## Abstract

Free space optical (FSO) communication is a promising technology for future deep space exploration, but it encounters coronal turbulence during superior solar conjunction. In this paper, the bit-error rate (BER) performance of deep space FSO communication systems is evaluated for optical waves propagating in the non-Kolmogorov coronal turbulence. By virtue of its high energy efficiency, the pulse position modulation (PPM) technique is adopted to mitigate the influence of turbulence under lognormal distribution channels. The effects of the parameters of the coronal turbulence and of the FSO system, such as the turbulence outer scale, spectral index, symbol number, data bit rate, equivalent load resistor, and average gain, on the BER are investigated and discussed in this paper. In addition, the performance improvement from M-ary PPM and binary phase-shift keying subcarrier intensity modulation is studied. The results of a numerical evaluation illustrate that a deep space FSO communication system with PPM scheme can well be used to mitigate the impact of coronal turbulence.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Due to its advantages of large transmission capacity, narrow beam size, high transmission rate, cost-effectiveness, etc., free space optical (FSO) communication has become a promising scheme and a technology complementary to radio frequency (RF) communication [1, 2]. Recently, FSO communication has exhibited its superiority in many scenarios, such as oceanic communication [3, 4], atmospheric communication [5, 6], and deep space communication [2, 7], and has gained plenty of attention. However, the performance of FSO communication systems is susceptible to degradation by the scintillation effect occurring when the optical waves pass through turbulent channels. The intensity fluctuations of the received signal caused by the adverse turbulence condition further induce a large bit-error rate (BER). Therefore, the FSO communication performance is severely degraded and the communication link can even be interrupted.

Much attention has been devoted in recent years to investigate the BER of FSO communication systems [8–11]. On the one hand, plenty of statistical distribution models have been introduced to understand the turbulence-induced fading process in FSO communication system. The lognormal (LN) distribution [8], Gamma-Gamma (GG) distribution [9], and exponential Weibull (EW) distribution [10] are the most extensively adopted fading models for characterizing the turbulence channels under weak scintillation, weak-to-moderate scintillation, and weak-to-strong scintillation, respectively. On the other hand, various modulation technologies have been used to reduce the influence of the turbulence fading effect on FSO communication systems and to achieve a high transmission rate and excellent BER [12, 13].

An expression of the link BER for FSO systems with on–off keying (OOK) modulation over atmospheric turbulence has been proposed in [8] and was further validated with the experimental data. Considering the effect GG distributed atmospheric turbulence on satellite-to-ground optical communication, the BER with binary phase-shift keying (BPSK) modulation was studied in [9]. In [10, 13–15], the BER of both atmospheric and underwater FSO systems with subcarrier intensity modulation (SIM) under LN, GG, and EW distributed channels were compared and it was demonstrated that the EW distribution has an advantage in characterizing the turbulence intensity for weak-to-strong scintillation.

After that, the BER of optical waves in EW fading channels with OOK and BPSK was further investigated comprehensively in [10]. Due to its simplicity, OOK has been the most studied technique for FSO communication in much of the literature. In addition, the BPSK modulation scheme has also had a wide range of application to FSO systems since it does without an adaptive threshold. However, the lower power efficiency of these modulation schemes restrains their further development. In [16], pulse position modulation (PPM) with high power efficiency was first proposed and it has received much attention. Since PPM offers a high power efficiency at the expense of increased bandwidth, some modified PPM schemes have been proposed. The *M*-ary PPM techniques are used in many communication system to transmit *M* different signals in one time slot by non-coherent detection at the receiver [17]. Up to now, several ideas for superimposing different coding and modulation techniques, such as multiple PPM (MPPM)-coded multiple-input multiple-output FSO systems and hybrid M-ary quadrature amplitude modulation, have been proposed to improve both the power efficiency and spectral efficiency simultaneously [18, 19].

