Phase fluctuations-induced bit error ratio of deep-space optical communication systems during superior solar conjunction

Guodong Sun; Guanjun Xu; Guanjun Xu; Yanli Shao; Yanli Shao; Qinyu Zhang; Zhaohui Song

doi:10.1364/OE.511003

1. Introduction

Due to the advancement in aerospace technology and the broad prospect of space resource utilization, deep-space exploration has captured the attention of numerous nations and space agencies. However, radio frequency (RF)-based deep-space communication, the only bridge between the Earth and the probes, cannot meet the requirements of high transfer rates for large amounts of high-resolution images/videos [1,2]. Fortunately, optical communication using laser beam offers a new alternative due to its advantages of broader bandwidth, higher transmission rate, and lower power consumption compared to RF-based communication [3]. Nevertheless, the deep-space environment exhibits a high plasma density caused by frequent solar activities. Optical waves passing through this environment experience fast amplitude and phase fluctuations due to solar wind irregularities, which result in beam attenuation, broadening, and flicker. These adverse effects further cause difficulties in recovering the phase and frequency, and technical issues in beam acquisition, tracking, and pointing [1]. Therefore, deep-space optical communication is a challenging topic that requires urgent developments.

In previous studies, the effects of different turbulent environment, such as marine, atmospheric, and ionospheric turbulence, on the radio and optical waves have been detailed scrutinized [4–6]. Lee et al. proposed a novel atmospheric scintillation index considering the aperture averaging effect under the Kolmogorov turbulence spectrum [7]. In [8], Cui et al. derived analytical expressions for the irradiance scintillation temporal power spectra, considering infrared and optical plane and spherical waves propagating through weak marine turbulence. They demonstrated that humidity affects amplitude scintillation, compared to atmospheric turbulence. Gudimetla et al. derived the expression for the log-amplitude correlation function for a spherical wave passing through anisotropic non-Kolmogorov atmospheric turbulence [9]. Cheng et al. developed a non-Kolmogorov atmospheric turbulence spatial power spectrum model and scintillation index for Gaussian-beam waves propagating through weak maritime atmospheric turbulence. Moreover, they analyzed the average bit error rate (BER) of the free-space optical (FSO) system under on-off-keying (OOK) modulation based on the derived scintillation index [10]. Based on the plasma power spectrum (which follows a power law), Tereshchenko et al. derived the logarithmic relative amplitude fluctuations of very high-frequency (VHF) waves passing through the ionosphere [11].

Pioneering works have analyzed the influence of ionospheric and atmospheric turbulence on RF and FSO communication systems. However, for deep-space optical communication, due to coronal plasma irregularities, the signals suffer from beam wander, intensity scintillation, angle-of-arrival fluctuations, and phase fluctuations, leading to rapid quality deterioration with high BER. More importantly, the communication link may even be interrupted when the distance between the Sun and the communication path is minimal. Ho et al. developed a complete theoretical approach to derive the variance of angular fluctuations of radio signals, during superior solar conjunction. They derived an angle-of-arrival fluctuations power spectrum, based on the power spectrum of the phase scintillations [12]. Morabito et al. analyzed Cassini data, acquired during superior solar conjunction, and proposed models to define intensity scintillations, spectral broadening, and phase scintillations using a Kolmogorov turbulence spectrum [13]. Efimov et al. developed a scintillation index model to reveal relationships between radio wave intensity scintillation and radially dependent variables, such as outer scale and spectral index, under a non-Kolmogorov turbulence spectrum [14].

In addition, several studies have investigated the effects of some undesired factors, such as absorption, scattering, and turbulence, on the communication performance, such as average BER, channel capacity, and outage probability. Lim et al. derived a closed-form expression for the average BER of FSO communication systems under gamma-gamma turbulence channels [15]. Navidpour et al. investigated the effects of spatial diversity on the BER performance of both plane and spherical waves over log-normal atmospheric turbulence channels [16]. Uysal et al. provided error performance bounds for FSO communication systems under strong atmospheric turbulence, assuming the channel to be a temporally correlated K channel, and employed an error control algorithm to alleviate turbulence impairments [17]. Ferdinandov et al. developed a novel comprehensive model to measure the combined influence of multiple stochastic factors, such as quantum noises, mechanical vibrations of the transmitting laser antenna, and atmospheric turbulence, on the BER of FSO communication systems [18].

As discussed above, many works have analyzed the impacts of turbulence on optical wave propagation. However, studies focusing on FSO communication under coronal plasma turbulence are still deficient, owing to the complicated implementation of optical communication systems and the largely unexplored deep-space environment [1]. Moreover, most studies merely focused on the scintillation index model derived from the expression for the amplitude fluctuations [19]. Utilizing the Rytov approximation, we previously analyzed the amplitude and angle-of-arrival fluctuations of optical waves passing through the non-Kolmogorov coronal turbulence [20,21]. However, the phase fluctuations of such waves and their influence on deep-space optical communication are yet to be discussed.

