## 3 Results  
By applying the self-consistent method already presented elsewhere [46; 47; 48], it is possible to extract the shot-by-shot number of detected photons from the output voltages of the detection chain. For the employed HPDs, the method consists of modeling the detection process in two steps: photodetection by the photocathode and amplification. The first process is given by a Bernoullian convolution, whereas the second one can be well approximated by the multiplication by a constant gain factor, whose value can be obtained by measuring the light at different values of the overall detection efficiency of the apparatus. This allows us to calculate all the relevant statistical quantities that characterize an optical state of light, such as those mentioned in the previous section. In particular, we have already noticed that the statistical distribution depends on the number of modes coming from the two diffusers and is symmetric with respect to \(\mu_{f}\) and \(\mu_{s}\). We decided to extract the number of modes coming from the second disk by the calculation of the statistical quantities mentioned in the previous section and to determine the number of modes coming from the first disk by the direct measurement of the size of the speckles and of the size of the first pin-hole. To this aim, we first placed a coupled-charged device (CCD) camera (mod. DCU223M, Thorlabs, 1024 \(\times\) 768 pixels, 4.65-\(\upmu\)m pixel size) exactly in the position of the first pin-hole (PH1) and recorded some single-shot images (see Figure 3b for a typical single-shot image). From the analysis of the autocorrelation function of the images, we extracted the typical speckle size. Indeed, we estimated an area within 2\(\sigma\) equal to \(A_{\rm sp}=0.84\pm 0.07\) mm\({}^{2}\). Second, we removed the CCD, placed the first pin-hole, and realized an imaging system to determine its aperture (see Figure 3c) by means of the same CCD camera. In this case, we obtained that the diameter of the first pin-hole was equal to \(d_{\rm PH1}=0.47\pm 0.04\) mm and the corresponding area to \(A_{\rm PH1}=0.18\pm 0.05\) mm\({}^{2}\). At first glance, the number of modes passing through this first pin-hole could be evaluated as the ratio of the area of the pin-hole to the area of the speckles. However, this definition of modes is only valid when a large number of speckles passes through the pin-hole. The opposite limit is the case, in which the pin-hole is much smaller than the typical speckle size. In this case, the value of \(\mu_{f}\) can be considered equal to 1 [29]. The intermediate situation is very complex to deal with since the speckles are not rigid spheres but coherence areas propagating through a hole. Thus, assuming that the pin-hole is uniform and circular, and that the optical intensity pattern on the scattering spot is Gaussian-shaped, the following closed formula can be used to extract the effective number of modes passing through the pin-hole [29]  
\[\mu_{f}=\bigg{(}\frac{A_{\rm PH1}}{A_{\rm sp}}\bigg{)}\bigg{\{}1-\exp\bigg{[} -2\frac{A_{\rm PH1}}{A_{\rm sp}}\bigg{]}\bigg{[}I_{0}\bigg{(}\frac{2A_{\rm PH1 }}{A_{\rm sp}}\bigg{)}+I_{1}\bigg{(}\frac{2A_{\rm PH1}}{A_{\rm sp}}\bigg{)} \bigg{]}\bigg{\}}^{-1}, \tag{11}\]where \(A_{\rm sp}\) and \(A_{\rm PH1}\) are the areas of speckles and pin-hole, respectively, while \(I_{0}\) and \(I_{1}\) are the zero- and first-order modified Bessel functions of the first kind. Applied to the measured values of speckles and pin-hole, Equation (11) returns a value \(\mu_{f}=1.22\pm 0.07\). Thanks to the independent determination of \(\mu_{f}\), by the calculation of the quantities in Equations (4), (7), and (10), it is possible to extract and compare the values of \(\mu_{s}\). First of all, we consider the reconstruction of the detected-photon distribution for two different mean values.