Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence

Abstract

Focusing on the three critical factors influencing laser communication systems operating in marine environments: atmospheric turbulence disturbances, atmospheric attenuation, and optical aberration effects, in this paper, we employ numerical simulation methods to systematically investigate the influence of four typical Zernike aberrations (defocus, y-coma, spherical aberration, and y-secondary quadrupole) on laser atmospheric transmission characteristics and system bit error rates. A comparison of their atmospheric transmission performance with that of the aberration-free state is also presented. The results show that reducing turbulence strength or increasing receiver aperture radius can effectively mitigate the scintillation effect of intensity fluctuations. Among the four typical aberrations, the fluctuation range of the relative change rate of the scintillation index for y-coma aberration relative to the aberration-free state is the largest. In weak turbulence and short-distance laser transmission over the sea, the beam drift caused by these four aberrations is not significant, and stronger turbulence strength or higher weight coefficients lead to more severe beam expansion. The on-axis logarithmic intensity probability density distribution of laser beams with different aberrations approximately follows a log-normal distribution. The skewness (S) and kurtosis (K) of the logarithmic intensity distribution are negatively correlated and always satisfy S < 0 and K > 0. Additionally, we found that as turbulence strength increases, turbulence effects significantly raise the required signal-to-noise ratio (SNR) values to achieve a bit error rate of 10⁻⁹. When turbulence strength reaches a certain level, the impact weights of different aberrations on system performance may undergo changes. These results can provide theoretical references for the design and optimization of laser system parameters in marine laser communication.

1. Introduction

Aberration refers to the deviation between real and ideal image points within a practical optical system [1,2]. Typically, due to the non-ideal characteristics of the optical imaging system, the paths of light rays are altered during transmission through the system’s various surfaces, leading to various types of aberrations. These aberrations can result in image blur, size variations, and morphological distortions, among other defects.

When a laser propagates through the maritime atmosphere, it inevitably encounters both static aberrations caused by optical system imperfections and dynamic aberrations due to marine atmospheric turbulence. This results in intensity scintillation [3,4], beam wandering [5,6], and beam spreading [7,8], all of which degrade the quality of the laser beam reaching the target [9,10]. Additionally, molecules and particles in the atmosphere cause energy attenuation of the laser beam, further exacerbating these problems and severely limiting the application of lasers in marine environments, particularly in critical fields such as laser communication, LiDAR, and laser weapons [11,12]. Therefore, it is important to conduct research on the propagation behavior of lasers in the marine atmosphere.

In recent years, research on the propagation characteristics of ideal laser beams in maritime atmospheric turbulence has become quite mature [13,14,15,16,17,18,19,20]. However, to better align with the practical applications in optical engineering, scholars have increasingly focused on the atmospheric transmission characteristics of non-ideal lasers that incorporate system aberrations. Roddier [21] simulated dynamic wavefront errors caused by atmospheric turbulence based on Noll’s [22] Zernike aberration decomposition method, which is suitable for adaptive optical correction dominated by low-order Zernike aberrations. Dai et al. [23] simulated the wave aberrations of the annular pupil using a Zernike polynomial expansion within the ring domain, obtaining the statistical distribution of the wavefront aberration decomposition coefficients under different turbulence statistical models. Furthermore, Ye et al. [24] equated the actual wavefront aberrations to the combination of dynamic aberrations introduced by atmospheric turbulence and static aberrations introduced by the system, concluding that Zernike spherical aberration has a significant impact on the beam quality factor. Adrian et al. [25] used two numerical methods and found that for focused Gaussian beams in specific conditions, correcting each new Zernike mode does not diminish gains uniformly, with certain modes, notably coma, showing increased importance, impacting adaptive optics system design. Additionally, Xiang et al. [26] studied the impact of aberrations in the emitting optical system on the on-axis gain in the far field, offering guidance for optimizing optical system design. Jiang et al. [27] employed the Fourier transform method to simulate the far-field spot distribution in the presence of aberrations and analyzed the impact of key parameters such as angular displacement, divergence angle, and gain on the light field distribution. Recently, Chemist et al. [28] further investigated how aberrations affect Laguerre–Gaussian beam quality, deriving analytical expressions for astigmatism and spherical aberration, and validating their validity through numerical simulations.

In summary, Zernike polynomials have been extensively utilized in simulating and analyzing systematic aberrations in the propagation characteristics of non-ideal laser beams within atmospheric turbulence. Despite significant advancements, further in-depth research is necessary to fully comprehend the combined impacts of diverse aberration types and varying turbulence intensities on laser beam propagation. Specifically, in marine atmospheric environments, the intricate interactions between the ocean and the atmosphere result in turbulence structures that differ markedly from those found on land, thereby augmenting the complexity of laser propagation. Consequently, accurately portraying and simulating the effects of marine atmospheric turbulence on laser transmission has emerged as a pivotal research focus.