As we have mentioned above, many researchers have studied the link performance of optical waves in both atmospheric and oceanic turbulence with different modulation techniques and have achieved abundant meaningful results. However, as far as our knowledge extends, the BER of FSO communication in deep space has not been studied comprehensively. With the development of deep space exploration, FSO communication is an alternative scheme to overcome the long transmission distance [2]. Nevertheless, the link will inevitably pass through the turbulent solar corona during superior solar conjunction when the probe moves on the opposite side of the Sun from to the Earth [20]. Therefore, the BER of optical waves propagating in coronal turbulence is a challenging problem.

In contrast to atmospheric and oceanic turbulence, the solar corona is filled with high density plasma that erupts from the Sun and further expands into interplanetary space (IPS). According to the mounting evidence of astronomical observation, the statistics of these plasma irregularities obey non-Kolmogorov’s spectrum model [21, 22]. Since the refractive index of the solar wind plasma changes, the signal intensity will be distorted when the radio frequency (RF) and optical signal pass through the turbulent coronal regions [20, 24].

Motivated by the above analysis, in this paper, we formulate and study the BER of optical waves propagating through coronal turbulence. The PPM technique is adopted since it is predominantly used for deep space FSO communication, due to its high power efficiency and lack of an adaptive threshold. We investigate the effect on the average BER of different coronal parameters, such as the turbulent outer scale, the non-Kolmogorov spectral index, and FSO system parameters, such as the symbol number, bit rate, equivalent load resistor, and average gain.

In addition, a comparison of the performance of M-ary PPM and BPSK modulation in deep space FSO communication system is made. Numerical evaluation are helpful for understand how coronal turbulence will affect FSO communication, and will also provide a theoretical basis for the design of future deep space FSO links.

The rest of this paper is organized as follows. In Section 2, the formulation of solar scintillation is introduced. After that, the BER with PPM and M-ary PPM techniques under coronal channels with LN distribution is derived. Numerical results are discussed in Section 3, and the conclusions are summarized in Section 4.

## 2. System model and formulation

#### 2.1. System model and the formulation of solar scintillation

A diagram for a deep space FSO communication system during superior solar conjunction is given in Fig. 1. The relay communications technique is adopted here, where a relay satellite around the Earth first receives the signal from the Earth station via RF and then transmits it by an optical wave. Here, we assume that the photo–detectors are located in both the receiver and the transmitter of the relay satellite and the probe. The distance between the Earth and the Sun is *L _{se}* and the distance between the probe and the Sun is

*L*. The distance between the surface of the Sun and the communication link is denoted by

_{sp}*r*, which is generally in a unit of the solar radius, ${R}_{sun}$. Therefore, both the Sun–Earth–Probe (SEP) angle,

*α*, and the Sun–Probe–Earth (SPE) angle,

*β*, have a direct relation with

*r*, since $\alpha =\mathrm{arcsin}\text{}\left(r/{L}_{se}\right)$ and $\beta =\mathrm{arcsin}\text{}\left(r/{L}_{sp}\right)$, as shown in Fig. 1.

As mentioned in Section I, the solar wind is a turbulent plasma erupting from the Sun and ultimately pervading in the entire IPS. The solar wind density and its irregularities are very high in the chromosphere and decrease along its radial direction, as is intuitive. Therefore, the optical waves transmitted between the Earth station and the probe will suffer severe fluctuations due to the irregularities in the solar wind when the FSO communication link passes through those coronal irregularities with strong turbulent channels. These fluctuations are generally referred to as solar scintillation [23]. Consequently, the performance of the link is degraded during this period, as depicted in Fig. 1. Note that the solar wind plasma permeates the whole IPS and its density decreases along its radial direction. Besides, the system model of FSO communication during superior solar conjunction has a short duration but will frequently occurrence. Therefore, we mainly consider the influence of the coronal turbulence on the optical waves that propagate along the transmission link.

