1.

INTRODUCTION

Free Space Optical (FSO) communication, as a wireless optical communication technology characterized by high bandwidth, high speed, high security, abundant spectrum resources, and strong resistance to electromagnetic interference, is expected to be applied in new communication scenarios such as space-ground integration networks, satellite networks, and autonomous driving networks [1-3]. Traditional FSO communication systems use lasers as carriers to transmit the signals in free space, requiring precise alignment of the transmitter and receiver to establish a reliable full-duplex communication link. As the communication distance and the movement speed of terminals increase, the technical difficulty, hardware cost, and power consumption of high-precision alignment equipment also rise, which prevents the weight reduction of FSO transceivers and practical applications of FSO communication [4]. To address this issue, FSO systems using modulating retroreflector (MRR) have been developed by replacing one transceiver in the traditional FSO system with a semi-passive MRR terminal to construct an asymmetric full-duplex link. The MRR terminal mainly consists of an optical retroreflector and a spatial light modulator, and its complexity, power consumption, and cost are considerably reduced due to no requirement of laser source and high-precision alignment equipment, which makes it suitable for space and mobile communication scenarios [5-8]. In fact, MRR FSO communication systems need to reuse the laser carrier to achieve full-duplex communication. For the uplink communication that brings back the MRR terminal’s information to the transceiver, the laser carrier undergoes a round-trip transmission in the atmospheric channel, and experiences quadruple beam expansion and double atmospheric turbulence effects. The consequent severe signal attenuation and disturbance limit the communication quality and distance [9-11].

Traditional FSO systems usually employ intensity modulation/direct detection (IM/DD), but coherent detection can improve receiver sensitivity to remedy the turbulence-induced fading, enhance optical signal frequency selectivity to suppress background light interference, and increase multi-dimensional signal modulation methods to enhance communication capacity. and hence it gains wide attention and research [12-17]. Kiasaleh et al. first used the K-distribution to describe atmospheric coherent channels and provided an analytical expression for average bit error rate of Differential Phase-Shift Keying (DPSK) [18]. Subsequent studies focused on different channel models, such as Lognormal, Gamma-Gamma, M-distribution, Lognormal-Rician, and modified Rician [19-23], analyzed various modulation formats, such as On-Off Keying (OOK), Pulse Position Modulation (PPM), Binary Phase Shift Keying (BPSK), M-ary Phase Shift Keying (MPSK), and Multiple Quadrature Amplitude Modulation (MQAM) [20, 24-27], and considered different diversity reception methods, such as Selection Combining (SC), Maximal Ratio Combining (MRC), and Equal Gain Combining (EGC) [27-29]. These extensive theoretical and experimental studies have basically confirmed that coherent detection offers better performance than direct detection in FSO communications. However, due to the strict frequency and phase matching requirements between the local oscillator (LO) light and the signal light, the control system becomes complex, and the computational burden of signal processing is heavy, thus limiting practical applications [13].

For MRR FSO communication systems, the transceiver transmits a laser beam towards the MRR terminal, and the echo beam reflected by the retroreflector returns to the transceiver [30]. It is evident that the echo beam and the transmitted beam are produced by the same laser and recombined at the transceiver, making the MRR FSO communication system structure naturally suitable for coherent detection, especially for homodyne detection [10]. However, the round-trip MRR FSO channel is different from the traditional single-trip FSO channel. So far, research on coherent detection for MRR FSO communication systems is still rare, with some studies focusing on reflective coherent fiber communication [31] and experimental studies on fiber-based coherent MRR FSO systems [32, 33].

This paper presents a theoretical calculation of the bit error rate (BER)performance of coherent detection MRR FSO systems, and provides analytical expressions for bit error rate (BER) performance of different modulation schemes in the MRR FSO double-trip channel, including OOK modulation under direct detection and OOK, BPSK, MPSK, and DPSK modulation under coherent detection. By comparing different detection methods under OOK modulation, the advantages of coherent detection are determined.The performance advantages and disadvantages of different modulation schemes under coherent detection are compared toprovide design references and optimization guidance for practical application system construction.

