# Tomographic Diffractive Microscopy: A Review of Methods and Recent Developments
## 4 The Optical Setup of Tomographic Diffractive Microscopy
An off-axis setup is a simple configuration for measuring the complex diffracted field from a single hologram and can avoid the image conjugation and enhance the imaging quality, but it is more sophisticated to calibrate compared to the on-axis system, see Figure 10b [23; 73; 74]. The main difference of the setup between on-axis and off-axis is the removal of the phase modulator. In off-axis, the references are propagated with a carefully chosen angle. For the sake of simplicity, considering the interference in one dimension, i.e., in \(x\)-axis, the intensity on the charge coupled device (CCD) camera could be written as:  
\[\text{S}=\text{I}_{\text{sample}}+\text{I}_{\text{reference}}+\text{e}^{- \text{i}\alpha\alpha}\text{E}_{\text{sample}}+\text{e}^{\text{i}\alpha\alpha} \text{E}_{\text{sample}}\text{*} \tag{15}\]By applying 2D Fourier transform of S, the Dirac function \(\delta_{\pm\alpha}\) introduced by \(\mathrm{e}^{\mathrm{i}\alpha\mathrm{x}}\) helps us to retrieve the complex field of the sample. Moreover, the benefit from the nature of single-shot measurement in the off-axis method is that it could significantly reduce the setup sensitivity to external fluctuations. However, it is worth noting that single-shot characteristics are measured at the cost of the camera's available pixels; a minimal off-axis angle to separate the different interference terms in the Fourier space and a maximal angle to successfully distinguish the interference fringes must be satisfied simultaneously [75]. To the best of our knowledge, the highest resolution of TDM reported as one-tenth wavelength of the illumination was using off-axis to yield the phase and amplitude of the diffracted field [10]. Besides the interferometric setup, a Shack-Hartmann wavefront sensor can be used to measure the interference fringes in the \(\mathrm{e}^{\mathrm{i}\alpha\mathrm{x}}\) plane. Figure 10: Schematic images of three different methods for complex diffracted field detection. (**a**) Phase shifting interferometry setup. (**b**) Off-axis interferometry setup. Compared to (**a**), the beam splitter at the right upper corner in (**b**) is rotated with a small angle so that the reference (blue) does not superimpose on the diffracted field (red). (**c**) Quadri-wave lateral shearing interferometry, wave front sensors setup. to measure the complex diffracted field without the reference beam, see Figure 10c, to eliminate the external influences and minimize the measurement period [76; 77; 78; 79]. In this sensor, a modified Hartmann mask (MHM) was closely set in front of the detector to create replicas of the incident wavefront in several identical but tilted waterfronts; their mutual interference patterns were then recorded on the detector. The phase recovered by applying a Fourier transform to the interferogram. [PERSON] et al. reported the principle of recovering the intensity and phase of a field with multi-wave interferometry [80]. A successful employment of a wavefront sensor based on quadri-wave lateral shearing interferometry (QWISI) in TDM measurement has also been presented [37], but the disadvantage of this sensors is the limitation of resolution compared to charge coupled device (CCD) or CMOS) cameras [37; 81]. Notice that most tomographic diffractive microscopy (TDM) setups have been used with a laser beam that was polarized in one direction--vertically or horizontally--for both the illumination and the reference wave if they are present; however, for the resolution of the measured sample, with respect to the TMD system far beyond the Rayleigh criterion, the diffracted field close to the edge of the NA of the objective is no longer parallel to the polarization of the illumination, a modified setup developed to retrieve the full vectorial diffracted field; the resolution was thus significantly improved [82; 83]. To conclude the development of the TDM setup, we present the recent remarkable progress of TDM setups in Table 1.