| Citation: | SONG Ji-kun, LI Yuan-yang, CHE Dong-bo, GUO Jin, WANG Ting-feng, LI Zhi-lai. Influence of turbulent atmosphere on the effect of coherent beam combining[J]. Chinese Optics, 2020, 13(4): 884-898. doi: 10.37188/CO.2019-0197
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Coherent beam combining of fiber laser array is the coherent superposition of sub-beams, which can greatly improve the output optical power and ensure good beam quality. It has a broad application prospect in laser radar, laser atmospheric transmission, free-space laser communication and other fields[1-4]. Therefore, coherent beam combining has become a research hotspot in the field of laser technology in recent years. The coherent combination technology based on Master Oscillator Power Amplifier (MOPA) structure adopts the modular structure, and realizes coherent beam combination through active phase control[5-6]. The commonly used active phase control methods mainly include heterodyne detection method[7], jitter method[8], simulated annealing algorithm[9] and Stochastic Parallel Gradient Descent (SPGD) algorithm[10-11]. The coherent combination system based on SPGD algorithm has low complexity and simple control strategy, and is expected to be a practical coherent combination scheme. The SPGD algorithm was first proposed by Voronstov et al. from the US Army Research Lab (ARL), and applied in the field of adaptive imaging[12]. In 2005, the ARL applied the SPGD algorithm to the fiber-laser optical phased array system to develop high-energy laser weapons[13]. In 2011, the University of Dayton in the United States cooperated with the ARL to control the beam phase by using the SPGD algorithm, and realized the coherent combination of seven 100-mW class fiber-laser targets in the loop with a transmission distance of 7 km[14]. In 2016, the research group realized the coherent combination of 21-beam fiber laser array with a transmission distance of 7 km by using the SPGD algorithm, and analyzed the system correction effect under different atmospheric turbulences[15]. In China, the National University of Defense Technology (NUDT), the Institute of Optics and Electronics, Chinese Academy of Sciences and other institutions have done in-depth research on SPGD-based coherent beam combination. In 2018, GENG Chao et al from the Institute of Optics and Electronics, Chinese Academy of Sciences carried out the coherent combination experiment of a 7-cell fiber laser array with a transmission distance of 200 m. In this experiment, the target-in-loop method was combined with SPGD optimal control algorithm to suppress the turbulence effect very well and obtain a good combination effect[16]. In 2019, SU Rong-tao et al. from the NUDT realized the coherent combination of 60 fiber laser beams by using the SPGD algorithm. This is the highest number of fiber laser beams in coherent combination that has been reported in China, and is also the highest number of phase-controlled beams in a fiber laser array that has been reported internationally and calculated with an optimization algorithm[17]. In the same year, ZHI Dong et al. in that research group realized the efficient coherent combination of 6 fiber laser targets in the loop with a transmission distance of 0.8 km by using the SPGD algorithm (tilted phase control) and multi-jitter method (phase-locked control)[18]. At present, the analyses of the coherent combination system based on SPGD algorithm are mostly limited to the correction of static aberration or the study of remote coherent combination effect through off-site experiments. Few literatures have studied the correction effect of coherent combination system at different atmospheric Greenwood frequencies. In this paper, a set of rotating phase screens are used to simulate turbulent atmosphere and the correction effect of SPGD-based coherent combination system at different atmospheric Greenwood frequencies is studied by numerical simulation and experiment.
In the gradient descent algorithm, if the objective function J(u) drops at a fast speed, the change of the independent variable u should occur in the direction of current negative gradient:
|
u(n+1)=u(n)−λ∇J(u(n)), |
(1) |
where n is the number of iterations; and λ is the iteration step. λ is positive when the objective function J(u) is at the minimum and negative when the objective function J(u) is at the maximum. For each component:
|
uj(n+1)=uj(n)−λ∂J(u(n))∂uj(n). |
(2) |
SPGD algorithm is a special gradient descent algorithm, which uses stochastic parallel perturbation to estimate the gradient[19]. The evaluation function J = J(u) of the system, which is the function of the phase control voltage u = {u1,u2,…,uN} applied to the beams, is taken as the optimization object of this algorithm. A set of small-perturbation signals δu = {δu1, δu2,... δuN} with a zero mean and fixed variance are actively applied to the phase control voltage. Then the variation of evaluation function δJ caused by stochastic disturbance voltage can be expressed as:
|
δJ=J(u+δu)−J(u). |
(3) |
The Taylor series expansion of equation (3) is:
|
δJ=N∑j=1∂J∂ujδuj+12N∑j,k=1∂J∂uj∂ukδujδuk+.... |
(4) |
By multiplying both sides of equation (4) by δuj and calculating the mathematical expectation, we obtain:
|
⟨δJδui⟩=N∑j∂J∂uj⟨δujδui⟩+12N∑j,k∂J∂uj∂uk⟨δujδukδui⟩+.... |
(5) |
Stochastic parallel perturbation δu = {δu1, δu2,... δuN} is a statistically independent random variable with zero mean and equal variance. So
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⟨δui⟩=0,⟨δuiδuj⟩=σ2δij, |
(6) |
Moreover, the probability density distribution of {δuj} is symmetric with respect to the mean value, that is,
|
⟨δuiδujδuk⟩=0. |
(7) |
Therefore, the equation (5) can be written as:
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⟨δJδui⟩=∂J∂uiσ2+o(σ4). |
(8) |
Statistically averaged, δJ/δui can be used as an estimate of the gradient component ∂J/∂ui. By substituting equation (8) into equation (2), the iterative formula of SPGD algorithm can be obtained:
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uj(n+1)=uj(n)−γδJ(n)δuj(n), |
(9) |
where γ = λ/σ2 is the gain coefficient of SPGD algorithm.
