Amplitude fluctuations for optical waves propagation through non-Kolmogorov coronal solar wind turbulence channels

Guanjun Xu; Zhaohui Song

doi:10.1364/OE.26.008566

1. Introduction

Optical communication is a promising technology and has attracted plenty of attention with increasing deep space exploration activities, as it has advantage in providing high data rate and large capacity [1, 2]. For the future deep space optical communication, the optical waves transmitted between the transmitter and the receiver will pass through the atmosphere, the interplanetary space, and the solar corona. The link performance will be severely degraded not only by the attenuation due to gaseous, rain, density absorption, etc., but also by the fast amplitude fluctuations due to scattering and diffraction effects caused by the turbulence and irregularities [3,4].

These phenomena mentioned above have been investigated for a long time. In particular, both influence and physical mechanism of atmosphere turbulence on optical waves propagation have been extensively studied [5–7]. Nevertheless, the effect of solar corona plasma on the propagation of optical waves during superior solar conjunction, when the probe revolves to the other side of the Sun relative to the Earth, is rarely explored yet. The coronal solar wind plasma imposes large degradation effects on the optical waves, induces amplitude fluctuations, phase fluctuations, angle-of-arrival fluctuations, spectral broadening, etc., and even causes communication failed. Therefore, investigating the effectes of solar wind plasma on optical waves propagation, especially, developing a model of amplitude fluctuations, has been a challenging topic for deep space optical communication.

In essence, the scattering and refraction effects on the optical waves are the problems of wave propagation in stochastic media, which are collectively referred to as scintillation. The atmosphere scintillation has been investigated in the previous research. In particular, the variance of amplitude fluctuations was well studied. Lee et al. developed a new scintillation index model for optical waves in atmosphere with Kolmogorov spectrum model [8]. Based on the Rytov approximation, Toselli et al. proposed a new amplitude fluctuations model for gaussian wave beam in weak atmospheric turbulence, and gave a comprehensive analysis of the long term beam spread, mean signal to noise (SNR) ratio and mean BER with the amplitude fluctuations [9]. Since the atmospheric turbulence sometime will deviate from the standard isotropic Kolmogorov spectrum, Gudimetla et al. developed the analytical expression for the log-amplitude correlation function for both plane wave and spherical wave passing through the anisotropic non-Kolmogorov atmosphere [10]. Considering that the traditional Kolmogorov theory is sometimes incomplete to describe atmospheric statistics properly, Cheng et al. studied the variance and power spectrum of the amplitude fluctuations for optical waves in non-Kolmogorov maritime atmospheric turbulence channels [7,11].

According to the above analyses, the amplitude fluctuations model of optical waves propagation in atmospheric turbulence has been well developed. However, the amplitude fluctuations model for optical waves propagation in coronal plasma has seldom been reported to the best of our knowledge. Therefore, the BER performance for optical waves propagation in coronal trubulence also has not been well studied. Recently, we have devoted ourself to understanding the solar scintillation effects on optical waves propagation [12]. The solar corona is filled with solar wind plasma and its irregularities spectrum model is different from the atmosphere’s spectrum model. Besides, the coronal turbulence statistics obey non-Kolmogorov’s spectrum model according to the increasing astronomical observation evidences. All of these prompt us to derive a new amplitude fluctuations model for the optical waves in coronal turbulence.

In this paper, both the coronal irregularities model and the non-Kolmogorov generalized coronal turbulence spectrum model were introduced and employed. After that, the closed-form amplitude fluctuations expression for optical waves propagation throuth the non-Kolmogorov coronal turbulence was derived based on the Rytov theory. Therefore, the computation time for amplitude fluctuations can be reduced and the physical mechanism for optical waves in coronal turbulence can be achieved. We also study the associated average BER performance of the free space optical (FSO) communication link during superior solar conjunction. The effect of various parameters on the amplitude fluctuations and the BER performance were further analyzed based on the derived amplitude fluctuations model.

2. Theoretical model

2.1. Solar wind density model and its irregularity model

In order to analyze the impact of coronal plasma on the optical communication link, the geometric diagram of superior solar conjunction between the Earth, the probe, and the Sun is shown in Fig. 1. Note that, the distance between the communication link and the Sun is the heliocentric distance, r, which is usually in units of the solar radius R_sun. According to the triangle relationship, the link distance can be obtained by the Sun-Earth-probe (SEP) angle, α and the Sun-probe-Earth (SPE) angle, β,

L = L_{se} \cos α + L_{sp} cos β,

where L_se and L_sp denote the distance between the Sun and the Earth, and the distance between the Sun and the probe, respectively. As can be seen from Fig. 1, the optical communication link will pass through the solar corona and degrades more seriously when the SEP angle get smaller.

Fig. 1 Geometric diagram for deep space optical communication during superior conjunction and the coordinate used for analyzing amplitude fluctuations.

