Abstract

In this manuscript, the influence of perturbative phase noise on active coherent polarization beam combining (CPBC) system is studied theoretically and experimentally. By employing a photo-detector to obtain phase error signal for feedback loop, actively coherent polarization beam combining of two 20 W-level single mode polarization-maintained (PM) fiber amplifiers are demonstrated with more than 94% combining efficiency. Then the influence of perturbative phase noise on active CPBC system is illustrated by incorporating a simulated phase noise signal in one of the two amplifiers. Experimental results show that the combining efficiency of the CPBC system is susceptible to the frequency or amplitude of the perturbative phase noise. In order to ensure the combining efficiency of the unit of CPBC system higher than 90%, the competence of our active phase control module for high power operation is discussed, which suggests that it could be worked at 100s W power level. The relationship between residual phase noise of the active controller and the normalized voltage signal of the photo-detector is developed and validated experimentally. Experimental results correspond exactly with the theoretically analyzed combining efficiency. Our method offers a useful approach to estimate the influence of phase noise on CPBC system.

© 2013 Optical Society of America

1. Introduction

Fiber lasers and amplifiers have attracted intensively attention due to their great potential in scaling to high power level with near diffraction-limited beam quality. However, the output power of the monolithic amplifier/laser will be ultimately limited by several obstacles, including high brightness pump diodes, fiber damage, thermal effects and non-linear effects [1, 2]. Coherent beam combination (CBC) technique provides a prominent approach to break through the limitations above-mentioned. Nowadays, active CBC of continuous-wave configuration had been extended to multi-channel [3] and the 4 kW combined output power had been demonstrated by Massachusetts Institute of Technology (MIT) [4]. Nevertheless, in most of the general CBC configurations, all the emitters were tiled into an array, which induced a portion of power encircled into the side-lobes in the far-field pattern [312], thus inevitably degrades the beam quality and power concentration of the coherently combined beam [5]. As a new technique to overcome the insufficiency of side-lobes in CBC system, coherent polarization beam combining (CPBC) was proposed and validated both in passive and active phasing techniques [1318]. By passive phasing technique, Phua et al. demonstrated CPBC of two watt-level Nd:YVO4 laser with near-perfect combining efficiency (>99%) [13]. Tan et al. reported CPBC of two Ho: YAG lasers to overcome thermal issues and preserve beam quality, and 10 W-level output power was produced with 96% combining efficiency [14]. As for active phasing technique, Uberna et al. validated the feasibility of CPBC technique in four fiber lasers and scaled the output power to 25 W with the combining efficiency of 94% by heterodyne phase control technique [15, 16]. Recently, active CPBC of two 200 W-level PM fiber amplifiers were presented by our group with more than 90% combining efficiency by single-frequency dithering method [17].

Due to the influence of some factors, such as thermal effects, ambient vibrations and nonlinear effects, the optical phases in fiber lasers and amplifiers are continuously changed along with time [1922]. As for active and passive phasing CBC architectures, the property of perturbative phase noise plays a significant impact on the phase locking effect and the power encircled in the central-lobe [23, 24]. In the previous publications, the phase noise properties of amplifier chains in active CBC configurations were measured [4, 1921], and differential pump power control method was proposed to mitigate its adverse effect [21]. Recently, Goodno et al. theoretically examined the phase noise influence on the actively aperture-filled CBC configuration by perturbative analytical method [23]. Li et al. clarified the impact of phase noise on a specific passive CBC architecture experimentally [24]. However, as far as we known, the inefficiency of phase noise on CPBC system has not been studied up to now. Therefore, it is necessary to clarify the influence of perturbative phase noise on the combining efficiency of the CBPC system theoretically and experimentally.

In this manuscript, the impact of perturbative phase noise on the active CPBC system is investigated theoretically and experimentally by active CPBC of two 20 W-level PM single mode amplifiers. By utilizing a phase modulator and a signal generator to generate an artificial phase noise, the influence of perturbative phase noise on the combining efficiency of the CPBC system is studied. Without artificial phase noise, the combining efficiency of the two-channel CPBC system can be higher than 94%. When the artificial phase noise is implemented, the combining efficiency of the system decreases with the increase of the frequency or amplitude of the artificial noise. In order to ensure the combining efficiency of the unit of CPBC system higher than 90%, the competence of our active phase control module for high power operation is also discussed in the end of the manuscript. The relationship between residual phase noise of the active controller and the normalized voltage signal of the photo-detector is developed and validated experimentally, which offers an approach to estimate the influence of phase noise on the combining efficiency of CPBC system.

