ATLANTIS is a joint project-team between Inria, CNRS and Université Côte d’Azur thanks to its association with the J.A. Dieudonné Mathematics Laboratory (UMR 7351). It is a follow-up of the NACHOS project-team from which it inherits a well identified expertise on theoretical and methodological aspects of the solution of PDE (Partial Differential Equation) systems modeling electromagnetic and elastodynamic wave propagation. Relying on this expertise, our research activities in Atlantis will aim at studying and impacting more deeply some scientific and technological challenges raised by physical problems involving waves in interaction with nanostructured matter. A crucial component in the implementation of this scientific endeavor lies in a close networking with physicists who bring the experimental counterpart of the proposed research.

Nanostructuring of materials has paved the way for manipulating and
enhancing wave-matter interactions, thereby opening the door for the
full control of these interactions at the nanoscale. In particular,
the interaction of light waves (or more general optical
waves) with matter is a subject of rapidly increasing scientific
importance and technological relevance. Indeed, the corresponding
science, referred to as nanophotonics37,
aims at using nanoscale light-matter interactions to achieve an
unprecedented level of control on light. Nanophotonics encompasses a
wide variety of topics, including metamaterials, plasmonics, high
resolution imaging, quantum nanophotonics and functional photonic
materials. Previously viewed as a largely academic field,
nanophotonics is now entering the mainstream, and will play a major
role in the development of exciting new products, ranging from high
efficiency solar cells, to personalized health monitoring devices able
to detect the chemical composition of molecules at ultralow
concentrations. Plasmonics 42 is a field closely
related to nanophotonics. Metallic nanostructures whose optical
scattering is dominated by the response of the conduction electrons
are considered as plasmonic media. If the metallic structure presents
an interface with a positive permittivity dielectric, collective
oscillations of surface electrons create waves (called surface
plasmons) that are guided along the interface, with the unique
characteristic of subwavelength-scale confinement. Nanofabricated
systems that exploit these plasmon waves offer fascinating
opportunities for crafting and controlling the propagation of light in
matter. In particular, it can be used to channel light efficiently
into nanometer-scale volumes. As light is squeezed down into
nanoscale volumes, field enhancement effects occur resulting in new
optical phenomena that can be exploited to challenge existing
technological limits and deliver superior photonic devices. The
resulting enhanced sensitivity of light to external parameters (for
example, an applied electric field or the dielectric constant of an
adsorbed molecular layer) shows also great promise for applications in
sensing and switching.

In ATLANTIS, our short-term and medium-term activities mainly focus on research fields related to nanophotonics. Driven by a number of nanophotonics-related physical drivers, our overall objectives are on one hand to design and develop innovative numerical methodologies for the simulation of nanaoscale light-matter interactions and to demonstrate their capabilities by studying challenging applications in close collaboration with physicist partners. On the methodological side, the Discontinous Galerkin (DG) family of methods is a cornerstone of our contributions, and our research directions in ATLANTIS aim at extending our former achievements on these methods in order to deal with more complex PDEs, including nonlinear problems and coupled systems stemming from multiphysics problems, relevant to the study of nanoscale wave-matter interactions. Moreover, mathematical modeling is a central activity of the team, in particular for shaping initial and boundary value problems in view of devising accurate, efficient and robust numerical methods in the presence of multiple space and time scales or/and geometrical singularities. Additional methodological topics that are considered in close collaboration with colleagues from other Inria teams or external applied mathematics research groups are model order reduction, inverse design, uncertainty analysis and high order geometry approximation. New contributions on these topics in the context of the physical problems studied in ATLANTIS will ultimately be implemented in the DIOGENeS software suite, which is a unique software plaform dedicated to computationl nanophotonics.

As a more prospective and longer term objective, we will also explore the possibility of exporting our methodological contributions to the much more recent scientific field of nanophononics that exploit the dynamics of phonons (quanta of lattice vibrations) at the nanoscale. These vibrations, occuring in a large variety of material systems, solids or liquids, can manifest as sound or heat. In contrast with electrons and photons, the launched efforts to achieve control of phonons are quite recent and the physics (and subsequent mathematical modeling) are not completely understood yet. These efforts came, as for nanophotonics, from the reduction of the size of electronic devices that opened up new possibilities for phonon propagation and interaction. By way of consequences, new thrilling perspectives to enhance the properties of nanodevices have appeared with technological applications encompassing nanoelectronics, renewable energy harvesting, nano- and optomechanics, quantum technologies, as well as medical therapy, imaging and diagnostics. The mathematical modeling of nanoscale sound and heat transport control raises several questions and it is not clear so far that it can rely on classical physics (mechanics) differential models as it is the case with nanophotonics, although the study of microscale phononic devices relies on PDE models that are relevant to our research activities.

Our activities ultimately materialize as innovative computational techniques for studying concrete questions and applications that are tightly linked to specific physical fields related to nanophotonics and plasmonics, and which will be considered in collaboration with physicists. These driving physical fields aim at exploiting the peculiarities of nanoscale light-matter interactions.

