High order integral equation method for diffraction gratings

Abstract

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green's functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green's functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

“…are needed to retrieve the fields by means of the representation formulas ( 19), (22), and (21). Indeed, by the transmission conditions (2c) we have…”

“…19. There and in subsequent related contributions, 20,21 Neumann-to-Dirichlet operators are combined with the quasi-periodic boundary conditions on the unit-cell walls to reduce the problem to a BIE expressed in terms of free-space Green function kernels. No explicit mention of issues associated with RW anomalies are reported in these works.…”

PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

“…If β j is quite close to 0, accurate boundary conditions can be developed; we refer readers to [26,29,34] for details. Besides, ũsc ref satisfies the quasi-periodic conditions (96) and (97) and the surface condition (98), but with u replaced by ũ.…”

“…If the surface has no defects, the total wave field for the plane-wave incidence is quasiperiodic so that the original scattering problem can be formulated in a single unit cell, bounded laterally but unbounded vertically. According to UPRC, the scattered wave at infinity can then be expressed in terms of upgoing Bloch waves, so that a transparent boundary condition or PML of a local/nonlocal boundary condition can be successfully used to terminate the unit cell vertically; readers are referred to [3,10,26,34] and the references therein, for related numerical methods as well as theories of exponential convergence due to a PML truncation. But, if the incident wave is nonquasi-periodic, e.g., the cylindrical wave, or if the surface is locally defected, much fewer numerical methods or theories have been developed as it is no longer straightforward to laterally terminate the scattering domain.…”

Improving accuracy of the numerical solution of Maxwell's equations by processing edge singularities of the electromagnetic field

“…It should be noted that in the considered approach the DtN map matrices of individual subdomains can be calculated by using an arbitrary method, not necessarily by the spectral method. In the case of a uniform dielectric constant inside each element, it is very convenient to calculate the DtN map matrix by using the boundary integral method [17,23]. Since in this method the calculated subdomain does not need to be mapped into a rectangle, we get considerable freedom in choosing the partition into subdomains.…”