Symmetric gradient-index media reconstruction.

Ray tracing in gradient-index (GRIN) media has been thoroughly studied and several ray tracing methods have been proposed. Methods are based on finding the ray path given a known GRIN. In recent decades, the inverse problem, which consists of finding the GRIN distribution for a given light ray path, has been gaining attention. Given that it is not an easy task, the methods proposed in the literature vary in degrees of difficulty. In this work, an alternative method is presented to derive symmetric GRIN distributions whose implementation can be considered the simplest to date. Since it is based on invariants, which result from the symmetries of the system as stated by Fermat's principle, it is an exact numerical method, i.e., the physical system is not approximated. The robustness of the method permits the reconstruction of the GRIN distribution from a ray propagating in three-dimensions. In order to demonstrate its operation, different known symmetric GRIN media are reconstructed using rays that propagate in two and three dimensions.

Design of a compact off-axis four-mirror objective system for a thermal camera.

The optical design of a compact off-axis four-mirror objective system (OFOS) of a thermal camera for a syndromic surveillance system that captures bioclinical signals, like temperature, directly related to the physical symptoms of the COVID-19 disease through thermal images is presented. The OFOS design works on wavelengths of 7.5-14 µm, with an f-number of 4, and a field of view (FOV) of 24×4 deg.

Testing lenses and reflecting concave surfaces using a knife-edge interferometer.

In a previous paper by Cebrian et al. [Opt. Eng.53, 092006 (2014)OPEGAR0091-328610.1117/1.OE.53.9.092006], the knife-edge interferometer (KEI) was used for testing lenses with a collimated beam illuminating the lens. In this paper, a 2f scheme is used to test lenses, eliminating the use of a collimation beam, and testing, for the first time, concave reflecting (spherical and parabolic) surfaces at the center of curvature with KEI.

Measurement of three-dimensional wavefronts using the Ichikawa-Lohmann-Takeda solution to the irradiance transport equation.

In this paper, we use the irradiance transport equation and the Fourier transform-based experimental solution given by Ichikawa-Lohmann-Takeda. We analyze experimental factors such as the digital filter, the introduced error for the rotation and period of the Ronchi ruling, and a new method is demonstrated for the measurement of 3D wavefront information.

Chalmers interferometric test using a reflective spatial light modulator.

In this paper, we propose the use of a reflective spatial light modulator (RSLM) controlled by a PC, instead of a metal plate with holes, to produce the interference patterns in Chalmers interferometric test. The main advantage of the proposed method is that with an RSLM, it is possible to test and obtain an interference pattern for any zone of a surface or lens by opening two appropriate apertures. This increases the accuracy of the results and reduces the time required to obtain them.

Ronchi test can detect piston by means of the defocusing term.

We present simulated results on piston detection applying the classical Ronchi test to a segmented surface. We have found that a piston error in a test segment, induces a change in the transversal aberration, that can be analyzed by mutually comparing the fringes frequency in each segment. We propose that the piston term of the segmented surface can be recovered by geometrically relating the change in transversal aberration with the piston term. To test this, we have simulated some ronchigrams for a known piston error, and we have been able to recover this term for a dynamic range comprised among 57nm and 550 microm. For piston errors > 550 microm a change in the transversal aberration can be appreciated and measured in the ronchigrams although these large pistons are now classical defocusings. Thus we have demonstrated that the Ronchi test can be an alternative method for the piston detection with a large dynamic range.

Correlation algorithm to recover the phase of a test surface using phase-shifting interferometry.

A correlation algorithm to recover the phase in phase-shifting interferometry is presented. We make numerical simulations to test the proposed algorithm and apply it to real interferograms with satisfactory results.

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.

Design of projective zone plates and their focusing properties.

Transmittances are designed by generation of a family of curves whose local geometry is similar to that of zone plates. The family is obtained by means of projective mapping in the normal direction of an initial curve, so that all the curves of the family have a common evolute. This property allows the prediction of the geometry of the focusing region of the transmittance. Experimental results with an ellipse as the initial curve are shown. We show that the proposed zone plates can be considered a generalization of conventional zone plates.

Multiple-path telescope: third-order design.

Exact results for the magnification and effective focal length of a multiple-path telescope are presented. We present analytical results for the third-order design of this system, using the Seidel sums for the five aberration terms. Numerical calculations are used throughout to corroborate these results. Third-order aberrations remain small or at least stable as the number of paths is increased.

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 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.

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.