Dynamical analysis of free tool rotation on a glass rotating at a constant angular velocity.

This work presents a dynamical analysis of a tool rotating freely due to the action of glass rotating at a constant angular velocity is presented. Preston [J. Soc. Glass Tech.XI, 214 (1927) JSGLAI0368-4105] analyzed tool movement neglecting the tool thickness, and only one torque component was obtained along the tool's normal direction. From this analysis, Preston concluded that angular velocities for tool and glass must be equal, even when this is not true experimentally. In this paper, a tool thickness different from zero was assumed. Then, two new torque components along the interface plane were found, which produced a non-homogenous pressure between the surfaces, and then an erratic tool angular velocity, meaning that the tool's known random rotation could be understood. If the tool's center is fixed on glass, a point with coordinates (x G,0) and a torque along the X G direction can be obtained. This new torque component predicts a non-synchronic and random rotation, while slowing down the tool angular velocity, which was confirmed experimentally.

Exact formula to relate optical path differences and transversal aberration components.

An exact equation to relate the optical path differences (OPD) with its transversal aberration components (TAC) is determined. The OPD-TAC equation reproduces the Rayces formula and introduces the coefficient for the longitudinal aberration. The defocus orthonormal Zernike polynomial (Z DF) is not a solution for the OPD-TAC equation since the obtained longitudinal defocus depends on the ray height on the exit pupil, meaning that it cannot be interpreted as a defocus. In order to find an exact expression for OPD defocus, first, a general relationship between the wavefront shape and its OPD is established. Second, an exact formula for the defocus OPD is established. Finally, it is proved that only the exact defocus OPD is an exact solution of the exact OPD-TAC equation.

Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils.

In a previous paper [Appl. Opt. 61, C20 (2022)] it was proven that for a circular exit pupil and any optical path differences, the border of any spot diagram is integrated by the caustic surface and/or marginal rays. In this paper, the previous results are extended to annular (circular an elliptical) as well as hexagonal (single and segmented) exit pupils. Several examples of wavefronts affected by linear combinations of orthonormal Zernike aberrations are shown.

Borders of spot diagrams are always caustic surfaces and/or marginal rays.

In this paper, it is proven that for any optical system with a circular exit pupil and wavefronts affected by any aberrations, the borders of all leaving rays are caustic surfaces and/or marginal rays. Several examples are shown for wavefronts affected by linear combinations of Zernike aberrations.

Lensometer with autocollimation and a square Ronchi grid.

A lensometer based on an autocollimation system and a square Ronchi grid was designed, constructed, and tested. Refractive powers of monofocal, astigmatic, bifocal, and progressive lenses were measured. The focal plane was identified when no spots, or a minimum number of fringes, are observed on the bironchigram (pattern with a square Ronchi grid). For cylindrical lenses, the spots were transformed in fringes along the $ x $x and ${y}$y directions from which the cylindrical and spherical powers were obtained. For the progressive lenses, a zero spots circle moved on the bironchigram plane along the umbilic zone while the square Ronchi grid was moved along the optical axis. This lensometer is compact, cheap, and precise. Our measurements and errors were very similar to those obtained with a commercial lensometer.

Exact calculation of conic constants and baffles for any two-mirror aplanatic telescope.

To calculate the conic constants of the primary and secondary mirrors of any aplanatic two-mirror telescope, nonexact equations have been used considering third-order approximations, although it was not considered that there is an infinite number of conic constants. In this paper, exact nonlinear equations are obtained; with them, the two conic constants of the mirrors are calculated without approximations. We also find that the conic constants depend on the position at the stop where the calculation is carried out, and there are always residual aberrations. In addition, a procedure is established for the exact calculation of the baffles. Finally, examples show the differences between exact and third-order calculations.

Small petal tools performance for parabolizing optical surfaces.

Small rigid petal tools, driven by a traditional polishing machine, were used to parabolize 20 mirrors 14 cm in diameter and 192 cm of curvature radius. Small rigid circular tools (SCTs), driven manually, were used to parabolize another 20 identical surfaces. A Ronchi test with a square grid was used to evaluate the performance of both techniques. If small rigid petal tools are used, the surface quality, the reproducibility in the production process, and the time spent required to generate the surfaces are markedly better than using SCTs.

Local and global surface errors evaluation using Ronchi test, without both approximation and integration.

