Solving conical diffraction grating problems with integral equations

Abstract

Off-plane scattering of time-harmonic plane waves by a plane diffraction grating with arbitrary conductivity and general surface profile is considered in a rigorous electromagnetic formulation. Integral equations for conical diffraction are obtained involving, besides the boundary integrals of the single and double layer potentials, singular integrals, the tangential derivative of single-layer potentials. We derive an explicit formula for the calculation of the absorption in conical diffraction. Some rules that are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive gratings, surfaces with edges, real profiles, and gratings working at short wavelengths.

Integral equation methods from grating theory to photonics: an overview and new approaches for conical diffraction

“…Only recently an integral method for real conical diffraction for a single grating has been published and implemented (35,36) leading to an inherently coupled integral equation system. An outline of this method and an extension to multi-profile diffraction gratings is presented in Sect.…”

“…Moreover, trigonometric collocation methods for solving system (15) converge under the assumptions made above. Some numerical results for gratings with continuous interface are reported in (36). For profiles with corners the algorithm uses a hybrid spline-trigonometric method on graded meshes.…”

Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings

“…A multilayer grating can have a fairly large number of boundaries, up to a few thousand for applications in the hard X-ray range. In the case of classical diffraction, when the wavevector of the incident wave is perpendicular to the z direction, the system breaks up into two independent problems for the two main polarization states, whereas for conical diffraction the boundary values of the z components of the fields and their normal and tangential derivatives are coupled (Goray & Schmidt, 2010. The grating diffracts the incident wave into a finite number of outgoing plane waves, the so-called reflected and, possibly, transmitted modes (orders).…”

Conical diffraction by multilayer gratings: a recursive integral equation approach

“…But off-plane diffraction has not been tackled for a long time, which was one of the real deficiencies of the method. Only recently, in [4], a numerical method for one-profile gratings has been proposed, which solves the integral equations using a hybrid piecewise-trigonometric polynomial collocation method very efficiently, including certain scenarios with unfavorably large ratio period over wavelength and non-smooth profile.…”

“…So the analysis of the recursive algorithm, involving the inversion of operator matrices is performed similarly to [11]. Moreover, the discretization methods from [4] can be used for the numerical realization of the algorithm. Although the inversion of discretization matrices is required, the actual demand for computer memory is comparatively small, which makes the conical diffraction problem tractable with standard PC even for a large number of layers.…”