PDE). In a first axis, I am studying ultrashort laser propagation in nonlinear crystals and the ionization of dielectrics. This is a domain where classical models are no longer relevant and new models have to be derived with their associated numerical schemes. I have tried as much as possible to validate the mathematical models with experimental data.

On the other hand, I am interested in modeling cancer growth. Contrary to the case of nonlinear optics, there is no master equation in cancer modeling. The role of the applied mathematicians is therefore not only to simplify or adapt models but also truly to write from scratch mathematical models behaving more or less in accordance with the biological or medical knowledge. Most of my models are based on fluid mechanics and level set approaches. They are written at the macroscopic scale to be useful for clinical applications. Through these works, I try to write accurate, simple and flexible methods and to compare the models with experimental data in order to have realistic or clinical applications.

Historically, my first research topic concerns the propagation of ultrashort laser pulses which I started studying during my PhD. Ultrashort pulses produce limited collateral damages in terms of stress waves, thermal conduction or melting to the materials they travel in. Their enormous power allows them to be used for material ablation or nano-machining but do not prevent their applications to biological materials. Ultrashort beams also permit to observe very short phenomena (chemical reaction in gases…). We try to validate the models with experimental data whenever it is possible.

I am interested in understanding the propagation of ultrashort laser pulses in nonlinear dielectrics. For such short pulses, the classical models are no longer accurate. I have developed a semi-classical model based on Maxwell-Bloch equations able to describe the propagation in a nonlinear crystal. Using this model, I was able to study various effects (harmonic generation, Raman scattering, optical rectification,…) with a single model whereas in many cases these phenomena were studied separately. I am now also interested in understanding the early ionization process to describe the ablation of a material for instance. It has many applications in nanotechnologies (data storage), biotechnologies (laser ablation of tumor, bioprinting) or for damage studies of optical materials. When studying laser ablation with femtosecond beams, the electromagnetic part can be safely decoupled from the hydrodynamical processes.