Coherent Beam Combining of Fiber Amplifiers Using ... - IEEE Xplore

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 2, MARCH/APRIL 2009

Coherent Beam Combining of Fiber Amplifiers Using Stochastic Parallel Gradient Descent Algorithm and Its Application Pu Zhou, Zejin Liu, Xiaolin Wang, Yanxing Ma, Haotong Ma, Xiaojun Xu, and Shaofeng Guo

Abstract—We present theoretical and experimental research on coherent beam combining of fiber amplifiers using stochastic parallel gradient descent (SPGD) algorithm. The feasibility of coherent beam combining using SPGD algorithm is detailed analytically. Numerical simulation is accomplished to explore the scaling potential to higher number of laser beams. Experimental investigation on coherent beam combining of two and three W-level fiber amplifiers is demonstrated. Several application fields, i.e., atmosphere distortion compensating, beam steering, and beam shaping based on coherent beam combining using SPGD algorithm are proposed. Index Terms—Beam shaping, beam steering, coherent beam combining, fiber amplifier, stochastic parallel gradient descent (SPGD) algorithm.

I. INTRODUCTION HERE is a continuing desire to scale fiber lasers to higher output powers via coherent beam combining. On one hand, recent advances have shown that fiber lasers have the capability to generate kilowatt output power with good beam quality [1], [2], but scaling up the output power from a singlefiber laser faces significant challenges such as nonlinear effects, thermal loading, fiber damage, as well as brightness of pump LDs. Coherent beam combining of fiber lasers can solve the power limitation while maintaining good beam quality [3]. On the other hand, the coherent combined beams can be densely packed together and transmitted by multiple small apertures such as Cassegrain telescopes or laser collimators [4]–[7]. Multiple small-aperture systems offer a number of important advantages over single apertures for high-power laser systems. The total effective aperture size is not limited by constraints in the process of mirror manufacturing, and the multiple smallaperture transmitting telescope is cost-effective compared with a large-aperture one [8]. Another desirable feature of multiple small-aperture transmitting systems is the potential of allelectrical fast beam steering [9], [10]. It had also been demonstrated that in thermal blooming environments peak irradiances can be improved using two- and four-aperture systems by using wave-optics computer codes [11]. Those advantages have boosted intense research and investigation on coherent beam combining recently.

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Manuscript received September 21, 2008; revised November 17, 2008. Current version published April 8, 2009. The authors are with the College of Optoelectric Science and Engineering, National University of Defense Technology, Changsha 410073, China (e-mail: [email protected]; [email protected]; chinawxllin@163. com; [email protected]; [email protected]; [email protected]; sfguo@ nudt.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTQE.2008.2010231

In coherent combining, several fibers are packaged together into an array, all the array elements operate with the same spectrum, and the relative phases of the elements are controlled [3]. Several approaches like multicore fiber laser [12], intracavity coupling [13]–[16], and master oscillator power amplifier (MOPA) configuration [17]–[26] have been proposed and robust coherent combining of a small number of fiber lasers has been demonstrated. Up to now, the highest-power demonstrations of CBC have involved active phase controlling with MOPA configuration [17], [18], [23]. There are mainly three kinds of phase controlling methods, i.e., heterodyne phase detecting [17]–[21], self-referenced phase detecting [22], [23], and no phase detecting with phase controlling signal generated by using stochastic parallel gradient descent (SPGD) algorithm [24]–[26]. Coherent beam combining using the SPGD algorithm has great potential due to the advantage of less complexity while maintaining the capability in scaling to a large number of beamlets that have been proposed for usage in building a new architecture of highenergy laser system [6]. In a previous study, coherent beam combining using the SPGD algorithm has been demonstrated by Ling Liu [24], [25] and Jan E. Kansky [26]. Nevertheless, to the best of our knowledge, the coherent beam combining experiment using SPGD algorithm has involved seed laser only and no amplifiers, phase distortions are static or quasi-static due to the fiber optical path length differences between the different beam channels, fiber optical path length slow variations and so on, but no fast time-varying phase distortions that often come up in high-power fiber amplifiers [8], [14]. In this paper, we present our recent research on coherent beam combining of fiber amplifiers using the SPGD algorithm and its applications. The rest of this paper is organized as follows. Section II details the feasibility of coherent beam combining using SPGD algorithm analytically. Section III presents numerical simulation to explore the scaling potential to a higher number of laser beams. Experimental investigation on coherent beam combining of two and three W-level fiber amplifiers will be demonstrated in Section IV. Subsequently, Section V details several application fields, i.e., atmosphere distortion compensating, and beam steering and beam shaping based on coherent beam combining using the SPGD algorithm. Conclusions are given in Section VI.

