Improved Ronchi tester.

We used a LED in a Ronchi tester to obtain two main improvements: (i) We can choose one of two wavelength bands to illuminate and record the ronchigram, and (ii) we can change the irradiance of the illumination source according to the optical system under test and the detector array. This can be done by use of an adequate electronic circuit.

Exact Calculation of the Circle of Least Confusion of a Rotationally Symmetric Mirror. II.

We compute the radius and the position of the center of the circle of least confusion, in an exact way and by using the third-order approximation, of a rotationally symmetric mirror when the point source is located at any position on the optical axis. For the spherical case we find that for some positions of the point source there is a considerable difference between the exact computations and those obtained by working up to third-order aberrations.

Only one fitting for bironchigrams.

We present an algorithm that uses a square grid in a Ronchi test. We assume that the point coordinates of this pattern (termed a bironchigram) are affected by Gaussian errors. To calculate the optical path difference, we apply only one nonlinear least-squares fit to the dot coordinates. The relevant equations are deduced, and experimental results are shown.

Optical alignment of segmented mirrors to the fluorescence detectors of the Pierre Auger observatory.

Segmented mirrors will be used in the telescopes of the Pierre Auger Fluorescence detector. To align the segments, we have developed four methods in which (a) the image of the stop border, (b) the image of a screen with concentric circles, and (c) the Ronchi pattern are used. In addition to these, we have developed a new method, (d), in which instead of the Ronchi ruling, we have used a circular grid. In this case we obtain a moiré pattern for each segment by means of which the experimental setup is simplified, and the sensitivity of the alignment is improved.

n-Hindle-sphere arrangement with an exact ray trace for testing hyperboloid convex mirrors.

We present calculations with an exact ray trace to determine the dimensions that define one or two Hindle spheres, since the paraxial theory is incongruent for convex hyperboloid mirrors with small f numbers. The equations are generalized to calculate the dimensions of n Hindle spheres, since in this way it is possible to reduce the dimensions of the spheres more. Actual calculations are done for the secondary mirrors of the Benemerita Universidad Autonoma de Puebla and Large Milimetric Telescopes; experimental results are shown for the latter.

Ronchi test with a square grid.

We use a square grid in the Ronchi test. This grid allows processing of both the X and the Y directions when calculating optical path difference. We use trapezoidal integration to analyze the new patterns, since it does not have the smoothing drawback at the edges of the wave front.

Exact Design of Aplanatic Microscope Objectives consisting of Two Conic Mirrors.

The necessary equations are derived for the design of aplanatic microscope objectives consisting of two mirrors, one concave and the other convex. The first-order parameters are calculated along with the conic constants of the mirrors, determined by means of an exact analysis to arrive at an aplanatic system.

Exact calculation of the circle of least confusion of a rotationally symmetric mirror.

The position and dimensions of the circle of least confusion (CLC) on axis for a lensless Schmidt camera telescope operating at F(0.82) are calculated. The camera is to be used in the fluorescence detector of the Pierre Auger Observatory. Our analysis was developed for an aspherical mirror for any on-axis position of the point light source. Our technique uses the intersection of the marginal ray from one side of the aperture with the caustic produced by the intermediate rays from the opposite side of the aperture to locate the CLC.

Polynomial Fitting of Interferograms with Gaussian Errors on Fringe Coordinates. III. Nonlinear Solution.

Assuming that the measured coordinates of the fringes of an interferogram have random errors and that they are considered Gaussian, the system of normal equations that is obtained on application of the least-squares method is converted into a nonlinear set of equations. We present an algorithm to estimate the coefficients of the nonlinear system by applying the Newton-Raphson method and starting the iteration from the standard classic solution. This algorithm is applied to a pattern of straight and equally spaced fringes, obtaining not only the right coefficients but also the adequate election of the terms to be included in the model, to show the contrast with the results of the classic method.

Inclined toroidal surface that fits an off-axis conic section.

We developed a formulation using the continuous least-squares method to determine the inclined toroidal surface that best fits a given off-axis conic section. A toroid with a known curvature is used to obtain an analytic equation for the angle of inclination of the axis with respect to the normal to the center of the off-axis section.

Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. 1: Computer simulations.

A series of simulations were made for an ideal Twyman-Green interferogram of equally spaced straight fringes having tilt only about x. It was found that fitting polynomials to the interferometric data resulted in biased estimates of some of the fitting coefficients to the optical path difference. The acceptance of the Seidel aberrations grows with the noise level and diminishes when the number of fringes is increased.

Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. II: Analytical study.

We consider an ideal Twyman-Green interferogram with equally spaced straight fringes parallel to the x axis and fringe coordinates that are affected by Gaussian errors. We adjust the data points by polynomial fitting to the interferograms. We use a statistical analysis to obtain analytical formulas for the expected values of the aberration coefficients. The result of the analysis shows that the expected coefficients are zero, except for tilt about x and for the comatic term, and that such deviation increases with the noise level and decreases with the number of fringes. Formulas are also obtained for the expected values of the sum of squares of the residuals. We show that the problem of choosing the wrong polynomial order is a consequence of erroneous adjustment of the data points.

Least-squares estimators for the center and radius of circular patterns.

Applying the least-squares methods to data points of a pattern border, we have estimated the center and radius of a circular pattern.

Ronchi and Hartmann tests with the same mathematical theory.

A common mathematical model is established for the Ronchi and Hartmann tests and for interpretation of the Ronchigrams as level curves of the components of the transversal aberrations. With the same point of view, a Hartmanngram is regarded as two 90 degrees crossed null Ronchi gratings. A simple and direct method is also developed for calculating Ronchigrams for the cases of centered and off-axis conic sections with the point light source at any location.

Null Hartmann and Ronchi-Hartmann tests.

Assuming the Ronchi and the Hartmann tests to be null tests, we were able to design special screens for each test that produce aligned straight fringes and a square array for the observed patterns. It also became clear that the screen filter and observing planes for both tests can be interchanged.

Comparison between toroidal and conic surfaces that best fit an off-axis conic section.

A mathematical treatment is developed to establish the difference in the sagitta between toroidal and off-axis conic surfaces. The best fit betwen these surfaces is found by optimizing the curvatures of the toroid, and a comparison is made between these results and those obtained previously.