BDCOA: Wavefront Aberration Compensation Using Improved Swarm Intelligence for FSO Communication

Abstract

Free Space Optical (FSO) communication is extensively utilized in the telecommunication industry for both ground and space wireless links, as well as last-mile applications, as a result of its lesser Bit Error Rate (BER), free spectrum, and easy relocation. However, atmospheric turbulence, also known as Wavefront Aberration (WA), is considered a serious issue because it causes higher BER and affects coupling efficiency. In order to address this issue, a Sensor-Less Adaptive Optics (SLAO) system is developed for FSO to enhance performance. In this research, the compensation of WA in SLAO is obtained by proposing the Brownian motion and Directional mutation scheme-based Coati Optimization Algorithm, BDCOA. Here, the BDCOA is developed to search for an optimum control signal value of actuators in Deformable Mirror (DM). The incorporated Brownian motion and directional mutation are used to avoid the local optimum issue and enhance search space efficiency while searching for the control signal. Therefore, the dynamic control signal optimization for DM using BDCOA helps to enhance the coupling efficiency. Thus, the WAs are compensated for and optical signal concentration is enhanced in FSO. The metrics used for analyzing the BDCOA are Root Mean Square (RMS), BER, coupling efficiency, and Strehl Ratio (SR). The existing methods, such as Simulated Annealing (SA) and Stochastic Parallel Gradient Descent (SPGD), Advanced Multi-Feedback SPGD (AMFSPGD), and Oppositional-Breeding Artificial Fish Swarm (OBAFS), are used for evaluating the performance of BDCOA. The RMS of BDCOA for iterations 500 is 0.12, which is less than that of the SA-SPGD and OBAFS.

1. Introduction

In the era of 6G, Free-Space Optical (FSO) communication is considered an effective solution in obtaining comprehensive signal coverage in airborne, terrestrial, and satellite networks via its free-space data transmission [1]. FSO is a technology used to broadcast a signal in wireless media by dispersing the light in free space, such as in vacuum and air. Therefore, the FSO transmission channel is a substitute solution in the field, which has geographical restrictions for installing fiber in roads, mountains, rivers, and so on [2]. FSO has similarities with optical fiber communication and offers various benefits such as accessibility of unlicensed spectrum, improved security, lesser atmospheric absorption, resistance against electromagnetic interference, higher channel capacity (terahertz bandwidth), and ease of installation in challenging places [3,4,5]. FSO is applicable in numerous fields, such as those of airborne, space, inter-satellite, and terrestrial links [6]. The typical wireless communication technology utilizing the Radio Frequency (RF) spectrum is applied for general purposes [7]. However, FSO is appropriate when there is a requirement for license free broadcasting, extensive bandwidth, and noise immunity with enhanced capacity [8,9,10].

In FSO, the data are initially converted to a modulated light signal and broadcasted through the air to the receiver, which then discovers the light signal and converts it back into data. If the requirement of data rate is to be several gigabits per second, FSO utilizes lasers or light-emitting diodes to generate the light signals and transmit information at rapid speeds over huge distances [11,12]. Additionally, FSO is utilized in blockage and last-mile applications, and is effective in the back-haul network of upcoming communication networks [13]. The propagation of a laser signal via an atmosphere channel is affected by atmospheric turbulence [14]. This atmospheric turbulence creates phase distortion, broadening laser signal amplitude scintillation, power, deep fade, and beam wandering in the receiver. The closed loop Adaptive Optics (AO) is utilized to minimize the impact of atmospheric turbulence by altering the Wavefront Aberrations (WAs) [15,16,17]. Accordingly, the control of wavefront correctors in the AO is handled by using the control algorithm by employing an optimization algorithm [18]. Moreover, the AO is helpful in overcoming the impact of random optical disturbance [19,20,21].

This section discusses the existing studies related to optical communication along with their advantages and limitations.

Ata et al. [22] presented the uplink and downlink communication system based on AO correction in the transmitter and Maximum Ratio Combining (MRC) diversity approach in the receiver. The AO correction in transmitter and MRC in receiver were used to enhance the BER performance in downlink communication rather than the uplink communication. The receiver diversity with MRC was effective in AO correction. The failure of concentrating over the phase compensation was subjected to signal distortion, which increased the BER.

Peng et al. [23] developed the Adagrad and Staged SPGD algorithm (AS-SPGD) for enhancing convergence in fiber coupling for FSO. The acceleration of SPGD was obtained by integrating the Adagrad’s adaptive gain approach of Adagrad SPGD to obtain wavefront tip-tilt aberration compensation based on FSM. Further, the local convergence of SPGD was enhanced by using the Staged SPGD approach. An optimum control signal was required to be selected to further minimize the WA.

