Solutions of the Diffusion Equation for the Slab Geometry

The following describes the light propagation through turbid media bounded by parallel planes.

The described derivations follow the methods from Contini 1997.^[1]

Diffusion Equation

\[(\frac{1}{\nu}\frac{\partial }{\partial t} - D\nabla^2 + \mu_a)\Phi(\vec{r}, t) = Q(\vec{r}, t)\]

Where $Q(\vec{r}, t)$ is the isotropic source term and D is the diffusion coefficient, $D = \frac{1}{3\mu_s'}$

Solution of the Diffusion Equation for homogeneous media in a slab geometry

The diffusion equation is a partial-differential equation that will require boundary conditions to solve for a specific geometry. Here we will utilize the extrapolated boundary conditions as described in Contini 1997.^[1] that assumes that the flux is equal to 0 on an extrapolated surface at a distance of ($2AD$). After the boundary conditions have been determined, utilizng the method of images will allow us to reconstruct the fluence inside the medium.

Utilizing Equation 33 from Contini 1997,^[1] the time-dependent Green's function for the fluence rate at ($\vec{r}$) can be described by:

\[\Phi(\vec{r}, t) = \frac{\nu}{(4\pi D \nu t)^\frac{3}{2}} exp(- \frac{\rho ^2}{4 D \nu t} - \mu_a \nu t) \times \sum_{m=-\infty}^{m=+\infty} \{exp(-\frac{(z-z_m^+)^2}{4 D \nu t}) - exp(-\frac{(z-z_m^-)^2}{4 D \nu t})\}\]

Where:

• $z_m^+ = 2m(s+ 2z_e)$
• $z_m^- = 2m(s +2z_e) - 2z_e - z_s$
• $m = 0, \pm 1, \pm 2, ...... \pm \infty$

To obtain solutions for the time-dependent transmittance and reflectance we can utilize Fick's law where:

\[R (\rho, t) = D\frac{\partial}{\partial z} \Phi(\rho, z = 0, t)\]

and

\[T (\rho, t) = - D\frac{\partial}{\partial z} \Phi(\rho, z = s, t)\]

Which yields Equation 36 from Contini 1997^[1] for the time-dependent reflectance on the surface:

\[R(\rho, t) = - \frac{exp(-\mu_a \nu t - \frac{\rho^2}{4 D \nu t})}{2(4\pi D \nu)^\frac{3}{2}t^\frac{5}{2}} \times \sum_{m=-\infty}^{m=+\infty} [z_{3,m}exp(-\frac{z_{3,m}^2}{4 D \nu t}) - z_{4,m}exp(-\frac{z_{4,m}^2}{4 D \nu t})]\]

and Equation 39 from Contini 1997.^[1] for the time-dependent transmittance on at the distance z=s where s is the thickness of the slab:

\[T(\rho, t) = \frac{exp(-\mu_a \nu t - \frac{\rho^2}{4 D \nu t})}{2(4\pi D \nu)^\frac{3}{2}t^\frac{5}{2}} \times \sum_{m=-\infty}^{m=+\infty} [z_{1,m}exp(-\frac{z_{1,m}^2}{4 D \nu t}) - z_{2,m}exp(-\frac{z_{2,m}^2}{4 D \nu t})]\]

Where:

• $z_{1,m} = s(1-2m) - 4mz_e - z_o$
• $z_{2,m} = s(1-2m) - (4m-2)z_e - z_o$
• $z_{3,m} = -2ms - 4mz_e - z_o$
• $z_{4,m} = -2ms - (4m-2)z_e - z_o$