In this context, based on the marine atmospheric turbulence power spectrum and Maxwell’s wave equation, we employ a numerical simulation method to systematically investigate the propagation characteristics of laser beams exhibiting several typical Zernike aberrations (including defocus, coma in the y-direction, spherical aberration, and secondary astigmatism in the y-direction) in various marine atmospheric turbulence condition. This study analyzes various parameters of the far-field beam spots, including the scintillation index, intensity probability density distribution, and statistical parameters such as mean, standard deviation, skewness, and kurtosis. Additionally, the average bit error rate <BER> of the system is calculated. Furthermore, we conduct a detailed analysis to compare the transmission outcomes of these aberrations with those in an aberration-free state.

2. Numerical Simulation Method

Figure 1 illustrates the specific steps and logic of the numerical simulation algorithm employed in this paper as follows:

According to Figure 1, the flowchart of this simulation algorithm mainly includes the following modules: the system initialization module, the laser emission light field setup module, the simulation environment setup module, the transmission equation solving module, the light field characteristic quantity calculation module, and the simulation results output module. Through the coordinated operation of these modules, the far-field spotlight intensity information and the characteristic physical parameters of the laser during its transmission in marine atmospheric turbulence can be obtained.

2.1. Numerical Simulation Method for Laser Atmospheric Transmission

The numerical simulation of light propagation in random media begins with the propagation equation of light, which is described by Maxwell’s wave equation [29]. The optical field E of a laser transmitting through atmospheric turbulence satisfies the following Maxwell’s wave equation:

$${\nabla }^{2}E+k^2{n}^{2}E+2\nabla \left(E·\nabla \ln \left(1+n_1\right)\right)=0$$

(1)

where ${\nabla }^2$ is the Laplacian operator, $n$ is the atmospheric refractive index, $k=2\pi /\lambda$ is the wave number, and $\lambda$ is the wavelength of the incident light. For light propagation in the atmosphere, $\lambda \ll l_0$ the polarization term on the left side of the equation is disregarded. It is assumed that only small-angle forward scattering occurs, and no backward scattering is present. Under the paraxial approximation along the z-axis, the optical field can be expressed as $E=u{e}^{ikz}$, and the propagation equation of the light wave in a medium with refractive index $n=1+n_1$ follows the standard parabolic equation, as shown below:

$${\displaystyle \frac{\partial u}{\partial z}}={\displaystyle \frac{i}{2k}}{\nabla }_{\perp }^{2}u+ik{n}_{1}u$$

(2)

where $n_1$ is the refractive index fluctuation, a small quantity. The standard parabolic equation forms the foundational basis for the numerical simulation and theoretical investigation of light propagation in random media.

As shown in Figure 2, The multi-layer phase screen numerical simulation method models a turbulent atmosphere as a series of very thin phase screens. When the phase variations induced by fluctuations in atmospheric refractive index are sufficiently small, the processes of vacuum transmission and atmospheric phase modulation can be treated as independent. This method divides the atmospheric medium into a sequence of parallel plates, each with a thickness of $\Delta z$. The optical field incident on each plate propagates to its rear surface according to the vacuum solution of the generalized parabolic equation, where it experiences phase modulation induced by the plate. The field then propagates through a vacuum and undergoes phase modulation again at the subsequent plate, thereby sequentially constructing the final optical field.

During the propagation process, we analyze the optical field traveling from the plane at $z=z_{i-1}$ through a parallel plate with thickness $\Delta z$ to the plane at $z_i=z_{i-1}+\Delta z$. This process can be expressed as a combination of the results from vacuum propagation over a distance $\Delta z$ and the phase modulation induced by the phase screen. The phase modulation is as follows [29]:

$$S(r,z_i)=k{\displaystyle {\int }_{Z_{i-1}}^{Z_i}{n}_1(r,z)dz}$$

(3)

i.e.,

$$u(r,z_i)\approx \exp \left[{\displaystyle \frac{i}{2k}}{\displaystyle {\int }_{Z_{i-1}}^{Z_i}{\nabla }_{\perp }^{2}dz}\right]\exp \left[iS(r,z_i)\right]u\left(r,z_{i-1}\right)$$

(4)

Due to the randomness of the magnitude of the phase modulation $S$, it is impossible to obtain an analytical solution for the above equation; numerical methods must be used. Common methods include the finite difference method, and the Fourier transform method, with this study employing the latter. Taking the Fourier transform of the above equation yields,

$$F\left[u\left(r,z_i\right)\right]=\exp \left[-{\displaystyle \frac{i\Delta z}{2k}}\left(K_x^2+K_y^2\right)\right]\times F\left[e^{is\left(r,z_i\right)}u\left(r,z_{i-1}\right)\right]$$

(5)

where $K_x^2$ and $K_y^2$ is the spatial wave numbers. Performing the inverse Fourier transform yields the final transmission equation.

$$u(r,z_i)=F^{-1}\left\{\exp \left[-{\displaystyle \frac{i\Delta z}{2k}}\left(K_x^2+K_y^2\right)\right]\times F\left[e^{is\left(r,z_i\right)}u\left(r,z_{i-1}\right)\right]\right\}$$

(6)

Repeat the above process for m times (where m is the number of phase screens) until the beam reaches the transmission distance L.

