## 2 Materials and Methods
### Performance Assessment  
In order to impartially assess the performances of the proposed setup, we made use of some parameters of two internationally agreed protocols for diffuse optics instrumentation (the \"Basic Instrumental Protocol\" (BIP) [20] and \"nEUROPt\" [21] ones). Their aim is to characterize the TR setup for brain imaging. For what concerns the BIP protocol, we acquired the IRF of the proposed setup at both wavelengths, coupling the launching and collection fiber with a diffusive medium (a thin film of Teflon(tm)) so as to excite all propagation modes of the detection fibers. In order to have a high-dynamic range measurement, we adopted the technique explained in [PERSON] et al. and [PERSON] et al. [12; 24], acquiring the re-emitted photons at different delays (from 0 up to 3450 ps at step of 25 ps) with an integration time of each delay of 10 s. Before rescaling the acquisitions and reconstructing the IRF, we subtracted the constant background (the so-called \"dark count\", which is not correlated with laser light) to all curves. We finally computed the FWHM and the dynamic range of the reconstructed IRF, as suggested by the BIP protocol. We then characterized the proposed setup at 820 nm using two features of the \"nEUROPt\" protocol (contrast and contrast-to-noise ratio (CNR)) since they are meant to define quantitatively the capability of detecting a small absorption change within the probed volume (e.g., absorption changes in the cortex due to brain activation). For both tests, we made use of a liquid phantom composed of a mix of water, Intralipid 20%, and Indian ink in such a proportion to have an absorption coefficient (\(\mu_{\text{a}}\)) of 0.1 cm\({}^{-1}\) and a reduced scattering one of 10 cm\({}^{-1}\) at 820 nm. In order to simulate the absorption changes related to a localized brain activation, we made use of solid black cylindrical inclusions (height: 4.5 mm; diameter: 5.32 mm). In [PERSON] et al. [25], the equivalence relation between a realistic absorption change in a finite volume and the perturbation produced by a small black object had been demonstrated. In our case, we performed all tests using the 100 mm\({}^{3}\) totally absorbing inclusion (which corresponded to a \(\Delta\mu_{\text{a}}\) of 0.15 cm\({}^{-1}\) over a 1 cm\({}^{3}\) volume). The inclusion was posed beneath the surface of the phantom and its position was moved from 2 up to 30 mm in depth (the distance between the centroid of the inclusion and the surface). For each depth, the time-resolved curve was acquired in difference slices, delaying the opening of the gate from 0 ns (situation where the first re-emitted photon was recorded) up to 3250 ps at step of 250 ps. The VOA was set in order to have nearly 1 Mcps for each delay (except for the latest where it was not possible to increase the injected power). For each delay, we acquired five repetitions (1 s each) of the distribution of time-of-flight of re-emitted photons. We computed contrast and CNR according the definition given in [PERSON] et al. [21] and briefly summarized in the following section. The contrast is a primary assessment of the effect of variation in absorption coefficient. In the case of time-resolved reflectance measurements, often the analysis is performed using time-windows, meaning that the distribution of time-of-flight is subdivided into portions and contrast (as well as other features)and can be computed within those subdivisions of the curve. In case of the so-called \"time-window\" analysis, the contrast in each temporal window (\(C(t_{W})\)) was computed by the following equation:  
\[C^{(t_{w})}=\frac{N_{0}^{(t_{w})}-N^{(t_{w})}}{N_{0}^{(t_{w})}}, \tag{1}\]  
where \(N_{0}^{(t_{w})}\) and \(N^{(t_{w})}\) represent the overall number of counts computed in the time-window (\(t_{w}\)) in the homogeneous or perturbed (e.g., when the inclusion mimicking the perturbation is inserted into the probed volume) state, respectively. In our case, each recorded curve was subdivided in time-windows of 500 ps in width.