2. Principle and Experimental Setup

In principle, the fulfillment of active CPBC can be illustrated as follows. Generally, when two linearly polarized beams are injected into a polarization beam combiner (PBC), the polarization state of the combined beam is not linearly polarized and fluctuates along with time due to phase difference between the two beams is uncertain, as is a mixture of the polarization state of two orthogonally polarized beams (as shown in Fig. 1(a)). However, when the phase difference (δ) between the two orthogonal polarizations is locked and set to δ = nπ, where n is an integer, the combined beam is a new pure linear-polarized one (see Fig. 1(b)), thus it can be completely transmitted through the next PBC by using a half wavelength plate (HWP) to rotate the polarization direction of the combined beam. Due to the linear property of the coherently combined beam, it can be further combined with another linearly polarized beam by a PBC, so multi-channel beams can be coherently combined straightforwardly by phase locking [15].

 figure: Fig. 1

Fig. 1 The polarization state of combined beam for (a) without phase control and (b) phase locked.

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The experimental setup to perform CPBC of two 20 W-level single mode PM fiber amplifier chains is demonstrated in Fig. 2. The seed is a linear-polarized and single-frequency (line-width <25 kHz) laser with a central wavelength of 1064.4 nm based on ultra-short-cavity configuration [25]. The seed laser is firstly split into two channels by a 50:50 splitter. After the splitter, each channel is connected by a LiNbO3 phase modulator. The electro-optical bandwidth and half wavelength voltage of the phase modulator are 150 MHz and 2.5 V, respectively. The laser power from the output of each phase modulator is about 5 mW, and the loss of laser power is induced by the insertion loss of the splitter and modulator. After the phase modulator, each fiber channel is amplified to 3 W level by a commercial all fiber single mode PM amplifier chain. As for the 20 W-level amplifier chains, the active fiber is double-clad single mode Yb-doped fiber, which had a 10 μm core diameter and 125 μm inner cladding diameter. The cladding absorption of the double-clad fiber is 5dB/m near 976 nm and 5 m active fiber is used in this stage. A laser diode (45 W maximum power) with 975 nm central wavelength, 105 μm pigtailed fiber is used to pump the active double-clad fiber in each amplifier chain via a PM (2 + 1) × 1 pump combiner. After amplification, the output power in each channel can be scaled to 20 W-level. After amplification, the two laser beams are collimated by two isolator-inserted collimators, and sent to free space for polarization combination.

 figure: Fig. 2

Fig. 2 Experimental setup of the two-channel CPBC system with artificial phase noise. PM: phase modulator; CO1-CO2: collimators; HWP: half wavelength plate; M1-M2: all- reflected mirrors; M3: high-reflected mirror (99.9:0.1); M4:96:4 splitter; PBC1-PBC2: polarization beam combiners; P: linear polarizer; PD-photo-detector. P1: Power meter 1; P2: Power meter 2.

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By rotating the HWPs in front of PBC1 and adjusting the two beams coaxially, they can be combined in PBC1 and injected into the followed components. M1 and M2 are all-reflected mirrors and M3 is a high-reflected (99.9:0.1) mirror. After M3, a fraction of the combined beam is sent to a 96:4 beam splitter (M4). After M4, 96% of the combined power is collected by the camera with some attenuators, and 4% of the transmitted power is injected into a linear polarizer (P) and then collected by a photo detector (PD). The extracted signal of the PD is transformed by the homemade active phase control module for phase control the two phase difference between two collimated beams based on field programmable gate array (FPGA). The majority power reflected by M3 is injected into a polarization extinction ratio test system (form by a HWP and a PBC) to measure the polarization extinction ratio of the coherently combined beam. P1 and P2 are two identical power meters (40 W range by Thorlabs Corporation), and used to measure the power of the two ports of PBC2. In Fig. 2, the phase modulator in the first amplifier chain is connected with a signal generator with the maximum modulation depth of 5 V and maximum modulation frequency of 240 MHz, which is used to generate the artificial phase noise in the CPBC system.

3. Theoretical Analysis of the Experimental Setup

In order to analyze the influence of phase noise on the CPBC system by combining the experimental results, a measurable parameter should be connected with the phase noise of the CPBC system. In this section, we establish the relationship between the residual phase noise of the active phase controller and the normalized energy in the PD. The principle diagram of the experimental setup is shown in Fig. 3. Due to the existence of residual phase noise, the combined beam is not pure linearly polarized, thus a small portion of power will still leak to port3.

 figure: Fig. 3

Fig. 3 The principle diagram of the experimental setup.

