# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 3 Results  
### Classification of the Length and Time Scales of Photon Transport  
In this subsection, we investigated the temporal profiles of the fluence rate \(\Phi(r,t)\) using the analytical solutions of the RTE (Equation (10)), dE0 (Equation (11) with \(M=0\)), and PDE (Equation (12)) for 3D infinite homogeneous media. Also, we investigated the peak time of \(\Phi(r,t)\), \(t_{peak}\), because it is one of characteristic times of photon transport, and a difference in \(\Phi(r,t)\) is correlated with a difference in \(t_{peak}\). Based on results of \(t_{peak}\) at different SD distances \(r_{sd}=r\), we classify length and time scales of photon transport into following three kinds of characteristic regimes; (1) ballistic regime: \(t_{peak}\sim t_{peak}^{ball}\), (2) scattering regime (few-scattering-event regime): \(t_{peak}\sim t_{peak}^{ball}\), \(t_{peak}^{diff}\), and (3) diffusive regime: \(t_{peak}\sim t_{peak}^{diff}\), where \(t_{peak}^{ball}\) and \(t_{peak}^{diff}\) are defined by  
\[t_{peak}^{ball}=\frac{r}{v}\,\ \ \ \ \ \ \ \ \ t_{peak}^{diff}=\frac{-3+2\sqrt{\frac{9}{4}+ \frac{r^{2}\mu_{a}}{D}}}{4\mu_{a}v}. \tag{15}\]  
\(t_{peak}^{ball}\) represents the arriving time of photons to the detector position without scattering, and \(t_{peak}^{diff}\) the peak time for the diffusive photons calculated from the PDE (Equation (12)), respectively. Figure 4 show examples of the temporal profiles of \(\Phi(r,t)\) at the three characteristic length regimes. Here, we used the analytical solutions with the optical properties for the parameter set A as listed in Table 1 corresponding to the highly forward scattering of photons. While the RTE can appropriately treat the highly forward scattering, the dE0 and PDE approximate the highly forward scattering to the isotropic scattering. As shown in Figure 4a corresponding to the ballistic regime, the \(\Phi\)-profiles using the RTE and dE0 are sharp, although both profiles are different from each other. The sharp profiles mean that photons are little scattered and travel almost straight. The peak times \(t_{peak}\) for the RTE and dE0 are almost the same as the arriving time \(t_{peak}^{ball}\), whose value is approximately 8.4 ps. Meanwhile, the broad profile of \(\Phi\) using the PDE is observed, indicating the unphysical multiple scattering of photons. In addition, the \(t_{peak}\)-value for the PDE is shorter than the \(t_{peak}^{ball}\)-value, meaning the violation of the causality in the PDE. As shown in Figure 4b corresponding to the scattering regime, the profiles using all the equations have smooth peaks. These peaks are formed by scattering of photons, so that these \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter Sets: [\(\mu_{a}\), \(\mu_{s}\), \(g\), \(n\), \(r_{sd}\)]** & **Range** & **Color** \\ \hline Set A: [0.10, 100, 0.90, 1.40, \(r_{sd}\)] & \(r_{sd}\): 0.15–10.0 & Black \\ Set B: [0.35, 58.0, 0.80, 1.56, \(r_{sd}\)] & \(r_{sd}\): 0.15–10.0 & Blue \\ Set C: [\(\mu_{a}\), 100, 0.90, 1.40, 0.60] & \(\mu_{a}\): 0.10–10.0 & Red \\ Set D: [0.10, \(\mu_{s}\), 0.90, 1.40, 0.60] & \(\mu_{s}\): 30.0–200 & Green \\ Set E: [0.10, 100, \(g\), 1.40, 0.60] & \(g\): 0.10–0.95 & Yellow \\ Set F: [0.10, 50.0, \(g\), 1.40, 0.60] & \(g\): 0.10–0.95 & Purple \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical properties: \(\mu_{a}\)[cm\({}^{-1}\)], \(\mu_{s}\)[cm\({}^{-1}\)], \(g[-]\), and \(n[-]\); and SD distance \(r_{sd}\)[cm]. Corresponding colors are used in Figures 5, 6 and 8–11. peak times are longer than \(t_{peak}^{ball}\), in other words, the paths of photons are longer than the SD distance. In Figure 4b, the peak times for the dE0 and PDE are different from that for the RTE, indicating that the forward scattering of photons is not approximated to the isotropic scattering in this regime. As shown in Figure 4c corresponding to the diffusive regime, all the profiles are almost the same to each other, meaning the photon scattering becomes diffusive and the isotropic scattering approximation holds.