1. Introduction

Recently, space optical communication has attracted considerable attention due to its higher data rate, higher optical bandwidth available and better security compared with space microwave communication [1–4]. Compared to the traditional intensity modulation direct detection (IM/DD) scheme, the phase modulation coherent detection scheme can suppress background noise and thermal noise, which will significantly improve the sensitivity of the receiver [5]. So far, many experiments on space coherent optical communication with phase modulation have been deployed [6–9]. However, phase modulation coherent communication is inevitably affected by the laser linewidth induced phase noise [10,11]. Meanwhile, in a practical space coherent optical communication system, a laser with appropriate linewidth is used to compensate for the phase noise. Thus, it is important to identify the laser linewidth tolerance of the space coherent optical communication system. Until now, the linewidth tolerance in coherent communication has been investigated by many researchers as follows.

In 2015, a novel two-stage carrier phase estimation (CPE) algorithm for 64-ary quadrature amplitude modulation (64-QAM) coherent optical systems is proposed in Ref. [12]. The proposed CPE algorithm can obtain a combined laser linewidth symbol duration product as high as 5.6 × 10⁻⁵ at a target BER of 1 × 10⁻² with 1 dB signal-to-noise ratio (SNR) sensitivity penalty. It means that more than 1.7 MHz combined laser linewidth can be tolerated with an industry standard symbol rate of 32 GBaud. In 2018, based on the extended quadrature phase- shift keying (QPSK) partitioning and quasi-linear approximation, Ref. [13] proposes and verifies an innovative multi-format CPE scheme for coherent optical m-QAM flexible transmission systems. The simulation results with experimental parameters are presented for comparison, giving the transmitter and LO the same linewidth of 2.4 MHz for QPSK and 1MHz for 16QAM. In 2020, Ref. [14] investigates the laser linewidth tolerance for nonlinear frequency division multiplexing transmission system and the results showed that under the condition of 2 GBaud symbol rate, the laser linewidth tolerances for QPSK, 8-PSK, and 16-APSK formats are within the range of 2.3 MHz, 1.05 MHz, and 250 kHz, respectively. In 2021, Ref. [15] proposes a joint and modulation transparent frequency offset (FO) and phase noise tracking scheme using a self-learning Kalman filter. The experiment demonstrates that with 1dB OSNR penalty, the proposed scheme can tolerate linewidth up to 1.5 MHz and 300 kHz for 16QAM and 64QAM, respectively. The above researches on linewidth tolerance, however, are all based on optical fiber channel, which is quite different from the space channel. In the real space optical communication system, the situation is much more complicated in the presence of atmospheric turbulence, pointing error, and intensity scintillation [16]. Under these factors, laser linewidth tolerance will further decrease. Therefore, some researches on laser linewidth tolerance in space optical communication system have been carried on. In 2017, Ref. [17] introduced a 1 km link length coherent free-space optical system with QPSK modulation using a two-stage M-th power CPE scheme, and the effects of log-normal amplitude fluctuations and Gauss phase fluctuations are considered. The simulation result showed that symbol error rate (SER) could achieve 10⁻⁸ at the rate of 20 Gbps and the linewidth tolerance can reach 10 MHz. Reference [18] demonstrates an 8-QAM coherent free-space optical communication system using amplitude compensation and phase recovery and the combined linewidth tolerance of transmitter and local oscillation is 10 kHz.

