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search.filters.applied.f.access: openAccess × Author: Rathsfeld, Andreas × Start date: 2018 × WGL Institute: WIAS ×

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    Radiation conditions for the Helmholtz equation in a half plane filled by inhomogeneous periodic material
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hu, Guanghui; Rathsfeld, Andreas
    In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method.
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    Chirped 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, Mindaugas
    We 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.
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    Fast 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, Lahbib
    To 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.
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    On a half-space radiation condition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Rathsfeld, Andreas
    For 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.
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    PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Yu, Xiuchen; Hu, Guanghui; Lu, Wangtao; Rathsfeld, Andreas
    This paper studies the perfectly-matched-layer (PML) method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges to the true solution in the physical subregion of the strip with an error bounded by the reciprocal PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem over the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot converge exponentially to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip.