1. Introduction

The study of laser beam propagation through oceanic turbulence has garnered significant attention due to its wide range of applications, including wireless optical communication and imaging systems [1–5]. Underwater imaging technology is crucial in exploring and understanding the underwater environment [6]. This technology is utilized for mapping the surroundings of naval ships and scientific research for marine life detection and marine mineral exploration [6]. The performance of underwater optical imaging systems and underwater channels for laser and quantum communications is heavily influenced by the properties of the medium through which light propagates [7,8].

Natural water, as opposed to pure water, possesses distinct optical properties. In addition to pure water, natural seawater contains soluble salts such as NaCl, MgCl2, Na2SO4, CaCl2, and KCl. Furthermore, ocean and natural water contain impurities like chlorophylls, folic and humic acid, and suspended particles [9].

Among the various factors affecting underwater optical beam propagation through seawater, oceanic turbulence stands out as a significant phenomenon. Oceanic turbulence arises from fluctuations in refractive index caused by temperature and salinity fluctuations [10]. Temperature fluctuations are the primary factor influencing underwater optical beam propagation through seawater [9]. These fluctuations in refractive index, occurring in both time and space domains, lead to wavefront distortions, beam broadening, and beam wandering [11].

Random wavefront distortions result in a decrease in image quality. Measuring the extent of degradation can be done by assessing parameters such as Modulation Transfer Function (MTF) and Point Spread Function (PSF). To mitigate this degradation, Adaptive Optics (AO) techniques can be utilized. AO systems are equipped with the ability to estimate the turbulence parameters and perform corrections accordingly.

One of the most commonly used sensors in AO systems for measuring wavefront distortions is the Shack-Hartmann Sensor (SHS) [12]. Centroid displacements are measured using focal spots produced by microlens arrays, which are then compared to reference positions. These measurements enable the reconstruction of the wavefront on the aperture of the receiver. To perform wavefront decompositions and reconstructions, orthogonal polynomials such as Zernike polynomials are necessary. By combining the Shack-Hartmann sensor and Zernike polynomials, wavefront decomposition, reconstruction, and aberration estimation can be achieved [13].

In the following sections of this article, we will elaborate on the wavefront reconstruction method and the reasoning behind utilizing Zernike polynomials. We will then describe the laboratory setup for data collection. Using the chosen wavefront reconstruction method, we will calculate Zernike polynomial coefficients and the average mean square value of turbulence-induced phase aberration ($\sigma ^2$ or variance) for each frame of the acquired images. In the following section, we will define and calculate the residual phase error ($\psi$) parameter, representing the variance of aberrations after correction. We will also calculate and correct errors in the obtained data caused by measurement errors (noise) and correlation between Zernike polynomials to obtain the actual coefficients of turbulence-induced aberrations. Finally, we will examine the results obtained from the variance and residual phase error curves in each section mentioned above.

2. Wavefront reconstruction and underwater turbulence-induced phase aberration

The wavefront aberrations of an optical system can be indirectly calculated by evaluating the wavefront in the exit pupil of the system. Various wavefront reconstruction methods can perform this calculation.

These methods include zonal methods, modal methods, modal Fourier reconstruction, and direct determination of Zernike coefficients from slope measurements [14]. This study utilizes the latter approach. Specifically, the Shack-Hartmann sensor determines the wavefront gradients and wavefront expansion can be done using a series of orthogonal functions. The aberration value in the wavefront is then determined using the coefficients of these functions.

One of the most well-known and widely used functions in this field is the Zernike polynomials [15–20]. These polynomials are orthogonal in circular apertures and are typically expressed in polar coordinates (r, $\theta$) [16,21]. However, it is essential to acknowledge that the correlation analysis of Zernike polynomial coefficients has revealed their interdependence, indicating that a wavefront cannot be fully decomposed using these polynomials alone [13,16,21]. Consequently, this research addresses this issue by calculating the coefficients of aberrations using Zernike polynomials. Subsequently, we adjusted the correlation between these coefficients after accounting sensor noise.

