# Tomographic Diffractive Microscopy: A Review of Methods and Recent Developments
## 4 The Optical Setup of Tomographic Diffractive Microscopy  
The TDM can be implemented in either transmission configuration or reflection configuration. The need of quantitative phase measurement is the significant difference between TDM and conventional wide-field microscopes [15]. The TDM setup consists of illuminating the sample with a collimated beam from controlled angles of incidence and recording the field diffracted by the sample in phase and in amplitude. Varying the illumination of TDM: In principle, two methods are available for varying illumination angles, rotation of the collimated beam [37] and rotation of the sample [58]; the former is easier to realize. Keeping the sample static is easy to manipulate for a TDM user. A rotating mirror permits control of the deflection of the collimated beam [37; 59]; however, the missing parts of non-captured frequencies present a strong anisotropic resolution along the optical axis, especially in transmission configuration of TDM. To obtain an improved and isotropic resolution, [PERSON] et al. used a mirror-assisted setup combining the transmission and reflection configuration together [57] or used an ellipsoidal mirror for expanding the angular coverage of diffracted field detection [60]. Another way for varying the illumination is rotation of the sample and keeping incident light static. In TDM under transmission configuration, fixing the setup, the sample along the \(x\)-axis was successfully rotated to give a quasi-isotropic resolution [61]. Using optical tweezers is another promising method [62; 63; 64; 65; 66]. To achieve complete configuration, a combination of specimen rotation and illumination rotation simultaneously is also feasible by adopting an integrated setup [67]. Phase and amplitude detection of TDM: An interferometric arrangement of setup--phase shift interferometry--is commonly used in TDM for the detection of complex diffracted fields, see Figure 10a [38; 68]; it involves shifting the phase relative to the reference beam in steps, recording of the interference between the diffracted field and reference beam for each phase step, and calculation of the phase from four or more detected intensities on a CCD camera. For example, the intensity measured on the CCD camera can be written as S, adding the phase shift by phase modulator, see Figure 10a; we could detect four successive intensities on the camera as:  
\[ \begin{split}\text{S}_{1}&=\text{I}_{\text{sample}} +\text{I}_{\text{reference}}+2\sqrt{\text{I}_{\text{sample}}\text{I}_{\text{ reference}}}\text{cos}(\varphi+\frac{\pi}{2})\\ \text{S}_{2}&=\text{I}_{\text{sample}}+\text{I}_{ \text{reference}}+2\sqrt{\text{I}_{\text{sample}}\text{I}_{\text{reference}}} \text{cos}(\varphi)\\ \text{S}_{3}&=\text{I}_{\text{sample}}+\text{I}_{ \text{reference}}+2\sqrt{\text{I}_{\text{sample}}\text{I}_{\text{reference}}} \text{cos}(\varphi-\frac{\pi}{2})\\ \text{S}_{4}&=\text{I}_{\text{sample}}+\text{I}_{ \text{reference}}+2\sqrt{\text{I}_{\text{sample}}\text{I}_{\text{reference}}} \text{cos}(\varphi-\pi)\end{split}  \tag{14}\]  
where \(\varphi\) is the original phase differences between the diffracted field and the reference. The diffracted field of the sample I\({}_{\text{sample}}\) could be retrieved as \(\text{E}_{\text{sample}}=\frac{(\text{S}_{2}-\text{S}_{4})+\text{i}(\text{ S}_{3}-\text{S}_{1}))}{4\sqrt{\text{I}_{\text{reference}}}}\). Such a configuration is usually called an on-axis (or inline) system; since the phase information is obtained by making consecutive image subtractions, it provides a precise measurement of the phase of the diffracted field [69; 70]. However, performing several phase steps is time consuming, especially if a large number of illumination angles have to be applied successively in TDM. Phase fluctuations due to thermal and/or mechanical drift during acquisition can interfere with the measurement [71; 72].