## Abstract

We explore, by means of experiments and simulation, the power combining efficiency and power fluctuation of coherently phased 2, 4, 6, 8, 10, 12, 14, 16-channel fiber-laser arrays using fused 50:50 single-mode couplers. The measured evolution of power combining efficiency with array size agrees with simulations based on a new propagation model. For our particular system the power fluctuations due to small wavelength-scale length variations are seen to scale with array size as ${N}^{3}$. Beat spectra support the notion that a lack of coherently-combined supermodes in arrays of increasing size leads to a decrease in combined-power efficiency.

© 2010 OSA

## 1. Introduction

There has been much interest in passive coherent phasing of fiber laser arrays as a possible path for multi-kW power scaling. In principle, passive beam combining of an *N*-channel fiber laser array can be regarded as an interferometric system of *N* coupled amplifiers in a composite cavity. The multiple longitudinal modes of individual fiber lasers of varying lengths are superposed to form coherently-combined modes (or supermodes) of the composite cavity whenever there is a coincidence in the individual frequency combs. As the number of elements in the array increases, the probability of finding such an accidental coincidence in the resonances of the array system is decreased, and thus the combined-power efficiency drops.

Several methods have been proposed for passive coherent phasing, including distributed evanescent coupling [1], discrete directional coupling [2,3], and the use of self-Fourier cavities [4]. However, the most important question associated with this beam combining approach is how the coherent-combing efficiency scales with the array size. The initial experimental explorations using a fixed 8-channel array by Shirakawa et al [2] indicate that combining efficiency is expected to decrease with the increase of the array size. This appears to be supported by theoretical estimates as well [5–8]. However, due to the limited experimental data and the approximate character of the first theoretical estimates [6–8], this passively-phased array size scaling is still not sufficiently understood.

In this paper we present a systematic experimental and simulational study of 2- to 16-channel fiber-laser array coherent phasing. We find good agreement between the experimental combining efficiencies and the results of simulations using a new propagation model. We also explore for the first time the important question of the dependence of power fluctuations on array size. Finally, the beat spectra are studied to provide supportive evidence for the diminishing probability of finding supermodes in a larger array size.

## 2. Experimental configuration

As a model system for exploring fiber-laser array passive-coherent phasing we choose an all-single-mode-fiber configuration, where the combining is accomplished using 50:50 single-mode fiber couplers [2]. This enables a simple and easily scalable experimental implementation, with unambiguous beam-combining interpretation. The experimental setup is shown in Fig. 1 as an example for 16-channel combining. Each single-mode fiber laser channel consists of a 980/1064-nm WDM, connected to a 3.5-m long Yb-doped single-mode fiber with a 1064-nm Faraday mirror at one end of the cavity. These laser channels are combined into various-sized arrays using 50:50 single-mode couplers. The basic building block is a 2-laser array, thus all array sizes between 2 and 16 are explored as multiples of 2 (2, 4, 6, 8, 10, 12, 14, and 16). The individual configurations are arranged as in Fig. 2 with the total lengths of the 2, 4, 8, 16-channel lasers being 8.5m, 10.5m, 12.5m, and 14.5m, respectively. The output-end of the cavity of this array is formed by a single straight-cleaved fiber-end, providing ~4% back reflection. All the other remaining output leads of 50:50 fiber couplers are angle-cleaved to prevent feedback from these ends. During experiments an optimized coherent combination has been attained by balancing pumping power for each 2-channel building block such that power equality of two inputs of each fused coupler is achieved. Due to the broad-band nature of Faraday mirrors, each laser channel was operating at ~8nm spectral bandwidth.

## 3. Power combining efficiency

Taking ${P}_{out}$ as the output of the straight-cleaved end and ${P}_{i}$ as the power from the ${i}^{th}$ single laser if uncoupled, we define the power combining efficiency for an *N*-channel array as

Number of Channels | Measured Combining Efficiency (Fluctuation) | Calculated Efficiency |
---|---|---|

2 | 0.98 ( ± 1.5%) | 0.997 |

4 | 0.963 ( ± 2%) | 0.974 |

6 | 0.913 ( ± 2.5%) | 0.94 |

8 | 0.888 ( ± 4%) | 0.89 |

10 | 0.816 ( ± 8%) | 0.81 |

12 | 0.752 ( ± 12%) | 0.74 |

14 | 0.665 ( ± 16.75%) | 0.662 |

16 | 0.542 ( ± 27.5%) | 0.527 |

The calculated power combining efficiencies (green solid dots) are obtained from a recently published model that accounts for the multiple longitudinal modes of individual fiber lasers, the formation of the composite-cavity modes, and the natural selection of the resonant arrays modes that have minimum loss [9]. Since it is based on the amplifying nonlinear Schrödinger equation, effects such as gain saturation, fiber nonlinearity, group velocity dispersion, and loss dispersion of bandwidth limiting elements in the cavity can be readily taken into account (see Appendix). This new model also exhibits self-adjustment process of beam combining suitable for describing the dynamic features such as power fluctuation and beat spectra.

