## 3 Results
In Figure 4, we show the experimental data in semilogarithmic scale together with the theoretical fitting functions according to Equation (4), in which the value of \(\mu_{f}\) is fixed, while that of \(\mu_{s}\) is left as a fitting parameter. The values of \(\mu_{s}\) obtained at the two different mean numbers of photons (see panels (a,b)) are very similar to each other, thus proving that the number of modes is independent of the mean value of light. The good quality of the fitting procedure is proved by the high values of the fidelity, which is defined as \(f=\sum_{m=0}^{\overline{m}}\sqrt{P(m)P_{\rm theo}(m)}\), where \(P(m)\) and \(P_{\rm theo}(m)\) are the experimental and theoretical distributions, respectively, and the sum is extended up to the maximum detected-photon number, \(\overline{m}\), above which the two distributions become negligible. To further investigate the independence of the number of modes from the mean value, we can also calculate the second-order autocorrelation function for detected photons. In Figure 5, we show the behavior of \(g^{2}\) as a function of the mean value for the two BS outputs. By fitting the data measured in the two BS arms according to the theoretical model in Equation (7), it is possible to extract the values of \(\mu_{s}\). Note that such values are constant in the investigated intensity range, as it is well evident from Figure 6, where the calculated values of \(\mu_{s}\) as obtained by inverting Equation (7) applied to experimental data, are compared to the fitting values of Figure 5.  
The small difference between the values of \(\mu_{s}\) extracted from the two fits (see caption of Figure 5) may be ascribed to a non-perfect balancing of the two BS arms, which slightly differ from each other at low mean values. In particular, we note that the value extracted from panel (a) is similar (within \(1\sigma\)) to the values extracted from the photon-number distributions. The slight discrepancy is due to the fact that \(\mu_{s}\), as calculated from \(g^{2}\), is obtained by together considering the measurements performed at different mean values, while that obtained from each photon-number distribution is referred to a specific mean value. Figure 4: Detected-photon number distributions reconstructed at one BS output for two mean values of the speckled-speckle field. Black dots and error bars: experimental data; gray curves: theoretical fitting functions according to Equation (4), in which \(\mu_{f}=1.22\), while \(\mu_{s}\) is the only fitting parameter. The obtained values are: \(\mu_{s}=1.74\pm 0.02\) in panel (**a**) and \(\mu_{s}=1.76\pm 0.02\) in panel (**b**). The corresponding values of fidelity are: \(f=0.9999\) and \(f=0.9999\), respectively. Figure 5: Second-order autocorrelation function \(g^{2}(m)\) as a function of the mean number of detected photons reconstructed at the two BS outputs. Black dots and error bars: experimental data; gray curves: theoretical fitting functions according to Equation (7), in which \(\mu_{f}=1.22\), while \(\mu_{s}\) is the only fitting parameter. The obtained values are: \(\mu_{s}=1.72\pm 0.05\) in panel (**a**) and \(\mu_{s}=1.48\pm 0.05\) in panel (**b**). Figure 6: Values of \(\mu_{s}\) obtained by inverting Equation (7) applied to the experimental data as a function of the mean number of detected photons reconstructed at the two BS outputs. Black dots and error bars: experimental data; gray lines: values of \(\mu_{s}\) obtained from the fitting procedure applied to the data in Figure 5, namely, \(\mu_{s}=1.72\) in panel (**a**) and \(\mu_{s}=1.48\) in panel (**b**). As a final comparison, we consider the case of the cross-correlation coefficient calculated between the two BS outputs. The experimental data are shown in Figure 7a together with the theoretical expectation according to Equation (10). Additionally, in this case, we kept the value of \(\mu_{f}\) fixed and extracted that of \(\mu_{s}\) from the fitting procedure. The good superposition of data on the theoretical curve proves again the correctness of the theoretical model.