Direct and inverse discrete Zernike transform

Contiene: 4 ilustraciones, 2 tablas y fórmulas.An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.Supported by the Comisión Interministerial de Ciencia y Tecnología, Spain, under Grant FIS2008-00697. R. Rivera acknowledges support by Alban, the European Union Program of High Level Scholarships for Latin America, scholarship Nº E07D402088CL. - This paper was published in Optics Express, and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-24269. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.Peer reviewe

Positioning of diffuse metallic target with a point diffractive interferometer

Proton acceleration obtained by focusing an ultraintense ultrafast laser beam presents the technological and metrological challenge of correctly placing diffuse metallic target at the focus of the laser beam. In this work we present the use of the Point diffractive interferometer for solving this problem. We studied the accuracy and precision of the system at repositioning the metallic target after displacing and horizontally tilting the target out of its reference position. We achieved an accuracy at repositioning the target at its reference position of 1.50 µm and 6 arcmin with a precision of 1.40 µm and 5.7 arcmin. Our work shows the high accuracy provided by a system as simple as the Point Diffraction interferometer even at positioning surfaces with diffuse reflectionS

Jacobi–Fourier phase mask for wavefront coding

In this work we propose Jacobi–Fourier phase masks for wavefront coding-based imaging systems. The optical properties of the phase mask is study in detail and numerical simulation are shown. Pixel size and noise are taken into account for the deconvolution of images. Numerical simulations indicate that overall performance is better than of the well-known and commonly used trefoil phaseThis work was supported by the Spanish Ministry of Economía y Competitividad FIS2016-77319-C2-1-R, and FEDER, Xunta de Galicia/FEDER ED431E 2018/08. E. González Amador thanks to Consejo Nacional de Ciencia y Tecnología (CONACyT); with CVU no. 714742. Also, we thank by the support to PADES program; Award no. 2018-13-011-047S

Wavefront sensing with critical sampling

Different types of nonredundant sampling patterns are shown to guarantee completeness of the basis formed by the sampled partial derivatives of Zernike polynomials, commonly used to reconstruct the wavefront from its slopes (wavefront sensing). In the ideal noise-free case, this enables one to recover double the number of modes J than sampling points I (critical sampling J=2I ). With real data, noise amplification makes the optimal number of modes lower I<J<2I . Our computer simulations show that optimized nonredundant sampling provides a significant improvement of wavefront reconstructions, with the number of modes recovered about 2.5 higher than with standard sampling patterns.Peer reviewe

A calibration set for ocular aberrometers: Manufacture, testing and application

Purpose. To manufacture and test a set of phase plates for the calibration of ocular aberrometers and eventually to apply it to the calibration of an ocular laser ray tracing aberrometer. Methods. The set of phase plates is made by a grey-scale single-mask photosculpture in photoresist method. Each plate induces a given amount of a particular aberration (Zernike) mode. The set contains two subsets: (1) different pure Zernike modes to test the accuracy among different orders (from 3rd to 7th, about 0.3 - 0.4 microns); and (2) plates having different amounts of the same mode, 3rd order coma ranging from 0.11 to 0.47 microns. Right after manufacturing, the plates were tested twice, as a cross-check, measuring the aberration pattern of each plate with a Mach-Zehnder interferometer and a single-pass Hartmann-Shack wavefront sensor. The set was then applied to the calibration of an ocular double-pass laser ray tracing aberrometer. Results. We found a close agreement between the three types of measurement. The maximum difference between H-S and LRT measurements was 0.032 μm (that is about λ/20, half of the typical measuring error in human eyes). This permitted us to detect a small bias in the ocular laser ray tracing aberrometer. Conclusions. The calibration set may be a powerful tool for the assessment of accuracy and reliability in ocular aberrometry. It allowed us to discover a small bias, that is almost impossible to detect working with human eyes or trial lenses. This type of calibration tool is especially important in clinical environments.This work has been partially supported by the Comisión Interministerial de Ciencia y Tecnología (Spain), under grant DPI2002-04370-C02.Peer reviewe

