An Approach to Direct 3D Imaging with Coherent Light

We consider the 3D coefficient inverse problem for parabolic wave equation. It involves determining the spatial distribution of refractive and absorption indices by processing phase diffraction patterns obtained by irradiating an object with a set of Gaussian beams. Unlike tomography and ptychography, rotation or scanning of the sample is not required. The problem is solved by expanding the wave field and the complex dielectric constant epsilon r ->\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \left(\overrightarrow{r}\right)$$\end{document} over the full set of Gaussian beam functions. To determine epsilon r ->,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \left(\overrightarrow{r}\right),$$\end{document} we obtain a nonlinear matrix equation. The condition of its solvability allows the selection of sampling frequencies by coordinates in accordance with the practical task.