Sridhar et al. [24] evaluated the FSO’s performance by using the M-ary Phase Shift Keying (MPSK), Binary Phase Shift Keying (BPSK), On-Off Keying (OOK), and M-ary Quadrature Amplitude Modulation (M-QAM) modulation approaches in transmitters and avalanche photodiode (APD) in receivers. The examination of FSO broadcasting was considered the normal (turbulence free) channel model, with the atmospherically turbulent gamma channel model used for the condition of weak/medium and strong air turbulence over air channels. However, the small routing error affected the performance of the FSO in atmospheric turbulence. In general, the small routing error was the misalignment of the route of transmitted optical beam corresponding to the intended receiver.

Zhang et al. [25] presented the combination of adaptive moment estimation (Adam) from deep learning and SPGD, AdamSPGD, to ensure an effective wavefront correction. The development of AdamSPGD was used to enhance the correction speed and robustness of SLAO. In this approach, huge Zernike polynomial modes were utilized for precisely denoting the WAs. An inappropriate control signal in the DM of SLAO caused signal distortion during communication, thereby increasing the BER.

Lv and Hong [26] developed an adaptive threshold return-to-zero on-off keying based on self-pilot tone, namely RZ-OOK decision for FSO communication. The RZ-OOK had impulse series features in the spectrum and these impulses were used as transmitted signal’s pilot tones for transferring the Channel State Information (CSI). Further, an Adaptive Threshold Decision (ATD) was obtained by allocating optimum weight factors in the obtained CSI signal to enhance the CSI signal’s power. The signal distortion was aroused in RZ-OOK when the WA compensation was not effectively performed in FSO.

Liu et al. [27] developed the hybrid algorithm for enhancing coupling efficiency and mixing efficiency by improving the performance indicators of a coherent communication system. In this hybrid approach, the combination of SA and SPGD algorithms were used for error correction. The hybrid approach employed rapid convergence speed of course correction from SA along with SPGD for modifying the residual error. This process minimized the probability of local optimization in the SPGD algorithm and ensured reliability in correction. In optical communication, the phase compensation was required to be obtained for further enhancing the performances.

Li et al. [28] presented the AMFSPGD for enhancing the correction of WA. The spot array was obtained by using a multiplexed computer-generated hologram. Accordingly, the consistent Strehl Ratio (SR) was acquired to provide multi-feedback, which suggested highly extensive optimized information. The search gradients were effectively adjusted while obtaining an optimal value with effective probability based on the integration of global and local gradient in AMF-SPGD. The usage of hard max recognition was subjected to make corrections unstable in FSO.

Kumar and Khandelwal [29] presented the OBAFS approach for correcting the WA in SLAO system. The utilization of OBAFS was used to obtain an optimum control signal for DM’s actuators. Next, the discovery of control parameters was used to compensate the WA and to enhance the FSO communication. Effective exploration and exploitation were required to be obtained in order to search for appropriate control signals.

The summarization of related works is provided along with the advantages and limitations in

Table 1. Literature review.

Author	Methodology	Advantages	Limitations
Ata et al. [22]	An uplink and downlink communication system was developed by using AO correction in the transmitter and MRC diversity approach in the receiver.	The performance of BER was minimized based on the AO correction and MRC receiver.	Signal distortion was caused during communication because of the failure in performing phase compensation.
Peng, J et al. [23]	The convergence in the fiber coupling of FSO was enhanced by using the AS-SPGD.	The incorporation of Staged SPGD was used to enhance the local convergence in FSO.	Reduction of WA was required to be done based on the selection of optimum control signal.
Sridhar et al. [24]	The performance of data broadcasting in FSO was examined with different modulation approaches such as MPSK, BPSK, OOK and M-QAM in transmitters and APD in receivers.	The M-PSK was designed for maximizing the data rate to enhance the transmission quality and BER.	The FSO performance in atmospheric turbulence was affected due to small routing error.
Zhang et al. [25]	The correction of wavefront was achieved by integrating the Adam from deep learning and AdamSPGD in FSO.	A precise depiction of WA was obtained by utilizing the Zernike polynomial modes.	In communication phase, the signal distortion occurred due to processing of DM with an inappropriate control signal in SLAO.
Lv and Hong [26]	An adaptive threshold RZ-OOK decision was developed according to the self-pilot tone to obtain the CSI of FSO.	The power of CSI signal was enhanced by assigning the optimum weight factors to achieve ATD.	In this work, there was a probability of signal distortion occurrence when the WA compensation was not done in FSO.
Liu et al. [27]	A hybrid approach of SA with SPGD was developed to perform error correction.	The integration of SA and SPGD reduced the probability of local optimization, supporting the reliability in correction.	Phase compensation was required to be done for further minimizing the errors in communication.
Li et al. [28]	The AMFSPGD was presented for improving the correction of WA.	The integration of global and local gradient in AMF-SPGD was used to effectively adjust the search gradient and obtain an optimal value for WA correction.	Corrections in FSO were unstable due to hard max recognition.
Kumar and Khandelwal [29]	The WA correction in SLAO system was developed by using OBAFS.	The control parameters selection was utilized to compensate the WA which improved FSO performances.	However, the selection of appropriate control signals was required to be obtained for additionally minimizing the error performances.

