Electromagnetism with Extreme Materials

Content. We propose to model the electromagnetic waves generated by linear and eventually nonlinear polarizations created by small scaled particles enjoying extreme values of their electric permittivity or magnetic permeability. Our aim is to model such composites, estimate the generated fields and apply such results to the following selected directions: (1) Material Sciences (Linear and Non-Linear Volumetric as well as low dimensional metamaterials), (2) Imaging using Contrast Agents (as Optical Imaging as well as Photo-Acoustics) and (3) Therapy with Heat Generated by Nanoparticles. Hypotheses. The models we propose are related to the electromagnetism in the presence of nano-scaled particles enjoying extreme values of their electric permittivity or magnetic permeability. Examples of such materials are related to magnetic (or plasmonic) nanoparticles, electric (or dielectric) nanoparticles and e -near-zero (or -near zero) particles. Regarding nonlinear materials, we mainly deal with the second or third harmonic generation particles (SH or THG). Methods. Such small-scaled/large-contrast materials generate resonances. Exciting such material with incident frequencies close to such resonances, generates local modes that amplifies the response of the composite (i.e. background-particles composite) to the incident fields. These resonances and the created local modes are characterized by geometric and material properties of the nanoparticles. As these particles are at our disposal, we can tune them in such a way to generate needed resonances and local modes. In short, we can perturb given materials by clustering nano-scaled particles to show needed or new behaviors. Originality of the project. The project is at the cutting edges in the different, but mathematically interconnected, fields of Material Sciences, Imaging as well as Therapy by Heat. We tackle these problems with a unified way in the framework of estimating the electromagnetic field generated by extreme materials. At the mathematical level, a particular originality is to combine effective medium theory (homogenization) with inverse problems theory to solve mathematical imaging and therapy problems.

Motivation. The project was stated in the framework of the electromagnetic wave propagation in the presence of extreme materials embedded in a moderate background. The extreme materials are supported in nano-scaled particles (as Dielectric or Plasmonic nanoparticles). The goal is to understand the effect of such resonating objects on the generated wave fields keeping in mind motivations coming from different branches of applied sciences, as the material sciences, imaging using contrast agents and therapy modalities using heat generation or acoustic cavitation. Approach. We have identified the correct regimes under which the nanoparticles generate proper resonances and quantified the main dominating wavefields in different scenarios related to the three branches mentioned above. We based our analysis on robust mathematical tools ranging from integral equations, spectral theory and deep understanding of asymptotic analysis taking into account the needed scales. Outcome. We cite few highlights as outcomes of the project: A. Material sciences. We focused on deriving the effective medium theory for the following models: 1. Composites made as arrangements of mixed dimers dielectric/plasmonic nanoparticles. Both the effective permittivity and permeability can be tuned to be positive or negative. Therefore we can design single or double negative material. 2. The time-domain acoustic waves propagating in a bubbly media. The effective model is dispersive with a time-convolution term highlighting the resonant effect created by the bubbles. B. Imaging modalities based on contrast agents. We have identified two scenarios: 1. In the first scenario, we inject the contrast agents one after another. In the time-harmonic regimes, we can recover the resonances. In the time domain regime, we can recover the internal values of the travel time function. In both cases, we can recover the speed of propagation which can be used for applications. 2. In the second scenario, we inject the contrasts agents all at once. We proposed an approach how to linearize the related boundary maps, as the Dirichlet to Neumann maps. This allows us to {get read of the high nonlinearity of the traditional imaging problems. C. Therapy methods using heat generation or acoustic cavitation. 1. Exciting the medium with frequencies near the Plasmonic or Dielectric resonances, we can generate a desired amount of heat in close proximity to the injected nanoparticle, while the heat diminishes as we move away from it. 2. We can generate any desired level of pressure in the close proximity to the bubble which decays as we move away from it. These results offer a wide range of potential applications in the areas of photo-thermal therapy and non-invasive sonotherapy. Our findings give a solid mathematical background to these modalities and provide with results that go beyond the known results that were obtained based on the traditional modalities.