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In light of the above descriptions, the coronal solar wind is the main factor to amplitude fluctuations. Aiming at evaluating the amplitude fluctuations, it is necessary to develop a better solar wind density model. As the electrons is lighter than ions, the solar wind plasma density in this paper is denoted by the electron density. The solar wind electron density model has been proposed in amount of literatures [13,14], all of these models are founded based on long term observations rather than a solid foundation of theory. Here, we use a widely applicable empirical formula to describe the electron density [15]

N_{e} (r) = 2.21 \times 10^{14} {(\frac{r}{R_{sun}})}^{- 6} + 1.55 \times 10^{12} {(\frac{r}{R_{sun}})}^{- 2.3} .

Note that the solar wind fluctuates both in spatial and temporal, therefore, its exact density model remains a mystery. This model is different from the other density models since we mainly consider about the amplitude fluctuations caused by coronal plasma irregularities when the SEP angle is less than 1 degree. It is generally indicated that all the solar wind speed, solar magnetic field, the temperature and the pressure play an important role on solar wind turbulence. Therefore, the turbulence is not stable but irregular without fixed rules. According to Taylor’s frozen theory [16], the solar wind plasma irregularities are frozen in and the plasma density does not change remarkably in a short period of time. This assumption is justifiable because the optical signal transmission speed is faster than the solar wind speed. Therefore, only the plasma density in spatial is considered in this paper.

As the solar corona is located near the solar surface with the higher electron density, it is generally believed that there is a corresponding increased density fluctuations [15]. Besides, the irregularities of solar wind plasma density are usually assumed proportional to the plasma density as

δ N_{e} (r) = η N_{e} (r),

where η is solar wind plasma density relative fluctuation factor [17]. Previous observing results indicate that the relative level of solar wind density fluctuations is larger than 0.01 when the heliocentric distance is smaller than 4R_sun [18]. According to the geometric diagram in Fig. 1, we have r² = z² + L_se² − 2zL_se cos α. Therefore, δN_e(r) also can be recast as a function of z.

Since the amplitude fluctuations are mainly caused by the fluctuations of the media dielectric constant, therefore, it is necessary to investigate the fluctuations of the dielectric permittivity. According to [19,20], we have $δ ε (r) = - \frac{r_{e} λ^{2}}{π} δ N_{e} (r)$ , where r_e and λ are the classical electron radius and the wavelength of the optical signal, respectively. In fact, the permittivity should be varied both in spatial and temporal. Nevertheless, we only consider its spatial variation for simplicity according to the Taylor’s frozen theory. Therefore, the variance between the dielectric permittivity and the plasma density fluctuations can be described as

〈 δ ε (r) δ ε (r^{'}) 〉 = \frac{{r_{e}}^{2} λ^{4}}{π^{2}} 〈 δ N_{e} (r) δ N_{e} (r^{'}) 〉 .

2.2. Generalized modified coronal turbulence spectrum model

In addition to the solar wind plasma density and its irregularities, the spatial spectrum is its another characteristic. The power spectrum of fluctuations in various solar wind plasma has been measured from many space explorations and is different from the other spectrum models in various turbulent medium, such as, the atmosphere, ocean, and ionosphere. Bastian has pointed out that the spatial spectrum is consistent with a power law Φ_N (κ) ∝ κ^−p [21]. Here, p and κ denote the spectral index and the magnitude of the spatial frequency vector, respectively. According to the astronomical observation [17], the spectral index is restricted in the range of 3 to 4 in this paper.

Since the solar wind erupt from the Sun and its density decreases along the radial direction, the coronal plasma turbulence environment is inhomogeneous and isotropic in essence. For the sake of simplicity, the coronal turbulence normally is normally treated as locally homogeneous under the Taylor’s frozen theory and has been wildly applied in radio astronomy [17,19].

Astronomical observations also pointed that, the power spectrum is approximate to non-Kolmogorov spectrum model when the heliocentric distance is less than 50R_sun [17, 22]. In order to investigate the amplitude fluctuations for optical waves propagation in non-Kolmogorov coronal solar wind turbulence, a theoretical spectrum model for the refractive index fluctuations is considered and can be assumed that all the spectral index values range from 3 to 4 [17],

Φ_{N} (κ, z) = C_{N}^{2} (z) κ^{- p},

where

C_{N}^{2}

is the factor characterizing the strength of the solar wind fluctuations. According to the astronomical research, it is normally written as [23]

C_{N}^{2} (z) = {\begin{matrix} {(2 π)}^{- 3 / 2} \frac{(p - 3) Γ (\frac{p}{2}) κ_{o}^{p - 3}}{Γ (\frac{p - 1}{2})} 〈 δ N_{e}^{2} 〉, for 3 < p < 4 \\ 4 π ln (\frac{2 κ_{o}}{κ_{i}}) 〈 δ N_{e}^{2} 〉, for p = 3 \end{matrix},

Note that, the angle brackets 〈〉 represent the ensemble average of realizations. $δ N_{e}^{2}$ denotes the variance of the the solar wind density fluctuations and can be obtained by Eqs. (2) and (3). Where, Γ() is the gamma function. $κ_{o} = \frac{2 π}{l_{o}}$ , $κ_{i} = \frac{2 π}{l_{i}}$ . l_o and l_i represent the outer scale and inner scale of the solar wind turbulence, respectively.

