## 1 Introduction  
Proton acceleration up to energies of tens of MeV using high intensity laser pulses (I \(>\)10\({}^{18}\) W/cm\({}^{2}\)) that interact with thin solid targets has been an issue of considerable interest over the last two decades, both experimentally and theoretically [1; 2; 3; 4; 5; 6]. Most studies on laser-driven proton acceleration were performed under experimental conditions which are characteristic of the so-called Target Normal Sheath Acceleration (TNSA) mechanism. TNSA represents the main rear surface ion acceleration regime, where femtosecond laser pulses with intensities higher than 10\({}^{18}\) W/cm\({}^{2}\) interact with thin solid targets, which are a few microns thick [1; 2]. As the laser pre-pulse interacts with the solid target, a pre-plasma is generated at the front target surface, shortly before the main pulse arrival. The electrons generated at the front side of the solid target foil, as an outcome of the main laser pulse interaction with the pre-plasma, absorb a percentage of the laser energy. These hot electrons, characterized by a mean free path larger than the target thickness, are transported through the target and then reach its rear surface where they generate a dense electron sheath. Following the expansion of this electron sheath an electric field of TV/m magnitude occurs, normal to the target surface. Due to this strong electric field the water and organic molecules contained within the adsorbed surface contaminants present at the rear side of the target can get dissociated and ionized. The protons generated by such mechanism are accelerated in a direction normal to the target [1; 2]. The analysis of proton beams characteristics and of the associated energy distribution is usually performed with radiochromic films or solid-state nuclear track detectors, most often coupled with Thomson Parabola spectrometers [3; 4]. The CR-39 nuclear track detector [polyallyl-diglycol-carbonate (PADC), C\({}_{12}\)H\({}_{18}\)O\(\gamma\)], has demonstrated its capabilities to detect protons as it is insensitive to electromagnetic noise [3; 7; 8], while showing high uniformity and high sensitivity to ions, as well as excellent etching properties. As reported in [3], once the laser accelerated protons are analyzed by means of CR-39 detectors, the output data can be used to characterize accelerated ion beams without requiring an additional spectrometer. A high energy charged particle (for example a proton) travelling through a CR-39 detector loses energy along its trajectory in matter by electronic collisions, by scattering, then stops when it has completely deposed its energy into the detector material, according to the Bragg curve. The point where the proton entirely deposes its energy as ionizing radiation and comes to rest is known as the Bragg peak [7; 9]. It is assumed [3] that a proton leaves a track in matter when it has totally deposed its energy. As the stopping power scaling law of charged particles in matter is dE/dx \(\sim\)1/E, the track diameter can be related to the proton energy [10]. Thus, as the proton energy increases the track diameter decreases. The geometrical characteristics of the damaged zone left by the proton into the detector can be investigated by means of optical and atomic force microscopy after etching the plastic detector in very well-defined conditions of time, temperature, and concentrations of etching solution [3]. Recent literature in the field [2; 4] emphasizes the advantages of using radiochromic films to reveal proton densities higher than 10\({}^{7}\) particles/cm\({}^{2}\). Proton beams with lower densities can be detected using solid-state nuclear track detectors [7; 11; 12]. With respect to late experimental results in the field [3; 7; 13; 14], our paper brings a new approach on the spectral characterization of laser accelerated protons with energies higher than 10.5 MeV extracted from the front side of a second CR-39 detector (arranged in a stack of 2 CR-39 detectors).