Mathematical Analysis of Multi-dimensional Topological Edge Modes
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
The goal of this proposal is to develop novel mathematical techniques for analysing topological edge modes that will inform rapid advancements in metamaterial design. This proposal will yield new analytic methods capable of describing the rich variety of multi-dimensional metamaterial geometries being developed. This will lead to rigorous results relating the properties of topological edge modes (localization strength, robustness, eigenfrequency) with those of the underlying materials (topological indices, symmetry) in multi-dimensional systems. This will reveal fundamental new insight into the fundamental physics and the deliverables (theorems, formulas, codes) will be used to rapidly and accurately guide metamaterial design. It will also overcome the reliance on perturbative methods. This will be achieved by developing approaches for modelling scattering by almost periodic (semi-infinite, randomly perturbed, quasi-periodic) structures with general geometries in dimension greater than one. These will combine my extensive expertise in boundary integral methods with the expertise of the host group (led by Richard Craster, at Imperial College London) in the Wiener-Hopf method. Additionally, input from the unique, world-leading expertise in topological waveguide design possessed by the host group and other members of the Imperial College Centre for Plasmonics and Metamaterials will help guide the project towards achieving high-impact, cutting-edge results. This fellowship will further my career by giving me the opportunity to conduct an independent program of research and to develop high-profile collaborations within the Imperial College Centre for Plasmonics and Metamaterials. I will develop the management, communication and technical skills needed for a career of applied mathematical research and to achieve my goal of becoming a European leader on the mathematical analysis of metamaterials.
Organisations
People |
ORCID iD |
| Richard Craster (Principal Investigator) | |
| Bryn Davies (Fellow) |
Publications
Davies B
(2023)
Graded Quasiperiodic Metamaterials Perform Fractal Rainbow Trapping.
in Physical review letters
Davies B
(2023)
Landscape of wave focusing and localization at low frequencies
in Studies in Applied Mathematics
Putley H
(2023)
Effective properties of periodic plate-array metacylinders
in Physical Review B
Davies B
(2023)
On the problem of comparing graded metamaterials
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Ammari H
(2023)
Convergence Rates for Defect Modes in Large Finite Resonator Arrays
in SIAM Journal on Mathematical Analysis
Chaplain G
(2023)
Tunable topological edge modes in Su-Schrieffer-Heeger arrays
in Applied Physics Letters
Ammari H
(2024)
Stability of the Non-Hermitian Skin Effect in One Dimension
in SIAM Journal on Applied Mathematics
Dunckley L
(2024)
Hierarchical band gaps in complex periodic systems
in Comptes Rendus. Mécanique
Ammari H
(2024)
Mathematical Foundations of the Non-Hermitian Skin Effect.
in Archive for rational mechanics and analysis
| Description | This award has developed methods for characterising the spectra of truncated periodic structures using the truncated Floquet transform. This allows the discrete spectra of finite-sized multi-dimensional metamaterials to be related to their infinitely periodic counterparts. A quantitative convergence theory has been established, which reveals how large a finite-sized metamaterial needs to be for its spectrum to be predicted by Floquet-Bloch theory with a desired accuracy [Bull. London Math. Soc. 2025]. For the particular case of localized edge modes, this theory reveals how the convergence rate depends on the dimension of the dimension of the lattice [SIAM J. Math. Anal. 55 (6) 5993-7761, 2023]. This has fulfilled the objectives of work package 1. This intuitive approach allows for 'approximate' band structures of finite-sized metamaterials to be constructed in a precise way and for the efficacy of finite-sized topological waveguides to be quantified [Stud. Appl. Math. 153 (4), e12765, 2024]. As a result, this method is useful to the wider multi-disciplinary metamaterials community. In collaboration with colleagues from Exeter and Turin, we developed a novel strategy for dynamically tuning the operating frequency of topological edge modes. An important feature of such systems is that their wave guiding is single frequency. To maximise their usefulness, one