arXiv subjects

Xinlin Cao

Publications and source records attributed to Xinlin Cao.

14 recordsLinked to original sources

arXiv2026

Electromagnetic Scattering by a Cluster of Hybrid Dielectric-Plasmonic Dimers

We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in $\mathbb{R}^3$. Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose size $a$ is small compared to the wavelength, with intra-dimer and inter-dimer distances scaling like $a^{t_1}$ and $a^{t_2}$, and the frequency is tuned near suitable electric and magnetic resonances of the associated Newtonian and magnetization operators on the reference shapes. Under these geometric, contrast and spectral assumptions, we derive a Foldy--Lax type approximation for the Maxwell system. We show that the scattered field and its far field admit asymptotic expansions in terms of four moments attached to each dimer, which solve an explicit finite-dimensional linear system. We prove invertibility of this system under quantitative smallness conditions on the contrast and the dimer density, and we obtain error estimates uniform in the number of dimers. By extracting the dominant components, we further show that each hybrid dimer behaves, at leading order, as a co-located electric and magnetic dipole driven by the local fields, and we identify the corresponding $6\times 6$ polarizability matrix. This provides a discrete model for clusters of hybrid dimers that is suitable for fast forward simulations, inverse schemes, and as input for effective-medium descriptions. In particular, it suggests parameter regimes where clusters of hybrid dimers can generate (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.

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arXiv2024

Electromagnetic waves generated by a hybrid dielectric-plasmonic dimer

We know that the electric field generated by a plasmonic nano-particle (with negative permittivity) is given as a polarization of the incident electric field. Similarly, the electric field produced by a dielectric nano-particle (with positive but high permittivity) is given as a polarization of the incident magnetic field. In this work, we demonstrate that a hybrid dimer composed of two closely coupled nano-particles, one plasmonic and the other dielectric can polarize both the incident electric and magnetic fields. Consequently, such hybrid dimers have the potential to modify both the electric permittivity and magnetic permeability of the surrounding medium. However, this dual modification occurs only when the two nano-particles share common resonant frequencies. We derive the asymptotic expansion of the fields generated by these hybrid dimers in the subwavelength regime for incident frequencies near their shared resonant frequencies.

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arXiv2024

Effective Medium Theory for Heat Generation Using Plasmonics: A Parabolic Transmission Problem Driven by the Maxwell System

The excitation of plasmonic nanoparticles by incident electromagnetic waves at frequencies near their subwavelength resonances induces localized heat generation in the surrounding medium. We develop a mathematical framework to rigorously quantify this heat generation in systems of arbitrarily distributed nanoparticles. 1. For an arbitrary discrete distribution of M nanoparticles within a bounded domain, the effective heat distribution is described by a coupled system: Volterra-type integral equations for the heat conduction and a Foldy-Lax-type system governing the self consistent electric field intensities. These equations are parameterized by the particle geometries and the local electromagnetic field interactions. The effective heat generation is computed by solving these coupled systems, with the computational complexity scaling as M^2. 2. In the case M >> 1, under natural scaling regimes, the discrete system converges to a continuum model, yielding an effective parabolic equation for the heat distribution. The source term in this homogenized parabolic model is characterized by the solution of the homogenized Maxwells equations, incorporating an effective permittivity distribution derived from the Drude model under resonance conditions. Our analysis utilizes advanced tools in potential theory, asymptotic analysis and homogenization. By leveraging layer potential representations, we derive point-wise field approximations. The coupling between the Maxwell and heat equations is resolved by analyzing the spectral properties of the nanoparticles and their scaling limits. This framework reduces the problem to two mathematical challenges: a control problem for the effective parabolic system and an internal phase-less inverse problem for the Maxwell system, thus providing a unified approach to modeling heat generation in nanoparticle clusters.

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arXiv2024

Optical Inversion Using Plasmonic Contrast Agents

We describe a new method to reconstruct the permittivity distribution, of an object to image, from the remotely measured electromagnetic field. We propose to use the remote fields measured before and after injecting locally in the medium plasmonic nano-particles. Such a technique is known in the framework of imaging using contrast agents where, in optical imaging, the nano-particles play the role of these contrast agents. The plasmonic nano-particles are known to enjoy resonant effects, as enhancing the applied incident field, while excited at certain particular frequencies called plasmonic resonances. These resonant frequencies encode the values of the unknown permittivity at the location of the injected nano-particles. The imaging methods we propose mainly use this resonant effect. We show that the imaging functional build up from contrasting the fields before and after injecting the nano-particles, measured at one single back-scattered direction, and in an explicit band of incident frequencies reaches its maximum values, in terms of the incident frequency, precisely at the mentioned plasmonic resonances. Such a behavior allows us to recover these plasmonic resonances from which we recover the point-wise values of the permittivity distribution. In this work, we describe the method and provide the mathematical justification of this resonant effect and its use for the optical inversion using plasmonic nano-particles as contrast agents.

