Measurement of Atmospheric Coherence Length from a Shack–Hartmann Wavefront Sensor with Extended Sources

Abstract

Free Space Optical Communication (FSOC) is a wireless communication method that utilizes laser beams for high speed and secure data transmission. Its performance is affected by various factors, among which atmospheric turbulence causes random fluctuations in the atmospheric refractive index, significantly impacting the reliability of communication links. The atmospheric coherence length is a key parameter describing the coherence properties of a laser signal as it propagates through the atmosphere, and accurately measuring it is crucial for assessing the quality of FSOC links. This paper proposes a novel strategy that utilizes extended sources directly as the information sources, combining the wavefront phase variance method with the extended source offset algorithm based on Shack–Hartmann wavefront sensors to directly measure atmospheric coherence length. Existing methods in extended scenarios typically rely on deploying laser beacons to aid in the calibration of atmospheric coherence length but setting up suitable beacons on horizontal communication links is challenging. Additionally, these approaches can be costly in terms of equipment and measurement expenses. Compared to traditional measurement methods, the algorithm proposed in this paper can measure directly based on extended scenarios in horizontal links, thereby effectively reducing system complexity and equipment costs. To verify the feasibility and effectiveness of this method, targeted simulations and experiments were conducted, and the results show that the coherence length measured by the algorithm is highly consistent with that measured by the Differential Image Motion Monitor (DIMM), with a deviation of less than 2% from actual values, effectively demonstrating the algorithm’s feasibility in coherence length assessment.

1. Introduction

Free Space Optical Communication (FSOC) typically uses narrow bandwidth-modulated laser beams as carrier signals to achieve high speed and secure wireless data transmission between two fixed nodes. Compared to traditional transmission methods such as radio waves, laser beams are not subject to intercell interference, do not require frequency coordination, and are less susceptible to electromagnetic interference generated by other communication networks or electronic devices [1,2,3,4,5,6]. However, since the laser beam in FSOC partially or completely travels through the atmosphere, the system’s sensitivity to atmospheric conditions means that its performance is significantly affected by atmospheric turbulence disturbances. Particularly in near-ground horizontal laser communication scenarios such as maritime and urban communications, increasing turbulence intensity and communication distance may cause the laser beam to experience random jitter and high-frequency phase fluctuations, which can severely degrade communication quality [2,7,8]. Measuring turbulence intensity and understanding the characteristics and variations in turbulence provide effective references for assessing the stability of FSOC. Based on this, Fried proposed the atmospheric coherence length r₀ (also known as the Fried parameter) in 1965 as a characteristic scale for describing the coherence of a light beam during its propagation through the atmosphere [9]. The atmospheric coherence length not only reflects the intensity of turbulence but also characterizes the amplitude fluctuations of the wavefront phase and amplitude. Accurate measurements of the atmospheric coherence length can effectively assess the quality of FSOC links.

Currently, the main methods for measuring atmospheric coherence length include the Differential Image Motion Monitor (DIMM) method, the Shack–Hartmann Image Motion Monitor (SHIMM) method, wavefront structure function methods, and wavefront phase variance methods [10,11]. The DIMM method calculates the atmospheric coherence length r₀ by measuring the differential motion of target images from two sub-apertures defined by a telescope aperture mask. This approach effectively avoids errors caused by non-atmospheric factors, such as telescope tracking, vibrations, optical quality, and temperature effects [12]. The SHIMM method enhances the signal-to-noise ratio (SNR) by replacing the standard double-aperture mask in the DIMM with a Shack–Hartmann lens array [13,14,15]. The wavefront structure function method calculates the corresponding wavefront structure function from a single frame reconstructed by a Shack–Hartmann wavefront sensor, using the measured values to perform a least-squares estimation of r₀ against theoretical values [16,17]. The wavefront phase variance method derives r₀ by obtaining wavefront distortion information through a wavefront sensor and calculating the variance of Zernike coefficients [16,17,18,19,20,21].

