## 2 Materials and Methods  
When a coherent light beam, such as a laser beam, impinges on a diffuser, it produces a speckle field composed of many spatial modes, the speckles, that are the result of the constructive interference of the radiation coming from the small scattering centers within the area of the diffuser illuminated by the laser beam [29]. The statistical distribution of the speckle field is given by a thermal distribution \(p_{\mathrm{th}}(I)=(1/\langle I\rangle)\exp{(-I/\langle I\rangle)}\), where \(\langle I\rangle\) is the mean intensity of the speckle field. If we now select a portion of the speckle field with a pin-hole, so that only a single speckle passes through, and rotate the diffuser, the statistical distribution of the light at the exit of the pin-hole results thermal. Conversely, by selecting more than one speckle with the pin-hole, the statistical distribution becomes multi-mode thermal. By assuming that all the selected \(\mu\) modes, i.e., the speckles, in the light are equally populated, the probability density function corresponding to the collected light can be expressed as [43]  
\[p_{\mu}(I)=\frac{\mu^{\mu}I^{\mu-1}}{\Gamma(\mu)\langle I\rangle^{\mu}}\exp{ \left[-\mu\frac{I}{\langle I\rangle}\right]}, \tag{1}\]  
where \(\mu\) is the effective number of speckles, \(\langle I\rangle=\mu I_{0}\) is the mean intensity of the speckle field, \(I_{0}\) being the mean intensity of each speckle, and \(\Gamma\) is the \(\Gamma\)-function. By applying to Equation (1) the semiclassical theory of photodetection [43], the probability distribution of detected photons, \(m\), for a PNR detector  
\[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]  
can be obtained, in which \(\langle m\rangle\) is the total mean number of detected photons. If \(\mu_{f}\) speckles of this first speckle field are now selected through a pin-hole and sent to a second diffuser,at whose output a second pin-hole selects \(\mu_{s}\) speckles, the general expression of the density function corresponding to the obtained speckled-speckle field is [29]  
\[p_{sth}(I)=\frac{2(\mu_{f}\mu_{s})^{\frac{(\mu_{f}+\mu_{s})}{2}}}{\langle I \rangle\Gamma(\mu_{s})\Gamma(\mu_{f})}\left(\frac{I}{\langle I\rangle}\right)^ {\frac{(\mu_{f}+\mu_{s}-2)}{2}}K_{|\mu_{f}-\mu_{s}|}\left(2\sqrt{\mu_{s}\mu_{f }\frac{I}{\langle I\rangle}}\right), \tag{3}\]  
where \(K_{|\mu_{f}-\mu_{s}|}\) is the \(|\mu_{f}-\mu_{s}|\)-order modified Bessel function of the second kind. By again applying the semiclassical theory of photodetection to Equation (3), we obtain the probability distribution of detected photons  
\[P_{\mu_{f},\mu_{s}}(m) = \frac{1}{\Gamma(\mu_{f})\Gamma(\mu_{s})m!}\left(\frac{\mu_{f}\mu _{s}}{\langle m\rangle}\right)^{\frac{(\mu_{f}+\mu_{s}-|\mu_{f}-\mu_{s}|)}{2}}\] \[\times \Gamma\left(\frac{(2m+\mu_{f}+\mu_{s}-|\mu_{f}-\mu_{s}|)}{2} \right)\Gamma\left(\frac{(2m+\mu_{f}+\mu_{s}+|\mu_{f}-\mu_{s}|)}{2}\right)\] \[\times U\left(\frac{(2m+\mu_{f}+\mu_{s}+|\mu_{f}-\mu_{s}|)}{2},1-|\mu_ {f}-\mu_{s}|,\frac{\mu_{f}\mu_{s}}{\langle m\rangle}\right),\]  
where \(U(a,b,c)\) is the confluent hypergeometric function of the second kind. We notice that the expression in Equation (4) is more general than that reported on Ref. [26], where \(\mu_{s}\) is considered equal to \(1\). From Equation (4) we can obtain the first two moments of the distribution, from which we can calculate relevant statistical quantities, such as the second-order autocorrelation function. We obtain  
\[\langle m\rangle = \mu_{s}\langle m_{s}\rangle \tag{5}\] \[\sigma^{2}(m) = \left(\frac{1}{\mu_{f}}+\frac{1}{\mu_{s}}+\frac{1}{\mu_{f}\mu_{s} }\right)\langle m\rangle^{2}+\langle m\rangle, \tag{6}\]  
\(\langle m_{s}\rangle\) being the mean value associated with each speckle coming from the second diffuser.