According to [7, 24], the dielectric constant of the solar wind plasma, as a direct result, the amplitude fluctuations, is related to the plasma density by

where*r*is the classical electron radius and

_{e}*λ*is the wavelength. Note that the amplitude fluctuations are only related to

*λ*and the fluctuation of the density of the solar wind, $\delta {N}_{e}$. Therefore, research on the influence of ionospheric plasma on RF signals provides a potential way to investigate the impact of coronal turbulence on optical waves [24].

The performance level of deep space FSO system during superior solar conjunction due to intensity fluctuations is usually characterized by the variance of amplitude fluctuations, $\u3008{\chi}^{2}\u3009$, and scintillation index. Apart from the impact of coronal turbulence, atmospheric turbulence also has a significant influence on the whole FSO communication system, which has been extensively studied, as we elaborated on in Section I. However, we mainly focus on the impact of coronal turbulence on the deep space FSO system in this paper, for simplicity.

An expression of the amplitude fluctuations under weak solar scintillation has been successfully derived with Rytov’s approximation method in our previous research, and was introduced in [7].

Here, the amplitude fluctuations are the double integral over the magnitude of the spatial frequency vector, *κ*, and the optical waves propagation distance, *z*, since the coronal turbulence extends over the whole link. *L* represents the length of the FSO communication link, which is the distance between the relay satellite around the Earth and the probe. ${\mathrm{\Phi}}_{N}\left(\kappa ,z\right)$ is the spatial spectrum of the coronal turbulence. In the following description, the spectrum model is simplified as ${\mathrm{\Phi}}_{N}\left(\kappa \right)$ since *z* can be represented by *r* according to Fig. 1. Here, a non-Kolmogorov spectrum model is adopted [25].

Here, *p* represents the spectral index fluctuation in the range of 3 to 4 [26], $\mathrm{\Gamma}(\cdot )$ is the Gamma function, ${\kappa}_{o}=2\pi /{L}_{o}$, and *L _{o}* is the outer scale of the coronal turbulence. $\delta {N}_{e}$ denotes the irregularities of the solar wind density, and $\u3008\cdot \u3009$ means the ensemble average. Note that $\delta {N}_{e}$ is normally assumed proportional to the solar wind plasma density,

*N*, i.e. $\delta {N}_{e}\approxeq \eta {N}_{e}$, which has been extensively employed in astronomy. Here,

_{e}*η*represents the relative solar wind density fluctuations coefficient. Since we mainly consider the effect of coronal turbulence on the optical waves during superior solar conjunction, the following simplified, but widely used, solar wind density model for $1.5{R}_{sun}\le r\le 2{R}_{sun}$ is adopted in this paper [24]. Note that ${R}_{sun}$ is the solar radius.

By substituting (3) into (2), a closed form formula for the amplitude fluctuations can be obtained as

According to [28], the relation between the scintillation index, *m*^{2}, which is characterized as the standard deviation of the intensity of the waves normalized to the average received intensity, and the variance of the amplitude fluctuations can be written as ${m}^{2}=4{\chi}^{2}$.

#### 2.2. BER under PPM and M-ary PPM

The link performance of deep space FSO communication systems during superior solar conjunction is affected by the amplitude fluctuations resulting from turbulent irregularities in the solar wind. Theoretical and experimental research into the propagation of waves in coronal turbulence demonstrate that the communication link will encounter weak turbulence, moderate turbulence, and strong turbulence, successively, with a decrease in the heliocentric distance or SEP angle [7]. Consequently, the intensity of the solar scintillation increases during this period. For the sake of simplicity, only weak solar scintillation is considered in this paper.

In this section, we consider an avalanche photodiode (APD)-based direct-detection receiver. The current induced in the load resistance in the response of the APD is modeled as

*q*is the charge of the electron $\left(\approxeq 1.602\times {10}^{-19}C\right)$, and

*G*denotes the random gain of the APD for the

_{n}*n*-th detected photon. The response of the APD to a photon in the

*n*-th event time,

*t*, is represented by $x\left(t\right)$. $n\left(t\right)$ is the thermal noise at the receiver.

