We present a new model for studying the beam combining mechanism, spectral and temporal dynamics, the role of nonlinearity, and the power scaling issue of discretely coupled fiber laser arrays. The model accounts for the multiple longitudinal modes of individual fiber lasers and shows directly the formation of the composite-cavity modes. Detailed output power spectra and their evolution with increasing array size and pump power are also explored for the first time. In addition, it is, to our knowledge, the only model that closely resembles the real experimental conditions in which no deliberate control of the fiber lengths (mismatch) is required while highly efficient coherent beam combining is still attained.
©2009 Optical Society of America
There is much current interest in scaling up the output power of a single fiber laser by coherently combining the fields of several amplifying fibers into a high-brightness, diffraction-limited beam [1–6]. One approach that has been pursued with some success is the use of discrete 50:50 directional couplers to create an interferometric system of coupled amplifier pairs in a composite cavity. This pair-wise combining scheme forms the basis of a tree architecture that can, in principle, be scaled up to any even number 2xN of fiber lasers. Several groups have demonstrated highly efficient coherent beam combining using up to eight erbium-doped fiber lasers [2–6].
In principle, a two-channel fiber laser array is just a Michelson interferometer except that both arms are replaced by rare-earth-doped fibers. The 50:50 directional coupler acts like a beam splitter as shown in Fig. 1. Constructive or destructive interference occurs depending on the relative phase of the incident fields if their coherence is assumed. The waves generated from the individual active fibers then add on or cancel out with each other accordingly at the coupler outputs. Since uneven fiber lengths directly relate to the accumulated phase difference of the propagating waves, one might suppose that successful beam combination would require accurate control of fiber lengths. However, experimentation has verified the robust and reliable operation of power addition of two-channel fiber laser arrays even when their lengths are not carefully adjusted. Furthermore the combining efficiency has been seen to drop dramatically when the number of fiber amplifiers exceeds eight, thus limiting the scalability of this method.
Several theoretical analyses have been published aimed at explaining the limitation of power scaling and elucidating the nature of the beam combining process [5,7–13]. These include static calculations of the spectral response of passive multi-arm interferometers [5,7–10] and dynamic simulations based on iterative maps for the rate equations and a single-longitudinal-mode cavity field [11,12]. Currently there appears to be some debate as to whether the coherent phasing of multiple fiber amplifiers is a “self-organization” process involving coupled nonlinear oscillators [11–13] or the result of an accidental coincidence between the frequency combs of multiple resonators. Any attempt to resolve this debate must take into account the multiple-longitudinal-mode nature of fiber lasers and allow for arbitrary length differences of the amplifying fibers. Yet the only published dynamic studies include only a single mode, require a fixed phase difference, and yield no spectral information.
Here we present a model based on the amplifying Nonlinear Schrödinger Equation that incorporates the multiple longitudinal modes of a fiber laser and allows for the natural selection of the resonant array modes that experience the minimum loss. It is a propagation model that takes into account gain saturation, fiber nonlinearity, group velocity dispersion, and the loss dispersion of bandwidth limiting elements in the complex cavity. In agreement with experimental observations, the model shows that efficient coherent beam combining occurs without the need for interferometric control of fiber lengths so long as there is sufficient bandwidth available. It is the first model, to the best of our knowledge, that provides detailed spectral information on the output of coherently combined fiber lasers.
2. Model and benchmark
Figure 2 depicts two independent single mode fibers coupled discretely by a directional coupler. The continuous-wave pump beams are launched into each fiber by a wavelength division multiplexer (WDM) at z=0 and excite active ions that give rise to gain at longer wavelengths. Assuming single polarization, the coherent waves generated in each amplifying fiber are governed by the nonlinear Schrödinger equation in conjunction with the rate equation for the population inversion 
Ej(z,t) and ΔNj refer to the slowly varying envelope of the electric field and the population inversion in the first and second fiber for j=1,2 respectively. From left to right, the terms in Eq. (1) account for the effects of linear gain gj(ΔNj) fiber losses α, the inverse of the group velocity β1, the frequency-dependent losses b, the group velocity dispersion β2, and lastly the nonresonant Kerr nonlinearity γ. As for gain dynamics in Eq. (2), Rp(t) specifies the pumping rate. Its second and third terms describe the process of excited population relaxation with upper-state lifetime τ and laser gain saturation at high intensity fields. The electric field amplitudes are normalized such that |Ej|2 represents power distributions.
