Study on the Difference of Wavefront Distortion on Beams Caused by Wavelength Differences in Weak Turbulence Region

Abstract

In dual-wavelength free-space optical communication systems, the wavefront distortion differences caused by the wavelength difference of the signal and beacon lights cannot be completely corrected. In this study, according to the method of geometric optics, the relationship between the wavefront phases of the signal and beacon lights is analyzed by establishing a mathematical model of the cross-correlation ratio of the arrival-angle fluctuation and phase relationship of different Gaussian-beam wavelengths under weak turbulence near the surface. The numerical results show that after Gaussian beams of different wavelengths propagate through the same atmospheric channel, the cross-correlation number of the arrival-angle fluctuation decreases with an increase in the wavelength difference, waist radius, and turbulence intensity, and increases with an increase in propagation distance. The beam-wavefront phase difference increases with an increasing wavelength difference. Finally, a wavefront distortion correction experiment of a dual-wavelength adaptive optical system verified the correctness of the wavefront phase-difference relationship between the signal light and beacon light established in this study.

1. Introduction

In a free-space optical (FSO) communication system, when influenced by random fluctuations of the atmospheric refractive index after being transmitted through a free channel, a signal beam will experience optical turbulence effects, such as phase distortion, light-intensity fluctuation, beam drift, and arrival-angle fluctuation [1,2,3]. Adaptive optics (AO) technology can effectively suppress the influence of atmospheric turbulence on a signal light based on real-time detection and correction [4,5,6].

When the selected signal-light wavelength has poor penetrability or serious attenuation in the atmospheric channel, the system transmits a beacon light along the same optical path as the signal light [7,8]. Wavefront distortion data are obtained through beacon light detection, and the wavefront distortion of the signal light is corrected accordingly [9,10]. In dual-wavelength FSO communication systems, the signal light and beacon light have different wavelengths; hence, the refraction angle and optical path difference caused by turbulence differ when Gaussian beams are transmitted through an atmospheric channel. This causes a dispersion effect, which results in an incomplete correction of the signal light when correcting it according to the beacon light distortion data. A correction residual exists in AO caused by the signal light and the beacon light [11,12].

In recent years, considerable research has been carried out on wavefront aberrations between the signal light and beacon light in dual-wavelength FSO communication systems. From the beam propagation perspective, the Gaussian beam is modulated by atmospheric turbulence in the atmospheric channel, and the propagation direction of the local wave vector changes, resulting in random fluctuations in the phase plane of the beam and its angle of arrival (AOA) [13,14].

Toselli et al. established an arrival angle scintillation model under a non-Kolmogorov turbulence spectrum [15]. David et al. analyzed the arrival angles of beams of different wavelengths in strong and weak turbulent regions using ray tracing and wave optics [16]. Conan and Borgnino analyzed the AOA of a beam on the receiving plane with a theoretical derivation and numerical calculation, and explained the space–time correlation of the variance of the AOA fluctuation and the difference under the influence of different wavelengths [17]. The correlation between beams of different wavelengths in the weak turbulence region was described by Akira using the cross-correlation power spectrum. Ben-Yosef et al. experimentally measured the intensity flicker correlation coefficients of beams with different wavelengths [18].

Based on a study of the fluctuation of the beam AOA, and to correct the correction residual in a dual-wavelength FSO communication system, Gorelava et al. measured the wavefront distortion variables of two beams with 530 nm and 1060 nm wavelengths through a Shack–Hartmann wavefront sensor. By comparing the measurement results, they found a certain correlation between the wavefront distortion of two beams with different wavelengths. The degree of distortion increased with an increase in the transmission path distance [19].

Li et al. proposed an adaptive optics technique to correct the AOA fluctuation of beams of different wavelengths [20] and analyzed in detail the influence of the beam wavelength on the AOA fluctuation. Kibblewhite et al. designed deformable mirrors in an adaptive optics system according to the power spectrum of the AOA fluctuation with different beam wavelengths [21]. Lukin calculated the correction accuracy and efficiency of an adaptive optical system when the beacon light and signal light wavelengths were inconsistent [22].