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Due to that the emitting beam profiles of single-mode fibers are approximately Gaussian distributions, so the complex fields of the two injected beams can be expressed by the normalized amplitude profile of Gaussian beam as

E1(x,y)=ψ1(x,y)exp(iϕ1)E2(x,y)=ψ2(x,y)exp(iϕ2)

Where

ψ1(x,y)=2Pb1π.1w1.exp[(x2+y2)w12]ψ2(x,y)=2Pb2π.1w2.exp[(x2+y2)w22]
Pb1=|E1(x,y)|2dxdyPb2=|E2(x,y)|2dxdy

Where Pb1 and Pb2 are the output power of the two beams, w1 and w2 are the beam waists of the two fiber laser beams, Φ1 and Φ2 denote the phases of two fiber laser beams, respectively.

Due to that the phase noise in fiber amplifiers varies with time, so its influence on the combining efficiency of the CPBC system is an averaged effect. Therefore, we define the combining efficiency of the experimental setup as the ratio of the maximum average output power in port2 (Pc(t)) and the summation of the two injected laser beams (P0). Defining that δ(t) (δ(t) = Φ1-Φ2) is the phase deviation between the two channels, and T is the investigation time interval of the CPBC system, the phase noise related combining efficiency can be expressed by the equation:

η=<Pc(t)>tP0

Where

<Pc(t)>=12[ψ21(x,y)+ψ22(x,y)]dxdy+12T0T4ξ21+ξ22dt
ξ1(t)=ψ(x,y)1ψ(x,y)2cosδ(t)dxdyξ2=[ψ21(x,y)ψ22(x,y)]dxdy
P0=[ψ21(x,y)+ψ22(x,y)]dxdy

Defining that θ is the polarization angle of the linear polarizer (P), the relationship between the residual phase noise of the active phase control module and the normalized energy (denoted by voltage V(t)) in the PD can be expressed by

cosδ(t)=V(t)κ1cos2θκ2sin2θ2κ1κ2sin(θ)cos(θ)

Where

κ1=Pb1P0κ2=Pb2P0

In general operation, without phase control, by rotating the polarization angle of the linear polarizer (P), the peak-vale fluctuation energy collected in the PD can be maximized. This state is more advantageous for the active phase control system to demodulate the actual phase difference between the two amplifier chains. The active phase control module can lock the transformed voltage in either maximum state or minimum state, and the combining efficiency difference of the CPBC system in these two states can be negligible in the previous experiment.

As a special case that the output power of the first beam (Pb1) is approximately equal to the scaled power of second beam (Pb2), Eq. (4) can be simplified by

η={1T0TV(t)dtphaselockingatmaximumstate1T0T[1V(t)]dtphaselockingatminimumstate

By the analysis aforementioned, we established the relationship between the residual phase noise of the active controller and the normalized voltage signal of the photo-detector, which offers an approach to estimate the influence of phase noise on the combining efficiency of the CPBC system experimentally.

4. Experimental Results and Discussions

4.1 Experimental results and discussions without artificial phase noise

As shown in Fig. 2, after two collimators, the output powers of the two channels are scaled to 17.9 W and 18.2 W, respectively. The extinction ratios of the two amplifier chains are measured to be 26.4 dB and 25.4 dB respectively by a HWP and a PBC, and both of the M2 factors of the two amplifier chains are measured to be within 1.1. As mentioned above, the phase noise properties of the amplifier chains are crucial for active phasing effect. Therefore, in front of phase locking process, the spectral density of power of the phase noise in the two amplifier chains in 3 W level and 20 W level are measured based on coherent detection technique (as similar in Ref [19, 20].), which is shown in Fig. 4.

 figure: Fig. 4

Fig. 4 The spectral density of power of phase noise in the two amplifier chains. (a) First amplifier in 3 W level. (b) First amplifier in 20 W level. (c) Second amplifier in 3 W level. (d) Second amplifier in 20 W level.

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From the measurement phase noise information, we conclude that the characteristic frequencies of the two 3 W and 20 W level amplifier chains are below 260 Hz and 400 Hz, respectively. By investigation, the envelopes of the spectral density of power below 260 Hz are mainly induced by cooling fans inserted into the commercial amplifiers. When the 20 W level amplifiers performed, some relative small (compared with frequency below 260 Hz) envelopes emerged between 260 Hz and 400 Hz, which is mainly induced by the influence of implementation of water tank in the main amplifiers.

