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In order to improve the correction effect of the adaptive optical system in coherent optical communication, we investigate the relative distortion between the wavefronts of different wavelengths of the beams transmitted on the near-ground horizontal atmospheric turbulent links emitted by coherent optical communication system. And the situation is analyzed when the wavelength corresponding to the wavefront detected by the wavefront detector and the wavelength corrected by the deformed mirror are different, the influence of the wavelength factor on the adaptive optical system correction. We use a series of trigonometric functions and the Hankel transformation to derive the corrected residual variance and the Strehl ratio between the wavefront distortions of the wavelengths of the dual-wavelength combined beam in atmospheric turbulence. In relation to the parameters of the turbulent environment, the ensemble average of the wavefront difference corresponding to different wavelengths the derived is proposed as the coefficient to correct the dual-wavelength adaptive optical system. The results show that the statistic of the turbulence internal scale has a major influence on the difference between the wavefronts. By adding the correction coefficient, the signal light’s wavefront of the coherent optical communication system can be corrected more effectively by the dual-wavelength adaptive optical closure.

The coherent optical communication system uses a laser beam as information carrier to communicate between channels in space (including the near-Earth atmosphere, low orbit, medium orbit, synchronous orbit, interstellar space), it based on coherent detection has the advantages of high receiving sensitivity, strong anti background noise and so on. The most fundamental effect of atmospheric turbulence for the channel on the laser transmission is the degradation of the coherence of the transmitted beam. The random change of atmospheric pressure and density caused by the random change of atmospheric temperature leads to the random change of atmospheric refractive index. When the beam is transmitted in an area with the random change of the refractive index, an optical path difference is generated, which leads to the wavefront distortion. The wavefront distortion makes the beam energy of the beam irradiating the receiving end sensor be unevenly distributed, which seriously affects the received power of the signal light and reduces the communication quality of the system [

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This paper studies the relative variation between the wavefront distortions of the two-wavelength combined beam in horizontal transmission and the influence of wavelength factor on the adaptive optical system. In the first part, we derive the residual variance and Strehl ratio of the wavefront correction of the dual-wavelength beam with a circular aperture in the same transmission path. In the second part, we study the effects of the scale of the turbulence, and transmission distance on the residual wavefront variance that correction of the adaptive optical system of tow wavelengths. We also study the ratio of the two wavefronts fluctuation variances of the dual-wavelength beam in the third part, the role of this ratio is important for improving the quality the correction to the dual-wavelength adaptive optics closed-loop and the improving the signal light’s power received by the system in multi-wavelength laser communication systems.

The transmitting end of the coherent optical communication system emits two beams of different wavelengths. One is the beacon beam and the other the signal beam, two beams of light are emitted in the optical antenna and become combined beams. The combined beam passes through the same horizontal atmospheric turbulence path and arrives at the adaptive optics module in the receiving end of the system. The wavefront of the signal wavelength λ_{1} is compensated by the wavefront compensation of the beacon wavelength λ_{2} in the module. When the beam travels along the z axis and enters the aperture of radius R, Wavefront of λ_{1} and λ_{2} in the aperture showed in _{1} is the point of wavefront with λ_{1}, ρ_{2} is the point of wavefront with λ_{2}, ρ is the distance between ρ_{1} and ρ_{2}. The compensated residual wavefront variance can be expressed by Equations (1), (2), (3), and (4), which also represent the difference between the wavefront aberrations corresponding to the two wavelengths of horizontal transmission.

