For several years, scientific, industrial, and biological fields have benefited from knowledge of phase information, which allows for the revealing of hidden features of various objects. An alternative to interferometry is single-beam phase retrieval techniques that are based on the transport of intensity equation, which describes the relation between the axial derivative of the intensity and the phase distribution for a given plane in the Fresnel region. The estimation of the axial intensity derivative is obtained from a series of intensity measurements, where the accuracy is subject to an optimum separation between the measurement planes depending on the number of planes, the level of noise, and the actual object phase distribution. In this Letter, a quantitative analysis of the error in estimated axial derivative is carried out and a model is reported that describes the interdependence between these parameters. The results of this work allow for estimation of the optimum separation between measurement planes with minimal error in the axial derivative.

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Phase retrieval based on the Transport of Intensity Equation (TIE) has shown to be a powerful tool to obtain the phase of complex fields. Recently, it has been proven that the performance of TIE techniques can be improved when using unequally spaced measurement planes. In this paper, an algorithm is presented that recovers accurately the phase of a complex objects from a set of intensity measurements obtained at unequal plane separations. This technique employs multiple band-pass filters in the frequency domain of the axial derivative and uses these specific frequency bands for the calculation of the final phase. This provides highest accuracy for TIE based phase recovery giving minimal phase error for a given set of measurement planes. Moreover, because each of these band-pass filters has a distinct sensitivity to noise, a new plane selection strategy is derived that equalizes the error contribution of all frequency bands. It is shown that this new separation strategy allows controlling the final error of the retrieved phase without using a priori information of the object. This is an advantage compared to previous optimum phase retrieval techniques. In order to show the stability and robustness of this new technique, we present the numerical simulations.

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Deterministic single beam phase retrieval techniques based on the transport of intensity equation (TIE) use the axial intensity derivative obtained from a series of intensities recorded along the propagation axis as an input to the TIE-based solver. The common belief is that, when reducing the error present in the axial intensity derivative, there will be minimal error in the retrieved phase. Thus, reported optimization schemes of measurement condition focuses on the minimization of error in the axial intensity derivative. As it is shown in this contribution, this assumption is not correct and leads to underestimating the value of plane separation, which increases the phase retrieval errors and sensitivity to noise of the TIE-based measurement system. Therefore, in this paper, a detailed analysis that shows the existence of an optimal separation that minimizes the error in the retrieved phase for a given TIE-based solver is carried out. The developed model is used to derive analytical expressions that provide an optimal plane separation for a given number of planes and level of noise for the case of equidistant plane separation. The obtained results are derived for the widely used Fourier-transform-based TIE solver, but it is shown that they can also be applied to multigrid-based techniques.