3.2.1 Core research topics

Our research activities are organized around core theoretical and methodological topics to address the above-listed modeling challenges.

Multiscale modeling. The physical models that we consider may feature three different space scales. First, the size of the computational domain is fixed by the nanostructure under consideration and the required observables. Second, the solution wavelength depends on the operating frequency and on the light velocity in the constitutive materials. Finally, the finest scale involved corresponds to the nanostructuring length. These three space scales can differ by orders of magnitude, leading to unaffordable computational costs, if the discretization scheme must resolve the nanostructure details. We thus aim at providing multiscale numerical schemes, that can embed fine scale information into a coarser mesh. Such methodologies lead to embarrassingly parallel two-level algorithms, that are especially suited for HPC environments and produce accurate numerical approximations. These multiscale schemes will in particular be designed in the framework of MHM (Multiscale Hybrid-Mixed) formulations recently introduced in collaboration with Frédéric Valentin at LNCC, Petropolis, Brazil. In the MHM framework, the inherent upscaling procedure that is at the heart of the approach, allows to incorporate more physics in the numerical schemes themselves, as the upscaling principle is used to construct physical basis functions that resolve the fine scales. At the second level, one defines a set of boundary value problems, whose solutions call for adapted versions of classical finite element or DG methods, and yield the upscaled basis functions.

High order DG methods. Designing numerical schemes that are high order accurate on general meshes, i.e., unstructured or hybrid structured/unstructured meshes, will still represent a major objective of our core research activities in ATLANTIS. We will focus on the family of Discontinuous Galerkin (DG) methods that has been extensively developed for wave propagation problems during the last 15 years. We will investigate several variants, namely nodal DGTD for time-domain problems, and HDG (Hybridized DG) for time-domain and frequency-domain problems, with the general goal of devising, analyzing and developing extensions of these methods, in order to deal with the physical drivers described in section 3.1: nonlinear features, in particular in relation with generation of higher order harmonics in electromagnetic wave interaction with nonlinear materials, and nonlinear models of electronic response in metallic and semiconductor materials; multiphysic couplings such as for instance when considering PDE models relevant to thermoplasmonics, optoelectronics and nanoelectronics. There are to date very few works promoting DG-type methods for these situations. Concerning nonlinear effects, it is worth notice that DG methods have already proved to be attractive and have been extensively developed for computational fluid dynamics, and in particular for gas dynamics. The development of a high order DG method for situations involving nonlinear materials require to address several issues such as the construction approximate Riemann solvers for the evaluation of numerical traces associated with the evaluation of boundary integral terms in DG weak forms, the formulation of positivity-preserving schemes as continuous solutions may develop nonlinear discontinuities and the definition of linearization strategies in conjunction with time-integration schemes.

Error estimation and adaptivity. While standard theory ensures the convergence of the discrete approximation to the correct solution, it does not permit to quantitatively estimate the discretization error in actual applications. As a result, the selection of the discretization parameters is often carried out by the practitioners themselves, based on their experience and hence manual approaches to guarantee a sufficient accuracy level. We aim at providing reliable measurements of the discretization error and devising a more systematic and rigorous procedure to select and adapt discretization parameters — in particular the mesh size and the discretization order — through the use of a posteriori estimators. On the one hand, we propose to design a posteriori error estimators for the wave-matter interaction problems we consider in ATLANTIS, able to provide a fully reliable error estimation. On the other hand, such estimators can be employed to drive hp-adaptive algorithms, where the mesh and discretization order are iteratively improved to fit the complicated structure of the solution.

Time integration for multiscale problems. Multiscale physical problems with complex geometries or het- erogeneous media are extremely challenging for conventional numerical simulations. Adaptive mesh refinement is an attractive technique for treating such problems and will be developed in our research activities in Atlantis. Local mesh refinement imposes a severe stability condition on explicit time integration since the allowed maximal time step size is constrained by the smallest element in the mesh. We will focus on different ways to overcome this stability condition, especially by using implicit-explicit (IMEX) methods where a time implicit scheme is used only for the refined part of the mesh, and a time explicit scheme is used for the other part. Optimizing the CPU time requires coupling IMEX methods with hybridized DG (HDG) methods and non-conforming meshes and will be one of the main objectives of our research activities in ATLANTIS.

Dealing with complex materials. Physically relevant simulations deal with increasing levels of complexity in the geometrical and/or physical characteristics of nanostructures, as well as their interaction with light. Standard simulation methods may fail to reproduce the underlying physical phenomena, therefore motivating the search for more sophisticated light-matter interaction numerical modeling strategies. A first decisive direction consists in refining classical linear dispersion models and we shall put a special focus on deriving a complete hierarchy of models, that will encompass standard linear models to more complex and nonlinear ones (such as Kerr-type materials, non- linear quantuum hydrodynamic theory models, etc.). One approach will rely on an accurate description of the Hamiltonian dynamics with intricate kinetic and exchange correlation energies, for different modeling purposes. A second direction will be motivated by the study of 2D materials. A major concern is centered around the choice of the modeling approach between a full costly 3D modeling and the use of equivalent boundary conditions (the so-called GSTC), that could in all generality be completely nonlinear. Assessing these two directions will require efficient dedicated numerical algorithms that will especially be able to tackle several types of nonlinearities and scales.

