# Optical Characterization of Homogeneous and Heterogeneous Intralipid-Based Samples
## 2 Light Propagation in Turbid Media
### Static and Dynamic Light Scattering  
In the basic scattering geometry adopted for the SLS investigation, the incident laser radiation has a wave vector \(\overrightarrow{k_{i}}\), pointing in the direction of propagation of the incoming radiation and having an amplitude \(\left|\overrightarrow{k_{i}}\right|=2\pi n/\lambda\), where n is the refractive index of the medium and \(\lambda\) is the wavelength of the light in the vacuum. The scattered radiation has a wave vector \(\overrightarrow{k_{s}}\) that has an amplitude \(k_{s}=k_{i}=k\), and points in the direction of the outgoing radiation. The intensity of the scattered light is measured by a detector far from the scattering event by assuming the measurement is performed in the single-scattering limit, i.e., light is scattered no more than one time before exiting the sample. The light scattered from two particles a and b travels towards the detector with a different path length  
\[\Delta\overrightarrow{s}=\frac{\overrightarrow{k_{s}}}{k_{s}}\cdot\Delta \overrightarrow{r}-\frac{\overrightarrow{k_{i}}}{k_{i}}\cdot\Delta \overrightarrow{r} \tag{2}\]  
where \(\Delta\overrightarrow{r}=\overrightarrow{r_{a}}-\overrightarrow{r_{b}}\) is the relative distance between the two particles. When the size of the scattering volume is much smaller than the distance from the scattering volume to the detector, the phase difference is given by the following equation  
\[\Delta\varphi=k\Delta s=\left(\overrightarrow{k_{s}}-\overrightarrow{k_{i}} \right)\cdot\Delta\overrightarrow{r}=\overrightarrow{q}\cdot\Delta \overrightarrow{r} \tag{3}\]  
where  
\[\overrightarrow{q}=\overrightarrow{k_{s}}-\overrightarrow{k_{i}} \tag{4}\]  
\[q=2k\sin\frac{\theta}{2} \tag{5}\]  
The waves scattered from the particles a and b interfere constructively or destructively according to their relative distance \(\Delta\overrightarrow{r}\) and to the scattering angle \(\theta\), that is the angle between \(\overrightarrow{k_{i}}\) and \(\overrightarrow{k_{s}}\). The quantity measured in a scattering experiment is proportional to the scattered intensity I(\(\theta\)) from all the \(N\) particles in the scattering volume \(V_{s}(q)\) averaged over some periods of time. If the particles are identical spheres, the scattered intensity can be written as:  
\[I(q)\propto\left|\overrightarrow{E}(q)\right|^{2}S(q) \tag{6}\]  
with  
\[S(q)=\frac{1}{N}\sum_{i,j=1}^{N}<e^{i\overrightarrow{q}\cdot(\overrightarrow {r_{i}}-\overrightarrow{r_{j}})}> \tag{7}\]  
where \(\overrightarrow{E}(q)\) is the average scattered electric field that depends on the scattering properties of a single particle, \(S(q)\) is the structure factor that depends only on the relative particle positions \((\overrightarrow{r_{i}}-\overrightarrow{r_{j}})\), and \(<\!\!>\) denotes the ensemble average. If the quantity \(\left|\overrightarrow{E}(q)\right|\) is known a priori from the experiment or theory, the structure factor \(S(q)\) can be retrieved from SLS measurements to investigate the fluctuations in the particle density and the correlation between particles that depends on the relative distance [35]. Equation (6) is too simple for describing the scattering process from nonspherical particles whose size is not negligible. In this case, a more structured expression is considered for describing the angular scattered intensity:  
\[\frac{KC}{R(q)}=\frac{1}{M\,F(q)}+2A_{2}C \tag{8}\]where  
\[R(q)=\frac{I(q)}{I_{0}}\frac{r^{2}}{V_{s}(q)} \tag{9}\]  
The quantity \(R(q)\), called the Rayleigh ratio, is the normalized intensity, and I\({}_{0}\) is the intensity of the light beam that illuminates the sample. The quantity \(C\) is the weight concentration of the particles with molecular weight \(M\). The quantity \(F(q)\) is the form factor of the particles, and it is related to the size and shape of particles themselves. A\({}_{2}\) is the second virial coefficient and measures the nonideality of the solution where the particles are dissolved or the interaction forces between the particles.