II. BRIEF REVIEW OF THEORY The theory for phase distortion correction based on the SPGD algorithm in adaptive optics system has been reviewed in detail

1077-260X/$25.00 © 2009 IEEE

ZHOU et al.: COHERENT BEAM COMBINING OF FIBER AMPLIFIERS USING SPGD ALGORITHM AND ITS APPLICATION

Fig. 1.

System setup for coherent beam combining using the SPGD algorithm.

the rms value of each beamlet’s optical phase. Throughout the following, we define the various control signals of each beamlet as zero-mean Gaussian variables (independent out u ) = 0, comes of a Gaussian process). So, we have E( N k k =1 E(uk uj ) = 0, E( N δu ) = 0, E(δu δu ) = 0, D(δu ) = k k j k k =1 σφ2 , E( N (u + δu )) = 0, where E and D denote the k k k =1 mathematical expectation and variance, respectively, uk is the control signal applied to the phase modulator, and δuk is the perturbation voltage. After the control signal is applied to the phase modulator, the rms value of the optical phase of all the beamlets is obtained ) = σ 2 (u + ϕ

and presented in [27] and [28]. In this paper, we present the theory for optical phase control using SPGD algorithm specified in the coherent beam combining field. Consider a coherent beam combining system with N channels of beamlets (see Fig. 1). The laser beam from the master oscillator (MO) is split into N channels (in the figure, we show two channels as an example) and coupled to the phase modulators. The laser beams from the phase modulators (PM) are then, sent to the fiber amplifier (AMP) and optical isolator (ISO) and then sent into fiber collimators (CO). It is to be noted that for high-power application, multistage fiber amplifiers are usually required. The beam array is sent to free space via the collimators. The collimated output beams are sampled by a beam splitter. After the splitter, part of the beam is sent to a focusing lens that images the far-field pattern onto the detector. A measured cost function J = J (u) obtained or calculated from the signal collected by the detector, is a function of the control parameters u = {u1 , . . . uN }, which is often the voltages applied to the phase modulators. In the SPGD algorithm, small perturbations {δuj } (j = 1, . . . , N ) are applied simultaneously to the system control parameters {u1 , u2 , . . . , uN }. The perturbations are typically chosen as statistically independent variables having zero mean and equal variances: δuk = 0, δuk δul = σu2 δkl , where δk l is the Kronecker symbol. The resulting change in cost function δJ = J(u1 + δu1 , . . . , uN + δuN ) − J(u1 − δu1 , . . . , uN − δuN ) is used directly for gradient estimation: J¯j = δJδuj . The SPGD algorithm is performed and the control voltages are updated following the equation uj = uj + γδuj δJ, where γ is the update gain, and γ < 0 accords to the procedure of minimization; γ > 0 accords to the procedure of maximization. The iteration for updating the control voltages keeps the cost function J evolving to its extremum. This could be proved as follows. The Strehl ratio (SR), defined as the ratio of peak far-field intensity of a beam divided by the peak intensity from a uniformly illuminated aperture having the same total power, represents perhaps the most commonly used cost function characterizing the efficiency of beam projection [5]. In this section, we define SR as the cost function in the coherent beam combining system. SR will get maximizing optimized when the beamlets are ideally coherent combined and there is constructive interference. According to [29], SR can be written as J = exp(−σ 2 ) = 1 − σ 2 + O(σ 4 ), where σ 2 = σ 2 (ϕ) is

249

N

(uk + ϕk − u − ϕ )2 =

k =1

N

(uk + ϕk )2 .

k =1

(1) The resulting change in cost function after perturbations can be expressed as − δu) − σ 2 (u + ϕ + δu) δJ = J+ − J− = σ 2 (u + ϕ = −4

N

(uk + ϕk ) δuk .