Dealing with coupled models. Several of our target physical fields are multiphysics in essence and require going beyond the sole description of the electromagnetic response. In thermoplasmonics, the various phenomena (heat transfer through light concentration, bubbles formation and dynamics) call for different kinds of governing PDEs (Maxwell, conduction, fluid dynamics). Since, in addition, these phenomena can occur in significatively different space and time scales, drawing a quite complete picture of the underlying physics is a challenging task, both in terms of modeling and numerical treatment. In the nanoelectronics field, an accurate description of the electronic properties involves including quantum effects. A coupling between Maxwell’s and Schrödinger equations (again at significantly different time and space scales) is a possible relevant scenario. In the optoelectronics field, the accurate prediction of semiconductors optical properties is a major concern. A possible strategy may require to solve both the electromagnetic and the drift-diffusion equations. In all these aforementioned examples, difficulties mainly arise both from the differences in physical nature as well as in the time/space scales at which each physical phenomenon occurs. Accurately modeling/solving their coupled interactions remains a formidable challenge.

High performance computing (HPC). HPC is transversal to almost all the other research topics considered in the team, and is concerned with both numerical algorithm design and software development. We will work toward taking advantage of fine grain massively parallel processing offered by GPUs in modern exascale architectures, by revisiting the algorithmic structure of the computationally intensive numerical kernels of the high order DG-based solvers that we develop. We will mostly rely on the lastest features of the OpenMP standard for implementing these numerical algorithms in the DIOGENeS software suite, which already offers a coarse grain parallelization based on MPI. We will also investigate task-based fine grain parallelization for dealing with load balancing issues inherent to realistic wave-matter interaction problems and that are sourced by both numerical (e.g., locally adapted interpolation order, PML layers, volumic observables, etc.) and physical (e.g., multiple material models) characteristics.

3.2.2 Complementary topics

Beside the above-discussed core reserach topics, we have also identified additional topics that are important or compulsory in view of maximizing the impact in nanophotonics or nanophononics of our core activities and methodological contributions.

Inverse design. Inverse design has emerged rather recently in nanophotonics, and is currently the subject of intense research as witnessed by several reviews 44. Artificial Intelligence (AI) techniques are also increasingly investigated within this context 51. In ATLANTIS, we will extend the modeling capabilities of the DIOGENeS software suite by using statistical learning techniques for the inverse design of nanophotonic devices. When it is linked to the simulation of a realistic 3D problem making use of one of the high order DG and HDG solvers we develop, the evaluation of a figure of merit is a costly process. Since a sufficiently large input data set of candidate designs, as required by using Deep Learning (DL), is generally not available, global optimization strategies relying on Gaussian Process (GP) models are considered in the first place. This activity will be conducted in close collaboration with researchers of the ACUMES project-team. In particular, we investigate GP-based inverse design strategies that were initially developed for optimization studies in relation with fluid flow problems 35- 36 and fluid-structure interaction problems 49.

Uncertainty analysis and quantification. The automatic inverse design of nanophotonic devices enables scientists and engineers to explore a wide design space and to maximize a device performance. However, due to the large uncertainty in the nanofabrication process, one may not be able to obtain a deterministic value of the objective, and the objective may vary dramatically with respect to a small variation in uncertain parameters. Therefore, one has to take into account the uncertainty in simulations and adopt a robust design model 39. We study this topic in close collaboration with researchers of the ACUMES project-team.

Numerical linear algebra. Sparse linear systems routinely appear when discretizing frequency-domain wave-matter interaction PDE problems. In the past, we have considered direct methods, as well as domain decomposition preconditioning coupled with iterative algorithms to solve such linear systems 7 In the future, we would like to further enhance the efficiency of our solvers by considering state-of-the-art linear algebra techniques such as block Krylov subspace methods 26, or low-rank compression techniques 48. We will also focus on multi-incidence problems in periodic structures, that are relevant to metagrating or metasurface design. Indeed, such problems lead to the resolution of several sparse linear systems that slightly differ from one another and could benefit from dedicated solution algorithms. We will collaborate with researchers of the HIEPACS (Inria Bordeaux-Sud Ouest) project-team to develop efficient and scalable solution strategies for such questions.

5.1.1 DIOGENeS

• Name: DIscOntinuous GalErkin Nanoscale Solvers
• Keywords: High-Performance Computing, Computational electromagnetics, Discontinuous Galerkin, Computational nanophotonics
• Functional Description: The DIOGENeS software suite provides several tools and solvers for the numerical resolution of light-matter interactions at nanometer scales. A choice can be made between time-domain (DGTD solver) and frequency-domain (HDGFD solver) depending on the problem. The available sources, material laws and observables are very well suited to nano-optics and nano-plasmonics (interaction with metals). A parallel implementation allows to consider large problems on dedicated cluster-like architectures.
• URL: https://diogenes.inria.fr/
• Authors: Stéphane Lanteri, Nikolai Schmitt, Alexis Gobé, Jonathan Viquerat, Guillaume Leroy
• Contact: Stéphane Lanteri

6.1.1 Explicit and IMEX Hybridizable DG methods

Participants: Théophile Chaumont-Frelet, Stéphane Descombes, Stéphane Lanteri, Georges Nehmetallah.