In our experiment, we introduce wavefront aberrations using thermally induced turbulence on a laser beam. This turbulence alters the phase of the wavefront. We create the turbulence screen using a rod heater. This method is known as the phase screen method. By utilizing Zernike polynomials, we can expand the turbulence-induced phase aberration [21]. Eq. (1) gives the expansion,

3. Laboratory arrangement

To experiment, we constructed a water acrylic tank with dimensions of $4\times 0.5\times 0.5m^3$. The water in the tank underwent a two-stage purification process to eliminate any changes in the refractive index and minimize absorption and dispersion. This purification process reduced the water hardness to 20 ppm. We then pumped the purified water through a heater into the tank, where we heated it to match the laboratory conditions. Four identical thermometers were installed at equidistant intervals to ensure uniform temperature distribution within the tank [9].

Based on the literature, two significant factors influencing the creation of underwater turbulence are thermal gradients and salinity fluctuations. Considering that salinity fluctuations will be significant and impactful at specific locations such as river outflows or estuaries, this experiment has evaluated the thermal gradient factor. Creating turbulence-induced temperature changes (refractive index variations) along the optical path is possible in two ways: a. the use of a phase plate that is easier to control and requires less equipment and b. the use of hot and cold plates at the top and bottom of a small water cell. According to previous research, the use of a rod heater allows for the reliable and reproducible simulation of turbulent conditions caused by temperature gradients. Therefore, in this experiment, a rod heater was used to create a phase screen, which induced a thermal gradient in the water. By altering the voltage applied to the heater, the strength of turbulence could be adjusted, thereby affecting the resulting thermal fluctuations in the phase screen [9]. The experimental setup, including the arrangement and test elements, is depicted in Fig. 1. It is worth mentioning that the most important parameter defining the strength of turbulence is the scintillation index. According to our previous study and our current calculations, the experimental conditions considered in this work satisfy the weak to moderate turbulence regime [9].

 

Fig. 1. Demonstration of the experimental arrangement implemented in this examination. The diagrammatic sketch (a) offers an outline of the setup, while (b) displays the sources employed, namely the Ion-Argon laser (1) and the lenses of 5 mm (2) and 50 mm (3). The components of the detector, which comprise the Neutral Density (ND) filter (1), iris diaphragm (2), collecting lens (3), and Shack-Hartmann detector (4), are revealed in (c), (d) the phase screen near the transmitter, (e) phase screen near the receiver.

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The transmitter unit of the experiment employed an argon-ion laser with a wavelength of 457 to 514 nm, which falls within the green-blue transmittance window of water. We expanded and parallelized the laser beam using two lenses in a confocal configuration, resulting in a tenfold expansion. The expanded beam then passed through the phase screen created by the rod heater in the water. Before entering the Shack-Hartmann sensor, we used a diaphragm to limit and define the beam size in the receiving unit. Diaphragms with diameters of 5, 15, and 25 mm were utilized in this experiment. Finally, the beam entered the Shack-Hartmann sensor, which measured the data for 5 minutes. The Zernike polynomial coefficients were calculated to determine the wavefront aberrations [9].

Figure 2 presents the results obtained for the Y-tilt with an aperture diameter of 15 mm and a focal length of 50 mm. Due to the large volume of acquired data (81 Zernike coefficients for each frame in 300 seconds across 12 different configurations), we have elected to display only the tilt value along the Y-axis for 4 configurations in Fig. 2. The rationale for selecting the Y-tilt is based on our observation that, across all data, the first 4 Zernike coefficients (except piston) and particularly the Y-tilt are more dominant and, consequently, exert a greater impact than other coefficients.

 

Fig. 2. Illustration of the second Zernike coefficient, specifically the Tilt in Y (j = 1), in the absence of any correction for an aperture diameter of 15 mm and a focal length of 50 mm. The figure showcases the behavior of this coefficient at two distinct points with two different rod heater voltages. This coefficient is shown at (a)) point A and with a voltage of 50 V, (b) point A and with a voltage of 150 V, (c) point E and with a voltage of 50 V, and (d) point E and with a voltage of 150 V for a duration of 300 seconds.

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In Fig. 3(b), we present the complete set of the first 81 Zernike coefficients for the turbulence phase screen near the transmitter, with a focal length of 50 mm, an aperture diameter of 15 mm, and a voltage of 50 volts. As can be observed, only two initial Zernike coefficients exert a significant impact on the wavefront of the laser.

 

Fig. 3. Illustration of (a) the second Zernike coefficient, the Tilt in Y (j = 1). This coefficient is acquired without the presence of turbulence. Also, (b) displays the first 81 Zernike coefficients (without piston) for the presence of turbulence at point A with a voltage of 50 V and an aperture diameter of 15 mm for the optical receiver system.