Quantum plasmonics. The physical phenomena involved in
the deep confinement of light when interacting with matter opens a
major route for novel nanoscale devices design. Indeed, the recent
progress of fabrication at the nanoscale makes it possible to conceive
metallic structures with increasingly large size mismatch, in which
microscale devices can be characterized by sub-nanometer features
31. These advances have also allowed to
achieve spatial separation between metallic elements of only few
nanometers 29. At such sizes quantum
effects become non-negligible, producing huge variations in the
macroscopic optical response. Following this evolution, the quantum
plasmonics field has emerged, and with it the possibility of building
quantum-controled devices, such as single photon sources, transistors
and ultra-compact circuitry at the nanoscale.

Thermoplasmonics. Plasmonic resonances can be
exploited for many applications 42. In particular,
the strong local field enhancement associated with the plasmonic
resonances of a metallic nanostructure, together with the absorption
properties of the metal, induce a photo-thermal energy conversion.
Thus, in the vicinity of the nanostructure, the temperature increases.
These effects, viewed as ohmic losses, have been for a long time
considered as a severe drawback for the realization of efficient
devices. However, the possibility to control this temperature rise
with the illumination wavelength or polarization has gathered strong
interest in the nano-optics community, establishing the basis of
thermoplasmonics 27. By increasing temperature in
their surroundings, metal nanostructures can be used as integrated
heat nanosources. Decisive advances are foreseen in nanomedicine with
applications in photothermal cancer therapy, nano-surgery, drug
delivery, photothermal imaging, protein tracking, photoacoustic
imaging, but also in nano-chemistry, optofluidics, solar and thermal
energy harvesting (thermophotovoltaics).

Planar optics. Nanostructuring of matter can be
tailored to shape, control wavefront and achieve unusual device
operations. Recent years have seen tremendous advances in the
fabrication and understanding of two-dimensional (2D) materials,
giving rise to the field of planar optics. In particular, the concept
of quasi-2D metasurfaces has started to develop into an exciting
research area, where nanostructured surfaces are designed for novel
functionalities
38- 30- 34.
Metasurfaces are planar metamaterials with subwavelength thickness,
consisting of single-layer or few-layer stacks of nanostructures.
They can be readily fabricated using lithography and nanoprinting
methods, and the ultrathin thickness in the wave propagation direction
can greatly suppress the undesirable losses. Metasurfaces enable a
spatially varying optical response (e.g. scattering amplitude, phase,
and polarization). They mold optical wavefronts into shapes that can
be designed at will, and facilitate the integration of functional
materials to accomplish active control and greatly enhanced nonlinear
response.

Optoelectronics and nanoelectronics. Semiconductors
also play a major role in leveraging nanoscale light-matter
interactions. Emission or absorption of light by a semiconductor is
at the heart of optoelectronics, which is concerned with devices that
source, detect or control light. Photodiodes, solar cells, light
emitting diodes (LEDs), optical fibers and semiconductor lasers are
some typical examples of optoelectronic devices. The attractive
properties of these devices is based on their efficiency in converting
light into electrical signals (or vice versa). Using a structuration
with low dimensional materials and carrier-photons interaction,
optoelectronics aims at improving the quality of these systems. A
closeby field is nanoelectronics 43, i.e., the
physical field that, while incorporating manufacturing constraints,
tries to describe and understand the influence of the
nanostructuration of electronic devices on their electronic
properties. This area has quickly evolved with the increasing
fabrication capabilities. One striking motivating example is the
drastic increase of the number of transistors (of a few nanometer
size) per chip on integrated circuits. At the achieved
nanostructuration scales, inter-atomic forces, tunneling or quantum
mechanical properties have a non-negligible impact. A full
understanding of these effects is mandatory for exploiting them in the
design of electronic components, thereby improving their
characteristics.

The processes that underly the above-described physical fields raise a number of modeling challenges that motivate our research agenda:

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.

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.

Nanoscale wave-matter interactions find many applications of industrial and societal relevance. The applications discussed in this section are those that we address in the first place in the short- to medium-term. Our general goal is to impact scientific discovery and technological development in these application topics by leveraging our methodological contributions for the numerical modeling of nanoscale wave-matter interactions, and working in close collaboration with external partners either from the academic or the industrial world. Each of these applications is linked to one or more of the driving physical fields described in section 3.1 except nanoelectronics that we consider as a more prospective, hence long-term application.

Photovoltaics (PV) converts photon energy from the sun into electric energy. One of the major challenges of the PV sector is to achieve high conversion efficiencies at low cost. Indeed, the ultimate success of PV cell technology requires substantial progress in both cost reduction and efficiency improvement. An actively studied approach to simultaneously achieve both objectives is to exploit light trapping schemes. Light trapping enables solar cells absorption using an active material layer much thinner than the material intrinsic absorption length. This then reduces the amount of materials used in PV cells, cuts cell cost, facilitates mass production of these cells that are based on less abundant material and moreover can improve cell efficiency (due to better collection of photo- generated charge carriers). Enhancing the light absorption in ultrathin film silicon solar cells is thus of paramount importance for improving efficiency and reducing costs. Our activities in relation with this application field aim at precisely studying light absorption in nanostructured solar cell structures with the help of an adapted numerical procedure. We consider both the characterization of light trapping for a given texturing of material layers, and the goal-oriented inverse design of the nanostructuring.

