Remarks on the stability of Cartesian PMLs in corners

This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure-velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results

Transparent-Influx Boundary Conditions for FEM Based Modelling of 2D Helmholtz Problems in Optics

A numerical method for the analysis of the 2D Helmholtz equation is presented, which incorporates Transparent-Influx Boundary Conditions into a variational formulation of the Helmholtz problem. For rectangular geometries, the non-locality of those boundaries can be efficiently handled by using Fourier decomposition. The Finite Element Method is used to discretise the interior and the nonlocal Dirichlet-to-Neumann operators arising from the formulation of Transparent-Influx Boundary Conditions

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

Simulations of acoustic wavefields in inhomogeneous media are always performed on finite numerical domains. If contrasts actually extend over the domain boundaries of the numerical volume, unwanted, non-physical reflections from the boundaries will occur. One technique to suppress these reflections is to attenuate them in a locally reflectionless absorbing boundary layer enclosing the spatial computational domain, a perfectly matched layer (PML). This technique is commonly applied in time-domain simulation methods like finite element methods or finite-difference time-domain, but has not been applied to the integral equation method. In this paper, a PML formulation for the three-dimensional frequency-domain integral-equation-based acoustic scattering problem is derived. Three-dimensional acoustic scattering configurations are used to test the PML formulation. The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs themselves are virtually reflectionless. In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions

Perfectly Matched Layer based modelling of layered media: Overview and perspective

Whereas the Perfectly Matched Layer (PML) was originally conceived to serve as a reflectionless absorbing boundary condition, terminating the infinite simulation domain in finite difference or finite element electromagnetic field solvers, unexpected applications of the PML arose by using it to close down open layered waveguide configurations. As a tribute to our former colleague, Prof. F. Olyslager, in this contribution, the PML-paradigm for layered media is explained and an overview of this paradigm's applications is presented. A novel illustrative example, focussing on the modelling of periodic microstrip structures, is also provided

Reflectionless discrete perfectly matched layers for higher-order finite difference schemes

This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach

Birefringent left-handed metamaterials and perfect lenses

We describe the properties of birefringent left-handed metamaterials and introduce the concept of a birefringent perfect lens. We demonstrate that, in a sharp contrast to the conventional left-handed perfect lens at $\epsilon=\mu=-1$, where $\epsilon$ is the dielectric constant and $\mu$ is the magnetic permeability, the birefringent left-handed lens can focus either TE or TM polarized waves or both of them, allowing a spatial separation of the TE and TM images. We discuss several applications of the birefringent left-handed lenses such as the beam splitting and near-field diagnostics at the sub-wavelength scale.Comment: 4 pages 6 figure

A Radial-Dependent Dispersive Finite-Difference Time-Domain Method for the Evaluation of Electromagnetic Cloaks

A radial-dependent dispersive finite-difference time-domain (FDTD) method is proposed to simulate electromagnetic cloaking devices. The Drude dispersion model is applied to model the electromagnetic characteristics of the cloaking medium. Both lossless and lossy cloaking materials are examined and their operating bandwidth is also investigated. It is demonstrated that the perfect "invisibility" from electromagnetic cloaks is only available for lossless metamaterials and within an extremely narrow frequency band.Comment: 18 pages, 10 figure