_{n}According to the derivation in [28], the average number of the received photon in a specific PPM slot duration can be calculated as

where*ρ*means the detector’s quantum efficiency. ℏ and

*c*are the Planck’s constant and the light speed, respectively. $\u3008{P}_{r}\u3009$ is the received average power for one PPM slot. The PPM symbol duration is denoted by ${T}_{w}=2{T}_{s}$.

*T*is the slot duration for a specific PPM signal with the bit duration ${T}_{b}=1/{R}_{b}$, where

_{s}*R*is the data bit rate.

_{b}Without loss of generality, we also assume *K _{s}* obeys the LN statistics distribution, which is defined as [27, 28]

Here, *σ _{k}* and

*n*represent the standard deviation and mean value of ln$\left({K}_{s}\right)$. Therefore, the unconditional BER,

_{k}*P*, in an FSO communication system with PPM can be written as [29]

_{b}Note that the Gaussian *Q*-function is expressed as $Q\left(t\right)=1/2erfc\left(t/\sqrt{2}\right)$ and $erfc(\cdot )$ is the complementary error function. Note that $\mathrm{\Xi}\left({K}_{s}\right)$ denotes the signal-to-noise ratio (SNR) and can be further given by [30]

*G*denotes the average gain of the APD, which usually falls in the range of 50∼250. Note that ${\sigma}_{th}{}^{2}=2\sigma {T}_{e}{T}_{s}/{R}_{L}$, where

*σ*is Boltzmann’s constant $\left(\approxeq 1.38\times {10}^{-23}J/K\right)$,

*T*and

_{e}*R*represent the effective temperature of the receiver in degree Kelvin and the equivalent load resistor, respectively.

_{L}It also should be noted that the noise sources in the FSO communication system can be divided as the shot noise associated to the received signal, the background optical power, the dark current noise, the thermal noise and the total equivalent noise input current associated to the amplifier. For simplicity, only the influence of background radiation noise is considered in (10). ${K}_{bg}=\rho {P}_{bg}{T}_{s}\lambda /\left(\hslash c\right)$ denotes the average photon count owing to the background radiation of power *P _{bg}* in each PPM slot. Note that

*P*is assumed as a constant value in the following calculation. The excess noise factor of APD in (10) can be expressed as $F=2+G\varsigma $, where

_{bg}*ς*denotes the ionization factor.

Substituting (10) and (8) into (9) and using Hermite’s polynomials [30], the BER of the FSO communications system with PPM can be finally recast as

*m*

^{2}is the scintillation index of the optical waves in coronal turbulence and can be obtained by (5). In addition, ${\left[{x}_{i}\right]}_{i=1}^{N}$ and ${\left[{w}_{i}\right]}_{i=1}^{N}$ denote the zero points and weight factors of the

*N*-th order Hermite polynomial, respectively. It is imperative to note that the values of

*w*,

_{i}*x*, and $Q(\cdot )$ in (11) are well tabulated and can be easily found in [30]. Moreover, the value of

_{i}*N*is chosen depending on the desired value of accuracy, which is taken as 20 in the subsequent calculations. Furthermore, the ensemble average of the photon count,

*K*, has a direct relationship with the coronal turbulence as $E\left[{K}_{s}\right]=\mathrm{exp}\text{}\left({\sigma}_{k}^{2}/2+{n}_{k}\right)$. Note that

_{s}*n*is the mean value of $\mathrm{ln}\text{}\left({K}_{s}\right)$.

