# Correlation Plenoptic Imaging: An Overview
## 4 A Different Architecture of CPI with Chaotic Light
### Plenoptic Properties of the Correlation Function
On the other hand, the stationarity of \(\varphi\) with respect to \(\boldsymbol{\rho}_{o}\) yields the focusing of the source on the sensor \(\mathrm{D}_{b}\):  
\[\boldsymbol{\rho}_{s}=-\frac{\boldsymbol{\rho}_{b}}{M}. \tag{32}\]  
Therefore, in the geometrical optics limit, the correlation function of Equation (27) reduces to the product of the tilted and rescaled geometrical image of the object with the source profile  
\[\Gamma_{(z_{a},z_{b})}(\boldsymbol{\rho}_{a},\boldsymbol{\rho}_{b})\sim\left| A\left[-\frac{z_{b}}{z_{bF}}\frac{\boldsymbol{\rho}_{a}}{m}-\frac{\boldsymbol{ \rho}_{b}}{M}\left(1-\frac{z_{b}}{z_{bF}}\right)\right]\right|^{2}\left| \mathcal{F}\left(-\frac{\boldsymbol{\rho}_{b}}{M}\right)\right|^{2}. \tag{33}\]  
By properly rescaling the variable \(\boldsymbol{\rho}_{a}\), the correlation function gives the perfectly aligned geometrical images of the desired scene, one for each value of \(\boldsymbol{\rho}_{b}\), namely  
\[\Gamma_{(z_{a},z_{b})}^{\text{ref}}\left[\frac{z_{bF}}{z_{b}}\boldsymbol{\rho }_{a}+\frac{\boldsymbol{\rho}_{b}}{M}m\left(1-\frac{z_{bF}}{z_{b}}\right), \boldsymbol{\rho}_{b}\right]\sim\left|\mathcal{F}\left(-\frac{\boldsymbol{ \rho}_{b}}{M}\right)\right|^{2}\left|A\left(-\frac{\boldsymbol{\rho}_{a}}{m} \right)\right|^{2}. \tag{34}\]  
Such rescaling is formally identical to the one employed both in standard plenoptic imaging [3] and in correlation plenoptic imaging with chaotic light (see Equation (11)). Similar to standard plenoptic imaging, the signal-to-noise ratio is highly improved by integrating the result of Equation (34) over the whole sensor array \(\boldsymbol{\rho}_{b}\), thus employing light coming from the whole light source. The result  
\[\Sigma_{(z_{a}^{\prime},z_{b})}^{\text{ref}}(\boldsymbol{\rho}_{a})=\int d \boldsymbol{\rho}_{b}\Gamma_{(z_{a}^{\prime},z_{b})}^{\text{ref}}\left[\frac{ z_{bF}}{z_{b}}\boldsymbol{\rho}_{a}+\frac{\boldsymbol{\rho}_{b}}{M}m\left(1-\frac{z_{ bF}}{z_{b}}\right),\boldsymbol{\rho}_{b}\right] \tag{35}\]  
represents the refocused incoherent ghost image of an object placed at a generic distance \(z_{b}\) from the source. Figure 13: The unfolded version (Klyshko picture) of the setup represented in Figure 12 shows the combined focusing effect on the basis of CPI with entangled photons: on one hand, the lens \(L_{a}\) focuses the ghost image of the object (by correlation measurement) on the detector \(\mathrm{D}_{a}\); on the other hand, the lens \(L_{b}\) reproduces on \(\mathrm{D}_{b}\) the image of the source, filtered by the object transmission function. Adapted with the permission from [41], copyright MDPI, 2016. The solid and dashed lines represent two-photon amplitudes transmitted by the same object point, and eventually focused on the same pixel of \(\mathrm{D}_{a}\). The dashed and dotted lines are emitted by the same source point and focused in the same pixel of \(\mathrm{D}_{b}\). The accuracy of the quasi one-to-one correspondence of object and source points with their images on \(\mathrm{D}_{a}\) and \(\mathrm{D}_{b}\), respectively, is at the basis of the resolution properties of CPI with entangled photons. On one hand, the object PSF in Equation (30) is characterized by a spot size \(\Delta\rho_{a}\sim m\,c\,z_{bF}/(\Omega\,D_{s})\), with \(D_{s}\) the typical transverse size (effective diameter) of the pump profile. On the other hand, the PSF of the source image is determined by the size \(a\) of the smallest object details, through \(\Delta\rho_{b}\sim M\,c\,z_{b}/(\Omega\,a)\). Hence, for pixel size above the resolution limits, spatial and directional resolutions are decoupled, as in chaotic light CPI. Therefore, also CPI with entangled photons overcomes the intrinsic limitations of standard plenoptic imaging and achieves a larger depth of field (as defining by the angular resolution), with diffraction-limited resolution.