We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations.

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The SIESTA method for ab initio order-N materials simulation

Secret History: Return of the Black Death Channel 4, 7-8pm In 1348 the Black Death swept through London, killing people within days of the appearance of their first symptoms. Exactly how many died, and why, has long been a mystery. This Secret History documentary follows experts as they pick through the evidence and reveal why the plague killed on such a scale. And they ask, what might be coming next?

Medscape

A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Phase shifting algorithms for fringe projection profilometry: A review

Shape measurements by fringe projection methods require high-quality sinusoidal fringes. We present a sinusoidal fringe generation technique that utilizes slightly defocused binary fringe projection. The proposed method is a spatial version of the well-known pulse-width modulation (PWM) technique of electrical engineering. PWM is easy to implement using off-the-shelf projectors, and it allows us to overcome the gamma problem (i.e., the nonlinear projector response) in the output light intensity. We will demonstrate that, with a small defocusing level--lower than with other techniques proposed in the literature--a high-quality sinusoidal pattern is obtained. Validation experiments using a commercial video projector are presented.

Pulse-width modulation in defocused three-dimensional fringe projection.

Diffraction patterns of apertures on screens uniformly illuminated are standard calculations in undergraduate optics and acoustics courses. These calculations imply two-dimensional (2-D) integrations which are often performed in the far zone, at moderate angles of diffraction, i.e., using Fresnel and Fraunhofer approximations. In this note, the 2-D integration is reduced to a 1-D parametric integration over the perimeter of the aperture—resembling the Rubinowicz’s representation of the Kirchhoff diffraction integral—which allows numerical evaluation with few computational resources. The proposed formula allows mathematically exact calculation of the near-field, in the context of scalar wave theory. Explicit calculations for circular and elliptical apertures are shown.

Diffracted field by an arbitrary aperture

In order to overcome the limitations of the sequential phase-shifting fringe pattern profilometry for dynamic measurements, a color-channel-based approach is presented. The proposed technique consists of projecting and acquiring a colored image formed by three sinusoidal phase-shifted patterns. Therefore, by using the conventional three-step phase-shifting algorithm, only one color image is required for phase retrieval each time. However, the use of colored fringe patterns leads to a major problem, the color crosstalk, which introduces phase errors when conventional phase-shifting algorithms with fixed phase-shift values are utilized to retrieve the phase. To overcome the crosstalk issue, we propose the use of a generalized phase-shifting algorithm with arbitrary phase-shift values. The simulations and experimental results show that the proposed algorithm can significantly reduce the influence of the color crosstalk.

Color-fringe pattern profilometry using a generalized phase-shifting algorithm.

Single-element interferometer

The convergence of optical imaging acquisition and image processing algorithms is a fast-evolving interdisciplinary research field focused on the reconstruction of images with novel features of interest. We propose a method for post-capture perspective shift reconstruction (in the x, y, and z directions) of a three-dimensional scene as well as refocusing with apertures of arbitrary shapes and sizes from an optimal multi-focus image stack. The approach is based on the reorganization of the acquired visual information considering a depth-variant point-spread function, which allows it to be applied to strongly defocused multi-focus image stacks. Our method is performed without estimating the depth map or segmenting the in-focus regions. A conventional camera combined with an electrically tunable lens is used for image acquisition and does not require scale transformation or registration between the acquired images. Experimental results for both real and synthetic data images are provided and compared to state-of-the-art schemes.

José A. Ferrari

Papers

Pulse-width modulation in defocused three-dimensional fringe projection.

Diffracted field by an arbitrary aperture

Color-fringe pattern profilometry using a generalized phase-shifting algorithm.

Single-element interferometer

Reconstruction of perspective shifts and refocusing of a three-dimensional scene from a multi-focus image stack