arXiv subjects

Lucas Chesnel

Publications and source records attributed to Lucas Chesnel.

At least 19 recordsLinked to original sources

arXiv2025

Trapped modes in electromagnetic waveguides

We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.

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arXiv2025

A few techniques to achieve invisibility in waveguides

The aim of this lecture is to consider a concrete problem, namely the identification of situations of invisibility in waveguides, to present techniques and tools that may be useful in various fields of applied mathematics. To be more specific, we will be interested in the propagation of acoustic waves in guides which are unbounded in one direction. In general, the diffraction of an incident field in such a structure in presence of an obstacle generates a reflection and a transmission characterized by some scattering coefficients. Our goal will be to play with the geometry, the frequency and/or the index material to control these scattering coefficients. We will explain how to: - develop a continuation method based on the use of shape derivatives to construct invisible defects; - exploit complex resonances located closed to the real axis to hid obstacles; - construct a non self-adjoint operator whose eigenvalues coincide with frequencies such that there are incident fields whose energy is completely transmitted. Our approaches will mainly rely on techniques of asymptotic analysis as well as spectral theory for self-adjoint and non self-adjoint operators. Most of the results will be illustrated by numerical experiments.

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arXiv2025

Eigenvalue falls in thin broken quantum strips

We are interested in the spectrum of the Dirichlet Laplacian in thin broken strips with angle $\alpha$. Playing with symmetries, this leads us to investigate spectral problems for the Laplace operator with mixed boundary conditions in trapezoids of thickness $\varepsilon>0$ small. We give an asymptotic expansion of the first eigenvalues and corresponding eigenfunctions as $\varepsilon$ tends to zero. The new point in this work is to study the dependence with respect to $\alpha$. We highlight a curious phenomenon of diving eigenvalues: when the strip is more and more broken, at certain critical angles, that we characterize, an eigenvalue moves down very rapidly below the pack of other eigenvalues. We prove that this occurs more gently at $\alpha=0$ than at positive critical angles.

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arXiv2025

Maxwell's equations with hypersingularities at a negative index material conical tip

We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity $\varepsilon$ and the permeability $μ$ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in $\varepsilon$ and in $μ$, the corresponding scalar operators are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical $L^2$ framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the $\mathrm{T}$-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.

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arXiv2024

Examples of non-scattering inhomogeneities

We consider the scattering of waves by a penetrable inclusion embedded in some reference medium. We exhibit examples of materials and geometries for which non-scattering frequencies exist, i.e., for which at some frequencies there are incident fields which produce null scattered fields outside of the inhomogeneity. We show in particular that certain domains with corners or even cusps can support non-scattering frequencies. We relate the latter, for some inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also find situations where incident non-scattering fields solve the Helmholtz equation in a neighborhood of the inhomogeneity and not in the whole space. Finally, in relation with invisibility, we give examples of inclusions of anisotropic materials which are non-scattering for all real frequencies. We prove that corresponding material indices must have a special structure on the boundary.

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arXiv2024

Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $κ=(κ_1,κ_2)$ which characterizes the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases $κ_1\ge0$ and $κ_2\in[-κ_1,κ_1]$. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $κ$. In particular, we show that for a given $κ_1>0$, there is some $h(κ_1)>0$ such that discrete spectrum exists for $κ_2\in[-κ_1,0)\cup(h(κ_1),κ_1]$ whereas it is empty for $κ_2\in[0,h(κ_1)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.

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arXiv2024

Generalized impedance boundary conditions with vanishing or sign-changing impedance

We consider a Laplace type problem with a generalized impedance boundary condition of the form $\partial_νu=-\partial_x(g\partial_xu)$ on a flat part $Γ$ of the boundary. Here $ν$ is the outward unit normal vector to $\partialΩ$, $g$ is the impedance function and $x$ is the coordinate along $Γ$. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$ have been investigated. In this work, we address situations where $Γ$ contains the origin and $g(x)=\mathbb{1}_{x>0}(x)x^α$ or $g(x)=-\mbox{sign}(x)|x|^α$ with $α\ge0$. In other words, we study cases where $g$ vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of $α$. For $α\in[0,1)$, we show that the associated operators are Fredholm of index zero while it is not the case when $α=1$. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems.

