arXiv subjects

Ahcene Ghandriche

Publications and source records attributed to Ahcene Ghandriche.

16 recordsLinked to original sources

arXiv2026

On the detection of medium inhomogeneity by contrast agent: wave scattering models and numerical implementations

We consider the wave scattering and inverse scattering in an inhomogeneous medium embedded a homogeneous droplet with a small size, which is modeled by a constant mass density and a small bulk modulus. Based on the Lippmann-Schwinger integral equation for scattering wave in inhomogeneous medium, we firstly develop an efficient approximate scheme for computing the scattered wave as well as its far-field pattern for any droplet located in the inhomogeneous background medium. By establishing the approximate relation between the far-field patterns of the scattered wave before and after the injection of a droplet, the scattered wave of the inhomogeneous medium after injecting the droplet is represented by a measurable far-field patterns, and consequently the inhomogeneity of the medium can be reconstructed from the Helmholtz equation. Finally, the reconstruction process in terms of the dual reciprocity method is proposed to realize the numerical algorithm for recovering the bulk modulus function inside a bounded domain in three dimensional space, by moving the droplet inside the bounded domain. Numerical implementations are given using the simulation data of the far-field pattern to show the validity of the reconstruction scheme, based on the mollification scheme for dealing with the ill-posedness of this inverse problem.

math.NA↗

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.

math.AP↗

arXiv2025

The Calderon Problem Revisited: Reconstruction With Resonant Perturbations

The original Calderón problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that 1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenized medium. In addition, the equivalent coefficient, which consist in the sum of the original potential and the effective coefficient, is negative valued with a controlable amplitude. 2) as the equivalent coefficient is negative valued, then we can linearize the corresponding Neumann to Dirichlet map using the effective coefficient's amplitude. 3) from the linearized Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).

math.AP↗

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.

math.AP↗

arXiv2024

Characterization of the Eigenvalues and Eigenfunctions of the Helmholtz Newtonian operator N^k

The Newtonian potential operator for the Helmholtz equation, which is represented by the volume integral with fundamental solution as kernel function, is of great importance for direct and inverse scattering of acoustic waves. In this paper, the eigensystem for the Newtonian potential operator is firstly shown to be equivalent to that for the Helmholtz equation with nonlocal boundary condition for a bounded and simply connected Lipschitz-regular domain. Then, we compute explicitly the eigenvalues and eigenfunctions of the Newtonian potential operator when it is defined in a 3-dimensional ball. Furthermore, the eigenvalues' asymptotic behavior is demonstrated. To illustrate the behavior of certain eigenfunctions, some numerical simulations are included.

math.SP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

arXiv2023

Estimation of the eigenvalues and the integral of eigenfunctions of the Newtonian potential operator

We consider the problem of estimating the eigenvalues and the integral of the corresponding eigenfunctions, associated to the Newtonian potential operator, defined in a bounded domain $Ω\subset \mathbb{R}^{d},$ where $d = 2, 3$, in terms of the maximum radius of $Ω$. We first provide these estimations in the particular case of a ball and a disc. Then we extend them to general shapes using a, derived, monotonicity property of the eigenvalues of the Newtonian operator. The derivation of the lower bounds are quite tedious for the 2D-Logarithmic potential operator. Such upper/lower bounds appear naturally while estimating the electric/acoustic fields propagating in $\mathbb{R}^{d}$ in the presence of small scaled and highly heterogeneous particles.

math.SP↗

arXiv2022

Simultaneous Reconstruction of Optical and Acoustical Properties in Photo-Acoustic Imaging using plasmonics

We propose an approach for the simultaneous reconstruction of the electromagnetic and acoustic material parameters, in the given medium $Ω$ where to image, using the photo-acoustic pressure, measured on a single point of the boundary of $Ω$, generated by plasmonic nanoparticles. We prove that the generated pressure, that we denote by $p^{\star}\left(x, s, ω\right)$, depending on only one fixed point $x \in \partial Ω$, the time variable $s$, in a large enough interval, and the incidence frequency $ω$, in a large enough band, is enough to reconstruct both the sound speed, the mass density and the permittivity inside $Ω$. Indeed, from the behavior of the measured pressure in terms of time, we can estimate the travel time of the pressure, for arriving points inside $Ω$, then using the eikonal equation we reconstruct the acoustic speed of propagation. In addition, we reconstruct the internal values of the acoustic Green's function. From the singularity analysis of this Green's function, we extract the integrals along the geodesics, for internal arriving points, of the logarithmic-gradient of the mass density. Solving this (internal) integral geometric problem provides us with the values of the mass density function inside $Ω$. Finally, from the behavior of $p^{\star}\left(x, s, ω\right)$ with respect to the frequency $ω$, we detect the generated plasmonic resonances from which we reconstruct the permittivity inside $Ω$.

math.AP↗

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).