To fill this gap, we derive the variance of the phase fluctuations employing the Rytov theory and non-Kolmogorov spatial power spectrum model. Furthermore, we establish the phase fluctuations-induced average BER model under binary phase shift keying (BPSK) modulation. The framework of this paper is as follows: Section 2 introduces the background model for the solar corona. The phase fluctuations model is subsequently established as a function of the spatial and environmental variables in Section 3. Section 4 further derives the average BER under BPSK modulation in the weak turbulence regime. The simulations of the theoretical model are discussed in Section 5. Finally, Section 6 presents the conclusions.

2. Theoretical model

2.1 Geometrical model for superior solar conjunction

To reveal the influence of different parameters, such as the heliocentric distance, outer scale, spectral index, relative plasma density fluctuation factor, and wavelength, on the phase fluctuations and system error performance, we first establish the geometrical diagram of the deep-space optical communication system, during superior solar conjunction. As shown in Fig. 1, the planet orbiting around the Sun rotates to the opposite side of the Sun from the Earth’s perspective, creating an alignment where the Sun lies between the Earth and the planet. This phenomenon is of considerable significance in deep-space communication systems, as the solar wind from the Sun’s corona can potentially disrupt optical signals during superior solar conjunction, posing challenges for communication with spacecraft revolving around the involved planet. The distance between the Sun and the communication path is the heliocentric distance, $r$, represented in units of the solar radius, $R\rm {_{sun}}$, for simplicity. The Sun-Earth-Probe (SEP) angle, $\alpha$, and Sun-Probe-Earth (SPE) angle, $\beta$, help determine the link distance, $L$, between the Earth and the probe, expressed as

(1)$$L = {L_{se}}\cos \alpha + {L_{sp}}\cos \beta,$$

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. In addition, both $\alpha$ and $\beta$ tend to $0^{\circ }$ during superior solar conjunction.

2.2 Solar wind density model

Based on Taylor’s frozen-flow theory, the coronal turbulence freezes during a short measurement interval and the plasma is approximately stationary [19]. The optical wave transmission speed far exceeds the solar wind velocity, and therefore we only consider the spatial solar wind plasma density model, in this study. Since spatial and temporal permittivity fluctuations have significant impacts on the phase fluctuations, we only consider the former for simplicity.

Most previous studies modeled empirical electron density for the coronal plasma as a function of $r$, assuming either spherical or azimuthal symmetry for the corona and symmetry along the equatorial plane of the Sun [22,23]. To avoid this deficiency, we applied a more reasonable electron density derived by Guhathakurta et al., quantitatively using both $r$ and the angular distance of a point from the current sheet in a heliomagnetic coordinate system (heliomagnetic latitude), $\theta _{mg}$, defined as [24]

(2)$${\theta_{mg}} = {\sin ^{ - 1}}\left[ {\sin \theta \cos \alpha ' - \cos \theta \sin \alpha '\sin \left( {\phi-{\phi _0}} \right)} \right],$$

where $\theta$ and $\phi$ are the heliographic latitude and longitude, respectively, $\alpha '$ is the tilt angle between the dipole axis and the rotation axis, and $\phi _{0}$ is the angle between the heliomagnetic and heliographic equators. This model applies to the solar minimum period in a weak turbulence regime. Note that this work mainly considers the solar wind irregularities model during the solar minimum period. Therefore, the impacts of extreme factors, such as solar flares, high-speed solar wind streams, and coronal mass ejections, on the rapid variations in solar wind density during the solar maximum period are ignored. The electron number density can then be expressed as [24]

(3)$$N\left( {r,{\theta _{mg}}} \right) = {N_p}\left( r \right) + \left[ {{N_{cs}}\left( r \right) - {N_p}\left( r \right)} \right]{e^{ - \frac{{\theta_{mg}^{2}}}{{{w^2}\left( r \right)}}}},$$

where ${N_p}(r)$ and ${N_{cs}}(r)$ are electron densities at the current sheet and poles, respectively, and $w(r)$ is the half angular width of the current sheet. These variables can be represented as [24]

(4a)$$ {N_{cs}}\left( r \right) = \sum_{i = 1}^3 {{a_i}{r^{ - {d_i}}}}$$

(4b)$$ {N_p}\left( r \right) = \sum_{i = 1}^3 {{c_i}{r^{ - {d_i}}}}$$

(4c)$$w\left( r \right) = \sum_{i = 1}^3 {{\gamma _i}{r^{ - {\delta _i}}}}, $$

where $a_{i}$, $c_{i}$, $d_{i}$, $\gamma _{i}$, and $\delta _{i}$ are constants employed in the calculations. The values assumed for these parameters are shown in Table 1.

