Publications
30 results found
Ammari H, Barandun S, Cao J, et al., 2024, Mathematical Foundations of the Non-Hermitian Skin Effect, Archive for Rational Mechanics and Analysis, Vol: 248, ISSN: 0003-9527
<jats:title>Abstract</jats:title><jats:p>We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.</jats:p>
Davies B, Morini L, 2024, Super band gaps and periodic approximants of generalised Fibonacci tilings, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 480, ISSN: 1364-5021
We present mathematical theory for self-similarity induced spectral gaps in the spectra of systems generated by generalised Fibonacci tilings. Our results characterise super band gaps, which are spectral gaps that exist for all sufficiently large periodic systems in a Fibonacci-generated sequence. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We apply our analytic results to characterise spectra in three different settings: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. The theory is shown to give accurate predictions of the super band gaps, with minimal computational cost and significantly greater precision than previous estimates. It also provides a mathematical foundation for using periodic approximants (supercells) to predict the transmission gaps of quasicrystalline samples, as we verify numerically.
Davies B, Lou Y, 2024, Landscape of wave focusing and localization at low frequencies, Studies in Applied Mathematics, Vol: 152, Pages: 760-777, ISSN: 0022-2526
High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.
Ammari H, Davies B, Hiltunen EO, 2024, Anderson localization in the subwavelength regime, Communications in Mathematical Physics, Vol: 405, ISSN: 0010-3616
In this paper, we use recent breakthroughs in the study of coupled subwavelength resonator systems to reveal new insight into the mechanisms responsible for the fundamental features of Anderson localization. The occurrence of strong localization in random media has proved difficult to understand, particularly in physically derived multi-dimensional models and systems with long-range interactions. We show here that the scattering of time-harmonic waves by high-contrast resonators with randomly chosen material parameters reproduces the characteristic features of Anderson localization. In particular, we show that the hybridization of subwavelength resonant modes is responsible for both the repulsion of energy levels as well as the widely observed phase transition, at which point eigenmode symmetries swap and very strong localization is possible. We derive results from first principles, using asymptotic expansions in terms of the material contrast parameter and obtain a characterization of the localized modes in terms of generalized capacitance matrices. This model captures the long-range interactions of the wave-scattering system and provides a concise framework to explain the exotic phenomena that are observed.
Ammari H, Davies B, Hiltunen EO, 2023, Convergence rates for defect modes in large finite resonator arrays, SIAM Journal on Mathematical Analysis, Vol: 55, Pages: 7616-7634, ISSN: 0036-1410
We show that defect modes in infinite systems of resonators have corresponding modes in finitesystems which converge as the size of the system increases. We study the generalized capacitancematrix as a model for three-dimensional coupled resonators with long-range interactions and consider defect modes that are induced by compact perturbations. If such a mode exists, then thereare elements of the discrete spectrum of the corresponding truncated finite system that converge toeach element of the pure point spectrum. The rate of convergence depends on the dimension of thelattice. When the dimension of the lattice is equal to that of the physical space, the convergence isexponential. Conversely, when the dimension of the lattice is less than that of the physical space,the convergence is only algebraic, because of long-range interactions arising due to coupling withthe far field.
Putley HJ, Guenneau S, Craster RV, et al., 2023, Effective properties of periodic plate-array metacylinders, Physical Review B, Vol: 108, ISSN: 2469-9950
We use semianalytic methods to model a periodic structure of plate-array cylinders (metacylinders), and derive several of the medium's effective material properties in the quasistatic limit. Subject to s-polarized [transverse-electric (TE)] light, the anisotropic dispersion of the crystal manifests as a Maxwell Garnett equation for the effective permittivity at leading order. This is performed both for the case of no material contrast between interior and exterior regions, and a nonunity normalized refractive index. In each case, the leading order effective permittivity is a function of the difference between Bloch wave and plate-array angles. As such, we envisage the metamaterial as being mechanically tunable through uniform mechanical rotation of the constituent metacylinders.
Davies B, Chaplain GJ, Starkey TA, et al., 2023, Graded quasiperiodic metamaterials perform fractal rainbow trapping, Physical Review Letters, Vol: 131, ISSN: 0031-9007
The rainbow trapping phenomenon of graded metamaterials can be combined with the fractal spectra of quasiperiodic waveguides to give a metamaterial that performs fractal rainbow trapping. This is achieved through a graded cut-and-project algorithm that yields a geometry for which the effective projection angle is graded along its length. As a result, the fractal structure of local band gaps varies with position, leading to broadband "fractal" rainbow trapping. We demonstrate this principle by designing an acoustic waveguide, which is characterised using theory, simulation and experiments.