The system structure using the SPGD algorithm to realize the coherent combination of two fiber laser beams is shown in Fig.1. First, the beam from the seed source is divided into two beams by a beam splitter, each of which passes through a phase modulator (for beam phase adjustment) and then a fiber amplifier (for beam power amplification). After the collimator collimation and lens focus, each beam is divided into two parts by beam splitter after passing through turbulent atmosphere (simulated by rotating phase screen). One part of the beam is received by CCD and then is used to observe the far-field pattern. The other part of the beam is detected by a single-point detector after attenuation (the attenuator is not shown in the Fig.1). Initially, when the phase controller is not run, the coherent combination system is in an open-loop state. Due to the influence of external environment and optical fiber amplifier, the phase of each beam fluctuates randomly. This will cause the random movement of interference pattern in the far field as well as the random change of the evaluation index detected by the single-point detector. When the SPGD algorithm is executed, the system is in a closed-loop state. According to the evaluation index detected by the single-point detector, the phase controller generates a set of control voltage signals to be applied to the phase modulator. After several iterations, the evaluation function no longer fluctuates randomly and drastically, and the interference pattern in the far field becomes stable. In this way, the system can reach the phase-locked state and achieve a relatively stable coherent combination output. In the coherent combination, the execution process of SPGD algorithm can be expressed as follows:
(1) Set the gain coefficient and the initial phase control voltage u(0) = {u1,u2,…,uN};
(2) Generate a set of stochastic disturbed voltages δu(k) = {δu1, δu2,... δuN} that follow the Bernoulli distribution;
(3) Apply the disturbed voltage to the phase control voltage and obtain the evaluation functions J+(u(k) + δu(k)) and J-(u(k) - δu(k));
(4) Calculate the variation of evaluation function δJ;
(5) Update the phase control voltage by using the equation (9);
(6) Repeat the steps (2) to (5) until the algorithm is ended.
In order to analyze the correction effect of the system on turbulent atmosphere at different atmospheric Greenwood frequencies, the coherent combination processes in free space and in turbulent atmosphere are simulated numerically in this section. The system parameters required for simulation are shown in Table 1. The spot radius of the unit beam is 2.5 mm, the distance between beam centers is 10 mm, the laser wavelength is 1 064 nm, and the transmission distance is 2 m. The Power-In-Bucket (PIB) of the far-field spot is taken as the evaluation function J of the combination system. Considering that in practical work, the thermal effect of gain medium (optical fiber) and the influence of external environment will cause the phase inconsistency of unit beams, the initial phase of each beam in the numerical simulation is assumed to follow the Gaussian distribution with the mean value of zero and the variance of 3π.
| Parameter | Value |
| Distance: L/m | 2 |
| Wavelength: λ/m | 1 064 × 10−9 |
| Beam radius: w0/m | 2.5 × 10−3 |
| Number of samples: | 256 |
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In order to facilitate comparative analysis, this section normalizes the value of the system evaluation index PIB, and stipulates that the PIB value is 1 when the output of coherent combination system is stable under the free-space condition. Firstly, the influence of different turbulence intensities on coherent combination system was analyzed under static atmospheric conditions (the algorithm runs much faster than the atmospheric Greenwood frequency). The change of the system evaluation index PIB with turbulence intensity is shown in Fig.2. The numerical results show that with the increase of atmospheric turbulence intensity, the turbulence will aggravate the wavefront distortion, beam expansion, spot drift and other beam effects, so that the energy concentration of far-field spot will decrease and the combination effect of the coherent combination system will become worse.