In addition to the effect of the atmospheric turbulence, the space optical communication system is strongly affected by the pointing error. The pointing error refers to the vibration of the transmitter beam in the receiver plane and the vibration of the receiver telescope relative to the received beam direction [19]. Building sway due to wind loads, differential heating and cooling, or ground motion over time can result in an important misalignment error [20,21]. Besides, the mechanical vibration or tracking system error will also cause the pointing error [22]. As far as we know, many studies focus on the pointing error in the free space optical communication system. The pointing error is usually compensated by the acquisition, tracking, and pointing (ATP) system in the space optical communication system [23]. The ATP tracking accuracy of the inter-satellite communication system can be the order of microradians [24], while the accuracy of the satellite-to-ground communication system is lower than the case in inter-satellite communication link [25], which means the pointing error cannot be compensated completely. Besides, for the space communication system with a high tracking accuracy ATP system, the linewidth tolerance can large. That means a wider linewidth laser can also meet the requirements for the space optical communication system. As we know, the wider linewidth laser is much cheaper than the narrow linewidth laser. Considering that commercial space optical communication project is sensitive to cost, the introduction of the wider linewidth laser will be helpful to save budgets. In addition, for the space communication system with a low tracking accuracy ATP system, the linewidth tolerance will be relatively low, and a narrow linewidth laser is necessary to meet the BER requirement for the space optical communication system. Therefore, the effect of pointing error on linewidth tolerance of space optical communication cannot be ignored and the linewidth tolerance should be identified more precisely under the effect of pointing error. In this paper, the effect of pointing error on linewidth tolerance will be discussed in detail. Firstly, a closed-form BER model for BPSK modulation and heterodyne detection space coherent optical communication system under atmospheric turbulence, pointing error, and laser phase noise is derived. Here we give the closed-form expression for received power affected by both atmospheric turbulence and pointing error. Besides, we also give the expression for the average phase noise power. Then, we conduct the numerical simulation based on the model to explore the linewidth tolerance with and without pointing error. The analysis shows that the laser linewidth tolerance is greatly affected by pointing error. Considering the pointing error is composed of jitter and boresight, the effects of jitter and boresight error on linewidth tolerance are analyzed respectively. The results indicate that linewidth tolerance decreases more under jitter than boresight which is due to the larger reduction of SNR caused by jitter. In addition, through our analysis, compared with the small pointing error, the linewidth tolerance decreases faster as pointing error increases under the large pointing error. The results of this paper have a good reference value for the selection of laser in a space coherent optical communication system.

2. Principle

2.1 Configuration of a space coherent optical communication system

The configuration of the space coherent optical communication system is shown in Fig. 1. The emitter on the satellite is composed of the signal laser diode 1 (LD1), phase modulator, radio frequency (RF) amplifier, and erbium-doped optical fiber amplifier (EDFA). The baseband signal is loaded into the phase modulator by an RF amplifier. Then the light emitted from LD1 is modulated to BPSK signal light. Finally, the light will be amplified by EDFA and sent to the space channel. On the other side, the coherent optical receiver is located on the ground, which is composed of a local oscillation (LO) LD2, a 3-dB optical coupler that adds a 180° phase shift to either the signal field or the LO field between the two output ports [5], balanced photodetectors (BPD), carrier recovery circuit, low pass filter (LPF) and digital signal processor (DSP) module. When the signal light is received and coupled into the fiber, it will be mixed with LO through 3-dB coupler, which greatly improves the sensitivity of the coherent receiver [26]. The mixing light will be detected by balanced photodetectors and converted to electrical signal. Then the detected signal is handled with carrier recovery circuit and filtered by LPF. Finally, the DSP module will conduct the demodulation of the signal.

 

Fig. 1. The configuration of a space coherent optical communication system.

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The practical space coherent optical communication system is usually disturbed by pointing error, atmospheric turbulence, and detector noise, which will all affect laser linewidth tolerance. Therefore, based on the system in Fig. 1, the BER model will be analyzed and the expression of laser linewidth tolerance under these factors will be given in the following section.

2.2 BER model analysis

In our system, compared to homodyne detection, a heterodyne coherent BPSK scheme is applied, because an optical phase-locked loop is no longer needed. This feature makes the heterodyne-detection scheme quite suitable for practical implementation in coherent light wave systems [20]. As is shown in Fig. 1, the received signal and LO are coherently mixed through a 3-dB coupler. After being detected by balanced photodetectors, the optical signal is converted to the electric signal. Based on [5], the SNR of BPSK modulation/heterodyne detection can be given by

However, the detectors are not the only devices that affect the decision threshold and deteriorate the BER performance. The laser can be another impact because the laser linewidth induced phase noise will further affect the phase modulation coherent optical communication system [10]. Taking account into the laser phase noise, the receiving current of balanced photodetectors can be given by

Here the signal phase φ_s is 0 or π, so the trigonometric transformation is conducted to split the signal phase and phase noise. The previous method [27] to calculate BER does not separate the phase noise power from the signal power. However, to calculate the SNR more conveniently, the instant phase noise power is introduced as

The laser linewidth induced phase noise θ is independent and identically distributed random Gaussian variables with zero mean and variance. The probability density function (PDF) of phase noise is given by [28]

To make the analysis simpler, here we derive a closed-form expression for the average laser phase noise power

It can be known from the Ref. [29] that

Then Eq. (5) can be further simplified into

Here we get a close-form expression for the average phase noise power, it can be seen from Eq. (4) that phase noise is mainly related to the $\mathrm{\sigma }_{\textrm{PN}}^\textrm{2}$, which is decided by the linewidth tolerance Δv and symbol period T_s.

