arXiv subjects

Sanghyeon Yu

Publications and source records attributed to Sanghyeon Yu.

At least 37 records · Page 2Linked to original sources

arXiv2017

Double-negative electromagnetic metamaterials due to chirality

The aim of this paper is to provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in chiral materials. The design of double-negative metamaterials generally requires the use of two different kinds of subwavelength resonators, which may limit the applicability of double-negative metamaterials. Herein, we rely on media that consist of only a single type of dielectric resonant element, and show how the chirality of the background medium induces double-negative refractive index metamaterial, which refracts waves negatively, hence acting as a superlens. Using plasmonic dielectric particles, it is proved that both the effective electric permittivity and the magnetic permeability can be negative near some resonant frequencies. A justification of the approximation of a plasmonic particle in a chiral medium by the sum of a resonant electric dipole and a resonant magnetic dipole, is provided. Moreover, the set of resonant frequencies is characterized. For an appropriate volume fraction of plasmonic particles with certain conditions on their configuration, a double-negative effective medium can be obtained when the frequency is near one of the resonant frequencies.

math.AP↗

arXiv2017

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.

math.NA↗

arXiv2017

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, the stress may be arbitrarily large as the inclusions get closer. The 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, $ε^{-1/2}$ where $ε$ is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where $ε^{-1/2}$ is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up in the context of the Lamé system with hard inclusions. The singular functions introduced in this paper play essential roles to overcome the difficulties in applying the methods of previous works. 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.

math.AP↗

arXiv2017

A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites

We prove in a mathematically rigorous way the asymptotic formula of Flaherty and Keller on the effective property of densely packed periodic elastic composites with hard inclusions. The proof is based on the primal-dual variational principle, where the upper bound is derived by using the Keller-type test functions and the lower bound by singular functions made of nuclei of strain. Singular functions are solutions of the Lamé system and capture precisely singular behavior of the stress in the narrow region between two adjacent hard inclusions.

math.AP↗

arXiv2017

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.

physics.comp-ph↗

arXiv2017

Stress concentration for two nearly touching circular holes

We consider the plane elasticity problem for two circular holes. When two holes are close to touching, the stress concentration happens in the narrow gap region. In this paper, we characterize the stress singularity between the two holes by an explicit function. A new method of a singular asymptotic expansion for the Fourier series with slowing decaying coefficients is developed to investigate the asymptotic behavior of the stress.

math.AP↗

arXiv2017

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.

math.AP↗

arXiv2017

Shielding at a distance due to anomalous resonance in superlens with eccentric core

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 the conventional shielding device, our proposed structure can block the effect of external electrical sources even on 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 möbius transformation via which an eccentric annulus is transformed into a concentric one. We also present several numerical examples.

math-ph↗

arXiv2017

Subwavelength phononic bandgap opening in bubbly media

The aim of this paper is to show both analytically and numerically the existence of a subwavelength phononic bandgap in bubble phononic crystals. The key is an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. The main findings in this paper are illustrated with a variety of numerical experiments.

math.AP↗

arXiv2016

Cloaking by anomalous localized resonance for linear elasticity on a coated structure

We investigate anomalous localized resonance on the circular coated structure and cloaking related to it in the context of elasto-static systems. The structure consists of the circular core with constant Lamé parameters and the circular shell of negative Lamé parameters proportional to those of the core. We show that the eigenvalues of the Neumann-Poincaré operator corresponding to the structure converges to certain non-zero numbers determined by Lamé parameters and derive precise asymptotics of the convergence. We then show with estimates that cloaking by anomalous localized resonance takes place if and only if the dipole type source lies inside critical radii determined by the radii of the core and the shell.

math.SP↗

arXiv2016

Shape reconstruction of nanoparticles from their associated plasmonic resonances

We prove by means of a couple of examples that plasmonic resonances can be used on one hand to classify shapes of nanoparticles with real algebraic boundaries and on the other hand to reconstruct the separation distance between two nanoparticles from measurements of their first collective plasmonic resonances. To this end, we explicitly compute the spectral decompositions of the Neumann-Poincaré operators associated with a class of quadrature domains and two nearly touching disks. Numerical results are included in support of our main findings.

math.AP↗

arXiv2016

Theory of plasmonic metasurfaces

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-Poincare type operator, we investigate the dependency of the impedance with respect to changes in the nanoparticle geometry and configuration.

math.AP↗

arXiv2015

Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations

In this paper we use the full Maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles. We mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size, shape, and arrangement of the nanoparticles, using the layer potential techniques associated with the full Maxwell equations. We present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction under which the Maxwell-Garnett theory is valid at plasmonic resonances.

math.AP↗

arXiv2015

Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients

We consider well-posedness of the boundary value problem in presence of an inclusion with complex conductivity $k$. We first consider the transmission problem in $\mathbb{R}^d$ and characterize solvability of the problem in terms of the spectrum of the Neumann-Poincaré operator. We then deal with the boundary value problem and show that the solution is bounded in its $H^1$-norm uniformly in $k$ as long as $k$ is at some distance from a closed interval in the negative real axis. We then show with an estimate that the solution depends on $k$ in its $H^1$-norm Lipschitz continuously. We finally show that the boundary perturbation formula in presence of a diametrically small inclusion is valid uniformly in $k$ away from the closed interval mentioned before. The results for the single inclusion case are extended to the case when there are multiple inclusions with different complex conductivities: We first obtain a complete characterization of solvability when inclusions consist of two disjoint disks and then prove solvability and uniform estimates when imaginary parts of conductivities have the same signs. The results are obtained using the spectral property of the associated Neumann-Poincaré operator and the spectral resolution.

math.AP↗

arXiv2015

Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

We first investigate spectral properties of the Neumann-Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, its spectrum consists of eigenvalues which accumulates to two numbers determined by Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on disks and ellipses. We then investigate resonance occurring at eigenvalues and anomalous localized resonance at accumulation points of eigenvalues. We show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

math.AP↗

arXiv2015

Asymptotic analysis for superfocusing of the electric field in between two nearly touching metallic spheres

We consider the enhancement of electric field in the presence of two perfectly conducting spheres. When the two spheres get closer, the electric field have a much larger magnitude compared to the external field in the small gap region between the two spheres. The enhanced field can be arbitrary large with the generic blow-up rate $|ε\lnε|^{-1}$ in three dimensional space, where $ε$ is the distance between the spheres. In this paper we derive rigorously an asymptotic formula of the electric field consisting of elementary functions. The asymptotic formula explicitly characterizes superfocusing of the electric field in terms of the spheres radii, the distance between the spheres, and the external field. We illustrate our results with numerical calculations.

math.AP↗

arXiv2015

Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plasmon resonance

The purpose of this paper is to investigate the spectral nature of the Neumann-Poincaré operator on the intersecting disks, which is a domain with the Lipschitz boundary. The complete spectral resolution of the operator is derived, which shows in particular that it admits only the absolutely continuous spectrum, no singularly continuous spectrum and no pure point spectrum. We then quantitatively analyze using the spectral resolution the plasmon resonance at the absolutely continuous spectrum.

math.AP↗

arXiv2013

Characterization of the gradient blow-up of the solution to the conductivity equation in the presence of adjacent circular inclusions

We consider the conductivity problem in the presence of adjacent circular inclusions having arbitrary constant conductivity. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. We characterize the gradient blow-up by deriving an explicit formula for the singular term of the solution in terms of the Lerch transcendent function. This derivation is valid for inclusions having arbitrary constant conductivity. We illustrate our results with numerical calculations.

math-ph↗