Retrieving Curvature’s Equation of a Radially Symmetric Concave Surface Using Fizeau Ring Fringes

In this work, we attempt to precisely find out the equation describing the shape of a radially symmetric concave surface. An experimental interference pattern of Fizeau ring fringes in case of transmission is obtained using the semi-reflecting concave surface (under study) placed on a glass plate. Theoretically, we showed that concentric ring fringes can be obtained by a concave surface which is a part of an ellipsoid or a paraboloid in addition to a sphere by simulating the ring fringes in each case. The simulated interferograms were obtained considering the traced optical paths of the interfered rays which guarantee accurate calculations. From a comparison between the experimental and the estimated interference patterns, we knew which equation was adapted to represent the concave surface used in our experiment. The best surface’s equation is that whose simulated pattern satisfied the minimum percent error with the experimental pattern. Interestingly, the surface’s equation was retrieved from a single experimental interferogram which was produced by a simple well-known optical setup. The proposed method is fast, simple and accurate which make it suitable for quality control of concave surfaces manufacturing.