Therefore, considering the average laser phase noise power, Eq. (1) can be rewritten by

Based on the Ref. [30], the closed-form BER expression of the BPSK modulation with the laser phase noise is given as

Equation (10) shows that the ensemble average BER of the system is related to the laser linewidth. Therefore, the laser linewidth should be limited to a certain range to maintain a low BER. Generally, the communication system is in good condition when the BER is under 1×10⁻⁹ [31], so the laser linewidth tolerance can be defined as the required linewidth under BER of 1×10⁻⁹, which is given as

However, unlike the fiber optical communication on the ground, the space coherent laser communication system suffers from atmospheric turbulence and pointing error [32], which will greatly deteriorate the BER performance. When these disturbances are existing, the linewidth tolerance will further decrease. As far as we know, the effect of atmospheric turbulence on linewidth tolerance has been studied [17], whereas the effect of pointing error on linewidth tolerance is rarely realized. Therefore, in the following part, we will establish the channel model including atmospheric turbulence and pointing error and focus on the pointing error effect on linewidth tolerance.

Firstly, the atmospheric turbulence channel model is introduced, which is described as random fluctuations of refractive index, resulting in random fluctuations of optical power. There are many statistical models to describe the atmospheric turbulence channel in space optical communication literature. The Log-Normal, K, and Gamma-Gamma distributed channel models are used by most researchers [33]. The Log-Normal distribution model usually describes the weak turbulence [34,35], K distribution model well conforms to the stronger turbulence conditions [36,37], while the Gamma-Gamma distribution can be used to describe all the turbulence scenarios from weak to strong [38], which is more suitable for the long-distance satellite-ground communication links with a complex atmospheric turbulence situation. The atmospheric turbulence is depicted by the gamma-gamma model, which is suitable for both small and large scales atmospheric fluctuations. The Gamma-Gamma model PDF is given by [33]

In the presence of atmospheric turbulence, the received optical power P_s will decrease, which is can be given as

Using the expansion of the modified Bessel function of the second kind and order of n in Ref. [40], Eq. (18) can be modified as

The result of Eq. (19) is calculated as

Apart from the atmospheric turbulence, the received optical power will be affected by the pointing error. When a Gaussian beam propagates through distance L from the transmitter, the received optical power P_s is the function of r that is the instantaneous radial displacement between the beam centroid and the detector center [41], which is given as

The pointing error is composed of boresight error and jitter error. Boresight leads to the fixed deviation of the beam center, while jitter causes the random shift near the beam center. Both boresight and jitter will cause instabilities to the detected power in the receiving terminal. Generally, at the receiver plane, the radial displacement r follows the Beckmann distribution [42]. However, in satellite FSO communication systems, it is widely accepted that jitter variance is the same for both horizontal and elevation axes [43,44]. As a result, the PDF of radial displacement r will specialize to Rician distribution [42]

With the joint effect of atmospheric turbulence and pointing error, the average received power is expressed as

To simplify the derivation, let

Then Eq. (23) can be rewritten as

The result of Eq. (23) can be further given by

Equation (27) is the closed-form expression of the received power under the combined effect of atmospheric turbulence, pointing error and laser linewidth induced phase noise. When received power is affected, the expressions for the phase noise and shot noise will change. Thus, the SNR will be rewritten as

Considering the pointing error, the closed-form BER formula can be rewritten as

When the BER in Eq. (29) is equal to BER_floor (=1×10⁻⁹), the linewidth tolerance Δv_t under different boresight error and jitter error can be obtained through the numerical simulation.

