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The Theory of Diffraction Tomography

Tomography is the three-dimensional reconstruction of an object from images taken at different angles. The term classical tomography is used, when the imaging beam travels in straight lines through the object. This assumption is valid for light with short wavelengths, for example in x-ray tomography. For classical tomography, a commonly used reconstruction method is the filtered back-projection algorithm which yields fast and stable object reconstructions. In the context of single-cell imaging, the back-projection algorithm has been used to investigate the cell structure or to quantify the refractive index distribution within single cells using light from the visible spectrum. Nevertheless, these approaches, commonly summarized as optical projection tomography, do not take into account diffraction. Diffraction tomography with the Rytov approximation resolves this issue. The explicit incorporation of the wave nature of light results in an enhanced reconstruction of the object's refractive index distribution. Here, we present a full literature review of diffraction tomography. We derive the theory starting from the wave equation and discuss its validity with the focus on applications for refractive index tomography. Furthermore, we derive the back-propagation algorithm, the diffraction-tomographic pendant to the back-projection algorithm, and describe its implementation in three dimensions. Finally, we showcase the application of the back-propagation algorithm to computer-generated scattering data. This review unifies the different notations in literature and gives a detailed description of the back-propagation algorithm, serving as a reliable basis for future work in the field of diffraction tomography.