Abstract

Multi-aperture receiver with optical combining architecture is an effective approach to overcome the turbulent atmosphere effect on the performance of the free-space optical (FSO) communications, in which how to combine the multiple laser beams received by the sub-apertures efficiently is one of the key technologies. In this paper, we focus on the combining module based on fiber couplers, and propose the all-fiber coherent beam combining (CBC) with two architectures by using active phase locking. To validate the feasibility of the proposed combining module, corresponding experiments and simulations on the CBC of four laser beams are carried out. The experimental results show that the phase differences among the input beams can be compensated and the combining efficiency can be stably promoted by active phase locking in CBC with both of the two architectures. The simulation results show that the combining efficiency fluctuates when turbulent atmosphere is considered, and the effectiveness of the combining module decreases as the turbulence increases. We believe that the combining module proposed in this paper has great potential, and the results can provide significant advices for researchers when building such a multi-aperture receiver with optical combining architecture for FSO commutation systems.

© 2017 Optical Society of America

1. Introduction

With the development of the Space-Ground Integration Network (SGIN), the demand of a higher link capacity is indubitable. Compared to the conventional radio frequency (RF) links, the free-space optical (FSO) communications offers numerous advantages including: ultra-high data rates (at the order of multiple gigabits per second), excellent security and large, unlicensed bandwidth, relatively low power consumption, and immunity to the electromagnetic interference [1,2]. As the receiver sensitivity can be improved by up to 20 dB compared with that of non-coherent intensity modulation/ direct detection (IM/DD) scheme [3], the coherent FSO communications has great potential to be used in various applications. In 2008, a 5.6 Gbps optical communication link has been verified in-orbit based on homodyne binary phase shift keying (BPSK) with a bit error rate better than 10−9 between two low earth orbit (LEO) satellites, NFIRE (U. S.) and TerraASR-X (Germany) [4]. Besides, a 5.625 Gbps bidirectional laser communication with BPSK scheme has been demonstrated in satellite-to-ground link between the NFIRE satellite and an Optical Ground Station (OGS) [5]. Unfortunately, the turbulent atmosphere can have a tremendous impact on coherent FSO links, which will lead to serious degradation for the received signal-to-noise ratio, and severely hinder the performance of the coherent FSO communication systems [6].

Several types of mitigation techniques can be adopted to alleviate the effect of the turbulent atmosphere on the performance of the coherent FSO communications. The monolithic-aperture receiver with adaptive optics [7–9] reduces the effect of the turbulent atmosphere with wavefront aberration correction. As an alternative, the multi-aperture receiver [10,11] combines signals detected by aperture array to ease deep fades. Compared with the monolithic-aperture receiver with adaptive optics, the multi-aperture receiver has easier manufacture, lower costs, superior reliability, smaller sub-aperture sizes, and more flexible sub-aperture positions [12].

In the multi-aperture receiver, it is necessary to combine the signals from the aperture array, to merge these signals and enhance the received signal-to-noise ratio. An efficient means of accomplishing this goal is to combine the digitized signals after photodetection by digitally combining [13,14], as shown in Fig. 1(a). Another approach for merging the signals is to optically combine the multiple laser beams received by the aperture array efficiently before photodetection [12,15–17], as shown in Fig. 1(b). A unique feature of the multi-aperture receiver with optical combining architecture is that the mature coherent demodulation techniques in conventional FSO communications with single-aperture receiver can be used for reference. However, a more detailed report about the optical combining module can seldom be found.

 

Fig. 1 Multi-aperture receiver with (a) digital combining architecture, and (b) optical combining architecture. LO: local oscillator.

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All-fiber coherent beam combining (CBC) provides a feasible solution in the multi-aperture receiver with optical combining architecture, as it has been proven to be a promising approach to merge the outputs of multiple fiber lasers into a single output fiber, with the advantages of being reliable, compact, and stable, and able to connect with other fiber devices easily [18–21]. Two kinds of all-fiber devices, the fiber coupler [18,19] and the fiber combiner [20,21], are mainly adopted as the combiner in the all-fiber CBC. The fiber combiner can handle mutli-kW optical powers. However, the output fiber of the fiber combiner is always multi-mode fiber [20], which will result in a degraded beam quality. As an alternative, the fiber coupler can be manufactured by using polarization-maintaining fibers (PMFs) with good beam quality. Nevertheless, at the present time, research on CBC via fiber couplers mainly focuses on passive schemes [18,19], where phase matching is inherent to the geometry, which is hard to be used in coherent FSO communication systems. Hence, the all-fiber CBC via fiber couplers with active phase locking deserves deep regards.

In this paper, we report on all-fiber active CBC via fiber couplers, with two different architectures, which can be adopted in the optical combining module for FSO communications employing multi-aperture receivers. The rest of this paper is organized as follows. In section 2, the combining module based on fiber couplers, with two different architectures (the integrated architecture by maximizing the cost function and the distributed architecture by minimizing the cost function), is proposed and analyzed. In section 3, experiments on CBC of four laser beams with phase locking are carried out to validate the feasibility of the proposed combining module. In section 4, simulations on CBC of four laser beams under turbulent atmosphere are presented to study the effectiveness and capability of the proposed combining module to accommodate the power fluctuations.