.

The limitations found from the existing studies are as follows: failure to concentrate on WA correction, higher BER, inadequate coupling efficiency, and inappropriate searching for control signals. The solution provided by the proposed research is specified in the subsequent text. The Brownian motion and Directional mutation scheme-based Coati Optimization Algorithm, namely BDCOA, is proposed to discover the optimum control signal value for DM in SLAO to minimize the WA. The issue of local optimum is avoided, while the search space efficiency is enhanced by integrating the Brownian motion and directional mutation in BDCOA. This enhanced searching process helps find optimum control signals. Hence, the optimization of dynamic control signals in DM using BDCOA also improves the coupling efficiency. Further, the compensation of WA is used to enhance optical signal concentration and minimize BER.

The key contributions are summarized as follows:

• The SLAO system is used as an important module for compensating the losses caused by WA in the FSO;
• In SLAO, the BDCOA is used for searching optimum control signals for the actuators in the DM. The Brownian Motion (BM) and Directional Mutation Scheme (DMS) are integrated into the BDCOA for avoiding the suboptimal solutions and enhancing the search space efficiency while searching for an optimum control signal;
• The wavefront distortions are compensated by selecting the control signals using BDCOA, which further helps enhance the quality of the received signal and coupling efficiency. The enhanced quality of received signals aids in minimizing the BER and RMS.

The remainder of this paper is organized as follows: The BDCOA based WA compensation in FSO is detailed in Section 2, while Section 3 details the outcomes of the research. The research is concluded in Section 4.

2. BDCOA Based Optimum Control Signal

In modern communication systems using FSO, the WA in the receiver reduces the coupling efficiency of broadcasting a laser signal (i.e., optical signal). The SLAO system is one of the important modules used for avoiding the losses that occur due to the WA. In this research, the SLAO is utilized with BDCOA to choose the optimum control signal values for actuators of DM. According to the optimum control values from BDCOA, the phase compensation is obtained in DM by minimizing the phase distortion and enhancing the FSO communications.

2.1. FSO System Model

Figure 1 shows the architecture of the FSO-based communication model, which comprises different modules such as Single-Mode Fiber (SMF), SLAO system, receiving terminal, and laser point source. A laser beam is utilized as an optical point source to emit the Gaussian beam in FSO. Next, the discharged signal is broadcast via the atmosphere with the laser carrier signal. The phase consistency is affected by the atmospheric turbulence caused by the WA. The utilization of sensor-less AO is employed to compensate for the WA. The coupling over the calibrated laser beam is ensured by using the SMF. In the reception terminal, the laser beam’s coupling efficiency is highly enhanced based on WA compensation, which further enhances the FSO performance.

2.2. Model of SLAO System

The working module of the SLAO of FSO is given in Figure 2, which comprises of Charge Coupled Device (CCD), beam splitter, lenses, controller, and a Deformable Mirror (DM). As illustrated in Figure 2, the laser carrier signal with WA is divided into two signals by utilizing the beam splitter.

Here, one signal is used to perform the aberration compensation while the remaining signal’s intensity is recorded in the CCD. The compensation phase $C\left(r\right)$ made by DM is used to fix the wavefront aberration $A\left(r\right)$. Equation (1) represents the residual phase $R\left(r\right)$ computation based on the $A\left(r\right)$ and $C\left(r\right)$.

$$R\left(r\right)=A\left(r\right)-C\left(r\right)$$

(1)

The minimization of the mean square of $R\left(r\right)$ is used to obtain the optimum voltage or control signal $v=\left\{v_1, v_2,\dots ,v_{32}\right\}$. The compensation phase $C\left(r\right)$ is computed by using the optimal control signal.

