## Abstract

We introduce a novel algorithm used in coherent beam combining: double stochastic approximation based on logic comparison (DSAL). Both the theoretical investigation and the numerical simulation indicated that the DSAL method can be convergent efficiently. The coherent beam combining of two W-level fiber amplifiers experiment shows that the whole system in close-loop performs well for long-time observation and nearly 72% combining efficiency was realized. In the further experiment with some improvement, the combining efficiency increased to 81%. This algorithm has widely potential application in coherent beam combining field.

©2009 Optical Society of America

## 1. Introduction

Recently, coherent beam combining (CBC) of laser with a phase modulator arrays configuration becomes a fast developing technique, due to its potential use in laser radar and energy delivering system [1]. Because of their inherent compact size and high beam quality, Fiber lasers/amplifiers are particularly well-suited to beam combining [2]. Several approaches like multicore fiber laser, intra-cavity coupling and MOPA(master oscillator power amplifier) configuration have been proposed and robust coherent combining of a small number of fiber lasers have been demonstrated [2–13]. Up to now, the highest-power demonstrations of CBC have involved active phase controlling with MOPA configuration [5, 6]. There are mainly three kinds of phase controlling method: heterodyne phase detecting [5–9], self-referenced phase detecting [10, 11], which is an application of multi-dithering technique [12], and intensity detecting which generating phase-controlling signal based on stochastic parallel gradient descent (SPGD) algorithm [13, 14].

Comparing with complex phase detecting system, intensity detecting is more available. However, the feedback could not be used to control the phased array directly. In order to get the phase-controlling signal, special algorithm is necessary. By now, SPGD is the most popular algorithm used in coherent beam combining. In this thesis, we introduce a novel algorithm: double stochastic approximation based on logic comparison (DSAL). Different from the other SA (stochastic approximation) algorithm, like SPGD and SAA (simulated annealing algorithm) [15] used in CBC, the DSAL is totally based on the logical comparison and needs not to keep the gradient information, which makes it more simple and direct. Both the theoretical analysis and the numerical simulation indicated that the DSAL method used in CBC can be convergent, and the distribution of the far-field intensity can be optimized effectively. The experimental investigation on coherent beam combining of two W-level fiber amplifiers also shows that the whole system in close-loop performs well for long-time observation and nearly 72% combining efficiency was realized.

## 2. Theory

Figure 1 is a schematic diagram of the CBC system with MOPA configuration and active phase controlling. The master oscillator provides the seed laser with single frequency and linear polarization. After through a fiber splitter, the energy of the beam is equally distributed into some power amplification (PA) systems with phase modulator (PM) array. The amplified laser beams are collimated and output together. The output is sampled and then focused into a detector with a pinhole in front, size of which is a little less than the main lobe of the focused beam. By using special algorithm, the control signal can be computed and feed back to the PM array. The energy distribution will be optimized if the convergence process is fast enough to compensate the phase noise.

For *N* laser beams combing, the phase of the *k*
_{th} beamlet is a statistically independent random zero-mean Gaussian variables *ϕ _{k}* .We define vector Φ = {

*ϕ*}(

_{k}*k*= 1,2,…

*N*),

*σ*

^{2}

_{Φ}= 〈

*ϕ*

^{2}

_{k}〉 as the variance of

*ϕ*, with the initial

_{k}*σ*

_{Φ}= 2π, due to the uncertainty of the fiber length and the phase noise. The DSAL algorithm, based on parallel stochastic approximation technique, can be described as below:

