# Light Transmission in Fog: The Influence of Wavelength on the Extinction Coefficient
## 1 Introduction  
The development of full autonomous vehicles is a challenge, and LiDARs are central to upcoming developments. Adverse weather such as fog affects the range of target detection by the sensor. Both experimental and simulation approaches are needed to determine the performance of the sensors. Measurements need to be made in controlled conditions at various fog densities. A comparative study of various LiDAR technologies with various wavelengths (905 nm and 1550 nm) was undertaken [1]. The results did not show particularly large differences in the target detection distances between wavelengths. In order to better understand these initial results and propose new options, we examined the influence of wavelength on attenuation in fog by means of a theoretical approach (using two models: Mie and MODTRAN) and experimental validation (direct measurement with a spectroradiometer). Our study takes into account:  
* the real droplet size distribution (DSD) of fog;
* two fog types;
* the meteorological optical range (MOR), denoted by \(V\), from 15 m to 175 m;
* and wavelengths from 350 nm to 2450 nm. Scientific studies concerning the extinction coefficient of light in fog have already been carried out, without fully addressing the problem: either because the wavelength ranges are limited, or because the DSD is not taken into account, or because the comparison between models and experimental measurements is not addressed. An early piece of research work concerning the study of the extinction coefficient for different wavelengths was carried out for aeronautics [2]. The objective was to have a measurement of the extinction coefficient for the visible, near-infrared, and far-infrared domains (to detect people in the fog). This initial work [2] took the DSD and Mie theory into consideration. However, since these measurements were made on natural fogs, few data were collected: only 600 measurements over eight wavelengths between 350 and 10,000 nm were obtained, compared to our 4456 measurements over 213 wavelengths between 360 and 2490 nm. In addition, most of the observations made were in hazy conditions (very light fog) and not in heavy fog conditions. The conclusions of this work are mixed: some fogs have constant extinction for wavelengths between 350 and 1700 nm, while others are said to have lower extinction for higher wavelengths. It was therefore observed that, beyond the visible range, fogs with the same MOR can have different extinction coefficients for different wavelengths. In addition, it was also observed that the extinction coefficient depends on the wavelength. The main application of the research work on spectral light transmission through fog was free space optics (FSO). Much work has been done to compare extinction coefficients according to wavelength in the near-infrared domain. A literature review on the extinction coefficient relationship with MOR lists nine models [3]. Some of these models are purely theoretical, while others come from real measurements such as those of our study. However, all the calculations were carried out on the 850-nm wavelength only. Other research work was carried out in the visible and near-infrared domains using different simulation software for the spectral band from 690 nm-1550 nm [4]. The purpose of that work was to determine the monochromatic radiation least attenuated by fog among the wavelengths 690 nm, 750 nm, 850 nm, and 1550 nm. According to that study, the most useful wavelength in fog conditions is 690 nm. However, this study was based solely on modeling, and there was no experimental confirmation of the results presented. In addition, the spectral band examined by the model was much narrower than that processed in our study. The extinction of light radiation in fog and rain was also studied with the objective of seeking a relationship between MOR and signal attenuation by taking into account multiple scattering [5].