# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 4 Conclusions  
We investigated the temporal profiles of the fluence rate \(\Phi(\mathbf{r},t)\) and their peak times \(t_{peak}\) for various kinds of random media at different SD distances by analytical solutions of the RTE, dE0, and PDE for 3D infinite homogeneous media. We evaluated the crossover lengths and times from the ballistic to scattering (few-scattering-event) regimes, and from the scattering to diffusive regimes as approximately \(2.2/\mu_{t}^{\prime}\) and \(2.2/\mu_{t}^{\prime}v\); \(10/\mu_{t}^{\prime}\) and \(40/\mu_{t}^{\prime}v\), respectively.  
Next, we investigated \(\Phi(\mathbf{r},t)\) and \(t_{peak}\) for the dEM mainly in the scattering regime. We found that the results for the dE0 are quite similar to the results for the PDE because the dE0 and PDE approximate the forward scattering of photons as the isotropic scattering. We also found that at the expansion order of \(M\) larger than 2, the results for the dEM are almost the same as those for the RTE in the scattering and diffusive regimes, meaning the validity of the dEM with \(M>2\) in the regimes. We evaluated the crossover lengths and times for the dE1 and dE2 as approximately \(4.0/\mu_{t}^{\prime}\) and \(6.3/\mu_{t}^{\prime}v\); \(2.8/\mu_{t}^{\prime}\) and \(3.5/\mu_{t}^{\prime}v\), respectively.  
Finally, we investigated the boundary effects on the characteristic length and time scales of photon transport by comparing the results for the infinite media with those for the semi-infinite media. We found that the boundary conditions less influence on the crossover lengths for the dE0, dE1, and dE2, while the boundary conditions reduce the \(t_{peak}\)-values from those for bulk in the regime of the longer length scale than \(4.0/\mu_{t}^{\prime}\), which is the same as the crossover length for the dE1. Hence, the crossover  
Figure 11: Normalized peak times \(t_{peak}\) for the RTE in infinite media, and semi-infinite media at the \(z\)- and \(\rho\)-directions. The SD distances \(r_{sd}\) are given by \(r\) in infinite media and by \(r_{si}\) in semi-infinite media, respectively. The dashed and dotted lines represent the connection lines for the results in infinite and semi-infinite media at the \(\rho\)-direction, respectively, for the parameter set A. The other details are the same as Figure 10.  
Figure 10: Relative errors \(E_{peak}^{M}\) for the (**a**) dE0, (**b**) dE1, and (**c**) dE2 in the semi-infinite media (Figure 3b) as a function of the normalized SD distances \(r_{si}=|\mathbf{r}-\mathbf{r}^{+}|\) at the parameter sets A and B listed in Table 1. The SD distances are varied in the \(z\)-direction with \(\mathbf{r}=(0,0,z)\) and in the \(\rho\)-direction with \(\mathbf{r}=(\rho,0)\). The other details are the same as Figure 8a–c.  
times for the dE1 and dE2 are less influenced by the boundary conditions. Our findings are useful for constructions of accurate and efficient photon transport models especially in the scattering regime.  
Conceptualization, H.F.; investigation and analysis, H.F. and M.U.; writing--original draft preparation, H.F.; writing--review and editing, H.F., K.K. and M.W.; supervision, K.K. and M.W.; All authors discussed the results and contributed to the final manuscript. All authors have read and agreed to the published version of the manuscript.  
The first author ([PERSON]) acknowledges support from Grant-in-Aid for Scientific Research (18K13694) of the Japan Society for the Promotion of Science, KAKENHI.  
The authors would like to thank [PERSON], [PERSON], and [PERSON] for fruitful discussion. The first author ([PERSON]) appreciates [PERSON], his laboratory members, and staffs at University of Michigan for their kind supports on [PERSON]'s research work.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.