## 1 Introduction  
The existence of intensity fluctuations in light sources is a useful resource for imaging applications. Indeed, there are imaging techniques, such as ghost imaging, whose working principle relies on the spatial correlations coming from the intensity fluctuations [1; 2]. The most common classical light source exhibiting such a feature is the pseudo-thermal light, typically obtained by passing a laser beam through a rotating ground-glass disk [3], which shows the so-called speckle field, consisting of many spatial modes [4]. Many realizations of this type of light in different intensity domains [5; 6; 7; 8; 9] have been exploited to address imaging protocols, reconstructing correlation images of different kinds of objects, including faint and photolabile ones [10; 11]. Indeed, the use of non-invasive methods for observing various physical and biological activities employing structured patterns, such as speckle patterns, represents a hot topic in the imaging context [12; 13]. The quality of the reconstructed images is usually quantified in terms of some figures of merit, such as the contrast and the signal-to-noise ratio [10; 14; 15; 16]. The values of these estimators can be increased by the use of light sources characterized by intensity fluctuations higher than those of pseudo-thermal light [17; 18; 19; 20; 21]. We call these sources \"super-thermal\" since their intensity fluctuates more than that of thermal ones. There are different ways to produce super-thermal sources. Some years ago, we demonstrated that the second-harmonics of a pseudo-thermal light displays super-thermal statistics [22; 23]. However, the generation of this kind of light is not always very efficient since it is based on a nonlinear process in which energy- and phase-matching conditions must be fulfilled [24]. In this work, we address a different strategy to generate super-thermal light field based on linear processes in two diffusers instead of just one [25; 26; 27; 28]: the pseudo-thermal light produced by the first diffuser is properly selected by a pin-hole and sent to the second diffuser. Such a scheme has already been investigated in the past, even if a general characterization of the obtained field is still missing [29]. In our work, we derive the statistical properties for the general case in which \(\mu_{f}\) spatial modes are selected at the output of the first disk and \(\mu_{s}\) modes at the exit of the second one. This choice allows a better comparison of the super-thermal statistics with the pseudo-thermal one. At variance with the large part of the experimental works on doubly-scattered light present in the literature that has been realized at the single-photon level [26; 30; 31] with single-photon detectors, in our investigation, we operate in the mesoscopic intensity regime by employing photon-number-resolving (PNR) detectors. The choice of this regime allows us to thoroughly study the statistical properties of light in terms of measurable quantities. We can calculate the photon-number distribution for detected photons, which, in many applications, is the most qualifying feature of the employed optical states [32; 33; 34; 35; 36; 37; 38]. As an example, we can consider employing this kind of light in the field of quantum information science: thermal or super-thermal states, superimposed to quantum states of light [39], can be used to encode information, such as in the protocol we presented in Refs. [40; 41]. The mesoscopic intensity regime is also potentially interesting for applications to imaging thanks to the use of arrays of PNR detectors [42]. In the following, we derive the general expression of the photon-number statistics corresponding to the generated super-thermal light that we will call the \"speckled-speckle\" field hereafter. Then, we investigate some quantities based on the first two moments of the distributions to put in evidence the potentialities of this light source, especially in view of possible applications in the imaging context.