wants to be able to control and tune this single frequency. We developed a tuning strategy by 3D printing samples from a photo-responsive polymer. Hence, we showed that the sample can be remotely excited with a laser to change its mechanical properties (in a fully reversible process) [Appl. Phys. Lett. 122 (22), 221703, 2023]. As a result, the operating frequency of a topological wave guide can be adjusted dynamically to suit operating needs. This award has developed novel landscape function approaches for metamaterials. The fundamental mathematical problem when designing topological wave guides is to predict where eigenmodes will be localised in a material with a highly complex geometry. An emerging method for doing this is using a so-called "landscape function" (developed by Filoche and Mayboroda [PNAS 109 (37), 14761-1476, 2012]). However, these methods are not generally appropriate for metamaterial design as they typically provide a tight estimate of localisation only for high-energy modes, whereas the most interesting and useful localisation in metamaterials occurs at very low frequencies. We exploited an asymptotic framework for subwavelength resonant metamaterials to develop a novel landscape function that captures wave localisation and focusing at very low frequencies in high-contrast metamaterials [Stud. Appl. Math. 152, 760-777, 2023]. This simple and easy-to-use tool, for which the code is available online [https://doi.org/10.5281/zenodo.8134312], can be readily used by physicists and engineers to predict localisation in metamaterials with very complex geometries with minimal computational expense (in particular, without needing to compute the system's eigenmodes). This award has developed analytic tools needed to facilitate the development of quasicrystalline metamaterials. This was the objective of work package 4 and has already led to the successful implementation of graded quasicrystalline metamaterials that perform a fractal version of the so-called "rainbow trapping" effect [Phys. Rev. Lett. 131, 177001, 2023]. This has applications in facilitating broadband energy harvesting. More recently, we completed work on the convergence of methods for approximating the spectra of quasicrystals, using supercells or superspaces [https://arxiv.org/pdf/2410.15829]. We showed how, provided you choose the right discretisation method (which is a common mistake in the literature), these approaches yield efficient strategies for quantifying spectra (to the resolution required for a given application). In the final part of the project, we realised that the methods we had developed were useful for systems with non-reciprocity and temporal modulation, both of which are very active subjects of research right now. We developed analytic (multi-scale asymptotic) methods for characterising topological edge localisation and defect-induced localisation in non-reciprocal lattices [Archive for Rational Mechanics and Analysis 248 (3), 33, 2024; Physical Review B 111 (3), 035109, 2025]. We also proposed point-scatter approximations for scattering in time-modulated systems [Physical Review B 110 (18), 184102, 2024]. These breakthroughs show how classical mathematical methods can be adapted to describe cutting-edge, novel phenomena in wave physics. |
| Exploitation Route | Energy harvesting and wave guiding are practical examples. This work involves collaborations with a group of physicists who are making experiments based around the research in this project. |
| Sectors | Education Other |
| Description | The methods developed for predicting localisation in complex and finite-sized metamaterials (through the truncated Floquet transform and landscape functions) represent simple and versatile tools that are grounded in rigorous mathematical analysis and can be readily picked up by physicists and engineers to design novel metamaterials (especially topological waveguides). The analytic methods being developed for quasicrystalline metamaterials are helping to nucleate this promising area. Relaxing the reliance on periodic geometries for metamaterial design has significant potential to, for example, developing devices with broader operating bandwidths. However, it comes with substantial modelling challenges as Floquet-Bloch methods are no longer applicable. The analytic methods being developed in this project will alleviate the reliance on empirical and phenomenological studies and allow for more systematic explorations of this exciting area. |
| First Year Of Impact | 2023 |