Table 1. Key parameters for the solar wind density model

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The spatially dependent permittivity fluctuations, $\delta \varepsilon$, can be expressed as [12]

(5)$$\delta \varepsilon \left( r \right) ={-} \frac{{{r_e}{\lambda ^2}}}{\pi }\delta {N_e}\left( r \right),$$

where $r_e$ is the classical electron radius, $\lambda$ is the wavelength of the optical signal and $\delta N_e$ is the electron number density fluctuation (irregularities) defined as [25]

(6)$$\delta {N_e}\left( r \right) = \eta {N_e}\left( r \right),$$

where $\eta$ is the relative fluctuation factor, revealing the fluctuation extent of electron density (0.06 < $\eta$ < 0.15 for 16$R\rm {_{sun}}$ < $r$ < 26$R\rm {_{sun}}$) [26].

2.3 Non-Kolmogorov spatial power spectrum

Different to the Kolmogorov power spectrum (with spectral index p = 11/3), the non-Kolmogorov power spectrum is defined as [27]

(7)$${\phi _N}\left( {\kappa ,z} \right) = \frac{{\left( {p - 3} \right)\Gamma \left( {\frac{p}{2}} \right)}}{{{{\left( {2\pi } \right)}^{\frac{3}{2}}}\Gamma \left( {\frac{{p - 1}}{2}} \right){\kappa ^p}}}\kappa _o^{p - 3}\left\langle {\delta N_e^2} \right\rangle,$$

where 3 < p < 4 is the generalized spectral index, $\Gamma (\cdot )$ is the gamma function, $\kappa$ is the magnitude of the spatial frequency vector, $\kappa _{o} = 2\pi /l_{\rm {o}}$, $l_{\rm {o}}$ is the turbulence outer scale.

3. Variance of phase fluctuations

Phase fluctuations are key factors for measuring the uncertainty of the received signal. Moreover, the variance of the phase fluctuations can help us understand the adverse effects of solar wind irregularities on wave propagation under different variables, such as the heliocentric distance, outer scale, spectral index, relative plasma density fluctuation factor, and wavelength. Following previous studies [19,28], we expand the surrogate function adopted in the Rytov theory to solve the wave equation in terms of a series as

(8)$$\Psi \left( {\vec r } \right) = {\Psi _1}\left( {\vec r } \right) + {\Psi _2}\left( {\vec r } \right) + {\Psi _3}\left( {\vec r } \right) + {\Psi _4}\left( {\vec r } \right) +{\cdot}{\cdot} \cdot{+} {\Psi _n}\left( {\vec r } \right).$$

The analytical expression for the phase fluctuations of optical wave transmission in a deep-space weak turbulence channel can then be expressed as [19]

(9)$$\varphi ={-} \frac{{{k^2}}}{{4\pi }}\int_0^\infty {\frac{1}{z}{\rm{d}}z\int_0^{2\pi } {{\rm{d}}\phi \int_0^\infty {\delta \varepsilon \left( {\vec {r} ,t} \right)\sin \left( {k\frac{{{r^2}}}{{2z}}} \right){\rm{d}}r} } } ,$$

where $k = \frac {{2\pi }}{\lambda }$ is the wavenumber. Substituting Eq. (5) into Eq. (9), we then obtain the variance of the phase fluctuations in spherical coordinate as

(10)$$\begin{array}{l} \begin{aligned} \left\langle {{\varphi ^2}} \right\rangle & = \frac{{r_e^2{k^4}{\lambda ^4}}}{{16{\pi ^4}}}\int_0^\infty {\frac{{{\rm{d}}{z_1}}}{{{z_1}}}} \int_0^{2\pi } {{\rm{d}}{\phi _1}} \int_0^\infty \sin \left( {\frac{{k{r_1}^2}}{{2{z_1}}}} \right){{r_1}{\rm{d}}{r_1}} \int_0^\infty {\frac{{{\rm{d}}{z_2}}}{{{z_2}}}} \int_0^{2\pi } {{\rm{d}}{\phi _2}} \\ & \cdot \int_0^\infty \sin \left( {\frac{{k{r_2}^2}}{{2{z_2}}}} \right)\left\langle {\delta {N_e}\left( \rho \right)\delta {N_e}\left( {{\rho ^{'}}} \right)} \right\rangle {{r_2}{\rm{d}}{r_2}}. \end{aligned} \end{array}$$

