Application of transport of intensity and phase equations for quantitative phase imaging using a non-recursive method

This study presents an application of the non-recursive approach after combining the transport of intensity equation (TIE) and the transport of phase equation (TPE), as proposed by Zhou et al . [ Appl. Opt. 61 , B190 ( 2022 ) APOPAI 0003-6935 10.1364/AO.444454 ] for quantitative phase imaging. Their method introduced a spatially varying propagation constant k ( x , y )= k 0 × n ( x , y ), allowing the TIE to be reformulated as a Poisson-type equation, where n ( x , y ) is the effective (virtual) refractive index. By integrating the TPE into the TIE framework, the fast Fourier transform TIE-TPE method effectively corrects errors caused by the conventional assumption of uniform intensity at the image plane. The method is validated through numerical simulation and optical experiments. Gaussian phase functions with varying amplitudes under perturbation have been used for numerical simulation. The experimental validation using an onion peel demonstrates a significant improvement in phase resolution. Additionally, the robustness of the method is demonstrated through phase reconstruction using the glass microbead powder sample. The concept of the virtual refractive index is also explored for qualitative analysis.