(2)

k =1

If the control voltages are updated as follows (m +1)

uk

(m )

+ γδJδuk

(m )

− 4γ

= uk = uk

(m )

N

(m )

uk

(m )

+ ϕk

(m )

2

δuk

(3)

k =1

then, the resulting change in the cost function can be written as ∆J = J(u(m +1) ) − J(u(m ) ) =

N

(m )

(m )

uk

+ ϕk

2

−

k =1

=

N

(m +1)

uk

(m +1)

2

+ ϕk

k =1

8γσu2

N

(m ) uk

2

+

(m ) ϕk

− 16γ

2

σu4

k =1

= 8α(1 − J

N 2 (m ) (m ) u k + ϕk k =1

(m )

) − 16α (1 − J 2

= 8α (1 − 2α) + (1 − 8α) J

(m )

(m )

)

+ 16α2 J (m )

(4)

where α = γσu2 . If γ is properly selected to ensure 0 < α < 1/2, then ∆J > 0 according to (4). This means that the SPGD algorithm provides an increase in the cost function on average. So, in statistical meaning, the cost function will get its maximum after steps of iterations of the control voltages in the end. III. SIMULATION AND SCALING ANALYSIS In this section, we study the performance of the SPGD algorithm under static noise. Suppose that there are only static phase distortions due to the fiber optical path length differences between the different channels. Numerical calculations will be carried out to correct those phase distortions by finite number of iterations. We will consider three fiber amplifier arrays as an example. As shown in Fig. 2, all laser elements are arranged with equal distance between the elements. Fig. 2 denotes three

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Fig. 2.


Three hexagonal distributed laser beam array.

hexagonal distributed laser arrays containing 3, 7, and 19 lasers, respectively. We suppose that each fiber laser beam has a single Gaussian mode output. The beam waist of each laser is ω0 . Fiber lasers are arranged in an array with the nearest neighbor separated by a distance d. Vacancy factor t = (d − 2ω0 )/ω0 can be used to describe the compactness of the array. A smaller t corresponds to a more compact array. The optical parameters used in calculating are as follows. λ = 1.06 µm, ω 0 = 5 mm, d = 15 mm, and t = 1. In the simulation, we suppose the rms value of static phase distortions for the laser array is σp = 3π rad, which is large enough that the effect of beam combining almost becomes the result of incoherent combining, as according to the criterion proposed in [18]. The parameters used for the SPGD algorithm are chosen as γ = 4 and σu2 = 0.09 after selecting from the best solution from the lot of samples. It is to be pointed out that a phase modulation amplitude of 0.3 rad (corresponding to σu2 = 0.09) has an insignificant effect on the beam combining efficiency according to the criterion proposed in [29]. Also, γ might be a variable value controlled by a supervisory control loop [30]; using adaptive γ led to approximately a 20% improvement in the convergence rate for all the examined controlled algorithms [31]. In this paper, we choose fixed γ just for simplicity. Numerical calculation shows the transition curves from the minimization of SR to the maximization of SR state, as shown in Fig. 3. In this figure, the average curve over 100 transition curves is shown. From Fig. 3, one sees that the SPGD algorithm can converge to the extremum of the cost function for all the three laser arrays. For the 3-laser, 7-laser, and 19-laser arrays, it requires 25, 80, and 200 steps of iterations, respectively, before the convergence. The more lasers in the array, the more steps of iteration that the SPGD algorithm needs to converge to the extremum of the cost function. Formal analysis indicates that the convergence rate typically increases at least as N 1/2 in adaptive optics field, where N is the number of beamlets [27]. We take further calculation on coherent beam combining of several more hexagonal distributed laser beam arrays (i.e., laser array containing 37, 61 lasers) using the SPGD algorithm. The dependence of the convergence rate on the number of beamlets is presented in Fig. 4. It is revealed that in coherent beam combining of N laser beamlets, the convergence rate scales linearly with N , which is different from the N 1/2 relationship. It can be briefly explained that the N 1/2 relationship is reached only if the full channel capacity is exploited. In our calculation, the parameters used for the SPGD algorithm are kept unchanged for the different laser arrays; in fact, there may exist specified optimal parameter pairs for each array. In addition, the N 1/2 relationship originates by heuristic