Δ = 1 π R 2 ∬ w ( ρ ) ( φ λ 1 ( ρ ) − φ λ 2 ( ρ ) ) 2 d 2 ρ (1)

w ( ρ ) = { 1 , ρ < R 0 , ρ > R (2)

〈 φ λ i ( k i , ρ ) 〉 2 = 4 π 2 ∫ 0 L d z ∫ 0 ∞ κ d κ Φ n ( κ ) J 0 ( ρ κ ) cos 2 ( ( L − Z ) γ i κ 2 2 k i ) , i = 1 , 2 (3)

〈 φ λ 1 ( k 1 , ρ ) φ λ 2 ( k 2 , ρ ) 〉 = 4 π 2 ∫ 0 L d z ∫ 0 ∞ κ d κ Φ n ( κ ) J 0 ( ρ κ ) cos ( ( L − Z ) γ 1 κ 2 2 k 1 ) cos ( ( L − Z ) γ 2 κ 2 2 k 2 ) (4)

where φ λ 1 represents the signal wavefront on the receiving aperture, φ λ 2 represents the beacon wavefront, w(ρ) is the aperture function, R is the aperture radius, and ρ is the distance between the two points on the beam section, it is also the radius of aperture the beam enters. And where k_{1} is the wave number of signal waves, k_{2} is the wave number of beacons, J_{0} is the zero-order Bessel function, and L is the propagation distance. The Gaussian beam can be approximated as a plane wave after long-distance transmission, and γ i is one [_{1} and λ_{2} can be written as Equation (4) [

Φ n ( k ) = 0.033 C n 2 κ − 11 / 3 (5)

Equation (6) gives the ensemble average of wavefront variance and Equation (7) gives the wavefront correlation function corresponding to the derived wavelengths. L_{0} is the outer scale of turbulence and l 0 is the inner scale of turbulence, where Q is 0.132 π 2 C n 2 .

〈 φ λ i ( k i , ρ ) 〉 2 = Q { 2 − 2 6 L ρ − 8 3 Γ ( − 5 6 ) Γ ( 17 6 ) − L 3 12 k i 2 2 4 3 ρ 4 3 Γ ( 7 6 ) Γ ( 5 6 ) , ρ ≠ 0 − 3 L 5 [ ( 2 π L 0 ) − 5 / 3 − ( 2 π l 0 ) − 5 / 3 ] − 2 L 3 56 k i 2 [ ( 2 π L 0 ) 7 / 3 − ( 2 π l 0 ) 7 / 3 ] , ρ = 0 (6)

〈 φ λ 1 ( k 1 , ρ ) φ λ 2 ( k 2 , ρ ) 〉 = Q { L 2 − 8 / 3 ρ − 8 / 3 Γ ( − 5 / 6 ) Γ ( 17 / 6 ) − 7 L 3 3 k 1 2 + k 2 2 8 k 1 2 k 2 2 2 4 / 3 ρ 4 / 3 Γ ( 7 / 6 ) Γ ( − 5 / 6 ) , ρ ≠ 0 − 3 L 5 [ ( 2 π L 0 ) − 5 / 3 − ( 2 π l 0 ) − 5 / 3 ] − L 3 ( k 1 2 + k 2 2 ) 28 k 1 2 k 2 2 [ ( 2 π L 0 ) 7 / 3 − ( 2 π l 0 ) 7 / 3 ] , ρ = 0 (7)

When the wavefront of the beam with the wavelength λ_{2} is used to compensate for the wavefront of the beam with the wavelength λ_{1}, the residual variance of the two spatial points on the aperture of a circle with radius R at a distance of ρ can be written as Equation (8):

Δ = 4 π R 2 ∬ w ( ρ ) { ∫ 0 L z d z ∫ 0 ∞ κ d κ Φ n ( κ ) J 0 ( ρ κ ) [ cos ( ( L − Z ) κ 2 2 k 1 ) − cos ( ( L − Z ) κ 2 2 k 2 ) ] 2 } d 2 ρ (8)

By the same method, the residual variance of the two-wavelength wavefront correction of the horizontally transmitted beam is derived as Equation (9), (10), with c = 7.1962e−05.

