## 2 Physical Concepts
The contrast (C) produced by localized optical inhomogeneities--i.e., the sensitivity to optical changes, in formula: \(\mathrm{C}=\frac{N_{0}-N}{N_{0}}\) where \(N_{0}\) and \(N\) are the number of photons for the unperturbed and  
Figure 2: The null source-detector distance (\(\rho\)) increases the number of collected photons at all times with respect to a larger \(\rho\) (modified from [38]). perturbed case, respectively [60]--is enhanced due to the strongest confinement of photon trajectories, as shown in Figure 3. The depth sensitivity is dependent only on the photon arrival time and is independent of the source-detector distance \(\rho\). Conversely, for \(\rho\) = 0 the spatial spread of photon trajectories is minimized, leading to a better confinement of the explored region. The practical consequence of higher spatial confinement is the increase in contrast, as shown in Figure 4 and the improvement in spatial resolution, displayed in Figure 5. The spatial resolution is defined as the full width at half maximum--FWHM--of the spatial point spread function--PSF--obtained measuring the diffuse reflectance curve with the perturbation at the given depth \(z\) and using the source-detector distance \(\rho\) (in formula: \(FWHM[PSF(z,\rho)]\)) [60]. The former consequence leads to a better capability of \(\rho\) = 0 measurements to detect deep optical perturbations as in the case of functional brain imaging, discussed in Section 4. The latter would permit to improve spatial resolution when imaging through diffusive media. Still, the gain in resolution is significant for shallower regions (\(z\) < 10 mm), while at higher depths the improvement in spatial resolution is less significant. Figure 4: The \(\rho\) = 0 case improves contrast at all times with respect to a larger \(\rho\) (modified from [38]). Figure 3: The depth sensitivity of a TD reflectance measurements to localized optical perturbations is dependent on the photon arrival time and independent of \(\rho\). The limiting case of \(\rho\) = 0 corresponds to the most spatially confined sensitivity map (modified from [38]). The contour lines correspond to time ranging from 0.5 to 4 ns at steps of 0.5 ns. It is necessary to warn that all these statements hold true for an ideal case of a perfect time-resolved system with a delta-like instrument response function (IRF) coupled with a sharp hardware time-gating. Practical systems exhibit non-ideal behaviors which distort the theoretical picture and diminishes the advantages of the \(\rho\) = 0 approach, as will be duly discussed in Section 3. In practical cases, a sage compromise taking into account the real system performances must be sought. On the other hand, the ideal case helps in addressing directions for technological advancements. To draw the future research directions, it is useful to identify the ultimate limits of diffuse optics in terms of maximum depth and sensitivity [61]. Figure 6 displays the results of simulations on the relative contrast produced by a 1 cm\({}^{3}\) localized optical perturbation with twice absorption as the background buried at increasing depths in a tissue-like diffusive medium. As a criterion of detectability, we require to have a change in the detected signal higher than 1% of baseline (contrast \(>\) 1%) and a fluctuation in the signal due to the intrinsic shot noise lower than 1% (Poisson \(<\) 1%). This corresponds to the white region in the figure. Both CW and TD approaches are considered, where--in the latter case--the temporal response of the detector is assumed to be an ideal delta function and a perfect sharp hardware gating is applied to progressively suppress photons arriving at earlier time. In both cases, a 1 cm\({}^{2}\) ideal detector with 100% quantum efficiency is considered together with a uniform illumination over 1 cm\({}^{2}\) area with 4 mW/mm\({}^{2}\) intensity (chosen in [61] representative of typical maximum permitted skin exposure). An ideal CW acquisition can increase depth sensitivity enlarging the source-detector separation.