2. Combining module based on fiber couplers

The 3-dB fiber coupler is a kind of specific 2 × 2 fiber coupler with the coupling ratio of 50/50. Theoretically, when two laser beams are launched into the two input ports, the optical powers come out from the two output ports, and the distribution of the optical powers between the two output ports is closely related to the phase difference and the power ratio of the input beams. Based on the coupled mode theory, the optical powers emerged from the two output ports can be given by [18,22]:

Iout1=12[Iin1+Iin22Iin1Iin2cos(Δδ+π2)]
Iout2=12[Iin1+Iin2+2Iin1Iin2cos(Δδ+π2)]
in which Iin1 and Iin2 are the optical powers of the two input beams, respectively, and Δδ is the phase difference between the input beams. Δδ = δin1-δin2, where δin1 and δin2 are the phases of the two input beams, respectively.

It can be noted that the optical powers emerged from the two output ports are relatively complemented. Moreover, when the phase difference between the input beams is controlled to be π/2 + k·2π (where k is an integer), and the optical powers of the input beams are identical (Iin1 = Iin2), all of the optical powers come out from output port-1 and no optical power leaks from output port-2. The property of the 3-dB fiber coupler makes it available for combining individual laser beams to one PMF, and the combining efficiency is related to the phase difference and power ratio of the individual beams. We define the output port-1 to be beam combining port, and the output port-2 to be beam leaking port. In this paper, we propose the combining module based on fiber couplers with two architectures, the integrated architecture by maximizing the cost function acquired from the beam combining port and the distributed architecture by minimizing the cost function acquired from the beam leaking port.

The structural schematic diagram of the combining module with the integrated architecture is illustrated in Fig. 2. The input beams are combined using cascaded 3-dB fiber couplers, and the piezoelectric-ring fiber-optic phase compensator (PCs) are employed in the input paths to compensate for the phase differences among the input beams. The combined beam output from the beam combining port of the last-stage fiber coupler is split into two beams, a small portion for phase locking, and the others for communications. The optical power split for phase locking is sent to a servo photo detector (PD) to provide a control signal as the cost function. The stochastic parallel gradient descent (SPGD) algorithm [23–25] is employed here to maximize the cost function, and corresponding analog voltage signals are generated to control the PCs to compensate for the phase differences among the input beams. It can be noted that the combining module with the integrated architecture has a simple structure, with symmetrical configuration.

 

Fig. 2 Structural schematic diagram of the combining module with the integrated architecture.

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The structural schematic diagram of the combining module with the distributed architecture is illustrated in Fig. 3. The input beams are combined by cascaded sub-combined units. Each sub-combined unit consists of a PC, a 3-dB fiber coupler, and a servo PD located behind the tail of the beam leaking port of the 3-dB fiber coupler. Different from the combining module with the integrated architecture, the optical power output from the beam leaking ports will act as the cost functions for active phase locking, and the SPGD algorithm is employed here to minimize the cost functions simultaneously. Each cost function provides an independent servo signal, which will generate corresponding analog voltage to control the corresponding PC to compensate for the phase difference between the input beams in each sub-combined unit. A unique feature of the combining module with the distributed architecture is that the combined beam output from the beam combining port of the last-stage fiber coupler can be used for communications directly and no extra beam splitting is needed.

 

Fig. 3 Structural schematic diagram of the combining module with the distributed architecture.

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The scalability of the combining module is affected by the number of the input beams, due to that both of the combining efficiency and the convergence rate of the SPGD algorithm will decrease as the number of the input beams increases.

To analyze the performance of the combining module quantitatively, we assume that the phase differences among the input beams have been completely compensated, and the optical power of each of the input beams is normalized to be 1. According to the calculation method referred in ref [12], the performance of the combining module with the two architectures influenced by the devices’ losses can be analyzed, as shown in Fig. 4, where the insertion loss of the PC is set to be 0.22 dB and the excess loss of the 3-dB fiber coupler is set to be 0.6 dB in the calculation. Figure 4(a) shows the curves of the combining efficiency as the function of the number of the input beams, it can be noted that the combining efficiency decreases as the number of the input beams increases. Figure 4(b) shows the curves of the combined optical power as the function of the number of the input beams, it can be noted that although the combining efficiency decreases, the combined optical power increases as the number of the input beams increases in both of the two architectures.

 

Fig. 4 Performance of the combining module: (a) combining efficiency and (b) combined optical power.

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Besides, in the combining module with the integrated architecture, the combined beam output from the beam combining port of the last-stage fiber coupler needs to be split into two beams, a small portion for phase locking, and the others for communications. As a result, the combined optical power would be further degraded. Moreover, the received optical power in communications is weak after attenuation in long distance turbulent atmosphere. Hence, optical power split for phase locking may not be strong enough to be detected.

On the other hand, the basic difference between the two architectures is the relationship between the servo signals and the correction devices. In CBC of N beams with the integrated architecture, only one servo signal controls N PCs simultaneously. In CBC with the distributed architecture, there are multiple servo signals, and each PC is controlled by a corresponding and independent servo signal. Hence, when combining N input beams, the ratio of the convergence rate between the integrated architecture and the distributed architecture is approximately equal to be N [26], which indicates that the combining module with the distributed architecture has a faster convergence rate than the integrated architecture, and the difference between the two convergence rate will be greatly enlarged as the number of the input beams increases.