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arXiv2023

On the breathing of spectral bands in periodic quantum waveguides with inflating resonators

We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $Π^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $Ω$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $Ω$ that we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $Ω$, there holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of $A^\varepsilon$ is very sparse, made of bands of length at most $O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $Ω$ however, we have $\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$ which allow for wave propagation in $Π^\varepsilon$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $Ω$ around a particular $Ω_\star$ where $\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the spectrum of $A^\varepsilon$: when inflating $Ω$ around $Ω_\star$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold $π^2/\varepsilon^2$, stops breathing and becomes extremely short as $Ω$ continues to inflate.

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arXiv2023

Acoustic waveguide with a dissipative inclusion

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient $η$, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when $η\to0^+$ (small dissipation) and when $η\to+\infty$ (large dissipation). Surprisingly, at the limit $η\to+\infty$, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to $η$. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations.

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arXiv2023

Spectrum of the Dirichlet Laplacian in a thin cubic lattice

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\varepsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-δ/\varepsilon})$, $δ>0$, while the length of the next spectral segments is $O(\varepsilon)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^Ω$ in the geometry $Ω$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^Ω$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^Ω$, that is no non trivial bounded solution at the threshold frequency for $A^Ω$. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.

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arXiv2021

Acoustic passive cloaking using thin outer resonators

We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width $\varepsilon$ and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width $\varepsilon$. Various numerical experiments illustrate the theory.

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arXiv2021

Design of an acoustic energy distributor using thin resonant slits

We consider the propagation of time harmonic acoustic waves in a device with three channels. The wave number is chosen such that only the piston mode can propagate. The main goal of this work is to present a geometry which can serve as an energy distributor. More precisely, the geometry is first designed so that for an incident wave coming from one channel, the energy is almost completely transmitted in the two other channels. Additionally, tuning a bit two geometrical parameters, we can control the ratio of energy transmitted in the two channels. The approach is based on asymptotic analysis for thin slits around resonance lengths. We also provide numerical results to illustrate the theory.

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arXiv2021

Design of a mode converter using thin resonant ligaments

The goal of this work is to design an acoustic mode converter. More precisely, the wave number is chosen so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory.

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arXiv2020

Maxwell's equations with hypersingularities at a conical plasmonic tip

In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle.

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arXiv2020

Abnormal acoustic transmission in a waveguide with perforated screens

We consider the propagation of the piston mode in an acoustic waveguide obstructed by two screens with small holes. In general, due to the features of the geometry, almost no energy of the incident wave is transmitted through the structure. The goal of this article is to show that tuning carefully the distance between the two screens, which form a resonator, one can get almost complete transmission. We obtain an explicit criterion, not so obvious to intuit, for this phenomenon to happen. Numerical experiments illustrate the analysis.

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arXiv2020

Qualitative indicator functions for imaging crack networks using acoustic waves

We consider the problem of imaging a crack network embedded in some homogeneous background from measured multi-static far field data generated by acoustic plane waves. We propose two novel approaches that can be seen as extensions of linear sampling-type methods and that provide indicator functions which are sensitive to local cracks densities. The first approach uses multiple frequencies data to compute spectral signatures associated with artificially embedded localized obstacles. The second approach also exploits the idea of incorporating an artificial background but uses data for a single frequency. The indicator function is built using a similar concept as for differential sampling methods: compare the solution of the interior transmission problem for healthy inclusion with the one with embedded cracks. The performance of the methods is tested and discussed on synthetic examples and the numerical results are compared with the ones obtained using the classical factorization method.

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arXiv2020

A continuation method for building invisible obstacles in waveguides

We consider the propagation of acoustic waves at a given wavenumber in a waveguide which is unbounded in one direction. We explain how to construct penetrable obstacles characterized by a physical coefficient $ρ$ which are invisible in various ways. In particular, we focus our attention on invisibility in reflection (the reflection matrix is zero), invisibility in reflection and transmission (the scattering matrix is the same as if there were no obstacle) and relative invisibility (two different obstacles have the same scattering matrix). To study these problems, we use a continuation method which requires to compute the scattering matrix $\mathbb{S}(ρ)$ as well as its differential with respect to the material index $d\mathbb{S}(ρ)$. The justification of the method also needs for the proof of abstract results of ontoness of well-chosen functionals constructed from the terms of $d\mathbb{S}(ρ)$. We provide a complete proof of the results in monomode regime when the wavenumber is such that only one mode can propagate. And we give all the ingredients to implement the method in multimode regime. We end the article by presenting numerical results to illustrate the analysis.

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arXiv2020

Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

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