1.Introduction

Nowadays, Free-Space-Optical (FSO) communication links are used for short distances in the Local Area Network or Metropolitan Area Network. They are also being used for long-distance links, particularly for conventional satellite and quantum security applications ^1-3. However, the process of designing optical links in the classical and quantum domain requires, in addition to the traditional optical power budget, to take into account diverse phenomena that affect the overall performance, such as the atmospheric turbulence ⁴. The atmospheric turbulence modifies the refractive index of air masses depending on the pressure and temperature value in such a way that the optical field’s diverse parameters may be affected by this phenomenon. Among these parameters are its phase, amplitude, state of polarization, orbital angular momentum, and also some phenomena such as the quantum entanglement ^5,6. In general, the FSO link is designed and characterized, firstly, in the laboratory environment, then, a real implementation at different distances is needed for different applications ^7,8. However, the characterization of such links still lacks a rigorous prior analysis regarding atmospheric turbulence. Although there are some experimental proposals for atmospheric turbulence chambers, they present important limitations and trade-offs such as 1) static or sometimes dynamic design with reduced configuration parameters, 2) reduced limit of emulated link distances, and 3) reduced control of atmospheric turbulence levels, among others ^9,10. Several mathematical algorithms simulate the effects of atmospheric turbulence over an optical field ^11-14. However, to the best of our knowledge, such algorithms do not determine the actual received optical power (measured in Watts) for the different possible regimes of atmospheric turbulence based on characteristics of physical devices. This document presents a technique for the design and statistical evaluation of FSO links based on the simulation of the optical information signal affected by different atmospheric turbulence levels. For this purpose, we generate by simulation a data time series with a Gamma-Gamma probability density function that allows modeling optical turbulence from low to high turbulence regimes. We organized our paper as follows: Section 2 presents some theoretical aspects of atmospheric turbulence and its effect over the optical power and the spatial phase of the received signal. Section 3 presents the results obtained in our simulations. Finally, Sec. 4 presents the conclusion and important aspects of future work.

2.Theoretical background

2.1.Atmospheric turbulence theory

A parameter commonly used to indicate the level of atmospheric turbulence is the scintillation index σI2 related to the irradiance of the optical field traveling through the free space channel. But as the optical power and the irradiance are related, one instead may use the power scintillation index σP2 for a particular distance link (L) using Eq. (1):

σP2=⟨PRx2(L)⟩⟨PRx(L)⟩2-1, (1)

where PRx(L) denotes the optical power affected by turbulence at a distance L(m) from the optical transmitter, ⟨⟩ is the ensemble average (sample mean), and, the log power variance (Rytov variance) σR2 is related to σP2 as σP2=exp(σR2)-1, considering that σP2≈σR2 for σR2≪1, i.e., applicable for weak turbulence regime, as Fig. 1 shows. Thus, the σR2 value is described for long-distance links (i.e., for plane wavefront) as Eq. (2) ¹⁵:

σP2=σR2=1.23Cn2k7/6L11/6. (2)

Figure 1 Relationship between σP2 and σR2 for several turbulence levels. 

On the other hand, for spherical wavefront, σP2=0.4σR2=0.5Cn2k7/6L11/6. Here, Cn2(m-2/3) is the index of refraction structure parameter, k=2π/λ is the optical wavenumber (m), λ is the wavelength, and L(m) is the propagation path length between transmitter and receiver systems. Also, Cn2 may be considered constant for a given condition of optical turbulence in horizontal links. However, if the weather conditions change, the value of Cn2 will change accordingly ¹⁵. We will see this effect more clearly in vertical links where the Cn2 value will depend on each atmospheric layer through which the optical beam travels. In our paper, we calculate the Cn2 value considering only the transmitter and receiver locations in a horizontal link for a fixed condition of optical turbulence (i.e., we defined the full effect of the turbulence through the atmosphere as a black box).