Jolissaint used a Shack–Hartmann wavefront sensor to measure the wavefront distortion results of a two-color laser after transmission in an atmospheric path and revealed the correlation between different wavelength distortions [23]. Ke et al. analyzed the correction residuals of the wavefront fluctuation of an adaptive optical system with different wavelengths and determined the correction coefficients of the correction residuals between the beacon wavefront and target wavefront fluctuation [24].

The key to correcting signal wavefront distortions in dual-wavelength FSO communication systems is to determine the wavefront phase relationship between the signal light and beacon light. In view of the residual error in the correction of signal wavefront distortions in weak turbulent regions, this study analyzes the correlation function between the AOA fluctuation of different Gaussian beam wavelengths. Wavefronts of different wavelengths have a specific phase relationship, and the expression of the distortion phase of Gaussian beams with different wavelengths is provided. Finally, the relationship obtained in this study is verified experimentally.

2. Theory

2.1. Correlation Function of the AOA Fluctuation at Different Wavelengths

After Gaussian beams with wavelengths of λ₁ and λ₂ are transmitted through the same atmospheric channel, the AOA fluctuations of the beams at the receiving plane are ${\alpha }_{{\lambda }_1}$ and ${\alpha }_{{\lambda }_2}$, respectively. The correlation between the two can be described by a cross-correlation function [25]:

$$\begin{array}{ccc}\langle {\alpha }_{{\lambda }_1}{\alpha }_{{\lambda }_2}\rangle & =& \frac{F^2}{4k_1{k}_2{w}_0^2}{\displaystyle \int d^4\rho \exp (-\frac{2{\rho }^2}{w_0^2})}\times \langle \frac{\partial s_1(F,{\rho }_{1,}{\lambda }_1)}{\partial {\rho }_1}\cdot \frac{\partial s_2(F,{\rho }_{2,}{\lambda }_2)}{\partial {\rho }_2}\rangle \hfill \\ & & T(\stackrel{\rightharpoonup }{\rho })\hfill & ,\end{array}$$

(1)

$$T(\stackrel{\rightharpoonup }{\rho })=\{\begin{array}{cc}1& \rho \le D\\ 0& \rho >D\end{array}$$

(2)

where F = z/(1 + (z/f))² is the curvature radius of the beam wavefront, z is the beam transmission distance, f is the beam parameter, $k_1=2\pi /{\lambda }_1$ and $k_2=2\pi /{\lambda }_2$ are wave numbers of beams with different wavelengths, ${\omega }_0$ is the beam waist radius, and D is the pupil diameter of the receiving optical system. $\rho ={\rho }_1-{\rho }_2$ represents the distance between ${\rho }_1$ and ${\rho }_2$ on the wavefront; when $\rho =0$, the two points coincide. In Equation (1), $〈\cdots 〉$ represents the ensemble average; its specific expression is [26]:

$$\langle \frac{\partial s_1(F,{\rho }_{1,}{\lambda }_1)}{\partial {\rho }_1}\cdot \frac{\partial s_2(F,{\rho }_{2,}{\lambda }_2)}{\partial {\rho }_2}\rangle =D_{\mathrm{S}}^{″}(\rho ,{\lambda }_1,{\lambda }_2)\frac{{\eta }^2}{{\rho }^2}+D_{\mathrm{S}}^{\prime }(\rho ,{\lambda }_1,{\lambda }_2)(1-\frac{{\eta }^2}{{\rho }^2}),$$

(3)

where

$$D_{\mathrm{S}}=(L,\rho ,{\lambda }_1,{\lambda }_2)=\frac{1}{1-h^2}\left[D_{\mathrm{S}}(L,\rho ,\lambda )-\frac{1}{h}{D}_X(Lh,\rho ,\lambda )\right],$$