math.AP↗

arXiv2022

The Foldy-Lax approximation is valid for nearly resonating frequencies

The waves (including acoustic, electromagnetic and elastic ones) propagating in the presence of a cluster of inhomogeneities undergo multiple interactions between them. When these inhomogeneities have sub-wavelength sizes, the dominating field due to the these multiple interactions is the Foldy-Lax field. This field models the interaction between the equivalent point-like scatterers, located at the centers of the small inhomogeneities, with scattering coefficients related to geometrical/material properties of each inhomogeneities, as polarization coefficients. One of the related questions left open for a long time is whether we can reconstruct this Foldy-Lax field from the scattered field measured far away from the cluster of the small inhomogeneities. This is the Foldy-Lax approximation (or Foldy-Lax paradigm). In this work, we show that this approximation is indeed valid as soon as the inhomogeneities enjoy critical scales between their sizes and contrasts. These critical scales allow them to generate resonances which can be characterized and computed. The main result here is that exciting the cluster by incident frequencies which are close to the real parts of these resonances allow us to reconstruct the Fold-Lax field from the scattered waves collected far away from the cluster itself (as the farfields). In short, we show that the Foldy-Lax approximation is valid using nearly resonating incident frequencies. This results is demonstrated by using, as small inhomogeneities, dielectric nanoparticles for the 2D TM model of electromagnetic waves and bubbles for the 3D acoustic waves.

math.AP↗

arXiv2021

Photo-acoustic inversion using plasmonic contrast agents: The full Maxwell model

We analyze the inversion of the photo-acoustic imaging modality using electromagnetic plasmonic nano-particles as contrast agents. We prove that the generated pressure, before and after injecting the plasmonic nano-particles, measured at a single point, located away from the targeted inhomogeneity to image, and at a given band of incident frequencies is enough to reconstruct the (eventually complex valued) permittivity. Indeed, from these measurements, we define an indicator function which depends on the used incident frequency and the time of measurement. This indicator function has differentiating behaviors in terms of both time and frequency. First, from the behavior in terms of time, we can estimate the arrival time of the pressure from which we can localize the injected nano-particle. Second, we show that this indicator function has maximum picks at incident frequencies close to the plasmonic resonances. This allows us to estimate these resonances from which we construct the permittivity. To justify these properties, we derive the dominant electric field created by the injected nano-particle when the incident frequency is close to plasmonic resonances. This is done for the full Maxwell system. To this end, we use a natural spectral decomposition of the vector space (L^2(D))^3 based on the spectra of the Newtonian and the Magnetization operators. As another key argument, we provide the singularity analysis of the Green's tensor of the Maxwell problem with varying permittivity. Such singularity is unusual if compared to the ones of the elliptic case (as the acoustic or elastic models). In addition, we show how the derived approximation of the electric fields propagates, as a source, in the induced pressure with the time.

math.AP↗

arXiv2020

An Introduction To The Mathematics Of The Imaging Modalities Using Small Scaled Contrast Agents

In the recent years, we witness a great interest in imaging, in a wide sense, using contrast agents. One of the reasons is that many imaging modalities, as the ones related to medical sciences, suffer from several shortcomings. The most serious one is the issue of instability. Indeed, it is, nowadays, a common certainty that classical inverse problems of recovering objects from remote measurements are, mostly, highly unstable. To recover the stability, it is advised to create, whenever possible, the missing contrasts in the targets to image. In this survey paper, we follow this direction and propose an approach how to analyze mathematically the effect of the injected agents on the different fields under consideration. These contrast agents are small-sized particles modeled with materials that enjoy high contrasts as compared to the ones of the background. These two properties allow them, under critical scales of size/contrast, to create local spots when excited from far. These local spots can be remotely recovered in stable ways. The accessible information on the target are encoded in theses spots. After stating a class of such imaging modalities that enter into this framework, as the acoustic imaging, photo-acoustic imaging, optical imaging and more, we provide detailed analysis for first two modalities where the contrast agents are micro-bubbles and nano-particles respectively.

physics.optics↗

arXiv2020

Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves). We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region. 1. We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minneart resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of the Minnaert resonance in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way. 2. In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

math.AP↗

arXiv2020

Mathematical Analysis of the Photo-acoustic imaging modality using resonating dielectric nanoparticles: The 2D TM-model

We deal with the photoacoustic imaging modality using dielectric nanoparticles as contrast agents. Exciting the heterogeneous tissue, localized in a bounded domain $Ω$, with an electromagnetic wave, at a given incident frequency, creates heat in its surrounding which in turn generates an acoustic pressure wave (or fluctuations). The acoustic pressure can be measured in the accessible region $\partial Ω$ surrounding the tissue of interest. The goal is then to extract information about the optical properties (i.e. the permittivity and conductivity) of this tissue from these measurements. We describe two scenarios. In the first one, we inject single nanoparticles while in the second one we inject couples of closely spaced nanoparticles (i.e. dimers). This can be done using dielectric nanoparticles enjoying high contrasts of both its electric permittivity and conductivity. These results are possible using frequencies of incidence close to the resonances of the used dielectric nanoparticles. This allows us to solve the photoacoustic inverse problem with direct approximation formulas linking the measured pressure to the optical properties of the tissue. The results are justified in the 2D TM-model.

math.AP↗