In fitting the theoretical calculations to the experiment, we had to account for the fact that the individual fiber lengths are not precisely known, as a result of occasional fiber breakage during assembly and splicing and connector uncertainties. We estimate an uncertainty of about 2% in the nominal lengths of the individual fiber amplifier channels. In the simulation, for a given number *N* of amplifying channels, a set of lengths *N* were randomly generated that varied within 2% of the nominal length. The simulated combining efficiency is the best fit of several realizations of length distributions, averaged over the fluctuations described in Se. 4. From Fig. 3, the simulated and experimental results agree very well and both of them indicate a clear evolution of combined-power efficiency with array size: power combining efficiency decreases monotonically with array size.

Prior to this work there have been only three published experimental data points regarding the scalability of this particular scheme of passive beam combining [2]. Two of those points are for 2-and 4-element arrays with a spectral bandwidth of 0.6 nm imposed by fiber Bragg gratings. The third point is for an 8-element array with a broadband mirror and a sprectral bandwidth of 10 nm. In our experiments we hold the bandwidth constant and vary the array size in order to obtain a consistent picture of how combining efficiency scales with number of amplifiers. In Fig. 4 we plot the experimental results of Shirakawa et al [2] (red squares), our new measurements (blue dots), and the theoretical estimate proposed by Kouznetsov et al [7] (red line). The simple theoretical estimate appears to predict a faster drop off in combining efficiency with array size than what we observe in our experiments.

The drop of combining efficiency means that the power of individual fiber lasers is not always coherently combined at the straight-cleaved end with a null at the angle-cleaved end. The decrease in combining efficiency reflects the difficulty in finding congruencies among the frequency combs of the individual resonators that make up the overall interferometric cavity laser. From Fig. 3 it can be seen that the practically useful maximum number of laser channels that can be coherently combined in this manner is approximately 10-12.

For a given array size, the combining efficiency can be improved by increasing the spectral bandwidth [2]. In our system the bandwidth of 8 nm imposed by the Faraday mirrors is close to the maximum 10 nm of the broadband mirrors used by Shirakawa, et al. [2] The use of Faraday mirrors aids in polarization control.

## 4. Power fluctuation

The output of the coherently combined fiber laser array exhibits significant power fluctuations on a time scale of milliseconds. These fluctuations are due to environmental factors such as temperature and pressure changes, the interferometric nature of the fiber array resulting in an efficient sensor for these changes. In Fig. 3, the measured power fluctuation, indicated by the experimental error bars, is seen to increase with array size. Here,

*σ*denotes the statistical standard deviation. The extent of $\pm 3\sigma $ includes approximately the maximum range of power fluctuation.

To simulate the power fluctuations we assume that environmental factors lead to length changes on the order of a wavelength for each channel, or, equivalently, a phase shift of $2\pi $. We let the length of each fiber increase by 0.8nm per round trip so that after about 1250 roundtrips a length change of about one wavelength has accumulated. The power value per round trip is recorded until several thousand round trips later the overall accumulation of phase shift has reached $2\pi $ (~1$\mu \text{m}$), then all recorded power values are statistically analyzed to attain the aforementioned definition of power fluctuation range ($\pm 3\sigma $). From Fig. 3, the statistical simulation results, using the upper (downward triangles) and lower (upward triangles) limits to represent the maximum and minimum of calculated power combining efficiency, indicate that the fluctuation ranges increase with the increasing array size and they agree well with similar power fluctuation values in experiments. The results indicate that small fluctuations in fiber length can result in substantial power instabilities and fluctuations, especially for arrays with a large number of elements.

