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.

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.

Tilted conic fitting to an off-axis conic surface.

A better fitting to an off-axis conic section can be obtained if a tilted conic is used. The mathematical equation for the tilting angle is obtained as well as the influence over some of the aberration coefficients.

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.

Cyclic interferometers for optical testing.

In this paper, we describe two compact cyclic interferometers which can be used for the testing of right-angle prisms; with some modifications they can also be used for testing convergent wave fronts.