2.3. DM Model

In SLAO, the DM uses 32 actuators as the wavefront compensator. Figure 3 shows the normalized layout of 32 actuators. The Gaussian Model expressed in Equation (2) is used to define the DM’s impact function. Next, the phase compensation $c\left(x, y\right)$ is conducted and it is computed using 32 actuators, as expressed in Equation (3).

$$G_j\left(x,y\right)=exp\left\{\ln \varphi {\left[{\displaystyle \frac{1}{n}}\sqrt{{\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2}\right]}^{\beta }\right\}$$

(2)

$$c\left(x,y\right)={\displaystyle {\sum }_{j=1}^{32}}{v}_j{G}_j\left(x,y\right)$$

(3)

where, the coupling coefficient is evaluated by the DM sizes and electrode actuators, denoted as $\varphi$, the center coordinate of actuator $j$ is denoted as $\left(x_j,y_j\right)$, the normalized distance among the nearby actuators is denoted as $n$, the Gaussian index is specified as $\beta$, and voltage of $j$th actuator is denoted as $v_j$. According to Equation (3), it is found that the $c\left(x,y\right)$ is linear with $v_j$.

The optimization of actuator voltages is ensured by reducing the mean square error in $c\left(x,y\right)$. BDCOA is used for optimizing the control signals or voltages. The WA compensation using BDCOA is detailed in the following section.

2.4. Wavefront Aberration Using BDCOA

BDCOA is used to discover optimum control signals for achieving phase compensation by minimizing the phase distortion and enhancing FSO performances. To obtain optimum control values from BDCOA, the DM obtains the phase compensation by reducing the phase distortion and enhances the FSO communications. The typical Coati Optimization Algorithm (COA) [30] is a meta heuristic algorithm that replicates the coati behavior in nature. The COA uses the principle of cooperating among agents, mimicking an animal’s action. Then, the BM [31] and DMS [32] are integrated to avoid the suboptimal solutions and enhance the searching capacity, which leads to obtaining an effective balance among exploration and exploitation during the searching process. This results in enhanced global searching efficiency in optimum control signals discovery used to enhance the received signal quality. Accordingly, the minimization of phase distortion using BDCOA for DM minimizes the probability of bit errors and further improves the coupling efficiency. The searching process for optimum control signals is detailed in the following section.

2.4.1. Algorithm Initialization Process

The matrix $X$ has random initial population locations, as represented in Equation (4), where an amount of coatis is specified as $N$ and problem dimension is specified as $m$. Every coati $i$ location is expressed in Equation (5), while the objective of every population is computed and the vector $F$ is generated with the dimension $N\times 1$, as represented in Equation (6). Here, the initial solutions are the random values of voltages for the DM actuator.

$$X={\left[\begin{array}{c}{X}_1\\ ⋮\\ X_i\\ ⋮\\ X_N\end{array}\right]}_{N\times m}=\left[\begin{array}{ccccc}{x}_{1,1}& \dots & x_{1,j}& \dots & x_{1,m}\\ ⋮& \ddots & ⋮& .& ⋮\\ x_{i,1}& \dots & x_{i,j}& \dots & x_{i,m}\\ ⋮& .& ⋮& \ddots & ⋮\\ x_{N,1}& \dots & x_{N,j}& \dots & x_{N,m}\end{array}\right]$$

(4)

$$X_i:x_{i,j}=\left\{v_1, v_2, \dots ,v_{32}\right\}, i=1,2,\dots ,N, j=1,2,\dots ,m$$

(5)

$$F={\left[\begin{array}{c}{F}_1\\ ⋮\\ F_i\\ ⋮\\ F_N\end{array}\right]}_{N\times 1}={\left[\begin{array}{c}F\left(X_1\right)\\ ⋮\\ F\left(X_i\right)\\ ⋮\\ F\left(X_N\right)\end{array}\right]}_{N\times 1}$$

(6)

where, position of coati $x$ in the lookup field is denoted as $X_i$ and the estimation of the decision variable $j$ is denoted as $x_{i,j}$.