Researchers have performed a series of optimizations and iterations on these methods, such as using three-aperture and four-aperture DIMMs for turbulence measurement [22,23,24,25], compensating for short-exposure turbulence effects and varying turbulence heights using image motion correlation between adjacent frames or weighted functions [26,27], and combining multi-aperture scintillation sensors (MASS), scintillation detection and ranging (SCIDAR), and Slope Detection and Ranging (SLODAR) instruments for studying multi-layer turbulence under various atmospheric and environmental conditions [14,25,27,28]. The advancement of the aforementioned techniques has significantly improved the accuracy of coherence length measurements. However, most studies are still focused on measuring the atmospheric coherence length in vertical communication links in point source scenarios, and research on remotely analyzing horizontal turbulence characteristics in the surface layer remains insufficient. Currently, in extended source application scenarios such as maritime radar target detection, near-ground laser guidance, and aircraft detection, the atmospheric coherence length r₀ is typically calibrated by employing focused beams, such as Rayleigh beacons, transmitted by atmospheric turbulence profiling radars [29,30]. This approach results in high system complexity and cost compared to directly using the extended source itself as the signal source, and due to low scattering efficiency, it may inadequately reflect turbulence conditions in near-ground atmospheres over long distances.

Compared to point sources, extended sources are easier to obtain and allow for the selection of high-contrast targets to enhance the SNR of detection. In response to the complexity of accurately determining the atmospheric coherence length, this study introduces a new strategy combining the wavefront phase variance method with the Shack–Hartmann Wavefront Sensor (SHWFS) extended source offset algorithm, using the extended source as a direct source of detection information to measure the Zernike coefficients of distorted wavefronts for calculating r₀. Our investigation validates the practicability and superiority of the proposed method by benchmarking its measurement precision against the established Differential Image Motion Monitor (DIMM) method across both point sources and extended sources. In Section 2, we describe the foundational principles and methodologies for measuring atmospheric coherence length utilizing both the DIMM and the wavefront phase variance methods, as well as the principle of the offset algorithm for SHWFS in extended scenarios. Section 3 and Section 4 detail the design and execution of a controllable turbulence phase plate and SHWFS simulation to verify the feasibility of the algorithm. We also delineate the construction of a test system with a highly repeatable turbulence phase plate. The effectiveness of our method will be confirmed by comparing its measurement validity and precision with the DIMM method in both point source and extended scenarios. Section 5 and Section 6 will discuss and summarize the research findings and the algorithm’s performance, offering insight into prospective avenues for algorithmic enhancements.

2. Principles and Common Methods

2.1. DIMM

In 1960, Stock and Keller introduced a method for measuring atmospheric coherence length using DIMM, which consists of an observatory telescope and a dual-aperture mask connected to a CCD for imaging, as shown in Figure 1. The atmospheric coherence length, denoted as r₀, is derived by assessing the relative variance of the fluctuations in the angles of arrival of the distorted wavefront within the sub-apertures at the telescope’s pupil. This variance is manifested on the CCD as a random jitter of the positions of the two stellar images. The jitter is calculated separately for two orthogonal directions relative to the line connecting the centers of the two sub-apertures: one direction is radial, and the other is transverse. David L. Fried and Roddier provided a formula for calculating the relative variance of the angle of arrival fluctuations between the two stellar images, relating the variance along the longitudinal and transverse connections to r₀ as follows [31]:

$${\sigma }_{\mathrm{l}}^2=2{\lambda }^2{r}_0^{-5/3}\left[0.179D^{-1/3}-0.0986d^{-1/3}\right]$$

(1)

$${\sigma }_t^2=2{\lambda }^2{r}_0^{-5/3}\left[0.179D^{-1/3}-0.1450d^{-1/3}\right]$$

(2)

where D represents the diameter of the telescope’s sub-aperture, d denotes the separation between the sub-apertures, and λ is the wavelength. The terms σ_l² and σ_t² correspond to the variance of image jitter in the directions parallel and perpendicular to the line connecting the sub-apertures, respectively. Based on the Taylor frozen turbulence hypothesis and Kolmogorov turbulence statistical theory, r₀ is statistically independent of direction. By combining the two expressions, we can derive the formula for calculating r₀ as follows:

$$r_0={\left[2{\lambda }^2(0.358D^{-1/3}-0.242d^{-1/3})/({\sigma }_1^2+{\sigma }_t^2)\right]}^{3/5}$$

(3)

In measurements, the arrival angle variance is generally obtained by calculating the variance of the relative position of the centroids of the two sub-aperture images. The variance σ_l² and σ_t² satisfy:

$$V=({\sigma }_1^2+{\sigma }_t^2)f^2$$

(4)

where V represents the relative positional variance of the two sub-aperture images, and f is the focal length of the telescope. Substituting Equation (4) into Equation (3) yields:

$$r_0={[2{\lambda }^2{f}^2(0.358D^{-1/3}-0.242d^{-1/3})/V]}^{3/5}$$

(5)

Obviously, in point source scenarios, the estimation error of the atmospheric coherence length through Equation (5) mainly arises from the calculation accuracy of the relative position of the centroids of two spots. This is essentially the same as the calculation accuracy of the offset between two sub-aperture images in extended source scenarios.

