The wavelength dependence of the extinction coefficient in fog and haze is investigated using Mie single scattering theory. It is shown that the effective radius of drop size distribution determines the slope of the log-log dependence of the extinction on wavelengths in the interval between 0.2 and 2 microns. The relation between the atmospheric visibility and the effective radius is derived from the empirical relationship of liquid water content and extinction. Based on these results, the model of the relationship between visibility and the extinction coefficient with different effective radii for fog and for haze conditions is proposed.
© 2011 OSA
Free-space optical (FSO) communication systems are influenced by the propagation impairments due to adverse weather conditions. Particularly, the scattering on the atmospheric aerosols/hydrometeors may cause significant extinction of an optical signal resulting in system outage.
Relation between atmospheric visibility and optical extinction in atmospheric hydrometeors is often applied during link margin analysis to estimate the availability performance of FSO communication systems [1–3]. This relation is dependent on the physical characteristics of hydrometeors as well as on the wavelength. In the sections 1.1 and 1.2 below, several available models of the relationship between extinction and atmospheric visibility are summarized. Some models are based merely on the empirical data [4,5], the other ones were obtained using some theoretical considerations [6,7]. After analyzing these models however, there seems to be no general agreement on the wavelength dependence of optical attenuation due to scattering in hydrometeors such as haze, mist and fog. Furthermore the models do not offer any suitable means to adapt their behavior to local specific climate conditions.
It was pointed out by several authors, e. g., , that it is mainly the different radius of scatterers that is responsible for different wavelength dependence of light extinction in dense fogs in comparison with the extinction in haze conditions. Nevertheless no attempts to date are known to authors that would reveal this difference explicitly in the form of relatively simple model equations. Therefore the wavelength dependence is analyzed using Mie scattering theory and a new model that takes the radius of scatterers into account is proposed for a limited range of wavelengths.
1.1 Visibility and Kruse model
The atmospheric visibility is defined as a distance where the 550 nm collimated light beam is attenuated to a fraction (5% or 2%) of an original power. Several models relate an extinction coefficient γ (1/km) and visibility V (km). The specific optical attenuation is obtained from the extinction coefficient by A (dB/km) = 10 log(e) γ.
The model derived in  has served a long time to the infrared communication community  as the only model providing a wavelength dependent relation between the atmospheric visibility and the extinction coefficient. For visibilities lower than 6 km it reads as follows:4] where the model was introduced that the model is assumed to apply for wavelengths at least in the range from 0.55 to 6 um. However, as pointed out by several authors [6,9], the wavelength dependence of the Kruse model was derived from data gathered during haze conditions and probably not during fog. Therefore, for the lowest visibilities (V < 0.5 km), the formula (1) is less reliable.
1.2 Other models
Kim et al. argued in  that the recent empirical data suggests no wavelength dependence of optical attenuation during low visibility conditions at least for wavelengths in the range from 0.55 to 1.55 µm. The modification of the Kruse model was proposed such that a wavelength dependence is removed altogether (q = 0) at the lowest visibilities (V < 0.5 km).
Al Naboulsi et al. introduced in  a slightly different approach to fog attenuation modeling. They considered two specific types of fog – advection and convection (also called radiation) fog. Then using FASCOD software, they derived interpolation formulas applicable at the wavelengths from 0.69 µm to 1.55 µm for visibilities between 0.05 and 1 km. Al Naboulsi models were compared with experimental data measured on the site “La Turbie” at Nice, France where the transmission of light with the wavelengths 0.85 and 0.95 µm through dense fog was investigated.
Ferdinandov et al. recently introduced in  a model that is claimed to be valid for the visibilities 0.1 < V(km) < 50 and in the range of wavelengths between 0.3 and 1.1 µm. It was derived by fitting the experimental data published in literature. It seems however that the used experimental data was not obtained during dense fog conditions.
Nebuloni analyzed in  extinction data available in three infrared regions to obtain the coefficients of a power law model γ = aVb fitting best the measured data at 1.2, 3.7 and 10.6 microns.
The general trend of the wavelength dependence is different for different models. The Kruse and Ferdinandov models describe decreasing trend for all visibilities, the Al Naboulsi models describe increasing attenuation with wavelength for visibilities up to 1 km and the Kim model gives no wavelength dependence for the lowest visibilities V < 0.5 km and it is similar to the Kruse model for higher visibilities V > 1 km.
