Light Transport in Tissue

The Green's Function for an Infinite Slab

The Green's function in the cylindrical coordinate system with r and r' expressed in $(\rho,\varphi,\zeta)$ coordinates [50] is

$\begin{displaymath} G(\mathbf{r};\mathbf{r}')=\sum_{n=1}^\infty{Z_n(\zeta)Z_n(\z... ...r K_0(\lambda_n\rho')I_0(\lambda_n\rho )& if $\rho\le\rho'$.} \end{displaymath}$

(4.124)

where K₀ and I₀ are modified Bessel functions and $Z_n(\zeta)$ is an eigenfunction satisfying the differential Equation (4.112)

$\begin{displaymath} Z_n(\zeta)=\sin(k_n\zeta+\gamma_n) \end{displaymath}$

(4.125)

The eigenvalue $\gamma_n$ is obtained by substituting the Green's function into the boundary condition at $\zeta = 0$

$\begin{displaymath} \tan\gamma_n=A_{\mathrm{top}}h'k_n \end{displaymath}$

(4.126)

The eigenvalue k_n is obtained imposing the boundary condition (4.114) at $\zeta=\tau'$

$\begin{displaymath} \tan(k_n\tau'+\gamma_n)-A_{\mathrm{bottom}}h'k_n \end{displaymath}$

(4.127)

Using (4.126) and the sum of angles expansion for the tangent simplifies Equation (4.127)

$\begin{displaymath} \tan k_n\tau'={(A_{\mathrm{top}}+A_{\mathrm{bottom}})h'k_n\over A_{\mathrm{top}}A_{\mathrm{bottom}}(h'^2k_n)^2-1} \end{displaymath}$

(4.128)

Evaluation of the roots k_n of this equation are discussed in Appendix D. The normalization factor N_n² is given by

$\begin{displaymath} N_n^2=\int_0^{\tau'} [z_n(\zeta)]^2\,d\zeta \end{displaymath}$

(4.129)

Substituting (4.125) and simplifying,

$\begin{displaymath} N_n^2={\sin2\gamma_n-\sin2(k_n\tau'+\gamma_n)+2k_n\tau'\over4k_n} \end{displaymath}$

(4.130)

Finally, substituting the Green's function (4.124) into the diffusion Equation (4.112) results in a relation between $\lambda_n$ and $\kappa_n$

$\begin{displaymath} \lambda_n^2=k_n^2+\kappa_d^2 \end{displaymath}$

(4.131)

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