Eq. (10) could be further recast and simplified, using the Wiener-Khinchin theorem, as

(11)$$\begin{array}{l} \begin{aligned} \left\langle {{\varphi ^2}} \right\rangle & = \frac{{r_e^2{k^4}{\lambda ^4}}}{{16{\pi ^4}}}\int_0^\infty {\frac{{{\rm{d}}{z_1}}}{{{z_1}}}} \int_0^\infty {\frac{{{\rm{d}}{z_2}}}{{{z_2}}}} \int_0^\infty {\sin \left( {\frac{{k{r_1}^2}}{{2{z_1}}}} \right){r_1}{\rm{d}}{r_1}} \int_0^\infty \sin \left({\frac{k{r_2}^2}{2{z_2}}}\right){r_2}{\rm{d}}{r_2} \int_0^\infty {{\kappa ^2}{\rm{d}}\kappa } \\ & \cdot\int_{\rm{0}}^\pi {{e^{\left[ {i\kappa \left( {{z_1} - {z_2}} \right)\cos \Psi } \right]}\sin \Psi {\phi _N}\left( {\kappa ,\frac{{{z_1} + {z_2}}}{2}} \right)}}{\rm{d}}\Psi \int_{\rm{0}}^{2\pi } {{\rm{d}}\omega } \int_{\rm{0}}^{2\pi } {e^{\left[ {i\kappa {r_1}\sin \Psi \cos \left( {\omega - {\phi _1}} \right)} \right]}} {{\rm{d}}{\phi _1}}\\ & \cdot\int_{\rm{0}}^{{\rm{2}}\pi }{e^{\left[ { - i\kappa {r_2}\sin \Psi \cos \left( {\omega - {\phi _2}} \right)} \right]}} {\rm{d}{\phi _2}} . \end{aligned} \end{array}$$

Substituting Eq. (7) into Eq. (11) and utilizing the zeroth-order Bessel function, we can further rewrite the variance as

(12)$$\left\langle {{\varphi ^2}} \right\rangle = 8{\pi ^4}r_e^2L\frac{{\left( {p - 3} \right)\Gamma \left( {\frac{p}{2}} \right)}}{{{{\left( {2\pi } \right)}^{\frac{3}{2}}}{k^2}\Gamma \left( {\frac{{p - 1}}{2}} \right)}}\kappa _o^{p - 3}\left\langle {\delta N_e^2} \right\rangle \int_0^\infty {{\kappa ^{1 - p}\left[ {1 + \frac{{\sin \left( {\frac{{L{\kappa ^2}}}{k}} \right)}}{{\frac{{L{\kappa ^2}}}{k}}}} \right]}\rm{d}\kappa }.$$

From Ramanujan’s master theorem [29], we have the following integral equality

(13)$$\int_0^\infty {{x^{ - \mu }}} \left( {1 + \frac{{\sin \left( {ax} \right)}}{ax}} \right){\rm{d}}x = \frac{{\pi {a^{\mu - 1}}\left( {\frac{{\mu - 1}}{2}} \right)!}}{2{\varepsilon \left( {\frac{{\mu - 1}}{2}} \right)\Gamma \left( {\frac{{1 + \mu }}{2}} \right)\sin \left( {\frac{{\pi \left( {1 - \mu } \right)}}{2}} \right)\mu !}},$$

which we substitute into Eq. (12), to obtain the final expression for the variance of phase fluctuations in deep-space coronal turbulence as

(14)$$\left\langle {{\varphi ^2}} \right\rangle = \frac{{\pi r{_{e}^2}{{\left( {2\pi } \right)}^{p - \frac{1}{2}}}\sec \left( {\frac{{\pi p}}{4}} \right){L^{\frac{p}{2}}}\Gamma \left( {\frac{p}{2}} \right)\left( {p - 3} \right){\eta ^2}\left\langle {N_e^2} \right\rangle }}{8{\varepsilon \left( {\frac{{p - 2}}{4}} \right)\Gamma \left( {\frac{p}{2} + 1} \right)\Gamma \left( {\frac{{p - 1}}{2}} \right)l_{\rm{o}}^{p - 3}{k^{1 + \frac{p}{2}}}}}.$$

From Eq. (14), we know that $\left \langle {{\varphi ^2}} \right \rangle$ correlates with various stochastic factors, for example, positively with the link distance (L), plasma density fluctuation factor ($\eta$) and wavelength of the optical signal ($\lambda$), negatively with the non-Kolmogorov spectral index (p), and outer scale ($l_{\rm {o}}$). These factors make the model physically realistic. Further figurative and elaborate relationships among these variables are illustrated in Section 5.