By substituting Eq. (6) into Eq. (5), the spatial spectrum of irregularities of the electron density can be represented by the following equation when 3<p<4.

Φ_{N} (κ, z) = {(2 π)}^{- 3 / 2} \frac{(p - 3) Γ (\frac{p}{2})}{Γ (\frac{p - 1}{2})} κ_{o}^{p - 3} 〈 δ N_{e}^{2} 〉 κ^{- p} .

2.3. Amplitude fluctuations variance

As the amplitude fluctuations are normally distributed around their mean levels, the signal variance χ² has been an essential parameter for characterizing their variability. In the weak fluctuation regime, the log-amplitude fluctuations for optical waves propagating through the coronal solar wind turbulence can be obtained based on the Rytov theory and can be expressed as [16]

χ = - \frac{k^{2}}{4 π} ∭ d^{3} ρ δ ε (r) \frac{1}{z} cos (\frac{k r^{2}}{2 z}),

where

k = \frac{2 π}{λ}

. According to Fig. 1, we have d³ρ = dzdϕdr. It is noteworthy that the unit of log-amplitude in Eq. (8) is Np and can be turn into dB by the following equation.

χ (d B) = 4.34 χ (N p) .

We define the log-amplitude as amplitude fluctuations for simplicity. Therefore, the variance of amplitude fluctuations can be written as

〈 χ^{2} 〉 = \frac{k^{4}}{16 π^{2}} ∭ d^{3} ρ ∭ d^{3} ρ^{'} \frac{1}{z_{1} z_{2}} cos (\frac{k {r_{1}}^{2}}{2 z_{1}}) cos (\frac{k {r_{2}}^{2}}{2 z_{2}}) 〈 δ ε (ρ) δ ε (ρ^{'}) 〉 .

This formula can be further recast under the derivation in Appendix A.

\begin{array}{r} 〈 χ^{2} 〉 = \frac{1}{2 π} {r_{e}}^{2} λ^{4} k^{2} \int_{0}^{\infty} d z_{1} \int_{0}^{\infty} d z_{2} sin (\frac{κ^{2} z_{1} \sin^{2} ψ}{2 k}) sin (\frac{κ^{2} z_{2} \sin^{2} ψ}{2 k}) \\ \cdot exp [i κ (z_{1} - z_{2}) cos ψ] \int_{0}^{\infty} κ^{2} d κ \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, \frac{z_{1} + z_{2}}{2}) . \end{array}

Note that, the exponential in Eq. (11) changes quickly and will disappear when z₁ is close to z₂. According to the following relationship,

\int_{0}^{R} d x \int_{0}^{R} d x^{'} f (x) g (x^{'}) exp [i κ (x - x^{'})] = 2 π δ (κ) \int_{0}^{R} d x f (x) g (x),

the double integration about z₁ and z₂ can be simplified as

\int_{0}^{\infty} d z_{1} \int_{0}^{\infty} d z_{2} \exp [i κ (z_{1} - z_{2}) cos ψ] F (z_{1}, z_{2}) = 2 π δ (κ cos ψ) \int_{0}^{\infty} d z F (z, z),

where

z = \frac{z_{1} + z_{2}}{2}

. Inserting Eq. (13) into Eq. (11), we obtain

〈 χ^{2} 〉 = {r_{e}}^{2} λ^{4} k^{2} \int_{0}^{\infty} κ^{2} d κ \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, z) δ (κ cos ψ) \int_{0}^{\infty} d z {sin}^{2} (\frac{z κ^{2} \sin^{2} ψ}{2 k}) .

Using the Delta function, $\int_{0}^{π} d ψ f (ψ) δ (κ cos ψ) = \frac{1}{κ} f (\frac{π}{2})$ , we can get a further simplification of the variance of the amplitude fluctuations as

〈 χ^{2} 〉 = 16 π^{4} {r_{e}}^{2} k^{- 2} \int_{0}^{\infty} κ d κ \int_{0}^{\infty} d z {sin}^{2} (\frac{z κ^{2}}{2 k}) Φ_{N} (κ, z) .

