Modeling, fabrication, and metrology of 3D printed Alvarez lenses prototypes.

In this work, we present the fabrication of two 3D printed plano-freeform prototypes, designed in such a way that, when assembled, an Alvarez lens is formed. The freeform surface of each element was mathematically described using Zernike polynomials and verified by implementing an off-axis null-screen test. Additionally, a characterization by refraction of the assembled lens was performed. Experimental images show the suitability of additive manufacturing engineering for prototyping freeform optics by providing a practical demonstration of the Alvarez lens concept.

Fabrication of biconvex spherical and aspherical lenses using 3D printing.

In this work, we present the methods of fabrication and characterization of biconvex spherical and aspherical lenses with 25 and 50 mm diameters that have been created via additive technology using a Formlabs Form 3 stereolithography 3D printer. After the prototypes are postprocessed, fabrication errors ≤2.47% for the radius of curvature, the optical power, and the focal length are obtained. We show eye fundus images captured with an indirect ophthalmoscope using the printed biconvex aspherical prototypes, proving the functionality of both the fabricated lenses and the proposed method, which is fast and low-cost.

Null screens to evaluate the shape of freeform surfaces: progressive addition lenses.

We propose a method for measuring the shape of freeform surfaces such as Progressive Addition Lenses (PAL). It is based on optical deflectometry by considering a non-uniform pattern of spots computed by using the null-screen method. This pattern is displayed on a flat LCD monitor being reflected on the freeform under test and whose image is recorded on a CCD camera placed at a predefined off-axis position. We use one image to calibrate the experimental setup and another to measure the freeform surface. We develop an iterative algorithm to retrieve the surface under test and calculate the spherical and cylindrical dioptric powers of the frontal freeform of a commercial PAL under test.

Null-screen design for highly freeform surface testing.

A new alternative to calculate the null-screen for highly freeform or complex surfaces for any desired pattern to be observed on the detector is presented. To validate the proposed method, we used the Zernike polynomials to design complex surfaces with sagittas greater or equal to 40 mm, and peak to valley greater or equal to 30 mm, between the used surface and the best fit sphere. The freeform surfaces were fabricated using a 3D printer and a five-axis CNC machine. With the proposed method we can calculate the image that will be observed over the detector for any null-screen and any freeform surface that want to be analyzed. The results showed that the proposal works very well for extremely fast complex freeform surfaces (with slopes ≤ 80°), obtaining an error smaller than 0.66% in PV and 0.36% in rms in sagitta differences.

Exact equations to measure highly aberrated wavefronts with the Hartmann test.

An exact vector expression for the deformations of a wavefront from any chosen reference surface, as a function of the directions of the real and reference rays, is deduced. It can be used with slope measuring test methods, such as Hartmann or Ronchi tests, but the need for a spherical reference is removed. We present simulated and experimental results to show the feasibility of this proposal.

General equations for the null-screen test for aspherical surfaces with deformation coefficients.

A modified approach to calculate the null screen for testing fast convex/concave aspherical surfaces with deformation coefficients is presented. Introducing the aberration polynomial into the equation of the sagitta, the null screens can be generated in a simple way, in contrast to the conventional design method. This approach is easy to implement for the calculation of the null screen since the equations presented here are easy to program computationally. The validation of this approach was done by analyzing the fast aspherical surface of a convex condenser lens (f/0.18). We obtained a percentage error smaller than 1.3% in the recovery of the coefficients that describe the shape of the surface.

Design of Hartmann type null screens for testing a plano-convex aspheric lens with a CCD sensor inside the caustic.

A new method to design Hartmann type null screens to test either qualitatively or quantitatively fast plano-convex aspherical lenses is presented. We design both radial and square null screens that produce arrays of circular spots uniformly distributed at predefined planes, considering that the CCD sensor is solely placed inside the caustic region. The designs of these null screens are based on knowledge of the caustic by refraction and on exact ray tracing. The null screens also serve to improve the alignment in optical systems.

Analytic aspheric coefficients to reduce the spherical aberration of lens elements used in collimated light.

We provide closed-form formulas for aspheric terms for either plano-convex or convex-plano aspheric lenses as functions of the paraxial parameters involved in the process of refraction. These formulas are obtained through an expansion in Taylor's series from the exact caustic equation produced by aspheric lenses considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. A comparison of the aspheric coefficients obtained through our analytic formulas and commercial optical design software is presented, showing good agreement. This is useful in reducing spherical aberration.

Modal integration of Hartmann and Shack-Hartmann patterns.

Instead of measuring the wavefront deformations directly, Hartmann and Shack-Hartmann tests measure the wavefront slopes, which are equivalent to the ray transverse aberrations. Numerous different integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. In this work we describe a modal method to integrate Hartmann and Shack-Hartmann patterns using orthogonal wavefront slope aberration polynomials, instead of the commonly used Zernike polynomials for the wavefront deformations.

Gaussian beam radius measurement with a knife-edge: a polynomial approximation to the inverse error function.

A method for approximating the inverse error function involved in the determination of the radius of a Gaussian beam is proposed. It is based on a polynomial inversion that can be developed to any desired degree, according to an a priori defined error budget. Analytic expressions are obtained and used to determine the radius of a TEM(oo) He-Ne laser beam from intensity measurements experimentally obtained by using the knife edge method. The error and the interval of validity of the approximation are determined for polynomials of different degrees. The analysis of the theoretical and experimental errors is also presented.