(4)

where $D(\rho ,{\lambda }_1,{\lambda }_2)$ is a dual-frequency phase-structure function, $h=(k_1-k_2)/(k_1+k_2)$, $\lambda =2\pi /k$, $k=\left(2k_1{k}_2\right)/(k_1+k_2)$, $D_X$ is a logarithmic amplitude structure function, and $D_s(L,\rho ,\lambda )$ is the phase-structure function. Equations (2)–(4) are substituted into Equation (1) and solved for the correlation function of the arrival angle fluctuation of Gaussian beams with different wavelengths:

$$\begin{array}{rr}{G}_{\mathsf{\alpha }}(Z,{\lambda }_1,{\lambda }_2)=& {\sigma }_{\mathsf{\alpha }}^2({\lambda }_1,{\lambda }_2)=\frac{0.101{\pi }^2\mathrm{\Gamma }(1/6)}{7.963\sqrt{2}}{C}_n^2{F}^3{(zR)}^{-1/3}\times \frac{16}{17}\times \mathrm{Re}\\ & \hfill \left[\frac{F(1,1/6,23/6;\frac{1}{1+i{g}_1})}{{(1+i{g}_1)}^{1/6}}+\frac{F(1,1/6,23/6;\frac{1}{1+i{g}_2})}{{(1+i{g}_2)}^{1/6}}\right],\end{array}$$

(5)

where $g_1=\mathrm{F}(k_1+k_2)/k_1{k}_2{\alpha }_0^2$, $g_2=\mathrm{F}(k_1-k_2)/k_1{k}_2{\alpha }_0^2$, $F(a,b,c,z)$ is a Gaussian hypergeometric function, R is the equivalent curvature radius of the wavefront, and $C_n^2$ is the atmospheric refractive index structure constant.

2.2. Phase Relationship between Gaussian Beams of Different Wavelengths

The essence of arrival angle scintillation is the change of wave vector propagation direction caused by wavefront phase distortion, so calculating the relationship between different phases is essential.

As shown in Figure 1, the incident light is incident along the z-axis in the normal direction, and the entire wave surface is distributed in the XOY plane. The wavefront measuring plane S is perpendicular to the z-axis, and the phase-distribution function of the incident light at the measuring line position is $\phi =f\left(y\right)$. Owing to the phase distortion of the wavefront, there is a phase difference between point P on the wave surface and the surrounding adjacent position.

Assuming that the signal wavelength is ${\lambda }_1$ and the beacon wavelength is ${\lambda }_2$, we selected points N₁ and N₂ along the equal phase plane of the front of the two light beams near point P. The distance between the two points PN₁ and PN₂ is the optical path difference, and the beam-transmission distance is L.

Perpendicular lines of two wavelength beam equal-phase planes are made at points N₁ and N₂; that is, the propagation directions of the wave surface at points N₁ and N₂ are KN₁ and KN₂, respectively, and the two propagation vectors intersect the measuring surface S at points Q₁ and Q₂, respectively. When $\rho$ is very small and close to 0, triangles N₁Q₁P and N₂Q₂P can be considered as right triangles. The phase difference between points Q₁P and Q₂P is the wavefront phase difference corresponding to the beacon and signal lights. The corresponding signal light wavefront phase difference is ${\phi }_{{\lambda }_1}$, the beacon light wavefront phase difference is ${\phi }_{{\lambda }_2}$, and the signal light and beacon light wavefront distortion phases under the above conditions can be expressed as:

$${\phi }_{{\lambda }_2}={\phi }_{\mathrm{p}}-{\phi }_{\mathrm{Q}}=\frac{2\pi }{{\lambda }_2}{N}_2{Q}_2=\frac{2\pi }{{\lambda }_2}(n_{\mathrm{p}}-n_{\mathrm{Q}})L,$$

(6)

$${\phi }_{{\lambda }_1}={\phi }_{\mathrm{p}}-{\phi }_{\mathrm{Q}1}=\frac{2\pi }{{\lambda }_1}{N}_1{Q}_1=\frac{2\pi }{{\lambda }_1}(n_{\mathrm{p}}-n_{\mathrm{Q}1})L,$$

(7)

Because the signal light and beacon light are transmitted along the same path, the air refractive index variation between the corresponding point positions is the same:

$$n_{\mathrm{p}}-n_{\mathrm{Q}1}=n_{\mathrm{p}}-n_{\mathrm{Q}2},$$

(8)

