My research lies at the interface between optics and mechanics and is led under the umbrella of CNRS with collaborators at Liverpool University [visit the Liverpool Mathematics Department Home page].

Metamaterials are artificially structured composite materials which offer unique, previously unexplored material electromagnetic and elastic parameters (light and sound travel in the wrong direction) [visit Imperial College Home Page], [visit University California San Diego Home Page]. They can be also used to design invisibility cloaks!

This research monograph covers scattering and spectral problems for electromagnetic waves in periodic materials. The book focuses on a mathematical understanding of so-called photonic band gap phenomena which are range of frequencies for which no wave is allowed to propagate within a composite structure. These accurate mathematical models for electromagnetism are addressed both on spectral and diffraction aspects. The authors introduce basic tools of spectral theory (Floquet-Bloch theory for partial differential operators with periodic coefficients but also analysis of operators on unbounded domains). Some particular emphasis is made on numerical algorithms from basic survey on transfer matrix methods, to modern finite element approach, multiple methods and Cauchy integrals for operators. Furthermore, various asymptotic tools are provided to look at so-called homogenisation problems in photonic crystal fibres. Recent results on properties of these micro-structured fibres are provided together with references to specialised literature. Applications of these mathematical models are in optical fibres with ultra-flattened dispersion and photonic band gap guidance.