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It is important to note that before collecting data, we obtained reference data when we turned off the rod heater, representing zero turbulence and no induced phase screen. Then we subtracted the Zernike coefficients of these reference data from the data obtained when we turned on the rod heater and the phase screen was present. This subtraction process effectively removed the aberrations caused by the optical system of the receiver, leaving only the aberrations caused by turbulence.

As illustrated in Fig. 3(a), the tilt value along the Y-axis for the reference data is zero. This indicates that no tilt aberration has been introduced by the optical receiver system during the operation of the rod heater and the presence of turbulence. Consequently, it can be inferred that the tilt aberration observed in the data is solely attributable to the turbulence generated. We will provide further details on the data calculation method in the next section.

4. Averaged mean square turbulence-induced phase aberration

The averaged mean square turbulence-induced phase aberration can be determined by utilizing the Zernike polynomial coefficients and decomposing the wavefront phase error caused by beam propagation across the phase screen. This calculation provides insight into the level of turbulence-induced phase aberration ($\sigma ^2$) present across the entire aperture. A lower value of this parameter signifies superior image quality compared to a system with a higher value.

To determine this quantity, we compute the Zernike coefficients of the wavefront phase error by employing Eqs. (1) and (2), after subtracting the reference data for each configuration.

Through the utilization of wavefront decomposition using the Zernike polynomials, it becomes possible to compute the average mean square of the wavefront phase error measured on the aperture of the optical receiver system:

However, through the utilization of the orthonormal condition of Zernike polynomials on the optical system aperture of the receiver, Eq. (4) can be elegantly simplified as [21]:

Therefore, based on the above relation, it can be observed that $\sigma ^2$ represents the averaged mean square of the Zernike polynomial coefficients. Figure 4 presents the results obtained for the variance of the initial 81 coefficients of Zernike polynomials in each frame over 300 seconds, with different aperture diameters and a focal length of 50 mm.

 

Fig. 4. Illustration of the averaged mean square values of the first 81 coefficients of Zernike polynomial coefficients ($\sigma ^2$) over 300 seconds time period, for three aperture diameters (5, 15, and 25 mm) and a focal length of 50 mm: (a) at point A, with a rod heater voltage of 50 V; (b) at point A, with a rod heater voltage of 150 V; (c) at point E, with a rod heater voltage of 50 V; and (d) at point E, with a rod heater voltage of 150 V.

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In Table 1, we compare the mean square values of the Zernike coefficients for points A and E under turbulence strengths of 50 and 150 volts. These values were obtained with different diameters of the receiver aperture. The data are displayed without any correction, with noise correction, and with noise and alias correction.

Table 1. The averaged mean square values of the Zernike polynomial coefficients ($\sigma ^2$) obtained over time in a turbulence strength of 50 and 150 volts. The measurements were taken at points A and E, with varying focal lengths and aperture diameters.

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The data in Table 1(columns 2 and 3) demonstrate that the phase screen turbulence affects the wavefront more when it is situated at point E, which is nearer to the receiver (this result is confirmed in Fig. 8(a)). This is the case in the weak turbulence regime (50 volts).

Conversely, the data in columns 2 and 3 of the second section (for v=150 volts) of Table 1 demonstrate that the phase screen turbulence affects the wavefront more when it is situated at point A, which is nearer to the transmitter. This is the case in the strong turbulence regime (150 volts). This result is confirmed in Fig. 8(b). As we expected and can be seen in Fig. 4 and Table 1, the phase screen in the higher voltage of the rod heater (150 V), representing a more turbulent medium, exerts a significantly higher influence on the wavefront of laser beams. As a result, the averaged mean square of the Zernike polynomial coefficients is approximately five orders larger in this case.

As demonstrated in Table 1(columns 2 and 3) for original data in the weak turbulence regime (50 volts), we can observe that an increase in the diameter of the receiver optical system corresponds with an increase in the detected aberration variances. When the receiver is in proximity to the turbulent phase screen, the level of aberration variances due to an increase in the diameter is observed to be higher than when the receiver is far from the turbulent medium. This is evidenced by the observation that as the receiver is near the rod heater (turbulent phase screen), it induces an increased level of aberration variances.