2.2. Wavefront Phase Variance Method

The wavefront of a circular domain with radius r can be fitted using Zernike polynomials defined on the unit circle through the least squares method:

$$\phi (\rho ,\theta )={\displaystyle \sum _j{a}_j}{Z}_j(\rho ,\theta )$$

(6)

where ρ is the normalized radius, θ is the azimuthal angle, and a_j is the coefficient of the j-th Zernike polynomial.

$$\left\{\begin{array}{l}{Z}_j^{even}=\sqrt{n+1}{R}_n^m(r)\sqrt{2}\mathit{cos}m\theta ,m\ne 0\\ Z_j^{odd}=\sqrt{n+1}{R}_n^m(r)\sqrt{2}\mathit{sin}m\theta ,m\ne 0\\ Z_j=\sqrt{n+1}{R}_n^0(r),m=0\end{array}\right.$$

(7)

$$R_n^m(r)={\displaystyle \sum }_{s=0}^{(n-m)/2}{\displaystyle \frac{{(-1)}^s(n-s)!}{s!\left[(n+m)/2-s\right]!\left[(n-m)/2-s\right]!}}{r}^{n-2s}$$

(8)

According to R.J. Noll’s definition [32], j is the mode sequence number defined by the radial order n and the azimuthal order m. m ≥ 1, n ≥ m, and both n and m are integers.

Based on Kolmogorov turbulence statistical theory, Fried provided a table of variances for various orders of Zernike coefficients, as shown in

Table 1. Variance of Zernike Coefficients for Distorted Wavefronts Complying with Kolmogorov Spectrum.

Zernike Order	Variance (rad2)	Zernike Order	Variance (rad2)	Zernike Order	Variance (rad2)
1	0.4479 (D/r0)5/3	8	0.0062 (D/r0)5/3	15	0.0012 (D/r0)5/3
2	0.4480 (D/r0)5/3	9	0.0062 (D/r0)5/3	16	0.0012 (D/r0)5/3
3	0.0230 (D/r0)5/3	10	0.0024 (D/r0)5/3	17	0.0012 (D/r0)5/3
4	0.0230 (D/r0)5/3	11	0.0024 (D/r0)5/3	18	0.0011 (D/r0)5/3
5	0.0232 (D/r0)5/3	12	0.0024 (D/r0)5/3	19	0.0012 (D/r0)5/3
6	0.0061 (D/r0)5/3	13	0.0024 (D/r0)5/3	20	0.0012 (D/r0)5/3
7	0.0062 (D/r0)5/3	14	0.0024 (D/r0)5/3

.

The schematic sketch for measuring coherence length using the wavefront phase variance method is shown in Figure 2. Therefore, according to Table 1, the steps to calculate the atmospheric coherence length r₀ using the wavefront phase variance method are as follows: First, measure the wavefront slopes with a Shack–Hartmann wavefront sensor and obtain the coefficients of the incident wavefront expanded in the various orders of Zernike polynomials to perform wavefront reconstruction. Then, conduct temporal statistical analysis to compute the coefficient variances, which can be used to backtrack the value of r₀.

2.3. Extended Source Offset Algorithm

Based on the aforementioned discussion, the primary requirement for measuring the r₀ value using the DIMM method is to obtain the variance of the jitter from the images of two sub-aperture stars on the CCD. Similarly, the wavefront phase variance method also necessitates the use of a SHWFS to determine the offsets of the sub-aperture images, which are then used to compute the slope of the incident wavefront and subsequently perform wavefront reconstruction. Clearly, the most critical aspect of the algorithm for measuring atmospheric coherence length using an extended source as a direct signal source is the calculation of the offsets of the sub-aperture images in both the DIMM and SHWFS methods. The most widely used algorithm for determining sub-aperture offsets for extended sources is the cross-correlation interpolation algorithm [33]. The commonly used criterion function for detecting wavefronts from extended sources is the normalized cross-correlation function. This function utilizes the cross-correlation values between the reference image and the image to be measured as a measure of similarity. The corresponding correlation function is defined as follows:

$$C(u,v)={\displaystyle \frac{{\displaystyle \sum _{x,y}\left[I_s\left(x,y\right)-{\overline{I}}_{u,v}\right]\left[t(x-u,y-v)-\overline{t}\right]}}{\sqrt{{\displaystyle \sum _{x,y}{\left[I_s\left(x,y\right)-{\overline{I}}_{u,v}\right]}^2{\displaystyle \sum _{x,y}{\left[t(x-u,y-v)-\overline{t}\right]}^2}}}}}$$

