An extrapolation-based method for improving the accuracy of phase retrieval with the transport of intensity equation

Transport of Intensity Equation (TIE) is a simple and efficient method for phase retrieval by solving the equation between the intensity axial derivative and phase. In this method, the estimation of the axial derivative of intensity is very crucial. Simply, we use two defocused intensity images to estimate the axial derivative by finite difference method. However, the result is still unsatisfactory even though the optimal defocused distance is adopted. The reason lies in that the intensity’s axial change is not linear in the propagation of light. Simply using the finite difference between the two defocused images will ignore higher order axial derivatives. In other words, the estimation of the axial derivative of intensity will contain nonlinear errors. To solve this problem, we propose an extrapolation-based method to estimate the axial derivative of intensity using multiple intensity images. With Taylor expansion and a series of combination and eliminations on these images, high order terms of axial derivative errors are removed. As a result, the nonlinear errors in estimation of the axial derivative will be reduced. The performance of our proposed method for different types of phases under different illumination conditions is investigated. Compared with normal TIE, our method can obtain a much more accurate phase profile.