# Optical Camera Communication as an Enabling Technology for Microalgae Cultivation +
## 3 Communications Modelling
### Communication Channel  
The channel can be divided into different layers. Figure 2 details the layered version of the channel, which is composed of: the inner air gap layer, the inner glass layer, the microalgae suspension in the water layer and the outer glass layer that emits the light that reaches the camera. This work focuses on the analysis of these four primary layers of the channel. The effect of the link's air gap between the container and the camera was addressed previously in [22]. This preliminary analysis shows two critical interfaces: the inner glass/water and the outer glass/air interfaces that affect communications' performance and constrain harvesting optimisation. Light rays emitted in a specific direction by the light source (\(\vartheta_{0},\varphi_{0}\)) change their trajectory upon reaching the surface of the inner glass (\(\vartheta_{\text{ig}},\varphi_{\text{ig}}\)). Then, those rays reach the first critical interface (inner glass/water interface) and undergo partial or total internal reflection depending on the critical angle (\(\Theta_{\text{glass/water}}=51.9^{\circ}\)) ( considering refraction indices of \(n_{air}=1.0\), \(n_{glass}=1.69\) and \(n_{water}=1.33\) for the air, glass and water media, respectively). However, total internal reflection will never happen at this critical interface  
Figure 2: Layered version of the channel. in the considered conditions. The reason is that as soon as \(\theta_{0}\) approaches 90 degrees, \(\theta_{\text{ig}}\) tends to 31\({}^{\circ}\), which is considerably lower than \(\Theta_{\text{glass/water}}\). In the microalgae medium, light rays travel with direction (\(\theta_{\text{water}}\), \(\varphi_{\text{water}}\)). Those rays that reach the top, bottom, left or right side reflectors of the container are reflected. Figure 2 shows an example of one ray reaching the container's right side and being reflected. Finally, within the outer glass, light propagates with (\(\theta_{\text{og}}\), \(\varphi_{\text{og}}\)) until it arrives at the second critical interface. The outer glass-air interface has its corresponding critical angle (\(\Theta_{\text{glass/air}}=36.28^{\circ}\)). Light rays with incident angles greater than \(\Theta_{\text{glass/air}}\) are reflected, reducing the total optical power leaving the photobioreactor. This preliminary analysis aims to better understand the behaviour of the radiance \(L(\overrightarrow{r},\overrightarrow{\theta})\) of the extended outer glass surface (Equation (4)); in other words, the radiant intensity \(\phi\), emitted from an infinitesimal unit surface, \(d_{A_{\perp}}\), and contained within a unit of solid angle aligned normal to the direction of interest, \(\overrightarrow{\theta}=(\theta,\varphi)\), for a particular location, \(\overrightarrow{r}=(x,y)\). \[L(x,y,z=z_{0}\theta,\varphi)=\frac{d^{2}\phi(x,y,z=z_{0},\theta,\varphi)}{d \Omega dAcos\theta} \tag{4}\]  
After characterising the radiance of the surface, the total irradiance over a pixel is obtained by integrating the incoming radiance from any direction in the normal hemisphere that encloses the pixel area Equation (5). As an example, Figure 3 shows the irradiance per unit of area of an infinitesimal portion, \(d_{\text{V}}\), of a virtual surface that encloses the container keeping the same horizontal distance to the container's centre point. \[E(x,y,z)=\int_{\Omega}L(x,y,z=z_{0},\theta,\varphi)cos\phi d\Omega \tag{5}\]  
In the stated conditions, the radiance would be affected by different phenomena: the culture's absorption capacity, the shading effect between cells, and light scattering. When the amount of biomass is negligible compared to the volume of water in the container, these phenomena might be neglected, and radiance can be estimated using geometrical optics.