Atmospheric scattering and turbulence modeling for ultraviolet wavelength applications

Abstract

The recent proliferation of free-space optics (FSO) technologies and development of pertaining research have led to exploit the whole optical bandwidth from infrared and visible up to deep ultraviolet (UV) in communications. Within this context, we decided to focus on UV FSO communication and remote sensing potentials by presenting a physically based single-scattering channel model, UVatmoScat, aiming to include, with respect to previous analyses, most atmospheric variables like fog, precipitations, aerosols, and turbulence: indeed, they may affect the performance of UV links in short-range outdoor applications adopting non-line-of-sight (NLOS) configuration, as here considered. This analytical single-scattering model computes the temporal impulse response and path loss and has been validated through Monte Carlo. The former provides the frequency characteristic of UV-NLOS propagation links on varying of atmospheric conditions, with different NLOS geometries and receiver apertures. 3-dB bandwidth numerical results show UV-NLOS systems as more significantly affected by the link geometric features than weather perturbations, demonstrating supportive to the choice of frequency constant-envelope modulations. Nevertheless, meteorological perturbations have to be properly considered to better optimize the transmission power in UV telecommunications, as well as in ozone or aerosol ground-sensing applications. The proposed UVAtmoScat model is a suitable, self-consistent, and effective tool for this purpose.

Appendix. Microphysically oriented atmospheric particle scattering model parametrization

By standing at scattering coefficient parametrization of (Mori & Marzano, 2015) and reference therein, the UVAtmoScat hydrometeor size distribution follows a law called as gamma particle size distribution (PSD), that is a general model for water particles:

$$ {N}_p\left({r}_v\right)={N}_e{\left(\frac{r_v}{r_e}\right)}^{\mu_e}{e}^{-{\varLambda}_e\left(\frac{r_v}{r_e}\right)} $$

(44)

where r_v [mm] is the volume-equivalent spherical radius, r_e [mm] is the effective radius, and N_e [mm⁻¹ m⁻³] is the effective particle number concentration defined as

$$ {N}_e={10}^3{W}_p\frac{3{\varLambda}_e^{4+{\mu}_e}}{4\pi {\rho}_p}\frac{1}{r_e^4}\frac{1}{\varGamma \left({\mu}_e+4\right)} $$

(45)

where W_p [g m⁻³] is the particle mass concentration and ρ_p[g cm⁻³] its specific density, while Λ_e—adimensional—is the so-called slope parameter, which is related to the “shape” parameter μ_e as follows:

$$ {\varLambda}_e=\frac{\varGamma \left({\mu}_e+4\right)}{\varGamma \left({\mu}_e+3\right)} $$

(46)

By defining the particle mass [kg] as

$$ {m}_p\left({r}_v\right)=\frac{4}{3}\pi {\rho}_p{r}_v^3 $$

(47)

and the N_p moment of order n as

$$ {m}_n={\int}_{r_m}^{r_M}{r}_v^n{N}_p\left({r}_v\right)d{r}_v $$

(48)

where r_m [mm] and r_M [mm] are the minimum and maximum particle radius, respectively; we obtain the particle mass concentration (or water content)

$$ {w}_p=\frac{4}{3}\pi {\rho}_p{m}_3 $$

(49)

and the effective radius

$$ {r}_e=\frac{m_3}{m_2} $$

(50)

The corresponding particle fall rate R_p [kg h⁻¹ m⁻²], defined as the particle mass crossing a horizontal cross-section of unit area over a given time interval is

$$ {R}_p={\int}_{r_m}^{r_M}{a}_v{r}_v^{b_v}{m}_p\left({r}_v\right){N}_p\left({r}_v\right)d{r}_v=\frac{4}{3}\pi {a}_v{\rho}_p{m}_3+{b}_v $$

(51)

where a_v and b_v are empirical coefficients taking into account the height-dependent air density. The fall rate can be also expressed by an equivalent height per unit time, as in the case of rain precipitations: R = R_p/ρ_p.

N_e and Λ_e, that are respectively computed by Eqs. (45) and (46), can be derived by analyzing a set of 1000 simulated PSDs for each particle class (e.g., fog, rain droplets), produced by the uniform random generation of μ_e ∈ [μ_{e, m}, μ_{e, M}], r_e ∈ [r_{e, m}, r_{e, M}], and W_p ∈ [W_{p, m}, W_{p, M}], where all the respective minimum and maximum values depend on particle class and are provided by (Mori & Marzano, 2015).

The hydrometeor extinction coefficient in the UVAtmoScat channel, once set the transmission wavelength, is given by:

$$ {k}_e^{(H)}={k}_a^{(H)}+{k}_s^{(H)}={\int}_{r_m}^{r_M}{\sigma}_e\left({r}_v\right){N}_p\left({r}_v\right)d{r}_v $$

(52)

where σ_e(r_v) = σ_a(r_v) + σ_s(r_v) is the extinction cross-section [m²], composed by the sum of related absorption and scattering cross-sections, that follow the Mie modeling laws for small particles exposed in (Bohren & Huffman, 1983). The related hydrometeor asymmetry factor g′ is provided by the following expression:

$$ g^{\prime}\left({\varOmega}_{\mathrm{T}}\right)=g^{\prime }={\int}_{4\pi}\left({\varOmega}_{\mathrm{T}}\cdot {\varOmega}_{\mathrm{R}}\right)p\left({\varOmega}_{\mathrm{T}},{\varOmega}_{\mathrm{R}}\right)d{\varOmega}_R $$

(53)

where p is the phase function and Ω_T and Ω_R are the incident and scattering direction unit vectors.

In case of fog particles, we define the visibility V as the range to which the corresponding transmittance t, i.e., the ratio between the received intensity and the incident one, is computed according to Beer-Lambert law at a given threshold value and becomes t₀:

$$ V=-\frac{\ln \left({t}_0\right)}{k_e^{(H)}} $$

(54)

where t₀ is here considered equal to 0.05: indeed, visibility is defined as the distance to an object where the image contrast drops to 5% threshold value with respect to the proximity to the same; in other cases, a 2% threshold value is considered (Mori & Marzano, 2015).

According to power-law regression method provided by MAPS model and considering the coefficients shown in Tables 1 and 2, we can express the main optical parameters as follows:

$$ {\displaystyle \begin{array}{c}{k}_e^{(H)}={C}_{1,e}{X}_p^{C_{2,e}}\\ {}{k}_s^{(H)}={C}_{1,s}{X}_p^{C_{2,s}}\\ {}g\hbox{'}{k}_s^{(H)}={C}_{1,g}{X}_p^{C_{2,g}}\end{array}} $$

(55)

Table 1 Regression coefficients of case λ = 260 nm to estimate optical parameters ${k}_e^{(H)},{k}_s^{(H)}\;\mathrm{and}\;\mathrm{g}^{\prime}\left[{\mathrm{km}}^{-1}\right]$ from visibility range V [km]

Table 2 Regression coefficients of case λ = 260 nm to estimate optical parameters ${k}_e^{(H)},{k}_s^{(H)}\;\mathrm{and}\;\mathrm{g}^{\prime}\left[{\mathrm{km}}^{-1}\right]$ from prec. rate R [mm/h]

where X_p depends on the kind of information acquired about hydrometeors:

in our case, X_p = V (visibility) for advection and radiation fog (Table 1) and = R (fall rate) for light and heavy rain precipitations (Table 2).