# Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media
## 1 Introduction
To overcome the difficulty, one needs to investigate the highly forward scattering of photons and influences of the highly forward-peaked phase function on the RTE-results in the scattering regime. For this purpose, in this paper, we employ the \(M\)-th order delta-Eddington equation (dEM), which is obtained by the delta-Eddington approximation (dEA) to the RTE. The dEA decomposes the highly forward-peaked phase function into purely forward-peaked and other components [16; 17]. The purely forward-peaked component is given by the delta-function and contributes to a modification of the scattering length scale. The other components are expanded by finite series of Legendre polynomials and used as the phase function in the dEM. The dEM has been widely employed in the field of biomedical optics since [PERSON] and coworkers have introduced from the field of astrophysics [18; 19; 20; 21; 22]. It has been shown that the numerical solutions of the dEM agree with those of the RTE while the number of the discrete angular directions is reduced to approximately one-sixth of the required number for the RTE-calculations. Nevertheless, a validity of the dEM is still unclear; [PERSON] and coworkers showed that the second order dEM can provide the accurate results the same as the RTE [21], while [PERSON] and coworkers stated that the zeroth order dEM is sufficient [22], and [PERSON] and coworkers employed the zeroth and first order dEM [20]. These differences probably come from the fact that the validity of the dEM depends on the SD distances, times after light incidence, and the optical properties, like the validity of the PDE. Our objective is to examine photon transport especially in the scattering regime for various kinds of random media by using the time-dependent RTE, dEM, and PDE, and to evaluate a regime (length and time scales) for the dEM to be valid as a function of \(\mu_{t}^{\prime}\) and \(\mu_{t}^{\prime}v\). We employed the analytical solutions instead of the numerical solutions because the analytical solutions are attained much faster than the numerical solutions and allow us to investigate a wide range of the SD distances and optical properties. In addition, we can focus the modification of the scattering coefficient and the approximation of the phase function by the dEA without a discussion of numerical errors induced by the numerical discretization. To the best of our knowledge, no studies have reported to evaluate the regime for the dEM. In this paper, we investigate photon transport for infinite and semi-infinite 3D homogeneous media as a first step of investigations for realistic heterogeneous biological tissue volumes. The investigations for infinite and semi-infinite 3D homogeneous media have been widely discussed for the diffuse optics community [4]. The analytical solutions have been employed for evaluations of the optical properties of various kinds of random media such as biological tissue volumes, colloidal suspensions [23], and agricultural products [24]. The following section describes the three kinds of the photon transport models for 3D random media: the time-dependent RTE, dEM, and PDE; and analytical solutions of the three models for the temporal profiles of the fluence rate. Section 3 investigates the results of the analytical solutions and evaluates the regime for the dEM. Finally, conclusions are described.