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Optimum phase retrieval using the transport of intensity equation

The Transport of Intensity Equation (TIE) relates linearly the phase of an object to the intensity distribution in the Fresnel region and can be used as a phase retrieval technique. The key element in a TIE based solver is the calculation of the axial intensity derivative. This parameter is calculated from a series of captured intensities but its accuracy is subject to several parameters, such as e.g. the separation between the measurement planes, the Signal to Noise Ratio (SNR) in the captured intensities, the actual object phase distribution. Despite the importance of the estimation of this parameter, there is no general discussion how to optimize the axial intensity derivative. In this work, we developed the mathematical framework in which the retrieved phase can be obtained. An optimal separation is derived, which minimizes the error in the calculation of the axial derivative. Besides this, we study using a numerical analysis how the accuracy of the axial derivative influence the accuracy of the retrieved phase. Hence, we present a numerical procedure based in the Root Square Mean Error, which is able to minimized the error in the retrieved phase. It is later shown that this analysis is significant more accurate than available methods proposed in the literature. It is further shown, that the plane separation that minimizes the error in the axial intensity derivative is different to the plane separation that minimizes the error in the retrieved phase.