Time-dependent equation for the intensity in the diffusion limit using a higher-order angular expansion.
The first two terms in the spherical-harmonic expansion (the P(1) approximation) of the radiative transfer equation yield the diffusion equation. This approximation applies to multiple scattering and results in a solution for the energy density, the gradient of which is proportional to the light intensity. In this work a higher-order spherical-harmonic expansion of the radiative transfer equation is developed. This equation applies to the radiant intensity rather than the energy density. The equation can be decomposed into two terms: a propagator term obtained from the determinant of the coupled equations describing the individual components of the intensity, and a mixing matrix that describes the cross coupling between different orders of the expansion. Using the Fourier transform, an approximation based on expanding at small wave vectors k leads to an equation similar to the diffusion equation. The equation is expected to predict the intensity for multiple scattering at earlier times and shorter distances than the diffusion equation can. The notion of an equivalent wave field is introduced.