Craster RV, Davies B, 2023, Asymptotic characterisation of localised defect modes: Su-Schrieffer-Heeger and related models, SIAM: Multiscale Modeling and Simulation, Vol: 21, Pages: 827-848, ISSN: 1540-3459
Motivated by topologically protected states in wave physics, we study localized eigenmodes in one-dimensional periodic media with defects. The Su–Schrieffer–Heeger model (the canonical example of a one-dimensional system with topologically protected localized defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterize the existence of localized eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenization and transfer matrix analysis, with good agreement between the two methods.
Davies B, Fehertoi-Nagy L, Putley HJ, 2023, On the problem of comparing graded metamaterials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, ISSN: 1364-5021
Alexopoulos K, Davies B, 2023, A mathematical design strategy for highly dispersive resonator systems, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214
Chaplain GJ, Gliozzi AS, Davies B, et al., 2023, Tunable topological edge modes in Su–Schrieffer–Heeger arrays, Applied Physics Letters, Vol: 122, Pages: 1-6, ISSN: 0003-6951
A potential weakness of topological waveguides is that they act on a fixed narrow band of frequencies. However, by 3D printing samples from a photo-responsive polymer, we can obtain a device whose operating frequency can be fine-tuned dynamically using laser excitation. This greatly enhances existing static tunability strategies, typically based on modifying the geometry. We use a version of the classical Su–Schrieffer–Heeger model to demonstrate our approach.
Ammari H, Davies B, 2023, Asymptotic links between signal processing, acoustic metamaterials and biology, SIAM Journal on Imaging Sciences, Vol: 16, Pages: 64-88, ISSN: 1936-4954
Biomimicry is a powerful science that takes advantage of nature's remarkableability to devise innovative solutions to challenging problems. In this work,we use asymptotic methods to develop the mathematical foundations for theexchange of design inspiration and features between biological hearing systems,signal processing algorithms and acoustic metamaterials. Our starting point isa concise asymptotic analysis of high-contrast acoustic metamaterials. We areable to fine tune this graded structure to mimic the biomechanical propertiesof the cochlea, at the same scale. We then turn our attention to developing abiomimetic signal processing algorithm. We use the response of the cochlea-likemetamaterial as an initial filtering layer and then add additional biomimeticprocessing stages, designed to mimic the human auditory system's ability torecognise the global properties of natural sounds. This demonstrates thethree-way exchange of ideas that, thanks to our analysis, is possible betweensignal processing, metamaterials and biology.
Dekoninck Bruhin N, Davies B, 2022, Bioinspired random projections for robust, sparse classification, SIAM Journal on Imaging Sciences, Vol: 15, Pages: 1833-1850, ISSN: 1936-4954
Inspired by the use of random projections in biological sensing systems, we present a new algorithmfor processing data in classification problems. This is based on observations of the human brainand the fruit fly’s olfactory system and involves randomly projecting data into a space of greatlyincreased dimension before applying a cap operation to truncate the smaller entries. This leads toa simple algorithm that is very computationally efficient and can be used to either give a sparserepresentation with minimal loss in classification accuracy or give improved robustness, in the sensethat classification accuracy is improved when noise is added to the data. This is demonstrated withnumerical experiments, which supplement theoretical results demonstrating that the resulting signaltransform is continuous and invertible, in an appropriate sense.
Bruhin ND, Davies B, 2022, Bioinspired Random Projections for Robust, Sparse Classification, Publisher: Society for Industrial & Applied Mathematics (SIAM)
Davies B, Craster RV, 2022, Symmetry-induced quasicrystalline waveguides, Wave Motion, Vol: 115, Pages: 1-16, ISSN: 0165-2125
Introducing an axis of reflectional symmetry in a quasicrystal leads to thecreation of localised edge modes that can be used to build waveguides. Wedevelop theory that characterises reflection-induced localised modes inmaterials that are formed by recursive tiling rules. This general theory treatsa one-dimensional continuous differential model and describes a broad class ofboth quasicrystalline and periodic materials. We present an analysis of amaterial based on the Fibonacci sequence, which has previously been shown tohave exotic, Cantor-like spectra with very wide spectral gaps. Our approachprovides a way to create localised edge modes at frequencies within thesespectral gaps, giving strong and stable wave localisation. We also use ourgeneral framework to make a comparison with reflection-induced modes inperiodic materials. These comparisons show that while quasicrystallinewaveguides enjoy enhanced robustness over periodic materials in certainsettings, the benefits are less clear if the decay rates are matched. Thisshows the need to carefully consider equivalent structures when makingrobustness comparisons and to draw conclusions on a case-by-case basis,depending on the specific application.
