Mean Square Error-Based Approach for the Detection of Focal Position of a Lens

Abstract

Determination of focal length of a lens is a common problem encountered in optical instrumentation; however, there exist very few solutions which are simple, inexpensive and yield automated output. In this paper, we present the results of investigation undertaken towards determining the focal length a test lens with high accuracy and precision by undertaking digital processing of self-images obtained at two Talbot planes using single linear grating. Light from a collimating lens is incident onto a test lens and mirror combination; the back-reflected light is incident onto a beam splitter to draw the beam for interferometric evaluation. Single CCD camera is used to capture self-images at the first and highest feasible Talbot plane for different positions of a given test lens. Self-images have been processed digitally to find mean square error (MSE) corresponding to each position of the lens. MSE is lowest when the lens is positioned at its focal length, for all other positions of lens MSE remains higher. Hence, focal position can be exactly differentiated from all other positions of lens. Also, it has been observed that MSE increases as the distance between Talbot planes increases. Hence, if the MSE is calculated for the first and the farthest Talbot plane, sensitivity is improved. To achieve high sensitivity, Talbot self-images at the first and till the eight Talbot planes (due to practical limitations) are analysed. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.