As Eq. (1) showed, σP2 parameter depends on the receiver optical power PRx(L) affected by the stochastic variations of the optical turbulence in addition to diverse (static) losses introduced by the free-space optical channel. As Eq. (1) showed, σP2 parameter depends on the receiver optical power PRx(L) according to the free space loss (L_s) (i.e. Free-Space Path Loss law is shown in Eq. (3)). Although other kinds of losses are present in an FSO link (e.g., beam divergence, weather conditions, pointing) and laws, as Beer’s law. Hence, PRx=PTxLs ¹⁶.

Ls=λ4πL2 (3)

The stochastic behavior of atmospheric optical turbulence is commonly described using a lognormal probability density function (pdf) due to its relative simplicity. However, this description is valid just for low to medium turbulence values. On the other hand, there are other more complex probability density functions, such as the called Gamma-Gamma, whose results are valid for weak to strong turbulence levels. The Gamma-Gamma (f _GG) pdf describes turbulence as a function of small and large-scale variations. Equation (4) shows the relation of f _GG with PRx with α and β parameters being the effective number of small-scale and large-scale eddies of the scattering environment, respectively ¹⁶. Commonly, Eq. (4) is related to the irradiance (W/m²), but considering the active area (m²) of a suitable photodetector, Eq. (4) can be expressed using the optical power received.

fGG(PRx;α,β)∝2αβa+β2ΓαΓβ×PRxα+β2-1kα-β(2αβPRx) (4)

where Γ(⋅) is Gamma function, and Ka(⋅) is the modified Bessel function of second kind of order α. The parameters of the Gamma-Gamma function are related to the Rytov variance using Eqs. (5) and (6) under the assumption of plane wave and negligible inner scale, which corresponds to long propagation distance and small detector area ¹⁷.

α=gσR=exp0.49σR21+1.11σR12/57/6-1-1 (5)

β=hσR=exp0.51σR21+1.69σR12/55/6-1-1 (6)

For instance, if σR2=0.3 implies the values of α = 8.42 and β = 6.91. Figure 2 shows the behavior of g(σR) and h(σR) for the range 0<σR2≤1, from weak to strong turbulence regimes.

Figure 2 Relationship between σR2 and α, β. 

2.3.Phase fluctuation induced by atmospheric turbulence

Another essential aspect to consider in FSO links is the phase fluctuation in a spatial region of the detector. The atmospheric turbulence modifies the phase of the signal in different regions of the wavefront that can affect the communications link’s performance, particularly for coherent detection schemes. There are various phase drifts in an FSO link (e.g., optical phase drift caused by phase noise); however, we will only consider spatial phase fluctuation because the other phase noises are not caused by turbulence. It is possible to determine the power spectrum of these phase fluctuations using Eq. (13), representing a thin phase screen ^20-22.

φn(K)=0.033Cn2k2Lx-11/3=0.033r0-5/3K-11/3, (13)

where K is the transverse coordinate, considering that a single atmospheric layer (multiple atmospheric layers simulation requires a high-end computer system) based on the aperture diameter D_G of the telescope used. r₀ is the wavefront coherence diameter describing the spatial correlation of phase fluctuations in the receiver plane due to random inhomogeneities in the atmosphere’s refractive index. In our case, r0=(k2Cn2L)-3/5, and it is constant because we consider just a single atmospheric layer. In particular, the phase screen does not modify the amplitude; in fact, the optical phase is the only parameter affected. It is important to mention that Eq. (13) is based on Discrete Fourier Transform (DFT), which means that the phase screen is periodic. This argument imposes a wrong hypothetical scenario of small non- homogeneous regions inside a homogeneous large region (see Fig. 3) that have to be considered for large region analysis taking into account D_G and r₀ parameters.

Figure 3 Individual non-homogeneous phase screen as part of a large homogeneous phase screen. 

