# Tomographic Diffractive Microscopy: A Review of Methods and Recent Developments
## 1 Introduction  
Microscopy has revolutionized biological research and promoted the development of human health. Despite the number of microscopes that have been created and applied for different research objectives, such as the electron microscope [1; 2], the atomic force microscope [3; 4], etc., the optical microscope is still the most used tool in biological research and life science due to its non-invasive nature [5]. However, the spatial resolution of conventional optical microscopes is limited by Rayleigh criterion, which is the smallest separation distance between two point sources that can be resolved, 0.61\(\mathrm{\SIUnitSymbolMicro}\)/NA (\(\lambda\) is the wavelength of light and NA is the numerical aperture of the objective). For the range of visible light used in optical microscopes, this limitation is \(-250\) nm. Attracted by the non-contact mechanical property of optical imaging, enormous efforts have been made to find the way to overcome this diffraction barrier, collectively termed as optical super-resolution microscopes [5; 6]. By placing the light source or an optical probe near the sample at a distance shorter than the wavelength, the diffraction limit can be bypassed by exploiting the properties of evanescent waves. This has led to the so-called near-field super-resolution microscopes, such as scanning near-field optical microscopy (SNOM), whose resolution is limited by the aperture size of the probe (or source), typically of \(-25\) nm [7]. However, SNOM is operated at near-field distance and in special conditions; its application is thus limited. Another common and powerful optical imaging technique is far-field super-resolution microscopes, including the label (fluorescence) [8; 9] and label-free microscopes [10; 11], such as thestimulated emission depletion microscope [12], the stochastic optical reconstruction microscope [13], the photo-activated localization microscope [14], and non-label far-field diffractive microscopes [15]. They have successfully overcome the diffraction limit to the level of tens of nanometers resolution. Moreover, compared to near-field microscopes, these far-field ones greatly simplify the experimental setup and increase numerous possibilities of practical use in biomedical research. However, in principle, fluorescence-based methods rely on prior knowledge of the investigated sample, such as molecules and the availability of specific fluorophores; recognition of antibodies against the specific molecules. In the bio-medical usage of non-linear microscopies, such as multi-photon excitation (MPE) fluorescence microscopy [16] and second-harmonic generation microscopy (SHG) [17], the labeling procedure may also restrict the mobility of the molecules, affecting their functions. Moreover, photobleaching and phototoxicity of fluorescent microscopy play limiting roles in biology [18]. Optical coherence topography (OCT) has the ability to visualize the anatomic structures in three-dimensions and in high resolution [19], but the lateral resolution is not well developed [20]. The coherent anti-stokes Raman spectroscopy microscope (CARS) is a new imaging technique without sample labeling, but it requires laser sources with excellent intensity stabilization [21]. Holography microscopes are able to view the contrast difference, but the quantitative information of intrinsic properties of the sample, like index of refraction or permittivity, is difficult to retrieve [22; 23]. Consequently, tomographic diffractive microscopy (TDM) is a super-resolution microscopy technique, it is label-free and far-field and works in a close to physiological environment, one that is non-label, and principally at normal pressure and room temperature. By illuminating the sample from various directions with coherent collimated light and detecting the complex diffracted field from many scattered angles [15; 24; 25; 26], together with numerical inversion procedures, TDM has emerged to provide quantitative reconstruction of the opto-geometrical characteristics of the sample [27; 28; 29; 30; 31; 32].