Hybridizable discontinuous Galerkin (HDG) methods were first studied in the team for frequency domain applications for which they enable the use of static condensation, leading to drastic reduction in computational time and memory consumption. More recently, we have investigated the use of HDG discretization to solve time-dependent problems. Specifically, in the context of the PhD thesis of Georges Nehmetallah, we focused on two particular aspects. On the one hand, HDG methods exhibit a superconvergence property that allows, by means of local postprocessing, to obtain new improved approximations of the unknowns. Our first contribution is to apply this methodology to time-dependent Maxwell's equations, where the post-processed approximation converges with order k+1 instead of k in the H(curl)-norm, when using polynomial of degree k greater than 1. The proposed method has been implemented for dealing with general 3D problems.

Another interesting aspect of the HDG method is that it can be conveniently employed to blend different time-integration schemes in different regions of the mesh. This is especially useful to efficiently handle locally refined space grids, that are required to take into account geometrical details. These ideas have already been explored for standard discontinuous Galerkin discretization in the context of the PhD thesis of Ludovic Moya 3-2. Here, we focused on HDG methods and we introduced a family of coupled implicit-explicit (IMEX) time integration methods for solving time-dependent Maxwell's equations. We established stability conditions that are independent of the size of the elements in the fine part of the mesh, and are only constrained by the coarse part. Numerical experiments on two-dimensional benchmarks illustrate the theory and the usefulness of the approach.

6.1.2 Reduced-order modeling for time-domain electromagnetics

Participants: Stéphane Lanteri, Kun Li, Liang Li.

This study is concerned with reduced-order modeling for time-domain electromagnetics and nanophotonics. More precisely, we consider the applicability of the proper orthogonal decomposition (POD) technique for the system of 3D time-domain Maxwell equations, possibly coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. We introduce a discontinuous Galerkin (DG) approach for the discretization of the problem in space based on an unstructured tetrahedral mesh. A reduced subspace with a significantly smaller dimension is constructed by a set of POD basis vectors extracted offline from snapshots that are obtained by the global DGTD scheme with a second order leap-frog method for time integration at a number of time levels. POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto the low-dimensional space. The stability of the POD-based ROM equipped with the second order leap-frog time scheme has been analysed through an energy method. Numerical experiments have allowed to verify the accuracy, and demonstrate the capabilities of the POD-based ROM. These very promising preliminary results 6-5 have been confirmed by extending the proposed POD-based ROM to the simulation of 3D nanophotonic problems.

6.1.3 Stability and asymptotic properties of the linearized Hydrodynamic Drude model

Participants: Serge Nicaise, Claire Scheid.

We go one step further toward a better understanding of the fundamental properties of the linearized hydrodynamical model studied in 9. This model is especially relevant for small nanoplasmonic structures (below 10 nm). Using a hydrodynamical description of the electron cloud, both retardation effects and nonlocal spatial response are taken into account. This results in a coupled PDE system for which we study the linear response. In 16, we concentrate on establishing well-posedness results combined to a theoretical and numerical stability analysis. We especially prove polynomial stability and provide optimal energy decay rate. Finally, we investigate the question of numerical stability of several explicit time integration strategies combined to a Discontinuous Galerkin spatial discretization.

6.1.4 Toward thermoplasmonics

Participants: Yves D'Angelo, Stéphane Lanteri, Thibault Laufroy, Claire Scheid.

Although losses in metal are viewed as a serious drawback in many plasmonics experiments, thermoplasmonics is the field of physics that tries to take advantage of the latter. Indeed, the strong field enhancement obtained in nanometallic structures lead to a localized temperature increase in its vicinity, leading to interesting photothermal effects. Therefore, metallic nanoparticles may be used as heat sources that can be easily integrated in various environments. This is especially appealing in the field of nanomedecine and can for example be used for diagnosis purposes or nanosurgery to cite but just a few. Due to the various scales and phenomena that come into play, accurate numerical modeling is challenging. Laser illumination first excites a plasmon oscillation (reaction of the electrons of the metal) that relaxes to a thermal equilibrium and in turn excites the metal lattice (phonons). The latter is then responsible for heating the surroundings. A relevant modeling approach thus consists in describing the electron-phonon coupling through the evolution of their respective temperature. Maxwell's equations are then coupled to a set of coupled nonlinear hyperbolic equations for the temperatures of respectively electrons, phonons and environment. The nonlinearities and the different time scales at which each thermalization occurs make the numerical approximation of these equations quite challenging. In the context of the PhD of Thibault Laufroy, which has started this year, we propose to develop a suitable numerical framework for studying thermoplasmonics. As a first step, we have reviewed the models used in thermoplasmonics that are most often based on strong or weak (nonlinear) couplings of Maxwell's equations with nonlinear equations modeling heat transfer (hyperbolic or parabolic), in order to lay the foundations of the numerical approximation framework that we will consider in the sequel. This activity is planed to be carried out in close collaboration with physicists or chemists specialized in the field, in particular from the Fresnel Institute in Marseilles (Guillaume Baffou).