- Step 1 : For the
*m*_{th}loop, get the cost function*J*=*J*(Φ^{(m)}) from the detector and generate statistically independent zero-mean Gaussian stochastic disturbance*δ*^{(m)}= {*δ*^{(m)}_{k}, (*k*= 1,2,…*N*), All the |*δ*^{(m)}_{k}| are small values with zero mean 〈*δ*^{(m)}_{k}〉 = 0 and equal variances*σ*much less than_{δ}*σ*_{Φ}. - Step 2 : Apply the control signal with the positive disturbance, get the cost function
*J*^{+}=*J*(Φ^{(m)}+*δ*^{(m)}) if*J*^{+}<*J*, update the control signal as Φ^{(m+1)}= Φ*(*+ Φ*m*)^{(m)}and go to the next loop, else, go to next step. - Step 3 : Apply the control signal with the negative disturbance, get the cost function
*J*^{?}=*J*(Φ^{(m)}?*δ*^{(m)}) If*J*^{?}>*J*, update the control signal as Φ^{(m+1)}= Φ^{(m)}?*δ*^{(m)}and go to the next loop, else, go to next step. - Step 4 : As both the positive and the negative perturbation could not improve the energy distribution, it may get an extremum. If the system remains stable for a few loops, compare the cost function
*J*with the optimized value*J*. Remain the signal and go to next loop if_{opt}*J*>_{opt}*J*, else, generate zero-mean statistically independent random perturbations*φ*whose variance is much bigger than*δ*, update the control signal and go to the next loop.

The purpose of this algorithm is reducing the phase variances *σ*
_{ϕ} to the extremum all the
time until the peak far-field intensity get the maximum, meanwhile the far-field energy distribution is optimized.

In order to prove the effect and convergence of DSAL method, we must refers to the work of Mr. C.D. Nabors[16]: the Strehl ratio *S*, defined as the ratio of
intensity of a beam divided by the peak intensity from a uniformly illuminated aperture having the same total power, may be decomposed as below:

In which the *N* emitter array is taken to be *Nx* × *N _{y}* rectangular grid of elements. To a certain system, the second term of formula (1) can be neglect while

*N*=

*N*

_{x}*N*≫ 1. Then we can simplify our cost function

_{y}*J*as formula (2):

Using the unit step function, we can describe the control signal of the *k*
_{th} element generated in step 2 and step 3 for the *m* + 1_{th} loop as formula (3):

Where

In which *ρ* is the correlation coefficient of Φ and *δ* case (a) For the orthogonal or totally independent *δ* and Φ,*ρ* = 0

Because of the limited *N* and phase modulated range (0~2*π*), the two could hardly be totally
independent or orthogonal.

case (b) For |*ρ*| ≤ *σ _{δ}*/

*σ*

_{Φ}case (c) For |*ρ*| > *σ _{δ}*/2

*σ*,

_{Φ}*u*(

*J*

^{±}?

*J*) ≈

*u*(∓2

*ρ*

*σ*

_{Φ}

*σ*

_{δ}) define

*σ*=

_{m}*σ*

^{m}_{Φ}:

From the case(a)(b)(c), we can see that the principle of DSAL is generating small random variables and choosing the useful ones directly, which makes the beams′ phase variance a monotonic decreasing quantity, until the system is already in the well situation (when *σ _{δ}* δ ≡ 2

*π*, initial

*σ*

_{Φ}= 2π, for the maximum

*ρ*= 1, the optimum

*S*≈ 0.92).

Some extreme situation was found during the simulation, while the system would be stable but not optimized. To jump out of those extremum, the series of large random perturbation *φ* = {*φ _{k}*}(

*k*= 1,2…

*N*) and the suitable criterion

*J*in the step 4 are necessary.

_{opt}## 3. Simulation

The simulation of the CBC system in Fig. 1 has been done with different beamlet number *N*. The emitter array is taken to be *N _{x}* ×

*N*rectangular grid of elements, in the centre of which are the circle laser beams with Gaussian intensity distribution, (Fig. 2(a)). Initially, each element had a random statistically independent zero-mean Gaussian phase noise with variance

_{y}*σ*

_{Φ}= 2

*π*, which led to a chaotic far-field intensity distribution (Fig. 2(b)).

Figure 2 exhibits the effect of the DSAL algorithm with different laser array. The perturbation function in the simulation has the same variance *σ _{δ}* = 2

*π*/20, a

*σ*= 2

_{φ}*π*/3.

Obviously, the far-field intensity distribution after 300 loops (Fig. 2(c)) is much more optimized than the initially, all of which are normalized by the ideal far-field maximum intensity with no phase noise.

Figure 3 shows the efficiency of the DSAL algorithm, in which we give the simulation result of 100 converging curve for different arrays with random initial phase noise. The x-coordinate is the number of loop times, while the y-coordinate is the energy in the pinhole, normalized by the ideal maximum without phase noise.