Fig. 3.

Convergence curve for different laser arrays.

Fig. 4.

Dependence of steps required for convergence for different laser arrays.

reasoning, not strict mathematical deduction [32]. So, in previous numerical and experimental studies on adaptive optics system employing the SPGD algorithm, the N 1/2 relationship is not always necessarily fulfilled (see [[28], Fig. 6], [33], Fig. 2, and [34], Fig. 6). However, the issue that convergence rate will decrease as an increase in the laser number brings great challenge to apply the SPGD algorithm to coherent combining of a relatively large number of high-power fiber amplifiers. Besides, the question remains open as to how rapidly the phase noise worsens as the fiber output power is increased [21]; thus, the bandwidth of phase controlling system should improve when the fiber laser is scaled to kilowatt power level. This means that the effective control bandwidth of a coherent beam combining system using SPGD algorithm decreases with an increase in the laser beam power and the number of beamlets. We think that the solution to this issue can be found in three aspects. First, more advanced SPGD controllers that have a faster updating rate should be investigated. Second, corresponding countermeasures should be taken to suppress the influence of phase distortions.


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ent cost functions. In the experiment, we will take laser power encircled in the pinhole as the cost function used in the SPGD algorithm. A. Coherent Beam Combining of Two Fiber Amplifiers

Fig. 5.

Dependence of different cost functions on phase error.

Fig. 6. Experimental results when system is in open loop. (a) A longtime observation of J . (b) Long-exposure far-field intensity distribution.

Third, make modification to the SPGD algorithm to accelerate the convergence rate when scaled to large-number lasers. IV. EXPERIMENTAL SETUP AND RESULTS In Sections II and III, SR is chosen as the cost function for analytical analysis and numerical simulation. Nevertheless, SR is more complex to obtain in the experiment than the power-inthe-bucket (PIB) target-plane cost function JPIB [4], [5], [25], which is defined as the integral of the target-plane intensity distribution over the on-axis circular area of radius bT that describes the amount of laser energy encircled in this area. In the experiment, JPIB can be easily obtained by locating a pinhole of radius bT before the photoelectricity detector. The radius bT is often at the level of the main-lobe radius b0T = 1.22λ (f /D), where f is the lens focus length, D is defined by a circle with the smallest diameter that contains all the beamlets within it (see Fig. 2, the dashed curves). The dependence of SR and JPIB with different radius on the rms of each beamlet’s optical phase is calculated and presented in Fig. 5. The parameters 1 , are the same as used in Fig. 3 for the 19-laser array. JPIB 2 3 JPIB , and JPIB correspond to pinhole radius equal to 25%, 33% and 50% of b0T , respectively. One sees that dependence on the phase error is on the whole nearly the same for differ-

