Theoretical and experimental studies of polarization fluctuations over atmospheric turbulent channels for wireless optical communication systems

Jiankun Zhang; Shengli Ding; Huili Zhai; Anhong Dang

doi:10.1364/OE.22.032482

1. Introduction

Wireless optical communication (WOC) is a promising candidate for data transmission due to its flexibility, wide-bandwidth and license-free [1]. However, the random variation of the refractive index (RI) caused by atmospheric turbulence can degrade the performance of WOC system severely. When transmitting through atmosphere, WOC signals suffer from intensity scintillation, phase fluctuation, as well as polarization alteration caused by turbulence [2–6].

The variation of RI is caused by the random variation of the temperature [7]. The longitudinal wind velocity associated with the turbulent atmosphere fluctuates randomly about its mean value, resulting in the changing of RI. When polarized light passes through the turbulent atmosphere, intensities in the two orthogonal polarizations are impacted differently, leading to the alteration of the direction and degree of polarization.

In WOC systems, polarization techniques are widely applied, such as polarization-multiplexing systems [8–11], coherent polarization shift keying (POLSK) systems [12,13], and so on. However, these systems are quite sensitive to polarization directions, requiring polarization agreement in both transmitting and receiving ends. Polarization fluctuations may cause a crosstalk between the signals in different polarizations, resulting in the degradation of optical signal-to-noise ratio (OSNR) and bit error rate (BER) performance [14]. Polarization fluctuations may also increase the complexity of polarization control feedback in the systems. Hence an applicable model for polarization fluctuations is important and necessary for performance estimations.

Up to now, many researchers have studied the polarization fluctuations caused by atmospheric turbulence [15–23]. However, their results are limited for some specific conditions to some extent. For example, an early theoretical model proposed in [15,16] is based on slight RI variations, and it shows two orders of magnitude smaller than experimental results presented in [17]. A space-to-ground propagation was carried in [18], but the turbulent parameters of which is unknown. Hence the measurement results are not compared with any theory. Recently, polarization fluctuations for partially coherent beams were studied [19–21], and polarization in non-Kolmogorov turbulence [22] and oceanic turbulence [23] was researched. However, a widely accepted model is not yet proposed to describe turbulence-induced polarization fluctuations. Moreover, most of the accomplished experiments are measured under a single turbulent condition, and the distribution of the polarization is seldom discussed.

This paper presents a novel and widely applicable theory to evaluate polarization fluctuations in turbulence. Firstly, the relationship between the polarization fluctuations and the index of refraction structure parameter is introduced and the probability distribution function (PDF) of received polarization angle is obtained through theoretical derivations. Then, a related experiment is carried out to measure the distribution of the polarization angle. In the experiment, different turbulent conditions are realized to confirm our theory in a wide range. Finally, the experimental results are compared with the theoretical conclusions.

2. Theoretical model of turbulence on polarization

Suppose the incident light is polarized along the z-axis and propagates along the x-axis. The time-varying light-field can be expressed as

E_{0} (t) = A_{0} \exp [j (ω_{0} t + S_{0})] \hat{x},

where A₀, ω₀ and S₀ denote the amplitude, frequency and phase of transmitting wave respectively. When propagating through turbulence, polarization alteration leads to the amplitude variation both in x and y directions. Then the receiving light-field is

E = (A_{x} \hat{x} + A_{y} \hat{y}) \exp {j [ω_{0} t + S (t)]},

where A_x and A_y are the polarization components of x and y direction separately, which satisfy

| A | = \sqrt{{| A_{x} |}^{2} + {| A_{y} |}^{2}}

; S(t) is the phase fluctuations caused by turbulence. The polarization angle is defined as φ = arctan(A_y / A_x), with A_x = Re{Ae^j^φ} and A_y = Im{Ae^j^φ}. When the polarization angle is quite small, there is φ ≈M where M = |E_y| / |E₀| is the polarizability. According to the perturbation theory, M is given by [14]

〈 M^{2} 〉 = \frac{4 \sqrt{π} L σ_{n}}{k^{2} l^{3}},

where σ_n is the root mean square of refractive index and satisfies

σ_{n} = {〈 Δ n^{2} 〉}^{1 / 2}

; l is the scale factor; k = 2π / λ is the wavenumber, where λ is the wavelength; L is propagation length. Then the root mean square σ_φ of the polarization angle φ caused by atmospheric turbulence is [17]

σ_{φ} = \frac{σ_{n} λ L^{1 / 2}}{2 π^{3 / 4} l^{3 / 2}} .