3. Experiments with phase locking

As the key to the proposed combining module is to compensate for the phase differences among the input beams, the experiments on CBC of four laser beams with phase locking are carried out in this section, to validate the feasibility of the proposed combining module based on fiber couplers, in which only phase difference is considered and power imbalance is neglected. All the fibers and fiber devices in the experiments are polarization-maintained. As shown in the experimental setups, the fiber laser with 1 × 4 fiber splitter and variable optical attenuators (VOAs) is applied to simulate the aperture array with four sub-apertures. An output from a linearly polarized single-mode fiber laser at 1064 nm (NKT photonics) is split into four beams using a 1 × 4 fiber splitter. Each of the beams is then sent through a VOA to adjust the optical powers of the input beams to be identical. In the experiments, the PCs made by our laboratory [12] have a half-wave voltage of 1.3 V and a first resonant frequency of about 32 kHz, with the insertion loss of no more than 0.22 dB. The 3-dB fiber couplers are polarization maintaining fused fiber couplers with the excess loss of no more than 0.6 dB at 1064 nm without the connector, produced by Advanced Fiber Resources (AFR) Corporation. The PDs are PDA36A silicon amplifier detectors with a 350 nm-1100 nm response wavelength and 12.5 MHz bandwidth when the gain is at 10 dB, produced by THORLABS Corporation.

3.1 CBC with the integrated architecture

The experimental setup of CBC of four laser beams with the integrated architecture is illustrated in Fig. 5. Four input beams are combined using cascaded 3-dB fiber couplers. Four PCs are employed in the input paths to compensate for the phase differences among the input beams. For simplicity, the optical power output from the beam combining port of the 3-dB fiber coupler 3, denoted as P4, is directly sent to a servo PD to provide a control signal as the cost function J for active phase locking, without splitting.

 

Fig. 5 Experimental setup of CBC of four laser beams with the integrated architecture.

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The control loop of the SPGD algorithm implemented here is presented as below:

  • 1) Generate a group of random voltage perturbations ΔUi(i=1,2,3,4), which obey the Bernoulli probability distribution with zero mean.
  • 2) Apply the signals Ui+ΔUi(i=1,2,3,4) on PCs simultaneously to get the corresponding cost function J+; then apply the signals UiΔUi(i=1,2,3,4) on PCs simultaneously to get the cost function J-.
  • 3) Update the control voltage signals on the PCs to maximize the cost function:
    Ui=Ui+γΔUi(J+J)/2

    where γ is the gain coefficient of the SPGD algorithm.

  • 4) Go to step 1 and continue the process, until the control procedure is stopped manually.

To quantitatively evaluate the combining performance, three extra detected PDs, which are not necessary in real applications, are located behind the tails of the beam leaking ports of the three 3-dB fiber couplers to detect the optical powers in the experiment, denoted as P1, P2, and P3, respectively.

The experimental results are shown in Fig. 6. The iteration rate of the SPGD algorithm is about 5 kHz. The durations of the open and closed states are both 12 s. It can be noted that when loop closed, almost all of the optical powers come out from the beam combining port of the last-stage fiber coupler.

 

Fig. 6 Experimental results of the output optical powers (a) P1; (b) P2; (c) P3; (d) P4.

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Without considering the excess losses of the 3-dB fiber couplers, the combining efficiency of the CBC of four laser beams can be measured by comparing the optical powers output from the beam combining and beam leaking ports of the 3-dB fiber couplers in the experiments, as shown in Eq. (4):

ηCBC=P4P1+P2+P3+P4

The combining efficiency calculated by using Eq. (4) is shown in Fig. 7. The average combining efficiency increases from 22.04% in the open loop to 97.83% in the closed loop, and the mean square error (MSE) decreases from 0.2028 in the open loop to 0.0117 in the closed loop. The closed loop of phase locking is achieved after about 27 iterations, equivalent to 5.4 ms, of SPGD optimization. A residual phase error is evaluated to be less than λ/23 by using the expression [23]:

Δϕrms=2Jrms/Jmax
where J(t) is the combining efficiency evolution when coherent beam combining achieved.

 

Fig. 7 Results of the calculated combining efficiency.

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3.2 CBC with the distributed architecture

The experimental setup of CBC of four laser beams with the distributed architecture is illustrated in Fig. 8. Two pairs of the input beams are first combined using two 3-dB fiber couplers, coupler-1 and coupler-2, to produce two new sub-combined beams, which are then sent to the 3-dB fiber coupler-3 for the second stage combination. Three servo PDs are located behind the tails of the beam leaking ports of the three 3-dB fiber couplers to detect the optical powers of the servo beams, which will act as the cost functions Ji(i = 1,2,3) for phase locking, denoted as P1, P2, and P3, respectively. Three PCs are employed in one of the input paths of each of the 3-dB fiber couplers, respectively.

 

Fig. 8 Experimental setup of the CBC of four laser beams with the distributed architecture.

