# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 3 System Applications
### Super-resolution Imaging by Coded FINCH Technique
If SLM\({}_{1}\) is illuminated by a beam with a diameter of \(D\), the resolution limit in the sensor plane is given as \(\lambda d_{1}z_{h}/(Df_{0})\cong M_{\rm T}\lambda/(2\mathrm{sin}\theta_{m})\), where \(\theta_{m}\) is the maximal angle between the optical axis and the marginal ray originated from the center of the object, scattered into the system and recorded by the camera. The resolution limit on the object plane is \(\lambda/(2\mathrm{sin}\theta_{m})\), and the angle \(\theta_{m}\) is given by [64],  
\[\sin\theta_{m}=\frac{\lambda\sigma}{2f_{0}\Delta}(f_{0}-d_{1})+\frac{w}{2f_{0}} \tag{15}\]  
Figure 23: Optical configuration of C-FINCH with a scattering mask. CPM—Coded phase mask; \(L_{0,1,2}\)—Refractive lenses; \(P_{1,2}\)—Polarizers; SLM—Spatial light modulator; \(\theta_{m}\)—is the maximal angle diffracted from the object acquired by the system; Blue arrows indicate polarization orientations. Figure 24: The magnitude of the PSH for different values of the scattering degree \(\sigma\). where \(\Delta\) is the pixel size of the CPM, \(w\) is the diameter of the input aperture of the FINCH system. Since sin\(\theta_{m}\) is directly dependent on the scattering degree \(\sigma\), the scattering mask indeed extends the effective NA of the C-FINCH from the initial value \(w/2f_{0}\) of FINCH. Therefore, the scattering mask has expanded the effective NA and improved the resolution of the imaging system. While the scattering mask improves the lateral resolution, there is a loss of FOV in the object plane. The FOV in the object plane in the absence of the CPM is \(V_{0}\)\(\times\)\(V_{0}\). On the image sensor, the magnification factor projects the FOV to \(M_{\text{T}}(V_{0}\)\(\times\)\(V_{0})\). In Figure 24, it is seen that with an increase in the scattering degree \(\sigma\), the area of the interference pattern on the hologram plane increases resulting in a decrease of the FOV. The area of the intensity distribution on the camera is about \(D_{I}\)\(\times\)\(D_{I}\), where \(D_{I}\)\(=\)\(\lambda d_{1}\sigma z_{h}/(f_{0}\Delta)\). Therefore, the FOV of C-FINCH at the object plane is \(V_{\text{c}}\)\(\times\)\(V_{\text{c}}\), where \(V_{\text{c}}\)\(=\)\(V_{0}\)\(-\)\(\lambda d_{1}\sigma z_{h}/(f_{0}\Delta\)\(\cdot\)\(M_{T})\)\(=\)\(V_{0}\)\(-\)\(\lambda d_{1}\sigma/\Delta\). Therefore, with an increase in the scattering degree, the lateral resolution is improved while the FOV decreases. After recording the PSH, the experiment was repeated for a USAF target located at the same axial location as the pinhole and the object holograms were recorded. The image of the target is reconstructed by cross-correlating the object and the point object holograms. The improved resolution of the reconstructed images for different values of the scattering degree \(\sigma\) is shown in Figures 25 and 26. The loss of FOV with an increase in the scattering degree \(\sigma\) is also clearly shown in Figures 25 and 26. In conclusion, the techniques of COACH and FINCH can be combined such that the lateral resolution of FINCH is controllably enhanced. A side effect of the resolution enhancement is some loss of the FOV.