For experiments, Δn is usually difficult to measure directly. Since the atmospheric structure function $D_{n} (r) = 〈 {[n (r_{1}) - n (r_{2})]}^{2} 〉 = 〈 {[n (r_{1})]}^{2} 〉 + 〈 {[n (r_{2})]}^{2} 〉 - 2 〈 n (r_{1}) n (r_{2}) 〉$ ([2], Sec 3.2), we can get $D_{n} (0) = 2 〈 n^{2} 〉 - 2 {〈 n 〉}^{2} = 2 C_{n}^{2} d_{0}^{2 / 3}$ in the correlation range. By substituting ${〈 n 〉}^{2} \approx C_{n}^{2} d_{0}^{2 / 3}$ [16] and $d_{0} \approx {(λ L)}^{1 / 2}$ [24,25], where d₀ is the correlation length, $σ_{n}^{2}$ can be approximated as

σ_{n}^{2} = 0.5 C_{n}^{2} λ^{1 / 3} L^{1 / 3},

where

C_{n}^{2}

is the index of refraction structure parameter which can be easily measured and compared with the theoretical results. Thus the RMS of depolarization can be written as

σ_{φ} = \frac{{(C_{n}^{2})}^{1 / 2} λ^{7 / 6} L^{2 / 3}}{2 \sqrt{2} π^{3 / 4} l^{3 / 2}} .

The PDF of polarization angle $Δ φ$ after atmospheric turbulence can be deduced according to Rytov theory. The perturbation decomposition method applying to wave equation converts its free space solution into perturbation multiplication. We have $ψ = ψ_{0} + ψ_{1}$ when the higher order terms of $ψ (r)$ can be neglected, where $ψ_{0}$ is the certain solution and $ψ_{1}$ is the perturbation term. The formal solution of scalar wave equation can be expressed as $\tilde{u} = \exp (ψ_{0} + ψ_{1})$ which satisfies Riccati equation ([2], Sec 5.3)

\nabla^{2} ψ (r) + {[\nabla ψ (r)]}^{2} + k^{2} n^{2} (r) = 0.

In the Born approximation, any component of the propagating light field in turbulent medium can be expressed as the sum of free space perturbation solutions. If we put A_x and A_y together to form complex amplitude as $\tilde{A} = A_{x} + j A_{y} = A e^{j φ}$ , where its real and image parts denote the x and y components respectively, then the complex amplitude without disturbance is ${\tilde{A}}_{0} = A_{0}$ accordingly. Combining with Eq. (1),(2), we can get

ψ_{1} = ψ - ψ_{0} =ln \frac{A}{A_{0}} + j φ + j (S - S_{0}) .

With the help of scalar diffraction theory and forward scattering approximation, we suppose the wave scattering angle caused by inhomogeneity of refractivity satisfies θ₀ = λ/l₀<<1, where θ₀ is the scattering angle and l₀ is the inner scale of turbulence. Then the solution of Eq. (7) is the convolution of the field source and Green function, that is

Δ_{A, S} + j φ = \frac{1}{4 π} \underset{V}{∭} \frac{\exp (j k | r - r^{'} |)}{| r - r^{'} |} 2 k^{2} n_{1} (r^{'}) {\tilde{u}}_{0} (r^{'}) d r^{'},

where Δ_A_,_S = ln (A / A₀) + j(S – S₀) is the variation of logarithmic amplitude and phase,

n_{1} (r^{'})

is the perturbation of

n (r^{'})

which means

n_{} (r^{'}) = 1 + n_{1} (r^{'})

, and

{\tilde{u}}_{0}

is the definite part of

\tilde{u}

. In a rectangular coordinate system, the logarithmic amplitude, phase and polarization angle of plane wave when propagating to z = L satisfy

