# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 3 Results
### Length and Time Scales for the dEM to Be Valid in the Scattering Regime  
In this subsection, we investigated the temporal profiles of the fluence rate \(\Phi(r,t)\) and their peak times \(t_{peak}\) for the RTE and dEM in the scattering regime for the purpose of evaluations of the length and time scales for the dEM to be valid. As mentioned in Section 2.1.3, for the dEM, the scattering length scale is modified to the inverse of \(\mu_{s}^{M}\) from the inverse of \(\mu_{s}\), and the highly forward scattering of photons described by \(p_{HG}\) is approximated to \(p_{\delta 2}^{M}\). These modifications influence the results of \(\Phi(r,t)\). Firstly, we calculated \(\Phi(r,t)\) and \(t_{peak}\) for the dEM at different \(g\)-values of 0.3, 0.6, 0.9, and 0.95 in the scattering length regime, and investigated the dependence of the results of \(\Phi(r,t)\) and \(t_{peak}\) on the expansion order \(M\) and on the \(g\)-value. The optical properties and SD distance are set as \(\mu_{a}=0.10\) cm\({}^{-1}\), \(\mu_{s}^{\prime}=\mu_{s}(1-g)=10.0\) cm\({}^{-1}\), \(n=1.40\), and \(r=0.32\) cm, where the normalized SD distance is constant as \(\sim 10^{0.5}/\mu_{t}^{\prime}\). Figure 7a shows the temporal profiles of \(\Phi(r,t)\) for the RTE and dEM with \(M=0\), 1, and 2 (dE0, dE1, and dE2) at \(g=0.9\), corresponding to the highly forward scattering. While the \(\Phi\)-profile for the dE0 largely deviates from that for the RTE, the profiles for the dE1 and dE2 are similar to that for the RTE. Moreover, we investigated the relative error \(E_{peak}^{M}\) of the peak time for the dEM:  
\[E_{peak}^{M}=\left|\frac{t_{peak,dEM}-t_{peak,RTE}}{t_{peak,RTE}}\right|\times 1 00, \tag{17}\]  
where \(t_{peak,dEM}\) represents the peak time of \(\Phi(r,t)\) for the dEM. As shown in Figure 7b, the \(E_{peak}^{M}\)-values are quite small when the expansion order \(M\) is larger than 2 at all the \(g\)-values. This result means that the dEM with \(M>2\) can provide the same accuracy as the RTE, almost independently of the \(g\)-values. Meanwhile, as shown in Figure 2, the \(g\)-dependences are observed in the ratios of \(\mu_{s}^{M}/\mu_{s}\) and the errors of the phase function \(E_{phase}^{M}\). These results suggest that for the accurate calculations of \(\Phi(r,t)\) using the RTE and dEM, it is sufficient to satisfy a few orders moment conditions of the phase function and the exact agreement of the profiles of the phase functions is not required. It is noted that the analytical results of the small \(E_{peak}^{M}\)-values with \(M>2\) are quite different from the numerical calculations. The numerical results for the dEM are usually accurate within a finite range of \(M\), and when the \(M\)-value is larger than the maximum value of the finite range, the numerical results for the dEM largely deviate from those for the RTE because of the numerical errors induced by angular discretization. On the above discussion, we investigated \(E_{peak}^{M}\) for the dEM at the constant normalized SD distance. Next, we investigated \(E_{peak}^{M}\) for the dE0, dE1, and dE2 at the normalized SD distances varied from the scattering to diffusive regimes. Then, we evaluated the crossover lengths for the dE0, dE1, and dE2; in a regime of a smaller length scale than the crossover length, the \(E_{peak}^{M}\)-values are larger than 2%, corresponding to the invalidity of the dEM, while in a regime of a longer length scale, the errors are smaller than 2%, corresponding to the validity of the dEM. Figure 8a-c show the \(E_{peak}^{M}\)-values as a function of the normalized SD distances at the parameter sets A to F listed in Table 1. While the max value of \(E_{peak}^{M}\) for the dE0 is over 20%, the max values for the dE1 and dE2 are around 10% and 4%, respectively. The length regime for the dE2 where the \(E_{peak}^{M}\)-values are larger than 2% is shorter than those for the dE0 and dE1.