The Effect of the Range of a Modulating Phase Mask on the Retrieval of a Complex Object from Intensity Measurements

The authors have been supported by the postdoctoral project (1.1.1.2/16/I/001, 1.1.1.2/VIAA/1/16/199), the CAMART2 project (grant agreement ID 739508), the Latvian Investment and Development Agency (LIDA) project (KC-PI-2017/105), and the grant for the Latvian State Emeritus Scientists.In many fields of science, it is often impossible to preserve the information about the phase of the electromagnetic field, and only the information about the magnitude is available. This is known as the phase problem. Various algorithms have been proposed to recover the information about phase from intensity measurements. Nowadays, iterative algorithms of phase retrieval have become popular. Many of these algorithms are based on modulating the object under study with several masks and retrieving the missing information about the phase of an object by applying mathematical optimization methods. Several of these algorithms are able to retrieve not only the phase but also the magnitude of the object under study. In this study, we investigate the effect of the range of modulation of a mask on the accuracy of the retrieved magnitude and phase map. We conclude that there is a sharp boundary of the range of modulation separating the successfully retrieved magnitude and phase maps from those retrieved unsuccessfully. A decrease in the range of modulation affects the accuracy of the retrieved magnitude and phase map differently. © 2021 V. Karitans et al., published by Sciendo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.Postdoctoral project (1.1.1.2/16/I/001, 1.1.1.2/VIAA/1/16/199); the Latvian Investment and Development Agency (LIDA) project (KC-PI-2017/105), and the grant for the Latvian State Emeritus Scientists; Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508, project CAMART2

Phase retrieval in arbitrarily shaped aperture with the transport-of-intensity equation

Phase is not easy to detect directly as intensity, but sometimes it contains the really desired information. The transport-of-intensity equation (TIE) is a powerful tool to retrieve the phase from the intensity. However, by considering the boundary energy exchange and the whole energy conversation in the field of view, the current popular Fast Fourier transform (FFT) based TIE solver can only retrieve the phase under homogeneous Neumann boundary condition. For many applications, the boundary condition could be more complex and general. A novel TIE phase retrieval method is proposed to deal with an optical field under a general boundary condition. In this method, an arbitrarily-shape hard aperture is added in the optical field. In our method, the TIE is solved by using iterative discrete cosine transforms (DCT) method, which contains a phase compensation mechanism to improve the retrieval results. The proposed method is verified in simulation with an arbitrary phase, an arbitrarily-shaped aperture, and non-uniform intensity distribution. Experiment is also carried out to check its feasibility and the method proposed in this work is very easy and straightforward to use in a practical measurement as a flexible phase retrieval tool.Published versio