The transport-of-intensity equation (TIE) is applied in the reconstruction of two interfering wavefronts by analyzing the interference patterns and their derivatives along their common propagation directions. The TIE is extended from one wave to two waves and is then applied to calculate the phase of the interference field. Finally, the phase shift concept is applied to reconstruct the phase distribution of two waves. The consistency of the method is verified by simulation.

In the last decade, the transport of intensity has been increasingly used in microscopy, wavefront sensing, and metrology. In this study, we verify by simulation and experiment the use of the transport of intensity equation (TIE) in the accurate testing of optical aspheric surfaces. Guided by simulation results and assuming that the experimental setup parameters and the conic constants are known, one can estimate an appropriate defocusing distance Δz that leads to an accurate solution of the TIE. In this paper, this method is verified through the construction of a non-nulled experiment for testing the 2D profile of an aspheric surface. The theoretical method and experimental results are compared to validate the results. Finally, to validate the TIE methodology, the phase distribution obtained by TIE is compared with the phase distribution obtained by a Shack-Hartmann sensor.

Recently the transport-of-intensity equation (TIE) has extended from one wave to two waves and then applied to calculate the phase of the interference field. In this work we will present the experimental verification of the application of TIE in the reconstruction of two interfering wavefronts. A Fizeau interferometer with phase shift capability is used for testing a flat surface. An error analysis is performed on the difference between the reconstructed wavefronts using TIE and the one measured wavefronts using phase shifting method. The error analysis shows some systematic errors with RMS value less than 0.5rad or λ/10. The issues such as computation time and spatial resolution of the reconstructed waves are discussed and possible applications of the presented method are given.

Nehmetallah, "A guide to properly select the defocusing distance for accurate solution of transport of intensity equation while testing aspheric surfaces,"