Additionally, in the process of laser transmission through the marine atmosphere, the laser beam is affected by solid particles, aerosols, and other substances, leading to scattering, refraction, and absorption phenomena, which result in the attenuation of the laser signal. In this paper, we assume uniform horizontal extinction, and the path loss can be expressed by the following empirical formula [30]:

$$h_l=\exp \left[-{\displaystyle {\int }_0^L\sigma \left(\lambda \right)dz}\right]$$

(7)

where $\sigma (\lambda )$ is the extinction coefficient, which is a parameter related to the wavelength $\lambda$ and the horizontal visibility $V_h$.

The extinction coefficient is given by [30]:

$$\sigma \left(\lambda \right)={\displaystyle \frac{3.912}{V_h}}{\left({\displaystyle \frac{\lambda }{550}}\right)}^{-q}$$

(8)

where q is the correction factor for the wavelength, and its value is determined as follows:

$$q=\left\{\begin{array}{ll}0.585V_h^{1/3}\hfill & V_h<6\mathrm{km}\hfill \\ 1.3\hfill & 6\mathrm{km}\le V_h<50\mathrm{km}\hfill \\ 1.6\hfill & V_h>50\mathrm{km}\hfill \end{array}\right.$$

(9)

Thus, the intensity distribution information on the target surface can be expressed by the following:

$$I\left(\mathit{r},z=L\right)={\left|u\left(\mathit{r},z=\mathrm{L}\right)\right|}^2\exp \left[-{\displaystyle {\int }_0^{\mathrm{L}}\sigma \left(\lambda \right)\mathrm{d}\mathrm{z}}\right]$$

(10)

2.2. Construction of Turbulent Phase Screens

To obtain reasonable simulation results from the aforementioned methods, a core issue that needs to be addressed is how to introduce a random phase φ that satisfies the characteristics of atmospheric turbulence in each iteration process. To tackle this problem, a commonly used method for constructing phase screens is based on Fourier transformation. The fundamental principle involves generating a complex spatial random field using the refractive index spectrum of turbulence and a complex Gaussian random number matrix. Then, an inverse Fourier transformation is performed to obtain the two-dimensional spatial distribution of phases. This process can be represented as follows [31]:

$${\phi }_H\left(x,y\right)={\displaystyle \sum _{mm}{\displaystyle \sum _{nn}{C}_{mm,nn}}}\exp \left[i2\pi \left(f_{x_{nn}}x+f_{y_{mm}}y\right)\right]$$

(11)

where $C_{mm,nn}=h\left(f_{x_{nn}},f_{y_{mm}}\right)\sqrt{2\pi k^2\Delta z{\Phi }_n\left(\kappa \right)}\Delta f_x\Delta f_y$, $f_{x_{nn}}$ and $f_{y_{mm}}$ represent the discrete spatial frequencies in the x-direction and y-direction, respectively; $\Delta f_x=1/L_x$, and $\Delta f_y=1/L_y$, where $L_x$ and $L_y$ represent the grid sizes in the x-direction and y-direction; $\Delta z$ is the spacing between phase screens along the z-axis; and ${\Phi }_n\left(\kappa \right)$ is the power spectral density function of the atmospheric refractive index fluctuation.

In the near-sea surface environment, the maritime atmospheric turbulence power spectrum can be represented as follows [32]:

$${\Phi }_n\left(\kappa \right)=0.033C_n^2\left[1-0.061\kappa /{\kappa }_H+2.836{\left(\kappa /{\kappa }_H\right)}^{7/6}\right]{\displaystyle \frac{\exp \left(-{\kappa }^2/{\kappa }_H^2\right)}{{\left({\kappa }^2+{\kappa }_0^2\right)}^{11/6}}}$$

(12)

where $\kappa$ indicates the spatial frequency magnitude, and $l_0$ and $L_0$ are the outer and inner scales of turbulence, respectively. ${\kappa }_0=2\pi /L_0$ is the outer scale wave number parameter, while ${\kappa }_H=3.41/l_0$ is the inner scale wave number parameter. $C_n^2$ is the atmospheric refractive index structure constant, also known as optical turbulence strengths.

It is worth noting that the phase screens generated by the Fourier transform method clearly exhibit a deficiency in low-frequency components. In this study, a higher-order harmonic compensation technique is employed to enhance the low-frequency components. In this study, a low-frequency sub-harmonic compensation technique has been employed to enhance the low-frequency components of the phase screens. Setting the specific harmonic order $N_p=3$, the new phase screen $\phi \left(x,y\right)$ can be expressed as follows [33]:

$$\begin{array}{cc}\hfill \phi \left(x,y\right)& ={\phi }_H\left(x,y\right)+{\phi }_L\left(x,y\right)\hfill \\ & ={\displaystyle \sum _{mm=-\infty }^{\infty }{\displaystyle \sum _{nn=-\infty }^{\infty }{C}_{mm,nn}\exp \left[i2\pi \left(f_{x_{nn}}x+f_{y_{mm}}y\right)\right]}}+{\displaystyle \sum _{p=1}^{N_p}{\displaystyle \sum _{mm=-1}^1{\displaystyle \sum _{nn=-1}^1{C}_{mm,nn}\exp \left[i2\pi \left(f_{x_{nn}}x+f_{y_{mm}}y\right)\right]}}}\hfill \end{array}$$

(13)

where ${\phi }_L\left(x,y\right)$ represents the low-frequency compensation component of the phase screen, P denotes the sub-harmonic order, and $\Delta f_p=1/\left(3^{p}L\right)$ is the frequency interval between the sub-harmonics.