6.1.5 Corner effects in nanoplasmonics

Participants: Camille Carvalho, Patrick Ciarlet, Claire Scheid.

In this work, we study nanoplasmonic structures with corners (typically a diedral/triangular structure); a situation that raises a lot of issues. We focus on a lossles Drude dispersion model and propose to investigate the range of validity of the amplitude limit principle. The latter predicts the asymptotic harmonic regime of a structure that is monochromatically illuminated, which makes a frequency domain approach relevant. However, in frequency domain, several well posedness problems arise due to the presence of corners (studied in the PhD thesis of Camille Carvalho). This should impact the validity of the limit amplitude principle and has not yet been addressed in the literature in this precise setting. Here, we combine frequency-domain and time-domain viewpoints to give a numerical answer to this question in two dimensions. We show that the limit amplitude principle does not hold for whole interval of frequencies, that are explicited using the well-posedness analysis. This work will be submitted in 2021.

6.1.6 DG methods for semiconductor device modeling

Participants: Stéphane Lanteri, Massimiliano Montone, Claire Scheid.

Semiconductors constitute the heart and soul of integrated electronic systems, thus of modern technology. The extraordinary progress in circuit integration, since Moore’s prediction from the early seventies, can certainly be majorly credited to the concurrent refinement of computer-aided design tools. Indeed, a higher scale of integration has punctually demanded a deeper understanding of semiconductor physics and more accurate mathematical models of devices to be numerically implemented and simulated. Charge transport is the starring phenomenon that needs to be predicted in order to build a mathematical model, based on higher-level quantities (e.g. electric current and voltage), that can be practically used for device simulation. A rigorous description demands statistics and quantum mechanics of electron/hole ensembles in crystal lattices, an approach that eventually leads to the Boltzmann transport equation, which solves for a state density function in the position-momentum space. The Boltzmann transport equation is extremely complex to deal with. In most applications, it is licit to call for simplifying assumptions to the degree where electron and hole ensembles are seen as gases of classical particles having an effective mass, which accounts for them being subjected to the electrostatic field produced by the atomic lattice. As a result, charge carrier transport is eventually described by a drift-diffusion model. This yields a system of partial differential equations (PDEs) which can be solved at two levels:

• Quasistatic approximation - The external force applied to the crystal is electrostatic, and drift-diffusion equations are coupled to a Poisson equation for the electrostatic potential. The goal is to determine the spatial distribution of carrier concentrations and the electric field (deduced from the potential).
• Fullwave model - The crystal is subject to an applied electromagnetic field, and Maxwell equations are coupled with transport equations for carrier dynamics. The goal is to determine the space-time evolution of carrier concentrations and the electromagnetic field.

The quasistatic approximation is rigorous when the steady state of the semiconductor has to be calculated. For example, this could be a preliminary step to the fullwave simulation of a device that is biased prior to responding to a (time-varying) electromagnetic excitation. The fullwave model is particularly relevant to electro-optics, i.e. when light-matter interaction is investigated. Indeed, such study is essential to understanding and accurately modeling the operation of photonic devices for light generation, modulation, absorption. In the context of the PhD of Massimiliano Montone that started thsi year, we study the formulation, analysis and development of a DGTD method for solving the coupled system of Maxwell equatiosn and drift-diffusion equations in the fullwave setting.

6.1.7 Advanced modeling of nanostructured CMOS imagers based on DG methods

Participants: Jérémy Grebot, Stéphane Lanteri, Denis Rideau, Claire Scheid.

In the context of the PhD of Jérémy Grebot that started this year, we study the photo-conversion process in CMOS imagers at micrometric wavelengths. If the majority of sensors operate on visible light, their modeling remains a difficult task. Indeed, part of the complexity of the modeling results from the very different shape and size factors between the whole pixel, the micrometric lenses and the nanometric structuring of the surface of the absorbing semiconductor. Moreover, an accurate estimation of light absorption is delicate because it depends on the absorbing material but also on the temperature of the latter and the density of free carriers. Finally, the transport of the photo-generated carriers can be tricky to simulate when they are in limited number. This last step requires, for example, a Monte-Carlo algorithm dedicated to stochastic phenomena. All in all, the ideal simulation chain will therefore have to deal with light transport, photon absorption in the SiGe (Germanium-doped Silicon) semiconductor material and the transport of the induced carriers to the electrodes. In this study, a true multiscale methodology will be set up to analyze several optimization strategies for imagers under development at STMicroelectronics. In particular, our goals are investigate, model and optimize the quantum efficiency and SNR (Signal-to-Noise Ratio) for infrared light in an image sensor. The impact of the absorbing material (SiGe or Ge possibly mechanically stressed) as well as the complex structuring of the surfaces will be systematically studied. For this purpose a modeling of the light absorption in the whole device, from curved and micrometric lenses to the pyramidal and nanometric texturing of the doped silicon, is necessary. This modeling of the absorption of photons is carried out with the help of a high order DGTD method. A preliminary task, which is underway, is to define an appropriate dispersion model of the SiGe for different doping rates, and taking into account the dependence to the temperature.