4. Average BER under coronal turbulence with BPSK modulation

In this section, we derive the closed-form expression of the average BER for deep-space communication systems, while taking into account the effects of solar wind turbulence, during solar conjunction. Considering that phase fluctuations caused by the Costas loop circuit are equally important, we have accounted for phase fluctuations induced by both turbulence and the Costas loop, denoted as $\kappa = \Delta \psi - \varphi$ [30], where $\Delta \psi$ is the phase deviation caused by the Costas loop. For the BPSK modulation scheme, the bit error probability considering the phase fluctuations can be expressed as [30]

(15)$${P_b} = {1 \over 2}{\rm{erfc}}\left[ {\sqrt \gamma \cos \left( \kappa \right)} \right],$$

where ${\rm {erfc}}(\cdot )$ is the complementary error function, defined as erfc$\left ( x \right ) = \frac {2}{{\sqrt \pi }}\int _x^\infty {\exp \left ( { - {u^2}} \right )} \rm {d}\it {u}$; and $\gamma = \frac {a^2}{2{\sigma _{n}^2}}$ denotes the signal-to-noise ratio. The current from the detector, a, can be calculated as [31]

(16)$$a = Ge\left( {{K_s} + {K_b}} \right) + {I_{dc}}{T_s},$$

where $G$ is the average gain of the avalanche photodiode (APD) at the receiver end, $e$ is the electron quantity; $K_s$ is the photon count, $K_b$ is the background photon count; $I_{dc}$ is the dark current and $T_s$ is the bit time. The variance of the noise, $\sigma _{n}^2$, can be calculated as [31]

(17)$$\sigma_{n}^{2} = {\left( {Ge} \right)^{\rm{2}}}F\left( {{K_s} + {K_b}} \right) + \sigma _T^2,$$

where $F = 2 + G\zeta$ is the excessive noise factor, where $\zeta$ is the ionization factor of APD; and $\sigma _T^2 = \frac {{2{k_c}T{T_s}}}{{{R_L}}}$ is the thermal noise, where T is the temperature at the receiver environment, $k_c$ is the Boltzmann constant, and $R_L$ is the load resistance. Based on the perturbation approximation theory [32], we model the probability density function (PDF) of the phase fluctuations, $f_g$($\varphi$), as a Gaussian distribution [33]

(18)$${f_g}\left( \varphi \right) = \frac{1}{{\sqrt {2\pi } {\sigma _\kappa }{e^{\left( { \frac{{\left\langle {{\varphi ^2}} \right\rangle }}{{2\sigma _{_\kappa }^2}}} \right)}}}},$$

where $\sigma _k^2$ is the variance of phase. As previously mentioned, we know that the propagating wave is also affected by intensity scintillation. The PDF for the light intensity, u, can be expressed as [20]

(19)$${P_I}\left( u \right) = \frac{{{e^{ - \frac{{{{\left[ {\ln \left( u \right) + \frac{m}{2}} \right]}^2}}}{{2m}}}}}}{{u\sqrt {2\pi m} }},$$

where m is the scintillation index. We can express the average BER due to phase fluctuations and light intensity as

(20)$$\left\langle {BER} \right\rangle {\rm =\ }\int_{ - \infty }^\infty {\int_0^\infty {{P_b}{P_I}\left( u \right){f_g}\left( \varphi \right)} } {\rm{d}}u{\rm{d}}\kappa.$$

Substituting Eqs. (15), (18) and (19) into Eq. (20), we rewrite the average BER as

(21)$$\left\langle {BER} \right\rangle {\rm =\ }\int_{ - \infty }^\infty {Q\left[ {{{a\cos \left( \kappa \right)} \over {{\sigma _n}}}} \right]} \cdot {{{e^{\left[ { - {{{{\left( {\kappa - \Delta \psi } \right)}^2}} \over {2\sigma _{_\kappa }^2}}} \right]}}} \over {\sqrt {2\pi } {\sigma _\kappa }}}{\rm{d}}\kappa \int_0^\infty {{{{e^{ - {{{{\left( {\ln u + {m \over 2}} \right)}^2}} \over {2m}}}}} \over {u \cdot \sqrt {2\pi m} }}} {\rm{d}}u.$$

By invoking a change of variable, $t = {{\ln u + {m \over 2}} \over {\sqrt {2m} }}$, in Eq. (21), it could be further simplified as

(22)$$\left\langle {BER} \right\rangle {\rm =\ }\int_{ - \infty }^\infty {Q\left[ {{{a\cos \left( \kappa \right)} \over {{\sigma _n}}}} \right]} \cdot {{{e^{\left[ { - {{{{\left( {\kappa - \Delta \psi } \right)}^2}} \over {2\sigma _{_\kappa }^2}}} \right]}}} \over {\sqrt {2\pi } {\sigma _\kappa }}}{\rm{d}}\kappa.$$