2.3. Construction of Zernike Polynomial Aberrations

Zernike polynomials, due to their form being consistent with that of aberration polynomials, are often used to fit wavefronts and analyze wavefront characteristics [22,34]. In wave optics, phase or wavefront is commonly used to quantitatively describe the aberrations of optical systems. The wavefront $\Phi$ is defined as the distance traveled by the light wave (with the smallest unit being the wavelength $\lambda$), and the phase is given by the following:

$$S={\displaystyle \frac{2\pi \Phi }{\lambda }}=k\Phi$$

(14)

In a circular domain, Zernike polynomials are orthogonal two-dimensional polynomials that include the radial coordinate $\rho$ and the angular coordinate $\theta \in \left[0,2\pi \right]$, typically expressed in polar coordinates $\left(\rho ,\theta \right)$. This study adopts Noll’s representation of Zernike aberrations and introduces a representation of Zernike polynomials that only includes a single order, which is given by [25]:

$$Z_i(\rho ,\theta )=\sqrt{n+1}\left\{\begin{array}{ll}{R}_n^m\left(\rho \right)\sqrt{2}\cos (m\theta )\hfill & (\begin{array}{cc}even& i\end{array},m\ne 0)\hfill \\ R_n^m\left(\rho \right)\sqrt{2}\sin (m\theta )\hfill & (\begin{array}{cc}odd& i\end{array},m\ne 0)\hfill \\ R_n^m\left(\rho \right)\hfill & (m=0)\hfill \end{array}\right.$$

(15)

Its orthogonal property is given by [35]:

$${\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^1{Z}_i}}\left(\rho ,\theta \right)\cdot Z_{i’}\left(\rho ,\theta \right)\rho d\rho d\theta =\pi {\delta }_{ii’}$$

(16)

where $R_n^m(\rho )$ is the radial function, ${\delta }_{ii’}$ is the Dirac delta function, and the integers satisfy the condition: $n\ge m\ge 0$ is an even integer.

The optical aberrations of the system are simulated using orthogonal Zernike polynomials within the unit circular domain. Within an entrance pupil of a given radius, the wavefront of the light wave is described by [25]:

$$\Phi \left(R\rho ,\theta \right)={\displaystyle \sum _{i=1}^{\infty }{a}_i\cdot Z_i(\rho ,\theta )}$$

(17)

Based on the orthogonal properties of Zernike polynomials, the coefficients of each term can be determined as follows [36]:

$$a_i={\pi }^{-1}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^1\Phi \left(R\rho ,\theta \right)}}\cdot Z_i(\rho ,\theta )\rho d\rho d\theta$$

(18)

The corresponding phase can be given by the following:

$$S_i={\displaystyle \frac{2\pi {\Phi }_i}{\lambda }}=k{a}_i\cdot Z_i(\rho ,\theta )$$

(19)

where $a_i$ represents the weight coefficient of the $i$-th order Zernike polynomial, and $Z_i$ is the $i$-th order Zernike polynomial. By setting different values for $a_i$, various types of aberrations and combinations of aberrations can be obtained.

The light field with the introduction of initial aberrations is given by the following:

$$u(\rho ,\theta ,z=0)=u_0(\rho ,\theta ,z=0)\cdot e^{S_i}$$

(20)

where $u_0$ and $u$ are the original emitted light field and the initial light field modulated by aberrations, respectively.

3. Analysis of Numerical Simulation Results

The conditions for Gaussian laser atmospheric transmission are as follows: wavelength $\lambda$ is 532 nm, initial beam diameter is 80 mm, emission aperture is 200 mm, radius of curvature R is 1100 m, emitted optical power is 1 W, zenith angle is 90°, horizontal transmission distance L is 1100 m, and $V_h$ = 10 km, According to Formulas (8) and (9), the extinction coefficient ${\sigma }_{\lambda =532\mathrm{nm}}=3.49\times {10}^{-5}{\mathrm{m}}^{-1}$. The simulation is performed 200 times with m = 22 phase screens, each having a grid size of 512 × 512 with a spacing of 1 mm. This study primarily investigates how single aberrations affect the intensity scintillation characteristics of Gaussian lasers during atmospheric transmission. Focusing on four typical aberrations (defocus (Z₄), coma in the y-direction (Z₇), spherical aberration (Z₁₁), and secondary astigmatism in the y-direction (Z₂₅)), we analyze their impact on intensity scintillation, beam wandering, beam spread, and logarithmic intensity fluctuation probability density distribution, respectively.

3.1. Intensity Scintillation Effect

In laser atmospheric transmission, the normalized intensity fluctuation variance ${\beta }_I^2$, also known as the intensity scintillation index, is commonly used to quantitatively describe the magnitude of intensity fluctuations. The on-axis scintillation index can be expressed as follows [37]:

$${\beta }_I^2={\displaystyle \frac{〈I^2〉-{〈I〉}^2}{{〈I〉}^2}}$$

(21)

In this formula, $I$ is the light intensity on the axis at the receiver, and <·> denotes the ensemble average.