In this paper, the rotating phase screen generated by calculation is used to simulate the dynamic atmospheric turbulence[20]. The generation of rotating phase screen is shown in Fig.3. Firstly, the method of power spectrum inversion plus subharmonic low-frequency compensation is used to simulate the generation of a set of large phase screens obeying the Kolmogorov statistical law. Secondly, the center distance R between sub-phase screen and large phase screen, the size d of the sub-phase screen as well as the rotation angle θ within the time τ are determined. Then, the center of the sub-phase screen at the time t is determined according to the center distance R, and a d×d rectangle phase screen is selected at the center of the sub-phase screen. At t+τ, the new sub-phase screen is determined according to the rotation angle. The relationship between the Greenwood frequency fG of turbulent atmosphere and the rotational speed of phase screen can be expressed as:
|
fG=(0.102k2v5/3C2n)3/5, |
(10) |
where k = 2π/λ is wave number; v = 2πnspR, where nsp is the rotational speed of phase screen.
In order to facilitate the comparison with the experimental results, it is assumed that the running frequency of a single iteration of the algorithm is 350 Hz, the center distance R=30 mm, and the atmospheric refractive index structure constant Cn2 = 5×10−11m−2/3. Under the condition of turbulent atmosphere, the change of the system evaluation index PIB with the number of iterations is shown in Fig.4. When the system is an open loop, the PIB value fluctuates randomly between 0.2 and 0.6 under the influence of turbulent atmosphere. When the system is a closed loop, the SPGD algorithm fails to make the combination system provide stable phase-locked output due to the continuous disturbance of turbulent atmosphere. However, the combination effect of the system using SPGD to control the beam phase is significantly better than that in the open-loop state. With the increase of atmospheric Greenwood frequency, the jitter amplitude and frequency of PIB value will increase, and the combination effect of the system will become worse. This shows that the coherent combination system based on SPGD algorithm has a significant correction effect on turbulent atmosphere, but the atmospheric Greenwood frequency has a great influence on the combination effect of the system.
To quantitatively analyze the influence of atmospheric Greenwood frequency on the combination effect of the system, the change of the evaluation index PIB of coherent combination system with atmospheric Greenwood frequency has been obtained through simulation calculation, as shown in Fig. 5. The simulation results show that when the iteration frequency of the algorithm is much higher than atmospheric Greenwood frequency (fG < 0.5 Hz), the convergence process of the algorithm will be less affected by turbulent atmosphere, and the amplitude of combination effect variation will be small. However, with the continuous increase of atmospheric Greenwood frequency, the turbulence of the outgoing beam by turbulent atmosphere will be quickened, and the PIB of the far-field spot will fluctuate more frequently. As a result, the SPGD algorithm can’t timely and effectively adjust the control voltage of the phase modulator, resulting in worse and worse combination effect of the system. When the atmospheric Greenwood frequency is greater than 2 Hz, the change of the system evaluation index PIB tends to be gentle with the increase of atmospheric Greenwood frequency. This is because the disturbance by turbulent atmosphere is aggravated, the sensitivity of the control algorithm to external disturbance is reduced, and thus the closed-loop control effect of the system is not obvious. The comparison between open-loop and closed-loop system states shows that the evaluation index of the system in the closed-loop state is higher than that in the open-loop state.
An experimental platform was built based on the structural diagram of coherent combination system shown in Fig. 1, and the SPGD algorithm was used to experimentally study the correction effect of coherent combination system on turbulent atmosphere. The central wavelength of the seed source is 1 064 nm, the line width is less than 20 kHz, and the maximum output power is 100 mW. The fiber amplifier is polarization-maintaining fiber amplifier with the maximum amplification factor of 10 times. The phase modulator is a lithium-niobate phase modulator made by French iXblue, with a half-wave voltage of 2 V. The detector is an InGaAs avalanche photodetector with an effective diameter of 0.2 mm. The SPGD algorithm controller is composed of NI acquisition card and LabView upper computer software. The time for each iteration operation in the algorithm is about 2.8 ms. In the experiment, Lexitek's static phase screen (its thickness is 22 mm, and the corresponding atmospheric coherence length is r0 = 4.8 mm) was selected. It is a pseudo-random phase plate with Kolmogorov statistical law made by using the near-refractive-index matching of optical polymer and acrylic plastic and the related algorithm. The turbulent atmosphere is simulated as the rotational speed of the phase screen is controlled by a control box. The atmospheric refractive index structure constant can be obtained by equation (11) to be
|
C2n=0.42r−5/30k−2. |
(11) |
In order to analyze the influence of turbulent atmosphere on coherent combination, the coherent combination experiment without phase screen is first carried out in this paper. The change of the detector’s output voltage with the running times (time) of the algorithm in the open-loop or closed-loop system is shown in Fig. 6. When the SPGD algorithm is not run, the system will be in an open-loop state and the evaluation index output by the detector will constantly change between 0.5 V and 0.7 V. When the SPGD algorithm is executed, the system will be in a closed-loop state. After several iterations, the voltage value output by the detector will basically stay around 0.8 V. It should be noted that in the closed-loop process of the system, the value of the system evaluation index PIB will fluctuate greatly for two reasons. First, the phase control voltage is out of the hardware voltage range, so the control voltage is reset, resulting in the loss of system locking. Second, the seed source or amplifier is unstable.