This issue is now converted to the double integration over the wave vector κ and the link path z. Considering the triangular transformation, ${sin}^{2} (\frac{z κ^{2}}{2 k}) = \frac{1}{2} [1 - cos (\frac{z κ^{2}}{k})]$ , the variance of amplitude fluctuations can be obtained by substituting the solar wind turbulence spectrum, Eq. (7), into Eq. (15).

〈 χ^{2} 〉 = \frac{{r_{e}}^{2} k^{- 2} {(2 π)}^{5 / 2}}{2} \frac{(p - 3) Γ (\frac{p}{2})}{Γ (\frac{p - 1}{2})} L 〈 δ N_{e}^{2} 〉 κ_{o}^{p - 3} \int_{0}^{\infty} κ^{1 - p} d κ [1 - \frac{\sin (\frac{L κ^{2}}{k})}{\frac{L κ^{2}}{k}}] .

In light of the following integral equation [24],

\int_{0}^{\infty} \frac{1}{x^{μ}} (1 - \frac{sin (a x)}{a x}) d x = a^{μ - 1} \frac{- π}{2 Γ (1 + μ) cos (\frac{π}{2} μ)}, 1 < μ < 3, a > 0 .

The variance of the amplitude fluctuations is of this form,

〈 χ^{2} 〉 = \frac{{r_{e}}^{2} {(2 π)}^{11 / 2 - p}}{8} \frac{(p - 3) Γ (\frac{p}{2})}{Γ (\frac{p - 1}{2}) Γ (1 + \frac{p}{2})} 〈 δ N_{e}^{2} 〉 {l_{o}}^{3 - p} L^{\frac{p}{2}} k^{- \frac{p}{2} - 1} \frac{- π}{cos (\frac{π}{4} p)} .

According to the relationship between the solar wind fluctuations and solar wind density in Eq. (3), we can finally obtain the close-form variance of the amplitude fluctuations as,

〈 χ^{2} 〉 = - \frac{(p - 3) Γ (\frac{p}{2}) {r_{e}}^{2} {(2 π)}^{11 / 2 - p} π}{8 Γ (\frac{p - 1}{2}) Γ (1 + \frac{p}{2})} η^{2} 〈 {N_{e}}^{2} 〉 {l_{o}}^{3 - p} L^{\frac{p}{2}} k^{- \frac{p}{2} - 1} sec (\frac{π}{4} p) .

It is necessary to be noticed that Eq. (19) is in concise close-form. The amplitude fluctuations of optical waves in coronal turbulence are with the dependence of the optical wavelength, λ, link distance, L, non-Kolmogorov spectral index, p, turbulence outer scale, l_o, and the relative density fluctuation factor η. Note that, our proposed amplitude fluctuation model is different from the other models not only because the generalized modified coronal turbulence spectrum model was applied in this paper, but also because the derived model has a more complicated relationship with these parameters. It can be found that the amplitude fluctuations rapidly decrease with increasing turbulence outer scale at a rate of $l_{o}^{3 - p}$ and also decrease with decreasing wavelength at a rate of λ^p/2+1. Amplitude fluctuations are further enlarged at a rate of L^p/2 as the communication link distance becomes larger. Since we consider about the optical waves propagation in the non-Kolmogorov turbulence channels with 3<p<4, therefore, these tendencies are more accuracy for calculation of the amplitude fluctuations.

In addition, the scintillation index is another parameter used for estimating the amplitude fluctuations. It is defined as the intensity standard deviation normalized to the average received intensity, $m = \sqrt{(〈 I^{2} 〉 - {〈 I 〉}^{2}) / {〈 I 〉}^{2}}$ , where I is the received intensity. Many studies have proved that [16,29],

m^{2} = 4 〈 χ^{2} 〉 .

Therefore, the scintillation index m can be further derived by Eq. (19).

2.4. Analysis of the average BER performance

Here, the BER performance of the free space optical system induced by the irregularities plasma on amplitude fluctuations when optical waves propagation through weak non-Kolmogorov coronal turbulence channels is further investigated. For the on-off keying modulation FSO communication system, the average BER has been well developed by many studies and can be written as [7,9,28,29]:

〈 BER 〉 = \frac{1}{2} \int_{0}^{\infty} f_{I} (u) erfc (\frac{〈 {SNR}_{0} 〉}{2 \sqrt{2}} u) d u,

where erfc (x) is the complementary error function,

erfc (x) = \frac{2}{\sqrt{π}} \int_{x}^{\infty} exp (- u^{2}) d u

. SNR₀ is the average signal to noise ratio in absence of turbulence. f_I (u) denotes the probalility density function. Without loss of generality, f_I (u) is modeled to be the log-normal distribution with unit mean under the weaken scintillation.

f_{I} (u) = \frac{1}{u \sqrt{2 π m}} exp {- \frac{{[ln (u) + \frac{1}{2} m]}^{2}}{2 m}} .

