# 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
We evaluated crossover lengths and times from the ballistic to scattering regimes, and from the scattering to diffusive regimes, respectively, by investigating \(t_{peak}\) for the RTE, dE0, and PDE. Then, we normalized the SD distance \(r\) by \(\mu_{t}^{\prime}\) and the peak time \(t_{peak}\) by \(\mu_{t}^{\prime}v\), respectively, with the reduced transport coefficient of \(\mu_{t}^{\prime}=\mu_{s}(1-g)+\mu_{a}\). The inverses of \(\mu_{t}^{\prime}\) and \(\mu_{t}^{\prime}v\) correspond to the characteristic length and time for the dE0 and PDE where photons experience single scattering and absorbing processes. The normalizations allow us to investigate general features of photon transport in various kinds of random media. Figure 5 shows the results of the normalized peak times as a function of the normalized SD distances for the parameter sets A to F listed in Table 1 in a regime of short length and time scales. Here, the colors of the figures correspond to the parameter sets, e.g., the black colored pluses, circles, and squares correspond to the results for the parameter set A. In the regime of \(\log_{10}r\mu_{t}^{\prime}\lesssim 0.35\) and \(\log_{10}t_{peak}\mu_{t}^{\prime}v\lesssim 0.35\), the \(t_{peak}\)-results for the RTE and dE0 are linearly related to \(r\), while the results for the PDE are not. This result suggests that the regime of the length and time scales correspond to the ballistic regime; and the crossover length and time from the ballistic to scattering regimes are evaluated as approximately \(10^{0.35}/\mu_{t}^{\prime}\sim 2.2/\mu_{t}^{\prime}\) and \(10^{0.35}/\mu_{t}^{\prime}v\sim 2.2/\mu_{t}^{\prime}v\), respectively. Figure 6a shows the \(t_{peak}\)-results in a regime of longer length and time scales than the ballistic regime. In the case of the same parameter set (the same colored figures), the \(t_{peak}\)-results for the dE0 and PDE deviate from those for the RTE in the regime of \(0.35\lesssim\log_{10}r\mu_{t}^{\prime}\lesssim 1.0\) and \(0.35\lesssim\log_{10}t_{peak}\mu_{t}^{\prime}v\lesssim 1.6\), while the results for all the equations are almost the same to each other in the regime of \(1.0\lesssim\log_{10}r\mu_{t}^{\prime}\) and \(1.6\lesssim\log_{10}t_{peak}\mu_{t}^{\prime}v\). For the further investigation of \(t_{peak}\), we evaluated the relative errors \(E_{peak}\) of \(t_{peak}\) against the RTE, defined by  
\[E_{peak}=\left|\frac{t_{peak}-t_{peak,RTE}}{t_{peak,RTE}}\right|\times 100, \hskip 14.226378 ptt_{peak}=\{t_{peak,dE0},\ t_{peak,PDE}\}. \tag{16}\]  
Although we calculated \(E_{peak}\) for all the parameter sets, we showed the results only for the parameter set A in Figure 6b because the results for the other parameter sets behave similarly to the results for the parameter set A. As shown in Figure 6b, the \(E_{peak}\)-values for the dE0 and PDE are larger than 2% in the regime of \(0.35\lesssim\log_{10}r\mu_{t}^{\prime}\lesssim 1.0\). Here the \(E_{peak}\)-value of 2% is considered as a thresh to determine whether the dE0 and PDE are valid or not. The large \(E_{peak}\)-values indicate the  
Figure 4: Analytical solutions of the RTE (Equation (10)), dE0 (Equation (11) with \(M=0\)), and PDE (Equation (12)) at the three characteristic length regimes of photon transport: (**a**) ballistic regime at the SD distance of \(r=0.18\) cm (\(\log_{10}r\mu_{t}^{\prime}=0.26\)), (**b**) scattering regime at \(r=0.40\) cm (\(\log_{10}r\mu_{t}^{\prime}=0.60\)), and (**c**) diffusive regime at \(r=1.60\) cm (\(\log_{10}r\mu_{t}^{\prime}=1.20\)). The optical properties at the parameter set A are used, which are listed in Table 1.  
strong influence of modeling the forward scattering on the \(\Phi\)-results in this regime. Meanwhile, the \(E_{peak}\)-values for the dE0 and PDE are less than 2% in the regime of \(1.0\lesssim\log_{10}n\mu_{t}^{\prime}\), meaning the isotropic scattering approximation holds.