Metasurfaces produce abrupt changes over the scale of the free-space wavelength in the phase, amplitude and/or polarization of a light beam. Metasurfaces are generally created by assembling arrays of miniature, anisotropic light scatterers (i.e. resonators such as optical antennas). The spacing between antennas and their dimensions are much smaller than the wavelength. As a result the metasurfaces, on account of Huygens principle, are able to mould optical wavefronts into arbitrary shapes with subwavelength resolution by introducing spatial variations in the optical response of the light scatterers. Designing metasurfaces for realistic applications such as metalenses 50 is a challenging inverse problem. In this context, the ultimate goal of our activities is to develop numerical methodologies for the inverse design of large-area metasurfaces 47.

Recent research on the interaction of short optical pulses with semiconductors has stimulated the development of low power terahertz (THz) radiation transmitters. The THz spectral range of electromagnetic waves (0.1 to 10 THz) is of great interest. In particular, it includes the excitation frequencies of semiconductors and dielectrics, as well as rotational and vibrational resonances of complex molecules. As a result, THz waves have many applications in areas ranging from the detection of dangerous or illicit substances and biological sensing to diagnosis and diseases treatment in medicine. The most common mecanism of THz generation is based on the use of THz photoconductive antennas (PCA), consisting of two electrodes spaced by a given gap and placed onto a semiconductor surface. The excitation of the gap by a femtosecond optical pulse induces a sharp increase of the concentration of charge carriers for a short period of time, and a THz pulse is generated. Computer simulation plays a central role in understanding and mastering these phenomena in order to improve the design of PCA devices. The numerical modeling of a general 3D PCA configuration is a challenging task. Indeed, it requires the simultaneous solution of charge transport in the semiconductor substrate and the electromagnetic wave radiation from the antenna 46- 52. The recently-introduced concept of hybrid photoconductive antennas leveraging plasmonic effects is even more challenging 41. So far, existing simulation approaches are based on the FDTD method, and are only able to deal with classical PCAs. In relation with the design of photonic devices for THz waves generation and manioulation, we intend to a multiscale numerical modeling strategy for solving the system of Maxwell equations coupled to various models of charge carrier dynamics in semiconductors.

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. Nanocubes are extensively studied in this context and have been shown to support such gap plasmon modes. At visible frequencies, the lossy behavior of metals will cause the progressive absorption of the trapped electromagnetic field, turning the metallic nanocubes into efficient absorbers. The frequencies at which this absorption occurs can be tuned by adjusting the dimensions of the nanocube and the spacer. Such metallic nanocubes can be used for a broad range of applications including plasmonic sensing, surface enhanced Raman scattering (SERS), metamaterials, catalysis, and bionanotechnology. We aim at devising a numerical methodology for characterizing the impact of geometrical parameters such as the dimensions of the cube, the rounding of nanocube corners or the size of the slit separating the cube and the substrate, on the overall performance of these absorbers. In practice, this leads us to address two main modeling issues. First, as the size of the slit is decreased, spatial dispersion effects 33 have to be taken into account when dealing with plasmonic structures. For this purpose, we consider a fluid model in the form of a nonlocal hydrodynamic Drude model 32, which materializes as a system of PDEs coupled to Maxwell's equations 9-8. The second issue is concerned with the assessment of geometrical uncertainties and their role in the development of spatial dispersion effects.

The field of thermoplasmonics has developed an extensive toolbox to
produce, control and monitor heat at the nanometer scale.
Nanoparticles are promising nano-sensing and nano-manipulating tools,
and recent studies yielded remarkable advances in design, synthesis,
and implementation of luminescent nanoparticles. Some applications
deal with bio-imaging and bio-sensing, like e.g. luminescent
nanothermometers, nanoparticles capable of providing contactless
thermal reading through their light emission properties
40. Also, bio-functionalized gold nanorods
are promising candidates for light-induced hyperthermia
45, to cause local and selective damage in
malignant tissue. At the same time, laser pulse interaction with
plasmonic nanostructures can also be exploited for cell nanosurgery
28, including plasmonic enhanced cell
transfection, molecular surgery and drug delivery. In parallel to all
these bio-oriented applications, plasmonic nanoparticles can also be
thought of as prototypic systems for understanding fundamental
aspects of nanoscale material as well as light-matter interaction.
Specific numerical modeling tools are essential to provide a good
insight in this understanding.

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.

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.

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.

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).

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.

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:

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.

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.

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)).

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.

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.

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.

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.

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.

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.

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.

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.

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).

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).

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.

Nom: DGTD solvers for time-domain elecromagnetics
with application to photovoltaics

Nom: DGTD solvers for semiconductor device modeling

Nom: DGTD solvers for modeling light absorption in CMOS imagers

We have produced a review of recent optimization techniques used in the inverse design of high performance metasurfaces. We believe that this review will push this exciting field towards realistic and practical applications, ranging from laser wavefront engineering to innovative facial recognition and motion detection devices, including augmented reality retro-reflectors and related complex light field engineering. This review has been published in Laser & Photonics Review 14.