Using the following Gauss-Hermite quadrature integration approximation

\int_{- \infty}^{\infty} f (x) exp (- x^{2}) d x ≅ \sum_{i = 1}^{n} w_{i} f (x_{i})

where

{x_{i} |}_{i = 1}^{n}

is the zeros of an nth-order Hermite polynomial He_n,

{w_{i} |}_{i = 1}^{n}

is the associated weight factors and is given by

w_{i} = \frac{2^{n - 1} n! x_{i}}{n^{2} {[{He}_{n} (x_{i})]}^{2}}

[27].

By substituting Eqs. (22) and (23) into Eq. (21), the average BER can be accurately approximated as

BER ≅ \frac{1}{π} \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{π}} \sum_{i = 1}^{n} w_{i} exp {- \frac{{SNR}_{0} exp (2 \sqrt{2 m} x_{i} - m)}{2 {sin}^{2} θ}} d θ

Therefore, an efficiently average BER formula can be further recast as

BER ≅ \frac{1}{2 \sqrt{π}} \sum_{i = 1}^{n} w_{i} erfc [\frac{1}{\sqrt{2}} {SNR}_{0} exp (\sqrt{2 m} x_{i} - \frac{m}{2})]

3. Simulation and discussion

In this section, simulations are conducted to illustrate the effects of different parameters on the variance of amplitude fluctuations. As we can see in Eqs. (1) and (19), the variance of amplitude fluctuations depends on the length of the communication link, which is constituted by SEP angle and SPE angle. Therefore, we first investigate the influence of SEP angle and SPE angle on the amplitude fluctuations. The simulation parameters are set as, p=11/3, λ=1550nm, l_o=5 × 10⁷m, respectively. The relative solar wind density fluctuation factor is fixed as η=5% unless otherwise specified. In this paper, the distance between the Earth and the Sun, L_se= 1.5 × 10¹¹m, and the distance between the spacecraft and the Sun, L_sp, are assurmed as constant for convenience. We also let L_sp=1.5L_se as appropriate for Mars’ orbit during superior solar conjunction.

Figure 2 shows the variance of the amplitude fluctuations and the color bar on the right side represents the normalized value. As shown above, with the decrease of the SEP angle and SPE angle, the variance of the amplitude fluctuations increases. This phenomenon can be interpreted directly from the function of link distance in Eq. (1). When SEP angle and SPE angle decrease, L raises, so that the variance of amplitude fluctuations also increases. It can also be explained from this point of view, the communication link pass through a more irregularity solar wind turbulence with the decrease of SEP angle and SPE angle, and as a result, the optical waves suffer more serious influence on amplitude fluctuations. In addition, the SEP angle induces more obviously effect on the amplitude fluctuations than the SPE angle. Therefore, we set SPE angle as a constant (β=0.5 degree) in the following simulations for the sake of simplicity.

Fig. 2 Normalized amplitude fluctuations dependence on both SEP angle SPE angle.

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In the following simulations, the effects of the outer scale of the turbulence, the non-Kolmogorov spectral index, and the optical wavelength on the amplitude fluctuations are analyzed as shown in Fig. 3, Fig. 4, and Fig. 5, respectively. In order to avoid mutual interferences between these parameters, only one parameter is varied at a time. The key parameters used in the simulations are listed in Table 3. Note that, with the limitations imposed by experimental measurements, it is very difficult to obtain the precise outer scale data, therefore, all of the parameters set here are a compromise between the assumption in [25] and the experimental results reported in [13].

Fig. 3 Normalized amplitude fluctuations dependence on SEP angle for various outer scales.

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Fig. 4 Normalized amplitude fluctuations dependence on SEP angle for various non-Kolmogorov spectral indices.

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Fig. 5 Normalized amplitude fluctuations dependence on SEP angle for various wavelengths.

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Table 1. Parameters used in the calculations.

View Table

Figure 3 shows the impact of the outer scale on the amplitude fluctuations of optical waves propagation in the coronal plasma by changing SEP angle from 0.5 degree to 0.8 degree. As we can see, the amplitude fluctuations decrease with the increase of SEP angle under different outer scales. This result further certificates the conclusion in Fig. 2. Simulation results also show that the turbulence outer scale has a significantly restrictive impact on the amplitude fluctuations. As the outer scale increases, the normalized amplitude fluctuations decrease. It is interesting that the amplitude fluctuations decrease sharply when the outer scale increases from 1×10⁷ m to 3×10⁷ m, and this tendency slows down when the outer scale increases from 4×10⁷ m to 7×10⁷ m. This phenomenon can be explained directly because the focusing effect is more prominent as the outer scale increases.

