Simultaneous estimation of phase and its pth order derivatives

One unaddressed challenge in optical metrology has been the measurement of higher order derivatives of rough specimens subjected to loading. In this paper, we investigate an approach that allows for the simultaneous estimation of the phase and its higher order derivatives from a noisy interference field. The interference phase is represented as a weighted linear combination of linearly independent Fourier basis functions. The interference field is represented as a state space model with the weights of the basis functions as the elements of the state vector. These weights are accurately estimated by employing the extended Kalman filter. The interference phase and phase derivatives are subsequently computed using the estimated weights. Since the Fourier basis functions are infinitely differentiable, phase derivatives of any arbitrary order can be estimated. Simulation and experimental results are provided to substantiate the effectiveness of the proposed method in the presence of high noise.