  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.
