# Optical Camera Communication as an Enabling Technology for Microalgae Cultivation +
## 3 Communications Modelling
### Communication Channel
Now, considering that the channel consists of symmetrically repeating parallel layers that start and end in the same air medium, light's incident angle \(\theta_{0}\), \(\varphi_{0}\) coincides with exiting radiance angle \(\theta_{\text{out}}\), \(\varphi_{\text{out}}\) without any hard restriction as there is no total internal reflection. Figure 2 shows light example paths in 2D dimensions. Now, considering that the bottom, top, left and right container's sides reflect light in a specular manner, the radiance of a point (\(x^{\prime},y^{\prime},z=0\)) of the external surface in any direction \(\overrightarrow{\theta^{\prime}}\) is a fraction of the radiance emitted by the light source in the same direction  
Figure 3: Virtual surface that encloses the container. (or in a shifted direction, if the ray comes from a reflection in the walls) (\(\overrightarrow{\vartheta^{\prime}}\)), but from a translated origin point, \(x\),\(y\), of the output point (\(x=x^{\prime}+a\),\(y=y^{\prime}+b\)). As a result, the output radiance from infinitesimal surfaces located at the container's centre tends to mimic the original lamp radiation pattern. However, the output radiance at the borders of the container tends to skew. This skewness occurs because there are no direct light contributions. This skewness effect of the radiance pattern significantly constrains the communications' performance, as the optical signal power recovered from specific viewpoints would not be enough to establish effective communication. As it can be extracted from preliminary experiments, this skewness can be partially reduced if the container's sides reflection has a perfect diffuse behaviour, causing the light to be uniformly spread in all directions. However, the skewness cannot be fully mitigated since the light rays that contribute to reducing this effect come from very steep entry angles and repeatedly bounce within the container boundaries, ultimately reducing its optical power. Regarding light power exiting the surface, it is important to consider the fraction of the incident light reflected at each interface. Considering non-polarised light the effective reflection coefficient, for each input angle, \(\vartheta_{i}\), can be expressed by Equation (6). \[R_{\text{eff}}=\frac{(R_{s}+R_{p})}{2} \tag{6}\]  
where \(R_{s}\) (Equation (7)) and \(R_{p}\) (Equation (8)) are the reflectances for s-polarized and p-polarized light, respectively. \[R_{s}=\left|\frac{n_{1}\cos\vartheta_{i}-n_{2}\sqrt{1-(n_{1}/n_{2}\cdot\sin \vartheta_{i})^{2}}}{n_{1}\cos\vartheta_{i}+n_{2}\sqrt{1-(n_{1}/n_{2}\cdot \sin\vartheta_{i})^{2}}}\right|^{2} \tag{7}\]  
\[R_{s}=\left|\frac{n_{1}\sqrt{1-(n_{1}/n_{2}\cdot\sin\vartheta_{i})^{2}}-n_{2} \cos\vartheta_{i}}{n_{1}\sqrt{1-(n_{1}/n_{2}\cdot\sin\vartheta_{i})^{2}}+n_{ 2}\cos\vartheta_{i}}\right|^{2} \tag{8}\]  
where \(n_{1}\) and \(n_{2}\) are the refractive indexes of both mediums. Hence, light rays with acute entry angles have a significant amount of power reflected. This has some implications. At first, not all the optical power reaches the microalgae culture, and it will depend on the radiance pattern of the surface and the effective emission angle of the source. Finally, not all the optical power leaves the container's surface, which is not desirable for establishing an optical communication link despite being suitable for cultivation. Under this initial configuration, when the biomass concentration is above a threshold level, the scattering and the absorption phenomena cannot be neglected. In that case, [PERSON]'s law can be used to describe the attenuation of light due to absorption by the biomass concentration (Equation (9)). This equation states that the attenuation of light over a distance is proportional to the light intensity, where \(C\) is the volumetric absorption coefficient. The latter is the product of the specific light absorption coefficient, \(\alpha_{x,\lambda}\), and the biomass concentration (\(\rho\)).