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The control loop of the SPGD algorithm implemented here is presented as below:

  • 1) Generate a group of random voltage perturbations ΔUi(i=1,2,3), which obey the Bernoulli probability distribution with zero means.
  • 2) Apply the signals Ui+ΔUi(i=1,2,3) on corresponding PC-1, PC-2, and PC-3, respectively, to get the corresponding cost function Ji+(i = 1,2,3); then apply the signals UiΔUi(i=1,2,3) on corresponding PC-1, PC-2, and PC-3, respectively, to get the cost function Ji-(i = 1,2,3).
  • 3) Update the control voltage signals on the PCs to minimize the cost functions:
    Ui=UiγΔUi(Ji+Ji)/2

    where γ is the gain coefficient of the SPGD algorithm.

  • 4) Go to step 1 and continue the process, until the control procedure is stopped manually.

The combined optical power output from the beam combining port of the 3-dB fiber coupler-3 is detected by an extra detected PD, denoted as P4, to quantitatively evaluate the combining performance.

The experimental results are shown in Fig. 9. The iteration rate of the SPGD algorithm is about 5 kHz. The durations of the open and closed states are both 12 s. It can be noted that when loop closed, the cost functions, P1, P2, and P3, reach the minimum simultaneously, and meanwhile the combined optical power output from the beam combining port of the 3-dB fiber coupler-3, P4, reaches the maximum.

 

Fig. 9 Experimental results of the output optical powers (a) P1; (b) P2; (c) P3; (d) P4.

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The combining efficiency calculated by using Eq. (4) is shown in Fig. 10. The average combining efficiency increases from 27.30% in the open loop to 96.48% in the closed loop, and the MSE decreases from 0.1790 in the open loop to 0.0088 in the closed loop. The closed loop of phase locking is achieved after about 14 iterations, equivalent to 2.8 ms, of SPGD optimization. A residual phase error is evaluated to be less than λ/25 by using Eq. (5).

 

Fig. 10 Results of the calculated combining efficiency.

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3.3 Comparison on the experimental results

The experimental results show that the average combining efficiency is enhanced to be 97.83% in the closed loop after about 27 iterations of SPGD optimization in CBC with the integrated architecture, and enhanced to be 96.48% after about 14 iterations in CBC with the distributed architecture, which indicate that the phase differences among the input beams can be compensated and the combining efficiency can be stably promoted by active phase locking in CBC with both of the two architectures.

As the optical powers of the four input beams are adjusted to be identical, the theoretical combining efficiency can reach 100% when phase controlled without considering the devices’ losses. The deviations of the combining efficiency in the experimental results may be caused by the coupling ratio deviation of the 3-dB fiber couplers, the residual phase difference among the input beams, the different insertion losses of the connectors and the PCs. Moreover, the CBC with distributed architecture is asymmetric, where the insertion loss of the PC will cause an extra power imbalance in combining. Hence, the combining efficiency in distributed architecture is less than that in integrated architecture. In fact, the combining efficiency can be further enhanced by using fiber fusion technology, improving the active phase locking algorithm, and employing fiber couplers with lower coupling ratio deviations and PCs with lower insertion losses.

On the other hand, it can be noted that the CBC with distributed architecture has a faster convergence rate than the CBC with the integrated architecture, as the iterations needed to achieve a close loop in the CBC with the distributed architecture is only half of that in the CBC with the integrated architecture, which fits the theoretical result in section 2.

4. Simulations under turbulent atmosphere

In this section, the combining performance of the CBC of four laser beams is analyzed in simulation to study the effectiveness and capability of the proposed combining module to accommodate the power fluctuations in each sub-aperture induced by the turbulent atmosphere, under the assumption that the control bandwidth is unlimited, and the devices’ losses are negligible.

Turbulent atmosphere can bring about not only the phase variations but also the irradiance fluctuations, which will affect the optical powers received by the sub-apertures and result in the power imbalance in the CBC.

The optical power received by the i-th sub-aperture Ii(i = 1,2,3,4) can be given by:

Ii=ηi×If_i
in which ηi is the coupling efficiency and If_i is the optical power in the front surface of the i-th sub-aperture.

The coupling efficiency of the i-th sub-aperture can be given by [27]:

ηi=|EO(r)FO(r)ds|2|EO(r)|2ds×|FO(r)|2ds
where EO is the focused optical field at the pupil plane and can be written as:
EO(r)=F[EA(r)]
where EA is the signal optical field after transmitting through the atmosphere.

In the FSO communications for which the distance is well larger than Fraunhofer distance, the incident optical field in the aperture plane of the receiver is a plane wave [28], and EA can be expressed by:

EA(r)=Esexp[jφ(r)]
where Es represents the amplitude and φ(r) denotes the phase variations.

FO is the PMF mode profile at the pupil plane and can be expressed as:

FO(r)=2πω02exp(r2ω02)
and normalized as |∫∫FO(r)ds|2 = 1, where ω0 represents the fiber mode-field radius.

The intensity in the front surface of the i-th sub-aperture can be given by:

If_i=π4D2×Irr
where D is the diameter of the sub-aperture receiver, and Irr is the received signal irradiance.

