# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 1 Introduction  
Elucidation of the photon scattering and transport in random media such as biological tissue volumes is crucial for biomedical optical imaging using the near-infrared light in the wavelength range from 700 to 1100 nm such as diffuse optical tomography [1; 2; 3], because the imaging technique uses light scattered by the media. There are mainly three kinds of systems of the imaging techniques: time-domain, steady-state, and frequency-domain systems. Among them, the time-domain system has several advantages over steady-state and frequency-domain systems thanks to the richest information of the time-domain measurement data [4]. Time-dependent photon transport is basically classified into the two kinds of characteristic regimes (length and time scales): the ballistic and diffusive regimes corresponding to short and long source-detector (SD) distances and times after light incidence, respectively [5]. In the ballistic regime, photon is little scattered and almost goes straight. In the diffusive regime, photon is diffusive due to multiple scattering of photons, and the scattering process can be approximated as isotropic. Between the ballistic and diffusive regimes, photon undergoes a fewscattering events in the highly forward direction and travels along the zig-zag paths while changing the direction [6; 7]. In this paper, we call this few-scattering-event regime between the ballistic and diffusive regimes as the scattering regime, so that we consider the three kinds of characteristic regimes for photon transport. The radiative transfer equation (RTE), which is the linear Boltzmann equation, can describe photon transport for random media in the three characteristic regimes because the phase function in the RTE can treat the highly forward scattering of photons. However, the numerical calculations of the RTE require high computational loads (times and memories) especially for three-dimensional (3D) random media because the RTE is an integro-differential equation with respect to the light intensity as a function of position vector, unit angular direction vector, and time. To reduce the computational loads, many researches have employed the photon diffusion equation (PDE), which is obtained by the diffusion approximation to the RTE and omits the angular direction. However, the PDE is valid only in the diffusive regime. A coupling model using the RTE and PDE has been constructed [8; 9]. For the coupling model, the RTE is calculated in the ballistic and scattering regimes, while the PDE is calculated in the diffusive regime. For constructions of the coupling model or appropriate uses of the PDE, many researches have devoted their work to evaluations of the crossover length and time from the scattering to diffusive regimes [9; 10; 11; 12; 13; 14; 15]. For example, a comparison study [10] between actual light measurements and the PDE-results for slab media showed that the crossover length is evaluated as approximately \(10/\mu_{t}^{\prime}\), where the reduced transport coefficient \(\mu_{t}^{\prime}\) is a sum of reduced scattering coefficient and absorption coefficient. A numerical study [9] using the time-dependent RTE and PDE for 2D media evaluated the crossover length the same as the above study [10], and in addition, evaluated the crossover time as approximately \(20/\mu_{t}^{\prime}v\), where \(v\) is the speed of light in the medium. Although the coupling model can provide accurate and efficient results, a further efficiency improvement of the photon transport model is necessary because the RTE-calculations in the scattering regime still require high computational loads. For modeling the highly forward scattering of photons, the phase function in the RTE is highly forward-peaked and changes exponentially with respect to the scattering angle. Basically, accurate numerical treatments of the highly forward-peaked phase function require a large number of discrete angular directions, resulting in high computational loads of the RTE-calculations.