Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture (original) (raw)

Wavefront analysis from its slope data

Current Developments in Lens Design and Optical Engineering XVIII, 2017

In the aberration analysis of a wavefront over a certain domain, the polynomials that are orthogonal over and represent balanced wave aberrations for this domain are used. For example, Zernike circle polynomials are used for the analysis of a circular wavefront. Similarly, the annular polynomials are used to analyze the annular wavefronts for systems with annular pupils, as in a rotationally symmetric two-mirror system, such as the Hubble space telescope. However, when the data available for analysis are the slopes of a wavefront, as, for example, in a Shack-Hartmann sensor, we can integrate the slope data to obtain the wavefront data, and then use the orthogonal polynomials to obtain the aberration coefficients. An alternative is to find vector functions that are orthogonal to the gradients of the wavefront polynomials, and obtain the aberration coefficients directly as the inner products of these functions with the slope data. In this paper, we show that an infinite number of vector functions can be obtained in this manner. We show further that the vector functions that are irrotational are unique and propagate minimum uncorrelated additive random noise from the slope data to the aberration coefficients.

Quantifications of error propagation in slope-based wavefront estimations

Journal of the Optical Society of America A, 2006

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 oddnumber 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.

Principles of Wavefront Sensing and Reconstruction

A variety of approaches to wavefront sensing and reconstruction are surveyed as they are used in adaptive optics and related applications. These include the Gerchberg-Saxton algorithm; shearing interferometry; and Shack-Hartmann, curvature, and pyramid wavefront sensing. Emphasis is placed on the relevant optics and mathematics, which are developed in some detail for Shack-Hartmann and curvature sensing (currently the two most widely-used approaches) and also to a lesser extent for pyramid sensing. Examples are given throughout.

Practical approach to modal basis selection and wavefront estimation

Adaptive Optical Systems Technology, 2000

The MPIA/MPE adaptive optics with a laser guide star system ALFA works excellent with natural guide stars up to 13th magnitude in R-band. Using fainter natural guide stars or the extended laser guide star, ALFA's performance does not entirely satisfy our expectations. We describe our efforts in optimizing the wavefront estimation process. Starting with a detailed system analysis, this paper will show how to construct a modal basis set which efficiently uses Shack-Hartmann measurements while keeping a certain number of low order modes close to analytical basis sets like Zernikes or Karhunen-Loève functions. We will also introduce various phase estimators (least squares, weighted least squares, maximum a posteriori) and show how these can be applied to the ALFA AO. A first test done at the Calar Alto 3.5m-telescope will be discussed.

Generalized Wave-Front Reconstruction Algorithm Applied in a Shack-Hartmann Test

Applied Optics, 2000

A generalized numerical wave-front reconstruction method is proposed that is suitable for diversified irregular pupil shapes of optical systems to be measured. That is, to make a generalized and regular normal equation set, the test domain is extended to a regular square shape. The compatibility of this method is discussed in detail, and efficient algorithms ͑such as the Cholesky method͒ for solving this normal equation set are given. In addition, the authors give strict analyses of not only the error propagation in the wave-front estimate but also of the discretization errors of this domain extension algorithm. Finally, some application examples are given to demonstrate this algorithm.

Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

Journal of the Optical Society of America A, 2005

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15ϫ 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than / 130-/ 150 after ϳ12 iterations and less than / 100 (both for = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.

Improvements on the optical differentiation wavefront sensor

Monthly Notices of the Royal Astronomical Society, 2005

We describe a novel concept for high-resolution wavefront sensing based on the optical differentiation wavefront sensor (OD). It keeps the advantages of high resolution, adjustable dynamic range, ability to work with polychromatic sources and, in addition, it achieves good performance in wavefront reconstruction when the field is perturbed by scintillation. Moreover, this new concept can be used as multi-object wavefront sensor in multiconjugate adaptive optics systems. It is able to provide high resolution and high sampling operation, which is of great interest for the projected extreme adaptive optics systems for large telescopes.

Analytical Study of Optical Wavefront Aberrations Using Maple

1997

This paper describes a package for analytical ray tracing of relatively simple optical systems. AESOP (An Extensible Symbolic Optics Package) enables analysis of the effects of small optical element misalignments or other perturbations. (It is possible to include two or more simultaneous independent perturbations.) Wavefront aberrations and optical path variations can be studied as functions of the perturbation parameters. The power of this approach lies in the fact that the results can be manipulated algebraically, allowing determination of misalignment tolerances as well as developing physical intuition, especially in the picometer regime of optical path length variations.