As depicted in Fig. 4, for constant SEP angle and SPE angle, the amplitude fluctuations decrease monotonically with the increase of spectral index. This is consistent with results of the amplitude fluctuations of radio wave propagation in ionosphere plasma [26] and optical wave propagation in atmosphere [10]. The variance of amplitude fluctuations goes to zero when the spectral index increase to 3.9. This is because the non-Kolmogorov spatial spectrum of the solar wind density decreases with the increase of spectral index. According to the astronomical observation, the spectral index is fluctuated between 3 and 4 when the SEP angle is less than 50 degree. It is noteworthy that there is no explicit correlation between the spectral index value and the SEP angle. Therefore, our proposed amplitude fluctuations model needs to be further studied in consideration of the distribution of the non-Kolmogorov spectral index.

Figure 5 demonstrates the evolution of the amplitude fluctuations of optical waves in coronal solar wind turbulence as a function of SEP angle with different wavelengths. It is obvious that the amplitude fluctuations are sensitive to wavelength and decrease with decreasing wavelength. This phenomenon is reasonable, because the optical wave with high frequency can mitigate the influence of solar wind turbulence more effectively. As a result, we can use high frequency optical waves to eliminate the effect of solar scintillation in the future deep space communication.

Figure 6 explores the influences of solar wind density relative fluctuation factor, η, on the amplitude fluctuations of the optical communication link during superior solar conjunction. The fluctuation factor is set as η=5%, 10%, 15%, and 20%, respectively. It is clear that the amplitude fluctuations are affected by turbulence fluctuation factor and increase with the increase of density fluctuation factor value. For the constant SEP angle, α=0.55 degree, the amplitude fluctuations reach 1.4 × 10⁻² dB when η=10%, and its penalty rises to 1.6 × 10⁻² dB for the case of η=15%. In addition, the amplitude fluctuations decrease more sharply with larger density fluctuation factor. For instance, when the SEP angle decreases from α=0.55 degree to α=0.65 degree, the amplitude fluctuations decrease from 1.9 × 10⁻² dB to 1.5 × 10⁻²dB as η=20% whereas from 1 × 10⁻² dB to 5 × 10⁻³dB as η=5%. This indicats that the larger turbulence relative fluctuation factor has a more evidently impact on amplitude fluctuations at small SEP angle. These variations trend reveals that the coronal solar wind turbulence’s effect on the optical waves propagation varies with the SEP angle and follows the rules derived in this study.

Fig. 6 Normalized amplitude fluctuations dependence on SEP angle for various density relative fluctuation factors.

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The effects of various parameters on amplitude fluctuations have been analyzed in the above simulations. Fig. 7 shows the average BER of the FSO communication system with optical waves passing through the non-Kolmogorov coronal turbulence channels under various outer scales, spectral indices, wavelengths, and density relative fluctuations factors. For the sake of conciseness and without loss of generality, we fix SEP angle and SEP angle are 0.4 degree and 0.5 degree, respectively. Other key parameters are set in the range of Table 3.

Fig. 7 BER performance against normalized SNR₀ for optical waves in weak coronal turbulence channel under different (a) outer scales, (b) spectral indices, (c) wavelengths, and (d) density fluctuation factors.

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It is shown in Fig. 7 (a) and (b) that the change in outer scale, l_o, and the spectral index, p, has a rather profound impact on the BER performance. The BER decreases with the increase of l_o and p. For instance, a receiver operating at SNR₀=18 dB when l_o = 5 × 10⁸ m will require SNR₀=22.5 dB when l_o decreases to 5 × 10⁷ m. Thus, it is an increase of about 4.5 dB in the required SNR₀ at the receiver. This is because that, for high values of l_o and p, e.g., 5×10⁸m and 3.8, the amplitude fluctuations are low as depicted in Fig. 3 and Fig. 4, respectively. Therefore, the optical communication system is less degraded in coronal turbulence.

Given the profound impact of the wavelength, λ, on amplitude fluctuations performance, the BER is further plotted versus λ in Fig. 7 (c) for various levels of SNR₀. As expected, the decrease in the optical wavelength results in an decrease in the required signal level SNR₀ to achieve the same BER performance. Therefore, decreasing the wavelength is a well alternative to enhance the BER performance of the future deep space FSO communication system in coronal turbulence environment. It is clear that the BER is affect by the density fluctuations factors, η, as shown in Fig. 7 (d). The SNR₀ for reaching the benchmark BER increases with the increasing of η. For instance, a normalized SNR₀ of about 18 dB is needed to achieve the BER of 10⁻⁹ in coronal turbulence with η = 5%, while it penalty rises to 23 dB when η = 20%.