Δ = 4 π R 2 ∬ w ( ρ ) { 2 16 / 3 Γ ( 19 6 ) C n 2 ρ 16 / 3 Γ ( − 7 6 ) ∫ 0 L ( L − Z ) 4 d z } d 2 ρ (9)

Δ = { 39.4 π R 2 L 5 C n 2 ( k 1 2 − k 2 2 ) 2 k 1 4 k 2 4 L 0 19 3 − l 0 19 3 ( L 0 l 0 ) 19 3 , ρ = 0 c π R 2 Γ ( 19 6 ) L 5 ρ − 10 / 3 Γ ( − 7 6 ) C n 2 ( k 1 2 − k 2 2 ) 2 k 1 4 k 2 4 , l 0 ≤ ρ ≤ L 0 (10)

When the distances ρ between the point ρ_{1} and ρ_{2} on the aperture of a circle with radius R are zero, means on the same point, the value of the Bessel function in the Equation (8) is one, the applicable range of the Kolmogorov turbulence spectrum is in the inertia sub-region, so the integral range of the spatial wavenumber should be changed from 2π/ l 0 to 2π/L_{0}. Re-integrate, and then obtain the two-wavelength wavefront corrected residual variance when ρ = 0. The Strehl ratio is calculated using the corrected residual variance. The Strehl ratio can then be used to evaluate the wavefront quality after wavefront correction in the dual-wavelength adaptive optics system. The reason for the correction in the residual variance is that each actuator of the deformable mirror moves according to the reconstructed wavefront φ λ 2 when the deformable mirror corrects the wavefront φ λ 1 , as shown in _{1} is 1.5 μm and ρ is zero, the corrected residual variance varies with wavelength λ_{2} as shown in

With the change of λ_{2} and the difference between the internal scale and the external scale, the trend of the corrected residual variance and the Strehl ratio at the same point between the two wavefronts after horizontal transmission for 10 km is shown in _{2} comes closer to λ_{1}, the two wavefront differences and the corrected residual variance gradually decrease, and the Strehl ratio gradually increases. In

_{1} and ρ_{2} in the aperture at a distance ρ of 10 cm when the two wavelengths are fixed, the corrected residual variance without the correction coefficient and the Strehl ratio changes with the transmission distance under different turbulent intensities. The corrected residual variance continues to increase as the transmission distance increases, the greater the turbulence intensity, the faster the increased speed, and the corrected Strehl ratio gradually decreases with the increase of the transmission distance, and the decrease slows down. _{2} at different distances of two points. As λ_{2} approaches λ_{1} and the distance between the point ρ_{1} and ρ_{2} gradually decreases, the Strehl ratio gradually increases and then decreases as the difference between the two wavelengths increases. The smaller the distance between the point ρ_{1} and ρ_{2} of the wavefront in the aperture, the faster the Strehl ratio rises.

In the process of horizontal transmission, the internal scale has a greater effect on the two wavefronts with different wavelengths than the size of the turbulent outer scale. When the internal scale increases, the turbulence is generally

stronger and has a greater impact on the Strehl ratio after dual-wavelength correction. The influence of the changes of external scale on the correction with wavefront of different wavelengths is very small, and the smaller the degree difference between the two wavelengths, the more two wavefronts tend to be the same spatial distribution at the same point in the aperture.

To reduce the residual variance of the wavefront φ λ 1 , when the wavefront φ λ 2 of the beacon beam detected in wavefront sensor, the ratio of the ensemble average of wavefront standard deviation of corresponding two wavelengths in Equation (3) is taken as the correction coefficient. That is φ λ 1 − G φ λ 2 , so the corrected residual variance containing the correction coefficient can be written as Equation (11).