2.4.2. Directional Mutation

In this phase, the direction of coatis is changed to enhance discovery in the search space efficiency. The DMS expressed in Equation (7) is derived by using the directed value $d$ for improving the search space efficiency.

$$X=X_{sbest}+d\times \rho \times \left(X_{r2}-X_{r3}\right)$$

(7)

where, the solution formulated from the uppermost $S\times N$ solutions is specified as $X_{sbest}$, two different solutions randomly chosen from population are denoted as $X_{r2}$ and $X_{r3}$, while $\rho$ denotes the constant and real-value factor chosen among $\left[0, 1\right]$ and $d$ is the directed value, as expressed in Equation (8).

$$d=\left\{\begin{array}{cc}1\hfill & \hfill if F\left(X_{r2}\right)<F\left(X_{r3}\right)\\ -1\hfill & \hfill Otherwise\end{array}\right.$$

(8)

2.4.3. Exploration Phase (Hunting and Attacking)

In this phase, the coatis are separated into two different groups. The first group climbs the tree to frighten the prey while the second group waits under the tree for the falling of frightened prey. This kind of motion pattern allows the coati to thoroughly search the problem space. The coati’s prey is referred to as Iguana. The location of the optimum member of the population is considered as the Iguana’s location. Equation (9) specifies the first half of the coatis that climb the tree. If the Iguanas falls to the ground, its location is randomly changed based on Equation (10). Equation (11) specifies the second half of coatis that wait under the tree. The BM is incorporated in this phase, while the searching escapes from suboptimal solutions and searches in the other regions with extensive capacity to discover the global optimum value. This adaptive and dynamic process enhances the BDCOA capacity for effective exploring and exploitation of the searching space. Each step of the BM is discovered by the probability function denoted by normal Gaussian distribution with mean $\left(\delta =0\right)$ and unit variance $\left({\sigma }^2=1\right)$. The Probability Density Function of BM i.e., $BM\left(y,\delta ,\sigma \right)$ at a certain point $y$ is denoted in Equation (12).

$$X_i^{P1}:x_{i,j}^{P1}=x_{i,j}+rand·\left({\mathrm{Iguana}}_j-I·x_{i,j}\right) \mathrm{for} i=1,2,\dots ,\left|{\displaystyle \frac{N}{2}}\right| \mathrm{and} j=1,2,\dots ,m$$

(9)

$$Iguan{a}^G:Iguan{a}_i^G=l{b}_j+rand·\left(u{b}_j-l{b}_j\right)$$

(10)

$$X_i^{P1}:x_{i,j}^{P1}=\left\{\begin{array}{cc}{x}_{i,j}+rand·\left(Iguan{a}_j^G-I·x_{i,j}\right),\hfill & \hfill F_{Iguana}<F_i\\ x_{i,j}+BM·\left(x_{i,j}-Iguan{a}_j^G\right),\hfill & \hfill else\end{array}\right.$$

$$\mathrm{for} i={\displaystyle \lfloor \frac{N}{2}\rfloor }+1,{\displaystyle \lfloor \frac{N}{2}\rfloor }+2,\dots N \mathrm{and} j=1,2,\dots ,m$$

(11)

$$BM\left(y,\delta ,\sigma \right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\exp \left(-{\displaystyle \frac{{\left(y-\delta \right)}^2}{2{\sigma }^2}}\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}\exp \left(-{\displaystyle \frac{y^2}{2}}\right)$$

(12)

where, the new location computed for coati $i$ is denoted as $X_i^{P1},x_{i,j}^{P1}$ is in dimension $j$, the new location’s fitness value is denoted as $F_i^{P1}$, real random value among $\left[0, 1\right]$ is denoted as $r$, upper and lower values of variable $j$ are denoted as $u{b}_j$ and $l{b}_j$, respectively, prey’s location is denoted as $Iguana,Iguan{a}_j$ is its $j$th dimension, $I$ is an integer value that is either 1 or 2, prey’s location in ground is denoted as $Iguan{a}^G$, its $j$th dimension is denoted as $Iguan{a}_j^G,Iguan{a}^G$’s fitness value is denoted as $F_{Iguan{a}^G}$, and the floor function is denoted as $\lfloor ·\rfloor$. If the fitness of $X_i^{P1}$ of coati $i$ has better fitness than the fitness of old location $X_i$, it is unaltered. Otherwise, the old location is preserved as shown in Equation (13)

$$X_i=\left\{\begin{array}{cc}{X}_i^{P1},\hfill & \hfill F_i^{P1}<F_i\\ X_i,\hfill & \hfill else\end{array}\right.$$

(13)

Each solution (coati) fitness is evaluated according to the RMS. It is computed by using Equations (14) and (15).

$$F_i=\max \left\{{\displaystyle \frac{1}{RMS}}\right\}$$

(14)

$$\begin{gathered} RMS={\iint }_S\left(R\left(r\right)-\overline{R\left(r\right)}\right)r·dr \\ ={\iint }_S{\left[\left(A\left(r\right)-C\left(r\right)\right)-\overline{\left(A\left(r\right)-C\left(r\right)\right)}\right]}^{2}r·dr \end{gathered}$$

(15)

where the residual phase is represented as $R\left(r\right)=A\left(r\right)-C\left(r\right)$ and the normalized circle is denoted as $S$.

