# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 3 Results
### Influence of the Boundary Conditions on the Crossover Lengths and Times  
In this subsection, we investigated the boundary effects on the fluence rate \(\Phi(\mathbf{r},t)\) and on the crossover lengths and times for the dEM with \(M=0\), 1, and 2 by comparing the analytical solutions for the semi-infinite media with those for the infinite media. We considered the parameter sets A and B listed in Table 1, and varied the SD distances, denoted by \(r_{si}=|\mathbf{r}-\mathbf{r}^{+}|\), in the two directions: the \(z\)-direction with \(\mathbf{r}=(0,0,z)\) and \(\rho\)-direction with \(\mathbf{r}=(\mathbf{\rho},0)\) as shown in Figure 3b. For the case of the detector position in the \(z\)-direction, as the \(z\)-value increases, the detector position becomes far from the boundary, so that the boundary effects probably become weak. Meanwhile, for the case of the detector position in the \(\rho\)-direction, a distance between the detector and the boundary is unchanged, so that the boundary effects are also unchanged. Figure 10a-c show the relative errors \(E_{peak}^{M}\) for the dE0, dE1, and dE2 in the semi-infinite media as a function of the normalized SD distance by \(\mu_{t}^{\prime}\). The crossover lengths for the dEM in the semi-infinite media are almost the same as those in the infinite media, meaning less influences of the boundary conditions on the crossover lengths.  
Figure 7: (**a**) Temporal profiles of \(\Phi(r,t)\) using the RTE, dE0, dE1, and dE2 at \(g=0.9\). (**b**) Relative errors \(E_{peak}^{M}\) of the peak time of \(\Phi(r,t)\) (Equation (17) as a function of the expansion order \(M\) at different \(g\)-values of 0.3, 0.6, 0.9, and 0.95. 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.  
Figure 11 shows the normalized peak times for the RTE in the infinite and semi-infinite media as a function of the SD distances \(r_{sd}\) for the parameter sets A and B. Here, the SD distances \(r_{sd}\) are given by \(r\) in infinite media, and by \(r_{si}\) in semi-infinite media, respectively. The \(t_{peak}\)-results in the infinite media and semi-infinite media at the \(z\)-direction behave similarly to each other on the whole regime. Meanwhile, the \(t_{peak}\)-results in the semi-infinite media at the \(\rho\)-direction (dotted line in Figure 11 for the results at the parameter set A) are smaller than those in the other two cases (dashed line) especially on the regime of the longer length scale than approximately \(10^{0.6}/\mu_{t}^{\prime}\sim 4.0/\mu_{t}^{\prime}\), which is almost the same as the crossover length for the dE1. These results suggest that the boundary conditions influence the crossover time for the dE0, while they have less influence the times for the dE1 and dE2.  
Figure 8: Relative errors \(E_{peak}^{M}\) (Equation (17)) for the (**a**) dE0, (**b**) dE1, and (**c**) dE2 as a function of the normalized SD distances by \(\mu_{t}^{\prime}\) at the parameter sets A to F listed in Table 1. The yellow, orange, and red colored areas correspond to the length regimes where the dE0, dE1, and dE2 are invalid, respectively. (**d**) Normalization of the SD distances by \(\mu_{t}^{M}\). The gray colored areas correspond to the crossover length for the dEM. The other details are the same as Figure 6.  
Figure 9: Normalized peak times of \(\Phi(r,t)\) for the RTE, dE0, dE1, and dE2 as a function of the normalized SD distances. The dashed line represents the connection line for the RTE-results at the parameter set A. The other details are the same as Figure 8a-c.