When the signal light and beacon light are transmitted in the same atmospheric channel, the random changes of the refractive index in atmospheric turbulence have different modulation effects on different wavelengths of Gaussian beams. There are also certain differences in the wavefront fluctuation and distortion, the phase difference between PQ₂ and PQ₁ is different, and the corresponding wavefront distortion phase difference is:

$$\begin{array}{ccc}\Delta & =& \left|{\phi }_{{\lambda }_1}-{\phi }_{{\lambda }_2}\right|\hfill \\ & =& ({\phi }_{\mathrm{p}}-{\phi }_{\mathrm{Q}})-({\phi }_{\mathrm{p}}-{\phi }_{\mathrm{Q}1})\hfill \\ & =& 2\pi (n_{\mathrm{p}}-n_{\mathrm{Q}})L\left|\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1{\lambda }_2}\right|,\hfill \end{array}$$

(9)

Furthermore, the relationship between the signal wavefront phase ${\phi }_{{\lambda }_1}$ and beacon wavefront phase ${\phi }_{{\lambda }_2}$ at point P is:

$${\phi }_{{\lambda }_1}=\frac{\frac{2\pi }{{\lambda }_2}(n_{\mathrm{p}}-n_{\mathrm{Q}})L\pm \Delta }{\frac{2\pi }{{\lambda }_2}(n_{\mathrm{p}}-n_{\mathrm{Q}})L}\cdot {\phi }_{{\lambda }_2},$$

(10)

After substituting Equations (8) and (9) into Equation (10), the relationship between the distortion phases of the Gaussian beams with different wavelengths is obtained as follows:

$${\phi }_{{\lambda }_1}=(\frac{{\lambda }_2}{{\lambda }_1}\pm \left|\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1{\lambda }_2}\right|){\phi }_{{\lambda }_2},$$

(11)

According to this relationship, the phase difference of the wavefront distortion of beams with different wavelengths can be calculated, based on the known wavelengths of the signal light and beacon light, beam transmission distance L, and refractive index change. The wavefront distortion measurement value of the signal light can also be calculated, based on the wavefront distortion measurement value of the beacon light.

3. Simulation

3.1. Propagation Characteristics of Gaussian Beams with Different Wavelengths in the Same Atmospheric Channel

The outer turbulence scale L₀ = 5 m, inner turbulence scale l₀ = 0.1 m, waist radius ${\omega }_0$ = 5 mm, and atmospheric refractive index structure constant $C_n^2$ = 1 × 10⁻¹⁸ m^−2/3. Figure 2 shows the simulation results of the wavefront phase distortion of Gaussian beams with 1550 nm, 850 nm, 632.8 nm, and 530 nm wavelengths after transmission in the same atmospheric channel, with transmission distance L = 10 km.

It can be seen from the figure that under the same transmission conditions, the degree of the phase distortion of the Gaussian beam varies significantly with the wavelength. This indicates that the atmospheric turbulence differently modulates Gaussian beams of different wavelengths.

Red represents the fluctuation of the phase to the transmission direction, and blue represents the phase fluctuation of the reverse transmission direction. The darker the color is, the more intense the fluctuation is. Figure 3 shows the relationship between the fluctuation variance of the wavefront phase distortion of the Gaussian beam and wavelength, under the simulation conditions in Figure 2. It can be seen from Figure 3 that the variance of the wavefront phase fluctuation gradually decreases with the increase in wavelength. This indicates that the atmospheric turbulence disturbs long waves less than it does short waves; that is, the phase-fluctuation degree of short waves is higher than that of long waves.

If a long-wave beacon light is used to correct a short-wave signal light, the correction will be insufficient, and the correction residual will increase with the increase in the wavelength difference between the two, until finally the compensation effect tends to disappear. If a short-wave beacon light is used to correct a long-wave signal light, the correction amount is greater than the actual phase-distortion variable, and the correction residual increases rapidly with the increase in wavelength. Finally, when the wavelength difference reaches a certain value, the correction residual will reach the phase distortion error without correction.

Figure 4 shows the relationship between the wavefront phase distortion difference and the wavelength difference of different Gaussian beams, in which the beacon light wavelength is ${\lambda }_2=632.8 \mathrm{n}\mathrm{m}$. It can be seen from the figure that the smaller the wavelength difference between the signal and beacon lights, the smaller the distortion phase difference. This indicates that the beacon light wavefront distortion variable can be approximated to the signal light wavefront distortion variable only when the signal and beacon light wavelengths are very close.

