Diffuse optics studies how light propagates through strongly scattering materials, including biological tissue, powders, and disordered solids. In these materials, repeated scattering events make it impractical to calculate the trajectory of every photon individually.
Yet, after many scattering events, the more exact but more complicated radiative transfer equation is usually replaced with the simpler diffusion equation, which should become increasingly accurate as light propagates.

The diffusion equation contains a coefficient describing how quickly light spreads out: the diffusion coefficient. For more than 30 years, however, researchers have disagreed over how this coefficient should be defined.

The version most commonly used includes an explicit dependence on the absorption of the material. This creates a fundamental problem because it violates a fundamental scaling property of the radiative transfer equation.

We therefore have a situation where the diffusion equation is used as an approximation to the radiative transfer equation, while at the same time disregarding one of its key characteristics. Researchers have known about this inconsistency for decades and although several alternative definitions of the diffusion coefficient have been proposed, no consensus has emerged.

A recent paper by André Liemert and colleagues aims to settle this issue once and for all.

Establishing which definition is correct has proved surprisingly difficult. The differences between competing expressions can be extremely small, making them difficult to resolve experimentally or through numerical simulations. Rather than relying on such comparisons, the researchers tackled the problem analytically, using solutions of the radiative transfer equation to examine both pulsed illumination and continuous light.

For a short pulse of light, they demonstrated that the long-time behaviour converges exactly to the diffusion description only if the diffusion coefficient is independent of absorption, as required by the radiative transfer equation.

For continuous illumination, the situation is different and a paradox emerges. Even at arbitrarily large distances, where diffusion would be expected to become increasingly accurate, no single diffusion coefficient can reproduce both the rate at which the light signal decays and its amplitude. This means steady-state diffusion remains a useful approximation, but it can never be the exact long-distance limit of light transport in an absorbing material.

Any future study in biomedical optics or materials physics that uses diffusion models to extract absorption and scattering properties will have to take these results into account.