High-order integral equation methods for problems of scattering by bumps and cavities on half-planes

This paper presents high-order integral equation methods for evaluation of electromagnetic wave scattering by dielectric bumps and dielectric cavities on perfectly conducting or dielectric half-planes. In detail, the algorithms introduced in this paper apply to eight classical scattering problems, namely: scattering by a dielectric bump on a perfectly conducting or a dielectric half-plane, and scattering by a filled, overfilled or void dielectric cavity on a perfectly conducting or a dielectric half-plane. In all cases field representations based on single-layer potentials for appropriately chosen Green functions are used. The numerical far fields and near fields exhibit excellent convergence as discretizations are refined--even at and around points where singular fields and infinite currents exist.Comment: 25 pages, 7 figure

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique

Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions

Windowed Integral Equation Methods for Problems of Scattering by Defects and Obstacles in Layered Media

This thesis concerns development of efficient high-order boundary integral equation methods for the numerical solution of problems of acoustic and electromagnetic scattering in the presence of planar layered media in two and three spatial dimensions. The interest in such problems arises from application areas that benefit from accurate numerical modeling of the layered media scattering phenomena, such as electronics, near-field optics, plasmonics and photonics as well as communications, radar and remote sensing. A number of efficient algorithms applicable to various problems in these areas are pre- sented in this thesis, including (i) A Sommerfeld integral based high-order integral equation method for problems of scattering by defects in presence of infinite ground and other layered media, (ii) Studies of resonances and near resonances and their impact on the absorptive properties of rough surfaces, and (iii) A novel Window Green Function Method (WGF) for problems of scattering by obstacles and defects in the presence of layered media. The WGF approach makes it possible to completely avoid use of expensive Sommerfeld integrals that are typically utilized in layer-media simulations. In fact, the methods and studies referred in points (i) and (ii) above motivated the development of the markedly more efficient WGF alternative.</p

Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

This paper presents a second-kind surface integral equation method for the numerical solution of frequency-domain electromagnetic scattering problems by locally perturbed layered media in three spatial dimensions. Unlike standard approaches, the proposed methodology does not involve the use of layer Green functions. It instead leverages an indirect M\"uller formulation in terms of free-space Green functions that entails integration over the entire unbounded penetrable boundary. The integral equation domain is effectively reduced to a small-area surface by means of the windowed Green function method, which exhibits high-order convergence as the size of the truncated surface increases. The resulting (second-kind) windowed integral equation is then numerically solved by means of the standard Galerkin method of moments (MoM) using RWG basis functions. The methodology is validated by comparison with Mie-series and Sommerfeld-integral exact solutions as well as against a layer Green function-based MoM. Challenging examples including realistic structures relevant to the design of plasmonic solar cells and all-dielectric metasurfaces, demonstrate the applicability, efficiency, and accuracy of the proposed methodology

Sideways adiabaticity: Beyond ray optics for slowly varying metasurfaces

Optical metasurfaces (subwavelength-patterned surfaces typically described by variable effective surface impedances) are typically modeled by an approximation akin to ray optics: the reflection or transmission of an incident wave at each point of the surface is computed as if the surface were "locally uniform", and then the total field is obtained by summing all of these local scattered fields via a Huygens principle. (Similar approximations are found in scalar diffraction theory and in ray optics for curved surfaces.) In this paper, we develop a precise theory of such approximations for variable-impedance surfaces. Not only do we obtain a type of adiabatic theorem showing that the "zeroth-order" locally uniform approximation converges in the limit as the surface varies more and more slowly, including a way to quantify the rate of convergence, but we also obtain an infinite series of higher-order corrections. These corrections, which can be computed to any desired order by performing integral operations on the surface fields, allow rapidly varying surfaces to be modeled with arbitrary accuracy, and also allow one to validate designs based on the zeroth-order approximation (which is often surprisingly accurate) without resorting to expensive brute-force Maxwell solvers. We show that our formulation works arbitrarily close to the surface, and can even compute coupling to guided modes, whereas in the far-field limit our zeroth-order result simplifies to an expression similar to what has been used by other authors