Δ 2 = 1 π R 2 ∬ w ( ρ ) ( φ λ 1 ( ρ ) − G φ λ 2 ( ρ ) ) 2 d 2 ρ (11)

G can be written as Equation (12), C n 2 2 is the refractive index structure constant corresponding to λ_{1}, C n 2 2 is the refractive index structure constant corresponding to λ_{2}, and A, B, C, and D are environmental factors given by Equations (13), (14), (15), and (16):

G = 〈 φ λ 1 ( k 1 , ρ ) φ λ 2 ( k 2 , ρ ) 〉 = { C n 1 2 ( 1 − B A k 2 2 ) C n 2 2 ( 1 − B A k 1 2 ) , ρ ≠ 0 C n 1 2 ( C − 1 k 1 2 D ) C n 2 2 ( C − 1 k 2 2 D ) , ρ = 0 (12)

A = 3 × 2 5 3 L ρ 4 3 k 1 2 k 2 2 Γ ( − 5 6 ) Γ ( 5 6 ) (13)

B = 2 4 3 L 3 ρ − 8 3 Γ ( 17 6 ) Γ ( 7 6 ) (14)

C = − 3 5 L [ ( 2 π L 0 ) − 5 / 3 − ( 2 π l 0 ) − 5 / 3 ] (15)

D = L 3 28 [ ( 2 π L 0 ) − 7 / 3 − ( 2 π l 0 ) − 7 / 3 ] (16)

On the horizontal link near the ground, the refractive index structure constant is generally considered to be the constant corresponding to the environment. It can be calculated in the range of a given band (17), out scale can be calculated in Equation (18) [

C n 2 = [ 77.6 e − 16 T 2 ( 1 + ( 7.52 e − 3 ) λ − 2 ) P ] 2 C T 2 (17)

L 0 = 0.5 + 5 exp [ − ( z − 7500 2500 ) 2 ] (18)

Δ 2 = 0.132 π C n 2 R 2 { − 1 2 L [ ( 2 π L 0 ) − 2 − ( 2 π l 0 ) − 2 ] + ( 4.19 e − 3 ) L 5 ( k 2 2 − G k 1 2 ) 4 k 1 4 k 2 4 × [ ( 2 π L 0 ) 19 / 3 − ( 2 π l 0 ) 19 / 3 ] − ( − L 3 7 ) ( 1 − G ) ( k 2 2 − G k 1 2 ) k 1 2 k 2 2 [ ( 2 π L 0 ) 7 / 3 − ( 2 π l 0 ) 7 / 3 ] } , ρ = 0 (19)

Δ 2 = 0.132 π 2 C n 2 R 2 [ L 2 − 8 / 3 ρ − 8 / 3 Γ ( 5 / 6 ) Γ ( 17 / 6 ) + 17 L 5 2 16 / 3 640 ρ 16 / 3 Γ ( 19 / 6 ) ( k 2 2 − G k 1 2 ) 4 Γ ( − 7 / 6 ) k 1 4 k 2 4 − ( − L 3 ) 2 4 / 3 24 ρ 4 / 3 Γ ( 7 / 6 ) Γ ( 5 / 6 ) ( 1 − G ) ( k 2 2 − G k 1 2 ) k 1 2 k 2 2 ] , l 0 ≤ ρ ≤ L 0 (20)

If the transmission distance is 10 km and λ_{1} is 1.5 μm, then for different λ_{2} the corresponding correction coefficients are calculated according to Equations (19) and (20), and the correction residual variance and Strehl ratio containing the correction coefficients are obtained. The change of the correction residual variance and Strehl ratio with λ_{2} with the correction coefficient at the same point that ρ equal to zero as in _{2} is less than λ_{1} for different internal scales, the larger the internal scale is, the faster the correction residual variance will decrease. At the same time the faster the corrected Strehl ratio will rise, and the correction effect is significantly better than that shown in

ratio with the wavelength λ_{2} are as shown in _{2}, the Strehl ratio starts to increase gradually and finally becomes saturated. The corrected residual variance increases with the increase of the distance between point ρ_{1} and ρ_{2} in the wavefront space, indicating that the existence of the correction coefficient makes the corrected residual variance less sensitive to the change of the distance ρ in wavefront space, and it is helpful to improve the quality of wavefront. When compared with the case where the correction coefficient is not added, the quality of the system obviously improves after the correction coefficient is added.