By performing a variable substitution, where $s = {{\kappa - \Delta \psi } \over {\sqrt 2 {\sigma _\kappa }}}$ in Eq. (22), it could be recast as

(23)$$\left\langle {BER} \right\rangle {\rm =\ }{1 \over {\sqrt \pi }}\int_{ - \infty }^\infty {Q\left[ {{{a\cos \left( {2\sqrt 2 {\sigma _\kappa }s + \varphi + \Delta \psi } \right)} \over {{\sigma _n}}}} \right]} \cdot {e^{ - {s^2}}}{\rm{d}}s.$$

With the help of the Gauss-Hermite quadrature integration approximation expressed as

(24)$$\int_{ - \infty }^\infty {f\left( x \right)} {e^{ - {x^2}}}\rm{d}\it{x} \cong \sum_{i = 1}^n {{w_i}} f\left( {{x_i}} \right),$$

where ${x_i}\left | {_{i=1}^n} \right.$ is the i$\rm {^{th}}$ root of the Hermite polynomial $\rm {H{e_{n}}}$, ${w_i}\left | {_{i=1}^n} \right.$ is the corresponding weight factor, defined as ${w_i} = \frac {{{2^{n - 1}}n!{x_i}}}{{{n^2}{{[\rm {H{e_{n}}}\left (\it { {{x_i}}} \right )]}^2}}}$, we finally obtain the average BER for a deep-space communication system experiencing coronal turbulence under BPSK modulation as

(25)$$\left\langle {BER} \right\rangle = {1 \over {\sqrt \pi }}\sum_{i = 1}^n {{w_i}} Q\left[ {{{a\cos \left( {2\sqrt 2 {\sigma _\kappa }{x_i} + \varphi + \Delta \psi } \right)} \over {{\sigma _n}}}} \right],$$

where $\it {Q}(\cdot )$ is the Gaussian function.

Fig. 1. Geometric diagram of deep-space optical communication system during superior solar conjunction.

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5. Simulation analysis

We investigated the impacts of different parameters, such as $L$, $l_{\rm {o}}$, $p$, $\lambda$, and $\eta$, over coronal turbulence using the analytical expressions for the phase fluctuations in Eq. (14) and average BER in Eq. (25), on the deep-space communication system. All calculations were performed on MATLAB, and the conclusions were further verified utilizing Monte Carlo simulations. All spatial and turbulent environmental variables for each simulation are specified in Table 2.

Table 2. Key variables employed in the coronal turbulence model

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In the following analysis, $\eta$ is set to 0.1 (unless otherwise specified). Other parameters concerning plasma density irregularities, taken from [24], are listed in Table 1. For non-Kolmogorov turbulence channels and empirical data, we have 3 < $p$ < 4 and 1$\times 10^{7}$ < $l_{\rm {o}}$ < 7$\times 10^7$ m. We limit the wavelength range within 850 to 1600 nm for the simulations.

From Fig. 2, we evaluate the effects of link distance on the normalized phase fluctuations variance, and the color bar on the right side of this figure denotes the normalized value of the phase fluctuations. The phase fluctuations intensify with decreasing $\alpha$ and $\beta$. With a fixed link distance, smaller angles imply that the transmission path is closer to the corona, leading to communication environment complexities due to intense plasma irregularities. Furthermore, when $\beta$ increases from 0.5$^\circ$ to 0.7$^\circ$, the phase fluctuations exhibit no significant changes in trend for a fixed $\alpha$, i.e. phase fluctuations are more sensitive to the variations in $\beta$. Therefore, we fix $\beta$ = 0.5$^\circ$ for the following analysis for simplicity.

Fig. 2. Normalized phase fluctuations as a function of SEP and SPE angles.

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Fig. 3 illustrates the effects of the spectral index on the phase fluctuations. From Fig. 3, phase fluctuations increase as p decreases from 3.6, reaching maximum intensity for p = 3.1. Fluctuations vary gradually for p in the range of 3.6 to 3.9, reducing rapidly for p = 4. This phenomenon is consistent with previous studies [20,21,34].

Fig. 3. Normalized phase fluctuations as a function of SEP angle and spectral index.

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The influence of plasma density relative fluctuation factor on the phase fluctuations is depicted in Fig. 4. As shown in Fig. 4, increasing $\eta$ and decreasing $\alpha$ result in high phase fluctuations. For $\alpha$ > 0.2, increasing $\eta$ has minimal impacts on the phase fluctuations. However, for 0.19 < $\alpha$ < 0.2, increasing $\eta$, and hence, increasingly intense solar activities have prominent effects on the phase fluctuations. Therefore, we conclude that the plasma density fluctuation factor, $\eta$, is critical in deep-space communication under a small SEP angle.