3.Results and analysis

In order to determine the temporal behavior of a hypothetical optical signal received, some technical settings were established, such as PTx=500 mW, L = 10,000 m, λ = 1550 nm, τ_c = 20 ms, 5 x 10⁵ samples with Δt=10-4s and bit rate at 350 KHz. All parameters are variable in the simulation, and, in fact, some parameters are simplified or normalized to show general results. Figure 4 shows the optical signal on the receiver side based on Eqs. (3) and (11) for different turbulence regimes. Besides, Fig. 5 shows the distribution of numerical data of the optical signal received considering weak, moderate, and strong atmospheric turbulence regime with particular values for σR2, α, and β. These distributions have a Gamma-Gamma probability density function shape. In particular, Cn2=2.22×10-16 m^-2/3 for σR2=0.3, and Cn2 can be calculated for a particular turbulence regime. Thus, the results showed in Figs. 4 and 5 can be modified according to atmospheric turbulence levels (i.e., modifying Cn2 parameter) based on the Rytov variance and Gamma-Gamma parameters.

Figure 4 Simulated Gamma-Gamma Time-series signal for different turbulence regimes. 

Figure 5 Probability density function of the simulated Gamma- Gamma signal for different turbulence regimes a)Weak, b) Moderate, c) Strong. 

Figure 5 shows the distribution of the samples and, regarding the strong turbulence regime, the mean value of PRx is less than the other regimes because PRx presents more fluctuations in the samples ensemble, i.e., the optical power received is increased for those samples that had less optical power in the weak and moderate turbulence regimes.

Thus, for a strong turbulence regime, the probability is increased for all samples that represent high power received (PRx>2), and decreased for PRx=1. On the other hand, Fig. 6 shows the probability density function considering a fixed ensemble data related to Δt=10 ms in order to visualize the particular behavior of the optical signal related to the temporal variation using a turbulence level condition close to the moderate turbulence regimes. It can be seen that the density function does not change; that is, it remains Gaussian, so it can be interpreted that the optical state has not changed due to the non-linearity of the atmospheric channel.

Figure 6 Probability density function of the simulated Gamma-Gamma signal received. 

Figure 7 shows the simulated phase screen considering the parameters mentioned and an aperture diameter D_G = 5 cm based on Eq. (13). The results show that the phase screen imposes a phase variation added to the phase noise of the optical signal for a particular spatial position on the receiver side. The maximum and minimum variations are 8 and -8 radians, respectively, for Cn2=2.22×10-16 m^-2/3 (i.e., meaning a moderate turbulence level, although this parameter can be modified). In order to clarify, the maximum and minimum variations do not imply generalized phase changes; in fact, the mean phase variation is less than 0.5 radians. It is important to clarify that the phase screen showed is a random numerical result according to (13), i.e., in general, different representations of phase screen can be generated maintaining the maximum and minimum phase variation similar. Also, Fig. 7 is represented using the physical dimension of the receiver aperture diameter, while Fig. 8 shows the same results in a different representation considering the spatial samples based on the sample resolution, i.e., 400 samples for x and y-axis, which means that 1 sample is equivalent to 0.0125 cm. This resolution is optimal in order to characterize small and large eddies present in the turbulence medium.

Figure 7 Simulated phase screen of the signal received. D _G = 5 cm. 

Figure 8 Simulated phase screen of the signal received. Wavefront phase aberrations due to Kolmogorov turbulence, D _G = 5 cm. 

Therefore, considering a Gaussian beam transmitted through the atmospheric turbulence and a single-phase screen, both represented before, it is possible to obtain a spatial phase representation as Fig. 9 shows. It is important to mention that the results can vary because they are evaluated based on random variables. Nevertheless, the results generated are very useful for FSO communication systems. Besides Fig. 10 shows the theoretical and numerical results for the mean SNR parameter considering different atmospheric turbulence levels (from weak to strong turbulence) and aperture diameters. In particular, the σP2 values for each simulated series-time showed in Fig. 4 are affected by different D_G values, as Eq. (17) showed. Therefore, the mean SNR is reduced for stronger atmospheric turbulence levels, i.e., 1.4 dB for σR2=0.33 and 0.36 dB for σR2=1. The latter, due to the increasing variation of the values generated and showed in Fig. 5. Besides it is important to mention that the mean SNR is also increased based on the aperture diameter. Hence, the atmospheric turbulence characterization in an FSO link has to be performed based on a particular and well-known photodetector and passive optical devices. The SNR0 value was considered constant (i.e., 10 dB), which is related to Eq. (14).

