*3.4.2. Scintillation*

Scintillation may be the most noticeable one for FSO systems. Light traveling through scintil‐ lation will experience intensity fluctuations, even over relatively short propagation paths. The scintillation index, σ<sup>i</sup> 2 describes such intensity fluctuation as the normalized variance of the intensity fluctuations given by [8,14]:

$$
\sigma\_i^2 = \frac{\langle (\iota \cdot \zeta \jmath)^\natural \rangle}{\langle \jmath \rangle^2} = \frac{\langle \iota \cdot \jmath \rangle}{\langle \jmath \rangle^2} \cdot 1 \tag{25}
$$

Where:

day. Close to ground, there is the largest gradient of temperature associated with the largest values of atmospheric pressure (and air density). Therefore, one should expect larger values

**Figure 10.** Scintillation or fluctuations in beam intensity at the receiver due to turbulence cells that is smaller than the

at sea level. As the altitude increases, the temperature gradient decreases and so the air

2 [8].

In applications that envision a horizontal path even over a reasonably long distance, one can

can be as little as 10-17m-2/3, while for a strong turbulence it can be up to 10-13m-2/3 or larger.

h )<sup>10</sup> ex p ( h

Ao

The most important variable in its change is the wind and altitude. Turbulence has three main

Scintillation may be the most noticeable one for FSO systems. Light traveling through scintil‐ lation will experience intensity fluctuations, even over relatively short propagation paths. The

<sup>1000</sup> ) +

= 1.7 × 10-14m-2/3.

describes such intensity fluctuation as the normalized variance of the

However, a number of parametric models have been formulated to describe the Cn

among those, one of the more used models is the Hufnagel-Valley [32] given by:

/ 27)2(10-<sup>5</sup>

h

<sup>1500</sup> )+Aoexp ( h

υ

exp( - <sup>2</sup> for a weak turbulence at ground level

<sup>100</sup> ) (24)

2

profile and

Cn 2

assume Cn

beam diameter.

Where:

: is the altitude in m].

*3.4.2. Scintillation*

scintillation index, σ<sup>i</sup>

h

v

A

density with the result of smaller values of Cn

182 Contemporary Issues in Wireless Communications

C n

2(h

: is the wind speed at high altitude m / s .

2

intensity fluctuations given by [8,14]:

0: is the turbulence strength at the ground level,

effects ; scintillation, beam wander and beam spreading.

<sup>2</sup> to be practically constant. Typical value of Cn

)= 0.00594(

2.7 × 10-<sup>16</sup>

I = | E | <sup>2</sup> : is the signal irradiance (or intensity).

The strength of scintillation can be measured in terms of the variance of the beam amplitude or irradiance σ<sup>i</sup> given by the following:

$$
\sigma\_i^2 = 1.23 C\_n^2 k^{7/6} L^{-11/6} \tag{26}
$$

Where Cn 2 is the refractive index structure, k = 2π / λ is the wave number (an expression suggests that longer wavelengths experience a smaller variance), and lis the link range (m).

Where the Eq. 26 is valid for the condition of weak turbulence mathematically corresponding to σi 2< 1. Expressions of lognormal field amplitude variance depend on: the nature of the electromagnetic wave traveling in the turbulence and on the link geometry [8].

### *3.4.3. Beam spreading*

Beam spreading describes the broadening of the beam size at a target beyond the expected limit due to diffraction as the beam propagates in the turbulent atmosphere. Here, we describe the case of beam spreading for a Gaussian beam, at a distance l from the source, when the turbulence is present. Then one can write the irradiance of the beam averaged in time as [33]:

$$I(l,\text{ }r) = \frac{2P\_o}{\pi \omega\_{\text{off}}^2\left\{l\right\}} \exp\left\{\frac{\cdot 2r^2}{\omega\_{\text{off}}^2\left\{l\right\}}\right\} \tag{27}$$

Where:

Po: is total beam power in W

#### r: is the radial distance from the beam center

The beam will experience a degradation in quality with a consequence that the average beam waist in time will be ωeff (l)>ω(l). To quantify the amount of beam spreading, describes the effective beam waist average as:

$$
\omega\_{\rm eff} \, \mathrm{(J)^2} = \omega \, \mathrm{(J)^2} \, \mathrm{(1+T)} \tag{28}
$$

Where:

ω(l) : is the beam waist that after propagation distance Lis given by:

$$(\omega(I)^2 = \left[\omega\_o^2 + \left(\frac{2L}{k\omega\_o}\right)^2\right] \text{(m}^2\text{)}\tag{29}$$

In which ωo is the initial beam waist at L = 0, T : is the additional spreading of the beam caused by the turbulence. As seen in other turbulence figure of merits, T depends on the strength of turbulence and beam path. Particularly, *T* for horizontal path, one gets:

$$T = 1.33 \,\sigma\_{\!\!\!\!}^2 \Lambda^{5/6} \tag{30}$$

While the parameter Λ is given by:

$$A = \frac{2\,L}{k\,\omega^2(l)}\tag{31}$$

The effective waist, ωeff (l), describes the variation of the beam irradiance averaged over long term.

As seen in other turbulence figure of merits, ωeff (l)<sup>2</sup> depends on the turbulence strength and beam path. Evidently, due to the fact that ωeff (l)>ω(l) beam will experience a loss that at beam center will be equal:

$$L\_{\
u BE} = 20 \log\_{10} \{ \omega(l) \mid \omega\_{\text{eff}}(l) \} \tag{32}$$

#### **3.5. Geometric Losses (GL)**

The geometric path loss for an FSO link depends on the beam-width of the optical transmitter θ, its path length L and the area of the receiver aperture Ar*.* The transmitter power, Pt is spread over an area of π(Lθ)<sup>2</sup> / 4. The geometric path loss for an FSO link depends on the beam-width of the optical transmitter *θ*, its path length *L* and the area of the receiver aperture Ar. The transmitter power, *P*r is spread over an area of *π*(*Lθ*) 2/4. Geometric loss is the ratio of the surface area of the receiver aperture to the surface area of the transmitter beam at the receiver. Since the transmit beams spread constantly with increasing range at a rate determined by the divergence, geometric loss depends primarily on the divergence as well as the range and can be determined by the formula stated as [2]:

$$\text{geometric loss} = \frac{d\_2^2}{[d\_1 + (\iota \omega \theta)]^2} \tag{33}$$

Where:

*d*2 is the diameter receiver aperture (unit: m);

*d*1 is the diameter transmitter aperture (unit: m);

*θ* is the beam divergence (unit: mrad);

*L* is the link range (unit: m).

Geometric path loss is present for all FSO links and must always be taken into consideration in the planning of any link. This loss is a fixed value for a specific FSO deployment scenario; it does not vary with time, unlike the loss due to rain attenuation, fog, haze or scintillation.
