17 results
Search Results
Now showing 1 - 10 of 17
- ItemBenutzer-Handbuch DIPOG-1.4(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2004) Schmidt, GuntherThis is the manual of the software package DIPOG, version 1.4, which can be used to simulate and optimize binary and multilevel optical gratings. The algorithms are based on the finite--element solution of a system of Helmholtz equations, which are equivalent to the timeharmonic electromagnetic field equations, and on gradient methods for solving optimization problems. The package offers several options to postprocess the calculated electromagnetic fields.
- ItemApproximation of solutions to multidimensional parabolic equations by approximate approximations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lanzara, Flavia; Mazya, Vladimir; Schmidt, GuntherWe propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions.We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions.
- ItemAnalytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bugert, Beatrice; Schmidt, GuntherThis paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a 2π-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
- 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.
- ItemOn a fast integral equation method for diffraction gratings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Rathsfeld, Andreas; Schmidt, Gunther; Kleemann, BerndThe 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 ratios 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 application we present the simulation of coated echelle gratings which demonstrates the efficency of our approach.
- 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.
- ItemChirped photonic crystal for spatially filtered optical feedback to a broad-area laser(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Brée, Carsten; Gailevicius, Darius; Purlys, Vytautas; Werner, Guillermo Garre; Staliunas, Kestutis; Rathsfeld, Andreas; Schmidt, Gunther; Radziunas, MindaugasWe derive and analyze an efficient model for reinjection of spatially filtered optical feedback from an external resonator to a broad area, edge emitting semiconductor laser diode. Spatial filtering is achieved by a chirped photonic crystal, with variable periodicity along the optical axis and negligible resonant backscattering. The optimal chirp is obtained from a genetic algorithm, which yields solutions that are robust against perturbations. Extensive numerical simulations of the composite system with our optoelectronic solver indicate that spatially filtered reinjection enhances lower-order transversal optical modes in the laser diode and, consequently, improves the spatial beam quality.
- 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.