The active CPBC of the two amplifier chains is implemented based on our single-frequency dithering algorithm processor [26]. The mathematical principle of single frequency dithering technique can be found in Ref [26]. and it is not repeated here to save place. The metric function of the phasing algorithm is related to the amplitude of the voltage signal transformed by the PD. The voltage signal transformed by the PD incorporates the information of phase difference between the two channels, which can be used to generate the phase control signal by modulation and demodulation technique. In the presented experimental system, each iteration cycle of the algorithm works as follows. Assuming that t denotes time, and t0, t1 and t2 denote different moment. In the experiment, a 1MHz sine wave phase modulation signal is employed in the phase control module. In the interval of t0 to t1, the phase modulation signal and control signal S1 is added to the phase modulator in the first channel, while the phase control signal of the second channel remains zero. By calculation, the control signal S1 is proportional to sin(δ), as is similarly described in Ref [26]. except for the proportional coefficient. At the interval of t1 to t2, the phase control signal S2 (proportional to sin(-δ)) are applied to the phase modulator in the second channel. During this period, the modulation signal of the first channel becomes zero, and the control signal retains the value at t = t1. Provided that the interval T = t2-t0 is short enough so that the phases of two channels experience negligible fluctuations but long enough for the feedback process to be operative, the phase difference between the two channels will be compensated by repeating the above-operations in turns. Finally, the phase difference between the two channels can be compensated to be zero by selecting the appropriate feedback gain in the phase control module.

Before utilizing the artificial phase noise in the experimental setup, we investigate the active phase control process and phase noise suppression process of the two amplifier chains. The phase control and phase noise suppression process are shown in Fig. 5. Without the phase control module, the normalized energy collected by the PD fluctuates randomly due to that the phase difference between two amplifier chains continuously changes with the influence of thermal effects, experimental vibrations and so forth (shown in Fig. 5(a)). However, when the phase control module performed, the normalized energy in the PD can be locked effectively, which indicates that the phase difference between two amplifier chains is locked to be . From Fig. 5(b), we conclude that the spectral density of power of the phase noise below 400 Hz is efficiently suppressed in the closed loop, and no significant phase noise increases compared to the spectral density of power in single amplifier chains.

 figure: Fig. 5

Fig. 5 Time series signals and spectral density of energy encircled in the pinhole in open loop and closed loop. (a) Time series signals. (b) Spectral density.

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The phase control process of the experimental CPBC system can also be reflected by the change of intensity profiles collected by the camera. In the experiment, when the CPBC system is in the open loop, the intensity profile at the camera is changed along with time and the encircled power values in P1 and P2 are unsteady due to the undefined phase difference between two beams. Figure 6(a) plots three snapshots of the intensity pattern when the system is in the open loop. When the single-frequency dithering algorithm is implemented and the whole system is in the closed loop, the intensity profile and the combined output power is steady. The intensity profile in closed loop is shown in Fig. 6(b). The power meter values of P1 and P2 are 34.1 W and 1.9 W, respectively. Due to the limited polarization extinction ratios of the two amplifier chains, a small portion of the power (86.2 mW in the experiment) is leaked out in the other port of the PBC1. Considering that the output power of the two amplifier chains are 36.1 W, so the combining efficiency of the whole system is more than 94.4%.

 figure: Fig. 6

Fig. 6 The intensity profiles with the system in open loop and closed loop. (a) Three intensity profiles in open loop. (b) Intensity profiles in closed loop with phase locking to minimum (left) and maximum (right) state.

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In the present CPBC system, inefficiency induced by some imperfections is about 5.6%. By Eq. (10), we estimated that inefficiency caused by phase noise is 3.5%. Inefficiency induced by other factors, such as polarization extinction ratio, beam size error, beam quality of the injected beams, and coaxial error, is 2.1%. Specifically, due to that the polarization extinction ratios of the two beams are 26.4 dB and 25.4 dB, so its influence on combining efficiency is within 0.4%. According to the M2 factor data, the beam widths of two amplifier chains are 7.4 mm and 7.8 mm, respectively. By calculation using previous theoretical analysis, the beam size error induced inefficiency is 0.9%. The residual inefficiency (<1%) may be induced by the beam quality and the coaxial error in the experimental setup. By the analysis above, we show that the phase noise is a significant adverse factor in single mode CPBC system.

