Automated collimation testing by determining the statistical correlation coefficient of Talbot self-images

Abstract

In this paper, we propose a simple, fast, and accurate technique for detection of collimation position of an optical beam using the self-imaging phenomenon and correlation analysis. Herrera-Fernandez et al. [J. Opt.18, 075608 (2016)JOOPDB0150-536X10.1088/2040-8978/18/7/075608] proposed an experimental arrangement for collimation testing by comparing the period of two different self-images produced by a single diffraction grating. Following their approach, we propose a testing procedure based on correlation coefficient (CC) for efficient detection of variation in the size and fringe width of the Talbot self-images and thereby the collimation position. When the beam is collimated, the physical properties of the self-images of the grating, such as its size and fringe width, do not vary from one Talbot plane to the other and are identical; the CC is maximum in such a situation. For the de-collimated position, the size and fringe width of the self-images vary, and correspondingly the CC decreases. Hence, the magnitude of CC is a measure of degree of collimation. Using the method, we could set the collimation position to a resolution of 1 μm, which relates to ±0.25 in terms of collimation angle (for testing a collimating lens of diameter 46 mm and focal length 300 mm). In contrast to most collimation techniques reported to date, the proposed technique does not require a translation/rotation of the grating, use of complicated phase evaluation algorithms, or an intricate method for determination of period of the grating or its self-images. The technique is fully automated and provides high resolution and precision.

Histogram error based algorithm for efficient collimation testing

“…A comparison regarding the sensitivity in the collimation setting of the proposed technique with the previously reported technique [18] needs discussion. The technique used in [18] condensed the comparison of two (often large) 2D images down to a single scalar value; however, calculation of CC is computationally intensive.…”

“…A comparison regarding the sensitivity in the collimation setting of the proposed technique with the previously reported technique [18] needs discussion. The technique used in [18] condensed the comparison of two (often large) 2D images down to a single scalar value; however, calculation of CC is computationally intensive. It is well established that in Talbot interferometry, the higher accuracy and sensitivity in measurement is obtained when the images being compared are widely separated Talbot planes [6,16].…”

“…To validate the above statement, the sensitivity of the parameters, CC [18] and HE, are compared for the collimated position of the lens, when the images are captured at the 1st and 8th Talbot planes. The computed parameters, HE and CC, corresponding to two images captured at Talbot planes are: HE=0.0040 (0.4% away from the ideal value, HE=0, which is the minimum value, and is indicative of the perfect matching of the pair of images), and CC=0.19 (81% away from the ideal value, CC=1, corresponding to the perfect matching of the pair of images).…”

“…Additionally, determining the period of self-image requires laborious experimental effort, rigorous computation, and complex algorithms; thus considerable effort is involved in achieving beam collimation. Recently, we proposed a technique based on correlation coefficient (CC) for efficient detection of beam collimation [18]. The collimation position is determined by calculating the statistical Pearson's CC between the two self-images of the grating, acquired by two identical cameras positioned at different Talbot planes.…”

Two-dimensional Fourier domain Ronchi ruling measurement using Talbot-based crossing point modeling

“…Also, it is important to note that the proposed technique can be applied for measuring a collimation state or evaluating various beam characteristics, which sets a broader context of the applications. [4][5][6] 2 Talbot-Based Crossing Point Modeling…”