The experimental setup is the same as shown in Fig. 1. The master oscillator is a distributed feedback (DFB) polarization maintaining Yb-doped fiber laser with 1083 nm wavelength. The linewidth of the oscillator is less than 1 MHz. The laser beam from the master oscillator is split into three channels and coupled to two LiNbO3 phase modulators. The laser beams from the phase modulators are then sent to two fiber amplifiers and an optical isolator and then sent into two fiber collimators placed in parallel. The distance between the centers of each collimator is about 8 mm, and the beam radius of each beamlet is about 1 mm. The output power from each fiber amplifier can be tuned to be more than 1 W. The two collimated output beams are sampled by a cubic beam splitter. After the splitter, part of the beam is sent to a focusing lens with 1 m focusing length that images the central lobe of the far field onto a home-made pinhole with 95 µm radius. An silicon detector is located immediately behind the pinhole. The width of the main lobe of the far-field pattern is calculated to be 160 µm; thus the width of the pinhole is 60% of the main lobe. The optical energy detected is defined as cost function J and will be used in the SPGD algorithm. The object of coherent beam combining is to maximize the energy encircled in the central lobe, i.e., maximize the cost function. Cost function J is shared by an oscilloscope and a digital signal processor (DSP) chip served as the control circult. The curve for cost function as a function of time can be shown in the oscilloscope. Cost function J is acquired into the DSP chip through an A/D converter. The main frequency of the DSP chip is 25 MHz and the updating rate of control signal generated by this chip is pretested to be 16 500 times per s. In our experiment, the SPGD algorithm is performed on a DSP chip and the phase controlling signal is sent to the two LiNbO3 phase modulators by two D/A converters. The other part of the beam after the splitter is also focused by a lens. The charge-coupled device camera is set at the focal plane and can be used to observe the far-field beam profile. In the experiment, the optical power of two amplifiers is tuned to around 1 W and the gain of the SI Amplified detector is tuned so that the averaged cost function J is 0.5V for each single amplifier. When all the three amplifiers are turned on, the whole system is in open loop and the SPGD algorithm is not implemented; J will fluctuate between 0 and 2 V randomly when the system is in open loop and no phase controlling is implemented, as shown in Fig. 6(a), with the data sample rate being 8 ms. The average voltage encircled in the pinhole is 1.09 V, and the long-exposure far-field intensity distribution is shown in Fig. 6(b). When the SPGD algorithm is implemented and the whole system is in close loop, the dependence of J on time is shown in Fig. 7(a), with the data sample rate being 4 ms. For most of the time, the energy encircled in the pinhole can be locked steadily to be more than 1.9 V, which denotes a remarkable increase of energy encircled in the main lobe. The average voltage encircled

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B. Coherent Beam Combining of Three Fiber Amplifiers

Fig. 8.

Spectral density for the intensity signals in open loop and close loop.

in the pinhole is 1.95 V when the system is in close loop. The energy encircled in the pinhole was enhanced by a factor of 1.79 (compared with 1.09 V in the open loop), which is 89% of the ideal coherent combining case (energy encircled in the pinhole was enhanced by a factor of 2 in the ideal case). The longexposure far-field intensity distribution is shown in Fig. 7(b). The contrast of the far-field coherent combined beam profile is calculated to be as high as 98%, where the fringe contrast is defined by the formula (Im ax − Im in )/(Im ax + Im in ), where Im ax and Im in are the maximum optical intensity and the adjacent minimum on the far-field pattern, respectively. One sees from Figs. 6 and 7 that although high-frequency phase noise exists in the experimental environment, coherent beam combining based on the SPGD algorithm is rather robust and the algorithm performs well most of the time. It can be estimated from energy fluctuation shown in Fig. 6(a) that the rate of phase distortion due to the two fiber amplifiers is at hertz level. Fig. 8 shows the spectral density of the intensity signal obtained by the photoelectricity detector in the open loop and close loop using a fast Fourier transform. It is shown that the fluctuations at hertz level are successfully suppressed using the SPGD algorithm.