In conclusion, the closed-form BER model under the combined effect of atmospheric turbulence, pointing error and laser linewidth is derived. In the following section, based on the analysis above, the numerical simulation is conducted to discuss how pointing error affects the linewidth tolerance.

3. Simulation results and discussion

Based on the model in the second part, the linewidth tolerance in space coherent optical communication will be numerically analyzed in detail. The previous researches have investigated laser linewidth tolerance in the communication system based on fiber channel. It is shown that the linewidth tolerance could reach the order of MHz [12]. However, unlike the fiber optical communication system, the space optical communication system has a much longer channel and will be affected by some disturbances like atmosphere turbulence and pointing error. Consequently, the linewidth tolerance in the fiber communication system is no longer proper in the space communication system and the linewidth tolerance needs to be reanalyzed in the space channel. Therefore, we firstly discuss the laser linewidth tolerance without pointing error. The analysis will focus on the phase noise distribution and phase noise power under different linewidths. Then, the effect of pointing error on receiver power will be discussed in detail. Finally, linewidth tolerance under the jitter and boresight will be analyzed respectively and jointly. The system parameter settings are shown in Table 1.

Table 1. System Parameter Settings

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3.1 Linewidth tolerance without pointing error

We firstly analyze the laser phase noise distribution under different linewidths without pointing error. According to Eq. (4), the phase of the detected signal is disturbed by a Gaussian phase noise with zero mean and variable $\mathrm{\sigma }_{\textrm{PN}}^\textrm{2}$. It is known that the phase noise distribution is decided by variance $\mathrm{\sigma }_{\textrm{PN}}^\textrm{2}\; $that is related to linewidth Δν. Therefore, the system performance is decided by linewidth Δν. According to Eq. (10), we can calculate the ensemble average BER under different linewidths and the results are shown in Fig. 2(a). In Fig. 2(a), the Δν is set from 0 kHz to 1200 kHz. It is shown that as the linewidth of the laser increases, the BER of the system also increases.

 

Fig. 2. (a) The lg(BER) versus linewidth. The respective lg(BER) is marked as linewidth is 100 kHz, 300 kHz, 500 kHz and 1 MHz. The linewidth tolerance Δv_t is marked with red circle. (b) The PDF curves of phase noise θ under linewidth Δv = 100 kHz, 300 kHz, 500 kHz, 1 MHz respectively (left ordinate). The instant phase noise power $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ versus phase noise θ (right ordinate).

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Furthermore, the BER varies more rapidly when the linewidth is narrow. For example, when linewidth increases from 100 kHz to 300 kHz, the lg(BER) changes from -13.1 to -9.8. While linewidth increases from 300 kHz to 500 kHz, the lg(BER) varies from -9.8 to -8.25. It suggests that the BER is more sensitive to the narrow linewidth. When the linewidth comes to 1MHz, the lg(BER) is -6.4 which cannot meet the needs of the communication system. Based on Eq. (11), we can calculate the linewidth tolerance Δν_t. As is shown in Fig. 2(a), the Δν_t marked with a red solid line is 386 kHz. To show the linewidth requirements clearly, the red area in Fig. 2 (a) is the linewidths when lg(BER) is under -9. Therefore, for an optical communication system, the larger the area is, the more tolerant the system is to the laser linewidth.

Although BER also increases as the laser linewidth increases in the fiber optical communication system, it is found that in the long-distance space coherent optical communication system, the laser linewidth available is limited within several hundred kHz rather than MHz. What’s more, Fig. 2(a) also shows that BER is more sensitive to the linewidth variation in the space channel. In the following paragraphs, the physical reasons of how linewidth affects the BER will be analyzed from the perspective of phase noise power and SNR. The amount of phase noise power and SNR changing with laser linewidth will be given.