4.2 Experimental results and discussions with artificial phase noise

As above depiction, the phase noise property of the amplifier chain has crucial impact on the CBC and CPBC system. In the following section, the influence of the frequency and amplitude of perturbative phase noise on CPBC system will be clarified specifically. The artificial phase noise is generated by a signal generator to impose modulation signals to the phase modulator in the first amplifier. The amplitude of the artificial phase noise can be adjusted by altering the modulation depth of the phase modulator, and the frequency of the artificial phase noise is controlled by the signal generator. In order to verify the theoretical analysis, the measurable combining efficiency (directly detected by power meters) is compared with the theoretical analyzed combining efficiency by normalized voltage in the PD in different cases.

From the theoretical analysis above, the combining efficiency of the experimental configuration can be obtained by the normalized voltage in the PD by the expression

ηtotal=ηη0

Where η is the combining efficiency just considering the phase noise, which is expressed by Eq. (10). η0 is the inefficiency caused by other imperfections (2.1% in the experiment), which is considered in the following experimental combining efficiency analysis. In the experiment, the voltage collected in the PD is a set of discrete values, so Eq. (10) should be discrete for analyzing the combining efficiency of the system, which can be simply expressed by

η={1Ni=1NV(i)phaselockingatmaximumstate1Ni=1N[1V(i)]phaselockingatminimumstate

Where N is the sampling number of the investigated voltage signal.

Firstly, we investigate the amplitude of artificial phase noise on CPBC system with different phase modulation frequencies. At the present time, the specific phase noise model of fiber lasers and amplifiers has not been established for its complexity. Due to that the sine signal is an elementary basic for all the time changing phase noise, we employ a signal generator to impose a sine signal to simulate the phase noise in the CPBC system in the experiment. This process can be fulfilled by altering the modulation depth added in the phase modulator. Due to that the half wavelength voltage of the modulator is 2.5 V, we denoted the modulation depth by sign d, and its unit is set to wavelength (λ). In the experiment, we measured the change of normalized voltage signals along with the modulation depths in 200 Hz and 400 Hz modulation frequencies, which are shown in Figs. 7 and 8, respectively. The sample interval of the voltage signal is 0.002 s in the experiment.

 figure: Fig. 7

Fig. 7 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 200 Hz.

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 figure: Fig. 8

Fig. 8 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 400 Hz.

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Figures 7 and 8 reveal that the normalized voltage signal in the closed loop gradually fluctuates sharply along with increase of the modulation depth and modulation frequency of the phase modulator. After obtaining the normalized voltage in the PD in closed loop, the combining efficiency change with the modulation depth can be estimated by Eqs. (11) and (12). In the experiment, the combining efficiency of the system is also measured by power meters. The contrast of the estimated and measured combining efficiency is shown in Fig. 9.

 figure: Fig. 9

Fig. 9 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation frequency: 200 Hz. (b) Modulation frequency: 400 Hz.

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According to Fig. 9, we see that the estimated combining efficiency of the CPBC system and the measured combining efficiency by power meters matches well (matching error <1%) along with the increase of modulation depth. By Fig. 9, we conclude that the combining efficiency of the CPBC system is sensitive to the amplitude of phase noise. Moreover, with the increase of the frequency of the artificial phase noise, the impact of the amplitude of perturbative phase noise on combining efficiency is aggravated gradually. Specifically, as for 200 Hz modulation frequency, in the increase of modulation depth, the combining efficiency decreases from 94.6% to 90.1%, while 400 Hz modulation frequency imposed, the combining efficiency decreases from 94% to 79.4%. The tiny difference of the two initial combining efficiencies may be induced by the instability of the adjustment platforms for long time operation.

It could be concluded that the combining efficiency of the CPBC system is also sensitive to the frequency of phase modulation. In this section, we investigate the frequency of artificial phase noise on CPBC system with different phase modulation depths. In the experiment, we also measured the change of normalized voltage signals along with the modulation frequency (denoted by f in the figures) in λ/4 and λ/2 modulation depths, which are shown in Figs. 10 and 11, respectively. The sample interval of the normalized voltage in the PD is also set to 0.002 s.

 figure: Fig. 10

Fig. 10 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/4.

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 figure: Fig. 11

Fig. 11 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/2.

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Figures 10 and 11 also reveal that the normalized voltage signal in the closed loop gradually fluctuates sharply along with increase of the modulation frequency and modulation depth of the phase modulator. By using the normalized voltage in the PD in closed loop, the combining efficiency change with the modulation frequency can be estimated by Eqs. (11) and (12), which can be also compared with the measured combining efficiency by power meters. The contrast of the estimated and measured combining efficiency is shown in Fig. 12.

 figure: Fig. 12

Fig. 12 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation depth: λ/4. (b) Modulation depth: λ/2.