Note that only forward propagating waves are considered in Fig. 2. Because the reflectivity at the output port of fiber laser arrays is typically about 4%, the backward wave is always much weaker than the forward wave and hence standing wave effects as well as cross-saturation by backward waves can be neglected. The unidirectional model describes quite accurately the behavior of a ring fiber laser  and is expected to yield useful insight into the beam combining properties of fiber lasers under the high-output coupling condition. We note that unidirectional fiber laser arrays have also been demonstrated and their phase-locked operation is reported in Refs [15,16].
The fields exiting the fibers at z=0 pass through the 50:50 directional coupler, which connects the inputs E1, E2 and the outputs A1, A2 by the linear matrix
The field A2 exits the cavity through the angle-cleaved end at the left while 4% of the power in A1 is split equally and fed back to the fiber inputs at the right as indicated by the yellow dotted line in Fig. 2. The remaining 96% serves as the output of that port.
Before verifying the numerical scheme on a single fiber laser as described above, we make a further simplification of the rate equation. Typical roundtrip time for a fiber of tens of meters long is of order hundreds of nanoseconds, while the population relaxation constant is roughly ten milliseconds for Er-doped and one millisecond for Yb-doped fiber lasers. Another important time scale is the gain recovery time, which is also quite long and is of order milliseconds for Er-doped fibers [17–19]. The difference in time scales permits us to solve for the gain dynamics by setting the time derivative in the rate equation to zero. Assuming Psat=hν/στ×Aeff,gj=2σΔNj and g 0j=2σRp(t)τ, Eq. (2) becomes
where σ is the sum of absorption and emission cross sections and T is the computational time window. Equations (1) and (4) are integrated numerically and iteratively together with the coupling matrix to model the laser behavior of this composite cavity. In this paper, we adopt standard split-step Fourier methods (SSFM), which have been used extensively for studying nonlinear pulse propagation in fibers, to handle the multi-longitudinal-mode nature of continuous-wave fiber lasers.
A single 24 m long unidirectional Er-doped fiber laser with 4% power feedback is simulated for the purpose of benchmarking. Table 1 lists the parameters and their corresponding values as taken from Ref . The process of spontaneous emission is represented by very small complex numbers which are generated randomly and incorporated into each roundtrip for initiating the lasing process. Figure 3 shows the steady state output power distributions in both temporal (left) and spectral (right) domains for two cases: (a) with a Kerr nonlinearity γ=0.003 W-1m-1 and (b) with γ=0 W-1m-1. The time window T is chosen to be eight times the roundtrip duration to ensure dense discretization and higher resolution in the frequency domain. An average of approximately 28 mW power is obtained by ∫T 0|Ej|2 dt/T for either case. Because the large output coupling coefficient leads to significant amplitude changes along the fiber, the step size parameter of SSFM needs to remain small in order to obtain accurate integrations. Here we choose six or more steps for each roundtrip. Growing out of incoherent random noise, the laser output is characterized by a time-varying output and its spectrum consists of irregular spikes. The irregular time series is the result of the complex beating between a large numbers of longitudinal modes with random phases. The steady states are therefore defined by measuring the average powers between consecutive roundtrips. The shape of the spectral envelope is determined by the loss dispersion. It is evident that the inclusion of the nonlinear refractive index broadens the power spectrum significantly, which was first verified and reported by Roy et al for fiber lasers . This is a result of four-wave-mixing which can be approximately phase-matched because of the dense nature of the longitudinal modes. It is clear that this propagation model should be capable of describing the spectral properties of fiber laser arrays with multiple longitudinal modes.
3. Simulation for two-channel fiber laser arrays
We begin with a two-channel fiber laser array by setting L 2 to 24.0 m and R1, R2 to 4%, and 0% respectively in Fig. 2. The length difference ΔL is arbitrarily selected to be 30 cm and hence L1 equals 24.3 m. Using parameters from Table 1, Fig. 4 illustrates the simulation results for both temporal (left) and spectral (right) domains. It is interesting to see that essentially all the power, 56.26 mW, emerges from the first output port while very little (less than 0.05 mW) escapes from the other, angle-cleaved, one. (Note the orders-of-magnitude difference in the ordinate scales between Fig. 4(a) and (b).) For the efficiency calculations, a simulation of individual fiber lasers of lengths L1, L2 and equal 4% output coupling generated 28.27, 28.02 mW respectively. Their sum gives a total power of 56.29 mW and it is used, together with the array output power 56.26 mW, to define the combining efficiency in this paper. Here, the efficiency is high and close to 100%.