(9)

where u and v represent the offsets of the image to be measured relative to the template image in the x and y directions, respectively, I_s represents the image to be measured, $\overline{t}$ and ${\overline{I}}_{u,v}$ denote the mean values of the template image and the I_s (x, y) in the template range, respectively. The maximum value of C (u, v) corresponds to the coordinates (x₀, y₀) of the image to be measured, which represents the integer pixel-level offset. At the integer pixel location (x₀, y₀), data points within a neighborhood surrounding the peak (for instance, a 3 × 3 or 5 × 5 pixels range) are selected for interpolation. Subsequently, a mathematical model is chosen for interpolation, such as parabolic interpolation, Gaussian interpolation, or pyramid interpolation. The extremum of the fitted curve is then solved, allowing for the calculation of its deviation from the integer pixel peak position (Δx, Δy). Accordingly, the sub-pixel offset is given by (x₀ + Δx, y₀ + Δy). A commonly used one-dimensional parabolic interpolation algorithm is expressed as follows [34]:

$$S_x=x_0+0.5\times {\displaystyle \frac{C\left[x_0-1,y_0\right]-C\left[x_0+1,y_0\right]}{C\left[x_0-1,y_0\right]+C\left[x_0+1,y_0\right]-2C\left[x_0,y_0\right]}}$$

(10)

3. Simulation

3.1. Turbulence Simulation

The simulation method employed in this paper for the turbulence is derived from the work of Paulson, Wu, and Davis. It is a turbulence phase simulation method that combines the random sampling concept based on the Sparse Spectrum (SS) technique with the computational efficiency of discrete Fourier transform (DFT). For details of the algorithm, please refer to reference [35]. In the PWD method, the complex phase is given by:

$${\psi }_{PWD}(j\mathrm{\Delta }x,l\mathrm{\Delta }x)={\displaystyle \sum _{m,n=-N_{DFT}/2^{-1}}^{N_{DFT}/2-1}{a}_{m,n}^{(PWD)}}exp({\displaystyle \frac{2i\pi }{N_{DFT}}}((m+\xi )j+(n+\eta )l))$$

(11)

where ξ and η are independent and identically distribute random variables following a uniform distribution U(−0.5, 0.5). The PWD method compensates for the PWD phase through the random sampling of subharmonic (SH) components, as shown below:

$$\begin{array}{l}{\psi }_{SH}(j\mathrm{\Delta }x,l\mathrm{\Delta }x)={\displaystyle \sum _{p=1}^{N_{SH}}{\displaystyle \sum _{m,n=-1}^1{a}_{p,m,n}^{(PWD)}}}exp\left(\left.{\displaystyle \frac{2i\pi }{3^P{N}_{PWD}}}((m+{\xi }_{p,m,n})j+(n+{\eta }_{p,m,n})l)\right)\left[1-{\delta }_{m,n}{\delta }_{n,0}\right]\right.\\ +a_{N_{SH},0,0}^{(PWD)}exp({\displaystyle \frac{2i\pi }{3^{N_{SH}}{N}_{PWD}}}({\xi }_{N_{SH},0.0}j+{\eta }_{N_{SH},0.0}l)).\end{array}$$

(12)

where ${\xi }_{p,m,n}$ and ${\eta }_{p,m,n}$ are independent and identically distribute random variables also following a uniform distribution U(−0.5, 0.5).

Under the conditions of N_DFT = 1024 for the number of spectral components and N_SH = 8 for the number of subharmonic layers, while maintaining an aperture diameter D = 1 m, values of r₀ = 0.1, 0.2, 0.25, and 0.5 m were set. The generated wavefront phase maps on a 1024 × 1024 pixels grid are shown in Figure 3a. Figure 3b shows the far-field diffraction pattern of the same beam after turbulence of different intensities. From the results, it can be observed that the turbulence phases produced by the PWD-SH method exhibit non-periodic characteristics, with the turbulence intensity gradually decreasing as r₀ increases, leading to significantly more focused far-field diffraction patterns and clearer spots.

