Robust and precise algorithm for aspheric surfaces characterization by the conic section

Journal of the European Optical Society-Rapid Publications, Apr 2017

Background A new algorithm for precise characterisation of rotationally symmetric aspheric surfaces by the conic section and polynomial according to the ISO 10110 standard is described. Methods The algorithm uses only the iterative linear least squares. It uses fitting the surface form in a combination with terms containing its spatial derivatives that represent infinitesimal transformations of form. Results The algorithm reaches sub-nanometre residuals even though the aspheric surface is translated and rotated in the space. Conclusion he algorithm is computationally robust and an influence of local surface imperfections can be easily reduced by use of a criterion for residuals.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1186%2Fs41476-017-0040-1.pdf

Robust and precise algorithm for aspheric surfaces characterization by the conic section

K?en Journal of the European Optical Society-Rapid Publications Robust and precise algorithm for aspheric surfaces characterization by the conic section Petr K?en 0 0 Czech Metrology Institute , Okruz?ni? 31, CZ63800 Brno , Czech Republic Background: A new algorithm for precise characterisation of rotationally symmetric aspheric surfaces by the conic section and polynomial according to the ISO 10110 standard is described. Methods: The algorithm uses only the iterative linear least squares. It uses fitting the surface form in a combination with terms containing its spatial derivatives that represent infinitesimal transformations of form. Results: The algorithm reaches sub-nanometre residuals even though the aspheric surface is translated and rotated in the space. Conclusion: he algorithm is computationally robust and an influence of local surface imperfections can be easily reduced by use of a criterion for residuals. Aspheric lens; Robust algorithm; Least squares fitting; Metrology; ISO 10110 - Background Aspheric surfaces are recently widely used in industry. One of their applications is aspheric lens that often needs its precise characterisation of form. The description of the aspheric surface by the conic section with a polynomial correction is common in ray tracing software and in producer specifications of aspheric lens. The conic section surface fitting with a polynomial correction was addressed by several authors [1?5]. Also alternative descriptions of aspheric surfaces were introduced e.g. in [6, 7]. Nevertheless, a simple and robust algorithm is still needed to evaluate the conic section from measurement data in the ISO 10110-12 form. The coordinate system is shown in the Fig. 1. The design shape of aspheric surfaces is often described by the z-coordinate as a function of the distance r from z-axis in the form X A2ir2i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffi2ffiffi! 1??1 ? k?R2 ? 21R r2 ? 18?R3k r4 where R is the radius of curvature at the vertex and k is the conic constant (k < -1 hyperbolic, k = -1 parabolic, k > -1 elliptical, k = 0 spherical surfaces). The correction of surface is given by the even-power polynomial with coefficients A. Method Radius of curvature The expansion of (2) shows that the radius of curvature R at the vertex (r is small) can be obtained from linear least squares (i.e. L2-norm) simply using the first coefficient of an even-power polynomial The higher-order terms of Taylor series at the vertex are negligible and the even-power polynomial with the 18th power ? The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. z ? q~2r2 ? q~4r4 ? q~6r6 ? q~8r8 ? q~10r10 ? q~12r12 is sufficient for reduction of estimation error for R from data with a given range of r. However, high degree of polynomial could introduce numerical errors. They can be reduced by the following way. The linearity of problem allows making the second fit for residuals z ? z?~z for initial estimate ~z obtained from q~2i as z ? q2r2 ? q4r4 ? q6r6 ? q8r8 ? q10r10 ? q12r12 with relatively small error because the first term of expansion of (2) also does not depend on k. The error is below 10?6 in relative for examples from [3] and the 18th power polynomial, except the case 1 with relative error 0.002 for R. Thus it is a robust way to evaluate radius of curvature at the vertex. The polynomial with coefficients q2i also describes the aspheric surface very well for medium precision applications (i.e. 1? flatness of wavefront). Nevertheless, the conic section describes the aspheric form better with less number of coefficients. For example, the Taylor series of (2) for hyperbolic surface with high k converges slowly and thus the even-power polynomial must have more terms for the corresponding precision. It corresponds to the parabolic solution from the previous section. The convergence of iterations is worse close to k = -1 because there is a small contribution from the conic section (the terms of expansion for function c). In the next step of algorithm, user selects between the hyperbolic region (k < -1) and the elliptic region (k > -1). If the algorithm output has large errors the second option could be selected automatically. The next values are then R1 = R0 and k1 = -201 or k1 = -0.5 respectively (The algorithm also works for oblate elliptical surfaces if k1 is set as a larger positive number.). The value of k1 for hyperbolic region can be selected closer to the value -1 (e.g. -3 because the most of commercial aspheric lenses have the conic constant above -3). Nevertheless, the value -201 was selected for dem (...truncated)


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1186%2Fs41476-017-0040-1.pdf

Petr Křen. Robust and precise algorithm for aspheric surfaces characterization by the conic section, Journal of the European Optical Society-Rapid Publications, 2017, pp. 11, Volume 13, Issue 1, DOI: 10.1186/s41476-017-0040-1