In order to demonstrate the significance of our proposed model in the future deep space optical communication system, we further show the application of the derived closed-form amplitude fluctuations in an imagined Earth-Mars optical communication link during superior solar conjunction. Fig. 8 describes the simulation results of scintillation index as a function of the wavelength and the BER performance as the function of the scintillation index. The variation of the scintillation index curves are more intuitive in Fig. 8 (a). When the SEP angle is 0.3 degree, the scintillation index is 0.18, 0.24, and 0.41 when the wavelength is 850 nm, 1060 nm, and 1500 nm, respectively. The scintillation index not only decreases with the increasing of the SEP angle but also decreases with the decreasing of the wavelength. The simulation results in Fig. 8 (b) displays that at BER=10⁻⁹, which is usual requirement for deep-space optical communication link, the link performance promotes 2.4 dB when m decreases from 0.2 to 0.1, whereas 2.9 dB when m decreases from 0.3 to 0.1. Therefore, the derived amplitude fluctuations model not only can be used to forecast the variation of the scintillation index for optical waves propagation through coronal turbulence during superior solar conjunction, but also provide a way to predict the link performance.

Fig. 8 (a) The scintillation index as a function of wavelength and (b) the BER performance against normalized SNR₀ under different scintillation index.

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4. Conclusion

In summary, the variance of amplitude fluctuations for optical waves propagation in the coronal turbulence during superior solar conjunction has been investigated by establishing the coronal irregularity model and the non-Kolmogorov coronal turbulent spectrum model. Both the optical wavelength, non-Kolmogorov spectral index, turbulence outer scale, relative solar wind density fluctuation factor, and the link distance impose large influence on the close-form amplitude fluctuations expression. It is found that the variance of amplitude fluctuations decrease along the increase of turbulence outer scale and spectral index. As the SEP angle and SPE angle decrease, the optical waves encounter stronger solar wind turbulence, therefore, it will induce amplitude fluctuations increase gradually. This phenomenon also emerges when the relative solar wind density fluctuation factor becomes larger. Moreover, the coronal effect on amplitude fluctuations will be alleviated with the increase of the optical wavelength.

Furthermore, we investigate the BER performance of the optical waves propagation in non-Kolmogorov coronal turbulence channels by the derived amplitude fluctuations model. It is concluded all the turbulence outer scale, non-Kolmogorov spectral index, optical wavelength, and the relative solar wind density fluctuation factor induce severely impacts on the BER performance. The BER decreases with the increase of the outer scale and the spectral index, while with the decrease of the wavelength and the density fluctuation factor. These theoretical results presented in this paper are helpful for us to investigate the optical waves propagation through coronal turbulence and finally are beneficial in the design of deep space FSO communication system.

A. Appendix

By substituting Eq. (4) into Eq. (10), the amplitude fluctuations for optical waves propagation in coronal turbulence can be written as

\begin{array}{l} 〈 χ^{2} 〉 = & \frac{{r_{e}}^{2} λ^{4} k^{4}}{16 π^{4}} \int_{0}^{\infty} d z_{1} \int_{0}^{2 π} d ϕ_{1} \int_{0}^{\infty} r_{1} d r_{1} [\frac{1}{z_{1}} cos (\frac{k {r_{1}}^{2}}{2 z_{1}})] \\ \cdot \int_{0}^{\infty} d z_{2} \int_{0}^{2 π} d ϕ_{2} \int_{0}^{\infty} r_{2} d r_{2} [\frac{1}{z_{2}} cos (\frac{k {r_{2}}^{2}}{2 z_{2}})] \cdot 〈 δ N_{e} (ρ) δ N_{e} (ρ^{'}) 〉 . \end{array}

According to the Wiener-Khinchin theorem, the spatial covariance of the plasma density irregularities can be represented by the turbulent spectrum of the solar wind turbulence.

〈 δ N_{e} (ρ) δ N_{e} (ρ^{'}) 〉 = ∭ d^{3} κ Φ_{N} (κ) exp [i κ \cdot (ρ - ρ^{'})] .

In light of the coordinate transformation in Fig. 1, we have

\begin{array}{l} κ \cdot (ρ - ρ^{'}) & = (κ_{z} i_{z} + κ_{r} cos ω i_{x} + κ_{r} sin ω i_{y}) \\ \cdot [\begin{array}{l} (z_{1} - z_{2}) i_{z} + (r_{1} cos ϕ_{1} - r_{2} cos ϕ_{2}) i_{x} \\ + (r_{1} sin ϕ_{1} - r_{2} sin ϕ_{2}) i_{y} \end{array}] \\ = κ_{z} (z_{1} - z_{2}) + κ_{r} cos ω (r_{1} cos ϕ_{1} - r_{2} cos ϕ_{2}) \\ + κ_{r} sin ω (r_{1} sin ϕ_{1} - r_{2} sin ϕ_{2}) \end{array},

By substituting Eq. (28) into Eq. (27), the following equation can be obtained.

\begin{array}{l} 〈 δ N_{e} (ρ) δ N_{e} (ρ^{'}) 〉 = & \int_{0}^{\infty} κ^{2} d k \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, \frac{z_{1} + z_{2}}{2}) \int_{0}^{2 π} d ω \\ \cdot exp {\begin{cases} i [κ_{z} z_{1} + κ_{r} r_{1} cos (ω - ϕ_{1})] \\ - i [κ_{z} z_{2} + κ_{r} r_{2} cos (ω - ϕ_{2})] \end{cases}} \end{array} .

