## 1 Introduction  
Diffuse optics (DO) is a branch of optics which aims to study the optical properties of a diffusive medium using light in the near-infrared spectral window, that is the wavelength range where the water absorption is negligible [1]. When light interacts with matter, it is typically attenuated because of the selective absorption of the various chromophores of the material investigated at different wavelengths. In diffusive media, the ballistic photons are also deviated in different directions (scattered) because of the structural heterogeneities of the medium. The retrieval of the optical properties, i.e., absorption (\(\mu_{a}\)) and reduced scattering (\(\mu_{s}\)) coefficients, of a diffusive medium is the main purpose of diffuse optical spectroscopy (DOS) [2]. Examples of diffusive media are fruits, vegetables [3], wood [4], plastic or resins [5], pharmaceutical tablets [6], fog and clouds [7], and biological tissues such as brain, muscles, breast, bones, thyroid and others [8; 9; 10; 11; 12]. The main advantages of DO techniques are: (i) non-invasiveness and non-destructiveness: in-vivo measurements on human and industrial applications are possible without damaging the tissue/sample; (ii) functional measurements: possibility to follow the temporal variations of the optical properties with a real-time feedback; (iii) possibility to derive other parameters from the optical properties, such as--for biological media--the main tissue components (e.g., lipid, collagen and water) or metabolic and oxidative state indicators (e.g., oxygen saturations, hemoglobin and flux). Due to its intrinsic non-invasive nature, DO has a strong potential in several applications both in laboratories and professional environments (industries and clinics). Thanks to the key advantages described above, the most widespread application of DO is on biological tissues, where the aim is the non-invasive and real time assessment of the concentration of the main chromophores--e.g., oxy-(O\({}_{2}\)Hb) and deoxy-hemoglobin (HHB)--and the microvascular blood flow. These parameters can be assessed using near infrared spectroscopy (NIRS) and diffuse correlation spectroscopy (DCS), respectively [13; 14]. The time-domain (TD) approach, based on the use of picosecond pulsed lasers and time-resolved single-photon detection, offers the possibility to discriminate photons on the basis of their arrival time and to retrieve absolute values of the optical properties [15]. The temporal profile of laser pulses travelling through a diffusive medium is strongly shaped by the medium optical properties. In particular, the distribution of time-of flight (DTOF) of photons re-emitted at a given distance from the injection point (source-detector distance) is temporally broadened because of scattering and is attenuated with an exponentially decreasing tail due to absorption. In Figure 1 of Ref. [16], the reader can find a graphical representation of the changes in the DTOF for different values of the optical properties for a homogeneous medium. Starting from the DTOF at a given wavelength, it is possible to retrieve the medium optical properties by fitting the experimental data (e.g., using the classical Levenberg-Marquardt approach) with a proper theoretical model, obtained by the solution of the diffusion equation (DE). From the optical properties, and applying the Lambert Beer Law together with the chromophores extinction coefficients, it is possible to retrieve the hemodynamic parameters of interest or the concentration of key tissue constituents. It goes beyond the scope of this review to describe the different solutions of the DE for different geometries as well as other more advanced solutions of the transport equation or to discuss the specific procedures to extract the hemodynamic parameters out of the optical properties. For further details and schematics on TD measurements the reader can see Refs. [15; 16].