Δ_{A, S} + j φ = \int_{0}^{L} d z \int \int \frac{k^{2}}{2 π (L - z)} \exp [j k \frac{{(x - x^{'})}^{2} + {(y - y^{'})}^{2}}{2 (L - z)}] n_{1} (r^{'}) {\tilde{u}}_{0} (r^{'}) d x^{'} d y^{'},

where r´ = (x´,y´,z). In Eq. (10), Δ_A_,_S and φ can be expressed as a superposition of many independent contributions. Note that

ψ_{1} = Δ_{A, S} + j φ

, and

ψ_{1}

is a perturbation. From central-limit theorem, they all satisfy normal distribution. Thus the PDF of

Δ φ

is

f_{Pol} (Δ φ) = \frac{1}{\sqrt{2 π} σ_{φ}} e^{\frac{- Δ φ^{2}}{2 σ_{φ}^{2}}} .

3. Experimental study of polarization fluctuations in a turbulent channel

3.1 Experimental setup

In order to verify the theoretical model in practical wireless optical links, an outdoor experiment has been carried. As shown in Fig. 1, a 1550nm high-power precise tunable laser source is used in the transmitting site. The laser propagates through an adjustable polarization controller and a polarizer to generate linearly polarized light. After amplified and filtered, the beam is sent to the free space by a transmitting collimator. In the receiving site, the light is received by another telescope. After filtering, a polarization beam splitter is applied to detect polarization. In the splitter, the optical signal is decomposed into two linearly polarized light beams with polarization directions perpendicular to each other. At first, the intensities of two received light beams are measured back to back,

Fig. 1 Block diagram of the experimental setup. PC, polarization controller; EDFA, erbium-doped fiber amplifier; BPF, band-pass filter; PBS, polarization beam splitter; DSP, digital signal processing.

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\begin{array}{l} I_{1} = {| E_{0} \cdot \hat{z} |}^{2} = {| A_{0} e^{j k x} |}^{2} = A_{0}^{2}, \\ I_{2} = {| E_{0} \cdot \hat{y} |}^{2} = 0. \end{array}

Then the devices are put into turbulent channel to observe the changes of the splitter output. Suppose the intensities of two light beams after turbulence are I₁´, I₂´separately, and I₁´<< I₂´, then polarization angle φ can be estimated by [15]

φ = \frac{I_{2}^{'}}{I_{1}^{'}} .

To reflect the affection of atmospheric turbulence on turbulence accurately, the measurement precision is of great importance. In our experimental system, the detecting power range of received beams is −13 dBm and 30 dBm. According to Eq. (13) the measured angular error is within 5 × 10⁻⁵ rad (0.0028°), which can meet the requirement of our experiments. What’s more, narrow band filters are applied to suppress background noise.

3.2 Experimental results

Figure 2 shows the polarization angle and received optical power during 120s duration in strong, medium and weak turbulent conditions. The total follow-up observation lasts for an hour, and the experiment was repeated under different scintillation indexes (SI). In Fig. 2(a), the polarization angle fluctuates within 0.05 rad under strong turbulence, and within 10⁻² rad and 10⁻³ rad under medium and weak turbulence respectively. In Fig. 2(b), the received power fluctuates because of intensity scintillation. Considering from Eq. (13), little intensity may cause large angular error, data with low optical power were removed in order to estimate the polarization data correctly. All data except for low power ones were used to calculate polarization angle, and the angular RMS was computed under different SI’s separately.

Fig. 2 A measurement sample of (a) polarization angle and (b) light intensity during 120s under different SI’s.

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SI can be obtained by fitting with the distribution of intensity scintillation. As is well known [2,26,27], intensity scintillation fits with gamma-mamma (GG) distribution in a wide turbulent range. GG distribution is described as

f_{GG} (\hat{I}) = \frac{2 {(α β)}^{\frac{α + β}{2}}}{Γ (α) Γ (β)} {\hat{I}}^{\frac{α + β}{2} - 1} K_{α - β} (2 \sqrt{α β \hat{I}}), \hat{I} > 0, 〈 \hat{I} 〉 = 1,

where

\hat{I}

is the normalized intensity, Γ(∙) is the gamma function, K_v(∙) is the vth-order modified Bessel function of the second kind, α and β are the effective parameters related to the large-scale and small-scale factors [26]. SI is given by

S I_{G G} = \frac{1}{α} + \frac{1}{β} + \frac{1}{α β} .