In addition, it can also be concluded that when the signal light wavelength is larger than that of the beacon light, the minus sign should be considered in the phase relationship between the two wavelength beams. When the signal light wavelength is smaller than the beacon light wavelength, the plus sign should be considered in the relationship.

3.2. Correlation of Variance of the Arrival Angle Fluctuation of Gaussian Beams with Different Wavelengths

The correlation of the AOA fluctuation is related to multiple factors, such as the transmission distance, atmospheric refractive index structure constant, waist radius, and wavelength. This section analyzes the influence of the transmission distance, beam wavelength difference, and turbulence intensity on the correlation. Unless otherwise specified, the simulation parameters are as follows: transmission distance, L = 10 km; signal optical wavelength, 1550 nm; beacon optical wavelengths, 632.8 nm, 750 nm, 950 nm, and 1280 nm; and the atmospheric refractive index structure constant $C_n^2$ ranges from 1 × 10⁻¹⁷ m^−2/3–1 × 10⁻¹⁹ m^−2/3.

When the beam waist radius is 6.2 × 10⁻⁵ m and the atmospheric refractive index structure constant $C_n^2$ is 1 × 10⁻¹⁹ m^−2/3, Figure 5 shows the change in correlation number between the arrival angle fluctuations of different wavelengths with the transmission distance. As shown in the figure, as the transmission distance increases, the correlation between the arrival angle fluctuations of the different wavelengths gradually increases. The reason for this phenomenon is a statistical correlation between the wavefront distortions of different wavelengths in the same turbulence disturbance [26]. With the same distance, the smaller the wavelength difference between the two beams, the greater the cross-correlation coefficient between the corresponding AOA fluctuations.

Figure 6 shows the variation curve of the correlation number of the arrival angle fluctuations of Gaussian beams of different wavelengths with the atmospheric turbulence intensity. It can be seen from Figure 6 that the correlation number between the arrival angle fluctuations of the Gaussian beams of different wavelengths gradually increases with the decrease in the atmospheric refractive index structure constant. The smaller the wavelength difference between the beacon and signal lights, the larger the cross-correlation coefficient between the corresponding arrival angle fluctuations of beams of different wavelengths. This indicates that the distortion information between the arrival angle fluctuations of different wavelengths has a higher consistency.

Figure 7 shows the influence of the waist radius ${\omega }_0$ on the correlation between the arrival angle fluctuations of Gaussian beams of different wavelengths. The vertical axis represents the cross-correlation coefficient after translation into logarithm 10. It can be seen from the figure that the number of mutual relations between the angle-of-arrival fluctuations of Gaussian beams of different wavelengths decreases gradually with an increase in the waist radius. In addition, the smaller the wavelength difference between the two beams, the higher the number of mutual relations.

From the above analysis, it can be observed that the correlation between the arrival angle fluctuations of Gaussian beams of different wavelengths decreases with an increase in the atmospheric refractive index structure constant. For the same transmission distance and atmospheric refractive index structure constant, the arrival angle fluctuation of a short wave is higher than that of a long wave. In addition, for the same wavelength, the sensitivity of the AOA fluctuation to the transmission distance is much higher than that to the atmospheric refractive index structure constant.

4. Experiment

4.1. Wavefront Distortion Correction Compensation for a Dual-Wavelength FSO Communication System

The dual-wavelength FSO communication system uses a Shack–Hartmann wavefront sensor to measure the wavefront distortion phase of the beacon light, generate a wavefront reconstruction matrix, and correct the signal optical distortion wavefront, according to the reconstruction matrix. The correction matrix is calculated according to the quantitative relationship between the wavefront distortion phases of Gaussian beams of different wavelengths, as described in Section 2.2. The measured beacon light wavefront-reconstruction matrix is corrected to obtain the wavefront reconstruction matrix of the signal light to compensate for the chromatic difference between the signal and beacon lights and reduce the correction residual.