Fig. 4. Normalized phase fluctuations as a function of SEP angle and plasma density fluctuation factor.

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Figure 5 presents effects of outer scale and wavelength on the phase fluctuations. Increasing outer scale can mitigate the phase fluctuations. The contour lines are smoother for $l_{\rm {o}}$ in the range of 4$\times 10^7$ to 7$\times 10^7$ m, and start bending for $l_{\rm {o}}$ in the range of 1$\times 10^7$ to 3$\times 10^7$ m. This phenomenon can be attributed to the focusing effect, which mitigates the impacts of coronal turbulence by enhancing light intensity while sacrificing beam width, an observation that is more evident with increasing $l_{\rm {o}}$. Additionally, an apparent trend of increasing phase fluctuations is observed as $\lambda$ increases from 850 to 1600 nm, suggesting that smaller wavelengths can effectively mitigate phase fluctuations.

Fig. 5. Three-dimensional (3D) representation of normalized phase fluctuations as a function of outer scale and wavelength. Two-dimensional (2D) projected contour lines indicate the intensity of phase fluctuations.

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In this section, we further examine the impacts of various parameters on the average BER performance, while taking into account phase fluctuations and intensity scintillation. We fix $\beta$ = 0.5$^\circ$, $T$ = 300 K, $R_L$ = 50 $\Omega$, $I_{dc}$ = 1 nA, $G$ = 125, $\zeta$ = 0.028, $L$ = 2.5 AU, transmitted power, $P_T$, as 10 W, and the divergence angle, ${\bar \theta }$, as 30 $\mu$rad. Moreover, we set the modulation parameters as $K_b$ = 10, $K_s$ = 300, and $T_s$ = 5 ns. Other system and environmental parameters used in the simulations are listed in Table 3.

Table 3. System and environmental parameters employed for average BER simulations

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In Fig. 6, we aim to demonstrate effects of SEP angle and wavelength on BER. It is obvious that BER increases with decreasing $\alpha$ and increasing $\lambda$. The BER under different wavelengths become similar for $\alpha$ > 0.2, and is the poorest in particular for $\lambda$ = 1550 nm. These suggest that phase fluctuations could be effectively mitigated for optical waves with shorter wavelengths, corresponding to higher frequencies. Therefore, high-frequency optical signals are more desirable for future deep-space optical communication systems. Note that shorter wavelengths lead to lower BER, but these wavelengths also result in increased path loss. Despite the minimal shift in the SEP angle, from 0.19 to 0.21 degrees, it profoundly impacts the average BER. This is because, with a fixed but extensively long communication distance, even minor angle changes can lead to substantial variations in heliocentric distances. These variations mean the signal is affected by significant solar wind density fluctuations, resulting in a considerable impact on the BER. Monte-Carlo simulations further verify the above analytical results.

Fig. 6. Average BER as a function of SEP angle for different optical wavelengths under BPSK modulation.

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As a critical variable in the non-Kolmogorov power spectrum, we further investigate the influence of the spectral index on the average BER. In Fig. 7, the average BER performance improves with $l_{\rm {o}}$ increasing from 1$\times 10^7$ to 7$\times 10^7$ m, due to the focusing effect, which reduces the influence of coronal turbulence by amplifying light intensity. The BER deteriorates for a fixed L, when p < 11/3. Thus, for smaller p values, the BER performance degrades as the phase fluctuations increase significantly (as discussed in Fig. 3). The Monte-Carlo simulations perfectly match the analytical results.

Fig. 7. Average BER as a function of outer scale for different spectral index values under BPSK modulation.

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We quantitatively explore the effects of the relative fluctuation factor, $\eta$, which intuitively describes the irregularities in the background plasma model, on the BER performance. Figure 8 exhibits the BER for the deep-space optical communication system increases with decreasing $l_{\rm {o}}$ and increasing $\eta$, i.e., a larger fluctuation factor results in stronger turbulence. The accuracy of these analytical solutions is validated by the corresponding Monte-Carlo simulations. We further examine the impacts of the SEP angle and transmission distance on the average BER. As demonstrated in Fig. 9, the average BER performance is enhanced with the increased SEP angle and the decreased transmission distance. This improvement can be attributed to the larger SEP angles resulting in the longer heliocentric distances. Thus, the signal encounters less plasma irregularities. Also, the decreased transmission distances lead to fewer phase fluctuations of signals, improving the communication performance.

