arXiv subjects

Mourad Sini

Publications and source records attributed to Mourad Sini.

At least 37 records · Page 2Linked to original sources

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

Time-Dependent Acoustic Waves Generated by Multiple Resonant Bubbles: Application to Acoustic Cavitation

We analyse the ultrasound waves reflected by multiple bubbles in the linearized time-dependent acoustic model. The generated time-dependent wave field is estimated close to the bubbles. The motivation of this study comes from the therapy modality using acoustic cavitation generated by injected bubbles into the region of interest. The goal is to create enough, but not too much, pressure in the region of interest to eradicate anomalies in that region. Here, we derive the dominant part of the generated acoustic field by a cluster of bubbles taking into account the (high) contrasts of their mass density and bulk as well as their general distribution in the given region. As consequences of these approximations, we highlight the following features: 1. If we use dimers (two close bubbles), or generally polymers, then we obtain a remarkable enhancement of the whole echo in the whole time. 2. If we distribute the bubbles every where in the region of interest,then we can derive the effective acoustic model which turns out to be a dispersive one. We show that, for a given desired pressure, we can tune the effective model to generate it.

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arXiv2024

Extraction of the mass density using only the ${\mathtt{p}}$-parts of the elastic fields generated by injected highly dense small inclusions

We propose a reconstruction method to extract the variable mass density from the elastic farfields, with a single incident direction, measured before and after injecting highly dense small scaled inclusions. We take as a model, the Lamé system where the mass density is the unknown in $Ω$ and the Lamé parameters are known constants. The injected small/dense inclusion, $D:=z +a B\, (\subset\subset Ω)$ with $z$ as its location, $a\ll 1$ as its maximum radius and $B$ of unit volume, generates a sequences of resonant frequencies. These special frequencies are related to the eigenvalues of the Lamé volume integral operator defined on the domain of the inclusion and thus are, in principle, computable. After injecting the small inclusion at a location point $z$, we send an elastic incident plane wave at an incident frequency close to one of the mentioned resonant frequencies, say $ω_{n_0}$. Contrasting the ($\mathtt{p}$-parts of the) farfields generated, at one incident direction, before and after injecting this small inclusion, we provide an explicit formula that allows us to recover the total field $V^{t,\mathtt{p}}(z,-\hat{x})$ corresponding to $\mathtt{p}$-incident waves at the location $z$. This total field is generated in the absence of the inclusion. Then we repeat the experiment by injecting more inclusions inside $Ω$. Using this reconstructed field in the Lamé PDE system, via a numerical differentiation, we recover the values of the mass density inside $Ω$. It is worth mentioning that, we use measurements of dimension 3 to recover a function of 3 dimensions freedom. This makes the inverse problem not over determined. In addition, we use only the pressure wave and the $\mathtt{p}$-part of the farfield, for the reconstruction. To our best knowledge, this is the first result using only one type of elastic waves for the parameter identification.

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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

Dispersive Effective Model in the Time-Domain for Acoustic Waves Propagating in Bubbly Media

We derive the effective medium theory for the linearized time-domain acoustic waves propagating in a bubbly media. The analysis is done in the time-domain avoiding the need to use Fourier transformation. This allows considering general incident waves avoiding band limited ones as usually used in the literature. Most importantly, the outcome is as follows: 1. As the bubbles are resonating, with the unique subwavelength Minnaert resonance, the derived effective wave model is dispersive. Precisely, the effective acoustic model is an integro-differential one with a time-convolution term highlighting the resonance effect. 2. The periodicity in distributing the cluster of bubbles is not needed, contrary to the case of using traditional two-scale homogenization procedures. Precisely, given any $C^1$-smooth function $K$, we can distribute the bubbles so that locally the number of such bubbles is dictated by $K$. In addition to its dispersive character, the effective model is affected by the function $K$. Such freedom and generality is appreciable in different applied sciences including materials sciences and mathematical imaging.

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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.

math.AP↗

arXiv2024

Elastic fields generated by multiple small inclusions with high mass density at nearly resonant frequencies

We derive the elastic field generated by multiple small-scaled inclusions distributed in a bounded set of $\mathbb{R}^3$. These inclusions are modeled with moderate values of the Lamé coefficients while they have a large relative mass density. These properties allow them to enjoy a sequences of resonant frequencies that can be computed via the eigenvalues of the volume integral operator having the Navier fundamental matrix as a kernel, i.e. the Navier volume operator. The dominant field, i.e. the Foldy-Lax field, models the multiple interactions between the inclusions with scattering coefficients that are inversely proportional to the difference between the used incident frequency and the already mentioned resonances. We show, in particular, that to reconstruct remotely the scattered field generated after $N$ interactions between the inclusions, one needs to use an incident frequency appropriately close to the proper resonance of the inclusions. We provide an explicit link between the order $N$ of interactions and the distance from the incident frequency to the resonance. Finally, if the cluster of the inclusions is densely distributed in a given bounded domain, then the expression of the induced dominant field suggests that the equivalent homogenized mass density can change sign depending if the used incident frequencies is smaller or larger than a certain threshold (which is explicitly given in terms of the resonant frequencies of the inclusions).

