In wavefront propagation through homogeneous media, the amplitude and phase of a wave are related by a partial differential equation first derived by A. Sommerfeld and I. Runge in 1911.1
The paraxial approximation to this equation is called the transport-of-intensity equation (TIE). In the early 1980s, Teague2
proposed using TIE as a phase retrieval technique, and today, TIE is widely used in applications such as wavefront sensing in astronomy4
and optical metrology,5,6
or as a phase-imaging technique in optical,7
and neutron beams.10
The theory of partial differential equations establishes that TIE is a well-posed technique for phase retrieval (i.e., a unique solution is known to exist), provided the irradiance is a smooth and positive defined function with appropriate boundary conditions. In practice, however, TIE may become ill-conditioned due to experimental errors (e.g., measurement errors, misalignment, photodetection noise). One way to reduce the effects of some of these errors is to increase the separation between planes of measurement of irradiance that provides the input data for TIE calculations. Recently, an analytical procedure for determining the optimum location of measurement planes has been proposed.11
However, it is important to consider two important errors introduced by separating the planes long distances. First, the partial derivative of the irradiance is evaluated using the finite differences approximation, which associated error increases with the separation between planes. Second, the assumption (implicit in TIE) that the wavefront propagates paraxially between planes might be not valid for large propagation distances in the case of highly aberrated wavefronts. Therefore, in practice, the accuracy in solving TIE is limited by this error trade-off in the separation between planes.
In this work, we study systematically the accuracy of phase reconstruction due to numerical and experimental errors. We show that image denoising techniques applied to irradiance measurements reduce the need for increased separation between planes, thereby avoiding the associated errors. Quantitative error analysis is presented for experimental data and simulations.
The paper is organized as follows: Section 2 introduces the numerical method used to solve TIE and its associated error. Section 3 describes the experimental setup. Section 4 summarizes the experimental error analysis and modeling. Section 5 describes the image denoising technique and our quantitative results of its application. Section 6 gives a brief discussion of the general results.