Jorge Bouza-Domínguez

Validity conditions for the radiative transfer equation.

We compare the radiative transfer equation for media with constant refractive index with the radiative transfer equation for media with spatially varying refractive indices [J. Opt. A Pure App. Opt. 1, L1 (1999)] and obtain approximate conditions under which the former equation is accurate for modeling light propagation in scattering media with spatially varying refractive indices. These conditions impose restrictions on the variations of the refractive index, the gradient of the refractive index, the divergence of the rays, the frequency of modulation, and the widths of light pulses, which are related to the mean refractive index, the absorption coefficient, and the reduced scattering coefficient of the medium. Each condition is geometrically interpreted. Some implications of the results are discussed.

Testing of radiative transfer equations for biomedical applications

The radiative transfer equation (RTE) is an important theoretical tool in biomedical optics for describing light propagation in tissues. The solutions to its derived diffusion equation (DE) are used, for example, for dose calculation in photodynamic therapy and for optical tomography. The RTE is valid for constant refractive index and zero ray divergence. These conditions limit its applicability in biomedical optics. To eliminate these drawbacks three new RTEs have been proposed. In this paper we test the standard RTE and the new RTEs by solving them for the irradiance of rays propagating in an infinite medium with no scattering, no absorption and no amplification. The solutions to this problem must coincide with the irradiance laws of geometrical optics. We show that only one of those equations gives solutions, which are consistent with irradiance laws of geometrical optics due to its ability to model, the effect of spatially varying refractive index and non-negligible ray divergence. Consequently that equation gives a better description of light propagation in scattering media with spatially varying refractive index and near sources, a physical situation occurring frequently in biomedical optics.

Study of the failure of the time-independent diffuse equation near a point source

The diffusion equation (DE) is widely used in biomedical optics for describing light propagation in tissue. However, the DE yields inaccurate results near sources. This drawback is important in practical situations, when it is of primary interest to calculate the dose of light applied or to retrieve the optical properties of the tissue near the light source, e.g., the distal end of an optical fiber. To study this problem we derived a diffusion equation for constant refractive index and rays of arbitrary divergence (DErad) from a modified radiative transfer equation for spatially varying refractive index. We solve the DErad for a time-independent point source in near field and far field, which are defined by a parameter Rcrit. The far-field solution is the solution to the time-independent DE, the near-field solution agrees well with Monte Carlo simulation results and the Rcrit coincides with the reported radius of inaccuracy of the DE. These results suggest that the inaccuracy of the time-independent DE near a point source is due to a non-negligible ray divergence.

Diffusion equation and ray divergence near a point source

We solve a diffusion equation with nonzero ray divergence for a time-independent point source. The solutions suggest that non-negligible ray divergence causes the failure of the standard diffusion equation near a point source.

Deriving the inverse square law from radiative transfer equations

The radiative transfer equation (RTE) is the fundamental equation of the radiative transfer theory and one of more important theoretical tools in biomedical optics for describing light propagation in biological tissues. The RTE assumes that the refractive index of the medium is constant and the ray divergence is zero. These assumptions limit its range of applicability. To eliminate this drawback three new RTE have been proposed recently. Obviously, those equations must be carefully studied and compared. With that aim we solve the standard RTE and the new radiative transfer equations for the specific case of a time-independent isotropic point source in an infinite non-absorbing non-amplifying non-scattering linear medium with constant refractive index. The solution to this problem is the well-known inverse square law of geometrical optics. We show that only one of those equations gives solutions consistent with the inverse square law for the irradiance, due to its ability to model non-negligible ray divergence near a point source.