Fried Geometry Research Articles

Stability evaluation and improvement of adaptive optics systems by using the Lyapunov stability approach

In this research, investigations on the closed-loop control stability of adaptive optics systems are conducted by using the Lyapunov approach. As an direct metric of the control stability, the error propagator includes the effects of both the integral gain and the influence matrix and is effective for control-stability evaluation. An experimental 97-element AO system is developed for the control-stability investigation, and the Southwell sensor-actuator configuration rather than the Fried geometry is adopted so as to suppress the potential waffle mode. Because filtering out small singular values of the influence matrix can be used to improve the control stability, the effect of the influence matrix and the effect of the integral gain are considered as a whole by using the error propagator. Then, the control stability of the AO system is evaluated for varying the integral gains and the number of filtered-out singular values. Afterwards, an analysis of the evaluations of the error propagator is made, and a conclusion can be drawn that the control stability can be improved by filtering out more singular values of the influence matrix when the integral gain is high. In other words, the error propagator is useful for trading off the bandwidth error and the fitting error of AO systems in a control-stability approach. Finally, a performance measurement of the experimental AO system is conducted when 13 smaller singular values of the influence matrix are filtered out, and the results show that filtering out a small fraction of the singular values has a minor influence on the performance of this AO system.

Cumulative Reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes

The Cumulative Reconstructor (CuRe) is a new direct reconstructor for an optical wavefront from Shack-Hartmann wavefront sensor measurements. In this paper, the algorithm is adapted to realistic telescope geometries and the transition from modified Hudgin to Fried geometry is discussed. After a discussion of the noise propagation, we analyze the complexity of the algorithm. Our numerical tests confirm that the algorithm is very fast and accurate and can therefore be used for adaptive optics systems of Extremely Large Telescopes.

Preface: numerical and applied linear algebra

This special issue of Advances in Computational Mathematics is devoted to Numerical and Applied Linear Algebra, which is an important area of mathematics and a fundamental part of engineering and computational science problems. One of the main tools of this area is matrices, properties of which are studied and computations of them are performed, with different methods, in a variety of applications. The special issue contains 12 papers reflecting some specialized themes on numerical and applied linear algebra and at the same time gives an idea of the broadness of this area, which has grown spectacularly in the last decades. The work entitled “Structured linear algebra problems in adaptive optics imaging”, by Johnathan M. Bardsley, Sarah Knepper, and James Nagyy, discusses an adaptive optics image problem to remove the effects of the phase error. For solving the Kronecker product structured, rank-deficient least-squares problem on the Fried geometry, two types of regularization are considered: a truncated singular value decomposition type and a Tikhonov type. The paper “On the numerical solution of large-scale sparse discrete-time Riccati equations”, by Peter Benner and Heike Fasbender, deals with the numerical solution of discrete algebraic Riccati equations. Since the systems considered are large-scale, the solution process for the computation of a

Quantifications of error propagation in slope-based wavefront estimations

We discuss error propagation in the slope-based and the difference-based wavefront estimations. The error propagation coefficient can be expressed as a function of the eigenvalues of the wavefront-estimation-related matrices, and we establish such functions for each of the basic geometries with the serial numbering scheme with which a square sampling grid array is sequentially indexed row by row. We first show that for the wavefront estimation with the wavefront piston value determined, the odd-number grid sizes yield better error propagators than the even-number grid sizes for all geometries. We further show that for both slope-based and difference-based wavefront estimations, the Southwell geometry offers the best error propagators with the minimum-norm least-squares solutions. Noll's theoretical result, which was extensively used as a reference in the previous literature for error propagation estimates, corresponds to the Southwell geometry with an odd-number grid size. Typically the Fried geometry is not preferred in slope-based optical testing because it either allows subsize wavefront estimations within the testing domain or yields a two-rank deficient estimations matrix, which usually suffers from high error propagation and the waffle mode problem. The Southwell geometry, with an odd-number grid size if a zero point is assigned for the wavefront, is usually recommended in optical testing because it provides the lowest-error propagation for both slope-based and difference-based wavefront estimations.

