Anomalous Localized Resonance Research Articles

Elastodynamical resonances and cloaking of negative material structures beyond quasistatic approximation

AbstractGiven the flexibility of choosing negative elastic parameters, we construct material structures that can induce two resonance phenomena, referred to as the elastodynamical resonances. They mimic the emerging plasmon/polariton resonance and anomalous localized resonance in optics for subwavelength particles. However, we study the peculiar resonance phenomena for linear elasticity beyond the subwavelength regime. It is shown that the resonance behaviors possess distinct characters, with some similar to the subwavelength resonances, but some sharply different due to the frequency effect. It is particularly noted that we construct a core–shell material structure that can induce anomalous localized resonance as well as cloaking phenomena beyond the quasistatic limit. The study is boiled down to analyzing the so‐called elastic Neumann–Poincaré (N‐P) operator in the frequency regime. We provide an in‐depth analysis of the spectral properties of the N‐P operator on a circular domain beyond the quasistatic approximation, and these results are of independent interest to the spectral theory of layer potential operators.

Spectral analysis of scattering resonances with application on high-contrast nanospheres

In this paper, we provide further spectral analysis of the general asymptotic scattering resonances formula of small 3D dielectrics of arbitrary shape with high contrast, already derived in other works to a first-order approximation. To investigate the components of a full expansion of such resonances, a breakdown is presented for the case of high-contrast nanospheres, in terms of its radius h in the interval [0, 1]. We also derive, for radially symmetric fields, an exact resonance formula for a spherical scatterer in terms of its radius, not necessarily small, and dielectric susceptibility coefficient, not necessarily high. This formula is further developed and simplified in the case of high contrast nanospheres. Such formulas are useful in imaging applications to identify objects’ properties from frequency measurements. Another application is the study of negative refractive index materials, such as metamaterials, and the anomalous localized resonance phenomenon (ALR) that is associated with cloaking and superlensing.

Spectral Properties of Neumann-Poincaré Operator and Anomalous Localized Resonance in Elasticity Beyond Quasi-Static Limit

This paper is concerned with the polariton resonances and their application for cloaking due to anomalous localized resonance (CALR) for the elastic system within finite frequency regime beyond the quasi-static approximation. We first derive the complete spectral system of the Neumann-Poincare operator associated with the elastic system in $\mathbb{R}^{3}$ within the finite frequency regime. Based on the obtained spectral results, we construct a broad class of elastic configurations that can induce polariton resonances beyond the quasi-static limit. As an application, the invisibility cloaking effect is achieved through constructing a class of core-shell-matrix metamaterial structures provided the source is located inside a critical radius. Moreover, if the source is located outside the critical radius, it is proved that there is no resonance.

Recent progress on the mathematical study of anomalous localized resonance in elasticity

<p style='text-indent:20px;'>We consider the anomalous localized resonance induced by negative elastic metamaterials and its application in invisibility cloaking. We survey the recent mathematical developments in the literature and discuss two mathematical strategies that have been developed for tackling this peculiar resonance phenomenon. The first one is the spectral method, which explores the anomalous localized resonance through investigating the spectral system of the associated Neumann-Poincaré operator. The other one is the variational method, which considers the anomalous localized resonance via calculating the nontrivial kernels of a non-elliptic partial differential operator. The advantages and the relationship between the two methods are discussed. Finally, we propose some open problems for the future study.</p>

The invisibility via anomalous localized resonance of a source for electromagnetic waves

We investigate the invisibility via anomalous localized resonance of a general source in anisotropic media for electromagnetic waves. To this end, we first introduce the concept of doubly complementary media in the electromagnetic setting. These are media consisting of negative-index metamaterials in a shell and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core–shell structure. We then provide criteria for establishing the invisibility of a source in these media. We show that (i) a source is invisible if the power is blown up; (ii) a source is invisible if it is sufficiently close to the plasmonic shell structure, and it is visible if it is far from this plasmonic structure; (iii) if the plasmonic structure is complementary to an annulus of constant isotropic medium, there is a critical length that characterizes the cloaking phenomena, as first observed in the two-dimensional acoustic quasistatic setting by Milton and Nicorovici.

Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime

Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in [15]. A general setting in which cloaking a source via ALR takes place is the setting of doubly complementary media. This was introduced and studied in [20] for the quasistatic regime. In this paper, we study cloaking a source via ALR for doubly complementary media in the finite frequency regime. To this end, we establish the following results: (1) Cloaking a source via ALR appears if and only if the power blows up; (2) The power blows up if the source is “placed” near the plasmonic structure; (3) The power remains bounded if the source is far away from the plasmonic structure. Concerning the analysis, on one hand we extend ideas from [20] and on the other hand we add new insights into the problem. This allows us to not only overcome difficulties related to the finite frequency regime but to also obtain new information on the problem. In particular, we are able to characterize the behavior of fields far enough from the plasmonic shell, as the loss approaches 0 for an arbitrary source outside the core-shell structure in the doubly complementary media setting.

On anomalous localized resonance and plasmonic cloaking beyond the quasi-static limit

In this paper, we give the mathematical construction of novel core-shell plasmonic structures that can induce anomalous localized resonance and invisibility cloaking at certain finite frequencies beyond the quasi-static limit. The crucial ingredient in our study is that the plasmon constant and the loss parameter are constructed in a delicate way that are correlated and depend on the source and the size of the plasmonic structure. As a significant by-product of this study, we also derive the complete spectrum of the Neumann–Poincáre operator associated with the Helmholtz equation with finite frequencies in the radial geometry. The spectral result is the first one in its type and is of significant mathematical interest for its own sake.

On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances

This paper is concerned with the theoretical study of polariton resonances for linear elasticity governed by the Lamé system in R3, and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce polariton resonances. It is shown that if either one of the two convexity conditions on the Lamé parameters is broken, then we can construct certain polariton structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexity conditions is required. Indeed, the existing polariton structures are a particular case of the general structures constructed in our study. Furthermore, we consider the polariton resonances within the finite frequency regime, and rigorously verify the quasi-static approximation for diametrically small polariton inclusions. Finally, as an application of the newly found structures, we construct a polariton device of the core-shell-matrix form that can induce cloaking due to anomalous localized resonance in the quasi-static regime, which also includes the existing study as a special case.

Negative Index Materials: Some Mathematical Perspectives

Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones.

On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

This paper considers transmission problems for the Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero, the fields may blow up. When the speed of the blow up is suitable, this is called the anomalous localized resonance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak anomalous resonance (w-AR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-AR or ALR may or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary, neither ALR nor w-AR appears with nonzero frequencies (i.e., in the dynamic range) in dimensions ...

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 the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist.

The boundary integral equation (BIE) method ascertains explicit relations between localized surface phonon and plasmon polariton resonances and the eigenvalues of its associated electrostatic operator. We show that group-theoretical analysis of the Laplace equation can be used to calculate the full set of eigenvalues and eigenfunctions of the electrostatic operator for shapes and shells described by separable coordinate systems. These results not only unify and generalize many existing studies, but also offer us the opportunity to expand the study of phenomena such as cloaking by anomalous localized resonance. Hence, we calculate the eigenvalues and eigenfunctions of elliptic and circular cylinders. We illustrate the benefits of using the BIE method to interpret recent experiments involving localized surface phonon polariton resonances and the size scaling of plasmon resonances in graphene nanodiscs. Finally, symmetry-based operator analysis can be extended from the electrostatic to the full-wave regime. Thus, bound states of light in the continuum can be studied for shapes beyond spherical configurations.

Negative index materials and their applications: Recent mathematics progress

Negative index materials are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1946 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. Mathematically, the study of negative index materials faces two difficulties. Firstly, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Secondly, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this survey, the author discusses recent mathematics progress in understanding properties of negative index materials and their applications. The topics are reflecting complementary media, superlensing and cloaking by using complementary media, cloaking a source via anomalous localized resonance, the limiting absorption principle and the well-posedness of the Helmholtz equation with sign changing coefficients.

Shielding at a distance due to anomalous resonance**This work is supported by the Korean Ministry of Science, ICT and Future Planning through NRF grant nos. 2013R1A1A3012931 (for ML) and 2016R1A2B4014530 (for ML).