2.

MRR FSO COMMUNICATION SYSTEM

2.1

System Model

An typical coherent MRR FSO communication system using a single CCR is shown in Figure. 1. It mainly consists of the transceiver, the atmospheric turbulence channel, and the MRR terminal. In the transceiver, the homodyne detection is realized in this work.

Figure. 1

Schematic of coherent MRR FSO communication system.

The transceiver consists of a laser source, beam splitter 1, perforated reflector, reflector mirrors, beam splitter 2, and photodetector. The laser transmits the laser carrier, and the two beam splitters and two mirrors form the optical interference and homodyne coherent detection at the transceiver. The photodetector converts the received optical signal into an electrical signal. The MRR terminal includes a corner-cube retroreflector (CCR) and an optical modulator (OM). The CCR enables the incident light beam to be retroreflected, and the OM modulates the laser beam in various ways. The laser carrier is transmitted from the transceiver, passes through the atmospheric channel to the MRR terminal. A part of the beam is received by the photodetector at the MRR terminal, where the information transmitted from the transceiver is recovered which is referred to as the downlink. For clarity, the photodetector at the MRR terminal is not shown in Figure. 1. Meanwhile, the other part of the beam is modulated again by the MRR and retroreflected back to the transceiver, where it undergoes coherent detection to recover the information transmitted from the MRR terminal which is referred to as the uplink. Since the downlink is identical to a traditional FSO link, this paper only considers the uplink thatisalso referred to as the retroreflection link.

2.2

Double-pass Channel Model

In the retroreflection link of the MRR FSO system, the laser carrier first passes through the atmospheric turbulence channel from the transceiver to the MRR terminal, referred to as the forward channel. It then passes through the atmospheric channel from the MRR terminal back to the transceiver, referred to as the backward channel. Together, these are known as the doublepass channel. Intuitively, the double-passchannel can be viewed as a cascade of two single-pass FSO channels with spatial overlap between the two channels [34]. According to existing research, the fading coefficients of the forward and backward channels can be described by two correlated but different Lognormal (LN) distributions. This statistical model of the doublepass channel is known as the double LN model, and its probability density function (PDF) can be expressed as [35].

Where I_s is the normalized intensity of the received optical signal at the transceiver, R is a constant effective reflection coefficient of CCR. X₁ and X₂ are the normalized logarithmic received light amplitude for the forward and backward channels, respectively. μ_Xi and σ_Xi. are the average and standard deviation of X_i, as calculated by:

where I_i is the normalized received light intensity in the single-pass channel, I_i = exp(2X_i). μ_Ii and σ_Ii are the average and standard deviation of I_i. In (1), ρ_X is the correlation coefficient between X₁ and X₂, which is un able to be obtained directly, but it could be calculated from the correlation coefficient ρ_I between I₁ and I₂.

2.3

Coherent Detection Model

From the characteristics of the MRR FSO retroreflection link, it is evident that the reflected echo wave coherently combines with the LO light at the beam splitter 2 in the transceiver. The photodetector receives this coherent light intensity and converts it into the corresponding electrical signal. In this paper, we assume that the intensity of the reflected echo optical signal, denoted as I_s, follows the double LN model in (1). The intensity of the LO optical wave, denoted as I₀, is considered as a constant. The interferenceof the two optical waves can be expressed as [36]:

The detector receives the corresponding light intensity, and forms the following signal current:

Where η is the photoelectric conversion efficiency, and set to be 1. n(t) is the shot noise of the photodetector, which can be regarded as additive Gaussian white noise with zero mean variance . The phase component of the output signal contains the phase and phase noise of the signal, i.e. ϕ_so(t) = ϕ_si(t) + ϕ_sn(t), and the phase noise after demodulation is ϕ_n(t) = ϕ_sn(t) − ϕ₀(t). On the other hand, for the low-frequency component I_s + I₀, the frequency of the communication signal is significantly higher than that of the low-frequency component, so the final output signal is obtained by using simple high-pass filtering:

where I_s is the time-varying received light intensity under the influence of atmospheric turbulence, and ϕ_si(t) is the phase of signal modulation. The phase noise ϕ_n(t) has many ways of estimation and elimination, so we will not discuss it in this paper.

