Modal processing of Hartmann and Shack–Hartmann patterns by means of a least squares fitting of the transverse aberrations (original) (raw)
Related papers
Modal integration of Hartmann and Shack-Hartmann patterns
Journal of the Optical Society of America A
Instead of measuring the wavefront deformations directly, Hartmann and Shack-Hartmann tests measure the wavefront slopes, which are equivalent to the ray transverse aberrations. Numerous different integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. In this work we describe a modal method to integrate Hartmann and Shack-Hartmann patterns using orthogonal wavefront slope aberration polynomials, instead of the commonly used Zernike polynomials for the wavefront deformations.
This paper reports a method of wavefront sensing based on Hartmann-Shack (HS) centroid displacements and predicts the number of HS spots that can be successfully deleted without hampering the prediction of the defocus of the computed wavefront, described via Zernike polynomials. The deletion of the HS spots was randomized. The experiment was performed on real data acquired through a custom made aberrometer tested on a model eye with various axial lengths to simulate refractive errors (defocus) between *±1.50D. Estimates of defocus were made from each of 1000 runs at each axial length. The paper presents the standard deviation of error and mean error for 1000 trials. The results indicate that as high as 50 % of the HS spots can be deleted without affecting the estimation of spherical defocus, within typical clinically acceptable limits of AE0:25D.
Zonal wavefront reconstruction of Shack–Hartmann and Hartmann patterns with hexagonal cells
Optics Communications, 2018
In this article we will develop a method to integrate Shack-Hartmann and Hartmann pattern with hexagonal cells, using a polynomial representation (modal integration) over each hexagonal cell. Since each hexagonal has six sampling points, one at each vertex, instead of the typical four sampling points in square cells, it is possible to have a different representation of the wavefront in each cell, each with different aberration terms. The local curvatures and low order aberrations in each cell are calculated more accurately than for square cells. All the analytical functions over each hexagonal cell have a different unknown piston term, that is calculated with a method to be described here. As a result, wavefront retrieval and representation of freeform optical surfaces for some optical systems can be made, due to the calculation of aberrations in each hexagonal cell.
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.
Primary wavefront aberrations calculation from a defocused image or a Hartmanngram
Applied Optics, 2010
A wavefront aberration can be retrieved from a defocused image or a Hartmanngram by several different methods using diffraction theory and Fourier transforms. In this manuscript, we describe an alternate method for wavefront aberration determination from a defocused image or a Hartmanngram using a geometric l approximation. The main assumption is that the image is defocused, with the observation plane outside the caustic limits. The result will be applied to the retrieval of a wavefront with primary aberrations from a Hartmanngram or defocused image without the need for any transversal aberration integration.
Optics express, 2015
Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.
SPIE Proceedings, 2017
When using Shack-Hartmann wavefront sensors (SH) and Zernike coefficients (Zs) in applications where the position of the measurement and the point of interest are far apart, as it is common practice in ophthalmic optics, problems in the interpretation of the values of the Zs arise, related to how the shape of the wavefront propagates along the beam. One typical example is pupil conjugation where an auxiliary lens is added to match the size of the area of the interest of the beam with the size of the entrance pupil of the SH used for measurements. In the present work, we address this problem in the framework of a numerical scheme for modeling the beam propagation. We calculate the wavefronts with exact ray tracing plus the fitting of the impacts so as to match a rectangular grid. This procedure allows the subsequent calculation of the Zs or, similarly, the pupil function at an arbitrary plane perpendicular to the optical axis. All the numerical methods and procedures have been implemented in MATLAB code and can be illustrated by running the MATLAB script for the setup configuration that is being considered. Several examples are presented to illustrate the previous ideas and to show the real capabilities of our procedures. They will help to clarify the issues actually found in practical setups for beam manipulation, often encountered in ophthalmic optics.
Aberration Extraction in the Hartmann Test by Use of Spatial Filters
Applied Optics, 1999
Using a computer, we generated a set of filters to aid in the retrieval of aberration functions from Hartmanngrams. These filters consist of discrete two-dimensional data points, like the Hartmanngrams themselves, and are orthogonalized by the Gram-Schmidt procedure. The aberration coefficients are obtained by calculation of the scalar product of the Hartmanngram and each orthogonal filter.
Direct Ray Aberration Estimation in Hartmanngrams by use of a Regularized Phase-Tracking System
Applied Optics, 1999
The Hartmann test is a well-known technique for testing large telescope mirrors. The Hartmann technique samples the wave front under analysis by use of a screen of uniformly spaced array of holes located at the pupil plane. The traditional technique used to gather quantitative data requires the measurement of the centroid of these holes as imaged near the paraxial focus. The deviation from its unaberrated uniform position is proportional to the slope of the wave-front asphericity. The centroid estimation is normally done manually with the aid of a microscope or a densitometer; however, newer automatic fringe-processing techniques that use the synchronous detection technique or the Fourier phase-estimation method may also be used. Here we propose a new technique based on a regularized phase-tracking ͑RPT͒ system to detect the transverse aberration in Hartmanngrams in a direct way. That is, it takes the dotted pattern of the Hartmanngram as input, and as output the RPT system gives the unwrapped transverse ray aberration in just one step. Our RPT is compared with the synchronous and the Fourier methods, which may be regarded as its closest competitors.
Modal tomography of aberrations of the human eye
Laser Physics, 2006
The work is devoted to the research and development of a new method of modal phase tomography for diagnostics of aberrations of the human eye. Implementation of the method is based on a series of eye aberration measurements taken at different angles to the optical axis by means of a wave-front sensor. The technique of restoration permits the contributions of different elements of the eye to the total aberrations to be separated. The results of numerical research and a model experiment are presented.