The system setup for coherent beam combining of three fiber amplifiers is almost the same as that for two fiber lasers except for the number of fiber amplifiers. The three fiber collimators are placed in a symmetry triangle distribution. It is to be pointed out that coherent combining of three laser beams is realized by using only two control channels. This can be explained as follows: different from normal optimization problems, the solution to the optimization for the coherent beam combining results can be preknown, that is, all the beamlets have the same optical phase so that there is constructive interference in the far field. For the standard coherent beam combining procedure using the SPGD algorithm, all the beamlets are phase perturbed at the same time to achieve the ideal solution that all the optical phases are identical when the algorithm converges to a steady state. However, if the optical phase of one laser beamlet is kept not perturbed, for example, the N th laser in the N laser array, the SPGD algorithm can still converge to a steady state corresponding to the extremum of the cost function, when all the optical phases of the other N − 1 laser are the same as the N th laser, whose optical phase is kept constant for all steps of iteration. In the experiment, the optical power of three amplifiers are all tuned to be 800 mW and the gain of the SI-amplified detector is tuned so that the averaged cost function J is 1 V for each single amplifier. When all the three amplifiers are turned on, the whole system is in open loop and the SPGD algorithm is not implemented. The cost function J will be 0 V when the three laser beams are out of phase. On the other hand, J is 9 V when the three laser beams are in phase and coherent-combined. J will fluctuate between 0 and 9 V randomly when the system is in open loop and no phase controlling is implemented, as shown in Fig. 9(a), with the data sample rate at 8 ms. The average voltage encircled in the pinhole is 3.09 V, and the long-exposure far-field intensity distribution is shown in Fig. 9(b). They both denote an effect of incoherent beam combining. When the SPGD algorithm is implemented and the whole system is in close loop, the dependence of J on time is shown in Fig. 10(a), with the data sample rate being 8 ms. For most of the time, the energy encircled in the pinhole can be locked steadily to be more than 8 V. The average voltage encircled in the pinhole is 8.10 V when the system is in close loop. The energy encircled in the pinhole was enhanced by a factor of 2.62 (compared with 3.09 V in the open loop), which is 87% of the ideal coherent


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combining case (in the ideal case, the energy encircled in the pinhole was enhanced by a factor of 3). The long-exposure farfield intensity distribution is shown in Fig. 10(b). The contrast of the far-field coherent combined beam profile is calculated to be as high as 85%. It is to be noted that J will decrease to low values at intervals in the close loop for coherent beam combining of two or three fiber amplifiers. This can be explained by the fact that the optical phase of the laser beam may fluctuate several hundreds of waves. Thus, the phase controlling signal calculated by the DSP chip may exceed the actual range of the phase modulator, which is limited to three to four waves, thus leading to a slight bias in the corrected phase values and the need to reset the phase modulator [35]. The reset events make the laser beam combine incoherently at intervals. This problem can be solved if phase unwrapping is preprocessed to the phase controlling signal in our future endeavors. V. APPLICATIONS In this section, we will study three application fields of coherent beam combining system using the SPGD algorithm. One of the application fields, i.e., atmosphere distortion compensating, will be studied experimentally. The other two fields, i.e., beam steering and beam shaping, will be investigated numerically. A. Atmosphere Distortion Compensating Previous investigation has revealed that coherent beam combining can not only be used to compensate the phase fluctuations inside the laser system but also can compensate the turbulenceinduced piston-type aberrations that are encountered when the combined beam propagates in the turbulent atmosphere. Significant energy increasing in the far-field main lobe can be expected by the experimental results and theoretical analysis [5]. Piston-type turbulence aberration compensating has already been demonstrated based on multidithering approach [36], [37]. We will show in this paper that piston-type turbulence aberration compensating can also be compensated by using the SPGD algorithm. The experimental setup is shown in Fig. 11. In the light path from the collimator to the image system, the atmosphere can be disturbed by high-power fans and air conditions originally used for fresh air circulation of the clean room laboratory. The diameter of the collimator used in this experiment is 5 mm. The system

Fig. 11. Experimental setup for atmosphere distortion compensating using the SPGD algorithm.

