Integral equations for the modeling of plasmonic resonance of nanoparticles
 发布时间： 2017-08-08     09:58   【返回上一页】 发布人：\$curArticle.author

In the past decade, the modeling of metamaterials, and more generally of composite optical materials, has generated a considerable mathematical activity. Indeed, subwavelength composite structures made of metallic and dielectric materials may exhibit very interesting properties of enhancement or localization of the electromagnetic fields, that are very interesting for applications, such as nano-lithography, bio-sensing, medical imaging or cancer therapy.

In this course, we focus on the modeling of metallic nano-particles, that show remarkable resonant properties in the visible light. We present some of mathematical theory that has been designed to study the PDE’s (the conduction equation, the Helmholtz and the Maxwell equations) that describe such nano-particles, but are no-longer elliptic in this context.

In particular, we study the Neumann-Poincar/'e operator, an integral operator that naturally appears in the context of metamaterials as it may be used to represent the solutions of  transmission problems via potentiel theory. We are particular interested in its spectral properties, which are closely related to the well-posedness of these PDE's, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium.

The course is divided in 4 parts:

- An introduction on metamaterials, plasmonic resonances of nano-particles, the associated quasistatic asymptotic regime.

- The integral representations of solutions in the quasistatic regime, the Neumann-Poincar/’e operator and its spectral properties in the case of smooth nano-particles.

- The case of polygonal nano-particles in 2D : existence and characterization of the essential spectrum of the Neumann-Poincar/’e operator and its relation with elliptic corner singularities.

- A lecture on the spectral properties of a periodic collection of metallic nano-particles and its relation to homogenization.