# An Optical Differential Method for Underwater Wireless Communication in Turbid Environments
## 2 Methodology  
The underlying hypothesis of ODCM is that the intensity ratios between different light beams decay much slower than individual intensities themselves. Hereafter, we will reveal the fundamental reasons and quantify the application conditions for this concept. First, we consider that the light source is composed of two monochromatic beams. Each beam decays along the optical path following Beer-Lambert's rule as follows [17]:  
\[I_{i}=I_{0i}e^{-c_{i}d}, \tag{1}\]  
where \(i=1\), \(2\); \(I_{0i}\) is the initial intensity; \(I_{i}\) is the intensity after the light propagates a distance \(d\); and \(c_{i}>0\) is the exponential decay coefficient, also known as extinction coefficient (including the effects of scattering and absorption). Assuming \(c_{1}>c_{2}\), the intensity ratio decays in the same optical path as follows:  
\[R=I_{1}/I_{2}=I_{01}/I_{02}e^{-(c_{1}-c_{2})d}, \tag{2}\]  
Equation (2) indicates that the exponential decay coefficient of intensity ratio is \(c_{R}=c_{1}-c_{2}\). It is clear that \(c_{R}\) is smaller than \(c_{1}\). The hypothesis is valid; that is, \(c_{R}<c_{2}<c_{1}\) if the following inequation is satisfied:  
\[c_{2}<c_{1}<2c_{2} \tag{3}\]  
At first glance, the application condition (3) for ODCM is rather strict. However, for a real turbid underwater environment, for example, in a lake, river, or nearshore water, \(c_{1}-c_{2}\) approaches zero in most cases [18], so that the inequation is generally satisfied. The reason is that in turbid water, the scattering effect of seawater plays a major role in light attenuation [19]. The scattering of water bodies is divided into the scattering of particulate matter and the scattering of pure seawater itself. In 1994, [PERSON] et al. provided the scattering coefficient of pure seawater for light in different wavelength bands. Pure seawater has obvious scattering for short waves, but for visible light from 380 nm to 780 nm, the scattering coefficient is 0.0065-0.0003 m\({}^{-1}\)[20]. Therefore, the scattering effect of turbid water is mainly caused by suspended particulate matter. The size of particles is at a dimension spanning from several micrometers to tens of micrometers [21]. The scattering effect is not so sensitive to incidence light beams because the optical wavelength for underwater communication routinely falls within the visible range, especially the blue-green range, which is far beyond resonant excitation conditions [22]. Since the optical wavelength of the light source falls within the visible range, the intensity ratio between different wavelengths of light is equivalent to another familiar concept: color. Thus, the cardinal principle of ODCM is to measure the chromatic quantities in the turbid water instead of conventional photometric quantities, e.g., light intensity. As an example, a compound light source that includes two white LEDs with different colors is used in ODCM for underwater wireless communications. The emission spectra are shown in Figure 1a,b. The correlated color temperatures (CCTs) for the two kinds of LEDs are 6000 K (Figure 1a) and 3000 K (Figure 1b). The chromaticity coordinates for the 6000 K LED and 3000 K LED are (0.3198, 0.3315) and (0.4105, 0.3880), respectively. The application condition (3) is verified in theory first. The imaginary parts of the dielectric constant for the water, as well as the particles, are neglected; therefore, light attenuation is mainly caused by the scattering of particles. The refraction index of water and particles are 1.34 and 1.5, respectively [23]. The particles are regarded as spheres and homogeneously distributed in the water. The diameter of particles ranges from 2 um to 50 um.