6.2.1 Homogenization theory for metasurfaces

Participants: Agnès Maurel, Kim Pham, Nicolas Lebbe.

We applied the (surfacic) quasi-periodic homogeneization theory to derive equivalent transmission condition of deeply subwavelength microstructured metasurfaces. The resulting conditions are closely related to the so-called GSTC (Generalized Sheet Transition Conditions) that were originally derived in the 90s from physicial principles. Our approach provides a direct link between the coefficients of the susceptibility tensors driving surfacic polarization fields and the structural design of a microstructured metasurface. These equivalent boundary conditions are useful for numerical simulations since they reduce the complexity of the meshes at the interfaces and thus the number of degrees of freedom to be solved (see Fig. 1 (a)). The numerical implementation of these GSTCs was done both in 2D and 3D using the finite element method (FEM) in the frequency domain (see Fig. 1 (b)).

6.2.2 Equilibrated error estimators for high-order methods

Participants: Théohpile Chaumont-Frelet, Alexandre Ern, Patrick Vega, Martin Vohralík.

Error estimators are an important tool in the discretization of PDE poblems. On the one hand, they permit to estimate the accuracy of the discrete model. In addition, estimators typically provide local information about the error in parts of the discretization mesh, enabling local refinement algorithms.

A now well-established and widely used family of error estimators is called “residual-based”. These estimators have the advatange to be relatively easy to implement, and very efficient. Unfortunately, they become less accurate as the discretization order is increased. This leads in general to large error over-estimation, and decreased performance of adaptive algorithms.

In this collaboration with the SERENA project-team, we are interested in a modern family of error estimators, called “equilibrated-fluxes”. These estimators are slightly more costly to compute than residual-based ones, but they are much more efficient when combined with high-order discretization schemes. A key properties of such estimators is that they are p-robust, meaning that they do not suffer from an increase in the polynomial degree p employed in the discretization method.

A first contribution in this context is the work 1 already mentioned in the activity report of the last year. It is concerned with high-frequency Helmholtz problems, and that has just been accepted for publication in February 2021.

In 2020, we have focused on Maxwell's equations in the low-frequency limit. While the low-frequency limit is not directly of interest to the applications considered in the Atlantis project-team, it is mathematically interesting as a simplified foundation for later extension to the high-frequency case. Part of our work, presenting crucial abstract mathematical results as been published this year 10. We also submitted our main results 20, where the novel estimator is presented and analyzed, and its efficiency is numerically assessed.

A novel research direction we have started in 2020 in the context of the post-doctoral fellowship of Patrick Vega, concerns the use of equilibrated-fluxes for the Helmholtz equation combined with high-order absorbing boundary conditions. These boundary conditions are of great interest in applications as they lead to improved approximations of radiation conditions and more efficient domain-decomposition preconditioners. They are also challenging to deal with from the numerical point of view, since they combine “volume” PDEs with “surface” PDEs.

6.2.3 Residual error estimators for high frequency problems

Participants: Théohpile Chaumont-Frelet, Stéphane Lanteri, Patrick Vega.

Residual error estimators are widely employed in applications to drive adaptive mesh refinement algorithms. However, most of the work available in the literature deals with coercive problems and unfortunately, this setting does not cover high-frequency wave problems, which are of central interest in the Atlantis project-team.

The postdoctoral subject of Patrick Vega is to adapt, analyze and implement residual error estimators for high-frequency wave problems, and in particular, Maxwell's equations. So far, a rigorous analysis of high-frequency Maxwell's equations discretized with Nédélec finite elements or a hybridizable discontinuous Galerkin method have been performed in 3D 22. This theoretical analysis is also backed up by 2D numerical experiments, showing the ability of the estimator to efficiently drive adaptive mesh refinements.

We also started to investigate the use of residual error estimator for Maxwell's equations coupled with a linearized hydrodynamic Drude model. We covered the theoretical analysis and our first numerical experiments are promising. We are currently finalizing a publication on the subject, which is to be submitted in 2021.

6.2.4 Analysis of quasi-periodic boundary conditions

Participants: Théohpile Chaumont-Frelet, Zakaria Kassali, Stéphane Lanteri.

Quasi-perodidc boundary conditions arise in a number of applications considered in the team. Important examples are the simulations of solar cells and photonic crystals. These “unusual” boundary conditions correspond to an incident wave with an oblique incidence, and can strongly impact the stability and accuracy of numerical discretizations. We investigate these boundary conditions in the PhD of Zakaria Kassali. We have been able to derive sharp stability estimates for finite element discretizations involving quasi-periodicity. In particular, we have shown that a drastic accuracy loss can appear at certain angle of incidence, that mathematically correspond to “parabolically trapped” rays (see Fig. 2, blue curve). We propose to remedy this problem by using a multiscale hybrid-mixed (MHM) method (see Fig. 2, red curve). Our numerical investigations are rather promising, and current efforts are guided toward a rigorous error analysis for the MHM method, showing its robustness with respect to parabolically trapped rays.