A new calibration set of phase plates for ocular aberrometers

PURPOSE: To manufacture and test a set of phase plates for the calibration of ocular aberrometers and apply it to the calibration of an ocular laser ray tracing aberrometer. METHODS: The set of phase plates is made by a greyscale single-mask photosculpture in photoresist method. Each plate induces a given amount of a particular aberration (Zernike) mode. The set contains two subsets: 1) pure Zernike modes to test the accuracy among different orders (from 3rd to 7th, approximately 0.3 to 0.4 μm); and 2) plates having different amounts of the same mode, 3rd order coma ranging from 0.11 to 0.47 μm. After manufacturing, the plates were tested twice, as a crosscheck, measuring the aberration pattern of each plate with a Mach-Zehnder interferometer and a single-pass Hartmann-Shack wavefront sensor. The set was then applied to the calibration of an ocular double-pass laser ray tracing aberrometer. RESULTS: Close agreement was found between the three types of measurement. The maximum difference between Hartmann-Shack and laser ray tracing measurements was 0.032 μm (ie, approximately λ/20, half of the typical measuring error in human eyes). This permitted detection of a small bias in the ocular laser ray tracing aberrometer. CONCLUSIONS: The calibration set may be a powerful tool for the assessment of accuracy and reliability in ocular aberrometry. It discovered a small bias, which is almost impossible to detect when working with human eyes or trial lenses. This type of calibration tool is especially important in clinical environments.Peer Reviewe

A new calibration set of phase plates for ocular aberrometers.

PURPOSE: To manufacture and test a set of phase plates for the calibration of ocular aberrometers and apply it to the calibration of an ocular laser ray tracing aberrometer. METHODS: The set of phase plates is made by a greyscale single-mask photosculpture in photoresist method. Each plate induces a given amount of a particular aberration (Zernike) mode. The set contains two subsets: 1) pure Zernike modes to test the accuracy among different orders (from 3rd to 7th, approximately 0.3 to 0.4 μm); and 2) plates having different amounts of the same mode, 3rd order coma ranging from 0.11 to 0.47 μm. After manufacturing, the plates were tested twice, as a crosscheck, measuring the aberration pattern of each plate with a Mach-Zehnder interferometer and a single-pass Hartmann-Shack wavefront sensor. The set was then applied to the calibration of an ocular double-pass laser ray tracing aberrometer. RESULTS: Close agreement was found between the three types of measurement. The maximum difference between Hartmann-Shack and laser ray tracing measurements was 0.032 μm (ie, approximately λ/20, half of the typical measuring error in human eyes). This permitted detection of a small bias in the ocular laser ray tracing aberrometer. CONCLUSIONS: The calibration set may be a powerful tool for the assessment of accuracy and reliability in ocular aberrometry. It discovered a small bias, which is almost impossible to detect when working with human eyes or trial lenses. This type of calibration tool is especially important in ciinical environments.Peer reviewe

Using Shack-Hartmann wavefront sensors and Zernike coefficients for beam characterisation: numerical procedures

When using Shack-Hartmann wavefront sensors (SH) and Zernike coefficients (Zs) in applications where the position of the measurement and the point of interest are far apart, as it is common practice in ophthalmic optics, problems in the interpretation of the values of the Zs arise, related to how the shape of the wavefront propagates along the beam. One typical example is pupil conjugation where an auxiliary lens is added to match the size of the area of the interest of the beam with the size of the entrance pupil of the SH used for measurements. In the present work, we address this problem in the framework of a numerical scheme for modeling the beam propagation. We calculate the wavefronts with exact ray tracing plus the fitting of the impacts so as to match a rectangular grid. This procedure allows the subsequent calculation of the Zs or, similarly, the pupil function at an arbitrary plane perpendicular to the optical axisThe authors acknowledge the 'Agencia Estatal de Investigación' (AEI) and the 'Fondo Europeo de Desarrollo Regional' (FEDER), under project FIS2016-77319-C2-2-R of the Spanish 'Ministerio de Economia, Industria y Competitividad'.Postprint (published version

Low-cost method for manufacturing self-adherent PDMS lenses for presbyopia

We present an easy method for manufacturing elastomeric self-adherent lenses with optical quality. The lenses were manufactured in polydimethylsiloxane using ophthalmic and trial lenses of different base curve as master molds to generate different refractive powers. The diameter of the manufactured lenses ranged between 30 mm and 60 mm. We characterized the lenses alone and after being self-adhered to different spherocylindrical ophthalmic lenses, by measuring the refractive power and the Zernike aberrations. Besides, an artificial myopic eye was used to quantify the image quality provided by the lenses self-adhered to several ophthalmic lens. The manufactured lenses showed good optical quality with no aberrations. Low order refractive errors were found in one case after self-adhesion. The results confirm the suitability and simplicity of the proposed method to manufacture high quality elastomeric lenses that can be self-adhered to ophthalmic lenses to add power and help with presbyopiaThis work has been funded by Ministerio de Ciencia e Innovación PID2020-115909RB-I00, and by Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia (ED43B 2020/29). Ana I. Gómez-Varela acknowledges Consellería de Cultura, Educación e Ordenación Universitaria for a Postdoctoral fellowship (Xunta de Galicia, Spain; ED481D‐2021‐019)S