Ammari H, Davies B, Hiltunen EO, 2022, Robust edge modes in dislocated systems of subwavelength resonators, Journal of the London Mathematical Society, Vol: 106, Pages: 2075-2135, ISSN: 0024-6107
Robustly manipulating waves on subwavelength scales can be achieved by, first, designing a structure with a subwavelength band gap and, second, introducing a defect so that eigenfrequencies fall within the band gap. Such frequencies are well known to correspond to localized modes. We study a one-dimensional array of subwavelength resonators, prove that there is a subwavelength band gap, and show that by introducing a dislocation we can place localized modes at any point within the band gap. We complement this analysis by studying the stability properties of the corresponding finite array of resonators, demonstrating the value of being able to customize the position of eigenvalues within the band gap
Bell A, Davies B, Ammari H, 2022, Bernhard Riemann, the ear, and an atom of consciousness, Foundations of Science, Vol: 27, Pages: 855-873, ISSN: 1233-1821
Why did Bernhard Riemann (1826–1866), arguably the most original mathematician of his generation, spend the last year of life investigating the mechanism of hearing? Fighting tuberculosis and the hostility of eminent scientists such as Hermann Helmholtz, he appeared to forsake mathematics to prosecute a case close to his heart. Only sketchy pages from his last paper remain, but here we assemble some significant clues and triangulate from them to build a broad picture of what he might have been driving at. Our interpretation is that Riemann was a committed idealist and from this philosophical standpoint saw that the scientific enterprise was lame without the “poetry of hypothesis”. He believed that human thought was fundamentally the dynamics of “mind-masses” and that the human mind interpenetrated, and became part of, the microscopic physical domain of the cochlea. Therefore, a full description of hearing must necessarily include the perceptual dimensions of what he saw as a single manifold. The manifold contains all the psychophysical aspects of hearing, including the logarithmic transformations that arise from Fechner’s law, faithfully preserving all the subtle perceptual qualities of sound. For Riemann, hearing was a unitary physical and mental event, and parallels with modern ideas about consciousness and quantum biology are made. A unifying quantum mechanical model for an atom of consciousness—drawing on Riemann’s mind-masses and the similar “psychons” proposed by Eccles—is put forward.
Alexopoulos K, Davies B, 2022, Asymptotic analysis of subwavelength halide perovskite resonators, Partial Differential Equations and Applications, Vol: 3, ISSN: 2662-2963
Halide perovskites are promising materials with many significant applications in photovoltaics and optoelectronics. Their highly dispersive permittivity relation leads to a non-linear relationship between the frequency and the wavenumber. This, in turn, means the resonance of the system is described by a highly non-linear eigenvalue problem, which is mathematically challenging to understand. In this paper, we use integral methods to quantify the resonant properties of halide perovskite nano-particles. We prove that, for arbitrarily small particles, the subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. We also characterize the hybridized subwavelength resonant frequencies of a dimer of two halide perovskite particles. Finally, we examine the specific case of spherical resonators and demonstrate that our new results are consistent with previous works.
Davies B, Herren L, 2022, Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 478, ISSN: 1364-5021
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- Citations: 4
Davies B, Herren L, 2022, Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors, Publisher: ArXiv
The aim of this work is to derive precise formulas which describe how theproperties of subwavelength devices are changed by the introduction of errorsand imperfections. As a demonstrative example, we study a class ofcochlea-inspired rainbow sensors. These are devices based on a graded array ofsubwavelength resonators which have been designed to mimic the frequencyseparation performed by the cochlea. We show that the device's properties(including its role as a signal filtering device) are stable with respect tosmall imperfections in the positions and sizes of the resonators. Additionally,if the number of resonators is sufficiently large, then the device's propertiesare stable under the removal of a resonator.
Ammari H, Davies B, Hiltunen EO, et al., 2022, Wave Interaction with Subwavelength Resonators, Vol: 2308, Pages: 23-83, ISSN: 0075-8434
The aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation problems at subwavelength scales is demonstrated.
