arXiv subjects

Habib Ammari

Publications and source records attributed to Habib Ammari.

At least 73 records · Page 4Linked to original sources

arXiv2023

Spectral convergence in large finite resonator arrays: the essential spectrum and band structure

We show that resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states for the finite system converge in distribution to the (continuous) density of states of the infinite system. We achieve this by proving a weak convergence of the finite capacitance matrix to corresponding (translationally invariant) Toeplitz matrix of the infinite structure. With this characterization at hand, we use the truncated Floquet transform to introduce a notion of spectral band structure for finite materials. This principle is also applicable to structures that are not translationally invariant and have interfaces. We demonstrate this by considering examples of perturbed systems with defect modes, such as an analogue of the well-known interface Su-Schrieffer-Heeger (SSH) model.

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arXiv2023

Edge modes in subwavelength resonators in one dimension

We present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyse both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantised and the latter is not. Furthermore, we show the existence of localised edge modes arising from defects in the periodicity in both the Hermitian and non-Hermitian cases. In the non-Hermitian case, we provide a complete characterisation of the edge modes.

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arXiv2023

An Operator Theory for Analyzing the Resolution of Multi-illumination Imaging Modalities

By introducing a new operator theory, we provide a unified mathematical theory for general source resolution in the multi-illumination imaging problem. Our main idea is to transform multi-illumination imaging into single-snapshot imaging with a new imaging kernel that depends on both the illumination patterns and the point spread function of the imaging system. We thus prove that the resolution of multi-illumination imaging is approximately determined by the essential cutoff frequency of the new imaging kernel, which is roughly limited by the sum of the cutoff frequency of the point spread function and the maximum essential frequency in the illumination patterns. Our theory provides a unified way to estimate the resolution of various existing super-resolution modalities and results in the same estimates as those obtained in experiments. In addition, based on the reformulation of the multi-illumination imaging problem, we also estimate the resolution limits for resolving both complex and positive sources by sparsity-based approaches. We show that the resolution of multi-illumination imaging is approximately determined by the new imaging kernel from our operator theory and better resolution can be realized by sparsity-promoting techniques in practice but only for resolving very sparse sources. This explains experimentally observed phenomena in some sparsity-based super-resolution modalities.

eess.IV↗

arXiv2022

Super-resolution of positive near-colliding point sources

In this paper, we analyze the capacity of super-resolution of one-dimensional positive sources. In particular, we consider the same setting as in [arXiv:1904.09186v2 [math.NA]] and generalize the results there to the case of super-resolving positive sources. To be more specific, we consider resolving $d$ positive point sources with $p \leqslant d$ nodes closely spaced and forming a cluster, while the rest of the nodes are well separated. Similarly to [arXiv:1904.09186v2 [math.NA]], our results show that when the noise level $ε\lesssim \mathrm{SRF}^{-2 p+1}$, where $\mathrm{SRF}=(ΩΔ)^{-1}$ with $Ω$ being the cutoff frequency and $Δ$ the minimal separation between the nodes, the minimax error rate for reconstructing the cluster nodes is of order $\frac{1}Ω \mathrm{SRF}^{2 p-2} ε$, while for recovering the corresponding amplitudes $\left\{a_j\right\}$ the rate is of order $\mathrm{SRF}^{2 p-1} ε$. For the non-cluster nodes, the corresponding minimax rates for the recovery of nodes and amplitudes are of order $\fracεΩ$ and $ε$, respectively. Our numerical experiments show that the Matrix Pencil method achieves the above optimal bounds when resolving the positive sources.

