On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

A comparison-based diagnosis algorithm tailored for crossed cube multiprocessor systems

Comparison-based diagnosis is an effective approach to system-level fault diagnosis. Under the Maeng-Malek comparison model (NM* model), Sengupta and Dahbura proposed an O(N-5) diagnosis algorithm for general diagnosable systems with N nodes. Thanks to lower diameter and better graph embedding capability as compared with a hypercube of the same size, the crossed cube has been a promising candidate for interconnection networks. In this paper, we propose a fault diagnosis algorithm tailored for crossed cube connected multicomputer systems under the MM* model. By introducing appropriate data structures, this algorithm runs in O(Nlog(2)(2) N) time, which is linear in the size of the input. As a result, this algorithm is significantly superior to the Sengupta-Dahbura's algorithm when applied to crossed cube systems. (C) 2004 Elsevier B.V. All rights reserved.

Astigmatism in tapered slot antennas

A series of scale model measurements of transverse electromagnetic mode tapered slot antennas are presented. They show that the beam launched by this type of antenna is astigmatic. It is shown how an off-axis spherical mirror can be used to correct this astigmatism to allow efficient coupling to quasi-optical systems. A millimetre wave antenna and mirror combination is described and, with the aid of solid state noise diodes, the coupling of the launched beam to a quasi-optical spectrometer is shown to be in good agreement with that predicted by the scale model measurements.

Local gratings due to angular hole burning in a photorefractive polymer

We investigate the processes involved in writing real-time holographic gratings in a photorefractive polymer (PRP) that incorporates an azo-dye. In such systems there may be gratings due to mechanisms associated with trans–cis isomerization (angular hole burning (AHB) and/or angular redistribution), which appear in addition to those arising from the photorefractive (PR) effect. The work presented here helps to understand the interactions which may occur between these different gratings. The formation of local gratings due to mechanisms associated with photoisomerization is studied, in a new PRP based on the photoconductor poly(N-vinylcarbazole):2, 4, 7-trinitro-9-fluorenone, plasticized with N-ethylcarbazole. The polymer includes the azo-dye 4-nitro-4'-pentyloxy-azobenzene and we observe both PR and photoisomerization gratings. The gratings are shown to be both polarization-sensitive and reversible. The presence of the photoisomerization gratings (which diffract almost as strongly as the PR gratings) significantly affects the field-dependent diffractive behaviour of the composite. A measurement of the lifetime of the cis state is made (τcis = 38 s) using photoinduced dichroism. This is close to the decay time constant of the local gratings (τdecay = 42 s), and it is suggested that the local grating mechanism is AHB of the azo-dye. This is the first time (to the knowledge of the authors) that a local grating due to AHB has been demonstrated in a PRP.

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.