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 Möbius transformation, through which an eccentric annulus is transformed into a concentric one. We also present several numerical examples.

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 $k_0$ determined by Lame parameters such that two nonempty eigenvalue sequences of the Neumann--Poincare operator associated with the structure converge to $k_0$ and $-k_0$, respectively, and derive precise asymptotics of the convergence. We then show by qualitative estimates based on asymptotics of eigenvalues that cloaking by anomalous localized resonance takes place if and only if the dipole-type source lies inside critical radii determined by radii of the core and the shell. The critical radii corresponding to $k_0$ and $-k_0$ are different.

Cloaking an Arbitrary Object via Anomalous Localized Resonance: The Cloak is Independent of the Object

In this paper, we present various schemes of cloaking an arbitrary object via anomalous localized resonance and provide their analysis in two and three dimensions. This is a way to cloak an object using negative index materials in which the cloaking device is independent of the object. As a result, we show that in the two dimensional quasi-static regime an annular plasmonic structure of coefficient $-1$ cloaks small but finite size objects nearby. We also discuss its connections with superlensing and cloaking using complementary media. In particular, we confirm the possibility that a lens can act like a cloak and conversely. This possibility was raised about a decade ago in the literature.

Sensitivity of anomalous localized resonance phenomena with respect to dissipation

We analyze cloaking due to anomalous localized resonance in the quasistatic regime in the case when a general charge density distribution is brought near a slab superlens. If the charge density distribution is within a critical distance of the slab, then the power dissipation within the slab blows up as certain electrical dissipation parameters go to zero. The potential remains bounded far away from the slab in this limit, which leads to cloaking due to anomalous localized resonance. On the other hand, if the charge density distribution is further than this critical distance from the slab, then the power dissipation within the slab remains bounded and cloaking due to anomalous localized resonance does not occur. The critical distance is shown to strongly depend on the the rate at which the dissipation outside of the slab goes to zero.

Anomalous Localized Resonance Phenomena in the Nonmagnetic, Finite-Frequency Regime

The phenomenon of anomalous localized resonance (ALR) is observed at the interface between materials with positive and negative material parameters and is characterized by the fact that when a given source is placed near the interface, the electric and magnetic fields start to have very fast and large oscillations around the interface as the absorption in the materials becomes very small while they remain smooth and regular away from the interface. In this paper, we discuss the phenomenon of anomalous localized resonance (ALR) in the context of an infinite slab of homogeneous, nonmagnetic material (μ=1) with permittivityϵs=-1-iδfor some small lossδ≪1surrounded by positive, nonmagnetic, homogeneous media. We explicitly characterize the limit value of the product between frequency and the width of slab beyond which the ALR phenomenon does not occur and analyze the situation when the phenomenon is observed. In addition, we also construct sources for which the ALR phenomenon never appears.

On Anomalous Localized Resonance for the Elastostatic System

We consider the anomalous localized resonance due to a plasmonic structure for the elastostatic system in R^2. The plasmonic structure takes a general core-shell-matrix form with the metamaterial located in the shell. If there is no core, we show that resonance occurs for a very broad class of sources. If the core is nonempty and of an arbitrary shape, we show that there exists a critical radius such that anomalous localized resonance (ALR) occurs. Our argument is based on a variational technique by making use of the primal and dual variational principles for the elastostatic system, along with the construction of suitable test functions.

Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator

We investigate in a quantitative way the plasmon resonance at eigenvalues and the essential spectrum (the accumulation point of eigenvalues) of the Neumann–Poincaré operator on smooth domains. We first extend the symmetrization principle so that the single layer potential becomes a unitary operator from H−1/2 onto H1/2. We then show that the resonance at the essential spectrum is weaker than that at eigenvalues. It is shown that anomalous localized resonance occurs at the essential spectrum on ellipses, and cloaking due to anomalous localized resonance does occur on ellipses like on the core-shell structure considered in [19]. It is shown that cloaking due to anomalous localized resonance does not occur at the essential spectrum on three dimensional balls.