Gamma–Gamma distributed models [29] are considered to describe the intensity fluctuations caused by turbulent atmosphere, and the probability density function of the Gamma–Gamma model signal irradiance is given by:

f(Irr)=2(αβ)α+β2Γ(α)Γ(β)Irrα+β21Καβ(2αβIrr),Irr0
where α and β denote the effective numbers of large-scale and small-scale cells of the scattering process, respectively. Γ(·) is the gamma function, and Kαβ(·) is the modified Bessel function of the second kind and order αβ.

As this section focuses on the influence of the power fluctuations, we assume that the phase differences among the input beams are completely compensated in the simulation. According to Eq. (1), the combined output optical power Ic can be calculated by using the following formulas:

Isc1=12(I1+I2+2I1I2)
Isc2=12(I3+I4+2I3I4)
Ic=12(Isc1+Isc2+2Isc1Isc2)

The combining efficiency in the simulation is defined as the ratio of the combined output optical power to the total received optical power by the sub-apertures, and can be calculated by:

ηc=IcI1+I2+I3+I4

The combining loss of the CBC in the simulation can be given by:

CL=10log(IcI1+I2+I3+I4)

We consider a coherent FSO communication system with wavelength λ = 1064 nm, link distance Ld = 2 km, diameter of the sub-aperture D = 0.4 m, and fiber mode-field radius ω0 = 5.6 μm. In the simulation, the coefficients of the Zernike polynomials of the turbulent wavefront are generated using the method of the Zernike polynomials. The number of the Zernike modes used to generate the turbulent wavefront is 231 [30]. As the adaptive fiber coupler (AFC) [24,25] incorporates the functions of tip/tilt aberrations correcting and adaptive laser coupling, the tip/tilt errors in the sub-apertures are completely corrected to be zero in the simulation, and the turbulent wavefront errors are generated using the residual Zernike coefficients. We select D/r0 = 1, 3, 5, 7, and 9 (r0 is the atmospheric coherent length) as the parameter of the turbulent atmosphere strength. Accordingly, index of refraction structure parameters Cn2 can be calculated to be 1.56 × 10−16 m-2/3, 9.74 × 10−16 m-2/3, 2.28 × 10−15 m-2/3, 4 × 10−15 m-2/3, and 6.1 × 10−15 m-2/3, which can represent different intensity scenarios of turbulent atmosphere.

The 1000 simulation results of the combining efficiency, and the corresponding normalized probability distribution functions (PDF) of the combining loss of the CBC are calculated under different turbulent atmosphere conditions, as shown in Fig. 11. It can be noted that the combining efficiency fluctuates when turbulent atmosphere is considered. The average combining efficiency decreases as the turbulent atmosphere strength increases, as it is equal to be 99.75%, 98.34%, 95.87%, 93.03%, and 90.89% when D/r0 = 1, 3, 5, 7, and 9, respectively. Moreover, the combining loss is less than 0.22 dB (the combining efficiency is greater than 95% correspondingly) for 100%, 95.7%, 63%, 37%, and 25.4% of the time when D/r0 = 1, 3, 5, 7, and 9, respectively.

 

Fig. 11 Simulation results of the CBC of four input beams under the turbulent atmosphere conditions of (a) D/r0 = 1, (b) D/r0 = 3, (c) D/r0 = 5, (d) D/r0 = 7, and (e) D/r0 = 9.

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The results show that the effectiveness of the combining module decreases as the turbulence increases. We believe that the simulation results in this section can provide significant advices for researchers when building such a multi-aperture receiver with optical combining architecture for FSO commutation systems.

5. Conclusions

In conclusion, we have reported on the all-fiber CBC via 3-dB fiber couplers, with two architectures by using active phase locking. Owing to the property of the 3-dB fiber coupler that the optical powers emerged from the output ports are relatively complemented, we propose the combining module based on fiber couplers with two architectures, the integrated architecture by maximizing the cost function acquired from the beam combining port and the distributed architecture by minimizing the cost function acquired from the beam leaking port. The combining efficiency of the CBC is related to the phase difference and power ratio of the input beams. To validate the feasibility of the proposed combining module, experiments on CBC of four laser beams with phase locking are carried out based on SPGD algorithm. The experimental results show that the average combining efficiency is enhanced to be 97.83% in the closed loop after about 27 iterations of SPGD optimization in CBC with the integrated architecture, and enhanced to be 96.48% after about 14 iterations in CBC with the distributed architecture. It reveals that the phase differences among the input beams can be compensated and the combining efficiency can be stably promoted by active phase locking in CBC with both of the two architectures. Moreover, compared to the combining module with integrated architecture, the combining module with the distributed architecture has a faster convergence rate. To study the effectiveness and capability of the proposed combining module to accommodate the power fluctuations, simulations on CBC of four laser beams under turbulent atmosphere are presented. The simulation results show that the combining efficiency fluctuates when turbulent atmosphere is considered, and the effectiveness of the combining module decreases as the turbulence increases. We believe that the combining module proposed in this paper has great potential, and the results can provide significant advices for researchers when building such a multi-aperture receiver with optical combining architecture for FSO commutation systems.

Moreover, the combining module based on fiber couplers proposed in this paper will be further improved and expanded, CBC of more than four laser beams will be analyzed, and the experiments in real atmosphere with AFC array will be performed in the near future.