Figure 4a shows a sample set of 1000 independent turbulence phases generated under typical turbulence conditions using the PWD-SH algorithm. We calculated the average structure function and compared it with the expected values from Kolmogorov’s theory. The results demonstrate a high degree of agreement between the average values of the structure function and the theoretical values under different D/r₀ ratios as the aperture size changes, validating the effectiveness of this algorithm in turbulence parameterization. Furthermore, by utilizing the PWD-SH algorithm, we obtained the Zernike polynomial coefficients of various orders corresponding to wavefront distortions after passing through the turbulence. Based on the statistical results of the 1000 frames, Figure 4b presents the variance distribution of the first 20 Zernike coefficients and compares it with the theoretical variances. The experimental results indicate that the turbulence phase variance values closely match the theoretical values, and the variance rapidly decreases with increasing Zernike order, consistent with the theoretical expectations of the Kolmogorov spectrum. The simulation results confirm the feasibility and accuracy of the turbulence simulation algorithm.

3.2. Shack–Hartmann Sensor Simulation

In this section, we will discuss the implementation of the SHWFS simulation method and the related results. The simulation of the SHWFS is a prerequisite for the overall simulation of the wavefront phase variance method. The object plane size is set to 6.6 × 6.6 mm with a sampling point matrix of 1024 × 1024, the same as the simulated turbulence scale, illuminated by a uniformly coherent plane wave with a wavelength of 632.8 nm. In this study, we use randomly generated samples from the previously described PWD-SH method as the incident distorted wavefront before the microlens array. The microlenses then form an array of distorted spot images, which represents the imaging of a point source scenario. This array is convolved with the USAF1951 resolution target image to generate the image array for the extended sources SHWFS. The SHWFS simulation parameters are listed in

Table 2. Simulation Parameters for SHWFS.

SHWFS Parameters	Value
Incident Wavefront Diameter	6.6 mm
Microlens Number	11 × 11 (97 effective)
Microlens Focal Length	30 mm
Microlens Pitch	600 μm
Sub-aperture Pixels Number	60 × 60

.

A random turbulence phase sample is selected as the input distorted wavefront for the simulation. The cross-correlation parabolic interpolation algorithm is used to calculate the incident slopes, and the Zernike mode method is employed for wavefront reconstruction. As shown in Figure 5, the SHWFS can accurately detect the Zernike coefficients of the incident wavefront across various orders, with the reconstructed residual PV value approximately 1 rad, demonstrating the feasibility and effectiveness of the extended source offset algorithm and sensor simulation.

Existing research indicates that performing temporal statistical analysis on the atmospheric coherence length r₀ generated by turbulent phases requires at least 200 observed data frames of distorted wavefronts. To further investigate the temporal variation characteristics of the coherence length r₀, we pre-generated phases with different coherence lengths, including r₀ = 0.1, 0.2, 0.3, and 0.5 m, yielding a set of 1000 turbulence phase data frames. Subsequently, we used the extended source SHWFS to reconstruct the wavefronts for these four groups of turbulence phases. The variance of the defocus term for every 200 frames was utilized to calculate the corresponding r₀ values, and the relationship curve of r₀ values with the frame number was plotted as shown in Figure 6. For example, the r₀ data for Frame 201 is derived from the calculation of the defocus term variance from Frames 2 to 201. The reason for selecting the defocus term variance to calculate r₀ is that the defocus term has a high SNR, which effectively reflects the overall phase fluctuations of the wavefront. It is evident that the r₀ values measured using the wavefront phase variance method align well with theoretical expectations, demonstrating excellent reconstruction accuracy.

We also conducted an exploration of the impact of SNR on algorithm accuracy. When Gaussian noise with a standard deviation of sigma = 0.3 is added to the sub-aperture images, the measurement results of r₀ are noticeably affected. In this case, the blurry image characteristics due to the noise lead to significant measurement errors in r₀. Additionally, when the standard deviation sigma ≥ 0.5, we observed that the image features were almost completely lost, causing the measured r₀ values to no longer be reliable. This phenomenon indicates that low SNR significantly impacts the interference effects in the application of the wavefront phase variance method under extended sources, limiting the measurement accuracy of the system. In addition to the impact of low SNR on algorithm accuracy, strong turbulence conditions also significantly affect it. When severe turbulence causes sub-aperture images to undergo serious distortion and deformation, the normalized cross-correlation algorithm for extended sources sub-aperture offsets, which relies on the matching of correlation functions between reference images and template images, finds it challenging to accurately locate the position of shifted image points. Moreover, strong turbulence often results in blurred sub-aperture images and significant contrast reduction, and the normalized cross-correlation algorithm requires clear features, such as high-contrast edges or distinct high–low transition points, for accurate matching. This further increases computational errors. Therefore, in environments with low SNR and strong turbulence, the algorithm’s accuracy is notably reduced. Enhancing the algorithm’s robustness against these influencing factors will become the focus of our next research.

