A novel phase measuring deflectometry based on polar coordinate

Phase measuring deflectometry (PMD) is a method measure the surface of a mirror. However, when measuring convex mirrors, Cartesian coordinate fringes experience extreme compression. To address this issue, this paper proposes a novel PMD based on polar coordinate. This new method defines two linearly independent phase modulation directions, using the orthogonal basis of polar coordinate. It establishes polar coordinate fringes with rotational symmetry and radial pre-modulation, effectively reducing the impact of extreme compression at the edges. To correct the phase errors in polar coordinate fringes, a phase error compensation algorithm based on greyscale gradient is introduced. The algorithm calculates the influence factor of the eight connected domains around the error points to be compensated, utilising the phase grey gradient. The wavefront gradient data obtained through polar coordinate fringes are radial and tangential. Hence, a surface reconstruction method is proposed based on the Zernike partial derivative polynomial based on polar coordinate. In this method, the tangential partial derivatives and radial partial derivatives of the first 36 terms of Zernike in polar coordinate to construct Gram matrix equations. As a result, linearly independent Zernike recovery coefficients are obtained from coupled aliasing gradient data. Compared to the Cartesian coordinate system, the proposed method significantly reduces the fitting coefficient errors. Experimental measurements of convex mirrors with radii of curvature of 200 mm and 100 mm were conducted. The results demonstrate that compared to traditional PMD, this technique not only effectively suppresses extreme compression and increases the measurement area but also improves measurement accuracy by six times.