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Matthias Maier

Publications and source records attributed to Matthias Maier.

33 records · Page 2Linked to original sources
arXiv2020

Finite-size effects in wave transmission through plasmonic crystals: A tale of two scales

We study optical coefficients that characterize wave propagation through layered structures called plasmonic crystals. These consist of a finite number of stacked metallic sheets embedded in dielectric hosts with a subwavelength spacing. By adjustment of the frequency, spacing, number as well as geometry of the layers, these structures may exhibit appealing transmission properties in a range of frequencies from the terahertz to the mid-infrared regime. Our approach uses a blend of analytical and numerical methods for the distinct geometries with infinite, translation invariant, flat sheets and nanoribbons. We describe the transmission of plane waves through a plasmonic crystal in comparison to an effective dielectric slab of equal total thickness that emerges from homogenization, in the limit of zero interlayer spacing. We demonstrate numerically that the replacement of the discrete plasmonic crystal by its homogenized counterpart can accurately capture a transmission coefficient akin to the extinction spectrum, even for a relatively small number of layers. We point out the role of a geometry-dependent corrector field, which expresses the effect of subwavelength surface plasmons. In particular, by use of the corrector we describe lateral resonances inherent to the nanoribbon geometry.

cond-mat.mes-hall↗
arXiv2020

Homogenization of time-harmonic Maxwell's equations in nonhomogeneous plasmonic structures

We carry out the homogenization of time-harmonic Maxwell's equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale, $d$. Starting with the weak formulation of the requisite boundary value problem, we prove the convergence of its solution to a homogenization limit as $d$ approaches zero. The effective permittivity and cell problem express a bulk average from the host dielectric and a surface average germane to the 2D material (metallic layer). We discuss implications of this analysis in the modeling of plasmonic crystals.

math.AP↗
arXiv2020

The deal.II finite element library: design, features, and insights

deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, $hp$-adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse and worldwide community of developers and users, as well as an educational platform. We therefore also discuss some of the technical and social challenges and lessons learned in running a large community software project over the course of two decades.

cs.MS↗
arXiv2019

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation. This view entails the quasi-electrostatic approximation, by which the tangential electric field is replaced by the gradient of a scalar potential, $φ$. By the Wiener-Hopf method, we solve an integral equation for $φ$ in some generality. The EPP dispersion relation comes from the elimination of a divergent limiting Fourier integral for $φ$ at the edge. We connect the existence, or lack thereof, of the EPP dispersion relation to the index for Wiener-Hopf integral equations, an integer of topological character. We indicate that the values of this index may express an asymmetry due to the material anisotropy in the number of wave modes propagating on the sheet away from the edge with respect to the EPP direction of propagation. We discuss extensions such as the setting of two semi-infinite, coplanar sheets. Our theory forms a generalization of the treatment by Volkov and Mikhailov (1988 Sov. Phys. JETP 67 1639).

math-ph↗
arXiv2019

Graphene epsilon-near-zero plasmonic crystals

Plasmonic crystals are a class of optical metamaterials that consist of engineered structures at the sub-wavelength scale. They exhibit optical properties that are not found under normal circumstances in nature, such as negative-refractive-index and epsilon-near-zero (ENZ) behavior. Graphene-based plasmonic crystals present linear, elliptical, or hyperbolic dispersion relations that exhibit ENZ behavior, normal or negative-index diffraction. The optical properties can be dynamically tuned by controlling the operating frequency and the doping level of graphene. We propose a construction approach to expand the frequency range of the ENZ behavior. We demonstrate how the combination of a host material with an optical Lorentzian response in combination with a graphene conductivity that follows a Drude model leads to an ENZ condition spanning a large frequency range.

physics.app-ph↗
arXiv2019

Homogenization of plasmonic crystals: Seeking the epsilon-near-zero effect

By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell's equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host. This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. We demonstrate how our homogenization procedure may set the foundation for computational investigations of: effective optical responses of reasonably general geometries, and complicated design problems in the plasmonics of 2D materials.

math.NA↗
arXiv2019

Adaptive finite element simulations of waveguide configurations involving parallel 2D material sheets

We discuss analytically and numerically the propagation and energy transmission of electromagnetic waves caused by the coupling of surface plasmon polaritons (SPPs) between two spatially separated layers of 2D materials, such as graphene, at subwavelength distances. We construct an adaptive finite-element method to compute the ratio of energy transmitted within these waveguide structures reliably and efficiently. At its heart, the method is built upon a goal-oriented a posteriori error estimation with the dual-weighted residual method (DWR). Further, we derive analytic solutions of the two-layer system, compare those to (known) single-layer configurations, and compare and validate our numerical findings by comparing numerical and analytical values for optimal spacing of the two-layer configuration. Additional aspects of our numerical treatment, such as local grid refinement, and the utilization of perfectly matched layers (PMLs) are examined in detail.

physics.comp-ph↗
arXiv2018

Universal behavior of dispersive Dirac cone in gradient-index plasmonic metamaterials