A rather striking feature of the array output is the discrete nature of the power spectra compared to the quasi-continuous spectrum displayed by the single fiber laser. While the spectrum of the single laser is made up of the densely packed axial modes of a long cavity, the array resonances in Fig. 4 comprise a set of spikes equally separated by an interval of 0.667 GHz. This spectrum is the result of a Vernier effect involving the superposition of the frequency combs of the two coupled cavities with a length mismatch ΔL. For ring cavities it leads to a modulation of the comb spectrum with a beat frequency of Δv=c/n/ΔL. Using a refractive index of n=1.5 and 0.3 m for ΔL, we obtain 0.667 GHz which agrees exactly with the simulation result. For laser arrays with standing wave cavities, the optical path lengths double, so the mode separation becomes Δv=c/n/(2ΔL) . The Vernier effect results in the suppression of certain longitudinal modes and has been utilized in the Vernier-Michelson cavity to achieve single-frequency operation for gas lasers .
To further demonstrate the natural emergence of the array modes and the self-adjustment feature of our model regardless of the length differences, another simulation result is given with all the coefficients fixed as before except that L1 is changed to 24.08 m, so that ΔL=8 cm. Since Δυ is inversely proportional to ΔL, greater spacing is expected for a smaller length mismatch. Indeed the spectral intervals are measured to be 2.5 GHz in Fig. 5(a) and (b), which is consistent with the theoretical calculations. Note also that the main peak in the spectrum has shifted from 0.9074 GHz in Fig. 4(a) to -0.1848 GHz in Fig. 5(a) as the laser self-adjusts its frequency. This dependence of the beat spectrum on ΔL is routinely seen in experiments [3–6,16,20]. Further details of the spectrum can be seen by zooming in on one of the spikes in Fig. 5(a) (circled in green). It is seen that the spikes are actually the envelope of the individual cavity axial modes, which are equally spaced in the absence of the Kerr nonlinearity (Fig. 5 (c), γ=0 W-1m-1) and somewhat broadened and shifted in the presence of nonlinearity (Fig. 5(d), γ=0.003 W-1m-1). The shift of the peak due to nonlinearity is only about 1 MHz at these power levels. We remark that some frequency pulling of the individual modes has been observed in experiments and attributed to nonlinearity . Based on those results the authors suggested that the mechanism for spontaneous self-organization without cavity length control is a nonlinear process (in the sense of requiring an intensity-dependent refractive index). Our results however indicate that this spontaneous self-adjustment occurs even in the absence of nonlinearity as the laser seeks to operate on the lowest loss mode of the composite cavity. The presence of nonlinearity simply leads to a slight modification of the actual mode frequencies but cannot be seen as the fundamental mechanism leading to the coherent phasing of the two amplifying fibers.
The ability of two lasers to combine efficiently regardless of their length difference is a feature that emerges naturally from our model. It is merely a reflection of the fact that in the presence of a large number of longitudinal modes, the fields self adjust to select a new oscillation that corresponds to a common resonance of the combined cavity . This selfadjustment should be possible so long as there is sufficient bandwidth available to encompass at least one of these composite-cavity modes. In our model, the effect of bandwidth-limiting elements in the cavity is described by the parameter b. It represents a frequency-dependent quadratic loss term of the form -b(ω-ω 0)2 . To investigate the role of available bandwidth in beam-combining efficiency we consider a case where ΔL is small enough that the frequency spans are greater than the limited bandwidth imposed by a filter. First we assume 1 mm for ΔL and set the loss dispersion coefficient to zero. The simulation results are illustrated in Fig. 6(a). Note the modulation period is calculated to be 200 GHz. In the absence of bandwidth limiting the fiber lasers combine successfully with an efficiency close to 100%. The first two peaks near the center, pointed out with arrows in Fig. 6(a), are measured to be -117.9 GHz (left) and 82.02 GHz (right), the main peak. Again 200 GHz is verified by subtracting -117.9 from 82.02. Next the simulation is repeated with b=0.13 ps2m-1 which corresponds to a filter with a bandwidth of roughly 60 GHz. Since now higher frequencies experience more losses, only one peak with the least attenuation lases in Fig. 6(c) and the sum of P1 and P2 is also reduced. The combining efficiency decreases to 76% since there is now only a single mode within the gain bandwidth.