Fig. 8. Average BER as a function of outer scale for different plasma density fluctuation factor values under BPSK modulation.

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Fig. 9. Average BER as a function of SEP angle and transmission distance under BPSK modulation.

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6. Conclusion

In this study, we have derived the variance of the phase fluctuations and phase fluctuations-induced average BER under BPSK modulation for deep-space optical communication systems. Various variables, such as the heliocentric distance, outer scale, spectral index, relative plasma density fluctuation factor, and wavelength, are considered in the derived expressions for a realistic characterization. The simulation results demonstrate that the phase fluctuations increase with decreasing SEP and SPE angles. This is because smaller angles result in shorter heliocentric distances, with a fixed transmission distance, leading to the signal being affected by a higher density of solar wind. This condition intensifies the phase fluctuations of the signal. Moreover, the phase fluctuations increase and the BER performance of the communication system degrades for a high non-Kolmogorov spectral index and a more prominent relative plasma density fluctuation factor. Fortunately, a large outer scale and propagating waves with shorter wavelengths can mitigate the impacts of the coronal plasma turbulence. This study contributes to our understanding of deep-space optical communication systems performance under coronal turbulence during superior solar conjunction, and provides an important reference for the future of deep-space exploration.

Funding

National Natural Science Foundation of China (62027802, 62271202); Key Research and Development Program of Zhejiang Province (2023C01003); Open Foundation of State Key Laboratory of Integrated Services Networks Xidian University (ISN23-01); Pre-research Project on Civil Aerospace Technologies of CNSA (D020101); Frontier Scientific Research Program of Deep Space Exploration Laboratory (2022-QYKYJH-GCXD-001, 2022-QYKYJH-HXYF-018).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Sampled value	$a_{i}$ $(\times 1 0^{7})$	$c_{i}$ $(\times 1 0^{7})$	$d_{i}$	$γ_{i}$	$δ_{i}$
i = 1	1.07	0.14	2.8	16.3	0.5
i = 2	19.94	8.02	8.45	10	7.31
i = 3	22.1	8.12	16.87	43.2	7.52

Fig.	SEP, $α$	SPE, $β$	Outer scale,	Non-Kolmogorov	Wavelength,	Fluctuation
Fig.	(degrees)	(degrees)	$l_{o}$ ( $\times 10^{7}$ )	spectral index, p	$λ$ (nm)	factor, $η$
2	[0.19, 0.21]	[0.2, 0.5]	3	3.6	1550	0.1
3	[0.19, 0.21]	0.5	3	3.1-3.9	1550	0.1
4	[0.19, 0.21]	0.5	3	3.6	1550	0.001-0.02
5	0.2	0.5	1-7	3.6	850-1600	0.1

Fig.	SEP, $α$	Outer scale,	Non-Kolmogorov	Wavelength,	Fluctuation
Fig.	(degrees)	$l_{o}$ ( $\times 10^{7}$ )	spectral index, p	$λ$ (nm)	factor, $η$
6	[0.19,0.21]	5	3.5	[850, 1310, 1550]	0.1
7	0.2	1-7	[3.3, 3.5, 11/3]	1550	0.1
8	0.2	1-7	[0.1, 0.15, 0.2]	1550	0.1
9	[0.19,0.21]	5	3.5	1550	0.1

Sampled value	$a_{i}$ $(\times 1 0^{7})$	$c_{i}$ $(\times 1 0^{7})$	$d_{i}$	$γ_{i}$	$δ_{i}$
i = 1	1.07	0.14	2.8	16.3	0.5
i = 2	19.94	8.02	8.45	10	7.31
i = 3	22.1	8.12	16.87	43.2	7.52

Fig.	SEP, $α$	SPE, $β$	Outer scale,	Non-Kolmogorov	Wavelength,	Fluctuation
Fig.	(degrees)	(degrees)	$l_{o}$ ( $\times 10^{7}$ )	spectral index, p	$λ$ (nm)	factor, $η$
2	[0.19, 0.21]	[0.2, 0.5]	3	3.6	1550	0.1
3	[0.19, 0.21]	0.5	3	3.1-3.9	1550	0.1
4	[0.19, 0.21]	0.5	3	3.6	1550	0.001-0.02
5	0.2	0.5	1-7	3.6	850-1600	0.1

Phase fluctuations-induced bit error ratio of deep-space optical communication systems during superior solar conjunction

Abstract

1. Introduction

2. Theoretical model

2.1 Geometrical model for superior solar conjunction

2.2 Solar wind density model

2.3 Non-Kolmogorov spatial power spectrum

3. Variance of phase fluctuations

4. Average BER under coronal turbulence with BPSK modulation

5. Simulation analysis

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (3)

Equations (27)

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