Multigrid algorithm for least-squares wavefront reconstruction

Multigrid (MG) methods are presented for fast, efficient, flexible, and robust least-squares wavefront reconstruction in extremely high-resolution conventional adaptive optics, or ExAO. We demonstrate that MG can robustly handle a variety of sensor-actuator geometries, and it can accommodate deformable mirror influence function models that are more realistic than the common piecewise bilinear model. With MG one can also easily incorporate additional penalty, or regularization, terms to damp out the waffle mode in Fried geometry and to damp out instabilities due to actuators near the pupil boundary with poorly sensed influence. We present closed-loop simulation results that suggest that only one or two MG iterations per time step are needed to control an ExAO system.

Wavefront reconstruction using iterative discrete Fourier transforms with Fried geometry

A new kind of wavefront reconstruction algorithm is proposed in this paper, which can estimate the wavefront from its orthogonal gradients at different sample points. We model the gradients measured by wavefront sensor with the Fried geometry. Iterative discrete Fourier transforms and the correspondent Fried inverse filter in spatial frequency domain are introduced. No special boundary control is required to process the boundary condition in this algorithm. The stimulation and the experimental results show that the new algorithm has strong capability to fit the wavefront and it is more available if the sample points are few.

An Analysis of Fundamental Waffle Mode in Early AEOS Adaptive Optics Images11Based on observations made at the Maui Space Surveillance System, operated by Detachment 15 of the US Air Force Research Laboratory’s Directed Energy Directorate.

ABSTRACTAdaptive optics (AO) systems have significantly improved astronomical imaging capabilities over the last decade and are revolutionizing the kinds of science possible with 4–5 m class ground‐based telescopes. A thorough understanding of AO system performance at the telescope can enable new frontiers of science as observations push AO systems to their performance limits. We look at recent advances with wave‐front reconstruction (WFR) on the Advanced Electro‐Optical System (AEOS) 3.6 m telescope to show how progress made in improving WFR can be measured directly in improved science images. We describe how a “waffle mode” wave‐front error (which is not sensed by a Fried geometry Shack‐Hartmann wave‐front sensor) affects the AO point‐spread function. We model details of AEOS AO to simulate a PSF that matches the actual AO PSF in the I band and show that while the older observed AEOS PSF contained several times more waffle error than expected, improved WFR techniques noticeably improve AEOS AO performance. We estimate the impact of these improved WFRs on H‐band imaging at AEOS, chosen based on the optimization of the Lyot Project near‐infrared coronagraph at this bandpass.

Fast wave-front reconstruction by solving the Sylvester equation with the alternating direction implicit method

Large degree-of-freedom real-time adaptive optics (AO) control requires reconstruction algorithms that are computationally efficient and readily parallelized for hardware implementation. In particular, we find the wave-front reconstruction for the Hudgin and Fried geometry can be cast into a form of the well-known Sylvester equation using the Kronecker product properties of matrices. We derive the filters and inverse filtering formulas for wave-front reconstruction in two-dimensional (2-D) Discrete Cosine Transform (DCT) domain for these two geometries using the Hadamard product concept of matrices and the principle of separable variables. We introduce a recursive filtering (RF) method for the wave-front reconstruction on an annular aperture, in which, an imbedding step is used to convert an annular-aperture wave-front reconstruction into a squareaperture wave-front reconstruction, and then solving the Hudgin geometry problem on the square aperture. We apply the Alternating Direction Implicit (ADI) method to this imbedding step of the RF algorithm, to efficiently solve the annular-aperture wave-front reconstruction problem at cost of order of the number of degrees of freedom, O(n). Moreover, the ADI method is better suited for parallel implementation and we describe a practical real-time implementation for AO systems of order 3,000 actuators.

Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform.

Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack-Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster than, and can have noise propagation comparable with that of, traditional vector-matrix-multiply reconstructors. The boundary problem that prevented the accurate reconstruction of phase in circular apertures by means of square-grid Fourier transforms (FTs) is identified and solved. The methods are adapted for use on the Fried geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented with the use of both theory and simulation.

Evaluation of the performance of Hartmann sensors in strong scintillation.

A simulation study is presented that evaluates the performance of Hartmann wave-front sensors with measurements obtained with the Fried geometry and the Hutchin geometry. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and the weaknesses of Hartmann sensors used in each of the two geometries in the two-dimensional space of the Fried parameter r0 and the Rytov parameter. We found that the performance of Hartmann sensors degrades severely when the Rytov number exceeds 0.2 and the ratio l/r0 exceeds 1/4 (where l is the subaperture side length) because of the presence of branch points in the phase function and the effect of amplitude scintillation on the measurement values produced by the Hartmann sensor.