Note that the location of the main peak is now measured to be 55.87 GHz as opposed to 82.02 GHz in Fig. 6(b) without bandwidth limiting. In addition, a significant amount of power, 8.6 mW, appears at the lossy port. The occurrence of the frequency shift and the large output from the angle-cleaved port implies that the array, in the presence of frequencydependent losses, does not necessarily lase at the cold cavity composite resonances but at frequencies that minimize the overall cavity losses. It shows that the model, just like actual fiber laser arrays, does adjust itself and select the suitable resonant frequencies.
The final example given for two-channel fiber laser arrays is to examine the fine structure of the spectrum of amplified spontaneous emission for operation below threshold. Using realistic parameters from the experiment of Shirakawa, et al , the simulation results are illustrated in Fig. 7(a) with 12.682, 12.0 m for L1 and L2 respectively. Note that we double the fiber lengths since our model is based on the ring cavity configuration. The carrier wavelength is kept at 1.545 µm and the refractive index n is 1.45. The lasing threshold gth=α-ln(R)/L is calculated to be 0.312 m-1. We set g0 to 0.31 m-1 and the amplified spontaneous emissions spectrum from the first port agrees qualitatively with that of the experimental results  shown in Fig. 7(b). As in the experiment, the spectral packets are separated by about 302 MHz. Unlike the very narrow spectral packets (essentially spikes) in the previous plots where arrays are pumped above the threshold, they exhibit broader FWHM’s here and are seen clearly to comprise of small spikes separated by a free spectral range of 16.3 MHz. As for the second port, its spectrum is more complicated and features a split pattern around the peaks of the resonances. The spectra at the two ports are complementary in a manner similar to the reflection and transmission spectra of a Fabry-Perot. (For the second port the experimental spectrum of Ref . has some similarities to the theoretical one but is complicated by the presence of features attributed to the presence of an extraneous polarization component.) As the pump power increases the spectral packets are seen to narrow as shown in Fig. 7(c). This is a result of gain narrowing due to our assumption of homogeneous broadening in which most longitudinal modes are suppressed as a result of serious competition between adjacent frequencies.
4. Simulation for four-channel fiber laser arrays
To further demonstrate the capacity of this model, we apply the simulation to a four-channel fiber laser array with randomly chosen lengths of 24.0, 24.3, 23.73, and 24.63 meters. The results are shown in Fig. 8. Unlike two-channel arrays where Δυ is determined by the length differences, it is now determined by the greatest common divisor Lgcd of the four lengths . In this case, Lgcd equals 3 cm and indeed the amplified spontaneous emission spectrum (Fig. 8(a)) features a complicated interference pattern with a period of 6.67 GHz calculated from Δv=c/n/Lgcd using 1.5 for the refractive index n. As the pumping is increased above threshold there is a narrowing of the beat packets. Most of the power, 107.44 mW, emerges from the second output port as seen in Fig. 8(b). Since the four uncoupled fiber lasers produce a total of 112.63 mW, the combining efficiency is calculated to be 95.4%.
The reason for the efficiency drop is the decrease in the probability of finding an accidental coincidence in the resonances of mismatched cavities as the number of such cavities increases [7–10]. In the absence of an exact coincidence, which corresponds to a lossless mode, the system still finds the least lossy mode, which will generally have significant energy coupled through the lossy ports because of the residual phase mismatch at the couplers.
5. Discussion and conclusion
The self-adjustment process that leads to the efficient and robust combining of fiber lasers depends on the existence of a dense set of longitudinal modes from which the laser can select those that satisfy the minimum loss condition at the coupler. In our model, changes in fiber length differences are automatically compensated by changes in the lasing wavelength and the spectral signature of the combined lasers. The spectral changes seen in our simulations agree with experimental observations. We find that at these power levels the non-resonant nonlinear refractive index is not a significant factor in beam combining.
While the simulations presented here involved following the progress of a unidirectional wave as it propagates around a composite cavity, the model is easily extended to include counterpropagating waves as well as different polarizations. Because the solution scheme is the highly efficient split-step Fourier method, the model can be used to simulate the dynamics of many coupled amplifiers.
In conclusion, we have proposed a new model for studying discretely coupled fiber laser arrays. The model incorporates propagation effects, multiple longitudinal modes, unbalanced mirror reflectivities, uncontrolled fiber lengths, the intensity-dependent refractive index, and gain saturation. It lends support to the picture of coherent beam combining as simply the natural selection of the supermodes of a composite cavity that have the lowest loss.
This work was partially supported by the High Energy Laser Joint Technology Office (JTOMRI), the US Army Research Office and the Office of Naval Research under Award Nos. W911NF-05-1-0572 and N00014-07-1-1155.
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