# Light Transmission in Fog: The Influence of Wavelength on the Extinction Coefficient
## Theory and Definitions: Light and Fog
### Light Attenuation Due to Fog
The calculation of \(Q_{ext}\) requires the determination of the complex Lorenz-Mie coefficients \(a_{n}\) and \(b_{n}\)[16]:  
\[a_{n}=\frac{\gamma^{2}j_{n}(\gamma x)\left[xj_{n}(x)\right]^{\prime}-j_{n}(x) \left[\gamma xj_{n}(\gamma x)\right]^{\prime}}{\gamma^{2}j_{n}(\gamma x)\left[ xh_{n}^{(1)}(x)\right]^{\prime}-h_{n}^{(1)}(x)\left[\gamma xj_{n}(\gamma x) \right]^{\prime}} \tag{4}\]  
\[b_{n}=\frac{j_{n}(\gamma x)\left[xj_{n}(x)\right]^{\prime}-j_{n}(x)\left[ \gamma xj_{n}(\gamma x)\right]^{\prime}}{j_{n}(\gamma x)\left[xh_{n}^{(1)}(x )\right]^{\prime}-h_{n}^{(1)}(x)\left[\gamma xj_{n}(\gamma x)\right]^{\prime}} \tag{5}\]  
where \(x\) is the size parameter, \(\gamma=n_{1}/n_{2}\) is the index ratio, \(h_{n}(z)=j_{n}(z)+iy_{n}(z)\), and \(j_{n}(z)\) (resp. \(y_{n}(z)\)) is the Bessel function of the first kind (resp. second kind) of order \(n\) and argument \(z\). The coefficients \(a_{n}\) and \(b_{n}\) are then used to calculate the extinction efficiency factor \(Q_{ext}\):  
\[Q_{ext}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}(2n+1)Re(a_{n}+b_{n}) \tag{6}\]  
where \(Re(z)\) denotes the real part of the complex number \(z\). In practice, this infinite series can be truncated at a rank \(n_{max}\):  
\[Q_{ext}=\frac{2}{x^{2}}\sum_{n=1}^{n_{max}}(2n+1)Re(a_{n}+b_{n}) \tag{7}\]  
The proposed value for \(n_{max}\) is [16]:  
\[n_{max}=x+4x^{1/3}+2 \tag{8}\]  
The extinction efficiency factor \(Q_{ext}(\lambda)\) characterizes the extinction of a single particle in all directions. In order to take into account the extinction in fog resulting from the action of several particles, the extinction coefficient \(k(\lambda)\) is defined from the droplet size distribution of the fog. Mie theory makes it possible to determine the expression of this coefficient. Under the hypotheses of an  
Figure 2: Geometric representation of the diffusion from a spherical particle. In yellow, a schematic representation of the Mie diffusion by a spherical particle. unpolarized incident wave, the absence of multiple scattering of the wave on the scatterers present in the environment and independence between the scatterers (assumed to be quite distant from each other), the extinction coefficient \(k(\lambda)\) is expressed in m\({}^{-1}\) as:  
Figure 1:0  
where \(\lambda\) is the wavelength in the host environment, expressed in meters, \(x=2\pi r/\lambda\) is the size parameter, \(r\) is the radius of the droplet, expressed in meters, \(n(r)\) is the fog DSD, expressed in particles/cm\({}^{3}\), and \(Q_{ext}(x,\gamma)\) is the extinction efficiency factor expressed in m\({}^{-1}\) for the size parameter \(x\) and the refraction index ratio \(\gamma=n_{1}/n_{2}\). The extinction coefficient \(k(\lambda)\) therefore depends on the wavelength but also on the DSD. To study the influence of fog particle size and wavelength on the extinction phenomenon, \(Q_{ext}=f(\lambda)\) and \(Q_{ext}=f(d)\) with \(d\) the droplet diameter are plotted. The \(Q_{ext}=f(\lambda)\) plot (Figure 3a) shows a maximum extinction peak for each diameter \(d\). The position of this peak depends on the diameter. For particles with larger diameters, this extinction peak moves towards the infrared. Extinction is very important in the visible domain for water droplets with a diameter between 1 \(\upmu\)m and 2 \(\upmu\)m. For droplets with a diameter greater than 2 \(\upmu\)m, extinction is maximum in the infrared. This will help us to discuss the experimental results. The plot \(Q_{ext}=f(d)\) in Figure 3b also highlights the extinction peak and shows that curves tend to the value of \(Q_{ext}=2\) m\({}^{-1}\). Equations (3)-(9) were implemented in C++ to determine the extinction coefficient for each wavelength from the DSD measurements. We then refer to the Mie model in the following. As the Mie model remains partial (without molecular impact and in the context of single scattering), MODTRAN software [9] was also used to avoid these two latter assumptions.