To fit with Eq. (14), we can use the maximum likelihood method by finding α and β to satisfy the maximum likelihood function [28]. Then, α and β can be used to calculate the index of refraction structure parameter $C_{n}^{2}$ .

According to the above experimental method, we have measured a number of polarization rotation results in case of different turbulent conditions. Firstly, the transmission distance is fixed to 500m, and the RMS of polarization rotation angles is measured under different SI’s ranging from 0.01 to 0.1, as represented in Fig. 3(a). Then, the propagating length is changed from 100m to 1km with an interval of 100m. For each length, the RMS of polarization rotation angles is measured with $C_{n}^{2}$ = 2 × 10⁻¹⁵ m^-2/3 (for weaker turbulence) and 2 × 10⁻¹⁴ m^-2/3 (for stronger turbulence). The result is shown in Fig. 3(b).

Fig. 3 Theoretical (The.) and experimental (Exp.) polarization RMS in (a) different SI’s and (b) different distance.

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From Fig. 3 we can see that our theoretical model matches well with experimental results for parameters of both SI and ( $C_{n}^{2}$ , L). When turbulence is weak (corresponding to small SI or distance), the fluctuation of polarization is also weak, causing a small variation in the received polarization angle. Polarization fluctuations for weaker turbulence (SI<0.01) is hard to derive due to our measurement precision and usually could be neglected in practical communication systems so that they are not concerned in this paper. However, when turbulence is strong, the polarization fluctuations are quite large and may affect communications. For a kilometer-scale WOC link with $C_{n}^{2}$ = 2 × 10⁻¹⁴ m^-2/3, RMS could reach 0.05 rad (approximate 3°). What’s more, in our experiments, the SI values are not very large even in strong turbulence with aperture averaging [29], while in practical systems SI would be larger so that polarization changes would exceed the polarization error tolerance of communication systems.

Figure 4(a) plots the experimental PDF of the polarization deflection angle where SI takes 0.01, 0.05 and 0.1 respectively (curves with Exp.), which is compared with theoretical model Eq. (6) and (11). Figure 4(b) shows the corresponding distribution for intensity scintillation fitting with LN distribution. It can be seen that the PDF of both polarization rotation angle and normalized intensity match well with theoretical conclusions. Correlation coefficient r can be derived as $r^{2} = \underset{i}{Σ} {({\hat{p}}_{i} - \bar{p})}^{2} / \underset{i}{Σ} {(p_{i} - \bar{p})}^{2}$ where $p_{i}$ is the measured value of probability, ${\hat{p}}_{i} = f_{GG} (\hat{I})$ is theoretical value from Eq. (14), and $\bar{p}$ is the average of the sample. Although normal distribution seems simple, it shows good description for turbulence-induced polarization. For SI = 0.01, 0.05 and 0.1, its PDF has a correlation of 0.98, 0.98 and 0.99 to the normal distribution respectively.

Fig. 4 Distribution of (a) polarization angle and (b) corresponding normalized intensity under turbulence.

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4. Conclusion

A theoretical and experimental study of polarization fluctuations in turbulence channel is presented in this paper. The RMS and PDF model is employed to describe the polarization alteration, and further demonstrated through an outdoor experiment in different turbulent conditions. The experimental results have a good match with theory. This work possesses guiding significance to study the performance of WOC polarization systems.

Acknowledgments

This work is supported by the National Key Basic Research Program (973 Program) of China (Grant No. 2013CB329205) and the National Natural Science Foundation of China.

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Theoretical and experimental studies of polarization fluctuations over atmospheric turbulent channels for wireless optical communication systems

Abstract

1. Introduction

2. Theoretical model of turbulence on polarization

3. Experimental study of polarization fluctuations in a turbulent channel

3.1 Experimental setup

3.2 Experimental results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (15)

Optics Express