4. Experiment

To verify the effectiveness of the wavefront phase variance method, which incorporates the extended source offset algorithm, in measuring the atmospheric coherence length, we meticulously designed and conducted a series of precise experiments. The experimental optical system is shown in Figure 7, consisting of a light source module, a beam shaping module, and a wavefront detection module.

In the experimental optical path, the light source module uses a highly stable helium-neon laser (λ = 633 nm) to generate a point source for measuring in the same optical axis time-shared with the extended source, allowing for comparison of measurement results. An extended source is generated by transmitting an LED beam, which covers the visible spectrum, through a resolution plate (USAF 1951). The beam shaping module is composed of multiple optical elements, including lenses with diameters of 25.4 mm and 50.8 mm, and focal lengths of 100 mm, 200 mm, 300 mm, and 400 mm. These lenses form a beam expansion system capable of 1:4, 1:2, 3:4, and 1:1 scaling ratios. This system proportionally expands the beam size, and since turbulence-induced coherence length remains unchanged, the beam expansion itself does not introduce or alter existing wave aberrations. Additionally, a customized turbulence phase plate from Lexitek, with a coherent length of 0.3 mm (@633 nm), is used to generate turbulence phases, and its phase distribution conforms to the Kolmogorov turbulence theory. During the experiment, different combinations of lenses with varying magnifications and different aperture stop sizes are used to change the transmission size of the turbulence phase screen. Since the coherent length of the turbulence phase screen remains constant, this alters the D/r₀ value. The setup of the beam expansion system and the sizes of the aperture stop and field of view (FOV) stop ensure that the effective pupil size entering the SHWFS is always 4 mm, which is crucial for maintaining the accuracy of wavefront detection. The parameters of the SHWFS in the wavefront detection module are listed in

Table 3. Experiment Parameters for SHWFS.

Photodetector		Microlens Array
Model	EoSens1.1CXP2	Sub-aperture shape	Square
Sub-aperture pixels	28 × 28	Effective sub-apertures	89
Number of pixels	308 × 308	Microlens pitch	383.6 μm
Effective aperture diameter	4 mm	Focal Length	16.255 mm

.

To ensure reliable experimental data under both point sources and extended sources conditions, the laser beam and LED beam are directed through a beam splitter and collimating lenses to the turbulence phases, which are then transmitted to the SHWFS via the beam expansion system. As shown in Figure 7b, the experiment uses a time-sharing measurement method along the same optical axis, with careful adjustments made to the distances between the turbulence phases, SHWFS, and beam shaping system to ensure they are in conjugate positions. This setup assures consistency and comparability between the measurements of point sources and extended sources.

Before introducing the turbulence phases, the entire optical system was calibrated to ensure that both the laser and LED light sources could accurately focus on the same position of the Shack–Hartmann wavefront detector through the beam splitter. By artificially introducing defocus phases, the results from using the centroid method for point sources and the cross-correlation algorithm for extended sources indicated that the Zernike coefficient errors of the defocus term were below 4%. The calibration results demonstrated that the system possesses high precision measurement capabilities under the specified conditions, providing a reliable reference for subsequent experiments.

During the measurement of the coherence length r₀ based on a point source, the rotational speed at the center of the turbulence phase screen’s aperture is maintained at approximately 42 mm/s. This achieves a simulated Greenwood frequency of about 60 Hz, with the preset SHWFS exposure time set to 3 ms and an effective aperture size of 4 mm. After passing through the turbulence phase screen, the laser beam traverses an expansion system and passes through an aperture stop, a field stop, and a preset beam expansion ratio to ensure that the beam entering the detector always fills the effective sub-aperture. The detector monitors wavefront slope changes in real time and calculates the corresponding atmospheric coherence length r₀. Subsequently, the target is switched to an extended source, and the SHWFS exposure time is changed to 11 ms. The above measurement steps are repeated with the preset effective apertures set to 1 mm, 2 mm, 3 mm, and 4 mm, respectively. In total, 12 × 6000 frames of data were accumulated for both the point sources and extended sources measurements. Every 200 frames of Zernike coefficients were used to calculate an r₀ value, resulting in a statistical curve for r₀.