We have reproduced quantitatively the technique commonly used in optical shop to evaluate surface error from comparison between experimental and simulated Ronchigrams. We used this procedure to evaluate, from Ronchigrams of any number of fringes, the curvature radius and/or conic constant of conic surfaces. The error function is calculated without using integration (numerical or polynomial) so the corresponding problems were avoided. Furthermore, when the error function is described with cubic splines, then the local errors are very well reproduced, which is not the case with the polynomial description. We have described the error functions with conical surfaces or with cubic splines, and for the best reproduction of experimental Ronchigram we used genetic algorithms.

Correction of the Preston equation for low speeds.

According to Preston [J. Soc. Glass Technol. 11, 214 (1927)], the wear on a glass point in the polishing process is proportional to the work given by frictional force between glass and tool. He supposed that the frictional coefficient is a constant value. To verify this hypothesis, we measured the dragging forces applied to a tool as a function of the relative speed between a rotating glass and the tool center. To reproduce these experimental results, it was necessary to propose a new model, for which the frictional coefficient has a Gaussian dependence with relative speed. Therefore the wearing Preston equation has to be modified in order to include the frictional coefficient as a function of the relative speed.

Use of linear programming to calculate dwell times for the design of petal tools.

Two constraints in the design of a petal tool are, the angles that define it must all be positive, and wear must never be greater than the desired wear. The first constraint is equivalent to that of the positive dwell times of a small solid tool. In view of this foregoing, we present a design of petal tools that are used to generate conic surfaces from their nearest spheres and that correct the profile of a surface that is polished. We study optimal angular sizes of a petal tool, which are found after we use linear programming to calculate the optimal dwell times of a set of complete annular tools placed in different zones of the glass surface. We report numerical results of designed petal tools.

Calculating petal tools by using genetic algorithms.

To pass from a spherical surface to a conic one, it is possible to use a petal tool or a small solid tool that is placed at different time intervals at several radial zones of the glass. Genetic algorithms are applied to calculate the angular sizes of the incomplete annular tools that make up the petal tools. We also present the desired wear results carried out with the petal tool that was designed on the basis of the dwell times of complete annular tools. These dwell times are calculated by using base functions that are generated with annular tools and by applying the genetic algorithms.

Experimental results and wear predictions of petal tools that freely rotate.

It is difficult to calculate the wear produced by free-pinned tools because their angular movement is not entirely predictable. We analyze the wear produced with free-pinned ring tools, using both simulations and experiments. We conclude that the wear of an incomplete ring is directly proportional to the ring's angular size, independently of the mean radius of the ring. We present an algorithm for calculation of the wear produced by free-pinned petal tools, as they can be considered a linear combination of incomplete free-pinned ring tools. Finally, we apply this result to the enhancement of a defective flat surface and to making a concave spheric surface.

Algorithm for the simulation of ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates.

We present a simple algorithm that makes possible the simulation of ronchigrams for any optical system in which it is possible to make an exact ray tracing. We report the simulations for the following grids: the Ronchi classical, square, circular, radial, circular-radial, biparabolic, elliptic-hyperbolic, bipolar, and bielliptic ones.

Sensing a wave front by use of a diffraction grating.

A method is presented to sense the wave front at the exit of an optical surface. This method uses a set of diffracted rays generated when a He-Ne laser impinges on a rectangular diffraction grating. The grating was placed near the curvature center of the surface to be tested. After they are reflected in the test surface, the diffracted rays have the information of the slopes of the wave front, like in the Hartmann test. The Hartmann pattern was registered near the curvature center and captured with a CCD camera. The slopes for each ray are measured from the experimental pattern, and they are compared with the ideal ones simulated in a computer. The evaluation was carried out by use of Seidel polynomials to obtain the information of the aberrations of a mirror 53 cm in diameter.

Edge effects with the Preston equation for a circular tool and workpiece.

In a polishing process the wear is greater at the edge when the tool extends beyond the border of the workpiece. To explain this effect, we propose a new model in which the pressure is higher at the edge. This model is applied to the case of a circular tool that polishes a circular workpiece. Our model correctly predicts that a greater amount of material is removed from the edge of the workpiece.

Static and dynamic removal rates of a new hydrodynamic polishing tool.

A new tool for hydrodynamic radial polishing, HyDra, allows for the local polishing of optical surfaces with a controllable wear rate. The results of the removal rate for different polisher types and sizes, applied air pressures for slurry expulsion, and tool height with respect to the working surface, are reported. We present a numerical analysis of the volumetric removal rate for the dynamic experiments as well as a comparison with a similar technique.