6.3.1 Inverse design of metasurfaces using statistical learning methods

Participants: Mickael Binois, Régis Duvigneau, Mahmoud Elsawy, Patrice Genevet, Stéphane Lanteri.

Metasurfaces are flat optical nanocomponents, which are the basis of several more complicated optical devices. The optimization of their performance is thus a crucial concern as they impact a wide range of applications. Yet, current design techniques are mostly based on engineering knowledge, and may potentially be improved by resorting to inverse design. In this work, which is conducted in close collaboration with the group of Patrice Genevet at the Research Center for Heteroepitaxy and its Applications (CRHEA, CNRS), we study phase gradient metasurfaces. More precisely, we leverage a computational modeling strategy that combines a high order discontinuous Galerkin time-domain solver implemented in the DIOGENeS software suite with a statistical learning-based global optimization technique developed by researchers from the ACUMES project-team, to discover novel metasurface designs with high efficiency. Our numerical results reveal that rectangular and cylindrical nanopillar arrays can achieve more than respectively 88% and 85% of diffraction efficiency for TM polarization and both TM and TE polarization respectively, using less than 200 fullwave simulations. This year we extended our former work in 4 by considering multiobjective optimization settings as illustrated in Fig. 3.

6.3.2 Multiobjective optimization for large-scale achromatic metalenses

Participants: Mickaël Binois, Régis Duvigneau, Mahmoud Elsawy, Patrice Genevet, Stéphane Lanteri.

In this work, we present for the first time to the metasurface community a multiobjective optimization approach based on statistical learning. This method is based on a surrogate modeling, which replaces the high fidelity electromagnetic evaluation process with a simpler and cheaper model for the prediction of the new designs during the optimization process. Even though it converges to the global set of solutions, it requires fewer iterations compared to the classical global evolutionary strategies. We combine this approach with our in-house developed 3D electromagnetic solver based on the Discontinuous Galerkin Time-Domain (DGTD) method in order to optimize 3D achromatic metalens with numerical aperture NA above 0.5. Our metalens attempts to focus the RGB colours at the same focal plane with the maximum feasible focusing efficiency (see Fig. 4 (a)). In other words, we seek to get the most suitable compromise between the chromatic dispersion and the focusing efficiency for a given fixed focal distance. Our 3D lens is composed of concentric rings of cylindrical nanopillars represented by the red colour in Fig. 4 (b). In order to reduce the computational cost, we assume that the cylinders in each ring share the same diameter. In Fig. 4 (c), we present the intensity profile along the propagation direction z for one of the optimized designs, where the NA=0.56. As can be seen, the three colours are focused nearly at the predefined focal plane (represented by the vertical black line). The average focal error for the three wavelengths is less than 6%.

It is worth mentioning that the optimized designs have already been fabricated by our physical partners from CRHEA as depicted in Figs. 4 (d-f). The experimental characterization will be performed in the next weeks. In addition, the manuscript for our joint paper will be submitted soon to one of the prestigious scientific journals in the field of metasurfaces.

6.3.3 Optimization of metasurfaces accounting for fabrication uncertainties

Participants: Mickaël Binois, Régis Duvigneau, Mahmoud Elsawy, Patrice Genevet, Stéphane Lanteri.

Fabrication processes of metasurface components require dealing with geometrical features in the nanometer scale. Despite the advances of nanofabrication facilities developed during the past few years, uncertainties and systematic fabrication errors are usually non-negligible, and are the source of efficiency degradation compared to modeling and simulation results. In this study, we consider the optimization of metasurface designs taking into account uncertainties due to fabrication errors. Our approach is based on statistical learning optimization, where robust statistical metamodels replace expensive high fidelity numerical simulations. We apply this technique to optimize and design realistic phase gradient metasurfaces in the visible regime. Our preliminary results reveal that we obtain robust designs. which are less sensitivity to the geometrical fabrication errors as compared to the one optimized without considering uncertainties.

6.3.4 Simulation of 3D conformal metasurfaces using GSTCs

Participants: Sandeep Yadav Golla, Patrice Genevet, Nicolas Lebbe.

The development of metamaterials and the associated technologies have exploited transformation optics and its analogy with light propagation in curved space to manipulate light in arbitrary ways. In this collaboration, we addressed the problem of metasurface tranmission conditions at conformal interfaces using Generalized Sheet Transitions Conditions (C-GSTCs) derived following pioneering works of Idemen for planar interfaces using surfacic Dirac distributions. We also adapted the classical inversion procedure used in planar GSTCs to retrieve the susceptibilities to the conformal case by restricting the tensors values to their sole tangential components. These C-GSTCs were first implemented in both 2D and 3D using the FEM (see Fig. 15) and then used to study the impact of the interface geometry on different types of aberrations for metalenses.

6.3.5 3D metamaterial mode converter for integrated optics

Participants: Karim Hassan, Salim Boutami, Mahmoud Elsawy, Stéphane Lanteri.

