Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations

In this paper, consideration is given to how aliasing errors, introduced when evaluating nonlinear products, inexactly affect the solution of Galerkin spectral/hp element polynomial discretisations on triangles. A theoretical discussion is presented of how aliasing errors are introduced by a collocation projection onto a set of quadrature points insufficient for exact integration, and consider interpolation projections to geometrically symmetric ollocation points. The discussion is corroborated by numerica examples that elucidate the key features. The study is first motivated with a review of aliasing errors introduced in one-dimensional spectral-element methods (these results extend naturally to tensor-product quadrilaterals and hexahedra.) Within triangular domains two commonly used expansions are a hierarchical, or modal, expansion based on a rotationally non-symmetric collapsed-coordinate system, and a Lagrange expansion based on a set of rotationally symmetric nodal points. Whilst both expansions span the same polynomial space, the construction of the two bases numerically motivates a different set of collocation points for use in the collocation projection of a nonlinear product. The purpose of this paper is to compare these two collocation projections. The analysis and results show that aliasing errors produced using a collocation projection on the rotationally non-symmetric, collapsed-coordinate system are significantly smaller than those for a collocation projection using the rotationally symmetric nodal points. In the case of the collapsed coordinate projection, if the Gaussian quadrature order employed is less than half the polynomial order of the integrand, then it is possible for the aliasing error to modify the constant mode of the expansion and therefore affect the conservation property of the approximation. However, the use of a collocation projection onto a polynomial expansion associated with a set of rotationally symmetric nodal points within the triangle is always observed to be non-conservative. Nevertheless, the rotationally symmetric collocation will maintain the overall symmetry of the triangular region, which is not typically the case when a collapsed coordinate quadrature projection is used.     

Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence

The influence of birefringence caused by rotationally symmetric stress distribution in a high-resolution projection optical system is investigated. The general form of the pupil function is derived based on the Jones matrix calculation, expressing the wave front as a combination of the two orthogonal polarization components. Assuming a linearly polarized incident beam, it is found that the main polarization portion of the wave front exiting the projection lens has astigmatic aberration in the Seidel region and shows phase singularity at four pupil points at which the amplitude transmittance becomes zero.     

Nonnull testing of rotationally symmetric aspheres: a systematic error assessment

An optical setup for the testing of rotationally symmetric aspheres without a null optic is proposed. The optical setup is able to transfer the strongly curved wave fronts that stem from the reflection of a spherical testing wave front at a rotationally symmetric asphere. By simulation it is proved that the algorithms of the Shack-Hartmann sensor that is used can cope with the steep wave-front slopes (~110lambda/mm) in the detection plane. The systematic errors of the testing configuration are analyzed and separated. For all types of error, functionals are derived whose significance is proved by simulation. The maximum residual errors in the simulations are fewer than lambda/500 (peak to valley).     

Air-Coupled Transmission Coefficient Reconstruction Using a 3-D Complex-Transducer-Point Voltage Model

Results are presented on the reconstruction of the transmission coefficient in air-coupled ultrasound experiments where the excitation is either a Gaussian sheet beam or a rotationally symmetric Gaussian beam. The effects of nonideal limitations in the method are also reported. The angular spectrum of the transmitted signal is obtained by performing a spatial Fourier transform on data acquired in a coordinate scan of the receiving transducer, an operation that yields, in the 2-D case, exactly the plane-wave transmission coefficient. This equality is shown using an analysis based on the complex transducer point technique. The transmission function reconstructed from a 3-D rotationally symmetric Gaussian beam differs only slightly, in isotropic media, from the plane-wave transmission coefficient. In addition, the influence of a finite coordinate scan and finite step size are also studied. As a demonstration, elastic constants of isotropic and anisotropic materials are obtained from reconstructed transmission functions by inverting the experimental data. The reconstructed results are compared with independent measurements.     

Polarization aberrations. 2. Tilted and decentered optical systems

In paper 1 [Appl. Opt. 33, this issue (1994)] we examined the polarization aberrations of rotationally symmetric systems. In this paper we extend polarization-aberrration theory to include two types of tilted and decentered systems composed of rotationally symmetric elements. One type is systems with collinear centers of curvatures but with decentered pupils. The symmetry in such systems permits the analysis to proceed along lines similar to those in paper 1. The other type is systems with arbitrary tilts and decenters. In these systems the field dependencies of the aberrations from each surface are not concentric. The extension is made by use of a polarization-aberration matrix with vector, instead of scalar, arguments.     