Inserting Eq. (29) into (27) and making some simplification, we have

\begin{array}{l} 〈 χ^{2} 〉 = & \frac{{r_{e}}^{2} λ^{4}}{16 π^{4}} k^{4} \int_{0}^{\infty} \frac{d z_{1}}{z_{1}} \int_{0}^{\infty} \frac{d z_{2}}{z_{2}} \int_{0}^{\infty} r_{1} d r_{1} \int_{0}^{\infty} r_{2} d r_{2} cos (\frac{k {r_{1}}^{2}}{2 z_{1}}) cos (\frac{k {r_{2}}^{2}}{2 z_{2}}) \\ \cdot \int_{0}^{\infty} κ^{2} d κ \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, \frac{z_{1} + z_{2}}{2}) \int_{0}^{2 π} d ϕ_{1} \int_{0}^{2 π} d ϕ_{2} \\ \cdot \int_{0}^{2 π} d ω \cdot exp {\begin{cases} i [κ_{z} z_{1} + κ_{r} r_{1} cos (ω - ϕ_{1})] \\ - i [κ_{z} z_{2} + κ_{r} r_{2} cos (ω - ϕ_{2})] \end{cases}} \end{array},

According to coordinate relationship in Fig. 1, κ_z = κ cos ψ, k_r = κsinψ, therefore,

\begin{array}{l} 〈 χ^{2} 〉 = & \frac{{r_{e}}^{2} λ^{4}}{16 π^{4}} k^{4} \int_{0}^{\infty} \frac{d z_{1}}{z_{1}} \int_{0}^{\infty} \frac{d z_{2}}{z_{2}} \int_{0}^{\infty} r_{1} d r_{1} \int_{0}^{\infty} r_{2} d r_{2} cos (\frac{k {r_{1}}^{2}}{2 z_{1}}) cos (\frac{k {r_{2}}^{2}}{2 z_{2}}) \\ \cdot \exp [i κ (z_{1} - z_{2}) cos ψ] \int_{0}^{\infty} κ^{2} d κ \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, \frac{z_{1} + z_{2}}{2}) \\ \cdot \int_{0}^{2 π} d ω \int_{0}^{2 π} d ϕ_{1} exp [i κ r_{1} \sin ψ cos (ω - ϕ_{1})] \cdot \\ \cdot \int_{0}^{2 π} d ϕ_{2} exp [- i κ r_{2} \sin ψ cos (ω - ϕ_{2})] \end{array} .

In light of the following integrations in [27],

\int_{0}^{2 π} exp [i (a \sin x + b cos x)] d x = 2 π J_{0} (\sqrt{a^{2} + b^{2}}),

\int_{0}^{\infty} J_{0} (a x) cos (b x^{2}) x d x = \frac{1}{2 b} sin (\frac{a^{2}}{4 b}),

The variance of amplitude fluctuations of optical waves in coronal plasma can be expressed as

\begin{array}{l} 〈 χ^{2} 〉 = & \frac{1}{2 π} {r_{e}}^{2} λ^{4} k^{2} \int_{0}^{\infty} d z_{1} \int_{0}^{\infty} d z_{2} sin (\frac{κ^{2} z_{1} \sin^{2} ψ}{2 k}) sin (\frac{κ^{2} z_{2} \sin^{2} ψ}{2 k}) \\ \cdot exp [i κ (z_{1} - z_{2}) cos ψ] \int_{0}^{\infty} κ^{2} d κ \int_{0}^{π} d ψ sin ψ Φ_{N} (κ, \frac{z_{1} + z_{2}}{2}) \end{array} .

Funding

Shandong Space Innovation Fund (No. 2013JJ008).

References and links

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Fig.	Outer scale (m)	Spectral index	Wavelength (nm)
3	1 × 10⁷ < l_o < 7 × 10⁷	p=3.5	λ=1550
4	l_o=5 × 10⁷	3.1< p <3.9	λ=1550
5	l_o=5 × 10⁷	p=3.5	600< λ <1550

Amplitude fluctuations for optical waves propagation through non-Kolmogorov coronal solar wind turbulence channels

Abstract

1. Introduction

2. Theoretical model

2.1. Solar wind density model and its irregularity model

2.2. Generalized modified coronal turbulence spectrum model

2.3. Amplitude fluctuations variance

2.4. Analysis of the average BER performance

3. Simulation and discussion

4. Conclusion

A. Appendix

Funding

References and links

Cited By

Figures (8)

Tables (1)

Equations (34)

Optics Express