photonics hv Article Measurement of Focal Length and Radius of Curvature for Spherical Lenses and Mirrors by Using Digital-Grating Moiré Effect Chien-Yuan Han 1 , Wen-Tai Lo 2 , Kun-Huang Chen 3 , Ju-Yi Lee 4 , Chien-Hung Yeh 2 and Jing-Heng Chen 2, *   Citation: Han, C.-Y.; Lo,W.-T.; Chen, K.-H.; Lee, J.-Y.; Yeh, C.-H.; Chen, J.-H. Measurement of Focal Length and Radius of Curvature for Spherical Lenses and Mirrors by Using Digital-Grating Moiré Effect. Photonics 2021, 8, 252. https:// doi.org/10.3390/photonics8070252 Received: 25 May 2021 Accepted: 29 June 2021 Published: 1 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Electro-Optical Engineering, National United University, No. 2, Lienda, Nanshi Li, Miaoli 36063, Taiwan; cyhan@nuu.edu.tw 2 Department of Photonics, Feng Chia University, No. 100, Wenhwa Rd., Seatwen, Taichung 40724, Taiwan; fcum0405954@alumni.fcu.edu.tw (W.-T.L.); yehch@fcu.edu.tw (C.-H.Y.) 3 Department of Electrical Engineering, Feng Chia University, No. 100, Wenhwa Rd., Seatwen, Taichung 40724, Taiwan; chenkh@fcu.edu.tw 4 Department of Mechanical Engineering, National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan 320317, Taiwan; juyilee@ncu.edu.tw * Correspondence: jhchen@fcu.edu.tw; Tel.: +886-4-2451-7250 (ext. 5093); Fax: +886-4-2451-0182 Abstract: This paper proposes the use of digital-grating moiré effect for measuring the focal lengths and radius of curvatures of biconvex and biconcave spherical simple lenses and spherical mirrors. Based on Fresnel diffraction, the equation for the electric field of propagated light passing through the test samples was derived. Through digital image post processing, the recorded intensity on an observation screen was superimposed on a digital grating to generate a moiré pattern. On substituting the slant angle of the moiré pattern into the derived equation, the focal lengths and radius of curvatures could be determined. The experimental results successfully demonstrated the feasibility of the proposed method; the percent errors for focal length and radius of curvature measurement were less than 0.5%. The measurement uncertainty was analyzed and the correctness of the derived equation was confirmed through simulation. Because of the use of digital image post processing, the proposed method has advantages such as a simple set up, easy operation, high stability, high accuracy, and low cost. Thus, the method has considerable potential in relevant application. Keywords: moiré effects; diffraction theory; lenses; mirrors; digital image processing 1. Introduction Focal length and radius of curvature, the most important parameters of spherical lenses and mirrors, have been widely used to design optical systems and commercial electro-optical products. The measurement of focal length and radius of curvature is necessary to ensure the quality and performance of lenses and mirrors. Methods such as nodal slide and image magnification have been proposed for these measurements [1]. These methods have been successfully applied to measure the lenses with short focal lengths or large numerical apertures. Modern methods mainly use the Talbot effect and moiré technique to measure focal length [2–8]. In both these methods, two Ronchi gratings are required. After the test light passes through the test lens and mirror, the magnified self- image of the first grating is superimposed on the second grating to generate moiré fringes. By rotating one grating with respect to the other, the slant angle of the moiré fringes can be used to estimate to the curvature and focal length of the test samples. However, the rotation of the grating can introduce mechanical vibrations. In addition, two gratings are required; therefore, measurement using these two methods is complex and time-consuming. Digital image post processing, a recent trend in relevant applications, has several advantages [9,10]. The technique proposed by Angelis et al. for the automated analysis of moiré fringes for accurate measurement of the focal length of lenses is based on the fast Fourier transform Photonics 2021, 8, 252. https://doi.org/10.3390/photonics8070252 https://www.mdpi.com/journal/photonics