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- ItemDiffraction of stochastic point sets : exactly solvable examples(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Baake, Michael; Birkner, Matthias; Moody, Robert V.Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.
- ItemSolving conical diffraction with integral equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Goray, Leonid I.; Schmidt, GuntherOff-plane scattering of time-harmonic plane waves by a diffraction grating with arbitrary conductivity and general border profile is considered in a rigorous electromagnetic formulation. The integral equations for conical diffraction were obtained using the boundary integrals of the single and double layer potentials including the tangential derivative of single layer potentials interpreted as singular integrals. We derive an important formula for the calculation of the absorption in conical diffraction. Some rules which 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 surfaces, borders with edges, real border profiles, and gratings working at short wavelengths.
- ItemIntegral equations for conical diffraction by coated gratings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Schmidt, GuntherThe paper is devoted to integral formulations for the scattering of plane waves by diffraction gratings under oblique incidence. For the case of coated gratings Maxwell's equations can be reduced to a system of four singular integral equations on the piecewise smooth interfaces between different materials. We study analytic properties of the integral operators for periodic diffraction problems and obtain existence and uniqueness results for solutions of the systems corresponding to electromagnetic fields with locally finite energy.
- ItemElectromagnetic scattering by biperiodic multilayered gratings: A recursive integral equation approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bugert, Beatrice; Schmidt, GuntherIn this paper, we propose a new recursive integral equation algorithm to solve the direct problem of electromagnetic scattering by biperiodic multilayered structures with polyhedral Lipschitz regular interfaces. We work with a combined potential approach that involves one unknown density on each of the grating profiles of the multilayered scatterer. Justified by the transmission conditions of the underlying electromagnetic scattering problem, we assume that densities in adjacent layers are linearly linked by a boundary integral operator and derive a recursion for these densities. It comprehends the inversion of one boundary integral equation on each scattering interface. Our algorithm is shown to be equivalent to the biperiodic multilayered electromagnetic scattering problem. Moreover, we obtain new existence and uniqueness results for our recursive integral equation algorithm, which promises to lead to an efficient numerical implementation of the considered scattering problem. These solvability results depend on the regularity of the grating interfaces and the values of the electromagnetic material parameters of the biperiodic multilayered structure at hand.
- ItemConical diffraction by multilayer gratings : a recursive integral equations approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Schmidt, GuntherIn this paper we consider an integral equation algorithm to study the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in $R^2$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with $2 times 2$ operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.
- ItemIntegral methods for conical diffraction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Schmidt, GuntherThe paper is devoted to the scattering of a plane wave obliquely illuminating a periodic surface. Integral equation methods lead to a system of singular integral equations over the profile. Using boundary integral techniques we study the equivalence of these equations to the electromagnetic formulation, the existence and uniqueness of solutions under general assumptions on the permittivity and permeability of the materials. In particular, new results for materials with negative permittivity or permeability are established.
- ItemScattering of general incident beams by diffraction gratings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Schmidt, GuntherThe paper is devoted to the electromagnetic scattering of arbitrary time-harmonic fields by periodic structures. The Floquet-Fourier transform converts the full space Maxwell problem to a twoparameter family of diffraction problems with quasiperiodic incidence waves, for which conventional grating methods become applicable. The inverse transform is given by integrating with respect to the parameters over a infinite strip in R2. For the computation of the scattered fields we propose an algorithm, which extends known adaptive methods for the approximate calculation of multiple integrals. The novel adaptive approach provides autonomously the expansion of the incident field into quasiperiodic waves in order to approximate the scattered fields within a prescribed error tolerance. Some application examples are numerically examined.