However, we can see contrasting findings in the strong turbulence regime (150 volts). Here, we can observe that increasing turbulence on the phase screen, while simultaneously increasing the diameter of the receiver optical system, leads to a decrease in the induced aberration variances by the turbulence. This effect is similar to the phenomenon of aperture averaging [22]. Additionally, in contrast to the results depicted in Table 1, as the receiver is located near the turbulent phase screen, the level of induced aberration variances by the turbulence decreases.

Also, experimental results indicate that a significant increase in the averaged mean square of the Zernike polynomial coefficients is observed when the aperture diameter is increased from 5 mm to 15 mm. However, no substantial difference is observed when the aperture diameter is further increased from 15 mm to 25 mm.

This significant change between 5 mm to 15 mm and the minimal change between 15 mm to 25 mm bears resemblance to the changes observed in atmospheric optics, where the effects are influenced by the ratio of the receiver’s aperture diameter to the Fried parameter of the atmosphere. Even though these observations are rooted in underwater turbulence, it’s plausible that both scenarios display similar behavior due to the fundamental principles of wavefront propagation through a turbulent medium [21].

5. Residual phase error

Residual phase error plays a crucial role in determining Zernike polynomial coefficients within adaptive optics systems. It quantifies the level of distortion that persists on the wavefront even after the initial correction of M’s primary modes of the Zernike polynomials. By employing wavefront decomposition in conjunction with an adaptive optics system, the extent of wavefront correction can be accurately assessed by [21]:

We can define the remaining aberration on the wavefront as [21]:

Next, we present the aperture averaged mean square of wavefront phase error following the correction of M’s first modes of the Zernike polynomials [21]:

In Fig. 5, we present the plotted values of $\sigma _M^2$ for aperture diameters of 5, 15, and 25 mm in the receiver optical system. These measurements were taken under varying turbulent strengths for the first up to the 81st modes of Zernike polynomials. Additionally, we examined the performance at two specific points: point A, located near the transmitter, and point E, situated near the receiver.

 

Fig. 5. The residual phase error of coefficients of Zernike polynomials, averaged over a duration of 300 seconds, with measurements conducted at various aperture diameters and a focal length of 50 mm at points A and E. (a) corresponds to an applied voltage of 50 volts at point A. (b) to (e) are presented in a logarithmic scale: (b) represents the effect of an applied voltage of 50 volts at point A, (c) shows the response to 150 volts at point A, (d) illustrates the effect of 50 volts at point E, and (e) depicts the response to 150 volts at point E.

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It is important to note that, as the coefficients beyond the first 4 (excluding the piston) rapidly approach zero, as demonstrated in Fig. 5(a), it is necessary to display the residual phase error on a logarithmic scale. Figure 5(b) to (d) shows the residual phase error in all configurations after subtracting the first 4 coefficients. This indicates that most of the impact of underwater turbulence-induced aberrations on the laser beam, in both weak and strong turbulence, is concentrated within the first 4 coefficients, with the Y-tilt being particularly significant.

6. Correction of measurement errors

It is essential to acknowledge that the calculation of $\sigma _M^2$ in Eq. (9) is subject to measurement errors and errors arising from the correlation between the Zernike polynomials modes. Consequently, it becomes imperative to address and rectify these errors before further analysis. To mitigate the impact of measurement errors or noise present in the collected data, we can quantify the average mean square of the wavefront phase error on the aperture using Eqs. (10) and (11) [23,24]:

The subscript $i$ denotes the quantity that relies on the $i$th mode of the Zernike polynomials. The error variance in measuring the slope of the wavefront in Eq. (1), denoted as $\sigma _{i,N}^2$, is dependent on the $i$th mode of the Zernike polynomials. Additionally, the noise-free averaged mean square of turbulence-induced phase in the $i$th mode of the Zernike polynomials represented as $\sigma _{i,C}^2$, is the sum of the averaged mean square of the Zernike polynomials coefficients defined as

In order to determine the noise error, or more specifically, the noise variance, which exhibits temporal decorrelation on each of the Zernike polynomial modes, the temporal autocorrelation of the Zernike polynomial coefficients can be computed for the measured data using the following approach [23]:

Furthermore, the temporal autocorrelation of Zernike polynomials coefficients can be conveniently reformulated, owing to the absence of any discernible correlation between the noise and the acquired data [23]:

The temporal autocorrelation $C_{i,C}{\left (\tau \right )}$ and noise variance $\sigma _{i,N}^2\left (\tau \right )$ of the $i$th coefficient of Zernike polynomials play a crucial role in our analysis. To calculate $\sigma _{i,N}^2\left (\tau \right )$, we first determine $C_{i,M}\left (\tau \right )$. Subsequently, we apply a polynomial to the data obtained, excluding $C_{i,M}{\left (0\right )}$. By utilizing the obtained polynomial, we extrapolate $C_{i,C}{\left (0\right )}$, which is equivalent to $C_{i,M}{\left (0\right )}$ on the fitted polynomial curve. The discrepancy between $C_{i,M}\left (0\right )$ and $C_{i,C}{\left (0\right )}$ gives us the value of $\sigma _{i,N}^2\left (\tau \right )$. This calculation and its implications are further illustrated in Fig. 6 [23].