In order to more clearly illustrate the differences between the on-axis scintillation index of each aberration and the aberration-free state, we introduce the relative variation rate $R_{a_i}$, where $R_{a_i}$ = $\left({\beta }_{I,a_i}^2-{\beta }_{I,a_0}^2\right)/{\beta }_{I,a_0}^2$ and ${\beta }_{I,a_i}^2$ is the scintillation index for the weight coefficient $a_i$.

Here, we set the weight coefficient to be $a_i=0.00,0.06$ and 0.26, respectively. Figure 3 illustrates the variation of ${\beta }_{I,on}^2$ and $R_{a_i}$ for four typical single aberration conditions, as the turbulence strength changes from $2.5\times {10}^{-16}{m}^{-2/3}$ to ${10}^{-13}{m}^{-2/3}$.

From Figure 3, it can be observed that (1) in weak turbulence, compared to the no-aberration state, Z₄ aberration shows a suppressing effect on the on-axis scintillation index, especially at higher weight coefficients. When the turbulence strength grows up to a certain value, Z₄ aberration begins to have a promoting effect on the on-axis scintillation index, and the higher the weight coefficient, the more significant the promoting effect. Additionally, the maximum absolute value of the relative change rate of the scintillation index due to Z₄ aberration $\left|R_{a_4}\right|$ is 6%. (2) Compared to the no-aberration state, Z₇ aberration always has a promoting effect on the on-axis scintillation index, especially at stronger turbulence or higher weight coefficients. The maximum absolute value of the relative change rate of the scintillation index due to Z₇ aberration $\left|R_{a_7}\right|$ is 9.13%. (3) In weak turbulence, Z₁₁ aberration always has a promoting effect on the on-axis scintillation index, and the higher the aberration weight coefficient, the larger the relative change rate of the on-axis scintillation index. After the turbulence strength increases to a certain value, Z₁₁ aberration begins to have a suppressing effect on the on-axis scintillation index, and the larger the aberration weight coefficient, the smaller the relative change rate of the on-axis scintillation index. The maximum value of $\left|R_{a_{11}}\right|$ in this case is 5.8%. (4) Z₂₅ aberration always has a suppressing effect on the on-axis scintillation index, especially at higher weight coefficients. The maximum value of $\left|R_{a_{25}}\right|$ in this case is 5%. Thus, we can infer that compared to the no-aberration state, the range of fluctuations in the relative change rate of the scintillation index for each aberration satisfies Z₇ > Z₄ > Z₁₁ > Z₂₅.

In practical applications, it is necessary to consider situations with a certain receiving area, as laser beams typically have a defined area. The aperture scintillation index for a circular region with a specific receiving radius is given by [38]:

$${\beta }_I^2(r)={\displaystyle \frac{〈I{(r)}^2〉-{〈I(r)〉}^2}{{〈I(r)〉}^2}}$$

(22)

where $I(r)$ represents the total received light intensity over a circular region with a radius of $r$.

Figure 4 shows the variation in the scintillation index of a Gaussian beam as the turbulence strength changes from $2.5\times {10}^{-16}{m}^{-2/3}$ to ${10}^{-13}{m}^{-2/3}$, while all other parameters remain the same as those in Figure 3. The illustration includes conditions with no aberration, as well as with various aberrations: defocus (Z₄), y-direction coma (Z₇), spherical aberration (Z₁₁), and secondary astigmatism in the y-direction (Z₂₅), each tested at different receiver aperture sizes.

From Figure 4a–d, it can be observed that under identical turbulence conditions, the scintillation index corresponding to each system aberration (Z₄, Z₇, Z₁₁, Z₂₅) decreases as the receiver aperture radius increases. This suggests that the aperture averaging effect can reduce the scintillation index. Compared to the condition with no system aberration, the change patterns in the scintillation index for each aberration (Z₄, Z₇, Z₁₁, Z₂₅) differ as follows: (1) At the same weight coefficient, the stronger the turbulence strength or the smaller the receiver aperture radius, the higher the scintillation index for Z₄ ${\beta }_{I,Z_4}^2$. When $C_n^2\le 1.5\times {10}^{-14}{m}^{-2/3}$, ${\beta }_{I,Z_4}^2$ is always lower than that of the aberration-free condition ${\beta }_{I,no}^2$ at any receiver aperture radius, and the difference is so small as to be negligible. When $C_n^2>1.5\times {10}^{-14}{m}^{-2/3}$, ${\beta }_{I,Z_4}^2$ is always bigger than ${\beta }_{I,no}^2$, especially at stronger turbulence strength or a smaller receiver aperture radius. This means the promoting effect of Z₄ aberration on the scintillation index becomes more pronounced as turbulence strength increases or the receiver aperture size decreases. (2) Compared to the no-aberration state, Z₇, Z₁₁, and Z₂₅ aberrations always have a promoting effect on the on-axis scintillation index, especially at stronger turbulence or higher weight coefficients. However, this promoting effect is not significant, and the maximum differences in the scintillation index are all within 0.01. This means that the effects of Z₇, Z₁₁, and Z₂₅ on the scintillation index are limited and can be ignored. Thus, combined with Figure 3 and Figure 4, it is evident that increasing the receiver aperture radius can effectively reduce the promoting effect of aberrations on the scintillation effect.

