# Optical Camera Communication as an Enabling Technology for Microalgae Cultivation +
## 3 Communications Modelling
### Communication Channel
The integration of Equation (9) over the light path, taking into account the wavelength dependency, results in Equation (10). \[\frac{dI_{\lambda}(x)}{d(x)}=-C_{x,\lambda}\cdot I=-\alpha_{x, \lambda}\cdot\rho\cdot I \tag{9}\] \[I(x)=\sum_{\lambda=800}^{\lambda=300}I_{\lambda}(0)\cdot e^{- \alpha_{x,\lambda}\cdot\rho\cdot x} \tag{10}\]  
However, as light travels through the photobioreactor, it is absorbed and scattered by the microalgae. The light intensity \(I_{\lambda}(r,\overrightarrow{d})\) (\(\text{Wm}^{-2}\text{sr}^{-1}\)) at a given location within the container, \(r\), and in the direction \(\overrightarrow{s}\) can be determined by solving the radiative transfer equation (RTE) [23], which represents an energy balance on the radiative energy travellingalong a particular direction. In steady-state conditions, the RTE for non-collimated light, as in this case, can be expressed as Equation (11). \[\overrightarrow{s}\cdot\
abla I_{\lambda}(r,\overrightarrow{d})=-\beta_{ \lambda}I_{\lambda}(r,\overrightarrow{d})+\frac{\sigma_{\lambda}}{4\pi}\,I_{ \lambda}(r,\overrightarrow{d})\cdot\Phi_{\lambda}(\overrightarrow{s_{i}} \rightarrow\overrightarrow{s})d\Omega \tag{11}\]  
where \(\Phi_{\lambda}\) is the scattering phase function (SPF), which represents the angular distribution of the scattered light, in other words, the probability that radiation travelling in a given direction, \(\overrightarrow{s_{i}}\), will be scattered to the direction, \(\overrightarrow{s_{i}}\), of interest. This function is determined by the size, shape and refractive index distribution of the scattering particle. \(\beta_{\lambda}\) is the extinction coefficient (m\({}^{-1}\)) composed by the scattering coefficient, \(\sigma_{\lambda}\), and the absorption coefficient, \(\kappa_{\lambda}\) (m\({}^{-1}\)). These two parameters can be expressed in terms of the averaged scattering \(C_{\text{sca}}\) and absorption \(C_{\text{abs}}\) cross-sections (m\({}^{2}\)), respectively (Equation (12)). \[L(x,y,z=z_{0}\theta,\varphi)=\frac{d^{2}\phi(x,y,z=z_{0},\theta,\varphi)}{d \Omega dAcos\theta} \tag{4}\]0  
where \(N\) is the microorganism concentration expressed in number of cells per m\({}^{3}\) of water suspension. This RTE equation given by Equation (11) reveals that the scattering absorption cross-sections and the SPF have an important role in predicting light transfer in photobioreactors for simulation, design and optimisation purposes. However, these characteristics are interrelated and difficult to estimate from the electromagnetic wave perspective, given the microorganisms' complex morphology. Nonetheless, they can still be measured experimentally with more or less difficulty and related to some parameters of the cell, as detailed in [24]. Furthermore, these radiative properties of microalgae (the absorption cross-section, scattering cross-section and scattering phase function) vary significantly over time, depending on the strain's growth stage, as shown by the studies [25; 26]. As all these radiative parameters depend on the wavelength of light and because they vary in such a significant way through time, it can be considered that the values obtained by an RGB camera would correctly identify not only the strain, but its growth's state. Regarding the communications' performance, although the presence of these microorganisms attenuates the signal before it reaches the exposed container's surface, it may be beneficial in some cases because it distributes the optical power more evenly, uniforming the radiation pattern over the entire container's surface. In conclusion, microalgae particles' scattering phenomena can be exploited for communications in scenarios where the camera is not facing perfectly perpendicular to the transmitter surface.