Fig. 12. Experimental results for atmosphere distortion compensating. (a) Long-exposure far-field pattern in the open loop. (b) Long-exposure far-field pattern in the close loop.

configuration for atmosphere distortion compensating is shown in Fig. 11. Fig. 12(a) shows the long-exposure far-field intensity pattern when the whole system is in open loop. Note that the far-field pattern is different from that shown in Fig. 6(b) due to the distortions of the turbulence. A clear and stable interference pattern is obtained when the system is in close loop using the SPGD algorithm, as shown in Fig. 12(b). The energy encircled in the target pinhole is calculated to be 1.69 times higher than that in the open loop. The experimental results demonstrate the capability that both the phase fluctuations due to fiber amplifier and atmospheric turbulence on the laser beam propagation path can be compensated using the SPGD algorithm. The spectral density of the intensity signal obtained by the photoelectricity detector in the open loop and close loop is calculated and plotted in Fig. 13. It is shown that the rate of phase distortion is from several hertz to several hundreds of hertz due to distortions on account of turbulence. However, the fluctuations are successfully suppressed using the SPGD algorithm. It is to be noted that there exist limitations for atmosphere distortion compensating system employing system configuration shown in Fig. 11, in the case of which only piston-type phase distortion control can be achieved. However, a huge number of channels are required for practical use if no higher order phase distortion correction modules like tip-tilt-error control is incorporated [7]. The other limitation for the present configuration is the cooperative object (the photoelectricity detector) in the receiving plane, which transmits the cost function signal to the control circuit in real

254

Fig. 13.


Experimental results for atmosphere distortion compensating.

Fig. 14. Simulation of electronic beam steering of an N = 19 element laser array by steering angles of λ/D in each axis using the SPGD algorithm. (a) No steering. (b) Horizontal steering. (c) Vertical steering. (d) Both axes steering.

time. In practical application, target-in-the-loop strategy should be employed and cost function should be obtained in the absence of a cooperative beacon. B. Beam Steering One of the most popular applications of laser beam array would be to enable fast beam steering without mechanical control. This can be realized by controlling the relative optical phases between array elements [9], [10]. The location of the far-field main lobe will shift as the phase shift between array elements across the laser array linearly varies. Generally, the phase shift profile should be pregenerated and applied to each beamlet before the main lobe shifts. In coherent beam combining system using the SPGD algorithm, the phase shift profile is not necessarily pregenerated. The location where the main lobe shifts to will encircle most of the laser energy contained in the laser array. One can define the energy encircled in the target location as the cost function, and the SPGD algorithm can be performed to keep the energy encircled in that position increasing until convergence, and in that case, beam steering is realized. It can be simply said that the beam will steer to where one defines. We perform numerical simulation to validate the capability of beam steering using SPGD algorithm. We restrict our calculation to the 19-laser array as shown in Fig. 2. All the parameters are the same as that used in Fig. 3. Fig. 14 shows the simulation results for the electronic beam steering of an N = 19 element laser array by steering angles of λ/D in each axis. The main lobe steers to the location defined in the far-field plane after about 200 steps of iteration. It is to be noted that due to the length of this paper, the accuracy and range of beam steering using the SPGD algorithm will not be detailed here. C. Beam Shaping Coherent beam combining and beam steering in fact involves wavefront shaping in order to deliver a high brightness beam in

Fig. 15. Simulation of beam shaping of an N = 37 element laser array by using the SPGD algorithm. (a) Ideal hollow-dark beam. (b) Hollow-dark beam numerically obtained by using the SPGD algorithm.

specified location in the far-field plane. Moreover, beam shaping or intensity distribution can also be specified by coherent beam combining system using the SPGD algorithm. In fact, simulated annealing algorithm, which is also one of the optimization algorithms, has been successfully applied to the beam shaping field [38]. In this application, the cost function is defined as the correlation coefficient between the calculated beam profile and the ideal beam profile. The SPGD algorithm iteratively adjusts the control voltages of the independent phase modulators to increase the correlation coefficient until convergence. We perform numerical simulation to validate the capability of beam shaping using SPGD algorithm. We restrict our calculation to the 37-laser hexagonal distributed array. All the parameters are the same as that used in Fig. 3. Hollow-dark beam, which is shown in Fig. 15(a), is taken as an example here; the main goal of beam shaping being to obtain a hollow-dark beam profile by coherent beam combining of 37 lasers. The phase modulators are given randomly initialized control voltages in the simulation. After about 500 steps of iteration, the algorithm converge to its maximum. The final beam profile after beam shaping is shown in Fig. 15(b). The iteration curve for the cost function is shown in Fig. 16. One sees that the algorithm did not converge to the state that the correlation coefficient equals 1, and the calculated beam profile still has some difference from the desired one.