3.2. Beam Wandering Effect

The laser transmission through an oceanic turbulent atmosphere also induces a spot wander effect. In general, we use the variation of beam centroid position to describe the beam wandering effect, and the beam centroid can be expressed as follows [29]:

$${\mathit{\rho }}_c={\displaystyle \frac{{\displaystyle \iint \mathit{\rho }I\left(\mathit{\rho }\right) \mathrm{d}\mathit{\rho }}}{{\displaystyle \iint I\left(\mathit{\rho }\right) \mathrm{d}\mathit{\rho }}}}$$

(23)

where $\mathit{\rho }$ is the transverse vector, and $I$ is the light intensity. The beam centroid in the horizontal and vertical directions is as follows:

$$x_c={\displaystyle \frac{{\displaystyle \iint xI(x,y)\mathrm{d}x\mathrm{d}y}}{{\displaystyle \iint I(x,y)\mathrm{d}x\mathrm{d}y}}}$$

(24)

$$y_c={\displaystyle \frac{{\displaystyle \iint yI(x,y)\mathrm{d}x\mathrm{d}y}}{{\displaystyle \iint I(x,y)\mathrm{d}x\mathrm{d}y}}}$$

(25)

Then, we can obtain the root-mean-square (RMS) value of the beam centroid wandering as follows:

$${\sigma }_{\rho }=\sqrt{〈{\mathit{\rho }}_c^2〉}={\left[{\displaystyle \iint {\displaystyle \iint \left({\mathit{\rho }}_1·{\mathit{\rho }}_2\right)I\left({\mathit{\rho }}_1\right)I\left({\mathit{\rho }}_2\right)\mathrm{d}{\mathit{\rho }}_1\mathrm{d}{\mathit{\rho }}_2}}\right]}^{1/2}/{\displaystyle \iint I\left(\mathit{\rho }\right)\mathrm{d}\mathit{\rho }}$$

(26)

Here, we set the weight coefficient to be $a_i=0.00$, $0.06$, 0.26, and 0.45, respectively, Figure 5 illustrates the variation of the RMS of beam wandering for four typical single aberration conditions as the turbulence strength changes from $2.5\times {10}^{-16}{m}^{-2/3}$ to $7.5\times {10}^{-14}{m}^{-2/3}$.

From Figure 5, it can be observed that as the turbulence intensity increases, the centroid drift phenomenon of the beam spot becomes more pronounced, and different types of aberrations have varying effects on beam wandering. (1) In weak turbulence, the RMS of beam wandering of Z₄ aberration decreases as the weight coefficient increases. When the turbulence strength grows to a certain value, the RMS of beam wandering begins to increase with the higher weight coefficient of Z₄ aberration. Compared to the no aberration state, the RMS of beam wandering of Z₄ aberration is always larger, particularly under conditions of high weight coefficients and strong turbulence, with the maximum difference reaching 0.15 cm. (2) In weak turbulence, the variation of the weight coefficient of Z₇ aberration has a negligible impact on the beam wandering effect. As the turbulence strength increases, the fluctuation range of the RMS value of the beam wandering becomes more significantly affected by the weight coefficient. When the turbulence strength $C_n^2=7.45\times {10}^{-16}{m}^{-2/3}$, the fluctuation range of the RMS of the beam wandering for Z₇ aberration is 0.01 cm, while when the turbulence strength increases to $7.5\times {10}^{-14}{m}^{-2/3}$, the RMS fluctuation range of the beam wandering for Z₇ aberration grows up to 0.17 cm. Additionally, compared to the no-aberration state, the RMS of beam wandering for Z₇ aberration is always larger at any turbulence strength, with the maximum difference being 0.18 cm. (3) Compared to the no-aberration state, the RMS of the centroid drift for Z₁₁ aberration is consistently larger, and the difference increases as the turbulence intensity and the aberration weight coefficient become larger, with the maximum difference reaching 0.17 cm. (4) Under the same turbulence strength, the beam wander phenomenon becomes more severe as the weight coefficient of Z₂₅ aberration increases. Compared to the no-aberration state, the effect of Z₂₅ aberration on the far-field centroid drift is not significant, with the maximum difference being 0.11 cm. Therefore, in weak turbulence and short-range laser transmission over the sea, the beam wander caused by these four aberrations is not significant and can be neglected.

3.3. Beam Expansion

When a laser propagates through atmospheric turbulence, beam expansion occurs, which is typically described by the effective spot radius relative to the centroid of the spot [29].

$$W_{eff}^2=2{\displaystyle \iint r_1^{2}I\left(x,y\right)}\mathrm{dxd}y{\left[{\displaystyle \iint I\left(x,y\right)\mathrm{dxd}y}\right]}^{-1}$$

(27)

In this paper, to further explain the beam expansion, we introduce the relative vacuum expansion factor, which is the ratio of the beam expansion to the expansion in the vacuum. Its expression is given by the following:

$$K_W=W_{eff}/W_{eff0}$$

(28)

where $W_{eff}$ is the effective beam radius in turbulence, and $W_{eff0}$ is the effective radius in vacuum.

