Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry

The Zernike-polynomials-based modal reconstruction method is an important wavefront reconstruction method for lateral shearing interferometry. There are four typical Zernike-polynomial-based modal reconstruction methods: the Rimmer–Wyant method, the elliptical orthogonal transformation method, the numerical orthogonal transformation method (NOT), and the difference Zernike polynomial fitting method (DZF). In a previous paper [Appl. Opt.51, 5028 (2012)APOPAI0003-693510.1364/AO.51.005028], the overall performances of these four methods were comprehensively compared with each other. The conclusions showed that NOT and DZF have the highest reconstruction accuracies among these four methods. In addition, it was shown that the performance of NOT is identical to that of DZF; however, the reason behind this was not known until now, to our knowledge. In the present paper, we present a strictly mathematical proof for this highly significant result.