2.4.4. Exploitation Phase (Escaping from the Predators)

If a predator attacks the coati, the coati moves to a random location that is nearer to the location based on Equations (16) and (17). The new location is adequate while the fitness is enhanced according to the Equation (18).

$$l{b}_j^{local}={\displaystyle \frac{l{b}_j}{t}},\hspace{1em}\hspace{1em}u{b}_j^{local}={\displaystyle \frac{u{b}_j}{t}}, t=1,2,\dots ,T$$

(16)

$$X_i^{P2}:x_{i,j}^{P2}=x_{i,j}+\left(1-2rand\right)·\left(l{b}_j^{local}+rand·\left(u{b}_j^{local}-l{b}_j^{local}\right)\right)$$

$$i=1,2,\dots ,N,\hspace{1em} j=1,2,\dots ,m$$

(17)

$$X_i=\left\{\begin{array}{cc}{X}_i^{P2},\hfill & \hfill F_i^{P2}<F_i\\ X_i,\hfill & \hfill else\end{array}\right.$$

(18)

where $P2$ denotes the location and fitness of coati in the 2nd phase while the local upper and lower bounds of decision variable $j$ are denoted as $u{b}_j^{local}$ and $l{b}_j^{local}$, respectively.

The flowchart of BDCOA-based control signal optimization is shown in Figure 4.

3. Simulation Results

The BDCOA-based WA compensation is developed using MATLAB R2020b with the configuration of an i5 processor and 16 GB RAM.

Table 2. Simulation parameters.

Parameter	Names/Values
Communication channel	Free space
Optical power	10 dBm
Modulation	DPSK
Number of actuators	32
Wavelength of laser	1550 nm
Bit rate	1.25 Gb/s
Turbulence Model	Kolmogorov turbulence model
Noise	Atmospheric turbulence noise
Refractive structure parameter	$9\times {10}^{-15} {\mathrm{m}}^{-2/3}$
Length of FSO link	1000

shows the simulation parameters (i.e., Scenario 1) used for analyzing the BDCOA. The transmitter module of FSO has Continuous Wave (CW) laser source, where the data are produced using Pseudo-Random Bit Sequence (PRBS) generator, with a pulse generator based on NRZ and Differential Phase Shift Keying (DPSK) modulation. The SLAO has DM using Actuators with the SMF length of 500 m, while the receiver comprises a APD photodiode and Bessel–Thomson electrical filter. Here, Zernike polynomials are used to represent the WAs where refractive structure parameter is fixed as $9\times {10}^{-15} {\mathrm{m}}^{-2/3}$, that is turbulence intensity. This turbulence intensity leads to the moderate level of atmospheric turbulence. Further, the Kolmogorov model is considered as the turbulence model, which helps to simulate the random phase distortions generated by atmospheric turbulence. Accordingly, this Kolmogorov model considers an homogeneous and isotropic turbulent medium that interferes the broadcasting optical wavefront.

3.1. Performance Metrics

The metrics of Root Mean Square (RMS), Bit Error Rate (BER), coupling efficiency, and Strehl Ratio (SR) are considered for analyzing the BDCOA. RMS is the square root of changes in the input’s phase aberration $\left({\sigma }_A\right)$, which is denoted in Equation (19). BER is defined as the ratio of the total amount of errored bits and total amount of transmitted bits, as represented in Equation (20). Coupling efficiency is the proportion of the power existing in the transmitter to the power given to the receiver. SR, represented in Equation (21), is the quality computation metric for FSO. This is denoted as the proportion among WA’s peak intensity and higher intensity in optical system’s SR.

$$RMS=\sqrt{{\sigma }_A}$$

(19)

$$BER={\displaystyle \frac{Amount of bits received in error}{Total amount of transmitted bits }}$$

(20)

$$SR=\exp \left(-{\displaystyle {\sum }_{i=1}^N}{a}_i^2\right)$$

(21)

where, the coefficient of Zernike polynomial $i$ is denoted as $a_i$ which utilized for specifying the input WA, while the amount of Zernike modes is denoted as $N$.