_{k}In an M-ary PPM system, each optical pulse is divided into *M* adjacent time slots, which means each PPM symbol slot has $\mathrm{log}{\text{}}_{2}M$ bits. Therefore, the slot duration for the M-ary PPM FSO system can be defined as

Similarly, for a PPM system working in coronal turbulence conditions, the BER formula of the FSO system with M-ary PPM can be further expressed as

## 3. Numerical evaluations and discussion

In this section, the BER of a deep space FSO communication system with PPM under coronal turbulence during superior solar conjunction is investigated considering the effect of employing different values for the system parameters. All results were obtained by numerically evaluating (11) and (13) with the Matlab software package. In all the Figs., the receiver temperature is fixed at *T _{e}* = 100 K and we assume that the APD ionization factor $\zeta =0.028$. The solar wind density fluctuations coefficient is set to be $\eta =10\%$, which is reasonable in light of the astronomical observation [26]. It should be noted that the communication distance

*L*can be obtained by employing the triangle relation as showing in Fig. 1. In the following calculations, all of the other parameters have been selected from some actual FSO communication systems with deep space missions, as shown in Table 1. The antenna radius for the transmitter and receiver are 0.6 m and 0.15 m.

Figure 2 demonstrates the variation of BER as a function of the scintillation index, *m*^{2}, for different symbol numbers, *M*. The modulation-related parameters are set to be: *K _{s}* = 300, ${R}_{L}=60\mathrm{\Omega}$, $G=150$. According to deep space missions with optical wave communications [35], the data bit rate is normally lower than 30 Mb/s. However, we set it to be

*R*= 50 Mb/s in the subsequent calculations since this is the target for future optical communication systems. Once can see from Fig. 2 that the BER increases with an increase in the scintillation index. This is because the solar scintillation intensity increases from weak to moderate, and even up to strong, when the scintillation index increases. Therefore, the BER decreases along this tendency. In addition, Fig. 2 also shows that a higher modulation order with a large symbol number induces a larger BER. The BER increases from $1\times {10}^{-3}$ to $8.2\times {10}^{-2}$ when

_{b}*M*increases from 2 to 128 at ${m}^{2}=0.4$. Note that this phenomenon is more obvious when

*m*

^{2}gets larger. These phenomena are reasonable because the optical wave propagation through the coronal turbulence by M-ary PPM with a larger symbol number means the communication system needs a larger transmitted bandwidth, which induces more noise power, and finally gives a larger BER.

In Fig. 3 and Fig. 4, the influence of the average photodetector gain, *G*, and the photon count, *K _{s}*, are analyzed. The simulation parameters in Fig. 3 were set to be:

*M*= 8,

*K*= 300,

_{s}*R*= 50 Mb/s, ${R}_{L}=60\text{\Omega}$, and the parameters in Fig. 4 are set as:

_{b}*M*= 8, $G=150$,

*R*= 50 Mb/s, ${R}_{L}=60\text{\Omega}$. It can be seen in Fig. 3 that an increase in

_{b}*G*causes a decrease in the BER, especially when

*G*increases from 100 to 150. This can be attributed to the fact that the noise level increases along with an increase of average photodetector gain and finally contributes to the BER. The BER also decreases with an increase of

*K*, as shown in Fig. 4. It should be noted that this tendency becomes negligible when the scintillation index increases. For example, the BER decreases from $3\times {10}^{-2}$ to $5\times {10}^{-3}$ when

_{s}*K*increases from 180 to 260 at ${m}^{2}=0.4$. However, it decreases from $7\times {10}^{-2}$ to $2\times {10}^{-2}$ at ${m}^{2}=0.8$. This phenomenon can be directly explained from (13) in that a larger photon count dominates the BER degradation. Nevertheless, the scintillation index plays a dominant role in the BER, eclipsing that of the photon count when the optical waves are passing through moderate or even strong coronal turbulence.

_{s}As shown in Eq. (5), the coronal turbulence parameters, such as, the outer scale, *L _{o}*, spectral index,

*p*, also have a large influence on the BER. Therefore, both the effect of the coronal turbulence parameters and of the FSO system parameters will be investigated, comprehensively, in the following simulations. The simulation parameters have been set as follows unless otherwise specified:

*M*= 8,

*K*= 300, $G=150$,

_{s}*R*= 50 Mb/s, ${R}_{L}=60\text{\Omega}$.