Fig. 16.

Convergence curve for beam shaping using the SPGD algorithm.

However, this is the best solution that the SPGD algorithm can find. VI. CONCLUSION We have presented coherent beam combining of fiber amplifiers using the SPGD algorithm and its application from both theoretical and experimental aspects. The feasibility of coherent beam combining using SPGD algorithm is detailed analytically. Numerical simulation is accomplished to find the steps of iterations required for the algorithm convergent to its extremum scales almost linearly with the number of beamlets to be combined. Several application fields, i.e., atmosphere distortion compensating, beam steering, and beam shaping based on coherent beam combining using the SPGD algorithm are proposed. Experimental investigation on coherent beam combining of two and three W-level fiber amplifiers is demonstrated. Although challenges hold for coherent beam combining of large number and high-power fiber amplifiers using the SPGD algorithm, a solution to this issue may be found by investigating controllers with a faster updating rate and taking corresponding countermeasures to suppress the influence of phase distortions. These results and analyses demonstrate that a coherent beam combining system using the SPGD algorithm may finally lead to a practical multiple-aperture fiber laser system that delivers high-power fiber laser beam while maintaining good beam quality, embedded with the capability of atmospheric turbulence compensating, fast beam steering, and beam shaping. The advantage of beam combining using SPGD algorithm makes it so attractive that we believe it is of great potential in building new architecture of high-energy laser systems. We are presently construing an experimental system for coherent beam combining of more than three fiber amplifiers. REFERENCES [1] Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express, vol. 12, pp. 6088–6092, 2004.

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Pu Zhou received the B.S degree in optical engineering from National University of Defense Technology, Chang Sha, China, in 2005. He is now working toward the Ph.D. degree at the College of Optoelectric Science and Engineering, National University of Defense Technology. His curremt research interests include fiber laser/amplifier technology, and coherent combining of fiber lasers/amplifiers.

Zejin Liu received the Doctor’s degree in optical engineering from National University of Defense Technology, Chang Sha, China, in 1997. He is currently a Professor at the College of Optoelectric Science and Engineering, National University of Defense Technology. His current research interests include high-energy laser technology.

Xiaolin Wang received the B.S degree in optical engineering from the University of Electronic Science and Technology, Cheng Du, China, in 2006. He is now working toward the Ph.D. degree at the College of Optoelectric Science and Engineering, National University of Defense Technology, Chang Sha, China. His current research interests include fiber laser/amplifier technology, electronics technology, and coherent combining of fiber lasers/amplifiers.

Yanxing Ma received the B.S degree in optical engineering from Shanxi University, Tai Yuan, China, in 2006. He is now working toward the M.S. degree at the College of Optoelectric Science and Engineering, National University of Defense Technology, Chang Sha, China. His current research interests include fiber laser/amplifier technology, electronics technology, and coherent combining of fiber lasers/amplifiers.

Haotong Ma received the B.S degree in optical engineering from Shandong University, Ji Nan, China, in 2006. He is now working toward the Ph.D. degree at the College of Optoelectric Science and Engineering, National University of Defense Technology, Chang Sha, China. His current research interests include beam controlling.

Xiaojun Xu received the Doctor’s degree in optical engineering from the National University of Defense Technology, Chang Sha, China, in 2000. He is currently an Associate Professor at the College of Optoelectric Science and Engineering, National University of Defense Technology. His current research interests include solid state laser and adaptive optics.

Shaofeng Guo received the Doctor’s degree in optical engineering from the National University of Defense Technology, Chang Sha, China, in 2002. He is now an Associate Professor at the College of Optoelectric Science and Engineering, National University of Defense Technology. His current research interests include solid state laser and nonlinear optics.