| Description | Dynamo: Dynamic Spatio-Temporal Modulation of Light by Phononic Architectures |
| Amount | £236,690 (GBP) |
| Funding ID | 1033143 |
| Organisation | Innovate UK |
| Sector | Public |
| Country | United Kingdom |
| Start | 03/2022 |
| End | 02/2026 |
| Description | Next generation metamaterials: exploiting four dimensions |
| Amount | £7,731,660 (GBP) |
| Funding ID | EP/Y015673/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 03/2024 |
| End | 02/2029 |
| Title | Bloch spectra of one-dimensional halide perovskite photonic crystals |
| Description | Matlab code to compute the Bloch spectra of a one-dimensional photonic crystal. Transfer matrices are used to compute the spectra. The crystal has alternating layers of air and a material that has dispersive properties, given by a model based on the permittivity of halide perovskites. The material parameters can be modified by altering the parameters alpha, beta and gamma. See the paper The effect of singularities and damping on the spectra of photonic crystals by Alexopoulos and Davies for details of the problem formulation and some exemplar figures. |
| Type Of Technology | Software |
| Year Produced | 2023 |
| Open Source License? | Yes |
| URL | https://zenodo.org/record/8055546 |
| Title | Code for "Exponentially localised interface eigenmodes in finite chains of resonators" |
| Description | |
| Type Of Technology | Software |
| Year Produced | 2023 |
| Open Source License? | Yes |
| URL | https://zenodo.org/doi/10.5281/zenodo.10361316 |
| Title | Code for "Exponentially localised interface eigenmodes in finite chains of resonators" |
| Description | |
| Type Of Technology | Software |
| Year Produced | 2023 |
| Open Source License? | Yes |
| URL | https://zenodo.org/doi/10.5281/zenodo.10361315 |
| Title | Landscape theory for the generalised capacitance matrix |
| Description | Matlab code used to compute the capacitance matrix for a collection of spherical particles, which can have arbitrary positions in three-dimensional space. A multipole expansion is used, in terms of a basis consisting of spherical harmonics. After generating the capacitance matrix with the routine MakeC_mn.m, the eigenvectors are computed. Then, the landscape and the upper landscape, as defined in the reference [Landscape of wave localisation at low frequencies, Davies & Lou, 2023], are computed. The code can be executed by running the file RUN_example.m in Matlab, which distributes 15 unit spheres on the z=0 plane, as a demonstrative example. |
| Type Of Technology | Software |
| Year Produced | 2023 |
| Open Source License? | Yes |
| URL | https://zenodo.org/record/8134312 |
| Title | Statistics of a chaotic recursion relation with Van Hove singularities |
| Description | MATLAB code used to generate the numerical results in the paper "Van Hove singularities in the density of states of a chaotic dynamical system" by Bryn Davies. RUN_OrbitExamples.m can be used to generate plots of an orbit given suitable initial conditions. RUN_Histogram.m generates histograms of the distribution of orbits of the recursion relation, with initial conditions drawn from given distributions. |
| Type Of Technology | Software |
| Year Produced | 2024 |
| Open Source License? | Yes |
| URL | https://zenodo.org/doi/10.5281/zenodo.10986146 |
| Title | Statistics of a chaotic recursion relation with Van Hove singularities |
| Description | MATLAB code used to generate the numerical results in the paper "Van Hove singularities in the density of states of a chaotic dynamical system" by Bryn Davies. RUN_OrbitExamples.m can be used to generate plots of an orbit given suitable initial conditions. RUN_Histogram.m generates histograms of the distribution of orbits of the recursion relation, with initial conditions drawn from given distributions. |
| Type Of Technology | Software |
| Year Produced | 2024 |
| Open Source License? | Yes |
| URL | https://zenodo.org/doi/10.5281/zenodo.10986147 |
| Description | Popular science article in Plus magasine |
| Form Of Engagement Activity | A magazine, newsletter or online publication |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Public/other audiences |
| Results and Impact | A magasine article to communicate to the general public why multiple scattering problems in heterogeneous media are important, useful and challenging (and, hence, why we need to develop dedicated mathematical tools to be able to handle them effectively). |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://plus.maths.org/content/maths-dance |