Phase estimates from slope-type wave-front sensors
01 Jan 1978-Journal of the Optical Society of America (Optical Society of America)-Vol. 68, Iss: 1, pp 139-140
About: This article is published in Journal of the Optical Society of America.The article was published on 1978-01-01. It has received 151 citations till now. The article focuses on the topics: Phase (waves) & Wavefront.
Citations
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TL;DR: In this article, a robust method for 2D phase principal values (in a least-squares sense) by using fast cosine transforms was developed, which can be used to isolate inconsistent regions (i.e., phase shear).
Abstract: Two-dimensional (2D) phase unwrapping continues to find applications in a wide variety of scientific and engineering areas including optical and microwave interferometry, adaptive optics, compensated imaging, and synthetic-aperture-radar phase correction, and image processing. We have developed a robust method (not based on any path-following scheme) for unwrapping 2D phase principal values (in a least-squares sense) by using fast cosine transforms. If the 2D phase values are associated with a 2D weighting, the fast transforms can still be used in iterative methods for solving the weighted unwrapping problem. Weighted unwrapping can be used to isolate inconsistent regions (i.e., phase shear) in an elegant fashion.
1,031 citations
TL;DR: In this paper, the problem of wavefront estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view, and a new zonal phase gradient model is introduced and its error propagator, which relates the mean square wavefront error to the noisy slope measurements, has been compared with two previously used models.
Abstract: The problem of wave-front estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view. It is shown that the slope measurement sampling geometry influences the model selection for the phase estimation. Successive over-relaxation (SOR) is employed to numerically solve the exact zonal phase estimation problem. A new zonal phase gradient model is introduced and its error propagator, which relates the mean-square wave-front error to the noisy slope measurements, has been compared with two previously used models. A technique for the rapid extraction of phase aperture functions is presented. Error propagation properties for modal estimation are evaluated and compared with zonal estimation results.
958 citations
TL;DR: In this paper, the minimum Lp-norm solution is obtained by embedding the transform-based methods for unweighted and weighted least squares within a simple iterative structure, and the data-dependent weights are generated within the algorithm and need not be supplied explicitly by the user.
Abstract: We develop an algorithm for the minimum Lp-norm solution to the two-dimensional phase unwrapping problem. Rather than its being a mathematically intractable problem, we show that the governing equations are equivalent to those that describe weighted least-squares phase unwrapping. The only exception is that the weights are data dependent. In addition, we show that the minimum Lp-norm solution is obtained by embedding the transform-based methods for unweighted and weighted least squares within a simple iterative structure. The data-dependent weights are generated within the algorithm and need not be supplied explicitly by the user. Interesting and useful solutions to many phase unwrapping problems can be obtained when p< 2. Specifically, the minimum L0-norm solution requires the solution phase gradients to equal the input data phase gradients in as many places as possible. This concept provides an interesting link to branch-cut unwrapping methods, where none existed previously.
364 citations
TL;DR: In this article, a more realistic model of the wave-front measurements is used, and wave estimation and correction are analyzed as a unified process rather than being treated as separate and independent processes.
Abstract: In adaptive optical systems that compensate for random wave-front disturbances, a wave front is measured and corrections are made to bring it to the desired shape For most systems of this type, the local wave-front slope is first measured, the wave front is next reconstructed from the slope, and a correction is then fitted to the reconstructed wave front Here a more realistic model of the wave-front measurements is used than in the previous literature, and wave-front estimation and correction are analyzed as a unified process rather than being treated as separate and independent processes The optimum control law is derived for an arbitrary array of slope sensors and an arbitrary array of correctors Application of this law is shown to produce improved results with noisy measurements The residual error is shown to depend directly on the density of the slope measurements, but the sensitivity to the precise location of the measurements that was indicated in the earlier literature is not observed
237 citations
222 citations
References
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TL;DR: In this article, the problem of fitting a wavefront distortion estimate to a (single-instant) set of phase-difference measurements has been formulated as an unweighted least-square problem.
Abstract: The problem of fitting a wave-front distortion estimate to a (single-instant) set of phase-difference measurements has been formulated as an unweighted least-square problem. The least-square equations have been developed as a set of simultaneous equations for a square array of phase-difference sensors, with phase estimates at the corner of each measurement element. (This corresponds to the standard Hartmann configuration and to one version of a shearing interferometer of a predetection compensation wave-front sensor.) The noise dependence in the solution of the simultaneous equations is found to be expressible in terms of the solution to a particular version of the measurement inputs to the simultaneous equation, a sort of “Green’s-function” solution. The noise version of the simultaneous equations is solved using relaxation techniques for array sizes from 4 × 4 to 40 × 40 phase estimation points, and the mean-square wave-front error calculated as a function of the mean-square phase-difference measurement error. It is found that the results can be approximated within a fraction of a percent accuracy by 〈(δΦ)2〉=0.6558[1+0.2444 ln(N2)]σpd2, where 〈(δΦ)2〉 is the mean-square error (rad2) in the estimation of the wave-front distortion, for a square array consisting of N2 square subaperture elements over which two phase-difference measurements are made—one phase difference across the x dimension and the other difference across the y dimension. Here σpd2 is the mean-square error (rad2) in each phase-difference measurement.
548 citations
TL;DR: In this paper, the authors analyzed the conversion from phase differences to phases and derived the optimal linear estimator in terms of least noise propagation for a compensated imaging (CI) system.
Abstract: A critical component in a compensated imaging (CI) system is the wave-front sensor which measures the residual distortion of the wave front after reflecting off the active mirror. The sensor produces estimates of wave-front slopes or phase difference across the aperture. For many applications, the phase differences or slopes are not the most convenient form of data for processing or control, and they must be converted to absolute wave-front phases. This paper analyzes the conversion from phase differences to phases and derives the optimal linear estimator in terms of least noise propagation. Some remarks concerning hardware implementation are also made.
422 citations
19 Jul 1976
TL;DR: In this paper, the wavefront distortions produced by atmospheric fluctuations are discussed and the problem of finding the best way to process the measurements of these distortions so that appropriate corrections for them can be made is addressed.
Abstract: The wavefront distortions produced by atmospheric fluctuations are discussed in this paper. The problem at hand is: What is the best way to process the measurements of these distortions so that appropriate corrections for them can be made? If a set of N independent wavefront measurements are made, the measured wavefront can be established as some linear combination of these measurements. The measurements themselves need not be direct phase measurements but could be a set of wavefront slope measurements. Nevertheless, the problem is to find a procedure that gives a best estimate of the wave-front from the set of N measurements. With such a procedure, the system designer can make an estimate of the number of measurements required to achieve a certain desired level of performance as well as the dynamical system complexity required to process the data. What is considered here is an application and adaptation of the theory of optimal estimates to the problem of random wavefront estimation.© (1976) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
4 citations