## 2 Materials and Methods
It is worth noting that both in Equations (4) and (6), the expressions are symmetric with respect to the number of modes, regardless of whether they come from either the first diffuser or the second one. From Equations (5) and (6), it is possible to calculate the second-order autocorrelation function for detected photons [44]  
\[g^{2}(m)=\frac{\langle m^{2}\rangle}{\langle m\rangle^{2}}=\frac{\sigma^{2}(m )}{\langle m\rangle^{2}}+\frac{\langle m\rangle^{2}}{\langle m\rangle^{2}}= \left(1+\frac{1}{\mu_{f}}\right)\left(1+\frac{1}{\mu_{s}}\right)+\frac{1}{ \langle m\rangle}, \tag{7}\]  
where we note that the last term on the right side comes from taking detection into account. In fact, the second-order autocorrelation function of the incident photons, \(g^{2}(n)\), would only depend on the numbers of speckles \(\mu_{f}\) and \(\mu_{s}\)  and not on the mean value  
\[g^{2}(n)=\left(1+\frac{1}{\mu_{f}}\right)\left(1+\frac{1}{\mu_{s}}\right). \tag{8}\]  
Since the second-order autocorrelation function is the quantity that is evaluated in correlation-based imaging schemes, we investigate its dependence on the number of modes. In Figure 1, we plot \(g^{2}(m)\) as a function of the mean value for \(\mu_{f}=1\) and different choices of \(\mu_{s}\) (colored dots). In the same figure, we plot \(g^{2}(m)\) in the case of single-mode thermal light (dashed gray curve). We note that there are choices of \(\mu_{s}\) larger than \(1\) for which superthermal light is definitely more correlated than thermal light, thus proving that the speckled-speckle field can be a useful resource in all the situations in which having high fluctuations is important. We also note that even in the case \(\mu_{f}=\mu_{s}=2\), super-thermal light is still more correlated than a thermal light with \(\mu=1\). This fact can be emphasized by dividing the light beam at a balanced beam splitter (BS) and then calculating the correlation between the two BS outputs. We define the cross-correlation coefficient [45]  
\[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]0  
where \(\langle m_{1,2}\rangle\) and \(\sigma^{2}(m_{1,2})\) are the mean value and the variance in the \(j=1,2\) BS arm, respectively. Assuming a balanced BS, for the super-thermal light described by the distribution of detected photons in Equation (4), the correlation coefficient \(\Gamma\) reads  
\[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]1  
In Figure 2, we show \(\Gamma\) as a function of the mean value of detected photons for different choices of \(\mu_{f}\) and \(\mu_{s}\). For a direct comparison, in the same figure, we plot the behavior of \(\Gamma\) in the case of a single-mode thermal state. As already noticed for \(g^{2}\), superthermal light is more correlated than thermal light if the numbers of modes are small. Indeed, both \(\mu_{f}\) and \(\mu_{s}\) should be larger than 1 (i.e., \(\mu_{f}=\mu_{s}=5\)) to give values of \(\Gamma\) lower than those corresponding to a thermal field. Actually, in the case of \(\mu_{f}=\mu_{s}\), the boundary to a single-mode thermal field is reached for \(\mu_{f}=\mu_{s}=1+\sqrt{2}\). \[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]2 Second-order autocorrelation function \(g^{2}(m)\) as a function of the mean number of photons for \(\mu_{f}=1\) and different values of \(\mu_{s}\). Different colors refer to different choices of \(\mu_{s}\). From top to bottom: black to \(\mu_{s}=1\), red to \(\mu_{s}=2\), blue to \(\mu_{s}=5\), and green to \(\mu_{s}=10\). The dashed gray curve corresponds to a single-mode thermal state of light. To summarize, both Equations (7) and (10) depend on the number of modes coming from both disks.