Shape derivatives for the scattering by biperiodic gratings

Usually, 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 $H( m curl)$ 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

On a half-space radiation condition

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

On the Stability of Piecewise Linear Wavelet Collocation and the Solution of the Double Layer Equation over Polygonal Curves

In this paper we consider a piecewise linear collocation method for the solution of strongly elliptic operator equations over closed curves. The trial space is a subspace of the space of all piecewise linear functions defined over a uniform grid. This space is spanned by an arbitrary subset of the biorthogonal wavelet basis. To the subspace in the trial space there corresponds a natural subspace in the space of test functionals. This subspace is spanned by certain linear combinations of the Dirac delta functionals taken at the uniformly distributed grid points. For the resulting wavelet collocation method and a strongly elliptic operator equation, we prove stability and convergence. In particular, this general result applies to the double layer equation over a polygonal curve. We show that the wavelet collocation method with piecewise linear trial functions over a uniform grid converges with order O(n-2), where n is the number of degrees of freedom. Note that the step size of the underlying uniform partition is n-ɑ, ɑ ≥ 1. The stiffness matrix for the wavelet collocation method can be compressed to a matrix containing no more than O(n log n) non-zero entries such that the asymptotic convergence order is not effected

A Wavelet Algorithm for the Boundary Element Solution of a Geodetic Boundary Value Problem

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 become fast. 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

Simulating rough surfaces by periodic and biperiodic gratings

The scattering of acoustic and electro-magnetic plane waves by rough surfaces is the subject of many books and papers. For simplicity, we consider the special case, described by a Dirichlet boundary value problem of the Helmholtz equation in the half space above the surface. We recall the formulae of the far-field pattern and the far-field intensity. The far-field can be defined formally for general rough surfaces. However, the derivation as asymptotic limits works only for waves, which decay for surface points tending to infinity. Comparing with the case of periodic surface structures, it is clear that the rigorous model of plane-wave scattering is accurate for the near field close to the surface. For the far field, however, the finite extent of the beams in the planes orthogonal to the propagation direction is to be taken into account. Doing this rigorously, leads to extremely expensive computations or is simply impossible. Therefore and to enable the approximation of waves above the rough surface by waves above periodic and biperiodic rough structures, we consider a simplified model of beams. The beam is restricted to a cylindrical domain around a ray in propagation direction, and the wave is equal to a plane wave inside of this domain and to zero outside. Based on this beam model, we derive the corresponding asymptotic formulae for the wave and its intensity. The intensity is equal to the formally defined far-field intensity multiplied by a simple cosine factor. Under special assumptions, the intensity for the rough surface can be approximated by that for rough periodic and biperiodic surface structures. In particular, we can cope with the case of shallow roughness, where the reflected intensity includes, besides the smooth density function w.r.t. the angular direction, a plane-wave beam propagating into the reflection direction of the planar mirror. Altogether, the main point of the paper is to fix the technical assumptions needed for the far-field formula of a simple beam model and for the approximation by the far fields of periodized rough surfaces. Furthermore, using the beam model, we discuss numerical experiments for rough surfaces defined as realizations of a random field and, to get a more practical case, the Dirichlet condition is replaced by a transmission condition. The far-field intensity function for a rough surface is the limit of intensity functions for periodized rough surfaces if the period tends to infinity. However, almost the same intensity function can be obtained with a fixed period by computing the average over many different realizations of the random field. Finally, we present numerical results for an inverse problem, where the parameters of the random field are sought from measured mean values of the intensities

Edge Asymptotics for the Radiosity Equation over Polyhedral Boundaries

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

A quadrature algorithm for wavelet Galerkin methods

We consider the wavelet Galerkin method for the solution of boundary integral equations of the first and second kind including integral operators of order r less than zero. This is supposed to be based on an abstract wavelet basis which spans piecewise polynomials of order dT. For example, the bases can be chosen as the basis of tensor product interval wavelets defined over a set of parametrization patches. We define and analyze a quadrature algorithm for the wavelet Galerkin method which utilizes Smolyak quadrature rules of finite order. In particular, we prove that quadrature rules of an order larger than 2dT - r are sufficient to compose a quadrature algorithm for the wavelet Galerkin scheme such that the compressed and quadrature approximated method converges with the maximal order 2dT - r and such that the number of necessary arithmetic operations is less than 풪(N log N) with N the number of degrees of freedom. For the estimates, a degree of smoothness greater or equal to 2[2dT - r]+1 is needed

A Wavelet Algorithm for the Solution of a Singular Integral Equation over a Smooth Two-dimensional Manifold

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

Convergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures

Scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface structure in \R^3 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

Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings

Consider 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 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