arXiv subjects

Ory Schnitzer

Publications and source records attributed to Ory Schnitzer.

31 records · Page 2Linked to original sources

arXiv2020

Asymptotic modeling of Helmholtz resonators including thermoviscous effects

We systematically employ the method of matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometries. With an eye towards modeling acoustic metasurfaces, we consider resonators embedded in a rigid surface, each resonator consisting of an arbitrarily shaped cavity connected to the external half-space by a small cylindrical neck. The bulk of the analysis is devoted to the problem where a single resonator is subjected to a normally incident plane wave; the model is then extended using "Foldy's method" to the case of multiple resonators subjected to an arbitrary incident field. As an illustration, we derive critical-coupling conditions for optimal and perfect absorption by a single resonator and a model metasurface, respectively.

physics.flu-dyn↗

arXiv2018

Extraordinary transmission through a narrow slit

We revisit the problem of extraordinary transmission of acoustic (electromagnetic) waves through a slit in a rigid (perfectly conducting) wall. We use matched asymptotic expansions to study the pertinent limit where the slit width is small compared to the wall thickness, the latter being commensurate with the wavelength. Our analysis focuses on near-resonance frequencies, furnishing elementary formulae for the field enhancement, transmission efficiency, and deviations of the resonances from the Fabry-Perot frequencies of the slit. We find that the apertures' near fields play a dominant role, in contrast with the prevalent approximate theory of Takakura [Physical Review Letters, 86 5601 (2001)]. Our theory agrees remarkably well with numerical solutions and electromagnetic experiments [Suckling et al., Physical Review Letters, 92 147401 (2004)], thus providing a paradigm for analyzing a wide range of wave propagation problems involving small holes and slits.

physics.optics↗

arXiv2018

Geometric quantization of localized surface plasmons

We consider the quasi-static problem governing the localized surface plasmon modes and permittivity eigenvalues $ε$ of smooth, arbitrarily shaped, axisymmetric inclusions. We develop an asymptotic theory for the dense part of the spectrum, i.e., close to the accumulation value $ε=-1$ at which a flat interface supports surface plasmons; in this regime, the field oscillates rapidly along the surface and decays exponentially away from it on a comparable scale. With $τ=-(ε+1)$ as the small parameter, we develop a surface-ray description of the eigenfunctions in a narrow boundary layer about the interface; the fast phase variation, as well as the slowly varying amplitude and geometric phase, along the rays are determined as functions of the local geometry. We focus on modes varying at most moderately in the azimuthal direction, in which case the surface rays are meridian arcs that focus at the two poles. Asymptotically matching the diverging ray solutions with expansions valid in inner regions in the vicinities of the poles yields the quantization rule $$\frac{1}τ \sim \frac{πn }Θ+\frac{1}{2}\left(\fracπΘ-1\right)+o(1),$$ where $n\gg1$ is an integer and $Θ$ a geometric parameter given by the product of the inclusion length and the reciprocal average of its cross-sectional radius along its symmetry axis. For a sphere, $Θ=π$, whereby the formula returns the exact eigenvalues $ε_n=-1-1/n$. We also demonstrate good agreement with exact solutions in the case of prolate spheroids.

physics.optics↗

arXiv2018

Asymptotic modelling of phononic box crystals

We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a three-dimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbour interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs and crystals, that agree favourably with finite-element simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole- with dipole-type forcing; for the former the pressure field is asymptotically enhanced whilst for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann's Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array.

physics.app-ph↗

arXiv2018

Asymptotic approximations for the plasmon resonances of nearly touching spheres

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows [O. Schnitzer, \textit{Physical Review B}, \textbf{92} 235428 2015]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

physics.optics↗

arXiv2017

Spoof surface plasmons guided by narrow grooves

An approximate description of surface waves propagating along periodically grooved surfaces is intuitively developed in the limit where the grooves are narrow relative to the period. Considering acoustic and electromagnetic waves guided by rigid and perfectly conducting gratings, respectively, the wave field is obtained by interrelating elementary approximations obtained in three overlapping spatial domains. Specifically, above the grating and on the scale of the period the grooves are effectively reduced to point resonators characterised by their dimensions as well as the geometry of their apertures. Along with this descriptive physical picture emerges an analytical dispersion relation, which agrees remarkably well with exact calculations and improves on preceding approximations. Scalings and explicit formulae are obtained by simplifying the theory in three distinguished propagation regimes, namely where the Bloch wavenumber is respectively smaller than, close to, or larger than that corresponding to a groove resonance. Of particular interest is the latter regime where the field within the grooves is resonantly enhanced and the field above the grating is maximally localised, attenuating on a length scale comparable with the period.

physics.optics↗

arXiv2017

Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

We demonstrate that photonic and phononic crystals consisting of closely spaced inclusions constitute a versatile class of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystal form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous photonic (phononic) crystals are in fact asymptotically equivalent, at low frequencies, to discrete capacitor-inductor (mass-spring) networks whose lumped parameters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch, or branches when the discrete analogue is polyatomic, is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals. The insight gained from the network approach is used to show that, surprisingly, the continuum created by a closely packed hexagonal lattice of cylinders is represented by a discrete honeycomb lattice. The analogy is utilised to show that the hexagonal continuum lattice has a Dirac-point degeneracy that is lifted in a controlled manner by specifying the area of a symmetry-breaking defect.