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arXiv2024

Effective medium theory for Van-Der-Waals heterostructures

We derive the electromagnetic medium equivalent to a collection of all-dielectric nano-particles (enjoying high refractive indices) distributed locally non-periodically in a smooth domain $Ω$. Such distributions are used to model well known structures in material sciences as the Van-der-Waals heterostructures. Since the nano-particles are all-dielectric, then the permittivity remains unchanged while the permeability is altered by this effective medium. This equivalent medium describes, in particular, the effective medium of 2 dimensional type Van-der-Waals heterostructures. These structures are 3 dimensional which are build as superposition of identical (2D)-sheets each supporting locally non-periodic distributions of nano-particles. An explicit form of this effective medium is provided for the particular case of honeycomb heterostructures. At the mathematical analysis level, we propose a new approach to derive the effective medium when the subwavelength nano-particles are distributed non-periodically. The first step consists in deriving the point interaction approximation, also called the Foldy-Lax approximation. The scattered field is given as a superposition of dipoles (or poles for other models) multiplied by the elements of a vector which is itself solution of an algebraic system. This step is done regardless of the way how the particles are distributed. As a second step, which is the new and critical step, we rewrite this algebraic system according to the way how these nano-particles are locally distributed. The new algebraic system will then fix the related continuous Lippmann Schwinger system which, in its turn, indicates naturally the equivalent medium.

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arXiv2023

From all-dieletric nanoresonators to extended quasi-static plasmonic resonators

We derive the electromagnetic medium equivalent to a cluster of all-dielectric nanoparticles (i.e. enjoying high refractive indices), distributed periodically in a smooth domain $Ω$, while excited at nearly resonating dielectric incident frequencies (i.e. subwavelength Mie-resonant frequencies). This effective medium is an alteration of the permeability that keeps the permittivity unchanged. We provide regimes under which the effective permeability can be positive or negative valued. In addition, if the incident frequency is close to any of the subwavelength all-dielectric resonances, then the distributed cluster behaves as an extended quasi-static plasmonic resonator. Therefore, exciting the cluster of all-dielectric nanoresonators with nearly resonating incident frequencies, we can generate an extended quasi-static plasmonic resonator which creates giant electromagnetic fields in its surrounding.

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arXiv2022

The Electromagnetic Waves Generated by Dielectric Nanoparticles

We estimate the electromagnetic fields generated by a cluster of dielectric nanoparticles embedded into a background made of a vacuum. The dielectric nanoparticles are small scaled but enjoy high contrast of their relative permittivity. Such scales/contrasts can be ensured using the Lorentz model with incident frequencies chosen appropriately close to the undamped resonance (appearing in the Lorentz model). Under certain ratio between their size and contrast, these nanoparticles generate resonances, called dielectric resonances. These resonances are characterized and computed via the spectrum of the electric Newtonian operator, stated on the support of nanoparticles, projected on the space of divergence-free fields with vanishing boundary normal components. We characterize the dominant field generated by a cluster of such dielectric-resonating nanoparticles. In this point-interaction approximation, the nanoparticles can be distributed to occupy volume-like domains or low dimensional hypersurfaces where periodicity is not required. The form of these approximations suggests that the effective electromagnetic medium, equivalent to the cluster of such nanoparticles, is a perturbation of the magnetic permeability and not the electric permittivity. The cluster can be tuned such that the equivalent permeability has positive or negative values (while the permittivity stays unchanged).

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arXiv2021

The Effective Permittivity and Permeability Generated by a Cluster of Moderately Contrasting Nanoparticles

In a $3D$ bounded and $C^{1, α}$-smooth domain $Ω$, $α\in (0, 1)$, we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter $c_r:=\fracδ{a}$ where $a$ is the maximum diameter of the nanoparticles and $δ$ the minimum distance between them. The constant of the proportionality is given in terms of a priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parameter $c_r$, the used incident frequency and the domain $Ω$). A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in \cite{AM}, is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the Hölder spaces. As a by-product, this shows a Hölder regularity property of the electromagnetic fields up to the boundary of the inhomogeneity.

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arXiv2020

On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $π$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in [9] as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not $π$ when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in [9] are significantly relaxed in the present study for the conductive transmission eigenfunctions. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.

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arXiv2020

Determining a piecewise conductive medium body by a single far-field measurement

We are concerned with the inverse problem of recovering a conductive medium body. The conductive medium body arises in several applications of practical importance, including the modeling of an electromagnetic object coated with a thin layer of a highly conducting material and the magnetotellurics in geophysics. We consider the determination of the material parameters inside the body as well as on the conductive interface by the associated electromagnetic far-field measurement. Under the transverse-magnetic polarisation, we derive two novel unique identifiability results in determining a 2D piecewise conductive medium body associated with a polygonal-nest or a polygonal-cell geometry by a single active or passive far-field measurement.

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arXiv2019

On nodal and generalized singular structures of Laplacian eigenfunctions and applications in $\mathbb{R}^3$

This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian eigenfunctions is investigated at a point where two nodal or generalised singular lines intersect. It reveals a certain intriguing property that the vanishing order of the eigenfunction at the intersecting point is related to the rationality of the intersecting angle. In this paper, we consider the 3D counterpart of such a study by studying the analytic behaviours of the Laplacian eigenfunctions at places where nodal or generalised singular planes are intersect. Compared to the 2D case, the geometric situation is more complicated: the intersection of two planes generates an edge corner, whereas the intersection of more than three planes generates a vertex corner. We provide a comprehensive characterisation for all of those cases. Moreover, we apply the spectral results to establish some novel unique identifiability results for the geometric inverse problems of recovering the shape as well as the (possible) surface impedance parameter by the associated scattering far-field measurements.

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arXiv2019

On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is {\it irrational}, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.

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arXiv2018

Determining a fractional Helmholtz system with unknown source and medium parameter

We are concerned with an inverse problem associated with the fractional Helmholtz system that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequency-dependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.

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arXiv2017

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

Let $A\in\mathrm{Sym}(n\times n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(-\nabla\cdot(A(x)\nabla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $Ω$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s=1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

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