**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 selfimage 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

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

**Citation:** Han, C.-Y.; Lo, W.-T.; Chen,

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(FFT) and least-squares fit methods [11]. Lee proposed a Talbot interferometry-based digital image method for measuring the focal length of lenses without using moiré fringes. In this method, only one grating is required and the original Fourier transform is used to access the spectrum beyond the limitations of the usual fast Fourier transform [12]. However, using the original Fourier transform can be time-consuming.

In this paper, a method based on digital-grating moiré effect is proposed for measuring the focal lengths and radius of curvatures of spherical lenses and mirrors. The measurement system consists of a test light source, a Ronchi grating, a digital camera, and a personal computer. Based on Fresnel diffraction, the electric fields of the test light at various positions and the intensity at an observation screen were derived using the kernel equation for the method. After the test light passed through the grating and sample, the light intensity was projected on the observation screen and captured by the digital camera. This light intensity was superimposed on a digital grating to generate a moiré fringe. Using the slant angle of the moiré fringe, the focal length and radius of curvature of the test sample were determined. To demonstrate the feasibility of the proposed method, the focal lengths of three convex mirrors and two concave lenses were measured. The measurement percent errors for the focal lengths and radius of curvatures were less than 0.5%. Because the method applied digital image post processing, only a single digital image was required to be captured during measurement. Therefore, this method is time saving and free of mechanical vibrations. In addition, only one grating is required in this method. Consequently, this method has merits such as a simple establishment, easy operation, high stability, high accuracy, and low cost. Thus, it has considerable potential in relevant applications.
