We consider the solution of the Helmholtz equation −∆u(x)−n(x)²ω²u(x) = f(x), x = (x,y), in a domain Ω which is infinite in x and bounded in y. We assume that f(x) is supported in Ω⁰ := {x ∈ Ω |a⁻ < x < a⁺} and that n(x) is x-periodic in Ω\Ω⁰. We show how to obtain exact boundary conditions on the vertical segments, Γ⁻ := {x ∈ Ω |x = a⁻} and Γ⁺ := {x ∈ Ω |x = a⁺}, that will enable us to find the solution on Ω⁰ ∪Γ⁺ ∪Γ⁻. Then the solution can be extended in Ω in a straightforward manner from the values on Γ⁻ and Γ⁺. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. 

We introduce the equivalent sources for the Helmholtz equation and establish their connections to the naturally induced sources for the sound-soft, sound-hard, and impedance obstacles for the inverse scattering problems of the Helmholtz equation. As two applications, we employ the naturally induced sources to improve the boundary integral equation formulations for the obstacle scattering problems, and develop a unified, straightforward approach to establishing boundary conditions governing the domain derivatives of scattered waves for the soft, hard, and impedance obstacles. 

The integral equation method for the simulation of the diffraction by optical gratings is an efficient numerical tool if profile gratings determined by simple cross-section curves are considered. This method in its recent version is capable to tackle profile curves with corners, gratings with thin coated layers, and diffraction scenarios with unfavorably large ratio period over wavelength. We discuss special implementational issues including the efficient evaluation of the quasi-periodic Green kernels, the quadrature algorithm, and the iterative solution of the arising systems of linear equations. Finally, as an example we present the simulation of echelle gratings which demonstrates the efficiency of our approach.

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414, 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS^∗ method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS^∗ method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS^∗ method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS^∗ and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS^∗ method and illustrate the ill-conditioning that arises.

A hybrid finite element (FEM) and Fourier transform method is implemented to analyze the time domain scattering of a plane wave incident on a 2-D overfilled cavity embedded in the infinite ground plane. The algorithm first removes the time variable by Fourier transform, through which a frequency domain problem is obtained. An artificial boundary condition is then introduced on a hemisphere enclosing the cavity that couples the fields from the infinite exterior domain to those inside. The exterior problem is solved analytically via Fourier series solutions, while the interior region is solved using finite element method. In the end, the image functions in frequency domain are numerically inverted into the time domain. The perfect link over the artificial boundary between the FEM approximation in the interior and analytical solution in the exterior indicates the reliability of the method. A convergence analysis is also performed.

Optical wave-guiding structures that are non-uniform in the propagation direction are fundamental building blocks of integrated optical circuits. Numerical simulation of lightwaves propagating in these structures is an essential tool to engineers designing photonic components. In this paper, we review recent developments in the most widely used simulation methods for frequency domain propagation problems. 

In this paper we discuss scattering problems inherent in curved microstrip structures mounted on thin dielectric structures. We provide approximate boundary conditions for such structures in the framework of integral equations. 

Multidimensional tunneling appears in many problems at nano scale. The high dimensionality of the potential energy surface (e.g. many degrees of freedom) poses a great challenge in both theoretical and numerical description of tunneling. Numerical simulation based on Schrödinger equation is often prohibitively expensive. We propose an accurate, efficient, robust and easy-to-implement numerical method to calculate the ground state tunneling splitting based on imaginary-time path integral ('instanton' formulation). The method is genuinely multi-dimensional and free from any additional ad hoc assumptions on potential energy surface. It enables us to calculate the effects of all coupling modes on the tunneling degree of freedom without loss. We also review in this paper some theoretical background and survey some recent work from other groups in calculating multidimensional quantum tunneling effects in chemical reactions.