3.

BIT ERROR RATE (BER) PERFORMANCE ANALYSIS

Bit Error Rate (BER) is commonly used performance evaluation metric in digital communication systems. For the retroreflection link of MRR FSO systems, this paper theoretically calculates the average BER performance under the double LN channel for the first time. For any fading channel, the average BER can be expressed as the average of the conditional BER over the channel model for different modulation schemes [9]:

where P_c(I_s) represents the conditional BER. In the following, we will analyze the performance of On-Off Keying (OOK) modulation with direct detection and OOK, BPSK, MPSK, and DPSK with coherent detection. Herethe OOK modulation with direct detection is discussed as a benchmark for comparison.

3.1

OOK Modulation with Direct Detection

Forure fair performance comparison, the system parameters for OOK modulation with direct detection need to be consistent with those configured for the coherent detection models. Thus, the reflected echo optical signal intensity I_s follows the double LN model as defined in (1). The optical intensity received by the photodetector is converted into a photocurrent signal i_c (t) = ηI_s(t) + n(t), where the photodetector’s photoelectric conversion efficiency is assumed as η = 1, and the photodetector’s shot noise n(t) is modeled as additive white Gaussian noise (AWGN) with zero mean and variance . When a bit of 1 is transmitted, the transmitted optical intensity is 1, and when a bit of 0 is transmitted, the transmitted optical intensity is 0, which means the average transmitted optical intensity is 1/2. The instantaneous signal-to-noise ratio (SNR) can be defined by [35]:

were the average SNR is , then the conditional BER of direct detection of OOK modulation can be expressed as [9, 35]:

Where erfc(∙) and Q(∙) are complementary error function and Gaussian Q function. Combining (1) and (9) into (7), and replace the variable , then one have the average BER:

Where, y_i and w_i are zeros and weight coefficients of n-th Hermite polynomial [37].

3.2

OOK Modulation with Coherent Detection

In the SNR expression (8) for OOK modulation with direct detection, the variance of the photodetector’s shot noise n(t) can be expressed as [9]:

where e represents the electron charge and B represents the baseband bandwidth. The instantaneous SNR (8) can then be rewritten as:

In OOK modulation with coherent detection, the effective photocurrent of the homodyne coherent detection (6) increases to , which can significantly improve detection sensitivity while the required system bandwidth remains B. Generally, the LO opticalintensity is much stronger than the signal intensity, leading to shot noise predominantly caused by the LO wave. Therefore, the noise variance and SNR are:

From (12) and (14), when the LO light intensity is large, the homodyne detection SNR for OOK modulation is twice as large as that of direct detection, . The average BER of the system for OOK modulation homodyne detection is as follows:

3.3

BPSK and MPSK Modulation Coherent Detection

According to the coherent detection model described in section 1.3, the phase of the MPSK-modulated signal ϕ_si(t) can take values of β_i = 2iπ/M, i = 0,1,2, …, M − 1. For the typical cases of BPSK modulation, β_i = 0 and β_i = π, i.e. and . Using (6) and (14), the instantaneous SNR for the BPSK and MPSK can be expressed as [37]:

The corresponding conditional BERs are, respectively,

The average BER for coherent detection of BPSK and MPSK modulation can be calculated by:

Although the above formula is not in closed form, one can solve it by using numerical integration.

3.4

DPSK Modulation with Coherent Detection

Although DPSK modulation performs slightly worse than Phase-Shift Keying (PSK) modulation, it is not affected by carrier phase recovery ambiguity, making it commonly used in practical systems. The conditional Bit Error Rate (BER) for M-ary DPSK (MDPSK) is given as [37]:

In FSO communications, high-order MDPSK modulation signals are highly susceptible to atmospheric turbulence, making them difficult to be implemented. Therefore, this paper only considers Binary DPSK (BDPSK) modulation, for which the conditional BER can be simplified from (22):

The average BER is then given by:

Although the above expression has no closed-form solution, numerical integration can be used to solve it.

