# Correlation Plenoptic Imaging: An Overview
## 2 Correlation Plenoptic Imaging with Chaotic Light--First Scheme
### Depth-of-Field Improvement  
To analyze the fundamental limits defining the maximum achievable resolution and DOF of CPI, we shall compare the DOF of CPI and ghost imaging, which is in fact equivalent to any standard imaging (SI) obtained by a focusing element having the same numerical aperture as the chaotic source. For simplicity, we consider a 1D object \(A(\mathbf{\rho}_{o})=A(x_{0})\) and a source with a Gaussian profile, as in Equation (5). In Figures 5 and 6, we report the incoherent ghost image (left column)  
\[\Sigma_{(z_{a},z_{b})}\left(\frac{z_{a}}{z_{b}}x_{a}\right)=\int dx_{o}|A(x_{o} )|^{2}\mathcal{J}_{1}(x_{o}-x_{a}), \tag{13}\]  
which, for \(z_{a}=z_{b}\), coincides with \(\Sigma^{\text{foc}}\) of Equation (4), the coherent image from CPI (central column)  
Figure 3: **(a)** Theoretical prediction of the correlation between the intensity fluctuations retrieved by \(\text{D}_{a}\) and \(\text{D}_{b}\), as given by both Equation (2) and, in the geometrical optics limit, Equation (10). Plots have been obtained by considering the setup reported in Figure 1, with the parameters characterizing the experimental setup, in the case of a triple slit of width a and center-to-center distance \(d=2\,a\) as a transmissive object; **(b)** application to the data in panel (**a**) of the refocusing algorithm of Equation (11); **(c,d)** the solid (yellow) lines represent theoretical predictions obtained by integration of the data of panel (**a,b**), respectively, over the detector \(\text{D}_{b}\). In **(c)**, we report the ghost image, while **(d)** shows the CPI refocused image, as obtained by Equation (12). The (blue) points represent experimental data obtained in the setup that will be discussed in Section 3 after integration of the two-dimensional images over the coordinate \(y_{a}\). Reproduced with the permission from [42], copyright American Physical Society, 2017.  
and the refocused image from CPI (right column)  
\[\Sigma_{(z_{a},z_{b})}^{\text{ref}}\left(x_{a}\right)=\int dx_{b}\Gamma_{(z_{a},z _{b})}\left(\frac{z_{a}}{z_{b}}x_{a}-\frac{x_{b}}{M}\left(1-\frac{z_{a}}{z_{b}} \right),x_{b}\right), \tag{15}\]  
with \(\mathcal{J}_{1}(x)=\int dy\mathcal{J}(x,y)\) and \(\mathcal{C}_{1}(x)=\int dy\mathcal{C}(x,y)\), as defined in Equations (6) and (7). Single slits of different width \(a\) are considered in Figure 5, and a double-slit mask in Figure 6. The Gaussian profile of the source has width \(\sigma=1.08\,\text{mm}\). In line with the experiment that will be discussed in the following section, we fix the distance \(z_{a}\) between the source and the sensor \(\text{D}_{a}\), while changing the source-to-object distance \(z_{b}\). Comparison of panels (b)-(e)-(h) with panels (c)-(f)-(i) of Figure 5 indicates that the maximum achievable DOF in CPI is limited by diffraction at the object, which hinders the detection of directional information, and hence the ability to refocus. Such dependence can be understood in terms of Klyshko picture [68], as applied to ghost imaging with chaotic light [62]. In addition, Figure 6 shows that interference effects, typical of coherent imaging, also limit the maximum achievable DOF of the refocused image. On the other hand, the resolution of the focused image is only limited by the size of the focusing element. CPI thus reaches the fundamental limits imposed by the wave nature of light to both image resolution and DOF. Figure 4: Comparison between the out-of focus ghost image of a transmissive mask (**a**) and three attempts (**b–d**) of refocusing it by applying Equation (12). The distance between the source and sensor \(\text{D}_{a}\) is \(z_{a}=50\,\text{mm}\), while the object is placed at \(z_{b}=75\,\text{mm}\), namely at \(25\,\text{mm}\) from the object plane. The source is characterized by a Gaussian intensity profile with standard deviation \(\sigma=1\,\text{mm}\) and wavelength \(\lambda=500\,\text{nm}\).