Test optics error removal

Wave-front or surface errors may be divided into rotationally symmetric and nonrotationally symmetric terms. It is shown that if either the test part or the reference surface in an interferometric test is rotated to N equally spaced positions about the optical axis and the resulting wave fronts are averaged, then errors in the rotated member with angular orders that are not integer multiples of the number of positions will be removed. Thus if the test piece is rotated to N equally spaced positions and the data rotated back to a common orientation in software, all nonrotationally symmetric errors of the interferometer except those of angular order kNθ are completely removed. It is also shown how this method may be applied in an absolute test, giving both rotationally symmetric and nonsymmetric components of the surface. A general proof is given that assumes only that the surface or wave-front information can be described by some arbitrary set of orthognal polynomials in a radial coordinate r and terms in sin θ and cos θ. A simulation, using Zernike polynomials, is also presented.     

Limits of applicability of the model of paired forms in the investigation of the vibrations of rotationally symmetric systems

The article investigates the accuracy of application of the model of paired forms of vibrations for the calculation of resonance vibrations of rotationally symmetric systems with an asymmetry. It shows the effectiveness of the use of this model of paired forms of vibrations in the calculation of rotationally symmetric systems of the type of turbine rotors for the values of the parameters of the system most frequently encountered in practice.Translated from Problemy Prochnosti, No. 5, pp. 106–109, May, 1990.     

Image field distribution model of wavefront aberration and models of distortion and field curvature

The distribution model of wavefront aberrations, which takes on a significant role in the designs and alignments of imaging optical systems without vignetting, is newly presented. This model decomposes the complicated distributions into the characteristic components, which clarifies the alignment criteria. For the actual alignments, only small displacements (decentering, tilt, and surface distance) of rotationally symmetric surfaces in the system are assumed. Then, the model, which regards the aberration distributions of the system as the sum of the contributions of each surface, is extended for the system with surface displacements. As a result of the derivation, it is concluded that the aberration distributions in the rotationally nonsymmetric systems can be expressed as the sum of several folds of rotationally symmetric components. In addition, it is presented that, based on this model, suitable distribution models, even of the arbitrary higher order, can be constructed for any aberration coefficients in various optical systems.     

Weak formulation of Biot's equations in cylindrical coordinates with harmonic expansion in the circumferential direction

A weak symmetric form of Biot's equation in cylindrical coordinates with a spatial Fourier expansion in the circumferential direction is presented. The solid phase displacement and the pore pressure are used as the dependent variables. The original three‐dimensional boundary value problem is here, due to the orthogonality of the harmonic functions and the rotationally symmetric geometry, decomposed into independent two‐dimensional problems, one for each harmonic function. This formulation provides a computationally efficient procedure for vibroacoustic finite element modelling of rotationally symmetric three‐dimensional multilayered structures including porous elastic materials. By numerical simulations, this method is compared with, and verified against, full three‐dimensional Cartesian coordinate system finite element models. Copyright © 2009 John Wiley & Sons, Ltd.     

Rotational symmetry vs. axisymmetry in shell theory

This paper treats rotationally symmetric motions of nonlinearly axisymmetric shells that can suffer transverse shear as well as the usual flexure and base-surface extension and shear. It derives the governing equations in a convenient form and determines their mathematical structure. The complicated governing equations have the same virtue of the far simpler equations for axisymmetric motions that there is but one independent spatial variable. Consequently the constitutive equations enjoy convenient monotonicity properties. Besides deriving these equations, the main purposes of the paper are (i) to give a numerical illustration of the richness of rotationally symmetric motions caused by the coupling of the additional shearing modes with classical modes of deformation, and (ii) to discuss the subtle question of nonexistence of general axisymmetric motions of axisymmetric shells. The paper briefly treats spatially autonomous motions, which are governed by ordinary differential equations in time.     

Parametric characterization of rotationally symmetric hard-edged diffracted beams

The truncated second-order moments and generalized M2 factor (M(G)2 factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the M(G)2 factor take forms similar to those for the nontruncated case. The closed-form expression for the M(G)2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter beta and beam order N. For N --> infinity, the M(G)2 factor equals 4/square root of 3 corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.     

Exact eigenvalue calculations for structures with rotationally periodic substructures

The theory presented enables rotationally periodic (i.e. cyclically symmetric) three-dimensional substructures to be included when using existing algorithms to ensure that no eigenvalues are missed when the stiffness matrix method of structural analysis is used, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. A substructure can be connected in any required way to a parent structure which shares its rotational periodicity, or can be connected by nodes at each end of its axis of periodicity to any parent structure, i.e. the parent structure need not be periodic. The theory uses complex arithmetic, involves only one of the rotationally repeating portions of the substructure, allows nodes and members to coincide with the axis of rotational periodicity, permits efficient multi-level use of rotationally periodic substructures, and gives ‘exact’ eigenvalues if the member equations used are those obtained by solving appropriate differential equations. The competitiveness of the method is illustrated by approximate predictions of computation times and savings for two structures which contain rotationally periodic substructures.     

