The atmospheric turbulence is a critical issue in optical link experiments.
From Kolmogorov turbulence theory, to Hufnagel-Valley model, to scintillation the- ory, crucial concepts will be focused in the following sections, recalling the main features about turbulence topics, in order to create a knowledge base for a best in- terpretation of the next chapter. [94] [45]
Kolmogorov turbulence model
Light propagation in atmosphere (for visible and IR wavelengths) experiences several phenomena, mainly:
• Absorption and scattering by air molecules
• Clear air turbulence inducing phase fluctuations, focusing and defocusing ef- fects, local deviation in the direction of the electromagnetic propagation, scin- tillation (signal intensity fluctuations).
Owing to temperature and density variations in air atmosphere, the refractive index varies in random fashion in space and time, causing clear air turbulence effects. Considering the atmosphere as a fluid in continuous flow, the ratio between the fluid inertial forces and viscous forses is given by the Reynolds number,
Re = VscaleLscale
ν (1.48)
with Vscale the characteristic velocity scale, Lscale the length scale,ν the kinematics
viscosity. It permits to classify the nature (regime) of the flow: if laminar, the flow is stable, if turbulent, the flow becomes chaotic.
According Kolmogorov model small variations in temperature (< 1C) produce changes in wind velocity (eddies); so it is possible [51] [52] to describe the turbulent regime on the hypothesys that kinematic energy associated with large eddies is re- distributed without loss to eddies of smaller size, until finally dissipated by viscosity. The structure of turbulence according Kolmogorov theory is divided in three ranges:
• the input range characterized by bigger dimension eddies (L0), and it represents
the step when energy is injected into the turbulence and it depends on local conditions.
• the dissipation range which is characterized by smaller dimension eddies (l0
• In the inertial sub range the turbulent energy is transmitted from bigger di- mension eddies L0 to smaller dimension eddies l0.
In the inertial subrange, for a statistically homogeneous medium in turbulent regime, the longitudinal structure function of wind velocity between two observa- tional points r1 and r2 at distance r is:
Dν(r1, r2) = h[ν(R1) − ν(R2)]2i = Cν2R2/3 (1.49)
with l0 R L0 and Cν the structural constant of the wind velocity,
indicating the strength level of the turbulence.
Moreover, the three dimensional power spectrum of wind velocity is given by: Ψν(χ) = 0.033Cν2χ −11/3 (1.50) with 1 L0 χ 1 l0.
In refractive index terms, according the Gladstone relation: n − 1 ≈ 79 · 10−6P
T (1.51)
with T the Kelvin temperature and P the atmospheric pressure, the structure func- tion of the refractive index is given by:
Dn(n(r)) = h[n(R1) − n(R2)]2i = Cn2R2/3 (1.52)
with l0 R L0 and Cν and Cn2 the index of refraction structure parameter
indicating the strength of the turbulence.
Cn2 is related to the temperature structure parameter CT2 by the following rela- tion:
[n − 1
T ]
2C2
T = Cn2 (1.53)
Refractive index structure parameter
The crucial parameter in turbulence theory is the Refractive index Structure Param- eter, which depends on the site orography, location, altitude, time of day, season,... It is not a constant value: close to the ground, the largest temperature gradients associated with largest values of atmospheric pressures (and so air density), usually bring the refractive index structure parameter to high values.
At high altitudes, the decreasing temperature gradients (and so air density) usually bring the refractive index structure parameter to lower values. During the day, the refractive index structure parameter is usually stronger at noon, due to Earth surface
heating, while during the night it is usually expected lower.
Typical value for weak Refractive index Structure Parameter at ground level are expected as little as 10−17 m−3/2, while in strong turbulence regimes, are expected
values of 10−13 m−3/2 or larger.
Considering short horizontal links, the refractive index structure parameter could be considered quite constant, instead for ground-space links the path crosses different layers in totally different conditions of temperature gradient, air pressure and density along the altitude.
A large number of parametric models have been formulated in order to describe the profile of the refractive index structure parameter C2
n(h). One of the most used is
the Hufnagel-Valley profile, given by the following relation: [95] Cn2(h) = 0.00594( ν
27)
2(10−5h)10e−1000h + 2.7 · 10−16e−h/1500+ A
0e−h/100 (1.54)
with h the altitude [m],A0 the turbulence strength at the ground level, ν the wind
correlation factor at high altitude[m/s]. The wind correlation factor is expressed by:
W = [(1/15km) Z
v2(h)dh]1/2 (1.55)
this relation is valid in a range between 5 and 20 kilometers. For the wind velocity can be used the Bufton model
v(z) = 5 + 30exp−[(z − 9400)/4800]2 (1.56)
where z and v are expressed in [m] and [m/s].
Below it is reported a wind velocity profile as a function of the altitude (Figure 1.14 (right) ).
Figure 1.13: The figure shows Bufton velocity profile
The turbulence intensity is represented via the structural constant of the refrac- tive index C2
n. Typically, in day-time, the HV model (HV 5/7) is performed with
A0 = 1.7 · 10−14 m−2/3 and ν = 21m/s, realizing conditions for atmospheric coher-
ence of 5cm and isoplanatic angle of 7µrad at a wavelength of 500nm.
Turbulent effects
Optical quantum links in free space are prone to the many channel impairments, since the information to be transmitted and retrieved is encoded into some quantum state (e.g., polarization) of only a few photons.
The physical effects observed on the optical beam propagating through the atmo- sphere are:
• Beam wander which occurs when a laser beam is refracted by an eddy with size larger than the beam diameter, causing a displacement of the beam center. • Short term beam spread is due to the laser beam being refracted by an eddy smaller than the beam diameter. The short-term beam spread is an additional spread with respect to the standard spread due to the free space laser beam propagation (without turbulence effects).
Figure 1.14: The figures reports a Hufnagel Valley profile: the relation between C2 n
[m−2/3] and the altitude h[km]
• Long term beam spread is the combination of the beam wander and the short- term beam spread leads to the long-term beam spread. The notation “long term” refers to the observation time that is much longer than for the “short term” beam spread. While the short-term beam spread represents the addi- tional broadening at a certain instant, the long-term beam spread represents the area where at least 84 % of the intensity will be, as long as the turbulent environment parameters do not change. For the system loss calculations, the short term beam spread radius is more important than the beam wander, as the displacement of the beam center can be compensated by fast tracking systems but the short term turbulent spreading of the beam cannot.
• Scintillation the wave front is disturbed when it passes through turbulence, which leads to local changes of the electric field phase. When the wave impinges on the receiver, different parts of the wave interfere, resulting in a non-uniform distribution of the intensity that fluctuates about its average value. These spatial and temporal fluctuations in received irradiance are called scintillation.