To further explain how laser linewidth Δν affects the BER of the system, a quantitative analysis will be given based on Eq. (5). According to Eq. (5), the average laser phase noise power $\mathrm{\sigma }_\textrm{p}^\textrm{2}\; $is the integral of instant phase noise power $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ and phase noise PDF f(θ,Δν). So the figures of f(θ,Δν) and $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ are plotted in Fig. 2(b). In Fig. 2(b), the orange line, green line, yellow line and purple line represent corresponding PDF of phase noise θ with Δν set to be 100 kHz, 300 kHz, 500 kHz and 1 MHz respectively. It is shown that the phase noise is mainly limited in the range from -0.5 rad to 0.5 rad when laser linewidth is under 1MHz. Besides, Fig. 2(b) also indicates that the narrower laser linewidth is, the closer the PDF curve is centralized around 0. The blue line in Fig. 2(b) shows the instant $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ which is a fixed curve without any other influences. The plot shows that the instant $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ reaches valley value when phase noise is zero. In addition, the curve of $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ is symmetric about zero because the phase noise power is proportional to sin² (θ) according to Eq. (3).

Through the analysis above, we know that the results of Eq. (5) depend on the product of f(θ,Δν) and $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$. While the $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ is fixed, the f(θ,Δν) determined by linewidth is the key to integral result. To be more specific, the figures of the product of f(θ,Δν) and $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ under different linewidths are plotted in Fig. 3. Based on the integral theory, the areas bounded by f(θ,Δν)·$\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )}$ and the θ-axis are the average phase noise power caused by laser linewidth. In the Fig. 3(a), (b), (c) and (d), the phase noise powers are 3.7×10⁻⁹ mW, 1.1×10⁻⁸ mW, 1.8×10⁻⁸ mW and 3.6×10⁻⁸ mW corresponding to Δν = 100 kHz, 300 kHz, 500 kHz and 1 MHz respectively. The numerical simulation indicates that the phase noise power will get larger with the increase of linewidth. Thus, it can be concluded that when laser linewidth increases, the phase noise power will increase accordingly, and the BER of the system will increase as a result.

 

Fig. 3. (a) $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )\cdot}$f(θ,Δν) versus θ under Δν = 100 kHz. (b) $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )\cdot}$f(θ,Δν) versus θ under Δν = 300 kHz. (c) $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )\cdot}$f(θ,Δν) versus θ under Δν = 500 kHz. (d) $\mathrm{\sigma }_\textrm{p}^\textrm{2}\mathrm{(\theta )\cdot}$f(θ,Δν) versus θ under Δν = 1 MHz.

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In Table 2, the variations of received signal power P_s0, average phase noise power $\mathrm{\sigma }_\textrm{p}^\textrm{2}$, average SNR and BER with the increase of linewidths are shown. It is shown that as the linewidth increases, the phase noise will increase, and SNR will decrease correspondingly, which results in the deterioration of BER. When linewidth is below 386 kHz, the $\mathrm{\sigma }_\textrm{p}^\textrm{2}$ is below 1.4×10⁻⁸ mW and the BER is under 1.0×10⁻⁹. So, in our simulated space coherent communication system, the linewidth should be under 386 kHz to compensate for the phase noise power and ensure an appropriate SNR.

Table 2. Received Signal Power, Average Phase Noise Power, and Average SNR and BER under different linewidths.

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3.2 Pointing error effect on the BER performance

The laser phase noise error on BER performance without pointing error is analyzed above. And we find that the laser linewidth tolerance is 386 kHz to ensure the BER below 10⁻⁹. However, in the practical space optical communication, pointing error is an uncontrollable and unpredictable parament which cannot be ignored. Previous investigations have not considered the variation of the laser linewidth tolerance under the pointing error. Therefore, the effect of pointing error on linewidth tolerance will be analyzed in detail.

The pointing error is composed of jitter and boresight and both will affect the linewidth tolerance. Therefore, the following paragraphs will mainly focus on how jitter and boresight affect the linewidth tolerance respectively.

Both jitter and boresight error will cause the center of the beam to deviate from the detector center and reduce the received optical power P_s. According to Eq. (23), the received optical power P_s (σ_j, A_j) affected by pointing error is the integral of the products of the received power distribution P_s(r) and radial displacement r PDF f_p(r). Therefore, in Fig. 4(a) and (b), the curves of P_s(r) and f_p(r) are plotted.

 

Fig. 4. (a) PDF curves of radial displacement under different jitter σ_j (left ordinate) and instant received power P_s(r) (right ordinate). (b) PDF curves of radial displacement under different boresight σ_j (left ordinate) and instant received power P_s(r) (right ordinate). (c) Received optical power with and without jitter (red dotted-line and blue solid-line respectively). (d) Received optical power with and without boresight (red dotted-line and blue solid-line respectively). (e) lg(BER) with jitter and without pointing error when linewidth is 386 kHz. (f) lg(BER) with boresight and without pointing error when linewidth is 386 kHz.