The distorted wavefront $W(x,y)$ can be expressed as follows:

$$W(x,y)=W^{\prime }(x,y)-W_0(x,y)={\displaystyle \sum _{j=1}^{30}{W}_j}{Z}_j(x,y),$$

(12)

where $Z_j$ is a Zernike polynomial, $W_j$ is the coefficient of the Zernike polynomial, and j is the order of the Zernike polynomial, $W^{\prime }(x,y)$ is the measurement wavefront, $W_0(x,y)$ is the reference-plane wavefront, and the distorted wavefront is obtained from the difference between the two. A Gaussian beam with wavelength ${\lambda }_1$ is transmitted as a beacon light along with a signal light with wavelength ${\lambda }_2$ in the same optical path. The Shack–Hartmann wavefront sensor measures the beacon light wavefront to obtain the beacon light gradient matrix, including the distortion:

$$\left[\begin{array}{c}{b_{{\lambda }_2}(x,y)|}_1\\ {b_{{\lambda }_2}(x,y)|}_2\\ ⋮\\ {b_{{\lambda }_2}(x,y)|}_k\\ {c_{{\lambda }_2}(x,y)|}_1\\ {c_{{\lambda }_2}(x,y)|}_2\\ ⋮\\ {c_{{\lambda }_2}(x,y)|}_k\end{array}\right]=\left[\begin{array}{c}{g_1(x,y)|}_1{g_2(x,y)|}_1\dots {g_{j\max }(x,y)|}_1\\ {g_1(x,y)|}_2{g_2(x,y)|}_2\dots {g_{j\max }(x,y)|}_2\\ ⋮\\ {g_1(x,y)|}_k{g_2(x,y)|}_k\dots {g_{j\max }(x,y)|}_k\\ {h_1(x,y)|}_1{h_2(x,y)|}_1\dots {h_{j\max }(x,y)|}_1\\ {h_1(x,y)|}_2{h_2(x,y)|}_2\dots {h_{j\max }(x,y)|}_2\\ ⋮\\ {h_1(x,y)|}_k{h_2(x,y)|}_k\dots {h_{j\max }(x,y)|}_k\end{array}\right]\cdot \left[\begin{array}{c}{W}_1\\ W_2\\ ⋮\\ W_{j\max }\end{array}\right],$$

(13)

where ${\frac{\mathsf{\Delta }x(x,y)}{f}}_{{\lambda }_2}={\left.b_{{\lambda }_2}\right|}_k$ and ${\frac{\mathsf{\Delta }y(x,y)}{f}}_{{\lambda }_2}={\left.c_{{\lambda }_2}\right|}_k$. According to Equation (11), the following relationship can be obtained by calculating the offset gradient of the distortion of two Gaussian beams with different wavelengths on the measuring surface of the sensor:

$${{\frac{\Delta x(x,y)}{f}}_{{\lambda }_1}|}_k={\frac{1}{\gamma }\cdot {\frac{\Delta x(x,y)}{f}}_{{\lambda }_2}|}_k={b_{{\lambda }_1}(x,y)|}_k,$$

(14)

$${{\frac{\Delta y(x,y)}{f}}_{{\lambda }_1}|}_k={\frac{1}{\gamma }\cdot {\frac{\Delta y(x,y)}{f}}_{{\lambda }_2}|}_k={c_{{\lambda }_1}(x,y)|}_k,$$

(15)

where $\frac{1}{\gamma }=\frac{{\lambda }_2}{{\lambda }_1}\cdot (\cos A{\theta }_2+\frac{\sin A{\theta }_2}{\tan {\theta }_2})$. According to this relationship, a coefficient correction matrix can be obtained:

$$\gamma =\left[\begin{array}{cccc}{\gamma }_1& 0& \cdots & 0_k\\ 0& {\gamma }_2& \cdots & 0_k\\ & & \ddots & \\ 0& 0& \cdots & {\gamma }_k\end{array}\right],$$

(16)

The wavefront reconstruction matrix of the beacon light was obtained by multiplying the wavefront reconstruction matrix of the signal light by the coefficient correction matrix. In the adaptive optics control system, the beacon optical wavefront reconstruction matrices before and after correction are used to correct the signal optical wavefront distortion. The signal optical Strehl ratio (SR) is obtained, as shown in Figure 8.