eess.IV↗

arXiv2022

Quantum integrable systems and concentration of plasmon resonance

We are concerned with the quantitative mathematical understanding of surface plasmon resonance (SPR) when $d \geq 3$. SPR is the resonant oscillation of conducting electrons at the interface between negative and positive permittivity materials and forms the fundamental basis of many cutting-edge applications of metamaterials. It is recently found that the SPR concentrates due to curvature effect. In this paper, we derive sharper and more explicit characterisations of the SPR concentration at high-curvature places in both the static and quasi-static regimes. The study can be boiled down to analyzing the geometries of the so-called Neumann-Poincaré (NP) operators, which are certain pseudo-differential operators sitting on the interfacial boundary. We propose to study the joint Hamiltonian flow of an integral system given by the moment map defined by the NP operator. Via considering the Heisenberg picture and lifting the joint flow to a joint wave propagator, we establish a more general version of quantum ergodicity on each leaf of the foliation of this integrable system, which can then be used to establish the desired SPR concentration results. The mathematical framework developed in this paper leverages the Heisenberg picture of quantization and extends some results of quantum integrable system via generalising the concept of quantum ergodicity, which can be of independent interest to the spectral theory and the potential theory.

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arXiv2022

Asymptotic links between signal processing, acoustic metamaterials and biology

Biomimicry is a powerful science that takes advantage of nature's remarkable ability to devise innovative solutions to challenging problems. In this work, we use asymptotic methods to develop the mathematical foundations for the exchange of design inspiration and features between biological hearing systems, signal processing algorithms and acoustic metamaterials. Our starting point is a concise asymptotic analysis of high-contrast acoustic metamaterials. We are able to fine tune this graded structure to mimic the biomechanical properties of the cochlea, at the same scale. We then turn our attention to developing a biomimetic signal processing algorithm. We use the response of the cochlea-like metamaterial as an initial filtering layer and then add additional biomimetic processing stages, designed to mimic the human auditory system's ability to recognise the global properties of natural sounds. This demonstrates the three-way exchange of ideas that, thanks to our analysis, is possible between signal processing, metamaterials and biology.

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arXiv2022

Topological phenomena in honeycomb Floquet metamaterials

We dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation $\left\{\frac{d}{dt}X = A_α(t)X\right\}_{α\in \mathbb{T}^d}$, we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form $F,~P$ of the associated fundamental solution $\left\{X_α(t) = P(α,t)\exp(tF_α)\right\}_{α\in \mathbb{T}^d}$. This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix $X_α(T)$ and the Lyapunov transformation $P(α,t)$. The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of honeycomb Floquet metamaterials. We provide two interesting examples of topologically non-trivial time-modulated honeycomb structures.

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arXiv2022

Transmission properties of space-time modulated metamaterials

We prove the possibility of achieving exponentially growing wave propagation in space-time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k-gaps in the band structures of space-time modulated systems of subwavelength resonators.

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arXiv2022

Unidirectional edge modes in time-modulated metamaterials

We prove the possibility of achieving unidirectional edge modes in time-modulated supercell structures. Such finite structures consist of two trimers repeated periodically. Because of their symmetry, they admit degenerate edge eigenspaces. When the trimers are time-modulated with two opposite orientations, the degenerate eigenspace splits into two one-dimensional eigenspaces described by an analytical formula, each corresponds to a mode which is localized at one edge of the structure. Our results on the localization and stability of these edge modes with respect to fluctuations in the time-modulation amplitude are illustrated by several numerical simulations.

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arXiv2022

Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array

In this paper, we develop a new technique to obtain nearly optimal estimates of the computational resolution limits introduced in Appl. Comput. Harmon. Anal. 56 (2022) 402-446; IEEE Trans. Inf. Theory 67(7) (2021) 4812-4827; Inverse Probl. 37(10) (2021) 104001 for two-dimensional super-resolution problems. Our main contributions are fivefold: (i) Our work improves the resolution estimate for number detection and location recovery in two-dimensional super-resolution problems to nearly optimal; (ii) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems (or Direction of Arrival problems (DOA)). The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (iii) Our techniques pave the way for improving the estimate for resolution limits in higher-dimensional super-resolutions to nearly optimal; (iv) Inspired by these new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (v) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys many advantages compared to the conventional DOA algorithms. The coordinate-combination idea seems to be a promising way for multi-dimensional DOA estimation.