Davies B, Craster RV, 2022, Homogenisation of topologically protected edge states, Pages: X122-X124
We have developed a succinct approach for using homogenisation to derive explicit estimates for the properties of topologically protected edge states. Our approach uses transfer matrices to reduce the wave transmission problem to a set of difference equations, which can be handled concisely using high-frequency homogenisation. This gives estimates for the eigen-frequency and the decay rate of topologically protected edge states. We use a medium based on the Su-Schrieffer-Heeger model to demonstrate the method and show how it can be extended to more complex geometries.
Ammari H, Davies B, Hiltunen EO, et al., 2022, EXCEPTIONAL POINTS IN PARITY-TIME-SYMMETRIC SUBWAVELENGTH METAMATERIALS, SIAM JOURNAL ON MATHEMATICAL ANALYSIS, Vol: 54, Pages: 6223-6253, ISSN: 0036-1410
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- Citations: 2
Ammari H, Davies B, Hiltunen EO, et al., 2022, Wave Interaction with Subwavelength Resonators, APPLIED MATHEMATICAL PROBLEMS IN GEOPHYSICS, Editors: Chiappini, Vespri, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 23-83, ISBN: 978-3-031-05320-7
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- Citations: 2
Ammari H, Davies B, Hiltunen EO, et al., 2021, Bound states in the continuum and Fano resonances in subwavelength resonator arrays, Journal of Mathematical Physics, Vol: 62, ISSN: 0022-2488
When wave scattering systems are subject to certain symmetries, resonant states may decouple from the far-field continuum; they remain localized to the structure and cannot be excited by incident waves from the far field. In this work, we use layer-potential techniques to prove the existence of such states, known as bound states in the continuum, in systems of subwavelength resonators. When the symmetry is slightly broken, this resonant state can be excited from the far field. Remarkably, this may create asymmetric (Fano-type) scattering behavior where the transmission is fundamentally different for frequencies on either side of the resonant frequency. Using asymptotic analysis, we compute the scattering matrix of the system explicitly, thereby characterizing this Fano-type transmission anomaly.
Ammari H, Davies B, Hiltunen EO, et al., 2021, High‐order exceptional points and enhanced sensing in subwavelength resonator arrays, Studies in Applied Mathematics, Vol: 146, Pages: 440-462, ISSN: 0022-2526
Ammari H, Davies B, Hiltunen EO, et al., 2020, Topologically protected edge modes in one-dimensional chains of subwavelength resonators, Journal de Mathématiques Pures et Appliquées, Vol: 144, Pages: 17-49, ISSN: 0021-7824
The goal of this paper is to advance the development of wave-guiding subwavelength crystals by developing designs whose properties are stable with respect to imperfections in their construction. In particular, we make use of a locally resonant subwavelength structure, composed of a chain of high-contrast resonators, to trap waves at deep subwavelength scales. We first study an infinite chain of subwavelength resonator dimers and define topological quantities that capture the structure's wave transmission properties. Using this for guidance, we design a finite crystal that is shown to have wave localization properties, at subwavelength scales, that are robust with respect to random imperfections.
Ammari H, Davies B, 2020, Mimicking the active cochlea with a fluid-coupled array of subwavelength Hopf resonators, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 476, ISSN: 1364-5021
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- Citations: 15
Ammari H, Davies B, Yu S, 2020, Close-To-Touching Acoustic Subwavelength Resonators: Eigenfrequency Separation and Gradient Blow-Up, Multiscale Modeling & Simulation, Vol: 18, Pages: 1299-1317, ISSN: 1540-3459
Ammari H, Davies B, 2019, A fully coupled subwavelength resonance approach to filtering auditory signals, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 475, Pages: 20190049-20190049, ISSN: 1364-5021
<jats:p>The aim of this paper is to understand the behaviour of a large number of coupled subwavelength resonators. We use layer potential techniques in combination with numerical computations to study an acoustic pressure wave scattered by a graded array of subwavelength resonators. Using this approach, the spatial frequency separation properties of such an array can be understood. Our set-up is inspired by the graded structure of cochlear hair cells on the surface of the basilar membrane. We compute the resonant modes of the system and explore the model's ability to decompose incoming signals. We propose a mathematical explanation for phenomena identified with the cochlea's ‘travelling wave’ behaviour and tonotopic frequency map.</jats:p>
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