Determination of the phase in the center of a circular two-beam interference pattern to determine the displacement of a rough surface

We propose a fast algorithm to extract the phase in the center of an interferogram with circular fringes obtained in a simple Michelson interferometer setup with one rough surface. The phase in the center is a linear function of the displacement of an object, which is very advantageous. It avoids the uncertainty in displacement associated with intensity measurements near a maximum or minimum. Also the direction of movement is always clear. The method used also overcomes the problems associated with speckle, unavoidable with a rough surface, common in industrial applications, when using laser as a light source. Using a large number of points in the interferogram to obtain one value largely reduces noise and also increases the accuracy. Displacement is measured with an accuracy <10  nm.