4.

RESULTS AND DISCUSSION

Using the BER formulas for different modulation and detection methods mentioned above, the performance under different turbulence conditions are calculated and compared. Based on practical application scenarios, the MRR FSO communication system is configured accordingly. As shown in Figure. 1, the transceiver is configured with a laser with the wavelength of 1550 nm and beam waist of 16 mm, which is collimated towards the MRR terminal through an optical transceiver antenna with a 25 mm aperture. The CCR at the MRR terminal has a 50 mm aperture, and its effective reflection coefficient is assumed as R = 1. For the atmospheric turbulence channel, the distance between the transceiver and the MRR terminal is set to 1 km, and the retroreflection link length is equal to 2 km. Considering atmospheric turbulence with an inner scale of 6 mm and an outer scale of 1.5 m, three different atmospheric refractive index structure parameters , 10^–14, and 10⁻¹³m^−2/3 are set to consider the weak, moderate, and strong turbulence conditions. In the retroreflection link, the mean and variance (scintillation index) of the received optical intensity for the forward and backward channels under the three turbulence conditions are listed in Table 1. According to (2), the means μ_X1, μ_X2 and the variances σ_X1, σ_X2 of the normalized logarithmic received light amplitude for the forward and backward channels are calculated.

Table 1:

Received optical Intensity Indicators for Forward and Backward Channels under Different Turbulence Conditions.

First, we consider the case where the forward and backward channels are independent, i.e., ρ_I = 0, and then use the aforementioned BER formulas to calculate OOK, BPSK, 4PSK, BDPSK homodyne detection, and the BER curves for OOK with direct detection. The results are shown in Figure. 2. Regardless of the turbulence strength, the OOK with direct detection performs the worst, while the BPSK with coherent detection performs the best, and the performance of the 4PSK is lower than that of the BPSK. Additionally, the performance of the BDPSK is just lower than that of the BPSK. Moreover, at higher SNRs, the BER curve slope of BDPSK is steeper than that of BPSK, as shown in Figure. 2(a).

Figure. 2

BER performance for different detection: .

In the MRR FSO system, the forward and backward channels largely spatial overlap, and the time taken for the optical signal to pass through both paths is often shorter than the atmospheric turbulence correlation time (on the order of milliseconds). Therefore, based on the quasi-static atmospheric turbulence criterion, the forward and backward channels are affected by very similar atmospheric turbulence, leading to a high correlation in their channel fading [9,35]. Here, we use the BPSK with coherent detection to consider the impact of channel correlation on the BER performance.

By setting correlation coefficients of p_I = 0, 0.3, 0.6, and 0.9, we calculate the BER performance of the BPSK under different turbulence conditions, and the results are shown in Figure. 3. From the results, in any turbulence condition, as the correlation coefficient between the forward and backward channels increases, the BER performance degrades. This confirms that channel correlation inevitably reduces system performance.

Figure. 3

BER performance of BPSK with various channel correlation coefficient: .

5.

CONCLUSION

The MRR FSO system structure can effectively reduce the requirement for mutual tracking between the two communication terminals. However, since a single laser carrier needs to pass through the atmosphere twice, it is inevitablyaffected by more atmospheric turbulence, leading to a decline in system performance. Therefore, the use of the coherent detection to improve receiver sensitivity, signal-to-noise ratio (SNR), and overall system performance is of significant value.

In this paper, with the aid of the unique structure of MRR FSO communication systems, a zhomodyne coherence detection model is constructed for the first time, and the BER performance formula of the BPSK, MPSK and BDPSK with coherence

detection is given. Numerical calculations gave BER performance curves under different turbulence conditions for various modulation and detection methods. The results indicate that the BPSK offers the best performance, while the highly practical BDPSK exhibits the steepest BER curve slope. Additionally, the fading correlation between the forward and backward channels could impact the BER performance. The coherent detection model and BER calculation formulas developed in this paper constitute foundational work, which can provide theoretical support for future research and application design of more modulation methods, coding methods and communication structures of MRR FSO.