# Optical Characterization of Homogeneous and Heterogeneous Intralipid-Based Samples
## 2 Light Propagation in Turbid Media
### Diffusion Approximation
#### 2.2.2 Heterogeneous Slab
In Figure 2b, the scattering contrast (Equation (19)) in transmittance measurement configuration  
Figure 2: (**a**) The absorptive contrast of an inclusion with \(\Delta\mu_{a}/\mu_{a}\) = 0.2 located at different depth inside a slab with a thickness of \(d\) = 40 mm, reduced scattering coefficient \(\mu^{{}^{\prime}}s\)=1.0 mm\({}^{-1}\), absorption coefficient \(\mu_{a}\)=0.01 mm\({}^{-1}\). (**b**) The scattering contrast of an inclusion with \(\Delta\mu_{s}/\mu_{s}\)’ = 0.2 located at different depth inside a slab. The optical properties of the slab are the same as (**a**). as a function of time for a purely diffusive inclusion located at the same depth of the absorptive inclusion is shown. Conversely to the case of an absorber, the diffusive contrast increased as the time decreased toward to the ballistic time. As before, the diffusive contrast increased also as the inclusion moved close to the surfaces of the slab. A good agreement between FEM simulations and approximated solutions was observed, and, furthermore, the different temporal behaviour of the absorptive and diffusive contrast allowed one to discriminate between the optical properties of the inclusion. The accuracy of the first-order approximation method is limited to weak perturbations of a few percent. To overcome this limitation, a closed-form expression for the average diffuse intensity (Equation (12)) in the case of a steady-state diffusion equation for a turbid slab in the presence of an absorbing defect was derived through an iterative approximation scheme [39]:  
\[\Delta\varphi\left(\overset{\rightarrow}{r},\overset{\rightarrow}{r_{0}}; \Delta\mu_{a}\right)=\left(-\Delta\mu_{a}V+\frac{\Delta\mu_{a}{}^{2}V^{2} \varphi\left(\overset{\rightarrow}{r_{c}}\right)}{1+\Delta\mu_{a}V\varphi \left(\overset{\rightarrow}{r_{c}}\right)}\right)\frac{\varphi\left( \overset{\rightarrow}{r_{c}}-\overset{\rightarrow}{r_{0}}\right)\varphi \left(\overset{\rightarrow}{r}\right)}{4\pi} \tag{20}\]  
where \(\Delta\phi\) is the perturbation of the average intensity, and \(\varphi\) is the solution of the steady-state diffusion equation for the homogeneous slab and a point-like source located at \(\overset{\rightarrow}{r_{0}}\):  
\[\varphi\left(\overset{\rightarrow}{r}-\overset{\rightarrow}{r_{0}}\right)= \frac{1}{4\pi D}\sum_{m=-\infty}^{\infty}\frac{exp(-\sqrt{\frac{\mu_{a}}{D}} \rho_{+})}{\rho_{+}}-\frac{exp(-\sqrt{\frac{\mu_{a}}{D}}\rho_{-})}{\rho_{-}} \tag{21}\]  
with the quantities  
\[\rho_{\pm}^{2}=\rho^{2}+\left(z-z_{\pm m}\right) \tag{22}\]  
The square of the radial distance from the z-axis is \(\rho^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\). The quantities \(z_{+m}\) and \(z_{-m}\), given by  
\[I_{pert}(t;z_{incl})=I(t)+\delta I_{\mu_{a}}(t;z_{incl})+\delta I_{\mu_{s}} (t;z_{incl}) \tag{16}\]0  
\[I_{pert}(t;z_{incl})=I(t)+\delta I_{\mu_{a}}(t;z_{incl})+\delta I_{\mu_{s}} (t;z_{incl}) \tag{16}\]1  
define the locations of the positive and negative point-like sources along the z-axis. The quantity \(\ \overline{\varphi}\) is its average value over the volume V of the inclusion (Figure 1):  
\[I_{pert}(t;z_{incl})=I(t)+\delta I_{\mu_{a}}(t;z_{incl})+\delta I_{\mu_{s}} (t;z_{incl}) \tag{16}\]2  
Figure 3 shows the relative contrast in the case of a black defect (\(\Delta\mu_{a}>\infty\)) versus the inclusion radius: the red dotted curve is the model prediction as given by Equations (20) and (21), and the solid black curve is the result of the Monte Carlo (MC) simulation [39]. The results are obtained for an inclusion located midway between the source and the detector at \(z_{c}=\text{d}/2\) in a coaxial configuration. The host medium had a reduced scattering coefficient of \(\mu_{s}{}^{{}^{\prime}}=1.0\) mm\({}^{-1}\), absorption coefficient of \(\mu_{a}=0.01\) mm\({}^{-1}\), and thickness of \(d=40\) mm. As can be noticed, the theoretical predictions are in good agreement with MC simulations for the radius of the black defect ranging from 2 to 5 mm.