MSE-based analysis of circular grating self-images for testing beam collimation

Abstract

Mean square error (MSE) is used to detect variations in the period between a pair of self-images formed at two different Talbot planes of a circular grating (CG) using a beam splitter in a conventional collimation testing setup. By varying the position of the collimator with respect to the point source, the collimation state of the input beam is varied and the computed MSEs are analyzed to deduce the collimation state. The minimum value of the MSE indicates beam collimation. For equal sized images, the MSE relates to the sum of the squared difference between spatially correspondent pixel values of the images. Since comparison of the spatial information takes place at the pixels’ level, any small spatial shift between patterns of the two self-images due to collimation error is detected with precision. The CG, comprising concentric circular structures, offers added advantage in terms of error-free alignment, which otherwise is error prone and cumbersome with widely used linear gratings. It is well known that self-images formed with circular grating have good fidelity with less optical distortions and irregularities, especially at distant Talbot planes. Also, the self-images formed with circular gratings are less affected by lens aberrations, tilts, misalignments, etc. Higher sensitivity in beam collimation is achievable, as self-images of a CG can be recorded at widely separated Talbot planes, and analyzed using an algorithm which is more responsive toward any minute difference between them. The suggested method is promising for a quick collimation setting with good accuracy and enhanced sensitivity.

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“…To evaluate the effect of image reconstruction, the commonly used indicators are the mean-square error (MSE) and peak signal-to-noise ratio (PSNR) of the reconstructed image and the test image [ 17 , 18 ]. Among them, the mean square error calculation equation is …”