3.2. Performance Analysis

The performance of BDCOA is analyzed with three different algorithms, namely Mayfly Optimization Algorithm (MOA), Pelican Optimization Algorithm (POA), and COA. The reason for considering the MOA, POA, and COA algorithms is that they are all meta heuristic algorithms similar to BDCOA. The convergence analysis for BDCOA with other optimization algorithms is shown in Figure 5, where RMS value is considered as fitness. Figure 5 confirms that BDCOA has better convergence than MOA, POA, and COA. The BM incorporated in BDCOA escapes from the suboptimal solutions and ensures that BDCOA searches extensively to obtain the global optimum value. This adaptive and dynamic process helps enhance BDCOA’s ability to effectively explore and exploit space. The DMS is used to enhance global searching efficiency during the searching process. The aforementioned advantages ensure a reliable convergence.

3.3. Root Mean Square

Figure 6 presents the RMS analysis for BDCOA with different population sizes i.e., 20, 30, 40, and 50. The evaluation of RMS with different optimization methods is shown in Figure 7. This evaluation confirms that BDCOA with population 30 has enhanced performance when compared to the other variations. Additionally, BDCOA has a lower RMS than MOA, POA and COA due to its effectiveness when searching for optimum control signals. The control signals of DM are used by BDCOA to reduce the aberrations, thereby facilitating the minimization of RMS.

3.4. Coupling Efficiency

Figure 8 displays the coupling efficiency analysis for BDCOA with different population sizes i.e., 20, 30, 40, and 50. The evaluation of coupling efficiency with different optimization methods is represented in Figure 9. The evaluation demonstrates that BDCOA with population 30 demonstrates better performance than the other variations. Additionally, the coupling efficiency of BDCOA is higher than MOA, POA, and COA. The dynamic adjustment of control signals for DM from BDCOA helps enhance coupling efficiency. Accordingly, the WAs are compensated for and an optical signal concentration is enhanced in FSO.

3.5. Strehl Ratio

The SR of BDCOA is analyzed with different population sizes as shown in Figure 10. Moreover, the evaluation of SR with different optimization methods is shown in Figure 11. From the figures, it is confirmed that the SR of BDCOA is more improved than the SR of MOA, POA, and COA. Through the minimization of phase distortions by achieving optimum control signals from BDCOA for DM, the peak intensity and sharpness of focused light are enhanced in FSO. Accordingly, this leads to high SR, representing the FSO’s performance nearer to the ideal case.

3.6. Bit Error Rate

The BER of BDCOA is analyzed with different population sizes, as shown in Figure 12. The evaluation of BER with different optimization methods is shown in Figure 13. From the figures, it is confirmed that the BER of BDCOA is lesser when compared to MOA, POA, and COA. An appropriate control signal from BDCOA with DM is used to compensate for the wavefront distortions, thereby improving the quality of the received signal. The enhanced quality of the received signal minimizes the probability of bit errors, therefore also minimizing the BER.

3.7. Comparative Analysis

The existing methods, such as SA-SPGD [27], AMFSPGD [28], and OBAFS [29], are used to analyze the BDCOA used in the FSO. SA-SPGD [27] and OBAFS [29] are analyzed for Scenario 1 while the AMFSPGD [28] considers the wavelength of 632 nm i.e., Scenario 2. The analysis between the two different wavelengths emphasizes the sensitivity of various wavelengths to atmospheric turbulence i.e., a time-varying phenomenon. Scenario 1 is generally utilized in FSO communication due to its lower attenuation in the atmosphere and higher tolerance to turbulence. Scenario 2 is highly vulnerable to scattering effects, specifically adverse weather conditions. The analysis of two distinct scenarios is used to justify the effectiveness of BDCOA in practical settings, i.e., real-world wavelength options for FSO systems.

Table 3. Comparison of BDCOA for Scenario 1.

Performances	Methods	Iterations
		100	200	300	400	500
RMS	SA-SPGD [27]	1.65	0.9	0.45	0.2	0.15
	OBAFS [29]	1.7	0.42	0.42	0.41	0.41
	BDCOA	1.49	0.36	0.35	0.15	0.12
Coupling efficiency	SA-SPGD [27]	0.77	0.785	0.795	0.798	0.8
	OBAFS [29]	0.87	0.93	0.93	0.93	0.93
	BDCOA	0.89	0.94	0.94	0.94	0.95
SR	OBAFS [29]	0.66	0.83	0.86	0.86	0.86
	BDCOA	0.69	0.86	0.91	0.91	0.91
BER	SA-SPGD [27]	6.30 × 10−10	1.99 × 10−10	1.58 × 10−10	1.41 × 10−10	1.25 × 10−10
	BDCOA	5.21 × 10−10	1.71 × 10−10	1.49 × 10−10	1.23 × 10−10	1.07 × 10−10

and

Table 4. Comparison of BDCOA for Scenario 2.

Performances	Methods	Iterations
		50	100	150	200
SR	AMFSPGD [28]	0.83	0.88	0.89	0.89
	BDCOA	0.89	0.90	0.93	0.93

display the comparative analysis of BDCOA for Scenarios 1 and 2, respectively. This comparison confirms that BDCOA has superior performance when compared to SA-SPGD [21], AMFSPGD [28], and OBAFS [29]. The incorporation of BM and DMS in BDCOA made the FSO communication robust against the turbulence effects. Specifically, BM and DMS are used to avoid the risk of suboptimal solutions and improve the search space efficiency while searching for an optimum control signal for SLAO. From the outcomes, it is confirmed that the BDCOA has enhanced convergence when compared to the existing algorithms, specifically in reducing the RMS and increasing SR. This demonstrates that the BDCOA requires fewer iterations to achieve enhanced performance. The selection of dynamic control signals using BDCOA is used to compensate for the wavefront distortions in both the time-varying phenomenon used to improve the quality of the received signal and coupling efficiency. The improved quality of the received signal is used to reduce the RMS. Further, these optimum control signals minimize the phase distortions, which improves the peak intensity and sharpness of focused light in FSO for increasing the SR. Therefore, BDCOA demonstrates that it adapts to the different time-varying phenomena to offer enhanced performance in FSO.

3.8. Discussion

This section provides a discussion about the BDCOA-based WA correction in SLAO for FSO communication. The developed BDCOA is analyzed by using different optimization approaches such as MOA, POA, and COA with different population sizes of 20, 30, 40, and 50. This analysis shows that BDCOA outperforms MOA, POA, and COA. For example, the RMS of BDCOA at 500 iterations is 0.12, which is less than MOA, POA, and COA. The developed BDCOA is additionally compared with some existing studies focused on SA-SPGD [27], AMFSPGD [28], and OBAFS [29] for two different scenarios. The reason for the existing studies’ poor performance is that they fail to perform effective phase compensation and WA correction in SLAO. The proposed BDCOA discovers the optimum control signals for facilitating phase compensation, which reduces phase distortion and improves the FSO. The integration of BM and DMS in BDCOA are used to avoid the suboptimal solutions and enhance the global searching efficiency during the selection of optimum control signals for DM. This is beneficial for obtaining the WA correction to augment the received signal quality. The decrement in the phase distortion by using the BDCOA for DM reduces the probability of bit errors and further enhances the coupling efficiency. The comparison confirms that BDCOA outperforms the existing examples. BDCOA obtains a high performance over the iterations, confirming its reliability in maintaining the optimum performance in distinct scenarios i.e., a time-varying phenomenon. This confirms that BDCOA is suitable in handling the time-varying turbulence across different wavelengths.

4. Conclusions

In this research, BDCOA is developed to compensate for the WA in the SLAO over the free space optical communication. BDCOA is used to select optimum control signals for the actuators in the DM. The BM and DM incorporated in the BDCOA are employed to avoid suboptimal solutions and to improve search space efficiency to ensure balance among exploration and exploitation while searching for the control signal for the DM. The optimum control signals from BDCOA are used to enhance phase compensation in DM by reducing the phase distortion and improving the FSO’s performances. Therefore, the optimization of dynamic control signals for DM based on the BDCOA is used to upgrade the coupling efficiency. Moreover, the reduction in phase distortions is achieved by effectively selecting optimum control signals using BDCOA, which leads to advancement of the peak intensity and SR in time-varying phenomenon. Therefore, the simulations confirm that BDCOA offers superior outcomes compared to SA-SPGD, AMFSPGD, and OBAFS. The RMS of BDCOA for 500 iterations is 0.12, which is less than the SA-SPGD and OBAFS. Moreover, the BER of BDCOA for 500 iterations is 1.07 × 10⁻¹⁰, which is less than SA-SPGD. In the future, a novel optimization technique can be used to further enhance the WA compensation with the mitigation of routing error in FSO communication.