Visual instrument image-quality assessment with rotationally symmetric difference of Gaussians

We performed contrast-sensitivity measurements using rotationally symmetric, variable contrast difference-of-Gaussians targets, viewed through specially designed telescopes that presented various amounts of monochromatic aberrations. The contrast-sensitivity ratios between aberrated and unaberrated cases were correlated with instrument-based measures of image quality. The radius that encircles 84% of the energy of the point-spread function was found to give the best correlation with the subjective data.     

Generation of large-diameter diffractive elements with laser pattern generation

We describe the design and construction of a high-precision laser writing machine for the direct generation of large-diameter rotationally symmetric diffractive optics with continuous profiles in photoresist. The photoresist profile can be used as a replication master surface or etched into a silica substrate. Machine design methodology, as well as qualification of performance, is provided. Test results for an f/2 100-mm clear-aperture diffractive lens directly etched into a silica substrate are presented. Diffraction efficiency as a function of zone spacing and wave-front performance are given.     

Reference surface calibration of a Fizeau interferometer through even/odd synthesis

We propose an even-/odd-synthesis method for the elimination of additional aberration caused by misalignment or environmental vibration during the calibration of a Fizeau interferometer reference surface (RS). The odd and even parts of an RS can be obtained, because surface errors could be divided into rotationally symmetric and nonrotationally symmetric terms. We then propose a least-squares algorithm with a dual-objective optimization function for calibration of the measurement results at the confocal position. Finally, a complete RS can be eventually obtained by synthesizing the odd and even parts of the RS. It has been verified through experiments that the measurement repeatability of the PV value is better than 0.003λ, and the root-mean-square value is better than 0.0003λ.     

Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference

We have already reported a method for the quasi-absolute test of rotationally symmetric aspheres by means of combined diffractive optical elements (combo-DOEs). The combo-DOEs carry the information for the ideal shape of an aspheric surface under test as well as a spherical wave for the measurement at the cat's eye position. An experimental demonstration of the procedure is given. Measurements with two different designs of combo-DOEs have been conducted, and their relative advantages and disadvantages are discussed.     

Exact Calculation of the Circle of Least Confusion of a Rotationally Symmetric Mirror. II

We compute the radius and the position of the center of the circle of least confusion, in an exact way and by using the third-order approximation, of a rotationally symmetric mirror when the point source is located at any position on the optical axis. For the spherical case we find that for some positions of the point source there is a considerable difference between the exact computations and those obtained by working up to third-order aberrations.     

Design and scaling of monocentric multiscale imagers

Monocentric multi-scale (MMS) lenses are a new approach to high-resolution wide-angle imaging, where a monocentric objective lens is shared by an array of identical rotationally symmetric secondary imagers that each acquire one overlapping segment of a mosaic. This allows gigapixel images to be computationally integrated from conventional image sensors and relatively simple optics. Here we describe the MMS design space, introducing constraints on image continuity and uniformity, and show how paraxial system analysis can provide both volume scaling and a systematic design methodology for MMS imagers. We provide the detailed design of a 120° field of viewimager (currently under construction) resolving 2 gigapixels at 41.5 μrad instantaneous field of view, and demonstrate reasonable agreement with the first-order scaling calculation.     

Reflectors for uniform far-field irradiance: fundamental limits and example of an axisymmetric solution

We establish a fundamental bound on the field of view over which strictly uniform far-field irradiance can be achieved in symmetric two-dimensional (2D, troughlike) and three-dimensional (3D, conelike) illumination systems. Earlier results derived for particular 2D devices are shown to be special cases of the general formula. For a rotationally symmetric 3D luminaire with a Lambertian disk light source and a prescribed uniform core region half-angle theta(c), no more than tan(2)(theta(c)) can be projected within a uniform core region. Hence the efficiency with which such illuminators can produce uniform flux is severely limited for many problems of practical interest. Guided by the tailored edge-ray device formalism for the design of 2D luminaires, we develop a 3D reflector that produces extremely uniform far-field illuminance.     

Removing static aberrations from the active optics system of a wide-field telescope

The wavefront sensor in active and adaptive telescopes is usually not in the optical path toward the scientific detector. It may generate additional wavefront aberrations, which have to be separated from the errors due to the telescope optics. The aberrations that are not rotationally symmetric can be disentangled from the telescope aberrations by a series of measurements taken in the center of the field, with the wavefront sensor at different orientation angles with respect to the focal plane. This method has been applied at the VLT Survey Telescope on the ESO Paranal observatory.