Figure 8 shows the change curve of the system SR value during the correction process; (a) is the SR curve when the beacon wavefront distortion variable is directly used as the signal wavefront distortion variable for correction, and (b) is the SR curve when the beacon wavefront distortion variable is corrected using the wavefront phase matrix corrected by Equation (11) as the signal wavefront distortion variable.

It can be observed from the figure that the corrected SR ratio of the wavefront reconstruction matrix converges to 1. Compared to the situation in which the wavefront reconstruction matrix converges to 0.9 before the correction, the overall SR ratio increases by 10%. When the beacon wavefront distortion matrix and signal light are corrected, the system SR converges quickly. The SR tends to stabilize when the number of iterations is 80 and the SR jitter is significantly reduced. This verifies the correctness of the wavefront phase relationship between the signal light and the beacon light derived by Equation (10), and also explains the importance and practical significance of compensating the aberration color difference between the signal light and the beacon light.

4.2. Experimental Device

To verify the correctness of the theoretical analysis, the research team built a 1.2 km experimental optical path between the seventh floor of the North Discipline Building and the seventh floor of the Fifth Teaching Building of Xi’an University of Technology. A schematic diagram of the experimental setup is shown in Figure 9. The signal light and beacon light were transmitted by a beam combiner through the same atmospheric channel to the optical end on the seventh floor of the North Discipline Building and then transmitted to the receiving antenna on the seventh floor of the Fifth Teaching Building after being reflected by the plane reflector. The combined beam was reflected by the deformable mirror and the reflector, and then incident to the beam splitter.

The beam splitter separates the signal light from the beacon light. The wavefront sensor detects the wavefront distortions of the beacon and signal lights. Using an adaptive PID algorithm control, the wavefront distortion of the signal light is corrected [27]. In the experiment, the signal light wavelength was 1550 nm, and the beacon light wavelength was 650 nm. Figure 10 shows a physical image of the experiment, and

Table 1. Parameters of experimental equipment.

Device	Type	Parameters
Wavefront Sensor	THORLABS-HASO4-NIR	Wavelength range: 1500–1600 nm Number of micro-lenses: 32 × 40
Wavefront Sensor	THORLABS-HASO4-FIRST	Wavelength range: 400–1100 nm Number of micro-lenses: 32 × 40
Beacon Laser	MW-GX-650	Wavelength: 650 nm, linewidth: 0–3 nm, output power: 35 mW
Signal Laser	KOHERA-BASIK Module	Wavelength: 1550 nm, linewidth: 0.1 kHz, output power: 0–200 mW
Deformable Mirror	ALPAO-DM	Actuator number: 69, effective aperture: 10.5 mm, frequency: 800 Hz
Antenna	Maksutov–Cassegrain	Diameter: 25 mm, focal length: 300 mm
Laser Collimator	F810APC	Wavelength: 1550 nm/650 nm, focal length: 37.13 mm

shows the corresponding experimental instrument parameters.

5. Results and Discussion

The beacon light wavefront slope matrix size was 32 × 40, the partial matrix (size is 5 × 9) of the wavefront slope was selected for convenience, and the beacon light wavefront W_b(x,y) was:

$$W_s(x,y)=\left[\begin{array}{lllll}1.035649& -150.956345& -0.956818& 0.927685& 100.935940\\ 1.036523& -140.412547& -0.968546& 0.689754& 85.229863\\ 0.879471& -110.029023& -1.125849& 0.993528& 65.498232\\ 1.214295& -98.659436& -1.134262& 0.995635& 62.348757\\ 0.706035& -70.426181& -0.749565& 0.436325& 40.962742\\ 0.546754& -70.336354& -0.561424& 0.413361& 30.726241\\ 0.420319& -55.986219& -0.956341& 0.352981& 22.954864\\ 0.431875& -65.952561& -80.683419& -0.356591& -0.530355\\ 0.687421& -48.736929& -73.289416& 100.569631& 122.598361\end{array}\right]$$

