## 2 Materials and Methods
### Setup
The use of such TDC required software elaborations in order to overcome some intrinsic limitations, in particular: (i) high differential non-linearity (DNL) of the time scale (up to 100% of the least significant bit), and (ii) maximum acceptable rate for the synchronism pulse of 7 MHz, while the laser pulse rate is 40 MHz. \begin{table}
\begin{tabular}{c c} \hline \hline
**Building Block** & **Dimension (mm)** \\ \hline Time-to-Digital Converter (TDC) & \(237\times 310\times 97\) \\ Stack of 9 signal-translation boards & \(170\times 65\times 40\) \\ Photosensitive chip area & \(1.3\times 1.3\) \\ SiPM channel & Length = 40 mm; diameter = 6 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the main building blocks composing the proposed system. Figure 1: System architecture scheme (**left**) and photo (**right**). The acquired instrument response function of one channel (**top left**) shows a FWHM of 268 ps, while the tabled values highlights the compactness of the most innovative parts of the system. When the detectors are not exposed to the laser pulsed light, it is possible to acquire the background noise alone, which is expected to be completely flat over the TDC time scale, not being correlated with the laser synchronism. Thus, to overcome the issue of the high DNL, a dark count rate acquisition is used to calculate proper coefficients (one for each time bin), which allow us to linearize the TDC time scale. This procedure has to be done only once since the DNL is very stable over different days. As it concerns the maximum synchronism rate, we divide the 40 MHz synchronism of the laser by a factor of 16 thanks to a native start divider function that is embedded into the TDC module, obtaining 16 replicas of the DTOF. Since each replica conveys only 1/16 of the total number of detected photons, we sum all of the replicas by refolding the TDC time scale in order to have a single DTOF with all of the detected photons. Since the number of bins contained into a single replica of the DTOF is not integer (i.e., actual laser pulse rate is not a perfect multiple of the TDC bin size), before applying the refolding operation we resampled the TDC time scale by increasing the number of bins by a factor of 1000. Hence, the refolding operation that is applied on the re-binned timescale also allows for the reduction of the actual bin size in the final reconstructed waveform. The system is controlled by a home-made software, written in ANSI C, and compiled in the LabWindows(r) environment. The software controls all of the operations that are related to the measurements (e.g., automatic detection of optimal laser attenuation to cope with the single-photon statistics, data acquisition, real-time visualization, data save). This is a general purpose program to control time-resolved photonics systems, which has been evolving in the course of the last 25 years and is applied to more than 10 different instruments in various application fields (e.g., optical mammography, brain imaging, broadband diffuse spectroscopy, and time-resolved fluorescence spectroscopy). Although written in C language, it follows an object-oriented scheme, with new features being added as new objects of abstract classes. In particular, the new TDC functionalities were added as new acquisition board permitting a seamless operation as compared to standard TCSPC boards. TDC data are pre-processed for refolding over replicas and to compensate for non-linearity, with very little additional computation time (<20 ms), allowing us almost real-time reconstructions of a single curve. Figure 1 (top left) shows the Instrument Response Function (IRF) of one channel with a dynamic range of about three decades and a FWHM of 270 ps. The reported IRF is representative of all eight channels, which all show a similar IRF. For the key performances of the system, the reader can refer to the paper by [PERSON] et al.