# Light Transmission in Fog: The Influence of Wavelength on the Extinction Coefficient
## Theory and Definitions: Light and Fog
### Light Attenuation Due to Fog  
In foggy conditions, a luminous flux of wavelength \(\lambda\) is attenuated according to the Bouguer-Lambert law [10]:  
\[\Phi=\Phi_{0}.e^{-k(\lambda).d} \tag{2}\]  
where \(k(\lambda)\) is the extinction coefficient and \(d\) is the distance between the source (emitting the luminous flux \(\Phi_{0}\)) and the observer (receiving the luminous flux \(\Phi\)). This attenuation is due to the interaction of light radiation with microscopic water droplets [14; 15]. It is divided into two complementary phenomena: absorption and scattering. The absorption phenomenon is caused by the interaction between an electromagnetic wave and the electronic cloud of an atom. This phenomenon has the consequence of absorbing part of the incident energy. This part of the energy is conserved by the atom in interaction with the radiation. The scattering phenomenon is caused by the change in the trajectory of the luminous flux caused by fog microscopic water droplets. Scattering can be modeled using Mie theory [8]. This theory is a particular solution of Maxwell equations describing the elastic scattering of a plane electromagnetic wave by a spherical particle.This theory applies with the following assumptions:  
* a monochromatic incident light;
* a spherical, homogeneous, isotropic particle of radius \(r\), of refractive index \(n_{1}\);
* non-absorbing dispersion host environment of refractive index \(n_{2}\);
* a low concentration (single scattering). This assumption will be discussed further, compared with the MODTRAN model, which includes multiple scattering. The particle studied is a sphere of radius \(r\) and refractive index \(n_{1}\) (see Figure 2). This particle is placed in a host environment of refractive index \(n_{2}\) in which a monochromatic plane wave with wavelength \(\lambda\) propagates with a wave vector \(\overline{\ell}\) such that \(\|\overline{\ell}\|=2\pi/\lambda\). This incident wave meets the spherical particle, and the radiation is scattered in all directions as a result of this interaction. The polarization of the wave is identified by the scattering plane. The latter is defined by the direction of the wave vector \(\overline{\ell}\), as well as the observation direction \(\vec{OP}\). Figure 1: Droplet size distribution models for radiation and advection fogs with \(V=50\) m [11]. The size parameter \(x\) determines the diffusion and is defined by [8]:  
\[x=\|\overline{\ell}\|\;r=\frac{2\pi r}{\lambda} \tag{3}\]  
The angular distribution of the scattered radiation of spherical particles depends on their size. In the case of particles with a radius of \(r<<\lambda\), Mie theory and Rayleigh scattering give the same results. Rayleigh scattering is valid only for particles with a radius \(r\) less than \(\lambda/10\). Mie theory is valid for all the values of radius \(r\) provided that the particle is spherical. In the following, Mie theory will be applied to fog droplets in the Mie + PSA model, whereas the MODTRAN + PSA model will include Mie theory applied to fog droplets and Rayleigh theory to molecules. The amount of energy scattered forward is greater than in any other direction according to Mie theory, as shown in Figure 2.  
The extinction phenomenon is the combination of absorption and scattering. In order to study the extinction of radiation, it is necessary to calculate the extinction efficiency factor \(Q_{ext}\).