To further explore how the rate of fluctuation relates to array size, we plot in Fig. 5
the experimental (blue dots) and simulational (red squares) fluctuations versus number of channels in the array, *N*. We find that the power fluctuations scale with array size as ${N}^{3}$ (green fitting line). This scaling behavior of power fluctuations in coherent beam combining has never been reported. We do not yet have a simple explanation for this cubic dependence on array size but we note that ${N}^{3}$seems to describe the product of a coherent process (scaling as ${N}^{2}$) and an incoherent process (scaling as *N*). This rapid growth of fluctuations with arrays size is one of the factors that may limit the scalability of beam combining by passive coherent phasing. It is important to note, however, that these results are for a particular geometry of passive beam combining involving laser amplifiers in a composite cavity. The behavior of coupled laser oscillators may well be different.

## 5. Beat spectra

The decline of power combining efficiency with array size believed to be a consequence of the increasing scarcity of coherently combined modes within the laser gain bandwidth. We investigate this scarcity by measuring and calculating beat spectra in fiber-laser arrays. To make it easier to observe beat spectra within the limited spectral bandwidth of an RF spectrometer, an additional 37.5-m single-mode fiber is inserted at the output-end and a 2-m single-mode fiber added to one arm of the fiber-laser array. The greater optical in-fiber length leads to a smaller mode separation of longitudinal modes, and thus more longitudinal modes are expected to exist and beat with each other in this composite cavity. The schematic is shown in Fig. 6 as an example of 4-channel combining. During measurements, a fast photodetector and a 1-GHz RF Spectrum Analyzer are used to detect beat spectra.

According to 2-channel laser array theory, the free spectral range (FSR) of adjacent beat packets and mode separation (MS) of adjacent longitudinal modes are defined as:

where $\mathrm{\Delta}L$ and*L*are the length difference and average length of laser array, respectively. In 2-channel beat spectra, the roughly 56 MHz FSR in Fig. 7(a) and 2MHz MS in Fig. 7(b), based on Eqs. (6-7), correspond very well to the actual ~1.78-m in-fiber length difference and 46-m average length. In 4-channel spectra, ~2MHz MS in Fig. 7(d) is still observed but FSR is greatly increased up to 475-MHz in Fig. 7(c). The suppression of multiple beat packets in 2-channel to only one extra packet in 4-channel and zero extra packet in 8-channel or beyond within 1-GHz window directly indicates the number of the coherently-combined modes (supermodes) in the cavity is greatly reduced as array size multiplies, resulting in the drop of combined-power efficiency. In simulation, by selecting 2-channel in-fiber lengths as 47.82m and 46m; and 4-channel as 47.89m, 46m, 46.42m, and 46.21m, the calculated 2-channel beat spectrum in Fig. 8(a) exhibits multiple peaks whereas the 4-channel in Fig. 8(b) has only one extra peak. These length parameters used for simulation here are quite arbitrarily assigned since the suppression of supermodes from 2-channel to larger channels always holds. Therefore, the simulation result supports the experimental conclusion that the increase in the number of elements in the array leads to a greater suppression of supermodes, resulting in the decrease of power combining efficiency with larger array number.

## 6. Discussion and conclusion

The most important question associated with passive coherent phasing of fiber-laser arrays is how the coherent-combing efficiency scales with array size. In this paper, we have studied the detailed evolution of combined-power efficiency and the issue of power fluctuation versus array size from 2 to 16-channel passively coherently combined fiber-laser arrays.

For power combining efficiency, good agreement between our simulation model and experiments is demonstrated for arrays containing up to 16 channels. Small phase shifts resulting from wavelength-scale length variations are verified numerically to be an important factor resulting in fluctuations and instability in output power. The power fluctuations scale with array size as ${N}^{3}$. Investigation of array beat spectra supports the notion that the decrease of power combining efficiency with array size is a result of increasing scarcity of composite-cavity supermodes.

The work here has focused on a particular combining scheme involving separate amplifiers, 50:50 couplers, and uncontrolled fiber lengths. Other approaches to passive beam combining [10,11] may yield different results regarding scalability. In particular, our preliminary theoretical investigations indicate that the use of phase conjugate mirrors can significantly improve the power scaling behavior of passively combined fiber laser amplifiers.

## Appendix

The model used for the beam combining simulations is based on the amplified Nonlinear Schrödinger equation:

^{2}/m, g

_{0}(unsaturated gain) = 2.67 m

^{−1}, γ (nonlinear coefficient) = 0.003 W

^{−1}m

^{−1}, α (propagation loss) = 8 dB/km, and β

_{2}(phase dispersion) = 0.024 ps

^{2}/m. According to our study, nonlinearity has little effect on power efficiency, power fluctuation, and beat spectra.

## Acknowledgement

The authors acknowledge the financial support from Office of Naval Research (Grant No. N00014-07-1-1155).

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