The experiment used a combined beam of wavelengths of 1.55 μm and 0.65 μm that is emitted horizontally from the same transmitting antenna at a distance of 514 m from the receiving end. The structure of the receiving end is shown in

influence of sunlight the experiment was carried out at night, and to avoid the obstruction of buildings and trees and the vibration caused by the wind on the experimental equipment, the transmitter is placed on the 7th floor of the 5th teaching building of Xi’an University of Science and Technology, and the receiver is placed on the 7th floor of the discipline building. The transmitter and receiver are about 22 m away from the ground.

When the correction parameters are not added, the system performs a closed-loop correction, and the wavefront sensors A and B measure the root mean square (RMS) and peak valley (PV) values of the wavefront corresponding to the wavelengths of 1.55 μm and 0.65 μm respectively after removing the wavefront tilt. As shown in

The correction coefficient G is calculated from the wavelength according to Equations (17) and (18), the sub-mirror size of the phase-change mirror, and the transmission distance; and is loaded into the control algorithm of the deformable mirror, and then closed-loop correction is performed.

results of the wavefront sensors A and B again measuring the RMS and PV values of the corresponding wavefront. After closed-loop correction the wavefront RMS of the beam with the wavelength of 1.55 μm is between 0.052 and 0.171, the mean value is 0.121, and the wavefront RMS of the beam with the wavelength of 0.65 μm is between 0.071 and 0.252 with a mean value of 0.153.

By comparing the correction effect before and after the correction coefficient is added, we find that, after the correction coefficient is added, when the system is closed-loop, the mean value of the wavefront RMS and PV values in the measurement time is smaller than that without the correction coefficient. The wavefront distribution corresponding to 1.55 μm after correction is shown in

When the beam from the laser communication system travels horizontally near the ground, the ensemble average and wavefront correlation function of the wavefront difference corresponding to wavelength can be characterized by the turbulent environmental parameters and the beam wavelength at different minimum correction distances. When the spatial minimum correction distance is zero, the turbulence scale is the main factor for correcting the residual error; when the spatial correction distance is not zero, the spatial minimum correction distance is the main factor for correcting the residual error. In the transmission process, the size of the turbulent internal scale has a more significant effect than the external scale on the wavefront distortion of the wavelength. In the near-surface environment, the number of small-scale turbulent eddies is large, and the turbulent flow has a scale change more rapidly than the outer scale, so the wavefront corresponding to the wavelength in the combined light transmitted horizontally near the ground is more likely to produce spatial differences in the transmission, This makes it easy to drop the quality of the wavefront correction of the adaptive optics system, and thus affects the quality of the laser communication system. The ratio of the wavefront difference corresponding to the wavelength is used as the correction coefficient of the dual-wavelength adaptive optics in the laser communication. The system corrects the value after the beacon wavefront reconstruction and then loads it into the actuator movement amount of the deformable mirror, so that deformed mirror surface type is closer to the conjugate of the signal light’s wavefront, thereby improving the corrected Strehl ratio, improving the correction effect and signal to receive power, and improving the information transmission quality of the laser communication system.

We thank LetPub (http://www.letpub.com/) for its linguistic assistance during the preparation of this manuscript.

1) Natural Science Foundation of Shaanxi Provincial Department of Education (18JK0341); 2) Xi’an Science and Technology Innovation Guide Project (20110530YD8CG14 (12)); 3) Shaanxi Key Industry Innovation Project (2017 ZDCXL-GY-06-01).

The authors declared that they have no conflicts of interest in this work.

Ke, X.Z. and Chen, X.Z. (2020) Correcting Wavefront Distortion of Dual-Wavelength Beams Due to Atmospheric Turbulence with a Correction Coefficient. Optics and Photonics Journal, 10, 64-77. https://doi.org/10.4236/opj.2020.104006