Reliable and accurate testing methods are essential to guiding the polishing process during the figuring of optical telescope mirrors. With the natural advancement of technology, the procedures and instruments used to carry out this delicate task have consistently increased in sensitivity, but also in complexity and cost. Fortunately, throughout history, the Foucault knife-edge test has shown the potential to measure transverse aberrations in the order of the wavelength, mainly when described in terms of physical theory, which allows a quantitative interpretation of its characteristic shadowmaps. Our previous publication on this topic derived a closed mathematical formulation that directly relates the knife-edge position with the observed irradiance pattern. The present work addresses the quite unexplored problem of the wavefront’s gradient estimation from experimental captures of the test, which is achieved by means of an optimization algorithm featuring a proposed ad hoc cost function. The partial derivatives thereby calculated are then integrated by means of a Fourier-based algorithm to retrieve the mirror’s actual surface profile. To date and to the best of our knowledge, this is the very first time that a complete mathematical-grounded treatment of this optical phenomenon is presented, complemented by an image-processing algorithm which allows a quantitative calculation of the corresponding slope at any given point of the mirror’s surface, so that it becomes possible to accurately estimate the aberrations present in the analyzed concave device just through its associated foucaultgrams.
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