Funding

National Natural Science Foundation of China (61675205); Innovation Foundation of Chinese Academy of Sciences (CXJJ-15S096); CAS “Light of West China” program.

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18. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays,” Opt. Express 18(25), 25873–25886 (2010). [PubMed]  

19. C. Zhang, W. Chang, A. Galvanauskas, and H. G. Winful, “Simultaneous passive coherent beam combining and mode locking of fiber laser arrays,” Opt. Express 20(15), 16245–16257 (2012).

20. J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

21. R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

22. Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

23. C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013). [PubMed]  

24. F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

25. F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

26. B. Chen and X. Li, “Bandwidth of adaptive optics system based on stochastic parallel gradient descent algorithm,” Laser Optoelectron. Prog. 50, 030101 (2013).

27. W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

28. R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18434–18441 (2013). [PubMed]  

29. D. Bykhovsky, “Simple generation of Gamma, Gamma–Gamma, and K distributions with exponential autocorrelation function,” J. Lightwave Technol. 34(9), 2106–2110 (2016).

30. C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014). [PubMed]  

References

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  1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
  2. Z. Ghassemlooy, H. L. Minh, and M. Ijaz, “Free space optical communications,” Optical And Microwave Technologies for Telecommunication Networks, O. Strobel, ed. (Wiley, 2016), Chapter 10.4, pp. 349–377.
  3. G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002).
  4. B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).
  5. R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).
  6. M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).
  7. C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
    [PubMed]
  8. W. Liu, K. Yao, D. Huang, X. Lin, L. Wang, and Y. Lv, “Performance evaluation of coherent free space optical communications with a double-stage fast-steering-mirror adaptive optics system depending on the greenwood frequency,” Opt. Express 24(12), 13288–13302 (2016).
    [PubMed]
  9. J. Cao, X. Zhao, W. Liu, and H. Gu, “Performance analysis of a coherent free space optical communication system based on experiment,” Opt. Express 25(13), 15299–15312 (2017).
    [PubMed]
  10. A. Belmonte, “Capacity of coherent laser downlinks,” J. Lightwave Technol. 32(11), 2128–2132 (2014).
  11. J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
  12. Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
    [PubMed]
  13. D. J. Geisler, T. M. Yarnall, M. L. Stevens, C. M. Schieler, B. S. Robinson, and S. A. Hamilton, “Multi-aperture digital coherent combining for free-space optical communication receivers,” Opt. Express 24(12), 12661–12671 (2016).
    [PubMed]
  14. K. Li, J. Ma, L. Tan, S. Yu, and C. Zhai, “Performance analysis of fiber-based free-space optical communications with coherent detection spatial diversity,” Appl. Opt. 55(17), 4649–4656 (2016).
    [PubMed]
  15. A. Belmonte and J. M. Kahn, “Sequential optimization of adaptive arrays in coherent laser communications,” J. Lightwave Technol. 31(9), 1383–1387 (2013).
  16. A. Belmonte, “Digital equalization of time-delay array receivers on coherent laser communications,” Opt. Lett. 42(2), 310–313 (2017).
    [PubMed]
  17. Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).
  18. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays,” Opt. Express 18(25), 25873–25886 (2010).
    [PubMed]
  19. C. Zhang, W. Chang, A. Galvanauskas, and H. G. Winful, “Simultaneous passive coherent beam combining and mode locking of fiber laser arrays,” Opt. Express 20(15), 16245–16257 (2012).
  20. J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).
  21. R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).
  22. Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).
  23. C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
    [PubMed]
  24. F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).
  25. F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).
  26. B. Chen and X. Li, “Bandwidth of adaptive optics system based on stochastic parallel gradient descent algorithm,” Laser Optoelectron. Prog. 50, 030101 (2013).
  27. W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).
  28. R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18434–18441 (2013).
    [PubMed]
  29. D. Bykhovsky, “Simple generation of Gamma, Gamma–Gamma, and K distributions with exponential autocorrelation function,” J. Lightwave Technol. 34(9), 2106–2110 (2016).
  30. C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
    [PubMed]

2017 (5)

J. Cao, X. Zhao, W. Liu, and H. Gu, “Performance analysis of a coherent free space optical communication system based on experiment,” Opt. Express 25(13), 15299–15312 (2017).
[PubMed]

Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
[PubMed]

A. Belmonte, “Digital equalization of time-delay array receivers on coherent laser communications,” Opt. Lett. 42(2), 310–313 (2017).
[PubMed]

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

2016 (5)

2015 (3)

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

2014 (4)

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

A. Belmonte, “Capacity of coherent laser downlinks,” J. Lightwave Technol. 32(11), 2128–2132 (2014).

2013 (4)

2012 (2)

C. Zhang, W. Chang, A. Galvanauskas, and H. G. Winful, “Simultaneous passive coherent beam combining and mode locking of fiber laser arrays,” Opt. Express 20(15), 16245–16257 (2012).

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

2011 (1)

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

2010 (2)

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays,” Opt. Express 18(25), 25873–25886 (2010).
[PubMed]

2009 (1)

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

2006 (1)

Belmonte, A.

Boehmer, K.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Bykhovsky, D.

Cao, J.

Chae, C.

