# Tomographic Diffractive Microscopy: A Review of Methods and Recent Developments
## 5 The Advancements of Tomographic Diffractive Microscopy  
The purposes of the developments of TDM setup are to collect the diffracted field as much as possible. In the best case, the complex diffracted field, including the phase and amplitude from all possible angles of illumination and observation should be recorded for coverage of all frequency domains in Fourier space to achieve an isotropic resolution. Limited by the size and the precision, moving objectives and cameras are the challenges in a typical optical setup. Hence, illumination and sample are the only two moving components in TDM. However, varying illumination while keeping sample statics, whether in a transmission configuration or a reflection configuration, we only obtain the sample information from one side, and thus cannot get tomographic images of the sample with isotropic resolution. Rotation of the sample by optical tweezers seems promising; however, this approach is limited to certain types of samples and its controllability and measurability should be further proved [84]. A mirror-assisted method is thus a practical approach that detects the diffracted field from both sides of the sample simultaneously [57]. It is important to note that it has the similar principle of placing two opposing objectives as in a typical 4 pi microscope setup, but it provides much simpler practical implementation [85]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optical** & **Illumination** & **Measured** & **Lateral** & **Inversion** & **Reference** \\
**Schema/Configuration** & **Wavelength** & **Sample** & **Resolution** & **Inversion** & **Reference** \\ \hline Off-axis/Reflection & 475 nm & \begin{tabular}{c} nanofabricated \\ objects \\ \end{tabular} & 50 nm & Non-linear & [10] \\ \hline Off-axis/Completeness & 475 nm & \begin{tabular}{c} _Bellis perennis_ \\ pollen grain \\ _Betula_ pollen \\ grain \\ \end{tabular} & 200 nm & Linear & [25] \\ \hline On-axis/Reflection & 633 nm & \begin{tabular}{c} nanofabricated \\ objects \\ \end{tabular} & 400 nm & Non-linear & [38] \\ \hline On-axis/Completeness & 633 nm & \
\begin{tabular}{c} prolate \\ spheroid \\ \end{tabular} & \(\sim\)300 nm & Linear & [60] \\ \hline Wavefront sensors/Reflection & 633 nm &
\begin{tabular}{c} nanofabricated \\ objects \\ \end{tabular} & 1 \(\upmu\)m & Non-linear & [37] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Recent progress of tomographic diffractive microscopy TDM setups. The purpose of the inversion algorithm of TDM is to reconstruct the nature and the three-dimensional geometry of the investigated sample from the detected complex diffracted field and its corresponding illuminations. Most TDM applications are applied under Born approximation; the linear relationship between the sample permittivity and the diffracted field permits us to reconstruct the sample opto-geometry by using a simple inverse Fourier transform. However, to image high contrast sample, multiple scattering cannot be neglected and non-linear numerical inversion procedures are necessary. However, the main bottleneck of the non-linear inversion method is computation time. To overcome this disadvantage, a combination of linear and non-linear methods is helpful; for example, a fast Born approximation may provide noisy sample localization and reconstruction [10] that are useful for minimizing the investigation domain of the non-linear inversion procedure. One reported integrating DORT (decomposition de l'operateur de retournement temporal) and SVD (single value decomposition) methods to quickly localize the sample from a noisy environment. This not only improves the resolution but also ameliorates the reconstruction speed [53; 86]. Furthermore, if prior information of a sample, such as an estimation or range of sample permittivity, is available, the resolution can be improved [10].