In this study, we combined a statistical learning-based global optimization strategy with a high order 3D DGTD asolver to design a compact and highly efficient graded index photonic metalens. The metalens is composed of silicon (Si) strips of varying widths (in the transverse direction) and lengths (in the propagation direction) and operates at the telecommunication wavelength (see Fig.6 (a)) In our work 13, we tackled the challenging Transverse Electric case (TE) where the incident electric field is polarized perpendicular to strips direction. We reveal that the focusing efficiency approaches 80% for the traditional design with fixed strip lengths and varying widths. Nevertheless, we demonstrate numerically that the efficiency is as high as 87% for a design with varying strip lengths along the propagation direction. Our findings open numerous fascinating paths towards the realization of compact and highly efficient integrated photonic components.

6.3.6 Optimization of light-trapping in nanostructured solar cells

Participants: Stéphane Collin, Alexis Gobé, Stéphane Lanteri.

There is significant recent interest in designing ultrathin crystalline silicon solar cells with active layer thickness of a few micrometers. Efficient light absorption in such thin films requires both broadband antireflection coatings and effective light trapping techniques, which often have different design considerations. In collaboration with physicists from the Sunlit team at C2N-CNRS, We exploit statistical learning methods for the inverse design of material nanostructuring with the goal of optimizing light trapping properties of ultraphin solar cells. This objective is challenging because the underlying electromagnetic wave problems exhibit multiple resonances, while the geometrical settings are non-trivial. Such multi-resonant solar cell structures are attractive for maximizing light absorption for the full solar light spectrum as illustrated in Fig. 7. We exploit statistical learning methods for the inverse design of material nanostructuring with the goal of optimizing light trapping properties of ultraphin solar cells. This study is conducted in collaboration with the Sunlit headed by Stéphane Collin at the Center for Nanoscience and Nanotechnology (C2N, CNRS).

6.3.7 Plasmonic sensing with nanocubes

Participants: Antoine Moreau, Stéphane Lanteri, Guillaume Leroy, Claire Scheid.

The propagation of light in a slit between metals is known to give rise to guided modes. When the slit is of nanometric size, plasmonic effects must be taken into account, since most of the mode propagates inside the metal. Indeed, light experiences an important slowing-down in the slit, the resulting mode being called gap-plasmon. Hence, a metallic structure presenting a nanometric slit can act as a light trap, i.e. light will accumulate in a reduced space and lead to very intense, localized fields. We study the generation of gap plasmons by various configurations of silver nanocubes separated from a gold substrate by a dielectric layer, thus forming a narrow slit under the cube. When excited from above, this configuration is able to support gap-plasmon modes which, once trapped, will keep bouncing back and forth inside the cavity. We exploit statistical learning methods for the goal-oriented inverse design of cube size, dielectric and gold layer thickness, as well as gap size between cubes in a dimer configuration. This study is conducted in collaboration with Antoine Moreau at Institut Pascal (CNRS).

6.3.8 Multiple scattering in randon media

Participants: Stéphane Lanteri, Guillaume Leroy, Gian Luca Lippi.

Fluorescence signals emitted by probes, used to characterize the expression of biological markers in tissues or cells, can be very hard to detect due to a small amount of molecules of interest (proteins, nucleic sequences), to specific genes expressed at the cellular level, or to the limited number of cells expressing these markers in an organ or a tissue. Access to information coming from weaker emitters can only come from strengthening the signal, since electronic post-amplification raises the noise floor as well. While costly and molecule-specific biochemical processes are being developed for this purpose, a new mechanism based on the simultaneous action of stimulated emission and multiple scattering, induced by nanoparticles suspended in the sample, has been recently demonstrated to effectively amplify weak fluorescence signals. A precise assessment of the signal fluorescence amplification that can be achieved by such a scattering medium requires a electromagnetic wave propagation modeling approach capable of accurately and efficiently coping with multiple space and time scales, as well as with non-trivial geometrical features (shape and topological organization of scatterers in the medium). In the context of a collaboration with physicists from the Institut de Physique de Nice INPHYNI (Gian Luca Lippi from the complex photonic systems and materials group), we initiated this year a study on the simultaneous action of stimulated emission and multiple scattering by randomly distributed nanospheres in a bulk medium. From the numerical modeling point of view, or short term goal is to develop a time-domain numerical methodology for the simulation of random lasing in a gain medium.

8.1.1 FP7 & H2020 Projects

8.2.1 ANR project

9.1.1 Journal

9.1.2 Leadership within the scientific community

9.2.1 Teaching

• Claire Scheid, Analyse Fonctionnelle, Practical works, Master 1 MPA, 18 h, Université Côte d'Azur
• Claire Scheid, Optimisation et éléments finis, Master 1 MPA and IM, 54 h, Université Côte d'Azur
• Claire Scheid, Option Modélisation, Lectures and practical works, Master 2 Agrégation, 43 h, Université Côte d'Azur
• Yves D'Angelo, Modélisation Simulation Numérique, Master 1 IM/MPA, 50 h, Université Côte d'Azur
• Yves D'Angelo, Modélisation de la Turbulence dans l'Industrie, Master 1 MPA and IM, 16 h, Université Côte d'Azur
• Stéphane Descombes, Introduction to partial differential equations Master 1 MPA and IM, 30 h, Université Côte d'Azur
• Stéphane Descombes, Principal component analysis and supervised analysis, Master 2 SIRIS, Université Côte d'Azur
• Stéphane Descombes, Mathematics practical works, L1 SocioEco, 18 h, Université Côte d'Azur
• Engineering: Stéphane Lanteri, High performance scientific computing, 24 h, MAM5, Polytech Nice Sophia - Université Côte d'Azur

