The discrete eigenvalues of the radiative transfer equation with Henyey-Greenstein scattering

The radiative transfer equation (RTE) is fundamental in modelling photon transport across various scientific and engineering domains, including biomedical optics, atmospheric physics, and nuclear engineering. This study investigates photon propagation in a homogeneous medium using the Henyey-Greenstein (HG) scattering function, which is widely applied in biophysics to describe light scattering in biological tissues. The anisotropy parameter g characterizes scattering properties, playing a crucial role in photon movement. To solve the RTE for a steady-state homogeneous system, we employed two polynomial approximation methods: Legendre polynomial expansion (PN method) and second-kind Chebyshev polynomial expansion (UN method). The eigenvalues play a central role in classical problems such as the albedo problem, the Milne problem, and criticality analysis in slab geometry. We iteratively solve the RTE up to the fifth order and compare the convergence and accuracy of both methods. The results indicated that increasing the number of iterations improves accuracy, with both methods exhibiting stable convergence behaviour. A detailed comparison was conducted and the numerical results were validated against existing literature. Additionally, the tabulated results highlighted the forward and backward scattering components of the HG function, illustrating the influence of the anisotropy parameter on photon transport. The consistency between the proposed methods and the literature was confirmed by comparing discrete eigenvalues in tables. The maximum deviation between the P₉ and U₉ methods remained below 0.08, while both approaches closely matched the benchmark results in literature, supporting the reliability and numerical robustness of the polynomial expansions used. This study offers a valuable reference for future research, enabling comparisons and validation of computational approaches in radiative transfer modelling, particularly in biophysics and optical applications.