Free space optical communications (FSOC) is a promising technology for high-speed data transmission. Different from radio frequency (RF) communication, FSOC offers numerous advantages such as being license free, having a large bandwidth, and providing a secure communication [1], [2], [3]. Moreover, FSOC is characterized by its freedom from high cost associated with installation, maintenance, and repair of optical fibers [4], [5], [6]. With these advantages, FSOC has been expected to be widely applicable in various fields, including emergency response, temporary links for backhaul and fronthaul for next-generation cellular networks, disaster recovery, military and satellite applications, etc. [6], [7], [8].

Despite those advantages of FSOC, there are several shortcomings, especially, atmospheric turbulence-induced fading which is dependent on humidity, pressure, and temperature fluctuations, significantly deteriorating the performance of FSOC. As such, atmospheric turbulence causes fluctuations in the intensity of the received signal, impairing the link performance [9]. It also leads to beam spread where beamwidth at the receiver plane becomes wider due to the increased beam divergence [10], resulting in performance degradation. Moreover, pointing error problem arises from random misalignment between a transmitter and a receiver, which is caused by various factors such as beam wander, the movements of the transmitter and the receiver, and beam tracking error [1]. Those factors induce the fluctuating loss of signal to noise ratio (SNR), which deteriorates the performance of communications such as bit error rate (BER) and outage probability [10].

To address these aforementioned challenges, research on multiple-input multiple-output (MIMO) FSOC is actively underway, aiming to enhance the link performance by adopting the spatial diversity [11], [12], [13], [14], [15], [16], [17], [18], [19]. MIMO FSOC with spatial diversity can exploit the characteristic of high directivity of the beam, which can offer a reliable communications link even in scenario when the communications link transmitted from a single transmitter may be temporarily blocked due to obstacles or atmospheric conditions. Moreover, when the distance between a transmitter and a receiver is far enough, the impact of atmospheric turbulence can be mitigated, since multiple channels between transmitter and receiver lenses become independent [10], [11]. In addition, spatial diversity can exploit the property of wide beamwidth by combining the signals incoming from adjacent receiver [14], [16].

Most existing papers have primarily focused on transmitting the beams to the center of a receiver lens [20], [21], [22]. However, due to the presence of atmospheric turbulence and pointing error, the transmit beam coordinates maximizing the link performance such as diversity SNR may not be achieved by beam pointing to the center of its paired receiver lens. Especially, the optimal beam coordinates keep changing due to the randomness of atmospheric turbulence and pointing error. Thus, it is essential to employ a robust beam pointing algorithm for improving the link performance, adapting to the dynamics of FSOC channel environment.

Various studies are underway to analyze and improve the FSOC link performance adopting spatial diversity techniques, aimed at mitigating the adverse effects of atmospheric turbulence [11], [12], [13], [14], [15], [16], [17], [18], [19]. In [11], [12], [13], the exploration of the average achievable rate of MIMO FSOC with both equal gain combining (EGC) and maximal ratio combining (MRC) were studied. In [14], outage probability metric was used to demonstrate the effectiveness of turbulence mitigation with spatial diversity techniques. This work showed that EGC can achieve similar performance with optimal combining technique, but all these works did not account for the pointing error. On the other hand, the works in [15], [16], [17], [18], [19] considered the impact of atmospheric turbulence and pointing error, but these studies assumed that each transmitter aligns its beam towards fixed coordinates.

In addition, there are many studies to optimize the system parameters of FSOC [18], [19], [20], [21], [22]. The beamwidth optimization in a single-input single-output (SISO) FSOC was studied to minimize the outage probability [20] or maximize the network throughput [21], [22], while considering the impact of both atmospheric turbulence and pointing error. Furthermore, the optimal number of transmitter and beamwidth were analyzed for maximizing the average achievable rate under MIMO FSOC [18], and the beamwidth optimization to minimize BER was performed in [19]. However, all these works assume that each transmitter lens align its beam towards the center of its paired receiver lens. In FSOC with a single receiver lens, such as SISO or multiple-input single-output (MISO), transmitting the beam towards the center of the receiver lens will achieve the best performance. However, in a situation with multiple receiver lenses where diversity technique is used by exploiting the property of wide beamwidth, this beam pointing approach may not maximize the network throughput.

Most of existing research on MIMO FSOC for spatial diversity have been conducted with the fixed center coordinates of the beam footprint such as the center of its corresponding receiver lens [16], [17], [18], [19] and the centroid of receiver lenses [15]. To the best of our knowledge, the optimal transmit beam pointing to exploit the divergence of wide beamwidth has not been investigated yet. Moreover, it is crucial to obtain the robust beam pointing with respect to the dynamics of FSOC channel, i.e., atmospheric turbulence and pointing error, for providing the stable link performance.

Due to the beam divergence, the atmospheric turbulence, and the pointing error, it may not be optimal to align the center of beam footprint from a transmitter lens with the center of a receiver lens. In this paper, we propose the optimal beam pointing algorithm to maximize the received SNR at the receiver. Specifically, we investigate the beam pointing algorithm with an arbitrary number of transmitter and receiver lenses for EGC based MIMO FSOC. The main contributions of our work are summarized as follows.

We first analyze the convexity condition of the received SNR with respect to the beam coordinates under EGC based MIMO FSOC. We then propose an alternative algorithm to obtain optimal transmit beam pointing which maximizes the received SNR.

We analyze the average SNR with respect to atmospheric turbulence and pointing error. We then investigate the optimal beam pointing method for maximizing the averaged SNR, which is robust to atmospheric turbulence and pointing error in a dynamic network and does not require instantaneous channel state information (CSI).

We validate the superiority of the proposed algorithm through simulation results under various channel conditions. These simulations demonstrate that the optimal coordinates obtained from the proposed algorithm outperforms conventional methods where the beam is directed towards the center of the paired receiver lens.

We can offer guidance for designing beam alignment at a transmitter in a MIMO FSOC system by showing that the optimal beam coordinates, maximizing the average SNR, converge towards the centroid of the receiver lenses as the strength of atmospheric turbulence and pointing error increase.

The rest of this paper is organized as follows. We present our system model for MIMO FSOC in Section 2, and propose the algorithm in Section 3 to find the transmit beam coordinates for maximizing the network throughput. Then, we analyze the average SNR and optimize the transmit beam coordinates for maximizing the average SNR in Section 4. Section 5 presents simulation results to validate the proposed algorithm. Lastly, Section 6 concludes this paper.