# Light Transmission in Fog: The Influence of Wavelength on the Extinction Coefficient
## 1 Introduction
In both cases, it was found that the calculated extinction coefficient under the multiple scattering assumption (MODTRAN case) was lower than under the single scattering assumption (Mie theory). This difference was much greater for rain: the attenuation for multiple scattering was then half as great. In dense fog conditions, the extinction difference obtained by calculations based on these two assumptions was much smaller: about 1.5-times higher in the worst cases (\(V<100\) m) and equal in the best cases (\(V>300\) m). This study was done at specific wavelengths (550 nm, 850 nm, and 1550 nm) and not over an extended wavelength range, which justifies our work. Measurements similar to the ones we propose have already been made in the laboratory and compared to existing models [6]. However, these models do not take into account the DSD of the fog produced since they are of the form \(V=f(\lambda)\) where \(V\) is the MOR and \(\lambda\) the wavelength. As already stated, there exist different types of fog with small or large droplets, with the same MOR. It is therefore very difficult to know if the results obtained are reproducible with natural fog. In addition, the wavelength range studied was limited to 600 nm over 1750 nm. These measurements were preceded by other work, specifically at wavelengths of 830 nm and 1550 nm [7]. Those studies concluded by showing the importance of taking DSD into account in order to evaluate the exact extinction coefficient. The most recent research work focused on LiDAR technologies. The objective was to assess whether certain near-infrared wavelengths may be more relevant in fog. In [1], a comparison between two LiDARs of different wavelengths (905 nm and 1550 nm) in fog conditions was done. This previous study concluded that both wavelengths had the same behavior in fog conditions. However, this conclusion was deduced from a ratio calculation, because the two LiDARs did not use the same energy level at all. The one at 905 nm had an energy level 20-times lower than the LiDAR at 1550 nm for eye safety reasons. Our more complete approach will provide a direct measurement of the extinction coefficient under fog conditions for much wider wavelength ranges. To conclude, existing research work compared the extinction coefficient in fog for isolated wavelengths only. To our knowledge, no work has taken into account all wavelengths continuously. In addition, most of the research encountered has not taken into account the DSD, which nevertheless has a very strong impact on the extinction coefficient as described by the Mie theory. Finally, existing work focused either on the modeling or the experimental aspect. Here, we take all these aspects into account, in order to produce a more complete study. First of all, it is necessary to give some definitions about fog itself, at a macroscopic scale, by the reduction of visibility in the atmosphere (MOR), and also at a microscopic scale, by DSD. The interaction of light with fog is discussed in Section 2 considering two models for the wavelength range of 350 nm-2450 nm. The two models used are the Mie model [8] and the MODTRAN software model (which includes Mie diffusion) [9].The method used to study the influence of wavelength on light transmission in fog is given in Section 3. This method is based on a theoretical approach, based on DSD measurement, for calculating the extinction coefficient for different wavelengths, in comparison to measurements with a spectroradiometer in the range of 350 nm-2450 nm obtained in controlled fog conditions. The results are given in Section 4 and discussed in Section 5.