In this section, we set the weights to 0.00, 0.26, and 0.45. Figure 6 shows the variation of the effective beam radius $W_{eff}$ and the relative vacuum expansion factor $K_W$ for four typical single aberration conditions as the turbulence strength changes from $2.5\times {10}^{-16}{m}^{-2/3}$ to ${10}^{-13}{m}^{-2/3}$.

From Figure 6, we can clearly see that stronger turbulence results in more severe beam expansion. The influence patterns of the weight coefficients of different aberrations on the beam expansion characteristics in the far field are distinct: (1) When the weight coefficient of Z₄ aberration is higher, the effective beam radius in the far field is larger, and the relative vacuum expansion factor also increases, particularly in stronger turbulence conditions. Compared to the state without aberrations, Z₄ aberration always has a larger effective beam radius. In weak turbulence conditions, its relative vacuum expansion factor is consistently smaller, but as the turbulence strength increases, Z₄ aberration begins to enhance the beam expansion effect, especially under higher weight coefficients. (2) As the weight coefficient of Z₇ aberration increases, the effective beam radius and the relative vacuum expansion factor both increase. Compared to the state without aberrations, Z₇ always exhibits more severe beam expansion, particularly under higher weight coefficients. (3) The beam expansion phenomena of Z₁₁ and Z₂₅ aberrations are similar, in that both the effective beam radius and the beam expansion become larger as turbulence strength or the weight coefficient increases. However, compared to the state without aberrations, Z₁₁ aberration has a little greater promoting effect on beam expansion.

3.4. Probability Density Distribution of Light Intensity

The probability of distribution of light intensity fluctuations provides a fundamental statistical description of this random process. Oceanic atmospheric turbulence is typically weak to moderate [39], under which condition, the probability density distribution of log-intensity fluctuations for the received laser spot generally follows a normal distribution. Skewness ${\gamma }_1$ and kurtosis ${\gamma }_2$ are used to assess how the sample distribution deviates from a Gaussian distribution with the same mean and variance [29], calculated as follows:

$${\gamma }_1={\mu }_3/{\mu }_2^{3/2},{\gamma }_2={\mu }_4/{\mu }_2^2-3$$

(29)

where ${\mu }_3$ is the third moment of the logarithmic intensity probability density distribution, and ${\mu }_4$ is the fourth moment. Skewness reflects the asymmetry of the probability density distribution relative to the mean. If the probability distribution is symmetric about the mean (such as in a normal distribution), the skewness is zero; otherwise, the skewness is non-zero. Kurtosis reflects the concentration of the probability distribution relative to a normal distribution. If kurtosis is positive, the probability distribution is more concentrated than the normal distribution; otherwise, it is more dispersed.

Here, we set the weight coefficient to be $a_i$ = 0.00, 0.06, 0.16, 0.26, 0.36, 0.40, and 0.45, $C_n^2=7.45\times {10}^{-16}{m}^{-2/3}$, and r = 0 mm, and all the other parameters involved were the same as those in Figure 6. Figure 7a–d show the probability density distribution of the log-intensity of the spot on the target after the Gaussian beam has transmitted over 1.1 km under various aberration conditions. To provide a deeper analysis of the target intensity distribution, Figure 6e–h show the corresponding mean (Mean), standard deviation of log-intensity fluctuation (Std), skewness (S), and kurtosis (K).

As shown in Figure 7, the probability density distribution curves for each aberration exhibit different trends as the weight coefficient varies: (1) As the weight coefficient increases, the peak of the log-intensity distribution for Z₄ aberration gradually rises. Moreover, at lower weight coefficients, the absolute values of skewness and kurtosis are relatively small, and the probability density distribution curves approach a normal distribution. (2) As the weight coefficient of the aberration increases, the log-intensity probability density curve for Z₇ becomes more concentrated around the mean, exhibits better symmetry, and increasingly follows a normal distribution. (3) As the weight coefficient increases, the log-intensity probability density distribution curve for Z₁₁ aberration shifts to the right overall. Furthermore, as the weight coefficient grows higher, the deviation of the log-intensity probability density distribution curve from a normal distribution gradually increases as well. (4) For Z₂₅ aberration, when the weight exceeds 0.26, the logarithmic intensity probability density distribution curve gradually deviates from a normal distribution and shifts to the right overall. The peak logarithmic intensity gradually increases, and the concentration of energy continues to rise. Additionally, from Figure 7, we can also conclude that when $C_n^2=7.45\times {10}^{-16}{m}^{-2/3}$, the probability density distribution of the on-axis log-intensity for laser beams with different aberrations approximately follows a log-normal distribution. The skewness and kurtosis of the log-intensity distribution are negatively correlated and consistently satisfy S < 0 and K > 0.

3.5. 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]:

$$〈BER〉={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{\infty }{P}_I\left(u\right)}erfc\left({\displaystyle \frac{〈SNR〉\cdot u}{2\sqrt{2}}}\right)du$$

(30)

where $P_I\left(\cdot \right)$ is the probability density function, and $erfc\left(\cdot \right)$ is the complementary error function, which can be displayed by $erfc\left(x\right)=2{\pi }^{-1/2}{\displaystyle {\int }_x^{\infty }{P}_I\left(u\right)}erfc\left(SNR\cdot u/2\sqrt{2}\right)dt$. u represents unity normalized.

