Matias Ruiz

Mathematical and numerical framework for metasurfaces using thin layers of periodically distributed plasmonic nanoparticles

In this paper, we derive an impedance boundary condition to approximate the optical scattering effect of an array of plasmonic nanoparticles mounted on a perfectly conducting plate. We show that at some resonant frequencies the impedance blows up, allowing for a significant reduction of the scattering from the plate. Using the spectral properties of a Neumann–Poincaré type operator, we investigate the dependency of the impedance with respect to changes in the nanoparticle geometry and configuration.

Radar detection for non-stationary Doppler signal in one burst based on information geometry distance between paths

Classical Radar processing for non-stationary signal, corresponding to fast time variation of Doppler Spectrum in one burst, is no longer optimal. This phenomenon could be observed for high speed or abrupt Doppler variations of clutter or target signal but also in case of target migration during the burst duration due to high range resolution. We propose new Radar Doppler processing assuming that each non-stationary signal in one burst can be split into several short signals with less Doppler resolution but locally stationary, represented by time series of Toeplitz covariance matrices. In Information Geometry (IG) framework, these time series could be defined as a geodesic path (or geodesic polygon in discrete case) on covariance Toeplitz Hermitian Positive Definite matrix manifold. For this micro-Doppler analysis, we generalize the Fréchet distance between two curves in the plane to geodesic paths in abstract IG metric spaces of covariance matrix manifold. This approach is used for robust detection of target in case of non-stationary Time-Doppler spectrum (NS-OS-HDR-CFAR).

Reconstructing Fine Details of Small Objects by Using Plasmonic Spectroscopic Data

This paper is concerned with the inverse problem of reconstructing a small object from far field measurements. The inverse problem is severally ill-posed because of the diffraction limit and low signal to noise ratio. We propose a novel methodology to solve this type of inverse problems based on an idea from plasmonic sensing. By using the field interaction with a known plasmonic particle, the fine detail information of the small object can be encoded into the shift of the resonant frequencies of the two particle system in the far field. In the intermediate interaction regime, we show that this information is exactly the generalized polarization tensors associated with the small object, from which one can perform the reconstruction. Our theoretical findings are supplemented by a variety of numerical results. The results in the paper also provide a mathematical foundation for plasmonic sensing.

Reconstructing Fine Details of Small Objects by Using Plasmonic Spectroscopic Data. Part II: The Strong Interaction Regime

This paper is concerned with the inverse problem of reconstructing a small object from far field measurements by using the field interaction with a plasmonic particle which can be viewed as a passive sensor. It is a follow-up of the work [H. Ammari et al., Reconstructing fine details of small objects by using plasmonic spectroscopic data, SIAM J. Imag. Sci., to appear], where the intermediate interaction regime was considered. In that regime, it was shown that the presence of the target object induces small shifts to the resonant frequencies of the plasmonic particle. These shifts, which can be determined from the far field data, encodes the contracted generalized polarization tensors of the target object, from which one can perform reconstruction beyond the usual resolution limit. The main argument is based on perturbation theory. However, the same argument is no longer applicable in the strong interaction regime as considered in this paper due to the large shift induced by strong field interaction between the particles. We develop a novel technique based on conformal mapping theory to overcome this difficulty. The key is to design a conformal mapping which transforms the two particle system into a shell-core structure, in which the inner dielectric core corresponds to the target object. We show that a perturbation argument can be used to analyze the shift in the resonant frequencies due to the presence of the inner dielectric core. This shift also encodes information of the contracted polarization tensors of the core, from which one can reconstruct its shape, and hence the target object. Our theoretical findings are supplemented by a variety of numerical results based on an efficient optimal control algorithm. The results of this paper make the mathematical foundation for plasmonic sensing complete.

Heat Generation with Plasmonic Nanoparticles

In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and the cases of both a single and multiple particles. We clarify the strong dependency of the heat generation on the geometry of the particles as it depends on the eigenvalues of the associated Neumann--Poincare operator. For close-to-touching nanoparticles, we show that the temperature field deviates significantly from the one generated by two single nanoparticles. The results of this paper formally explain experimental results reported in the nanomedical literature. They open a door for solving the challenging problems of detecting plasmonic nanoparticles in biological media and monitoring temperature elevation in tissue generated by nanoparticle heating.

Field expansions for Systems of Strongly Coupled Plasmonic Nanoparticles

This paper is concerned with efficient representations and approximations of the solution to the scattering problem by a system of strongly coupled plasmonic particles. Three schemes are developed: the first is the resonant expansion which uses the resonant modes of the system of particles computed by a conformal transformation, the second is the hybridized resonant expansion which uses linear combinations of the resonant modes for each of the particles in the system as a basis to represent the solution, and the last one is the multipole expansion with respect to the origin. By considering a system formed by two plasmonic particles of circular shape, we demonstrate the relations between these expansion schemes and their advantages and disadvantages both analytically and numerically. In particular, we emphasize the efficiency of the resonant expansion scheme in approximating the near field of the system of particles. The difference between these plasmonic particle systems and the nonresonant dielectric particle system is also highlighted. The paper provides a guidance on the challenges for numerical simulations of strongly coupled plasmonic systems.

Mathematical analysis of plasmonic nanoparticles: the scalar case*

We analyze the plasmonic resonance which depends on the size and the shape of the nanoparticles. Moreover, we develop mathematical theories for plasmonic meta-surface which consists of periodic array of plasmonic particles and super-focusing by using plasmonic particles.