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- 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.
- ItemReflection of plane waves by rough surfaces in the sense of Born approximation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Arnold, Thomas; Rathsfeld, AndreasThe topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, i.e., by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns' formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions if the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms like those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough
- ItemScattering of time harmonic electromagnetic plane waves by perfectly conducting diffraction gratings(Berlin: Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Rathsfeld, AndreasConsider scattering of time-harmonic lectromagnetic plane waves by a doubly periodic surface in R^3. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while below is a perfectly conducting material. This paper is concerned with the existence of quasiperiodic solutions for any frequency of incidence. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves, under the assumption that the grating profile is a Lipschitz biperiodic surface. Our solvability result covers the resonance case where a Rayleigh frequency is allowed. Non-uniqueness examples are also presented in the resonance case and the TE or TM polarization case for classical gratings.
- ItemShape derivatives for the scattering by biperiodic gratings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Rathsfeld, AndreasUsually, the light diffraction by biperiodic grating structures is simulated by the time-harmonic Maxwell system with a constant magnetic permeability. For the optimization of the geometry parameters of the grating, a functional is defined which depends quadratically on the efficiencies of the reflected modes. The minimization of this functional by gradient based optimization schemes requires the computation of the shape derivatives of the functional with respect to the parameters of the geometry. Using classical ideas of shape calculus, formulas for these parameter derivatives are derived. In particular, these derivatives can be computed as material derivatives corresponding to a family of transformations of the underlying domain. However, the energy space $H(rm curl)$ for the electric fields is not invariant with respect to the transformation of geometry. Therefore, the formulas are derived first for the magnetic field vectors which belong to $[H^1]^3$. Afterwards, the magnetic fields in the shape-derivative formula are replaced by their electric counter parts. Numerical tests confirm the derived formulas.
- ItemOn a half-space radiation condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Rathsfeld, AndreasFor the Dirichlet problem of the Helmholtz equation over the half space or rough surfaces, a radiation condition is needed to guarantee a unique solution, which is physically meaningful. If the Dirichlet data is a general bounded continuous function, then the well-established Sommerfeld radiation condition, the angular spectrum representation, and the upward propagating radiation condition do not apply or require restrictions on the data, in order to define the involved integrals. In this paper a new condition based on a representation of the second derivative of the solution is proposed. The twice differentiable half-space Green's function is integrable and the corresponding radiation condition applies to general bounded functions. The condition is checked for special functions like plane waves and point source solution. Moreover, the Dirichlet problem for the half plane is discussed. Note that such a ``continuous'' radiation condition is helpful e.g. if finite sections of the rough-surface problem are analyzed.
- ItemConvergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Rathsfeld, AndreasScattering of time-harmonic electromagnetic plane waves by a doubly periodic surface structure in R3 can be simulated by a boundary value problem of the time-harmonic curl-curl equation. For a truncated FEM domain, non-local boundary value conditions are required in order to satisfy the radiation conditions for the upper and lower half spaces. Alternatively to boundary integral formulations, to approximate radiation conditions and absorbing boundary methods, Huber et al. [11] have proposed a coupling method based on an idea of Nitsche. In the case of profile gratings with perfectly conducting substrate, the authors have shown previously that a slightly modified variational equation can be proven to be equivalent to the boundary value problem and to be uniquely solvable. Now it is shown that this result can be used to prove convergence for the FEM coupled by truncated wave mode expansion. This result covers transmission gratings and gratings bounded by additional multi-layer systems.
- ItemFast scatterometric measurement of periodic surface structures plasma-etching processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Klesse, Wolfgang Matthias; Rathsfeld, Andreas; Groß, Claudine; Malguth, Enno; Skibitzki, Oliver; Zealouk, LahbibTo satisfy the continuous demand of ever smaller feature sizes, plasma etching technologies in microelectronics processing enable the fabrication of device structures with dimensions in the nanometer range. In a typical plasma etching system a plasma phase of a selected etching gas is activated, thereby generating highly energetic and reactive gas species which ultimately etch the substrate surface. Such dry etching processes are highly complex and require careful adjustment of many process parameters to meet the high technology requirements on the structure geometry. In this context, real-time access of the structures dimensions during the actual plasma process would be of great benefit by providing full dimension control and film integrity in real-time. In this paper, we evaluate the feasibility of reconstructing the etched dimensions with nanometer precision from reflectivity spectra of the etched surface, which are measured in real-time throughout the entire etch process. We develop and test a novel and fast reconstruction algorithm, using experimental reflection spectra taken about every second during the etch process of a periodic 2D model structure etched into a silicon substrate. Unfortunately, the numerical simulation of the reflectivity by Maxwell solvers is time consuming since it requires separate time-harmonic computations for each wavelength of the spectrum. To reduce the computing time, we propose that a library of spectra should be generated before the etching process. Each spectrum should correspond to a vector of geometry parameters s.t. the vector components scan the possible range of parameter values for the geometrical dimensions. We demonstrate that by replacing the numerically simulated spectra in the reconstruction algorithm by spectra interpolated from the library, it is possible to compute the geometry parameters in times less than a second. Finally, to also reduce memory size and computing time for the library, we reduce the scanning of the parameter values to a sparse grid.
- ItemComparison of numerical methods for the reconstruction of elastic obstacles from the far-field data of scattered acoustic waves(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hsiao, George C.; Rathsfeld, AndreasWe consider the inverse problem for an elastic body emerged in a fluid due to an acoustic wave. The shape of this obstacle is to be reconstructed from the far-field pattern of the scattered wave. For the numerical solution in the two-dimensional case, we compare a simple Newton type iteration method with the Kirsch-Kress algorithm. Our computational tests reveal that the Kirsch-Kress method converges faster for obstacles with very smooth boundaries. The simple Newton method, however, is more stable in the case of not so smooth domains and more robust with respect to measurement errors.
- ItemModeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Gross, Hermann; Henn, Mark-Alexander; Heidenreich, Sebastian; Rathsfeld, Andreas; Bär, MarkusWe investigate the impact of line edge and line width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The 2D rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method (FEM) providing a numerical solution of the two-dimensional Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. ...
- ItemFinite element method to fluid-solid interaction problems with unbounded periodic interfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hu, Guanghui; Rathsfeld, Andreas; Yin, TaoConsider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.