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In Fig. 4(a) and (b), the radial displacement r is set from 0 m to 1000 m, the left y-axis is the PDF of r and the right y-axis is the received power distribution P_s(r). Figure 4. (a) shows the PDF curves of radial displacement r under different jitter σ_j and fixed boresight A_j, the PDF curves shift right and become less concentrated when σ_j gets larger. P_s(r) is a Gaussian distribution and becomes smaller as r gets larger. While Fig. 4. (b) shows the PDF curves of radial displacement r under different boresight A_j and fixed jitter σ_j, the PDF curves also shift right when A_j gets larger. To sum up, both σ_j and A_j will affect the PDF of r and then deteriorate the received power.

According to Eq. (27), the received power under jitter error and boresight error are presented in Fig. 4(c) and (d) respectively. In Fig. 4(c), the blue line represents the relationship between received power P_s and σ_j. The red line P_s0 is the system received power without disturbance of pointing error. When σ_j varies from 1 µrad to 6 µrad, P_s deteriorates from -40.92 dBm to -42.98 dBm. Compared with P_s0, P_s deteriorates 2.12 dB when σ_j is 6 µrad. In Fig. 4(d), the blue line represents the relationship between received power P_s and A_j. Also, the red line P_s0 is the received power without disturbance of pointing error. P_s deteriorates from -40.92 dBm to -42.28 dBm as the A_j varies from 1 µrad to 6 µrad. However, the P_s will deteriorate 1.33 dB when A_j is 6 µrad. The P_s deteriorates less under the effect of boresight. That means the boresight error A_j has less effect on the received power than the jitter error σ_j.

Due to the pointing error, the received light power P_s decreases, which leads to the deterioration of the BER. To figure out how pointing error affect the linewidth tolerance, Fig. 4(e) depicts the curves of lg(BER) with jitter and without pointing error as the laser linewidth is 386 kHz and Fig. 4(f) depicts the curves of lg(BER) with boresight and without pointing error as the laser linewidth is 386 kHz. In Fig. 4(e), the lg(BER) increases from -8.97 to -7.68 when σ_j increases from 1 µrad to 6 µrad. Compared to the lg(BER) without pointing error, the lg(BER) deteriorates 1.32 dB. In Fig. 4(f), the lg(BER) will increase from -8.97 to -8.22 when A_j increases from 1 µrad to 6 µrad. The lg(BER) deteriorates 0.78 dB compared to the lg(BER) without pointing error. It can be concluded that both jitter and boresight will affect the system performance. That means under the effect of pointing error, lg(BER) cannot maintain -9 when linewidth is 386 kHz.

In conclusion, as the pointing error gets larger the receiving terminal will receive less optical power, which will lead to the penalty of the SNR. With the pointing error, the previous laser linewidth cannot maintain a lg(BER) of -9. Thus, a narrower linewidth laser is needed to compensate for the SNR penalty. In the next section, the linewidth tolerance Δv_t will be recalculated with the pointing error.

3.3 Linewidth tolerance with pointing error

In this section, the laser linewidth tolerance under the effect of pointing error will be identified. We will firstly analyze laser linewidth tolerance with jitter error and boresight error respectively. Then the joint effort of two errors will be discussed.

Firstly, Δv_t under different jitter and fixed boresight will be analyzed. According to Eq. (29), the BER under different jitter and linewidths are calculated. The Fig. 5(a) depicts the curves of lg(BER) versus linewidth when σ_j varies from 1 µrad to 6 µrad with A_j set to be 1 µrad. When σ_j varies from 1 µrad to 6 µrad, Δv_t are found to be 382 kHz, 367 kHz, 344 kHz, 305 kHz, 242 kHz and 80 kHz correspondingly. It is shown that Δv_t decreases as σ_j increases. Secondly, Δv_t under different boresight and fixed jitter will be analyzed. The Fig. 5(b) shows the curves of lg(BER) versus linewidth when A_j varies from 1 µrad to 6 µrad and σ_j is set to be 1 µrad. When A_j varies from 1 µrad to 6 µrad, the Δv_t are 382 kHz, 375 kHz, 363 kHz, 340 kHz, 315 kHz, and 266 kHz correspondingly. Similarly, Δv_t decreases as A_j increases. It can be concluded that with the growing pointing error, the laser linewidth tolerance will decrease a lot.

