Daniel Malacara-Hernández

Wavefront fitting with discrete orthogonal polynomials in a unit radius circle

Zernike polynomials have been used for some time to fit wavefront deformation measurements to a two-dimensional polynomial. Their orthogonality properties make them ideal for this kind of application. The typical procedure consists of first obtaining the fitting using x-y polynomials and then transforming them to Zernike polynomials by means of a matrix multiplication. Here, we present a new method for making this fitting faster by using a set of orthogonal polynomials on a discrete base of data points on a unitary circle.

A review of methods for measuring corneal topography

A review of some methods and their optical principles for measuring the corneal topography are presented in this paper. The concepts of principal curvatures and the ambiguity concerning the axial curvature of surfaces without symmetry of revolution are analyzed. These methods are divided into three groups according to the following optical principles: (1) specular reflection, which includes the Placido disk system, interferometry, and moiré deflectometry; (2) diffuse reflection, which includes moiré fringes, rasterstereography, and Fourier Transform Profilometry; and (3) scattered light, which includes the slitlamp system. We avoided describing the details of commercial instruments, only their working principles.

Two-step phase-shifting algorithm

A novel two-step phase-shifting algorithm for use in detecting wavefront phase is presented. This algorithm is very insensitive to detuning. The short time needed to acquire and process interferograms helps to minimize phase errors caused by vibration, temperature effects, and nonlinear errors of the piezoelectric transducer. The rms error due to detuning does not change as in classical phase-shifting algorithms with respect to the selected ideal phase shift. The resulting wrapped phase is modulo 2π. Finally the phase error is analyzed.

Object surface for applying a modified Hartmann test to measure corneal topography

A modified Hartmann test is proposed for measuring corneal topography. The plane screen with holes used in the typical Hartmann test is replaced with a curved object surface. This object surface yields a plane image for a spherical mirror surface. We show that the object surface is an oval of revolution that can be modeled by an ellipsoid. The plane image will be formed by a square array of circular spots, all with the same diameter. To obtain the square array in the image, we calculated the spatial distribution of the spots on the object surface.

Testing and centering of lenses by means of a Hartmann test with four holes

The Hartmann test is frequently performed with a screen with many holes to test the quality of a mirror or the aberrations of a lens. An application of the simplest form of this test with only four holes is described. This test cannot detect spherical aberration, but it may be useful to align optical systems, to detect and measure focus errors, or to detect lens decenterings and coma.

Wave-front retrieval from Hartmann test data

In the classical Hartmann test the wave front is obtained by integration of the transverse aberrations, joining the sampled points by small straight segments, in the so-called Newton integration. This integration is performed along straight lines joining the holes on the Hartmann screen. We propose a modification of this procedure, considering the cells of four holes of the Hartmann screen to fit a small second-power wave front recovering each square. This procedure has some important advantages, as described here.

What is a Hartmann test

In this paper we will review some of the many different practical arrangements that have been obtained to measure the transversal aberrations of optical systems based on the odd and vulnerable Hartmann test. There are many optical testing configurations that apparently are not related to the original Hartmann test. However, they are really the same thing and can be considered just a variation of the same basic arrangement, as will be described here.

Direct-phase detection of modulated Ronchi rulings using a phase-locked loop

A recently developed technique for fringe analysis based on a digital phase-locked loop is applied to detect the ray aberration of modulated Ronchi rulings. The Ronchi ruling is placed outside the caus- tic, and the shadow of the ruling is imaged directly into a two-dimensional CCD video array for further processing. The CCD array is placed at a

Iterative method to obtain the wrapped phase in an interferogram with a linear carrier

There is a great variety of techniques used to obtain the wrapped phase of a carrier frequency interferogram. We are interested in interferograms that have a bounded region data. Analyses of this kind of interferograms with the Fourier or synchronous method have an important source of error at the edge of the interferogram. In this paper we propose a new method that works out this boundary error effect. We describe an iterative synchronous method to obtain a very good phase estimation of a carrier frequency interferogram bounded by an arbitrary pupils shape. We also show that the proposed iterative method has a continuous and stable convergence towards the true modulating phase of the interferogram.

Wavefront fitting using Gaussian functions

Often the polynomial description of a wavefront shape is inaccurate because sharp local deformations are difficult to represent. In this case an analytical representation in terms of Gaussians may give better results. We have made a study of properties of Gaussians in fitting wavefronts. We analyzed an specific array of Gaussian functions and show their easy Fourier transformation. In Fourier space some criteria are proposed for setting the Gaussian width, the separation between these functions and making an estimation of the wavefront fitting error. Two simulated wavefronts are fitted and a comparison with a Zernike polynomial is made. It is well known that when fitting using Zernike polynomials one needs to find the optimal number of terms beyond which the errors in the approximation become larger. We show that using Gaussians the accuracy of the fit increases with the number of terms. We demonstrate some interesting properties, such as their facility to fit local deformations and fast parameter determination.