_{b}In Fig. 5, we illustrate the effect of the optical wavelength, *λ*, on the BER for different outer scales, *L _{o}*. Here, we set $p=3.6$. According to the long-time astronomical observations and analysis [26], the outer scale is set in the range of $5\times {10}^{7}\le {L}_{o}\le 2\times {10}^{8}$ m. As expected, decreasing the optical wavelength results in decreasing the BER for different

*L*. In addition, the BER also decreases with an increase of

_{o}*L*. The ability of

_{o}*L*to reduce the BER is more obvious when

_{o}*λ*is about 800 nm compared to $\lambda \approx 1200$ nm. Since we mainly consider the non-Kolmogorov spectrum model in this paper, the BER versus the wavelength for various

*p*values is presented in Fig. 6. An outer scale of ${L}_{o}=1.5\times {10}^{8}$ m was taken for this simulation. We can also observe that the BER increases with an increase of

*λ*. In addition, a smaller

*p*induces an higher BER.

The physical interpretation of the results shown in Fig. 5 and Fig. 6 is that, the integrated coronal turbulence strength decreases with an increase of outer scale and spectral index. Thus, it will finally degrade the value of the BER. Note that the decline of BER performance in Fig. 5 and Fig. 6 becomes more gradual with a decrease of outer scale and spectral index, respectively, when the wavelength is large. This phenomenon further indicates that the wavelength plays a vital role in the FSO communication system. Therefore, the communication performance can be significantly enhanced by decreasing the wavelength.

Figure 7 and Fig. 8 show the BER versus the heliocentric distance, *r*, for different data bit rates, *R _{b}*, and equivalent load resistors,

*R*, respectively. The heliocentric distance is set to be $1.75{R}_{sun}\le r\le 1.825{R}_{sun}$, therefore, the corresponding SEP angle is approximately ${0.467}^{\circ}\le \alpha \le {0.487}^{\circ}$. In addition, we set

_{L}*M*= 8,

*K*= 300, $G=150$, ${L}_{o}=5\times {10}^{7}$ m, $p=3.5$. The equivalent load resistor in Fig. 7 is ${R}_{L}=60\text{\Omega}$ and the data bit rate in Fig. 8 is

_{s}*R*= 50 Mb/s. Note that the data bit rate in Fig. 7 has been chosen in the range of 5∼50 Mb/s according to NASA’s future deep space project [35].

_{b}As shown in Fig. 7, the higher are the *R _{b}*, the larger the BER of the FSO communication system. As

*R*increases, a larger transmitted bandwidth is needed and finally induces more noise power, according to Eq. (12). In Fig. 8, it is clear that the BER decreases with an increase in

_{b}*R*. This is because a larger load resistor will improve the effective sensitivity of the photodetector, and as a result, bring about a better BER performance in the FSO communication system. It is worth noting that, increasing the load resistor is an alternative way to enhance the BER performance. However, it is not practical, due to the bandwidth constraints.

_{L}In Fig. 7 and Fig. 8, the BER decreases with an increase in the heliocentric distance or SEP angle for different *R _{b}* and

*R*. Note that this phenomenon is more obvious for a large heliocentric distance. For example, the BER gradually decreases from $8.9\times {10}^{-3}$ to $7.2\times {10}^{-3}$ when

_{L}*R*decreases from 50 Mb/s to 35 Mb/s at $r=1.78{R}_{sun}$. However, it decreases rapidly from $4.2\times {10}^{-4}$ to $8.6\times {10}^{-5}$ when $r=1.8{R}_{sun}$. This phenomenon is attributed to the fact that, the intensity of the coronal turbulence decreases with an increase in the heliocentric distance, and the scintillation index decreases at the same time. Therefore, the BER is reduced during this time, according to Eq. (13), along with the decrease in the bit rate and the increase in load resistor. In addition, we can conclude that the effect of solar wind irregularities on the propagation of the optical waves through the coronal turbulence can be negligible when $r\ge 1.8{R}_{sun}$ with ${R}_{b}\le 15$ Mb/s and ${R}_{L}\ge 60\text{\Omega}$.