 

Fig. 6. The calculation of noise variance for the second Zernike coefficient, specifically Tilt in the Y direction (j=1). The parameters used for this calculation include an aperture diameter of 5 mm, a focal length of 50 mm, and the point of interest being point A. Additionally, the rod heater voltage is set to 50 V.

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This noise correction is presented in columns 4 and 5 of Table 1. As can be observed, in weak turbulence (50 volts), this correction is minimal but noticeable. However, in strong turbulence (150 volts), the correction is insignificant and the data remain unchanged after noise correction. The impact of these corrections for both turbulence levels is illustrated in Fig. 8(a) and (b), where the effect of noise correction on the original, uncorrected data can be observed.

7. Correction of the correlations among Zernike polynomial modes

Upon rectifying the noise and measurement errors in the acquired data, it becomes imperative to address the correlation error between Zernike polynomial modes. To rectify the variance stemming from this error, it is essential to initially estimate the error resulting from the correlation of the Zernike polynomial modes. Subsequently, this estimated error must be subtracted from the variance of each of the calculated coefficients as follows [23]:

In the above equation, $\sigma _{i,A}^2$ represents the error variance resulting from the correlation between the modes of Zernike polynomials associated with the $i$th coefficient.

To accurately assess the magnitude of this error, it is necessary to determine the correlation between the modes of Zernike polynomials. This correlation leads to non-zero values in the non-diagonal elements of the covariance matrix or variance of the Zernike polynomial coefficients. Fortunately, there are various approaches available to mitigate this error and rectify the coefficients obtained for Zernike polynomials [21,25,26].

In this study, we have utilized a methodology that incorporates basis functions other than Zernike polynomials, capable of effectively expanding the turbulence-induced wavefront phase error while maintaining independence among their coefficients. This expansion results in a diagonal covariance matrix, thereby minimizing any potential issues. Moreover, the coefficients obtained from expanding the phase error using these new functions exhibit a more minor variance than those obtained for Zernike polynomials. One viable approach for calculating these new functions is to utilize the method proposed by Karhunen-Loeve [21,25,26]. This approach has demonstrated its efficacy in producing precise and dependable outcomes in analogous investigations.

Given the introduction of these novel functions, the expansion of the phase error after eliminating the piston can be expressed in the following manner [13,16,21]:

The vector comprises the coefficients $b_i$, while $M_{\text {cov}_b}$ represents the covariance matrix of these coefficients for the new functions. It is important to note that the modes $f_i\left (\rho,\theta \right )$ are not correlated, resulting in $M_{\text {cov}_b}$ being a diagonal matrix. Nevertheless, the coefficients $b_i$ can be determined using the coefficients of Zernike polynomials. We can perform the calculation in the following manner as a consequence [21]:

The vector $\mathbf {a}$ contains the coefficients, and $U^T$ represents the transpose of the unitary matrix $U$. Note that matrix $U$ is considered unitary when $U^{-1} = U^T$ equals to $U^T$. Under this condition, we define the columns of matrix to correspond to the eigenvectors of the covariance matrix of the coefficients $a_i$, as follows [21]:

Hence, the determination of the coefficients for the new basis functions $f_i\left (\rho,\theta \right )$ can be achieved by utilizing the eigenvectors of $M_{\text {cov}_a}$ and Eq. (18).

For calculating the variance of these coefficients, it is important to note that the modes of $f_i\left (\rho,\theta \right )$ are not correlated. We rectified the $\sigma _{i,A}^2$ error in Eq. (15), which allows us to treat the variance of these coefficients as:

Columns 6 and 7 of Table 1 show the results of applying alias correction to the data obtained under weak (50 volts) and strong (150 volts) turbulence conditions. The correction has negligible effects on the data, as the values have no significant variations after the correction.

