# Correlation Plenoptic Imaging: An Overview
## 4 A Different Architecture of CPI with Chaotic Light
In addition, when the sensor D\({}_{b}\) retrieves an out-of-focus image (\(S_{2}\
eq S_{2}^{f}\)), refocusing can be achieved through a simple rescaling procedure, and the final image is given by  
\[\Sigma^{\text{ref}}_{(S_{1},S_{2})}(\mathbf{\rho}_{b})=\int d\mathbf{\rho}_{a}\Gamma \left[\mathbf{\rho}_{a},\frac{S_{1}}{S_{2}^{f}}\mathbf{\rho}_{b}+\left(\frac{S_{1}}{S_ {2}^{f}}\right)\mathbf{\rho}_{a}\right]\simeq\left|A\left(-\frac{S_{1}}{S_{2}^{f} }\mathbf{\rho}_{a}\right)\right|^{2}\sim\Sigma^{\text{foc}}(\mathbf{\correctly taken into account. An outstanding example is given by entangled photon pairs, as produced by SPDC. Here, we will discuss the setup reported in Figure 12. In view of plenoptic imaging, this setup must enable the parallel acquisition of several images of the given scene, one for each propagation direction of light. We shall soon demonstrate that, also in this case, the sensor D\({}_{a}\) retrieves \(N\)_coherent ghost images_ of the object by means of correlation measurements with the \(N\) pixels of D\({}_{b}\), each one giving a different viewpoint on the desired scene. This is quite intuitive if one considers that, similar to the first CPI setup in Figure 1, D\({}_{b}\) reproduces the image of the light source. Like both previous CPI schemes, the lens \(L_{b}\) alone replaces the microlens array required in standard plenoptic imaging.  
As in the case of chaotic light, the intensity correlation measurement and, in the photon-counting regime, the coincidence detection, are described by the second order Glauber correlation function (see Equation (A1)). The expectation value is now taken over the two-photon signal-idler state produced by SPDC [70; 71; 72]  
\[|\Psi\rangle=\mathcal{N}\int dv\,s(LDv)\int d\mathbf{\kappa}_{i}d\mathbf{\kappa}_{s}h_{ \text{tr}}(\mathbf{\kappa}_{i}+\mathbf{\kappa}_{s})a_{\mathbf{k}_{i}}^{\dagger}a_{\mathbf{k}_{ s}}^{\dagger}|0\rangle, \tag{25}\]  
where \(v\) is the detuning of the signal and idler beams compared to their central frequencies \(\Omega_{s}=\Omega_{i}=\omega_{p}/2\), related with the pump laser frequency \(\omega_{p}\) by the phase-matching conditions, \(L\) is the longitudinal size of the SPDC crystal, \(D\) is the group velocity difference between the two beams,  
Figure 10:0 Scheme to perform CPI by exploiting the quantum correlations of entangled photon pairs emitted by spontaneous parametric down conversion (SPDC). Adapted with the permission from [41], copyright MDPI, 2016. The emitted photon beam is divided by a beam splitter. The reflected beam is collected, in arm \(a\), by the lens \(L_{a}\) of focal length \(f\), before being detected by the high-resolution detector D\({}_{a}\). The transmitted beam, in arm \(b\), impinges on an object at a distance \(z_{b}\) from the source, and is then refracted by a lens \(L_{b}\) of focal length \(F\) toward the high-resolution detector D\({}_{b}\). The distances \(z_{b},z_{b}^{\prime},z_{b}^{\prime\prime}\) between components in path \(b\) are chosen in such a way that an image of the source is formed on detector D\({}_{b}\) by the lens \(L_{b}\). The distances \(z_{a}\) and \(z_{a}^{\prime}\) are such that a ghost image of the object is retrieved on D\({}_{a}\) when intensity correlations with D\({}_{b}\) are measured; the ghost image is focused when the “two-photon thin-lens equation” \(1/(z_{b}+z_{a})+1/z_{a}^{\prime}=1/f\) is satisfied. Correlations between the intensities at the two sensors are retrieved either by coincidence counting or by software correlation of the registered intensity patterns. the function \(s(LD\
u)\) is the SPDC biphoton spectrum [73; 74], \(\mathcal{N}\) is an irrelevant normalization, and \(h_{\text{tr}}\) is the Fourier transform of the transverse amplitude profile of the pump laser:  
Figure 10:1  
and the \(a_{k,s}^{+}\) are the operators that create photons with a definite momentum out of the vacuum \(|0\rangle\).