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Modern high-power broad-area semiconductor laser diodes (BASLDs) deliver optical output powers of several ten Watts at high electro-optical conversion efficiencies, which makes them highly relevant for numerous industrial, medical and scientific applications. However, lateral multimode behavior in BASLDs due to thermal lensing turns out highly detrimental, as it results in poor focusability and decreased laser beam brightnesss. Approaches to overcome this issue include improved epitaxial layer design, the optimization of evanescent spatial filtering by tailoring the emitter geometry and facet reflectivity, or Fourier spatially filtered reinjection from an external resonator [1].

To satisfy the continuous demand of ever smaller feature sizes, plasma etching technologies in microelectronics enable the fabrication of device structures in the nanometer range. To control these processes, real-time access to the structure’s dimensions is needed. We develop a special method of optical critical dimension metrology and evaluate the feasibility of reconstructing the etched dimensions from experimental reflectivity spectra of the surface, taken about every second. For a periodic 2D model structure etched into a silicon, we develop and test a fast algorithm. To reduce the computing time, we generate a library of spectra before the etching. 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.

Consider 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 Lame constants. This article 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 quasiperiodic 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 and the convergence analysis is performed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water‐brass and water‐brass‐water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000‐000, 2015

Consider the scattering of time-harmonic electromagnetic 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 the material below is perfectly conducting. This paper is concerned with the existence of quasiperiodic solutions for any frequency. 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. The only assumption is that the grating profile is a Lipschitz biperiodic surface. Note that the solvability result of the present paper covers the resonance case where Rayleigh frequencies are allowed. Finally, non-uniqueness examples are presented in the resonance case and in the case of TE or TM polarization for classical gratings.

We consider the interaction between an elastic body and a compressible inviscid fluid, which occupies the unbounded exterior domain. The inverse problem of determining the shape of such an elastic scatterer from the measured far field pattern of the scattered fluid pressure field is of central importance in detecting and identifying submerged objects. Following a method proposed by Kirsch and Kress, we approximate the acoustic and elastodynamic wave by potentials over auxiliary surfaces, and we reformulate the inverse problem as an optimization problem. The objective function to be minimized is the sum of three terms. The first is the deviation of the approximate far field pattern from the measured one, the second is a regularization term, and the last a control term for the transmission condition. We prove that the optimization problem has a solution and that, for the regularization parameter tending to zero, the minimizers tend to a solution of the inverse problem. In contrast to a numerical method from...

We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gauss quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with O(n^2[log n]^2) resp. O(n^2) operations, where n is the dimension of the polynomial trial space.

In this paper, we consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, −1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]3) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N−(2−r)/2). Note that, in contrast to well-known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderon–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a part of the surface of the earth. This singular equation is the boundary integral equation corresponding to the oblique derivative boundary problem for Laplace's equation. We introduce special wavelet bases for the spaces of test and trial functions. Analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be reduced to sparse matrices such that the assembling of the matrices and the iterative solution of the matrix equations quicken. Though the theoretical results apply only to integral equations with ‘smooth’ solutions over ‘smooth’ manifolds, we present numerical tests for a geometry as difficult as the surface of the earth.

In this paper we consider a quadrature method for the numerical solution of a second-kind integral equation over the interval, where the integral operator is a compact perturbation of a Mellin convolution operator. This quadrature method relies upon a singularity subtraction and transformation technique. Stability and convergence order of the approximate solution are well known. We shall derive the first term in the asymptotics of the error which shows that, in the interior of the interval, the approximate solution converges with higher order than over the whole interval. This implies higher orders of convergence for the numerical calculation of smooth functionals to the exact solution. Moreover, the asymptotics allows us to define a new approximate solution extrapolated from the dilated solutions of the quadrature method over meshes with different mesh sizes. This extrapolated solution is designed to improve the low convergence order caused by the non-smoothness of the exact solution even when the transformation technique corresponds to slightly graded meshes. Finally, we discuss the application to the double-layer integral equation over the boundary of polygonal domains and report numerical results.

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, 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 where 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 such as 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 surfaces. Copyright © 2013 John Wiley & Sons, Ltd.

Light diffraction by biperiodic grating structures can be simulated by a boundary value problem of the equation ∇ × ∇ × u − k 2 u = 0 for the electric field u . To optimize the geometry parameters of the grating, a quadratic functional of u is defined. The minimization of this functional by gradient based optimization schemes requires shape derivatives of the functional with respect to the geometry parameters. However, a simple application of classical shape calculus is not possible since the energy space for the electric fields is not invariant with respect to the transformation of geometry. In a recent paper, Hettlich (2012) [15] has proposed to replace the electric field by a simple transform which leads to a differentiable vector field in the energy space. We follow here a different approach. For constant magnetic permeability, the magnetic field is piecewise in [ H 1 ] 3 . Applying the shape calculus to the magnetic field equation, substituting the magnetic field by the curl of the electric field, and employing some technical transformations, we derive stable formulas for the material derivatives depending on the electric field. Numerical tests confirm the formulas.

We consider the double layer potential operator W defined on the polyhedral boundary of an infinite cone and prove the invertibility of (I±2W) in the space of continuous functions. To do this we define an operator-valued symbol function for W and show that the spectral radii of its values are less than one half. In the last part of this paper we consider a piecewise constant collocation method for the numerical solution of the double layer potential equation over the boundary of a bounded polyhedron.

Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the domain occupied by the obstacle with the Helmholtz equation in the exterior. The far field pattern is obtained applying an integral operator to the scattered field function restricted to a simple smooth surface surrounding the obstacle. The subject of our paper is the inverse problem, where the shape of the elastic body represented by a parametrization of its boundary is to be reconstructed from a finite number of measured far field patterns. We define a family of objective functionals depending on a non-negative regularization parameter such that, for regularization parameter zero, the shape of the sought elastic obstacle is a minimizer of the functional. For any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem in its weak formulation may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.

Resist-surface bound air bubbles have been identified as a possible defect mechanism in immersion lithography. The general expectation is that the bubble will primarily cause local dose reductions, but no detailed simulations on this effect have been published. The work described in this paper is a first attempt to do so: we have simulated the effect of bubbles on 1:1 dense Line/Space patterning. Our results confirm that the major effect of the presence of a bubble is indeed underexposure - or in most cases even non-exposure - of the pattern in the area occupied by the bubble, but it also identifies a few more subtle characteristics of bubble-induced defects which can help identify defects observed on immersion wafers as being caused by a bubble. Apart from the simulation results, we also show a few experimentally observed immersion defects, which we believe are indeed generated by a bubble.

We consider the inverse diffraction problem to recover a two-dimensional periodic structure from scattered waves measured above and beneath the structure. The task is reformulated in the form of an optimization problem including special regularization terms. The solvability and the dependence on the parameter of regularization is analyzed. Numerical results for synthetic data demonstrate the practicability of the inversion algorithm.

This paper is concerned with the direct and inverse scattering problems in fluid-structure interaction. The scattering problem in the fluid-structure interaction can be simply described as follows: an acoustic wave propagates in the fluid domain of infinite extent where a bounded elastic body is immersed. The direct problem is to determine the scattered pressure and velocity fields in the fluid domain as well as the displacement fields in the elastic body, while the inverse problem is to reconstruct the shape of the elastic scatterer from a knowledge of the far field pattern of the fluid pressure or from the measured scattered fluid pressure field. As is well known, the inverse problems are generally nonlinear and highly ill-posed. For treating inverse problem of this kind, we reformulate the problem as a nonlinear optimization problem including special regularization terms. The precise formulation of the nonlinear objective functional will depend on the approaches of the direct problem. In this paper, the direct problem is reformulated by introducing an artificial boundary and the corresponding inverse problem will be analyzed. Some of the basic results are summarized without proofs. The latter are available in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

To control the design and manufacturing of lithographic masks a fast and non-destructive measurement technique is needed. In scatterometry the mask is illuminated by plane waves from various directions, the scattered light is measured, and the geometry of the mask surface is reconstructed from the measured data. The quality of reconstruction surely depends on the angles of incidence, on the wave lengths and/or the number of propagating scattered wave modes. Optimizing the conditioning of some Jacobian matrices, we develop a sensitivity analysis to find an optimal measurement configuration for scatterometry. Our first important test geometry is a simple periodic line-space structure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double-layer potential equation, the solution of the second kind integral equation with non-compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation.

In the present paper we give an overview on multiscale algorithms for the solution of boundary integral equations which are based on the use of wavelets. These methods have been introduced first by Beylkin, Coifman, and Rokhlin [5]. They have been developed and thoroughly investigated in the work of Alpert [1], Dahmen, Proessdorf, Schneider [16-19], Harten, Yad-Shalom [25], v.Petersdorff, Schwab [33-35], and Rathsfeld [39-40]. We describe the wavelet algorithm and the theoretical results on its stability, convergence, and complexity. Moreover, we discuss the application of the method to the solution of a two-dimensional scattering problem of acoustic or electromagnetic waves and to the solution of a fixed geodetic boundary value problem for the gravity field of the earth. The computational tests confirm the high compression rates and the saving of computation time predicted by the theory.

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a smooth surface. Using a fixed set of parametrizations, we introduce special wavelet bases for the spaces of test and trial functions. The trial wavelets have two vanishing moments only if their supports do not intersect the lines belonging to the common boundary of two subsurfaces defined by different parameter representations. Nevertheless, analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be compressed to sparse matrices such that the iterative solution of the matrix equations becomes fast. Finally, we present a fast quadrature algorithm for the computation of the compressed stiffness matrix.

In this paper we consider a piecewise linear collocation method for the solution of the double layer potential equation corresponding to Laplace's equation over polygonal domains. We give a wavelet algorithm for the computation of the corresponding stiffness matrix and for the solution of the arising matrix equation with no more than O(N·[log N]8) arithmetic operations. The error of the resulting approximate solution is of order O(N-2·[log N]6).Finally, we give some remarks on the generalization of the algorithm to the piecewise cubic collocation and present numerical tests.

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.

This paper is concerned with a quadrature method for singular integral equations on a finite interval. It is proved that the strong ellipticity is necessary and sufficient for this numerical procedure to be stable. The proof of this result is based on the symbol calculus on the algebra generated by Toeplitz matrices due to Gohberg and Krupnik [12].Estimates for the speed of convergence as well as numerical examples are given.