# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 3 Analysis of Numerical Simulation Results
### Bit Error Rate  
The bit error rate (BER) is commonly used as an important performance indicator for free-space optical communication quality [40; 41]. When the system utilizes on-off keying (OOK) intensity modulation, the average bit error rate <\(BER\)> is presented as follows [16]:  
\[\langle BER\rangle=\frac{1}{2}\int_{0}^{\infty}P_{I}(u)erfc\bigg{(}\frac{ \langle SNR\rangle\cdot u}{2\sqrt{2}}\bigg{)}du \tag{30}\]  
where \(P_{I}(\cdot)\) is the probability density function, and \(erfc(\cdot)\) is the complementary error function, which can be displayed by \(erfc(x)=2\pi^{-1/2}\int_{x}^{\infty}P_{I}(u)erfc\bigg{(}SNR\cdot u/2\sqrt{2} \bigg{)}dt\). \(u\) represents unity normalized.  
In marine atmospheric weak turbulence, the intensity possesses log-normal distribution is noted as follows [16]:  
\[P_{I}(u)=\frac{1}{\beta_{I}\sqrt{2\pi}}\exp\Bigg{\{}-\frac{\left[\ln(u)+0.5 \beta_{I}^{2}\right]^{2}}{2\beta_{I}^{2}}\Bigg{\}} \tag{31}\]  
where \(\beta_{I}^{2}\) is the scintillation index. In the presence of atmospheric turbulence, a BER value smaller than \(10^{-9}\) is generally used to quantify the potential improvement of the communication link performance [42; 43].  
Here, we set the weight coefficient to be \(a_{i}=0.26\), with \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\) and \(C_{n}^{2}=5.5\times 10^{-15}m^{-2/3}\), \(r=0\) mm, and all the other parameters involved were the same as those in Figure 7. Figure 8 shows the variation of <\(BER\)> with <\(SNR\)> for no aberration and four typical aberrations (Z4, Z7, Z11, Z25) at different turbulence strengths.  
As shown in Figure 8, under different turbulence conditions, the average \(BER\) decreases as the average \(SNR\) increases gradually. This aligns with theoretical expectations. Additionally, as the average \(SNR\) increases, the differences in average \(BER\) between various types of aberrations and the no-aberration Zno aberration condition become more pronounced. This indicates that under high <\(SNR\)> conditions, the impact of different aberrations on system performance becomes more distinct. Moreover, the <\(SNR\)> required for each type of aberration to achieve a <\(BER\)> of \(10^{-9}\) varies. In weak turbulence conditions, compared to the no-aberration state, each type of aberration requires a slightly lower \(<\)_SNR\(>\)_ to meet the requirement. Specifically, when \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\), the required \(<\)_SNR\(>\)_ values for achieving \(<\)_BER\(>\)_ of \(10^{-9}\) are in the following order: \(Z_{7}>Z_{11}>Z_{\text{no aberration}}>Z_{25}>Z_{4}\), with the \(<\)_SNR\(>\)_ ranging from 28 to 30 dB for each aberration type. However, as the turbulence intensity increases, the \(<\)_SNR\(>\)_ required to achieve the same \(<\)_BER\(>\)_ of \(10^{-9}\) significantly rises. When the turbulence strength increases to \(5.5\times 10^{-15}m^{-2/3}\), the required \(<\)SNR\(>\) values for achieving a \(<\)_BER\(>\)_ of \(10^{-9}\) change to the following order: \(Z_{11}>Z_{7}>Z_{\text{no aberration}}>Z_{25}>Z_{4}\), with the \(<\)_SNR\(>\)_ ranging from 210 to 230 dB for each aberration type. It can be inferred that as turbulence strength increases, the turbulence effects significantly raise the \(<\)_SNR\(>\)_ requirements, and when turbulence reaches a certain level, the impact weights of different aberrations on system performance undergo significant changes. Therefore, in the design and optimization of the system, it is essential to adjust the aberration correction strategies and manage the \(<\)_SNR\(>\)_ based on the varying turbulence strength to ensure efficient and stable communication performance under different turbulence conditions.