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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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arXiv2023

Wave Propagation in Pure-Time Modulated Step Media With Applications to Temporal-Aiming

We analyze the propagation of an incident electromagnetic wave in a purely-time modulated medium. Precisely, we assume that the permeability is unchanged while the permittivity has a multiple-step profile in time and uniformly constant in space. For this, we use time-dispersive Lorenz's model with time-dependent plasma frequency with highly concentrated values near the centers of the steps' intervals. Under certain regimes linking the number of steps to the contrasts of the permittivity, we can generate effective permittivity having positive and high values on a finite interval of time which behaves as a 'wall' that kills the forward waves and keeps only the backward waves (i.e. enabling full reflection). But this happens if these high values are away from a discrete set. If these high values are close to the elements of this discrete set, then the effective medium behaves as a 'well' that absorbs all the forward waves and hence there is no backward waves (i.e. enabling full transmission). Such results are reminiscent to the 'wall' and 'well' effects known in the quantum mechanics theory.

math-ph↗

arXiv2023

Acoustic Cavitation using Resonating Micro-Bubbles. Analysis in the Time-Domain

We study the time-domain acoustic wave propagation in the presence of a micro-bubble. This micro-bubble is characterized by a mass density and bulk modulus which are both very small as compared to the ones of the background vacuum. The goal is to estimate the amount of pressure that is created very near (at a distance proportional to the radius of the bubble) to the bubble. We show that, at that small distance, the dominating field is reminiscent to the wave created by a point-like obstacle modeled formally by a Dirac-like heterogeneity with support at the location of the bubble and the contrast between the bubble and background material as the scattering coefficient. As a conclusion, we can tune the bubbles material properties so that the pressure near it reaches a desired amount. Such design might be useful in the purpose of acoustic cavitation where one needs enough, but not too much, pressure to eliminate unwanted anomalies. The mathematical analysis is done using time-domain integral equations and asymptotic analysis techniques. A well known feature here is that the contrasting scales between the bubble and the background generate resonances (mainly the Minnaert one) in the time-harmonic regime. Such critical scales, and the generated resonances, are also reflected in the time-domain estimation of the acoustic wave. In particular, reaching the desired amount of pressure near the location of the bubble is possible only with such resonating bubbles.

math.AP↗

arXiv2023

Recovering both the wave speed and the source function in a time-domain wave equation by injecting contrasting droplets

Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the space-time dependent source function from the measurement of the wave, collected at a single point $x$ for a large enough interval of time, generated by a small scaled droplets, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel-time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small droplets) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation.

math.AP↗

arXiv2023

Electromagnetic waves generated by a dielectric moving at a constant speed

We consider a regular and bounded dielectric body moving at a speed $|V|$, following a constant vector field $V$, with respect to a reference frame. In this frame, the special relativity implies that the Maxwell system is derived through the constitutive equations linking the moving speed $|V|$ and the speed of light $c$ in the background medium (as the vacuum for instance). Based on this model, we derive the well-poseness of the related forward scattering problem in the natural regime where $\frac{|V|}{c}\leq C_{te}$ with an appropriate constant $C_{te} <1$ that we estimate. In particular, we show the invertibility of the related Lippmann-Schwinger system in this regime and state the corresponding Born series in terms of the ratio $\frac{|V|}{c}$. As an application, we state and show the unique identifiability of the inverse problem of detecting the dielectric body, without knowing the moving speed $|V|$ or $V$, by illuminating it with incident electromagnetic waves propagating at the speed $c$. Such identifiability result makes sense in the regime under consideration.

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↗

arXiv2023

Heat Generation Using Lorentzian Nanoparticles. The Full Maxwell System

We analyse and quantify the amount of heat generated by a nanoparticle, injected in a background medium, while excited by incident electromagnetic waves. These nanoparticles are dispersive with electric permittivity following the Lorentz model. The purpose is to determine the quantity of heat generated extremely close to the nanoparticle (at a distance proportional to the radius of the nanoparticle). This study extends our previous results, derived in the 2D TM and TE regimes, to the full Maxwell system. We show that by exciting the medium with incident frequencies close to the Plasmonic or Dielectric resonant frequencies, we can generate any desired amount of heat close to the injected nanoparticle while the amount of heat decreases away from it. These results offer a wide range of potential applications in the areas of photo-thermal therapy, drug delivery, and material science, to cite a few. To do so, we employ time-domain integral equations and asymptotic analysis techniques to study the corresponding mathematical model for heat generation. This model is given by the heat equation where the body source term comes from the modulus of the electric field generated by the used incident electromagnetic field. Therefore, we first analyse the dominant term of this electric field by studying the full Maxwell scattering problem in the presence of Plasmonic or All-dielectric nanoparticles. As a second step, we analyse the propagation of this dominant electric field in the estimation of the heat potential. For both the electromagnetic and parabolic models, the presence of the nanoparticles is translated into the appearance of large scales in the contrasts for the heat-conductivity (for the parabolic model) and the permittivity (for the full Maxwell system) between the nanoparticle and its surrounding.

math.AP↗

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↗

arXiv2022

Heat Generation using Lorentzian Nanoparticles: Estimation via Time-Domain Techniques

We analyze the mathematical model that describes the heat generated by electromagnetic nanoparticles. We use the known optical properties of the nanoparticles to control the support and amount of the heat needed around a nanoparticle. Precisely, we show that the dominant part of the heat around the nanoparticle is the electric field multiplied by a constant dependent, explicitly and only, on the permittivity and quantities related to the eigenvalues and eigenfunctions of the Magnetization (or the Newtonian) operator, defined on the nanoparticle, and inversely proportional to the distance to the nanoparticle. The nanoparticles are described via the Lorentz model. If the used incident frequency is chosen related to the plasmonic frequency $ω_p$ (via the Magnetization operator) then the nanoparticle behaves as a plasmonic one while if it is chosen related to the undamped resonance frequency $ω_0$ (via the Newtonian operator), then it behaves as a dielectric one. The two regimes exhibit different optical behaviors. In both cases, we estimate the generated heat and discuss advantages of each incident frequency regime. The analysis is based on time-domain integral equation techniques avoiding the use of (formal) Fourier type transformations.

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