# Tomographic Diffractive Microscopy: A Review of Methods and Recent Developments
## 2 Theoretical Background  
In conventional optical microscopes [33], the object is illuminated simultaneously by a sum of plane waves spatially incoherent with each other. Each plane wave propagates with a different illumination angle, so that the object is globally illuminated simultaneously with all possible angles, within a given numerical aperture noted \(\mathrm{NA_{inc}}\), which is the sine of the maximum illumination angle with respect to the optical axis of the microscope. For the detection, the most commonly used architecture consists in placing the object near the object focal plane of an objective lens; the diffraction field is detected by the detector, such as a CCD camera, which is placed at the image focal plane of the objective lens. However, limited by the \(\mathrm{NA_{inc}}\) of the objective lens, only parts of object diffracted field information could be collected by the objective lens; this leads to the well-known Rayleigh criterion, shown in Figure 1a.  
To demonstrate the link between \(\mathrm{NA_{inc}}\) of the objective lens and its achievable spatial resolution, we introduce the point-spread function (PSF) to describe the response of the imaging system to a point source, which bridges the original object O and resultant image \(\mathrm{G}\) as, \(\mathrm{G}=\mathrm{O}\) * PSF. The Fourier transform of PSF is defined as the optical transfer function (OTF), corresponding to a particular object in the Fourier transform domain. The direction of light propagation could be defined by the wave vector \(\mathbf{k}=\mathrm{k_{x}}\mathbf{x}+\mathrm{k_{y}}\mathbf{y}+\mathrm{k_{z}} \mathbf{z}\). We define \(\mathbf{k_{||}}=\mathrm{k_{x}}\mathbf{x}+\mathrm{k_{y}}\mathbf{y}\), where \(\mathbf{k}\) propagates on the (x,y) plane. As not all the light propagated through the sample is detectable in conventional microscopes, only the plane waves emerging from the sample with \(\left|\mathbf{k_{||}}\right|\leq\mathrm{k_{0}}\mathrm{NA}\) are collected, shown in Figure 2, where NA is the numerical aperture of the microscope objective; Rayleigh criterion is thus introduced limiting the resolution as \(\Delta\tau=0.61\lambda/\)NA. Moreover, by using conventional optical microscopes, one is unable to quantitatively obtain the opto-geometrical characteristics of the sample.  
Thus, two questions arise: Can we get the three-dimensional image of the sample and is it possible to investigate the material properties of the sample? The answer is tomographic diffractive microscopy, which was introduced by [PERSON] in the year of 1969 [PERSON] (1969 [PERSON] (1969). Upon recording complex diffracted fields (amplitude and phase) by coherently illuminating the sample, the index of refraction or the permittivity of the sample are retrievable by numerical inversion procedures, shown in Figure 1b  
Figure 1: Schemas of the wide-field microscopy and tomographic diffractive microscopy. (**a**) Wide-field microscopy. (**b**) Tomographic diffractive microscopy.  
Figure 2: A description of the accessible frequency domain of conventional microscopy. (**a**) The sample is illuminated in normal incidence and detected in the same direction. (**b**) The projection of the measurable diffracted wave vector **k** onto the transverse (xOy) in conventional microscopy.