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

Chan, V. W. S.

Chang, W.

Chang, W. Z.

Chen, B.

B. Chen and X. Li, “Bandwidth of adaptive optics system based on stochastic parallel gradient descent algorithm,” Laser Optoelectron. Prog. 50, 030101 (2013).

Chen, M.

C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
[PubMed]

Chen, S.

C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
[PubMed]

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

Chen, Z.

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

Czichy, R.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Dallmann, D.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Feldhaus, T.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Fields, R. A.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Freier, A.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Galvanauskas, A.

Geisler, D. J.

Geng, C.

Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
[PubMed]

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

Gregory, M.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

Greulich, P.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Gu, H.

Hamilton, S. A.

Heine, F.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Hildebrand, U.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Huang, D.

Huang, G.

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

Kaempfner, H.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Kahn, J. M.

Kämpfner, H.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

Kozlowski, D. A.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Lange, R.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Lee, E.

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

Li, F.

Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
[PubMed]

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

Li, J.

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

Li, K.

Li, X.

Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
[PubMed]

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

B. Chen and X. Li, “Bandwidth of adaptive optics system based on stochastic parallel gradient descent algorithm,” Laser Optoelectron. Prog. 50, 030101 (2013).

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

Lin, X.

Liu, C.

C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
[PubMed]

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

Liu, H.

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

Liu, W.

Liu, X.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Lunde, C. T.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Luo, Q.

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

Luo, W.

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

Lv, J.

Lv, Y.

Ma, J.

Ma, P.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Ma, Y.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Meyer, R.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Mu, J.

Muehlnikel, G.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Mueller, J.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Park, J.

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

Qiu, Q.

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

Reinhardt, M.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Robinson, B. S.

Saucke, K.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

Schieler, C. M.

Seel, S.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Smutny, B.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Sterr, U.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Stevens, M. L.

Su, R.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Tan, L.

Tan, Y.

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

Wandernoth, B.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Wang, J.

Wang, L.

Wang, X.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

Weichert, A.

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

Wicker, J. M.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Winful, H. G.

Wong, R. L.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Wu, T. W.

Wu, Y.

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

Xian, H.

C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
[PubMed]

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

Xiao, H.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Xu, X.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

Yang, Y.

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

Y. Yang, C. Geng, F. Li, and X. Li, “Combining module based on coherent polarization beam combining,” Appl. Opt. 56(7), 2020–2028 (2017).
[PubMed]

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

Yao, K.

Yarnall, T. M.

Yoon, G.

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

Yu, S.

Yura, H. T.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

Zhai, C.

Zhang, C.

Zhang, R.

Zhao, G.

Zhao, H.

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

Zhao, X.

Zhou, P.

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

Acta Opt. Sin. (1)

Y. Yang, C. Geng, F. Li, and X. Li, “Research of cascaded coherent combining of fiber lasers based on 3-dB fiber couplers,” Acta Opt. Sin. 35, s106005 (2015).

Appl. Opt. (2)

Chin. Phys. B (1)

W. Luo, C. Geng, Y. Wu, Y. Tan, Q. Luo, H. Liu, and X. Li, “Experimental demonstration of single-mode fiber coupling using adaptive fiber coupler,” Chin. Phys. B 23(1), 014207 (2014).

IEEE J. Sel. Top. Quantum Electron. (1)

R. Su, P. Zhou, X. Wang, Y. Ma, H. Xiao, P. Ma, X. Xu, and X. Liu, “High power narrow-linewidth nanosecond all-fiber lasers and their actively coherent beam combination,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0903913 (2014).

IEEE Photonics J. (1)

F. Li, C. Geng, G. Huang, Y. Yang, X. Li, and Q. Qiu, “Experimental demonstration of coherent combining with tip/tilt control based on adaptive space-to-fiber laser beam coupling,” IEEE Photonics J. 9(2), 7102812 (2017).

IEEE Photonics Technol. Lett. (3)

F. Li, C. Geng, X. Li, and Q. Qiu, “Co-aperture transceiving of two combined beams based on adaptive fiber coupling control,” IEEE Photonics Technol. Lett. 27(17), 1790–1797 (2015).

Y. Yang, C. Geng, F. Li, G. Huang, and X. Li, “Coherent polarization beam combining approach based on polarization controlling in fiber devices,” IEEE Photonics Technol. Lett. 29(12), 945–948 (2017).

J. Park, E. Lee, C. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).

J. Lightwave Technol. (4)

Laser Optoelectron. Prog. (1)

B. Chen and X. Li, “Bandwidth of adaptive optics system based on stochastic parallel gradient descent algorithm,” Laser Optoelectron. Prog. 50, 030101 (2013).

Opt. Commun. (2)

J. Li, H. Zhao, Z. Chen, X. Wang, and X. Xu, “All-fiber active coherent combining via a fiber combiner,” Opt. Commun. 286, 273–276 (2012).