9.2.2 Supervision

• PhD defended: Alexis Gobé, Discontinuous Galerkin methods for the simulation of multiscale nanophotonic problems with application to light trapping in solar cells, February 2020, Stéphane Lanteri
• PhD defended: Georges Nehmetallah, Hybridized discontinuous Galerkin methods coupled with hybrid explicit/implicit schemes for the unsteady maxwell equations, December 2020, Stéphane Descombes and Stéphane Lanteri
• PhD in progress: Zakaria Kassali, Multiscale finite element simulations applied to the design of photovoltaic cells, Théophile Chaumont-Frelet and Stéphane Lanteri
• PhD in progress: Massimiliano Montone, High order finite element type solvers for the coupled Maxwell-semiconductor equations in the time-domain, Stéphane Lanteri and Claire Scheid
• PhD in progress: Jérémy Grebot, Advanced modeling of nanostructured CMOS imagers, Stéphane Lanteri and Claire Scheid

9.2.3 Juries

• Stéphane Lanteri, reviewer: Minh Duy Truong, Aix-Marseille Univ., October 2020
• Stéphane Lanteri, examiner: Léo Martire, ISAE-Supaero, October 2020

International journals

• 10 articleThéophileT. Chaumont-Frelet, AlexandreA. Ern and MartinM. Vohralík. Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedronComptes Rendus Mathématique3589-10December 2020, 1101-1110
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• 11 article ThéophileT. Chaumont-Frelet, SergeS. Nicaise and JérômeJ. Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions Communications on Pure and Applied Analysis 2020
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• 12 articleVincentV. Darrigrand, DavidD. Pardo, ThéophileT. Chaumont-Frelet, IgnacioI. Gómez-Revuelto and Emilio LuisE. Garcia-Castillo. A Painless Automatic hp-Adaptive Strategy for Elliptic ProblemsFinite Elements in Analysis and Design178October 2020, 103424
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• 13 article Mahmoud M. R.M. Elsawy, KarimK. Hassan, SalimS. Boutami and StephaneS. Lanteri. Bayesian optimization and rigorous modelling of a highly efficient 3D metamaterial mode converter OSA Continuum June 2020
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• 14 articleMahmoudM. Elsawy, StéphaneS. Lanteri, RégisR. Duvigneau, JonathanJ. Fan and PatriceP. Genevet. Numerical optimization methods for metasurfacesLaser and Photonics Reviews1410October 2020, 1900445
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• 15 articleVan HieuV. Nguyen, AliouA. Diallo, PhilippeP. Le Thuc, RobertR. Staraj, StéphaneS. Lanteri and GeorgesG. F. Carles. A Miniature Implanted Antenna for UHF RFID ApplicationsProgress In Electromagnetics Research C99January 2020, 221-238
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• 16 article SergeS. Nicaise and ClaireC. Scheid. Stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics Computers and Mathematics with Applications 79 12 June 2020
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Conferences without proceedings

• 17 inproceedings Mahmoud M. R.M. Elsawy, StephaneS. Lanteri, RégisR. Duvigneau and PatriceP. Genevet. Statistical Learning Optimization for Highly Efficient Metasurface Designs SIAM Conference on Computational Science and Engineering 2021 Texas, United States March 2021
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Doctoral dissertations and habilitation theses

• 18 thesis AlexisA. Gobé. Discontinuous Galerkin methods for the simulation of multiscale nanophotonic problems with application to light trapping in solar cells Université Côte d'Azur February 2020
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Reports & preprints

• 19 misc CamilleC. Carvalho, PatrickP. Ciarlet and ClaireC. Scheid. Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation March 2021
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• 20 misc ThéophileT. Chaumont-Frelet, AlexandreA. Ern and MartinM. Vohralík. Stable broken H(curl) polynomial extensions and p-robust quasi-equilibrated a posteriori estimators for Maxwell's equations May 2020
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• 21 misc ThéophileT. Chaumont-Frelet, StéphaneS. Lanteri and PatrickP. Vega. A posteriori error estimates for finite element discretizations of time-harmonic Maxwell's equations coupled with a non-local hydrodynamic Drude model March 2021
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• 22 misc ThéophileT. Chaumont-Frelet and PatrickP. Vega. Frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell's equations September 2020
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• 23 misc ThéophileT. Chaumont-Frelet and BarbaraB. Verfürth. A generalized finite element method for problems with sign-changing coefficients August 2020
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• 24 misc ThéophileT. Chaumont-Frelet and MartinM. Vohralík. Equivalence of local-best and global-best approximations in H(curl) June 2020
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• 25 misc GeorgesG. Nehmetallah, ThéophileT. Chaumont-Frelet, StéphaneS. Descombes and StéphaneS. Lanteri. A postprocessing technique for a discontinuous Galerkin discretization of time-dependent Maxwell's equations October 2020
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