In marine atmospheric weak turbulence, the intensity possesses log-normal distribution is noted as follows [16]:

$$P_I\left(u\right)={\displaystyle \frac{1}{{\beta }_I\sqrt{2\pi }}}\exp \left\{-{\displaystyle \frac{{\left[\ln \left(u\right)+0.5{\beta }_I^2\right]}^2}{2{\beta }_I^2}}\right\}$$

(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 ${\mathrm{Z}}_{\mathrm{no} \mathrm{aberration}}$ 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: Z₇ > Z₁₁ > ${\mathrm{Z}}_{\mathrm{no} \mathrm{aberration}}$ > Z₂₅ > Z₄, 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: Z₁₁ > Z₇ > ${\mathrm{Z}}_{\mathrm{no} \mathrm{aberration}}$ > Z₂₅ > Z₄, 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.

4. Discussion and Future Work

This study systematically investigates the propagation characteristics of laser beams with four typical aberrations—defocus (Z₄), y-direction coma (Z₇), spherical aberration (Z₁₁), and secondary y-direction trefoil (Z₂₅)—after they propagate through maritime atmospheric turbulence over 1.1 km. Using the Zernike aberration fitting method and multi-layer phase screen simulation, we calculate the scintillation index, intensity probability density distribution, beam wandering and beam expansion effects at the target plane in maritime atmospheric turbulence under varying turbulence strengths, weight coefficients, and receiver aperture radii. The results show the following:

(1) Compared to the other three types of aberrations (i.e., defocus (Z₄), spherical aberration (Z₁₁), and secondary y-direction trefoil (Z₂₅)), the range of fluctuations in the relative change rate of the scintillation index for the Z₇ aberration is the largest relative to the no-aberration state. Thus, in practical system design, it may be necessary to pay particular attention to the correction of coma. This is similar to the result in reference [44], but what distinguishes our study is that we have added an analysis of the relative change rate compared to the no-aberration state, which provides a more comprehensive explanation of the trend regarding the influence of aberrations on the scintillation index. (2) Increasing the receiver aperture radius can partially mitigate the scintillation effects of intensity fluctuations, as also validated in [45]. Moreover, this study reveals that different aberrations respond differently to the aperture averaging effect. This observation provides a theoretical basis for targeted aberration correction strategies. (3) The on-axis log-intensity distribution of laser beams with different aberrations approximately follows a log-normal distribution, with skewness (S < 0) and kurtosis (K > 0) exhibiting a negative correlation. This is consistent with the empirical results in the land-based environment reported in Reference [29], indicating that the log-intensity distribution of laser beams under turbulent atmospheric conditions exhibits a universal log-normal characteristic. (4) It is important to note that in practical communication systems, the SNR typically operates within a limited range, making it challenging to achieve the extremely high values often assumed in theoretical analyses or simulations. To ensure efficient and stable communication in environments with complex turbulence, a multi-parameter optimization approach is necessary. This approach should consider factors such as aberration correction, aperture design, and turbulence mitigation to enhance system performance.

All the findings in this paper apply to short-distance (1.1 km) laser atmospheric transmission over the sea. In practical applications, the laser communication distance often needs to exceed 1.1 km [46,47], and the actual system involves complex combinations of aberrations as well as atmospheric non-uniformity patterns. By initially discussing the effects of various aberrations on short-distance laser atmospheric transmission over the sea, the next step is to comprehensively analyze their impact on long-distance laser transmission in a non-uniform atmospheric environment using the weighted summation approach provided in Reference [48]. Additionally, to enhance the reliability and practical value of our research results, we also plan to conduct relevant experimental validation. This aims to provide more theoretical and technical support for practical applications in free-space optical communications.

5. Conclusions

This study investigates four typical Zernike system aberrations and laser propagation characteristics in marine atmospheric turbulence through numerical simulation models, yielding the following key conclusions: Y-axis coma (Z₇) exhibits the most significant impact on the scintillation index, with the fluctuation range of its relative change rate surpassing other aberrations (Z₄, Z₁₁, Z₂₅), underscoring the critical need for coma correction in practical system design. Although beam wandering remains negligible under weak turbulence and short-range transmission, the synergistic interaction between turbulence strength and aberration weight coefficients significantly exacerbates beam expansion. The intensity distribution retains log-normal characteristics, consistently displaying persistent negative skewness (S < 0) and positive kurtosis (K > 0) in on-axis logarithmic intensity, a pattern aligned with terrestrial environments, thereby highlighting the universality of distortion mechanisms. Furthermore, as turbulence strength intensifies, its effects markedly elevate the required signal-to-noise ratio to achieve a bit error rate of 10⁻⁹. Notably, when turbulence strength exceeds a threshold, the relative impact weights of different aberrations on system performance may shift. Collectively, these findings provide a theoretical basis for the design and optimization of laser transmission systems in marine atmospheric turbulence and lay the foundation for future research.