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According to Fig. 12, we conclude that the estimated combining efficiency by the normalized energy in the PD and the measured combining efficiency by power meters matches well along with the increase of modulation frequency. From Fig. 12, we can conclude that the combining efficiency of the CPBC system is sensitive to the phase noise frequency. Similar to the prior trend, along with the increase of the amplitude of the artificial phase noise, the influence of the frequency of perturbative phase noise on combining efficiency is aggravated gradually. Specifically, as for λ/4 modulation depth, with the increase of modulation frequency, the combining efficiency declines from 93.8% to 77%. However, when λ/2 modulation depth implemented, the combining efficiency decreases from 94.3% to 58.4%.

By comparing the estimated combining efficiency and the measured combining efficiency in the circumstance of different phase noise frequency and amplitude above, we verified the validity of theoretical analysis in the previous section. More importantly, it offers a robust and simple technique to evaluate the influence of perturbative phase noise on the combining efficiency of the CPBC system by the measurable parameter (normalized voltage in the PD) in the experiment.

4.3 Evaluation the active phase control module for high power operation

In the previous analysis and discussion, we illustrate that the combining efficiency of the single mode CPBC system is mainly influenced by the amplitude and frequency property of the perturbative phase noise. In this section, we evaluate the active phase control module for high power operation. We choose 90% combining efficiency as the criteria to demarcate the performance of the active controller.

By adjusting the modulation depth of the artificial phase noise, the trend of combining efficiency of the experimental setup along with modulation frequency is obtained, which is shown in Fig. 13(a). As the specified (90%) criteria, the required finitude band frequency (fr) of the phase noise of the amplifier chain in the circumstance of different modulation depth is demonstrated, which is shown in Fig. 13(b).

 figure: Fig. 13

Fig. 13 (a) The trend of combining efficiency along with modulation frequency for different phase modulation depth. (b) The required finitude band frequency (fr) of the phase noise of the amplifier chain in the circumstance of different modulation depth.

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By the analysis, we show that the required finitude band frequency (fr) of the phase noise of the amplifier chain decreases rapidly with the increase of amplitude of phase noise. To ensure the combining efficiency of the system more than 90%, the large amplitude phase noise should be distributed below 200 Hz. According to the measuring results, the phase fluctuating frequency of a large-mode-area ytterbium-doped fiber amplifier at hundreds watts output power in a relatively quiet laboratory environment is well below several hundred Hz [4, 19], and the characteristic frequency is below 200 Hz [4]. Therefore, our active phase control module is competent in the circumstance that the output power of single amplifier chain is in hundreds level, which had been confirmed in the kW-level CBC configuration [26].

5. Conclusion

The impact of perturbative phase noise on active CPBC system is investigated in an all around way. Without artificial phase noise, two 20 W level single mode and PM fiber amplifier chains are actively coherent polarization beam combined with more than 94% combining efficiency. By imposing the artificial phase noise, the influence of amplitude and frequency property of perturbative phase noise on the active CPBC system is clarified theoretically and experimentally. The estimated combining efficiency by the normalized energy in the PD matches well with the experimentally measured results, which confirms the availability of the technique to appraise the influence of phase noise on the CPBC system. By discussing the competence of the active phase controller, it could be predicted that amplifiers with several hundred watts level output power can be employed for further power scaling the CPBC system.

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15. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010). [CrossRef]   [PubMed]  

16. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010). [CrossRef]   [PubMed]  

17. P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013). [CrossRef]  

18. P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012). [CrossRef]  

19. D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007). [CrossRef]   [PubMed]  

20. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004). [CrossRef]   [PubMed]  

21. H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012). [CrossRef]   [PubMed]  

22. M. Tröbs, S. Barke, T. Theeg, D. Kracht, G. Heinzel, and K. Danzmann, “Differential phase-noise properties of a ytterbium-doped fiber amplifier for the Laser Interferometer Space Antenna,” Opt. Lett. 35(3), 435–437 (2010). [CrossRef]   [PubMed]  

23. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010). [CrossRef]   [PubMed]  

24. Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012). [CrossRef]  

25. S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb³⁺-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011). [CrossRef]   [PubMed]  

26. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011). [CrossRef]   [PubMed]  

