Erez N. Ribak

Constrained realizations of Gaussian fields - A simple algorithm

A straightforward method for the construction of constrained realizations of Gaussian fields is presented. Consider a Gaussian random field and its ensemble mean field given a set of constraints. The residual of the field from its mean is statistically independent of the actual numerical value of the constraints. The algorithm is based on a simple construction of this residual field, which is then added to the analytically calculated mean field. This algorithm is exact and involves no iterations.

Phase retrieval by demodulation of a Hartmann–Shack sensor

Abstract We develop a method for retrieving the wavefront from a Hartmann–Shack sensor using a two-dimensional Fourier demodulation technique. This provides a means for analyzing the patterns produced by a Hartmann–Shack sensor with variable lenslet array pitch. The technique is based on applying a digital demodulation technique twice on the pattern, along two orthogonal axes. This provides both phase gradient components in real space. Later a Laplacian Fourier reconstruction is used to obtain the corresponding wavefront. We discuss the robustness of the technique to Poisson and white noise. Wavefront measurements of known aberrations are analyzed using this technique.

Turbulence-degraded wave fronts as fractal surfaces

We identify wave fronts that have passed through atmospheric turbulence as fractal surfaces from the Fractional Brownian motion family. The fractal character can be ascribed to both the spatial and the temporal behavior. The simulation of such wave fronts can be performed with fractal algorithms such as the Successive Random Additions algorithm. An important benefit is that wave fronts can be predicted on the basis of their past measurements. A simple temporal prediction reduces by 34% the residual error that is not corrected by adaptive-optics systems. Alternatively, it permits a 23% reduction in the measurement bandwidth. Spatiotemporal prediction that uses neighboring points and the effective wind speed is even more beneficial.

Simulated annealing in ocular adaptive optics

We present what is to our knowledge a first hardware realization of a simulated annealing algorithm in an adaptive optics system designed to image the retina of the human eye. The algorithm is applied to the retinal image itself without the need for wavefront sensors in the system. We find that this optimization algorithm can be an alternative to the traditional Hartmann-Shack sensing. We also compare the simulated annealing algorithm to the stochastic parallel gradient descent algorithm.

Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution

We have applied the technique of Fourier fringe analysis to microscopic interferograms of needle crystals that grow from a solution. We use a differential technique in which an empty field interferogram is compared with one that contains distortion and obscuration by the growing crystal, and we demonstrate both analytically and experimentally a phase shift sensitivity of 0.01 fringe with a spatial resolution of half of a fringe spacing (~1 µm). Following the analysis of the interferogram in two dimensions, we show that the three-dimensional refractive-index field around the crystal can be deduced, assuming that it is axially symmetric, by an iterative method.

Wavefront reconstruction from its gradients

Wavefronts reconstructed from measured gradients are composed of a straightforward integration of the measured data, plus a correction term that disappears when there are no measurement errors. For regions of any shape, this term is a solution of Poissons equation with Dirichlet conditions (V = 0 on the boundaries). We show that for rectangular regions, the correct solution is not a periodic one, but one expressed with Fourier cosine series. The correct solution has a lower variance than the periodic Fourier transform solution. Similar formulas exist for a circular region with obscuration. We present a near-optimal solution that is much faster than fast-Fourier-transform methods. By use of diagonal multigrid methods, a single iteration brings the correction term to within a standard deviation of 0.08, two iterations, to within 0.0064, etc.

Primordial Gaussian perturbation fields : constrained realizations

A novel method is devised in order to impose constraints on simulated realizations of random Gaussian fields. The power spectrum of the residual from the (ensemble) mean field under the given constraints is calculated. The Fourier components of the residual field have amplitudes set by the required power spectrum, and phases which are initially assumed to be uniformly distributed. A simple iterative procedure allows imposing constraints on the residual field while keeping its power spectrum intact. By transforming back and forth between real and Fourier spaces while constraining the field component in both domains, the required realizations are obtained within a few cycles

Direct demodulation of Hartmann–Shack patterns

Hartmann-Shack wave-front sensors produce a distorted grid of spots whose deviation from perfection is linear with the wave-front gradient. Usually, the centroid of each spot is calculated to provide that deviation, but it is also possible to perform the calculation by Fourier demodulation of the spot pattern [Opt. Commun. 215, 285, 2003]. We show that this demodulation can be performed directly on the grid, without reverting to Fourier transforms. Tracking the motion of each centroid individually is limited to well-defined spots with motions smaller than their pitch. In contrast, our method treats the image as a whole, is not limited to non-overlapping or sharp spots, and allows large spot motions. By replicating the array of spots slightly beyond the edge of the aperture, we reduce the chance for boundary phase dislocations in the reconstruction of the wave front. The method is especially suited to very large arrays.

Bimorph adaptive mirrors and curvature sensing

The applicability of wave-front correction by means of a bimorph mirror in conjunction with a curvature sensor is described. We use Zernike polynomials to describe the quality of the atmospheric-turbulence correction analytically. The match is limited by boundary conditions of the mirror and by the discreteness of the electrodes. The correction is limited by coupling of lower- and higher-order Zernike polynomials and necessitates an interfacing computer between the wave-front sensor and the bimorph mirror.

Offline, multidetector intensity interferometers – I. Theory

Stellar amplitude interferometry is limited by the need to have optical distances fixed and known to a fraction of the wavelength. We suggest reviving intensity interferometry, which requires hardware which is many orders of magnitude less accurate, at the cost of more limited sensitivity. We present an algorithm to use the very high redundancy of a uniform linear array to increase the sensitivity of the instrument by more than a 100-fold. When using an array of 100 elements, each almost 100 m in diameter, and conservative technological improvements, we can achieve a limiting magnitude of about m b = 14.4. Digitization, storage, and offline processing of all the data will also enable interferometric image reconstruction from a single observation run, and application of various algorithms at any later time. Coronagraphy, selectively suppressing only the large-scale structure of the source, can be achieved by specific aperture shapes. We conclude that after three decades of abandonment optical intensity interferometry deserves another review.