Figure 9 Simulated spatial phase distribution considering a single phase screen, D _G = 5 cm. 

Figure 10 Mean SNR for different D _G values and numerical results for mean SNR for simulated series-time data affected by atmospheric turbulence. 

Figure 11 shows the block diagram used in order to obtain the simulation results considering particular values for the different subsystems of the overall communication system. The simulation was performed using a homologated Matlab script based on the VPI photonics program, although the OptiSystem program is also useful. In particular, a bit sequence generator at 350 Kbps (related to τ_c = 20 ms) is used, a pulse generator based on Non-return-to-zero (NRZ) and Low Pass Filter (fc=0.7×bit rate) produces the input digital signal to the amplitude modulator. Concerning the optical source, a CW laser (500 mW) at λ =1550 nm (v = 193.41 THz) and Δv = 1 MHz is used. Thus, the electrical input signal modulates the optical signal. Next, the simulation considers some features of the optical transmitter, such as distance link (10 Km), attenuation (2.3 dB/Km), geometrical losses (10 dB), and transmitter losses (3 dB). The photoreceiver has a responsivity of 1 A/W and a dark current of 10 nA. Regarding the atmospheric channel, a multiplying point permits the introduction of the series-time turbulence based on Gamma-Gamma distribution function.

Figure 11 Block diagram used for simulation. 

Finally, Fig. 12 shows the results of E[BER] for different E[SNR] values based on Fig. 11. Remember that according to Eq. (17), the mean SNR value depends on some parameters such as D_G, σR2, and SNR₀. In this case, D_G = 5 cm, σR2=1, and SNR₀ are modified in relation to particular photodetectors. However, it is possible to use a particular photodetector and vary the Rytov variance, i.e., a particular mean SNR value can mean a combination of different parameter values presented in the free space link and receiver scheme. In fact, the results simulated based on series-time data affected by atmospheric turbulence described for Gamma-Gamma function are closely related to the theoretical results showed in ²¹.

Figure 12 Numerical results for mean BER of the series-time for different mean SNR. 

4.Conclusions

The accurate characterization of the optical signal transmitted through the atmospheric channel is necessary in order to offline research the possible effects in a future hypothetical FSO link to improve the overall performance. Thus, simulated series-time signals affected by atmospheric turbulence were generated based on two stochastic processes described by a Gamma probability density function in order to calculate the performance of a particular FSO link. The simulation permits to visualize an optical signal affected by different atmospheric turbulence regimes, i.e., from σR2=0.1 to σR2=1, and calculate the mean SNR and mean BER using a time-interval of 1 second. The simulation can be expanded to large time-intervals according to the performance parameters required (e.g., Bit Error Rate), transmission rates used in particular FSO links, and scheme modulations (e.g., Quadrature Amplitude Modulation (QAM), Phase Shift Keying (PSK), among others). Also, the spatial phase was simulated for particular σR2 value, although it is possible to modify the atmospheric condition parameters to determine the correct spatial phase perturbation. In particular, the correlation time value used is directly obtained based on the temporal covariance function for a particular condition of the FSO link, i.e., the time value for 1/e of the normalized temporal covariance function is the correlation time. This condition is described based on the distance of link, Rytov variance, and the transverse wind speed for Taylor’s frozen-turbulence hypothesis ²¹. This means that the simulation proposed is suitable to analyze only the FSO links that satisfy the conditions mentioned. Finally, the numerical results regarding the mean SNR and mean BER for different atmospheric turbulence regimes are highly related to the theoretical performance. In fact, if this simulation proposal is used for the design and proof of concept of FSO links previous to the real implementation, it is necessary to provide technical details of each optical and optoelectronic device to increase the accuracy of the numerical results.