From the mean curve (the red, dashed one) in Fig. 3, we could get the conclusion that, for *N* lasers array CBC (*N* ≤ 25), the average loop times of the DSAL algorithm converging to the optimum (*S* ≥ 0.92) is about 5*N*. Considering that the steps in a single loop may be less than 4, the real converging speed of the DSAL algorithm will be faster.

## 4.Experiment

The experimental setup is shown as Fig. 1 with 2 beamlets. The master oscillator is a distributed feedback (DFB) polarization maintaining Yb-doped fiber laser (made by KEOSPSYS) with 1083 nm wavelength. The linewidth of the oscillator is less than 1MHz. The laser beam from the master oscillator is split into two channels and coupled to two LiNbO_{3} phase modulator (made by Beijing Conquer Optical Technology CO., LTD). The laser beams from the phase modulators are sent to fiber amplifier (IPG photonics, YAD-1K) and optical isolator and then sent into two fiber collimators with 1.4mm aperture. The output power from each fiber amplifier can be tuned to be more than 1W. 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 1m focal length that images the central lobe of the far field onto a home-made pinhole with 95 *μ*m diameter, a SI Amplified detector (THORLABS, PDA36A-EC, with 1.25MHz bandwidth at 10dB gain) is located closely behind the pinhole. The optical energy detected is defined as cost function. *J* and will be used in the DSAL algorithm. Cost function *J* is shared by the Oscilloscope (Tektronix TDS 7154) and the Industrial Personal Computer (IPC, made by EVOC Intelligent Technology CO., LTD). The curve for cost function as a function of time can be shown in the Oscilloscope. Cost function *J* is acquired into the IPC using PCI6221 card (made by NI). The DSAL algorithm is performed on the IPC and the phase controlling signal is sent to the two LiNbO_{3} phase modulator by home-made control circuit.

When the DSAL algorithm is implemented and the whole system is in close-loop (*σ _{δ}* = 2

*π*/20 ,

*J*= 0.7

_{opt}*J*), the dependence of

_{idea}*J*on time is shown in Fig. 4 (b).

Comparing with the open-loop situation shown in Fig. 4 (a), the energy in the pinhole can be locked to be nearly the maximum, which denotes the result of ideal phase controlling and coherent combining. However, *J* will decrease to low values at intervals even in the close-loop. This can be explained as follows. On one hand, we must note that the whole close-loop system works in an ultra-clean lab, the round-the-clock working of the cooling fans at the ceiling of the room will induce high-frequency phase noise, which inevitably decrease sometimes the efficiency of phase controlling based on DSAL algorithm. On the other hand, for the laboratory environment, the optical phase of the laser beam maybe fluctuate several hundreds of waves, thus the phase controlling signal calculated by the IPC may exceed the actual range of the phase modulator, which is limited to several waves. This leads to a slight bias in the corrected phase values and the need to reset the phase modulator. The reset events also case phase noise.

The average voltage encircled in the pinhole is 0.21V when the system is in close-loop. The energy encircled in the pinhole was enhanced by a factor of 1.76 (compared with 0.12V in the open-loop), which is 88% of the ideal coherent combining case (in the ideal case the energy encircled in the pinhole was enhanced by a factor of 2). Considering that the improvement also including the movement of the far-field intensity peak, for the maximum of *J* increasing from 0.27V (open-loop) to 0.32 V (close-loop), both of the data should be normalized by their maximum. The real combining efficiency should be 72%.

Figure 5 plots the probability distribution of *J* by processing the data in Fig. 4. The probability when *J* is larger than 80% of its ideal value is calculated to be 41.2% in the close-loop, while in the open-loop it is only 19.8% from the data presented in Fig. 4. However, the system did not get completely optimized (the ideal combining efficiency is 92%). One probable reason is that the control bandwidth of the IPC (about 1 kHz) is at the same magnitude with the phase noise, so it is hardly to follow the changing.

