Gaussian quadrature to solve the wavefront in simultaneous Hartmann/Bi-Ronchi tests with SLM using directional derivative

Abstract We propose a technique for wavefront ( W ) sensing of optical systems using both Bi-Ronchi and Hartmann tests, modified by incorporating the classical tests by including a Spatial Light Modulator (SLM). To this aim, we replace the conventional Hartmann screens with structured apertures with a transmission Liquid Crystal Spatial Light Modulator (LC-SLM) using binary patterns of either circular or square apertures. We build overlapping horizontal and vertical fringe patterns to obtain equidistant square apertures (Bi-Ronchi). In the case of the Hartmann test, we substitute the square apertures with circles serving as binary masks, which we refer to as BRM (Bi-Ronchi Mask) and HM (Hartmann mask). We design an experimental setup in which a collimated laser illuminates the SLM, considering the diffractive effects of the SLM to optimize the identification of centroids, which yields transversal aberrations ( $$\bold{TA}({\rho},{\theta})$$ ). We assume a vector formulation suitable to interpret TA and thus directly solve $$\bold{W}({\rho }, {\theta})$$ by two-dimensional numerical integration applying directional derivative and Gaussian quadrature. We perform this process for both the Hartmann and Bi-Ronchi tests simultaneously, leveraging the spatial resolution and flexibility of the SLM. We present and discuss our results and compare them with those obtained using dedicated wavefront sensing software using the Ronchi test (under the conditions of our proposal) with the SLM.