We demonstrate analytically and numerically that the dispersive Dirac cone emulating an epsilon-near-zero (ENZ) behavior is a universal property within a family of plasmonic crystals consisting of two-dimensional (2D) metals. Our starting point is a periodic array of 2D metallic sheets embedded in an inhomogeneous and anisotropic dielectric host that allows for propagation of transverse-magnetic (TM) polarized waves. By invoking a systematic bifurcation argument for arbitrary dielectric profiles in one spatial dimension, we show how TM Bloch waves experience an effective dielectric function that averages out microscopic details of the host medium. The corresponding effective dispersion relation reduces to a Dirac cone when the conductivity of the metallic sheet and the period of the array satisfy a critical condition for ENZ behavior. Our analytical findings are in excellent agreement with numerical simulations.

physics.optics↗
arXiv2017

A duality-based optimization approach for model adaptivity in heterogeneous multiscale problems

This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators, that are derived in the general context of the Dual Weighted Residual (DWR) method. Based on the optimization approach a post-processing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, "goal-oriented" tuning of effective models with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with strong microscopic, random advection field.

math.NA↗
arXiv2017

Ultracompact amplitude modulator by coupling hyperbolic polaritons over a graphene-covered gap

The hyperbolic phonon-polaritons within the Reststrahlen band of hBN are of great interest for applications in nanophotonics as they are capable of propagating light signals with low losses over large distances. However, due to the phononic nature of the polaritons in hBN, amplitude modulation of its signal proves to be difficult and has been underexplored. In this paper, we propose theoretically a broadband efficient amplitude modulator for hyperbolic rays in hBN operating in the frequency range between 1450 cm$^{-1}$ and 1550 cm$^{-1}$. The modulating region comprises a few tens of nanometers wide gap carved within the hBN slab and covered by a graphene layer, where electrostatically gated graphene serves as a mediator that facilitates the coupling between phonon-polaritons on each side of the gap through plasmonic modes within graphene. We demonstrate that such an ultra compact modulator has insertion losses as low as 3 dB and provides modulation depth varying between 14 and 20 dB within the type-II hyperbolicity region of hBN.

cond-mat.mes-hall↗
arXiv2017

Flux Reconstruction for Goal-Oriented A Posteriori Error Estimation

We propose a new heuristic goal-oriented a posteriori error estimator that connects the dual weighted residual method with equilibrated a posteriori error estimation. Our numerical experiments demonstrate the practical reliability of the error estimator, confirming theoretical predictions, as well as optimally convergent adaptivity even over singular domains and coarse meshes. The central algorithm is a localized flux reconstruction, which has been implemented in the finite element library deal.II. For a solid preparation we assess the performance of the equilibrated a posteriori error estimator of the energy norm in numerical experiments. Moreover, we give what seems to be first rigorous discussion in the numerical literature of localized flux reconstruction over quadrilateral meshes with hanging nodes.

math.NA↗
arXiv2017

Dipole excitation of surface plasmon on a conducting sheet: finite element approximation and validation

We formulate and validate a finite element approach to the propagation of a slowly decaying electromagnetic wave, called surface plasmon-polariton, excited along a conducting sheet, e.g., a single-layer graphene sheet, by an electric Hertzian dipole. By using a suitably rescaled form of time-harmonic Maxwell's equations, we derive a variational formulation that enables a direct numerical treatment of the associated class of boundary value problems by appropriate curl-conforming finite elements. The conducting sheet is modeled as an idealized hypersurface with an effective electric conductivity. The requisite weak discontinuity for the tangential magnetic field across the hypersurface can be incorporated naturally into the variational formulation. We carry out numerical simulations for an infinite sheet with constant isotropic conductivity embedded in two spatial dimensions; and validate our numerics against the closed-form exact solution obtained by the Fourier transform in the tangential coordinate. Numerical aspects of our treatment such as an absorbing perfectly matched layer, as well as local refinement and a-posteriori error control are discussed.

math.NA↗
arXiv2017

Generation of surface plasmon-polaritons by edge effects

By using numerical and analytical methods, we describe the generation of fine-scale lateral electromagnetic waves, called surface plasmon-polaritons (SPPs), on atomically thick, metamaterial conducting sheets in two spatial dimensions (2D). Our computations capture the two-scale character of the total field and reveal how each edge of the sheet acts as a source of an SPP that may dominate the diffracted field. We use the finite element method to numerically implement a variational formulation for a weak discontinuity of the tangential magnetic field across a hypersurface. An adaptive, local mesh refinement strategy based on a posteriori error estimators is applied to resolve the pronounced two-scale character of wave propagation and radiation over the metamaterial sheet. We demonstrate by numerical examples how a singular geometry, e.g., sheets with sharp edges, and sharp spatial changes in the associated surface conductivity may significantly influence surface plasmons in nanophotonics.

physics.comp-ph↗
arXiv2017

On the Wiener-Hopf method for surface plasmons: Diffraction from semi-infinite metamaterial sheet

By formally invoking the Wiener-Hopf method, we explicitly solve a one-dimensional, singular integral equation for the excitation of a slowly decaying electromagnetic wave, called surface plasmon-polariton (SPP), of small wavelength on a semi-infinite, flat conducting sheet irradiated by a plane wave in two spatial dimensions. This setting is germane to wave diffraction by edges of large sheets of single-layer graphene. Our analytical approach includes: (i) formulation of a functional equation in the Fourier domain; (ii) evaluation of a split function, which is expressed by a contour integral and is a key ingredient of the Wiener-Hopf factorization; and (iii) extraction of the SPP as a simple-pole residue of a Fourier integral. Our analytical solution is in good agreement with a finite-element numerical computation.

math-ph↗