L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” Ch.# 12 in Gratings: Theory and Numeric Applications, Second Revisited Edition, E. Popov, ed. (Institut Fresnel, AMU, 2014)

 

This work is part of research that has been pursued by the authors over a long period of time for the purpose of developing accurate and fast numerical algorithms, including the commercial packages PCGrate and DiPoG [12.1, 12.2] designed to model multilayered gratings having mostly one-dimensional periodicity (1D), including roughness, and working in all, including the shortest, optical wavelength ranges at arbitrary optical mounts.

The boundary integral equation theory or, briey, integral method (IM) is presently universally recognized as one of the most developed and exible approaches to an accurate numerical solution of diffraction grating problems (see, e.g., Ref. 12.3 and Ch. 4 and references therein). Viewed in the historical context, this method was the rst to offer a solution to vector problems of light diffraction by optical gratings and to demonstrate remarkable agreement with experimental data. This should be attributed to the high accuracy and good convergence of the method, especially for the TM polarization plane. It does not involve limitations similar to those characteristic of the Coupled-Wave Analysis (CWA), and it provides a better convergence. The disadvantages of this method include its being mathematically complicated, as well as numerous “peculiarities” involved in numerical realization. In particular, quasi-periodic Greens functions and their derivatives appearing as kernels in the integral operators require sophisticated lattice sum techniques to evaluate. Moreover, application of the IM to cases of heterogeneous or anisotropic media meets with difculties; however, with the volume integral method it is possible to overcome these difculties easily. Nevertheless, it is on the basis of this theory that all the well-known problems of diffraction by periodic and non-periodic structures in optics and other elds have been solved. In many cases it offers the only possible way to follow up in research. The exibility and universality inherent in the IM, in particular, enable one rather easily to reduce the problem of radiation of Gaussian waves or of a localized source to that of plane-wave incidence, for which scientists all over the world have a set of numerical solutions. Generalizations of the IM have recently been proposed for arbitrarily proled 1D multilayer gratings [12.4], randomly-rough x-ray-extreme-ultraviolet (EUV) gratings and mirrors [12.5, 12.6], conical diffraction gratings including materials with a negative permittivity and permeability (metamaterials) [12.7, 12.8], bi-periodic anisotropic structures using a variation formulation [12.9], Fresnel zone plates and diffraction optical elements [12.10, 12.11], and two-dimensional (2D) [12.12, 12.13] and three-dimensional (3D) [12.14] photonic crystals (inclusions) of some geometries, among others.

The IM is so pivotal that one can indicate the few areas where it can be modied and improved to solve particular diffraction problems. By convention they are: (1) physical model—choice of boundary types, boundary conditions, layer and substrate refractive indices, and radiation conditions; (2) mathematical structure—integral representations using potentials or integral formulas and a multilayer scheme; (3) method of approximation and discretization—discretization schemes, choice of basis (trial) and test (weighting) functions, and treatment of coincident points and corners in boundary prole curves; (4) low-level details—calculations and optimization of kernel functions, mesh of discretization (collocation) points, quadrature rules, and solution of linear algebraic systems; (5) implementation enhancements—memory caching, other implementation details. A self-consistent explanation of the existing IMs is beyond the primary scope of the present study. The main purpose of this Chapter is to present a complete description in general operator form of the two IMs applied to 1D multilayer gratings working in conical diffraction mounts and in short waves. Our study also includes the calculus of grating absorption in the explicit form and scattering intensity of randomly-rough gratings using Monte Carlo simulations. For other formal IM treatments and their comparisons, one should rather look to the references of this Chapter as well as to Ch. 4 and to references therein.

Various kinds of electromagnetic features of different nature can exist and be explored in complex grating structures: Bragg and Brewster resonances, Rayleigh anomalies and groove shape features, waveguiding and Fano-type modes, etc. In conical diffraction, the inuence of possible types of waves can be mixed. For the purposes of this Chapter, we chose three important types, among many others, of diffraction grating problems to include them in Section 12.9 “Examples of numerical results”. They are: bare dielectric or metallic gratings of standard groove shapes working in conical diffraction in the resonance domain; shallow high-conductive or dielectric gratings of various boundary shapes, including closed ones, working in different mounts and supporting polariton-plasmon excitation or Bragg diffraction in the visible-infrared range; bare and multilayer gratings working in grazing-conical or near-normal in-plane diffraction in the soft x-ray-EUV range.

 

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