The wavelengths of signal light and beacon light were 650 nm and 1550 nm; according to Equation (16), the modified beacon light wavefront W_{m_b}(x,y) is:

$$W_{m\_b}(x,y)=\left[\begin{array}{lllll}0.898693& -123.478393& -0.813300& 0.727940& 79.309359\\ 0.813303& -112.091886& -0.898686& 0.642614& 61.417417\\ 0.727947& -100.748143& -0.984090& 1.051585& 55.646292\\ 0.642622& -89.316087& -0.813300& 0.472145& 46.990088\\ 0.557348& -77.802486& -0.898686& 0.387080& 38.445533\\ 0.472152& -66.203907& -0.642614& 0.341970& 28.135654\\ 0.387088& -54.513851& -0.727940& 0.256478& 19.754275\\ 0.341970& -52.658568& -82.783302& -0.331370& -0.387088\\ 0.256478& -40.823380& -75.636368& 91.499935& 82.331823\end{array}\right]$$

the cross-correlation coefficient between signal light wavefront W(x,y) and beacon light wavefront W_b(x,y) was 0.82, and the cross correlation coefficient between signal light wavefront W_x(x,y) and modified beacon light wavefront W_{m_b}(x,y) was 0.89, indicating that the similarity between the signal light and modified beacon light wavefronts increased. In this case, the wavefront distortion of the signal light could be corrected by measuring the wavefront distortion of the beacon light.

Figure 11 shows the variation curves of the signal wavefront peak-to-valley (PV) and wavefront root mean square (RMS) during the experiment.

Table 2. PV and RMS of signal beam wavefronts.

	Open-Loop Value	Closed-Loop Value	Mean Value	Variance Value
Unmodified wavefront PV	5.63 μm	1.78 μm	2.05 μm	0.46 μm2
Unmodified wavefront RMS	1.10 μm2	0.24 μm2	0.35 μm2	0.01 μm2
Modified wavefront PV	4.41 μm	0.51 μm	0.69 μm	0.08 μm2
Modified wavefront RMS	1.01 μm2	0.11 μm2	0.16 μm2	0.005 μm2

lists the PV and RMS values of the signal wavefront before and after the experiment. Considering Figure 11 and Table 2, when the beacon optical wavefront reconstruction matrix was directly used as the signal optical wavefront reconstruction matrix to correct the signal optical wavefront distortion, the signal optical wavefront PV and RMS values converged slowly during the correction process and fluctuated significantly after convergence.

When the correction matrix of the beacon optical wavefront reconstruction matrix was used as the signal optical wavefront reconstruction matrix to correct the signal wavefront distortion, the PV and RMS values of the signal wavefront converged faster and fluctuated less after convergence. The accuracy of the wavefront phase relationship of beams with different wavelengths derived in this study was further verified.

In the existing research on the distortion difference between beams of different wavelengths, plane waves are mainly used to describe the distortion difference; at the same time, the results of the distortion difference have not been applied to actual work.

In this paper, a description model of wavefront distortion difference is established for the Gaussian beam commonly used in optical signal transmission; and in the dual-wavelength adaptive optics system, an accurate calculation formula is given, and a method for compensating and correcting the distortion difference is proposed according to the working principle of the adaptive optics system, further reducing the correction residual, improving the correction accuracy, and applying the theoretical research results to practice.

6. Conclusions

The wavefront distortion correction residual of an adaptive optical system is a systematic error caused by the differences between the wavefront of the system signal and beacon lights. In this study, the correlation between the angle-of-arrival fluctuations at different wavelengths was analyzed. The expression of the cross-correlation function between the AOA fluctuations of different Gaussian-beam wavelengths and the relationship between the wavefront distortion phase of different Gaussian-beam wavelengths were derived.

On this basis, the effects of the propagation distance, atmospheric refractive index structure constant, Gaussian-beam waist radius, and wavelength difference on the angle-of-arrival fluctuation cross-correlation coefficient were analyzed. The wavefront distortion phase relations of beams with different wavelengths obtained in this study were verified experimentally.

The difference in wavefront distortion between beams of different wavelengths is caused by atmospheric dispersion. On the basis that there is no common optical path error, the correction of the wavefront distortion difference can improve the overall quality of the beam by 5%–10%.