References

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    [Crossref] [PubMed]
  5. P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
    [Crossref] [PubMed]
  6. G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
    [Crossref]
  7. T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
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    [Crossref]
  9. X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
    [Crossref] [PubMed]
  10. C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
    [Crossref]
  11. E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
    [Crossref]
  12. J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
    [Crossref] [PubMed]
  13. P. B. Phua and Y. L. Lim, “Coherent polarization locking with near-perfect combining efficiency,” Opt. Lett. 31(14), 2148–2150 (2006).
    [Crossref] [PubMed]
  14. L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
    [Crossref] [PubMed]
  15. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010).
    [Crossref] [PubMed]
  16. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010).
    [Crossref] [PubMed]
  17. P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
    [Crossref]
  18. P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
    [Crossref]
  19. D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
    [Crossref] [PubMed]
  20. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004).
    [Crossref] [PubMed]
  21. H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
    [Crossref] [PubMed]
  22. M. Tröbs, S. Barke, T. Theeg, D. Kracht, G. Heinzel, and K. Danzmann, “Differential phase-noise properties of a ytterbium-doped fiber amplifier for the Laser Interferometer Space Antenna,” Opt. Lett. 35(3), 435–437 (2010).
    [Crossref] [PubMed]
  23. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
    [Crossref] [PubMed]
  24. Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
    [Crossref]
  25. S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb³⁺-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
    [Crossref] [PubMed]
  26. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
    [Crossref] [PubMed]

2013 (1)

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

2012 (4)

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
[Crossref] [PubMed]

H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
[Crossref] [PubMed]

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

2011 (5)

2010 (4)

2009 (2)

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

2008 (2)

2007 (4)

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (1)

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

2004 (1)

Anderegg, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Augst, S. J.

Baker, J. T.

Barke, S.

Barthelemy, A.

Barty, C. P. J.

Beach, R. J.

Bellanger, C.

Benham, V.

Bochove, E. J.

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

Bourderionnet, J.

Bratcher, A.

Brignon, A.

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Chen, D. D.

Chen, Z. L.

Cheung, E. C.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Chua, C. F.

Corcoran, C. J.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

Culpepper, M. A.

Danzmann, K.

Dawson, J. W.

Desfarges-Berthelemot, A.

Dong, X.

Du, W.

Durville, F.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

Fan, T. Y.

Feng, Y.

Goldizen, K. C.

Goodno, G. D.

G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
[Crossref] [PubMed]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Guo, S.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Hammons, D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

He, B.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Heebner, J. E.

Heinzel, G.

Hickey, L. M. B.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Horley, R.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Injeyan, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Jeong, Y.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Jones, D. C.

Kermene, V.

Komine, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Kracht, D.

Leng, J.

Leng, J. Y.

Lhermite, J.

Li, Z.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Lim, Y. L.

Liu, C.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Liu, Z.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Liu, Z. J.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
[Crossref] [PubMed]

Long, W. H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Lou, Q.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Lu, C. A.

Ma, H.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Ma, P. F.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Ma, Y.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Ma, Y. X.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

McClellan, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

McNaught, S. J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Messerly, M. J.

Murphy, D. V.

Nelson, D. J.

Neumann, J.

Nilsson, J.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Pax, P. H.

Payne, D. N.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Peng, M. Y.

Phua, P. B.

Pilkington, D.

Primot, J.

Qi, Y.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Qian, Q.

Qiu, J. R.

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Redmond, S. M.

Rothenberg, J. E.

Sahu, J. K.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Sanchez, A.

Sanchez, A. D.

Scott, A. M.

Shakir, S. A.

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

Shay, T. M.

Shen, S. X.

Shih, C. C.

Shverdin, M. Y.

Si, L.

Siders, C. W.

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Sollee, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Spring, J.

Sridharan, A. K.

Stacey, C. D.

Stappaerts, E. A.

Su, R. T.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Tan, L. H.

Theeg, T.

Tiemann, B. G.

Tröbs, M.

Tünnermann, H.

Turner, P. W.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Uberna, R.

Wang, X.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Wang, X. L.

Ward, B.

Weber, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Wei, X. M.

Weiss, S. B.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Weßels, P.

Wickham, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Xiao, H.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

Xu, S. H.

Xu, X.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Xu, X. J.

Xue, Y.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Yang, Z. M.

Yu, C. X.

Zhang, Q. Y.

Zhang, W. N.

Zhao, Y.