Figure 7 compares the Y-tilt coefficients before and after noise and alias corrections. The original data, shown in blue line, are the same as in Fig. 2. The red and orange lines represent the Y-tilt coefficients after noise and alias corrections, respectively.

 

Fig. 7. The second Zernike coefficient, specifically the Tilt in Y (j=1), after being corrected for noise and alias. The measurements were taken for three different aperture diameters (5, 15, and 25 mm) and a fixed focal length of 50 mm. Subplots (a) corresponds to point A, where the rod heater voltage was set to 50 V, (b) point A, but with a rod heater voltage of 150 V, (c) the results obtained at point E, with a rod heater voltage of 50 V, and (d) the outcomes at point E, but with a rod heater voltage of 150 V.

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Figure 8(a) and (b) illustrate the impact of alias correction on the original data, and confirm that the alias error for the Zernike coefficients of underwater turbulence-induced aberrations is insignificant, especially for the first four coefficients excluding piston.

 

Fig. 8. The mean variance of the initial 81 coefficients of Zernike polynomials over 300 seconds time period, for three aperture diameters (5, 15, and 25 mm) and a focal length of 50 mm: (a) with a rod heater voltage of 50 V; (b) with a rod heater voltage of 150 V.

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As Table 1 and Fig. 7(b) and (d) indicate, the Y-tilt coefficient (and all other 80 Zernike coefficients) remains unchanged after correction under strong (150 volts) turbulence conditions. However, under weak (50 volts) turbulence conditions, the noise correction reduces this error in the Y-tilt coefficient, while the alias correction has no effect (as in both weak and strong turbulences).

As mentioned before, Fig. 8 show that the receiver position and aperture size affect the wavefront aberrations differently under low and severe turbulence conditions (Also, this issue is indicated in Table 1).

For low turbulence, the wavefront aberrations decrease as the receiver moves away from the phase screen and as the receiver aperture size decreases. For severe turbulence, the opposite is true: the wavefront aberrations decrease as the receiver moves closer to the phase screen and as the receiver aperture size increases.

8. Conclusion

In this study, we explored the impact of the initial four Zernike polynomial terms on underwater laser propagation and the effects of turbulence. Our findings emphasize a significant tilt aberration in the y-direction within the observed aberrations, a phenomenon directly linked to the thermal dynamics induced by the rod heater employed in our experimental configuration. The thermal energy discharged into the water generates a phase screen, propelling heat and resultant turbulence vertically toward the water’s surface. This upward movement aligns with the +Y direction in our setup, leading to the dominant presence of tilt aberrations in this specific orientation. Our analysis sheds light on the critical role of experimental setup and thermal effects in understanding and predicting aberrations in underwater laser propagation studies.

The relationship between the diameter of the receiver optical system, turbulence on the phase screen, and induced aberration variances was also examined. The findings show that increasing the receiver aperture size in low turbulence conditions leads to higher levels of wavefront distortions due to capturing more turbulence-induced aberrations. However, a larger receiver aperture decreases the induced aberration variances in strong turbulence conditions.

This suggests that the optimal approach to minimizing wavefront aberrations depends on the severity of turbulence. In low turbulence conditions, reducing wavefront aberrations can be achieved by moving the receiver away from the turbulence and reducing the receiver aperture size. In severe turbulence, increasing the aperture size can help mitigate wavefront distortions. The mean variance of the aberration coefficients under strong turbulence resembles the aperture averaging effect in atmospheric optics, which necessitates further investigation.

These findings have implications for designing and optimizing optical systems in different underwater conditions. It is also important to note that noise error exerts a more substantial influence than correlation error between Zernike polynomials when it comes to error correction.

Moreover, experimental results indicate a significant increase in the averaged mean square of Zernike polynomial coefficients when increasing the aperture diameter from 5 mm to 15 mm. Yet, no substantial difference is observed when further increasing from 15 mm to 25 mm. This change could be influenced by the ratio of the receiver’s aperture diameter to Fried parameter of atmospheric optics. Despite these observations being based on underwater turbulence, it is plausible that both scenarios exhibit similar behavior due to wavefront propagation principles through a turbulent medium. This statement requires further research to confirm its validity and implications.

In summary, this study contributes to understanding underwater laser propagation and turbulence and provides insights for designing and optimizing optical systems. Further research in this area can build upon these findings to enhance optical systems’ performance under various atmospheric conditions.