eess.IV↗

arXiv2022

Dynamic super-resolution in particle tracking problems

Particle tracking in biological imaging is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this paper, we provide a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for the dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates, we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles moving. This result is derived by an observation that the inherent cut-off frequency for the velocity recovery can be viewed as the total observation time multiplies the cut-off frequency of the imaging system, which may lead to a better resolution limit as compared to the one for each diffraction-limited frame. It is anticipated that this observation can inspire new reconstruction algorithms that improve the resolution of particle tracking in practice.

eess.SP↗

arXiv2021

Robust edge modes in dislocated systems of subwavelength resonators

Robustly manipulating waves on subwavelength scales can be achieved by, firstly, designing a structure with a subwavelength band gap and, secondly, introducing a defect so that eigenfrequencies fall within the band gap. Such frequencies are well known to correspond to localized modes. We study a one-dimensional array of subwavelength resonators, proving that there is a subwavelength band gap, and showing that by introducing a dislocation we can place localized modes at any point within the band gap. We complement this analysis by studying the stability properties of the corresponding finite array of resonators, demonstrating the value of being able to customize the position of eigenvalues within the band gap.

math.AP↗

arXiv2021

Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials

Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient matrix. Secondly, we apply such an asymptotic theory for the analysis of Floquet metamaterials. We provide a characterization of asymptotic exceptional points for a pair of subwavelength resonators with time-dependent material parameters. We prove that asymptotic exceptional points are obtained if the frequency components of the perturbations fulfill a certain ratio, which is determined by the geometry of the dimer of subwavelength resonators.

math.AP↗

arXiv2021

Non-reciprocal wave propagation in space-time modulated media

We prove the possibility of achieving non-reciprocal wave propagation in space-time modulated media and give an asymptotic analysis of the non-reciprocity property in terms of the amplitude of the time-modulation. Such modulation causes a folding of the band structure of the material, which may induce degenerate points. By breaking time-reversal symmetry, we show that these degeneracies may open into non-symmetric, unidirectional band gaps. Finally we illustrate our results by several numerical simulations.

math.AP↗

arXiv2021

On the validity of the tight-binding method for describing systems of subwavelength resonators

The goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbour approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbour approximations are not generally appropriate.

math.AP↗

arXiv2021

Localized sensitivity analysis at high-curvature boundary points of reconstructing inclusions in transmission problems

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarisation tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via the microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one steps depend on the global geometry, and one particular step depends on the mean curvature at a given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave super-resolution imaging.

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arXiv2021

Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices

In this work, we are concerned with the mathematical modeling of the electromagnetic (EM) scattering by arbitrarily shaped non-magnetic nanoparticles with high refractive indices. When illuminated by visible light, such particles can exhibit a very strong isotropic magnetic response, resulting from the coupling of the incident wave with the circular displacement currents of the EM fields. The main aim of this work is to mathematically illustrate this phenomenon. We shall first introduce the EM scattering resolvent and the concept of dielectric subwavelength resonances. Then we derive the a priori estimates for the subwavelength resonances and the associated resonant modes. We also show the existence of resonances and obtain their asymptotic expansions in terms of the small particle size and the high contrast parameter. After that, we investigate the enhancement of the scattering amplitude and the cross sections when the resonances occur. In doing so, we develop a novel multipole radiation framework that directly separates the electric and magnetic multipole moments and allows us to clearly see their orders of magnitude and blow--up rates. We prove that at the dielectric subwavelength resonant frequencies, the nanoparticles with high refractive indices behave like the sum of the electric dipole and the resonant magnetic dipole. Some explicit calculations and numerical experiments are also provided to validate our general results and formulas.

math-ph↗

arXiv2021

Time-dependent high-contrast subwavelength resonators

In the field of metamaterials, many intriguing phenomena arise from having a structure which is periodic in space. In time-dependent structures, conceptually similar properties can arise, which nevertheless have fundamentally different physical implications. In this work, we study time-dependent systems in the context of subwavelength metamaterials. The main result is a capacitance matrix characterization of the band structure, which generalizes previous recent work on static subwavelength metamaterials. This characterization provides both theoretical insight and efficient numerical methods to compute the dispersion relationship of time-dependent structures. We exemplify this in several structures exhibiting interesting wave manipulation properties.

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