Zhou, J.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Zhou, P.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
[Crossref] [PubMed]

Zhu, J.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (2)

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Laser Phys. Lett. (1)

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (11)

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb³⁺-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
[Crossref] [PubMed]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
[Crossref] [PubMed]

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

P. B. Phua and Y. L. Lim, “Coherent polarization locking with near-perfect combining efficiency,” Opt. Lett. 31(14), 2148–2150 (2006).
[Crossref] [PubMed]

L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
[Crossref] [PubMed]

D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
[Crossref] [PubMed]

S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004).
[Crossref] [PubMed]

H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
[Crossref] [PubMed]

M. Tröbs, S. Barke, T. Theeg, D. Kracht, G. Heinzel, and K. Danzmann, “Differential phase-noise properties of a ytterbium-doped fiber amplifier for the Laser Interferometer Space Antenna,” Opt. Lett. 35(3), 435–437 (2010).
[Crossref] [PubMed]

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Figures (13)

Fig. 1
Fig. 1 The polarization state of combined beam for (a) without phase control and (b) phase locked.
Fig. 2
Fig. 2 Experimental setup of the two-channel CPBC system with artificial phase noise. PM: phase modulator; CO1-CO2: collimators; HWP: half wavelength plate; M1-M2: all- reflected mirrors; M3: high-reflected mirror (99.9:0.1); M4:96:4 splitter; PBC1-PBC2: polarization beam combiners; P: linear polarizer; PD-photo-detector. P1: Power meter 1; P2: Power meter 2.
Fig. 3
Fig. 3 The principle diagram of the experimental setup.
Fig. 4
Fig. 4 The spectral density of power of phase noise in the two amplifier chains. (a) First amplifier in 3 W level. (b) First amplifier in 20 W level. (c) Second amplifier in 3 W level. (d) Second amplifier in 20 W level.
Fig. 5
Fig. 5 Time series signals and spectral density of energy encircled in the pinhole in open loop and closed loop. (a) Time series signals. (b) Spectral density.
Fig. 6
Fig. 6 The intensity profiles with the system in open loop and closed loop. (a) Three intensity profiles in open loop. (b) Intensity profiles in closed loop with phase locking to minimum (left) and maximum (right) state.
Fig. 7
Fig. 7 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 200 Hz.
Fig. 8
Fig. 8 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 400 Hz.
Fig. 9
Fig. 9 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation frequency: 200 Hz. (b) Modulation frequency: 400 Hz.
Fig. 10
Fig. 10 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/4.
Fig. 11
Fig. 11 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/2.
Fig. 12
Fig. 12 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation depth: λ/4. (b) Modulation depth: λ/2.
Fig. 13
Fig. 13 (a) The trend of combining efficiency along with modulation frequency for different phase modulation depth. (b) The required finitude band frequency (fr) of the phase noise of the amplifier chain in the circumstance of different modulation depth.

Equations (12)

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

E 1 (x,y)= ψ 1 (x,y)exp(i ϕ 1 ) E 2 (x,y)= ψ 2 (x,y)exp(i ϕ 2 )
ψ 1 (x,y)= 2 P b1 π . 1 w 1 .exp[ ( x 2 + y 2 ) w 1 2 ] ψ 2 (x,y)= 2 P b2 π . 1 w 2 .exp[ ( x 2 + y 2 ) w 2 2 ]
P b1 = | E 1 (x,y) | 2 dxdy P b2 = | E 2 (x,y) | 2 dxdy
η= < P c (t) > t P 0
< P c (t)>= 1 2 [ ψ 2 1 (x,y)+ ψ 2 2 (x,y) ] d x d y + 1 2T 0 T 4 ξ 2 1 + ξ 2 2 d t
ξ 1 (t)= ψ (x,y) 1 ψ (x,y) 2 cosδ(t) d x d y ξ 2 = [ ψ 2 1 (x,y) ψ 2 2 (x,y) ] d x d y
P 0 = [ ψ 2 1 (x,y)+ ψ 2 2 (x,y) ] d x d y
cosδ(t)= V(t) κ 1 cos 2 θ κ 2 sin 2 θ 2 κ 1 κ 2 sin(θ)cos(θ)
κ 1 = P b1 P 0 κ 2 = P b2 P 0
η={ 1 T 0 T V(t) d t phase locking at maximum state 1 T 0 T [ 1V(t) ] d t phase locking at minimum state
η total =η η 0
η={ 1 N i=1 N V(i) phase locking at maximum state 1 N i=1 N [ 1V(i) ] phase locking at minimum state

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