2. Scattering theory: extinction vs. effective radius
An extinction coefficient γ of the hydrometeor can be calculated (within the first order multiple scattering approximation) as:11]. A gamma type distribution is widely used for representing the microphysical properties of clouds and fogs :11] of the distribution that well characterizes the whole distribution from the point of view of an integrated geometrical cross section of all particles. The effective radius is proportional to a ratio of the total volume and the total geometrical cross section of particles in the distribution:
If re is fixed, the shape parameter α is inversely related to the dispersion of the distribution. Numerical experiments reveal that the extinction coefficient is not very sensitive to the shape parameter α when re is fixed. As an additional descriptive parameter of the distribution, a liquid water content LWC is chosen because it is proportional to the total volume of particles. For fixed LWC, the total geometrical cross section obtained from (5) is inversely proportional to the effective radius. In the small wavelength limit where λ → 0, the extinction efficiency factor Qext → 2 and so the extinction cross section Cext = πr 2 Qext is simply twice the geometrical cross section. Then the extinction coefficient is twice the total geometrical cross section:
The integral in (3) has been evaluated numerically (by an adaptive Simpson rule with relative accuracy better than 10−3) with the extinction efficiency factor Qext calculated using full Mie formulae . The complex refractive index of water was taken from . Figure 1 shows the wavelength dependence of the extinction coefficient for fixed LWC, α and different values of re. In the short wavelength limit, one can check in Fig. 1 that the extinction coefficient is given by γ(km−1) = 1500/re(µm) corresponding to (6) with LWC = ρ = 1. In the region where applicable (see Fig. 1), a simple relation between the extinction coefficient, the liquid water content and the effective radius is so obtained:Fig. 1 that there is no generally monotonic wavelength dependence except in some interval around 0.55 μm. It seems therefore that the exponent q in (1) may serve as some approximation only in the interval 0.2 μm < λ < 2 μm that is focused on in the following.
The wavelength dependence of the relative extinction coefficient in the mentioned interval is depicted in Fig. 2 . The whole interval was divided into two subintervals: λ < 0.55 μm and λ ≥ 0.55 μm and a linear approximation of the log-log wavelength dependence was found in both of them, see the straight lines in Fig. 2.
The slope of the log-log dependence s (the exponent in (10) below) as a function of the effective radius was obtained. The s(re) function can be approximated by the following expression:Table 1 .
Figure 3 shows the exponent s obtained for different re (points) and the interpolating functions (8) (lines) with the parameter values stated in Table 1. The function (8) was chosen so that the limit of s as re → 0 is s = −4 which corresponds to the Rayleigh scattering. The limit of s as re becomes large is s = 0 which corresponds to the geometrical optics approximation. Similarly as in (1), the wavelength dependence of the extinction coefficient can be expressed as:
3. Relation of the effective radius and atmospheric visibility
The model expressed by Eqs. (8)-(10) is based on the underlying physics as it takes the dependence on the effective radius of the particular hydrometeor into account. Both the visibility and the effective radius are the inputs of the model. Nevertheless it may be difficult in practice to get relevant information about the size of scattering particles. It is therefore desirable to estimate the relationship between the effective radius and visibility in order to express the extinction coefficient as a function of visibility only.
For that purpose, let us consider the following facts:
- a) The relation between the extinction coefficient γ and the liquid water content LWC is often (e.g .) empirically modeled by a power-law equation:
- b) The extinction coefficient γ is linearly proportional to LWC when both the effective radius re and the shape parameter α are fixed. It means c = 1 in such a case.
- c) For a fixed LWC (and fixed α) and λ = 0.55 μm, the effective radius re is inversely proportional to the extinction coefficient provided re > re min where re min is about 0.5 μm.
- d) It is from the definition of visibility that γ/γ 0 = V 0/V for the wavelength λ = 0.55 μm where V 0 stands for the visibility corresponding to γ 0 and LWC 0.
So using a), b) and c), one finds the following equation (see parameters in Fig. 4):14,15]), re 0 = 10 μm (), V 0 = 0.05 km. The Eq. (13) then reads:
4. Model summary
The new proposed model is applied as follows. First, the effective radius re is estimated from visibility using (13). Second, the exponent of wavelength dependence s is calculated using (8) and (9). Finally the extinction coefficient γ is obtained using (10).
5. Comparison with measured data
Figure 5 shows the example of a fog event observed in Prague, the Czech Republic. The two parameters of fog are measured : liquid water content LWC (g/m3) and particulate surface area PSA (cm2/m3). The effective radius, re (μm) is then calculated using its definition (5) as: re = 30000∙LWC/PSA. It can be seen in Fig. 5 that the re fluctuates around about 10 microns during the developed fog phase. It is expected, however, that the typical effective radius depends on local climatic conditions  and so the Eq. (14) is to be modified accordingly.
Figure 6 shows a comparison of different models applied at two typical FSO wavelengths. The measured (two-year) data  obtained on two FSO paths located in Prague are also shown in Fig. 6 as an empirical reference. A reasonable agreement of the proposed model with the measurement is achieved at least for visibilities lower than 2 km.
The wavelength dependent model of the relation between atmospheric visibility and the extinction coefficient of fog and haze was proposed based on the results of applied Mie scattering theory. The model is applicable in the interval of wavelengths 0.2 μm < λ < 2 μm and for visibilities lower than 10 km. The presented modeling approach reveals more explicitly the connection between the extinction coefficient and the microphysical parameters of fog and haze. It also provides some degree of flexibility to adapt its properties according to locally specific atmospheric conditions.
This research is supported by the Czech Science Foundation under the project No. P102/11/1376.
References and links
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