Light Transport in Tissue

Index mismatch, diffuse light incident

When diffuse light is incident on a slab having an index of refraction different from the medium directly above the slab, the boundary condition is given by Equation (4.36) with an extra term to account for the diffuse irradiance

$\displaystyle \int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega}$	=	$\displaystyle \int_{2\pi\,\,\mu\le0} r(\hat\mathbf{z}\cdot\hat\mathbf{s}) L_d(\mathbf{r},\hat\mathbf{s}) (-\hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega}$
	+	$\displaystyle \int_{2\pi\,\,\mu\ge0} t(\hat\mathbf{z}\cdot\hat\mathbf{s}) L_{di... ...hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega} \qquad\mathrm{at}\qquad z=0$	(4.50)

where $L_{di}(\mathbf{r},\hat\mathbf{s})$ represents the isotropic radiance incident on the slab and $t(\hat\mathbf{s}\cdot\hat\mathbf{z})=1-r(\hat\mathbf{s}\cdot\hat\mathbf{z})$ is the Fresnel transmission. Since the diffuse radiance $L_{di}(\mathbf{r},\hat\mathbf{s})$ is independent of angle, Equation (4.50) reduces to the form

$\begin{displaymath} {1\over4} \varphi_d(\mathbf{r}) +{1\over2}(\mathbf{F}_d(\mat... ...t\hat\mathbf{z}) +\pi L_{di}(\mathbf{r},\hat\mathbf{s})(1-R_1) \end{displaymath}$

(4.51)

The reflection coefficient R₁ is the same integral of the Fresnel reflection as defined in Equation (4.42) and is tabulated in Appendix B. Equation (4.51) is identical to Equation (4.44) except that the included diffuse light has been reduced by the light reflected at the boundary. This relation simplifies to

$\begin{displaymath} \varphi_d(\mathbf{r})-A_{\mathrm{top}}h{\partial\varphi_d(\m... ... L_{di}(\mathbf{r},\hat\mathbf{s}) \qquad\mathrm{at}\qquad z=0 \end{displaymath}$

(4.52)

The boundary condition for diffuse light incident from the bottom of the slab is similar to Equation (4.52). The only difference is the hemispheres over which the integrals are done. The boundary condition at z=d, where d is the depth of the slab is

$\displaystyle \int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) (-\hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega}$	=	$\displaystyle \int_{2\pi\,\,\mu\ge0} r(\hat\mathbf{z}\cdot\hat\mathbf{s}) L_d(\mathbf{r},\hat\mathbf{s}) (\hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega}$	(4.53)
	+	$\displaystyle \int_{2\pi\,\,\mu\le0} t(\hat\mathbf{z}\cdot\hat\mathbf{s}) L_{di... ...hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega} \qquad\mathrm{at}\qquad z=d$

where $L_{\mit di}'(\mathbf{r},\hat\mathbf{s})$ is the diffuse light incident from the bottom. Equation (4.53) simplifies to

$\begin{displaymath} \varphi_d(\mathbf{r})+A_{\mathrm{bottom}}h{\partial\varphi_d... ...L_{di}'(\mathbf{r},\hat\mathbf{s}) \qquad\mathrm{at}\qquad z=d \end{displaymath}$

(4.54)

Equations (4.52) and (4.54) are the boundary conditions for diffuse light incident on a slab with an index of refraction different from its non-scattering environment. If there is no light incident from the top or bottom then set $L_{di}(\mathbf{r},\hat\mathbf{s})$ or $L_{di}'(\mathbf{r},\hat\mathbf{s})$ to zero as appropriate.

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.