Sagittal and meridional radii of curvature for a surface with symmetry of revolution by using a null-screen testing method.

An algorithm to compute the sagittal and meridional radii of curvature for a surface of revolution is presented. The sagittal radius is obtained from the surface normal, and the meridional radius is calculated from a function fitted to the derivative of the sagittal curvature by using the surface-normals raw data. A calibration spherical surface is tested by using the null-screen testing method. Experimental results of the spherical surface show that the sagittal and meridional radii of curvature differ by 2.600% and 2.604%, respectively, with respect to the actual radius of the calibration spherical surface.

Improving fast aspheric convex surface tests with dynamic null screens using LCDs.

A method for testing fast aspheric convex surfaces with dynamic null screens using LCDs is shown. A flat null screen is designed and displayed on an LCD monitor with drop-shaped spots in such a way that the image, which is formed by reflection on the test surface, becomes an exactly square array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The position of the spots on the LCD can be changed in a dynamic way, to perform point-shifting of the image spots. The proposed procedure improves the dynamic point-shifting method. As has been shown previously, this process reduces the numerical error during the integration procedure, thereby improving the sensitivity of the test. The positioning accuracy for the screen spots is related to the LCD's spatial resolution. Results of the evaluation of a parabolic convex surface with f/#=0.22 are shown.

Classical Hartmann test with scanning.

In order to introduce many more evaluation points during the Hartmann test, the scanning of the screen across the pupil is proposed; after each step of the scan a different image of the bright spots is obtained. Basic ideas about how to design radial and square screens for the scanning are presented. Radial screens are scanned by rotation, whereas for square screens a linear inclined scan is enough to introduce many more evaluation points along two independent directions. For square screens it is experimentally shown that the lateral resolution of the test is improved.

Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen.

We report the testing of a fast off-axis surface based on the null screen principles. Here we design a tilted null screen with drop shaped spots drawn on it in such a way that its image, which is formed by reflection on the test surface, becomes an exact square array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The test surface has a radius of curvature of r = 20.4 mm (F/0.206). The surface departures from the best surface fit are shown; in addition, we show that the errors in the surface shape are below 0.4 mum when the errors in the determination of the coordinates of the centroids of the reflected images are less than 1 pixel, and the errors in the coordinates of the spots of the null screen are less than 0.5 mm.

Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal.

In general, a caustic by refraction at an arbitrary surface is commonly known as a diacaustic. We study the formation of the diacaustic in a plane interface between an isotropic medium and a uniaxial crystal, for both ordinary and extraordinary rays, when the crystal axis is perpendicular to the plane of incidence and when it lies in the plane of incidence. For the latter case two special positions of the crystal axis with respect to the normal to the refracting surface for the extraordinary rays are treated.

Profile and deformation coefficients measurement of fast optical surfaces.

In this work we show a method to obtain the profile of fast optical surfaces with symmetry of revolution, starting from experimental measurements of the longitudinal aberration and the angle between each normal line to the test surface and their optical axis. The method is based on a numerical and recursive integration, applied to a set of experimental measurements, the method gives a set of points on the test surface in orthogonal coordinates, from which we can determine the form of the surface, in independent way of a particular mathematical model that describes it. With its information is possible to adjust a polynomial to determine the paraxial radius curvature and the deformation coefficients. The method was applied to two surfaces and the results are compared with those obtained by a contact method, finding differences of some tenths of millimeter.

Testing fast aspheric convex surfaces with a linear array of sources.

We describe a noncontact test procedure with which to obtain the shapes of fast convex surfaces. For this, an array of sources is positioned in a straight line and separated in such a way that the image by reflection on the surface consists of a set of equally spaced bright spots. By rotating the surface, we test different meridians such that, after 360 degrees, the entire surface is measured. We present the source array design and the surface evaluation algorithm. We found that, to reduce numerical error in the evaluation of the shape of the surface, a numerical integration must be performed by a method that uses parabolic arcs instead of the traditional method that uses trapezoids. Through some numerical simulations we analyzed the accuracy of the method by introducing random displacements into the measured data. We found that to measure the quality of the surface with accuracy better than 5 microm, we have to measure the coordinates of the centroids on the image plane with an accuracy better than 0.5 pixel, and we to have measure the positions of the linear sources with an accuracy better than 0.5 mm. Experimental results for the testing of a carbon fiber convex sphere of 383.6-mm diameter (f/0.398) are shown.

Accuracy analysis in laser keratopography.

We analyze the accuracy of a laser keratopographer in the evaluation of corneal topography for non-smooth corneal surfaces and when some noise is introduced into measured data. Through some numerical simulations, cosinesoidal deformations with different amplitudes and spatial periods are introduced on theoretical surfaces. Gaussian noise is introduced on the simulated x and y position coordinates for the measured position of the reflected beam in order to simulate detection errors that are due to vibrations or electric and other noise on the position-sensing detector. We found that the topography of the surface could be obtained with reliable accuracy if the height-to-width ratio of the deformations of the surface is smaller than 0.02 and the error in the detection of position at the position-sensing detector is under 0.5 mm.