During the further experiment shown in Fig. 6, in stead of the IPC, we use a DSP (Digital Signal Processing) chip with 25MHz main frequency, which can generate control signal at the updating rate about 16,500 times per second. For clearly imaging, we took off the lens, and the sampled beam has been shared by the detector and a CCD camera (JAI, CV-A10CL). The CCD camera can memorizes 600 pictures for last 10 seconds, so we could get the long-exposure intensity distribution of the system both in open-loop and close-loop (shown in Fig.7(c) (d)). The distance between the collimators and the sampler was about 1m.

Comparing with the open-loop situation, both the long time observation of the cost function *J* and the long-exposure intensity distribution indicated that the system has been obviously optimized in the close-loop, as shown in Fig.7. For long time operation (more than 1 minute, shown in Fig.7 (a) (c)), the average voltage encircled in the pinhole enhanced from 0.53v (open-loop) to 0.86v (close-loop). The energy in the pinhole has been improved by a factor about 1.6, so that the combining efficiency should be nearly 81%.

Figure 8 is the vertical section of the long-exposure intensity distribution (10 seconds) in Fig.7 (b), (d). Both of them indicate that the fringe contrast has been enhanced in the close-loop, which means that the system stability would be improved and the two lasers could be well combined by the DSAL algorithm.

## 5. Conclusion

We introduce the DSAL algorithm as a parallel stochastic approximation method. The principle of this algorithm is reducing the phase variances to the extremum and then jumping out until the peak far-field intensity get the maximum, both of which are realized by stochastic disturbance. The simulation with different laser array shows that the DSAL algorithm can optimize the far-field intensity distribution efficiently. The coherent beam combing experiment of two W-level fiber amplifiers based on DSAL algorithm has been demonstrated. The whole system in close-loop performs well for long-time observation and 72% combining efficiency was realized. After the improvement of the control circuit, the combining efficiency could be increased to 81%.

Comparing with the popular algorithm used in coherent beam combining field, like SPGD and SAA, the DSAL algorithm needs not to keep the gradient or other information, so the preferences would be simple and direct. For N laser CBC, the convergence time of DSAL is about 5*N*, meanwhile, the convergence time of SPGD is a factor of *N*
^{1/2}[17]. In this opinion, we believe that the DSAL algorithm has widely potential use in the coherent beam combing with small number *N*.

## References and links

**1. **T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel.Top. Quantum Electron. **11**, 567–577 (2005). [CrossRef]

**2. **J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE. **6306**, 1–11 (2006).

**3. **Y. Huo, P. K. Cheo, and G. G. King, “Fundamental mode operation of a 19-core phaselocked Yb-doped fiber amplifier,” Opt. Express. **12**, 6230–6239 (2004). [CrossRef] [PubMed]

**4. **A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler, #x201D; Opt. Express. **10**, 1167–1172 (2002). [PubMed]

**5. **G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. **31**, 1247–1249 (2006). [CrossRef] [PubMed]

**6. **J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently Coupled High Power Fiber Arrays,” Proc. SPIE **6102**, 1–5 (2006).

**7. **R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt.Express. **16**, 2015–2022 (2008). [CrossRef] [PubMed]

**8. **E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wichham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt.Lett. **33**, 354–356 (2008). [CrossRef] [PubMed]

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

**10. **T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express. **14**, 12015–12021 (2006). [CrossRef] [PubMed]

**11. **T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quantum Electron. **13**, 480–486 (2007). [CrossRef]

**12. **T. R. O’Meara, “The multidither principle in adaptive optics,” J. Opt. Soc. Am. **67**, 306–315 (1977). [CrossRef]

**13. **L. Liu and M. A. Vorontsov, “Phase-Locking of Tiled Fiber Array using SPGD Feedback Controller,” Proc. SPIE. **5895**, 1–9 (2005).

**14. **L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive Phase-Locked Fiber Array with Wavefront Phase Tip-Tilt Compensation using Piezoelectric Fiber Positioners,” Proc. SPIE. **6708**, 1–12 (2007).

**15. **J. C. Spall, *Introduction to stochastic search and optimization* (John Wiley & Sons, Inc.2003). [CrossRef]

**16. **C. D. Nabors, “Effects of phase errors on coherent emitter arrays” Appl. Opt. **33**, 2284–2289 (1994). [CrossRef] [PubMed]

**17. **M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction” J. Opt. Soc. Am. A **15**, 2745–2758 (1998). [CrossRef]