In Figure 8a, the comparison between experimental measurements and reference values for point sources and extended sources at different D/r₀ ratios is presented. The data indicate that the measurement results remain relatively stable, with both the centroid method for point sources and the cross-correlation interpolation algorithm for extended sources producing accurately calculated r₀ values. Figure 8b,c show the spot arrays obtained by the SHWFS under conditions with and without the turbulence phases, respectively. Notably, as the effective aperture size increases, the sub-aperture images in the extended scene become progressively more diffuse and less defined in shape. This observation suggests a close relationship between the intensity of the incident light and the size of the template selection, as these factors significantly influence measurement accuracy. Therefore, future research will focus on the quantitative analysis of these elements and the optimization of their robustness. Such investigations will be crucial for enhancing the reliability and precision of measurements in varied experimental conditions, ensuring that the methodologies can be effectively applied in practical scenarios.

5. Discussion

In the DIMM method, atmospheric coherence length r₀ is estimated by calculating the variance of angles of arrival at two sub-apertures, which can be derived from the offset of centroids corresponding to the sub-aperture spots. In extended sources, the offset of the sub-aperture images obtained via the cross-correlation interpolation algorithm can be considered an equivalent representation of the spot centroid displacement. Under the experimental conditions of an effective aperture size D = 2 mm, we measured the coherence length of the turbulence phases using the DIMM method and the wavefront phase variance method in both point sources and extended sources. The experimental results shown in Figure 9 illustrate the calculation of coherence length r₀ using every 200 groups of offset values, plotted with a sliding window pattern. Throughout the measurement of 6000 frames, the r₀ values obtained via both the DIMM method and the wavefront phase variance method maintained significant stability. In the extended sources, the wavefront phase variance method yielded an average r₀ value of 0.2945 mm across 12 sets, while the DIMM method produced an average r₀ value of 0.3001 mm. The resulting error between the two methods was less than 2%.

Specifically, the results from the first 1000 frames show that the r₀ values from both methods fluctuated similarly, indicating favorable coherence between them. As the number of measurement frames increased, the variance of the measurement results gradually decreased, further demonstrating that the system’s stability and repeatability improved with extended measurement time. Notably, the average r₀ value in the point source scene using the DIMM method was 0.34 mm, compared to 0.31 mm in the extended sources. This confirms that the cross-correlation interpolation algorithm was successfully applied to DIMM in the extended sources, verifying the effectiveness and reliability of the method. Meanwhile, the wavefront phase variance method yielded an average r₀ value of 0.32 mm in the point source scene and 0.30 mm in the extended sources, both with relatively low standard deviations, further supporting the feasibility of combining the extended source offset algorithm with the wavefront phase variance method for calculating atmospheric coherence length.

In conclusion, both the DIMM method and the wavefront phase variance method demonstrated high accuracy and favorable consistency in measuring the coherence length of turbulence phases. Future research should delve deeper into factors affecting measurement accuracy, such as the stability of the light source and environmental noise, to further enhance measurement reliability. This is significant for effectively assessing the reliability of free-space optical communication systems.

6. Conclusions

We focus on the study of turbulence measurement methods based on extended sources, specifically by combining the extended sources offset algorithm of the SHWFS with the wavefront phase variance method to measure atmospheric coherence length r₀. Compared to traditional laser beacon methods, this approach effectively reduces equipment costs and complexity. By elucidating the principles of current measurement methods, we demonstrate how the wavefront phase variance method can derive coherence length by reconstructing the Zernike coefficients of the incident wavefront and statistically analyzing the temporal variance. Based on this foundation, we implemented a cross-correlation interpolation algorithm to achieve accurate offset calculations of the sub-aperture images for extended sources. Simulation analyses and experimental results indicate that this method can effectively measure atmospheric coherence length, with the measured r₀ values showing high consistency with theoretical predictions. This suggests the reliability and accuracy of the extended sources offset algorithm in practical applications.

Future research will aim to optimize the robustness and precision of the existing algorithms concerning their dependence on SNR. The goal is to provide effective support for achieving higher-quality evaluations of FSOC network links.