In this paper we provide a mathematical framework for localized plasmon resonance of nanoparticles. Using layer potential techniques associated with the full Maxwell equations, we derive small-volume expansions for the electromagnetic fields, which are uniformly valid with respect to the nanoparticle's bulk electron relaxation rate. Then, we discuss the scattering and absorption enhancements by plasmon resonant nanoparticles. We study both the cases of a single and multiple nanoparticles. We present numerical simulations of the localized surface plasmonic resonances associated to multiple particles in terms of their separation distance.

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.Mathematics Subject Classification (MSC2000): 35R30, 35C20.

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.© H. Ammari et al. / J. Math. Pures Appl. 103 (2015) 1390-1409 1391 de l'onde ultrasonore et dépend non linéairement de la conductivité électrique. Le problème est de reconstruire cette conductivité à partir des mesures de courant. En premier lieu, on utilise une fonction test (potentiel virtuel) pour quantifier le lien entre le signal et la conductivité. Ensuite, à l'aide d'une déconvolution et d'un filtrage, il est possible de ramener le problème à la reconstruction d'une carte de conductivité à partir de la donnée d'un courant électrique interne sur l'ensemble du domaine. On donne d'abord une méthode d'optimisation pour résoudre ce problème. Une seconde méthode de reconstruction directe, utilisant une méthode de viscosité et la résolution d'une équation de transport à coefficients discontinus, est ensuite proposée. On démontre que la résolution de ce problème donne une reconstruction exacte de la conductivité lorsque le paramètre de régularisation tend vers zéro. On illustre les deux méthodes numériquement et on compare leur performances (résolution et stabilité en présence de bruit de mesure).

This paper aims at understanding the nature of the subwavelength resonant frequencies of dielectric particles with high refractive indices. It is proved that for an arbitrary shaped particle, these subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. The enhancement of the scattered field at the resonant frequencies is shown. The hybridization of the subwavelength resonant frequencies of a dimer consisting of high refractive index dielectric nanoparticles is also characterized. KEYWORDS asymptotic expansions, dielectric nanoparticles, high refractive index, subwavelength resonances MSC CLASSIFICATION 35R30, 35C20

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