14 results found
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.
Bell A, Davies B, Ammari H, 2021, Bernhard Riemann, the ear, and an atom of consciousness, Foundations of Science, 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.
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
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>
Ammari H, Davies B, A fully-coupled subwavelength resonance approach to modelling the passive cochlea, Proc. R. Soc. A (2019) 475:20190049
The aim of this paper is to understand the behaviour of a large number ofcoupled subwavelength resonators. We use layer potential techniques incombination with numerical computations to study the acoustic pressure fielddue to scattering by a graded array of subwavelength resonators. Using thismethod, we study a graded-resonance model for the cochlea. We compute theresonant modes of the system and explore the model's ability to decomposeincoming signals. We are able to offer mathematical explanations for thecochlea's so-called "travelling wave" behaviour and tonotopic frequency map.
Ammari H, Davies B, Hiltunen EO, Robust edge modes in dislocated systems of subwavelength resonators
Robustly manipulating waves on subwavelength scales can be achieved by,firstly, designing a structure with a subwavelength band gap and, secondly,introducing a defect so that eigenfrequencies fall within the band gap. Suchfrequencies are well known to correspond to localized modes. We study aone-dimensional array of subwavelength resonators, proving that there is asubwavelength band gap, and showing that by introducing a dislocation we canplace localized modes at any point within the band gap. We complement thisanalysis by studying the stability properties of the corresponding finite arrayof resonators, demonstrating the value of being able to customize the positionof eigenvalues within the band gap.
Ammari H, Davies B, Hiltunen EO, et al., Wave interaction with subwavelength resonators
The aim of this review is to cover recent developments in the mathematicalanalysis of subwavelength resonators. The use of sophisticated mathematics inthe field of metamaterials is reported, which provides a mathematical frameworkfor focusing, trapping, and guiding of waves at subwavelength scales.Throughout this review, the power of layer potential techniques combined withasymptotic analysis for solving challenging wave propagation problems atsubwavelength scales is demonstrated.
Ammari H, Davies B, A biomimetic basis for auditory processing and the perception of natural sounds
Biomimicry is a powerful science that aims to take advantage of nature'sremarkable ability to devise innovative solutions to challenging problems. Inthe setting of hearing, mimicking how humans hear is the foremost strategy indesigning effective artificial hearing approaches. In this work, we explore themathematical foundations for the exchange of design inspiration and featuresbetween biological hearing systems, artificial sound-filtering devices, andsignal processing algorithms. Our starting point is a concise asymptoticanalysis of subwavelength acoustic metamaterials. We are able to fine tune thisstructure to mimic the biomechanical properties of the cochlea, at the samescale. We then turn our attention to developing a biomimetic signal processingalgorithm. We use the response of the cochlea-like structure as an initialfiltering layer and then add additional biomimetic processing stages, designedto mimic the human auditory system's ability to recognise the global propertiesof natural sounds.
Ammari H, Davies B, Hiltunen EO, et al., Exceptional points in parity-time-symmetric subwavelength metamaterials
When sources of energy gain and loss are introduced to a wave-scatteringsystem, the underlying mathematical formulation will be non-Hermitian. Thispaves the way for the existence of exceptional points, where eigenmodes arelinearly dependent. The primary goal of this work is to study the existence ofexceptional points in high-contrast subwavelength metamaterials. We begin bystudying a parity-time-symmetric pair of subwavelength resonators and provethat this system supports asymptotic exceptional points. These are points atwhich the subwavelength eigenvalues and eigenvectors coincide at leading orderin the asymptotic parameters. We then investigate further properties ofparity-time-symmetric subwavelength metamaterials. First, we study the exoticscattering behaviour of a metascreen composed of repeatingparity-time-symmetric pairs of subwavelength resonators. We prove that thenon-Hermitian nature of this structure means that it exhibits asymptoticunidirectional reflectionless transmission at certain frequencies anddemonstrate extraordinary transmission close to these frequencies. Thereafter,we consider cavities containing many small resonators and use homogenizationtheory to show that non-Hermitian behaviour can be replicated at themacroscale.
Davies B, Herren L, Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors
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, Functional analytic methods for discrete approximations of subwavelength resonator systems
We survey functional analytic methods for studying subwavelength resonatorsystems. In particular, rigorous discrete approximations of Helmholtzscattering problems are derived in an asymptotic subwavelength regime. This isachieved by re-framing the Helmholtz equation as a non-linear eigenvalueproblem in terms of integral operators. In the subwavelength limit, resonantstates are described by the eigenstates of the generalized capacitance matrix,which appears by perturbing the elements of the kernel of the limitingoperator. Using this formulation, we are able to describe subwavelengthresonance and related phenomena. In particular, we demonstrate large-scaleeffective parameters with exotic values. We also show that these systems canexhibit localized and guided waves on very small length scales. Using theconcept of topologically protected edge modes, such localization can be maderobust against structural imperfections.
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