 

Fig. 5. (a) BER versus linewidth Δv with different σ_j and fixed A_j. The intersections of dotted line and solid lines are the linewidth tolerances under different σ_j. (b) BER versus linewidth Δv with different A_j and fixed σ_j. The intersections of dotted line and solid lines are the linewidth tolerances under different A_j.

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To more directly show the effect of jitter error and boresight error on the linewidth tolerance, Δv_t versus jitter σ_j and boresight A_j is depicted in Fig. 6. The blue line represents how Δv_t varies under different σ_j and the orange line represents how Δv_t varies under different A_j. When σ_j increases from 0 to 6 µrad, Δv_t decreases from 386 kHz to 80 kHz. Similarly, Δv_t decreases from 386 kHz to 266 kHz as A_j increases from 0 to 6 µrad. The Δv_t reduces by 306 kHz as σ_j increases by 6 µrad, which is a decrease of 79.3%. While the Δv_t reduces by 120 kHz as A_j increases by 6 µrad, which is a decrease of 31.1%. It indicates that jitter has more impact on Δv_t than boresight. The reason why jitter has more impact on Δv_t than boresight will be explained in the following paragraphs.

 

Fig. 6. Linewidth tolerance Δv_t versus jitter σ_j and boresight A_j.

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In Fig. 4(c) and (d), the effect of jitter and boresight error on received optical power P_s is depicted. Figure 4(c) shows that the received optical power P_s deteriorates 2.18 dB as σ_j increases from 1 µrad to 6 µrad, while P_s deteriorates 1.45 dB as A_j increases from 1 µrad to 6 µrad in Fig. 4(d). It suggests that the same amount of jitter σ_j and boresight A_j have different penalty on P_s. The received optical power P_s decreases more under jitter than boresight.

Further, the variation of SNR is shown in Fig. 7 as jitter and boresight increase from 0 to 6 µrad. The blue line and orange line in Fig. 7 respectively represent the SNR curve under σ_j and A_j. When σ_j increases from 0 to 6 µrad, SNR decreases from 17.4 dB to 15.4 dB, which is a penalty of 2 dB, while when A_j is increasing from 0 to 6 µrad, SNR decreases from 17.4 dB to 16.1 dB, which is a penalty of 1.1 dB. As a result, jitter has a greater impact on SNR than boresight. When SNR is reduced, the system cannot achieve the BER of 10⁻⁹ under original linewidth tolerance Δv_t, so the phase noise power needs to be further compensated, which requires a narrower linewidth laser. Therefore, the more SNR is reduced, the more linewidth tolerance is reduced. Since jitter has a greater impact on SNR than boresight, linewidth tolerance Δv_t decreases more under jitter than boresight. The analysis above suggests that in the practical space coherent communication, jitter error should be controlled first to obtain a larger linewidth tolerance.

 

Fig. 7. SNR versus jitter σ_j and boresight A_j under linewidth tolerance Δv_t = 386 kHz.

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Besides, it can be seen in the Fig. 6 that linewidth tolerance Δv_t decreases more when jitter σ_j and boresight A_j error are large (≥ 3 µrad). In Fig. 6, when σ_j increases from 0 to 3 µrad, the Δv_t decreases from 386 kHz to 344 kHz, which is a reduction of 42 kHz. Similarly, when A_j increases from 0 to 3 µrad, the Δv_t decreases from 386 kHz to 363 kHz, which is a reduction of 23 kHz. However, when σ_j and A_j increases from 3 to 6 µrad, Δv_t decreases by 264 kHz and 97 kHz respectively. This phenomenon can also be explained from the perspective of SNR. As is shown in Fig. 7, when σ_j and A_j are increased from 0 to 3 µrad, SNR decreases by 0.6 dB and 0.4 dB, respectively. When σ_j and A_j increased from 3 to 6, SNR decreased by 1.4 dB and 0.9 dB, respectively. It indicates that the large pointing error has a greater impact on SNR than the small pointing error, so in the case of large pointing error, the Δv_t decreases more. In conclusion, in the practical space coherent optical communication., controlling the large pointing error can increase Δv_t a lot, while controlling the small pointing error has relatively tiny effect on increasing Δv_t.

