Sanghyeon Yu

Enhancement of near cloaking for the full Maxwell equations

Recently published methods for the quasi-static limit of the Helmholtz equation is extended to consider near cloaking for the full Maxwell equations. Effective near cloaking structures are described for the electromagnetic scattering problem at a fixed frequency. These structures are, prior to using the transformation optics, layered structures designed so that their first scattering coefficients vanish. As a result, any target inside the cloaking region has near-zero scattering cross section for a band of frequencies. Analytical results show that this construction significantly enhances the cloaking effect for the full Maxwell equations.

Generalized polarization tensors for shape description

With each domain, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper (Ammari et al. in Math. Comput. 81, 367–386, 2012), a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that high-frequency oscillations of the boundary of a domain are only contained in its high-order GPTs. Indeed, we provide an original stability and resolution analysis for the reconstruction of small shape changes from the GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We also propose an indicator of topology which could be used in some particular cases to test whether we have the correct number of connected components in the reconstructed image. We provide analytical and numerical evidence that GPTs can capture topology and high-frequency shape oscillations. The results of this paper clearly show that the concept of GPTs is a very promising new tool for shape description.

A New Optimal Control Approach for the Reconstruction of Extended Inclusions

The aim of this paper is to propose a new regularized optimal control formulation for recovering an extended inclusion from boundary measurements. Our approach provides an optimal representation of the shape of the inclusion. It guarantees local Lipschitz stability for the reconstruction problem. Some numerical experiments are performed to demonstrate the validity and the limitations of the proposed reconstruction method.

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.

Plasmonic interaction between nanospheres

When metallic (or plasmonic) nanospheres are nearly touching, strong concentration of light can occur in the narrow gap regions. This phenomenon has a potential application in nanophotonics, biosensing and spectroscopy. The understanding of the strong interaction between the plasmonic spheres turns out to be quite challenging. Indeed, an extremely high computational cost is required to compute the electromagnetic field. Also, the classical method of image charges, which is effective for dielectric spheres system, is not valid for plasmonic spheres because of their negative permittivities. Here we develop new analytical and numerical methods for the plasmonic spheres system by clarifying the connection between transformation optics and the method of image charges. We derive fully analytic solutions valid for two plasmonic spheres. We then develop a hybrid numerical scheme for computing the field distribution produced by an arbitrary number of spheres. Our method is highly efficient and accurate even in the nearly touching case and is valid for plasmonic spheres.

Shielding at a distance due to anomalous resonance

A cylindrical plasmonic structure with a concentric core exhibits an anomalous localized resonance which results in cloaking effects. Here we show that if the structure has an eccentric core, a new kind of shielding effect can happen. In contrast to conventional shielding devices, our proposed structure can block the effect of external electrical sources, even in a region which is not enclosed by any conducting materials. In fact, the shielded region is located at a distance from the device. We analytically investigate this phenomenon by using the Mobius transformation, through which an eccentric annulus is transformed into a concentric one. We also present several numerical examples.

Quantitative Characterization of Stress Concentration in the Presence of Closely Spaced Hard Inclusions in Two-Dimensional Linear Elasticity

In the region between close-to-touching hard inclusions, stress may be arbitrarily large as the inclusions get closer. This stress is represented by the gradient of a solution to the Lamé system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lamé system, and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions. As a consequence of the characterization, we are able to regain the existing upper bound on the blow-up rate of the gradient, namely, ɛ−1/2 where ɛ is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where ɛ−1/2 is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up and to obtain the optimal blow-up rate in the context of the Lamé system with hard inclusions. The singular functions introduced in this paper play essential roles in overcoming the difficulties in applying the methods of previous works. The main tools of this paper are the layer potential techniques and the variational principle. The variational principle can be applied because the singular functions of this paper are solutions of the Lamé system.

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.

Spectrum of Neumann--Poincaré Operator on Annuli and Cloaking by Anomalous Localized Resonance for Linear Elasticity

We investigate anomalous localized resonance on the circular coated structure and cloaking due to it in the context of elastostatic systems. The structure consists of the circular core with constant Lame parameters and the circular shell with negative Lame parameters proportional to those of the core. We show that there is a nonzero number