C. Liu, M. Chen, S. Chen, and H. Xian, “Adaptive optics for the free-space coherent optical communications,” Opt. Commun. 361, 21–24 (2016).
[PubMed]

Opt. Express (8)

W. Liu, K. Yao, D. Huang, X. Lin, L. Wang, and Y. Lv, “Performance evaluation of coherent free space optical communications with a double-stage fast-steering-mirror adaptive optics system depending on the greenwood frequency,” Opt. Express 24(12), 13288–13302 (2016).
[PubMed]

J. Cao, X. Zhao, W. Liu, and H. Gu, “Performance analysis of a coherent free space optical communication system based on experiment,” Opt. Express 25(13), 15299–15312 (2017).
[PubMed]

T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays,” Opt. Express 18(25), 25873–25886 (2010).
[PubMed]

C. Zhang, W. Chang, A. Galvanauskas, and H. G. Winful, “Simultaneous passive coherent beam combining and mode locking of fiber laser arrays,” Opt. Express 20(15), 16245–16257 (2012).

D. J. Geisler, T. M. Yarnall, M. L. Stevens, C. M. Schieler, B. S. Robinson, and S. A. Hamilton, “Multi-aperture digital coherent combining for free-space optical communication receivers,” Opt. Express 24(12), 12661–12671 (2016).
[PubMed]

C. Geng, W. Luo, Y. Tan, H. Liu, J. Mu, and X. Li, “Experimental demonstration of using divergence cost-function in SPGD algorithm for coherent beam combining with tip/tilt control,” Opt. Express 21(21), 25045–25055 (2013).
[PubMed]

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[PubMed]

R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18434–18441 (2013).
[PubMed]

Opt. Lett. (1)

Proc. SPIE (3)

B. Smutny, H. Kaempfner, G. Muehlnikel, U. Sterr, B. Wandernoth, F. Heine, U. Hildebrand, D. Dallmann, M. Reinhardt, A. Freier, R. Lange, K. Boehmer, T. Feldhaus, J. Mueller, A. Weichert, P. Greulich, S. Seel, R. Meyer, and R. Czichy, “5.6 Gbps optical intersatellite communication link,” Proc. SPIE 7199, 719906 (2009).

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, and C. T. Lunde, “5.625 Gbps bidirectional laser communications measurements between the NFIRE Satellite and an Optical Ground Station,” Proc. SPIE 8184, 81840D (2011).

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” Proc. SPIE 7587, 75970E (2010).

Other (2)

Z. Ghassemlooy, H. L. Minh, and M. Ijaz, “Free space optical communications,” Optical And Microwave Technologies for Telecommunication Networks, O. Strobel, ed. (Wiley, 2016), Chapter 10.4, pp. 349–377.

G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002).

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Figures (11)

Fig. 1
Fig. 1 Multi-aperture receiver with (a) digital combining architecture, and (b) optical combining architecture. LO: local oscillator.
Fig. 2
Fig. 2 Structural schematic diagram of the combining module with the integrated architecture.
Fig. 3
Fig. 3 Structural schematic diagram of the combining module with the distributed architecture.
Fig. 4
Fig. 4 Performance of the combining module: (a) combining efficiency and (b) combined optical power.
Fig. 5
Fig. 5 Experimental setup of CBC of four laser beams with the integrated architecture.
Fig. 6
Fig. 6 Experimental results of the output optical powers (a) P1; (b) P2; (c) P3; (d) P4.
Fig. 7
Fig. 7 Results of the calculated combining efficiency.
Fig. 8
Fig. 8 Experimental setup of the CBC of four laser beams with the distributed architecture.
Fig. 9
Fig. 9 Experimental results of the output optical powers (a) P1; (b) P2; (c) P3; (d) P4.
Fig. 10
Fig. 10 Results of the calculated combining efficiency.
Fig. 11
Fig. 11 Simulation results of the CBC of four input beams under the turbulent atmosphere conditions of (a) D/r0 = 1, (b) D/r0 = 3, (c) D/r0 = 5, (d) D/r0 = 7, and (e) D/r0 = 9.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

I out1 = 1 2 [ I in1 + I in2 2 I in1 I in2 cos( Δδ+ π 2 ) ]
I out2 = 1 2 [ I in1 + I in2 +2 I in1 I in2 cos( Δδ+ π 2 ) ]
U i = U i +γΔ U i ( J + J )/2
η CBC = P 4 P 1 + P 2 + P 3 + P 4
Δ ϕ rms =2 J rms / J max
U i = U i γΔ U i ( J i+ J i )/2
I i = η i × I f_i
η i = | E O ( r ) F O ( r )ds | 2 | E O ( r ) | 2 ds × | F O ( r ) | 2 ds
E O ( r )=F[ E A ( r ) ]
E A ( r )= E s exp[ jφ( r ) ]
F O ( r )= 2 π ω 0 2 exp( r 2 ω 0 2 )
I f_i = π 4 D 2 × I rr
f( I rr )= 2 ( αβ ) α+β 2 Γ( α )Γ( β ) I rr α+β 2 1 Κ αβ ( 2 αβ I rr ), I rr 0
I sc1 = 1 2 ( I 1 + I 2 +2 I 1 I 2 )
I sc2 = 1 2 ( I 3 + I 4 +2 I 3 I 4 )
I c = 1 2 ( I sc1 + I sc2 +2 I sc1 I sc2 )
η c = I c I 1 + I 2 + I 3 + I 4
CL=10log( I c I 1 + I 2 + I 3 + I 4 )

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