Bi-Ronchi-test proposal for wavefront sensing and aberration analysis based on the differential form of the Zernike polynomials

In this work, we show a proposal involving differential and vectorial equations to obtain the scalar surface of a wavefront (W) and the transversal aberration (TA) from the Bi-Ronchi test (BRT). The execution and computation times of this technique are significantly lower than those corresponding to the Ronchi test (RT). Moreover, it provides wavefront results along the axes X and Y by using bi-Ronchi ruling (bRR) with periods in XY. In the case of mirrors, this technique provides complete information of the whole surface along XY from a single test, with the potential to be applied in the other optical elements to measure their wavefronts. Hence, we propose the construction of square meshes, bRR type meshes, using Al and Au atomic depositions in glass substrates with different periods in XY, allowing us to observe bi-Ronchigrams. The captures of Bironchigrams enabled us to perform the analysis of the transversal aberrations in terms of the Zernike polynomials described by circular polynomials and an angular function with respect to the azimuthal term. We build a matrix solution from the vectors used to identify the centroids in the bi-Ronchigram higher-intensity zones, also considering the Rayces' equations from which the wavefront and the transversal aberrations are simultaneously obtained. We discuss and compare our results with the RT. Finally, we highlight an additional result related to the proposed technique, involving the contention of the Ronchigram in the Bi-Ronchigram, resembling the case where the BRT contains the RT.