_{b}To evaluate the BER performance under the M-ary PPM scheme, it is desirable to compare the numerical simulation with theoretical calculations. For an FSO communication system, the BER for M-ary PPM and BPSK over an LN channel model are shown in Fig. 9 as a function of SNR, *γ*, and scintillation index, *m*^{2}. We also set: *M* = 4, *K _{s}* = 300, $G=150$, ${R}_{L}=60\text{\Omega}$, ${L}_{o}=5\times {10}^{7}$ m, and $p=3.5$. For comparability, the symbol duration of the BPSK scheme is equal to the duration of the time slot of the 4PPM scheme. In addition, both schemes have the same data rate,

*R*= 50 Mb/s. Therefore, the bandwidth for BPSK and 4PPM are 100 MHz and 200 MHz, respectively.

_{b}As shown in Fig. 9, the results of numerical simulation of the 4PPM scheme are fairly consistent with the theoretical analysis, under different scintillation indices. Therefore, it is reasonable to believe that our theoretical result is correct. It can be seen that the required SNR for 4PPM is 12 dB, whereas for the BPSK it is about 13 dB to obtain a standard BER of ${10}^{-5}$ at a scintillation index of ${m}^{2}=0.1$. Similarly, at the higher scintillation index of ${m}^{2}=0.3$, the required SNR for 4PPM is about 15.8 dB, and for BPSK, it is around 17.3 dB to maintain the same BER. Therefore, this demonstrates that the 4PPM avoids the power penalty of 1 dB and 1.5 dB in the lower scintillation index and higher index, respectively, in comparison to the BPSK scheme. Of course, with both modulation schemes, the BER decreases as the scintillation index decreases. This can also be concluded from ^{Fig. 2, Fig. 3}, and Fig. 4, and indicates that the scintillation intensity has an intimate relation with the BER performance of FSO communication systems. Moreover, the 4PPM scheme achieves a better BER than the traditional BPSK scheme irrespective of the coronal turbulence conditions.

According to the above discussion, the main contribution of this paper is to understand the influence of various parameters on the performance of a deep space FSO communication link. Inspired by the simulation results, the BER can be significantly improved by increasing the equivalent load resistance, the average gain of the APD and decreasing the data bit rate. In addition, decreasing the wavelength would be an extremely effective way to enhance the BER, but needs investigation due to the system design. Moreover, both M-ary PPM and BPSK modulation schemes are alternative techniques for deep space FSO communication systems but should be further studied due to the variable solar corona environment. It should be noted that our results are strongly dependent on the simplified solar wind density model and fluctuations model, as depicted in Section 2. However, the actual coronal environment is very complicated. Therefore, more attention should be paid to improving the forecast accuracy of the effect of the coronal environment on an FSO communication system.

## 4. Conclusion

In this paper, we investigated the bit-error rate (BER) of free space optical (FSO) communication systems operating in the non-Kolmogorov coronal turbulence channels with a lognomal distribution. The pulse position modulation (PPM) technique is used to mitigate the BER caused by irregular solar wind turbulence under weak scintillation. In addition, the influence of various coronal parameters and FSO system parameters on the BER has been discussed. The results show that BER decreases with an increase of the turbulence outer scale, of the spectral index, and the heliocentric distance. Furthermore, increasing equivalent load resistance, photodetector gain, and decreasing the symbol number or the data bit rate can remarkably decrease the BER to some extent. Moreover, the M-ary PPM scheme surpasses traditional binary phase-shift keying modulation in improving the BER of an FSO system in coronal turbulence during superior solar conjunction. Therefore, our results have potential value for the design of future deep space FSO communication systems.

## Funding

National Natural Science Foundation (NSFC) (61801181, 61831008); Open Research Fund of Shanghai Key Laboratory of Multidimensional Information Processing.

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