During the process of space optical communication, jitter and boresight will both exist, therefore, the joint effort of σ_j and A_j on Δv_t is depicted in the Fig. 8. Both jitter and boresight are set from 1 µrad to 6 µrad. From the figure, the following conclusion can be drawn: firstly, when both jitter and boresight error are large, a narrow linewidth laser is needed. When σ_j and A_j are 6 µrad, a laser with linewidth below 8 kHz is needed to maintain a BER of 10⁻⁹. Secondly, jitter has more effect on Δv_t than boresight. When σ_j is 1 µrad, Δv_t reduces 116 kHz with A_j increasing from 1 µrad to 6 µrad. However, when A_j is 1 µrad, Δv_t reduces 302 kHz with σ_j increasing from 1 µrad to 6 µrad. Thirdly, from the trend of Fig. 8, large σ_j and A_j have more influence on Δv_t than small one, which has been mentioned in the above two paragraphs. To sum up, the effect of increasing the Δv_t by minimizing the pointing error is obvious when A_j or σ_j is large.

 

Fig. 8. The joint effort of jitter σ_j and boresight A_j on linewidth tolerance Δv_t

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Additionally, we will analyze the practical interest of simulation results of Fig. 8. Our analysis shows that the laser linewidth tolerance will decrease greatly with the increase of the pointing error. To enhance the linewidth tolerance, it is effective to control the pointing error. The pointing error is usually compensated by the acquisition, tracking, and pointing (ATP) system. However, the tracking accuracy of ATP systems is different. The result reminds us that for the space optical communication system with ATP systems of high accuracy, like the inter-satellite communication system [23], the laser linewidth tolerance can be enhanced to several hundred kHz. While for the space communication systems with ATP systems of low accuracy, like satellite-to-ground communication system [24], the linewidth tolerance can only reach several kHz. When pointing error cannot be controlled well, some other methods are needed to enhance the linewidth tolerance. For example, with the availability of high-speed digital signal processing (DSP), digital phase estimation provides an alternative for suppressing the phase noise and improving the laser linewidth tolerance. However, these carrier phase estimation (CPE) algorithms usually have a very high computational and system complexity to realize the real-time system [17]. Therefore, we think it is the combined effort of the ATP system, CPE algorithms, and other methods together to enhance the linewidth tolerance. And we provide an alternative for enhancing the linewidth tolerance from the perspective of reducing pointing error.

4. Conclusion

In this paper, the effect of pointing error on laser linewidth tolerance is analyzed in detail. First of all, we derive a closed-form BER model for BPSK modulation and heterodyne detection space coherent optical communication system under atmospheric turbulence, pointing error and laser phase noise. Based on the BER model, the numerical simulation of linewidth tolerance with and without pointing error is conducted. Due to the SNR reduction caused by long-distance space channel loss and atmospheric turbulence, the linewidth tolerance without pointing error is 386 kHz in our simulated system, which is much smaller than the case in the optical fiber channel. When pointing error is considered, the linewidth tolerance will further decrease. Our results show that when jitter σ_j and boresight A_j are both 6 µrad, the linewidth tolerance decreases from 386 kHz to 8kHz, which suggests that linewidth tolerance is greatly affected by pointing error. Besides, the results also show that as jitter σ_j and boresight A_j increase from 1 µrad to 6 µrad, Δv_t will reduce 116 kHz and 302 kHz respectively, which also indicates that jitter error has a greater impact on Δv_t than boresight. So, it should be taken more care of jitter error than boresight error when design space coherent optical communication system. In addition, our results indicate that compared with the small pointing error, under the large pointing error the linewidth tolerance decreases faster with the increasing of pointing error. That means when the pointing error is large, it is effective to suppress it to increase the linewidth tolerance. But when the pointing error is small, controlling the pointing error does not help increase the laser linewidth tolerance. The results of this paper have a good reference value for space coherent optical communication system design.