When the phase screen is added, the relationship between the output voltage of the detector and the running times (time) of the algorithm at different Greenwood frequencies can be shown in Fig. 7. It can be found that the SPGD algorithm can achieve stable phase-locked output when the phase screen is stationary, i.e. the atmospheric Greenwood frequency is zero. Due to the effect of phase screen on the beam expansion and beam distortion, the combination effect of the system under atmospheric conditions is lower than that in free space. When the atmospheric Greenwood frequency is 0.32 Hz, the atmospheric disturbance can cause the unable convergence of SPGD algorithm and thus the unable phase-locked output of the system. However, by comparing the output voltages of the detector in the open-loop and closed-loop states, it can be seen that the average output voltage with the SPGD algorithm running is significantly higher than that in the open loop. The experimental results show that the coherent combination system using SPGD algorithm has a certain correction effect on turbulent atmosphere. However, the atmospheric Greenwood frequency has a greater influence on the combination effect of the system.
The change of the detector’s output voltage with the Greenwood frequency of turbulent atmosphere when the system is in open and closed loops is shown in Fig. 8. It can be seen from the figure that when the Greenwood frequency is small, the continuous turbulence of the turbulent atmosphere causes the damage to the convergence of SPGD algorithm so that it cannot timely and effectively adjust the control voltage of the phase modulator. As a result, the voltage output by the detector decreases with the increase of Greenwood frequency. When the Greenwood frequency is greater than a certain value (0.5 Hz), the sensitivity of SPGD algorithm to external disturbance will decrease with the acceleration of atmospheric disturbance, and the output voltage of the detector will decrease slowly with the increase of Greenwood frequency. The changing trend of system combination effect relative to atmospheric Greenwood frequency basically matches the results of numerical simulation. In the low frequency (fG < 0.5Hz) part, there are two reasons for the difference in the changing trend of the synthetic effect of experiment and simulation. On the one hand, due to the influence of dynamic noise in the system and external environment, the number of iterations required for the convergence of SPGD algorithm in the hardware (about 100 iterations) is greater than that required in the numerical simulation (about 20 iterations). On the other hand, due to the limitation of experimental hardware, the iteration frequency of the algorithm is only 350 Hz, and the speed range of the control box is limited. Moreover, in this paper, there are insufficient samples for the experiment at lower atmospheric Greenwood frequency.
The random dynamic disturbance of turbulent atmosphere will cause the wavefront distortion and random phase change of light beam, which greatly limits the application of coherent combination system of fiber laser array in practice. Firstly, the influence of different atmospheric turbulence intensities on coherent combination system in static atmospheric environment (the algorithm runs much faster than the atmospheric Greenwood frequency) was analyzed in this paper. Then a set of rotating phase screens generated by numerical calculation were used to simulate turbulent atmosphere and to study the correction effect of the coherent combination system based on SPGD algorithm in turbulent atmosphere. The numerical results show that with the increase of atmospheric turbulence intensity, the far-field spot energy concentricity of the system will decline gradually. With the increase of atmospheric Greenwood frequency, the turbulent atmosphere will disturb the beam faster, and the convergence effect of the system will become worse and gradually level off. Finally, an experimental platform was built to simulate turbulent atmosphere by controlling the rotational speed of static phase screens, and to carry out the coherent combination of two laser beams. It can be concluded from the experimental results that under a certain iteration frequency (350 Hz) of the algorithm, with the acceleration of phase screen speed, the sensitivity of SPGD algorithm to atmospheric disturbance will continuously decline so that the average output voltage of the detector in the closed-loop system will decrease and tend to level off. The research on the correction effect of coherent combination system at different atmospheric Greenwood frequencies in this paper provides a reference for the application of the system in practice. In the future, we will further increase the iteration frequency of SPGD algorithm and the number of combined beams to study the impact of turbulent atmosphere on the system.
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| 2. | 赵辉，吕典楷，安静，邝凯达，余孟洁，张天骐. 空间光波前畸变校正中的元启发式SPGD算法. 红外与激光工程. 2022(07): 434-444 . |
Figures(8) / Tables(1)
SONG Ji-kun, LI Yuan-yang, CHE Dong-bo, GUO Jin, WANG Ting-feng, LI Zhi-lai. Influence of turbulent atmosphere on the effect of coherent beam combining[J]. Chinese Optics, 2020, 13(4): 884-898. doi: 10.37188/CO.2019-0197
| Parameter | Value |
| Distance: L/m | 2 |
| Wavelength: λ/m | 1 064 × 10−9 |
| Beam radius: w0/m | 2.5 × 10−3 |
| Number of samples: | 256 |
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