physics.optics↗

arXiv2016

Waves in slowly varying band-gap media

This paper is concerned with the asymptotic description of high-frequency waves in locally periodic media. A key issue is that the Bloch-dispersion curves vary with the local microstructure, giving rise to hidden singularities associated with band-gap edges and branch crossings. We describe an asymptotic approach for overcoming this difficulty, and take a first step by studying in detail the simplest case of 1D Helmholtz waves. The method entails matching adiabatically propagating Bloch waves, captured by a multiple-scale Wentzel-Kramers-Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localised about dispersion singularities. Within the latter regions the Bloch wavenumber is nearly critical; this allows their homogenisation, following the method of high-frequency homogenisation (HFH), over a naturally arising scale intermediate between the periodicity (wavelength) and the macro-scale. Analogously to a classical turning-point analysis, we show that close to a spatial band-gap edge the solution is an Airy function, only that it is modulated on the short scale by a standing-wave Bloch eigenfunction. By carrying out the asymptotic matching between the WKB and HFH solutions we provide a detailed description of Bloch-wave reflection from a band gap. Finally, we implement the asymptotic theory for a layered medium and demonstrate excellent agreement with numerical computations.

physics.optics↗

arXiv2016

Singular effective slip length for longitudinal flow over a dense bubble mattress

We consider the effective hydrophobicity of a periodically grooved surface immersed in liquid, with trapped shear-free bubbles protruding between the no-slip ridges at a $π/2$ contact angle. Specifically, we carry out a singular-perturbation analysis in the limit $ε\ll1$ where the bubbles are closely spaced, finding the effective slip length (normalised by the bubble radius) for longitudinal flow along the the ridges as $π/{\sqrt{2ε}}-(12/π)\ln 2 + (13π/24)\sqrt{2ε}+ o(\sqrtε)$, the small parameter $ε$ being the planform solid fraction. The square-root divergence highlights the strong hydrophobic character of this configuration; this leading singular term (along with the third term) follows from a local lubrication-like analysis of the gap regions between the bubbles, together with general matching considerations and a global conservation relation. The $O(1)$ constant term is found by matching with a leading-order solution in the "outer'" region, where the bubbles appear to be touching. We find excellent agreement between our slip-length formula and a numerical scheme recently derived using a "unified-transform" method (D. Crowdy, IMA J. Appl. Math., 80 1902, 2015). The comparison demonstrates that our asymptotic formula, together with the diametric "dilute-limit" approximation (D. Crowdy, J. Fluid Mech., 791 R7, 2016), provides an elementary analytical description for essentially arbitrary no-slip fractions.

physics.flu-dyn↗

arXiv2016

Surface-plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit

According to the hydrodynamic Drude model, surface-plasmon resonances of metallic nanostructures blueshift owing to the nonlocal response of the metal's electron gas. The screening length characterising the nonlocal effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained nonlocal description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary shaped nanometallic structures is then developed. The effect of nonlocality is not always a perturbation and we present a detailed analysis of the "bonding" modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalisation of those predicted assuming a local metal permittivity.

physics.optics↗

arXiv2016

Bloch waves in an arbitrary two-dimensional lattice of subwavelength Dirichlet scatterers

We study waves governed by the planar Helmholtz equation, propagating in an infinite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of the operating wavelength.

physics.class-ph↗

arXiv2015

Singular perturbations approach to localized surface-plasmon resonance: Nearly touching metal nanospheres

Metallic nano-structures characterised by multiple geometric length scales support low-frequency surface-plasmon modes, which enable strong light localization and field enhancement. We suggest studying such configurations using singular perturbation methods, and demonstrate the efficacy of this approach by considering, in the quasi-static limit, a pair of nearly touching metallic nano-spheres subjected to an incident electromagnetic wave polarized with the electric field along the line of sphere centers. Rather than attempting an exact analytical solution, we construct the pertinent (longitudinal) eigen-modes by matching relatively simple asymptotic expansions valid in overlapping spatial domains. We thereby arrive at an effective boundary eigenvalue problem in a half-space representing the metal region in the vicinity of the gap. Coupling with the gap field gives rise to a mixed-type boundary condition with varying coefficients, whereas coupling with the particle-scale field enters through an integral eigenvalue selection rule involving the electrostatic capacitance of the configuration. By solving the reduced problem we obtain accurate closed-form expressions for the resonance values of the metal dielectric function. Furthermore, together with an energy-like integral relation, the latter eigen-solutions yield also closed-form approximations for the induced-dipole moment and gap-field enhancement under resonance. We demonstrate agreement between the asymptotic formulas and a semi-numerical computation. The analysis, underpinned by asymptotic scaling arguments, elucidates how metal polarization together with geometrical confinement enables a strong plasmon-frequency redshift and amplified near-field at resonance.

cond-mat.mes-hall↗

arXiv2015

Asymptotics of surface-plasmon redshift saturation at sub-nanometric separations

Many promising nanophotonics endeavours hinge upon the unique plasmonic properties of nanometallic structures with narrow non-metallic gaps, which support super-concentrated bonding modes that singularly redshift with decreasing separations. In this letter, we present a descriptive physical picture, complemented by elementary asymptotic formulae, of a nonlocal mechanism for plasmon-redshift saturation at subnanometric gap widths. Thus, by considering the electron-charge and field distributions in the close vicinity of the metal-vacuum interface, we show that nonlocality is asymptotically manifested as an effective potential discontinuity. For bonding modes in the near-contact limit, the latter discontinuity is shown to be effectively equivalent to a widening of the gap. As a consequence, the resonance-frequency near-contact asymptotics are a renormalisation of the corresponding local ones. Specifically, the renormalisation furnishes an asymptotic plasmon-frequency lower bound that scales with the $1/4$-power of the Fermi wavelength. We demonstrate these remarkable features in the prototypical cases of nanowire and nanosphere dimers, showing agreement between our elementary expressions and previously reported numerical computations.

physics.optics↗