## Abstract

We present a dynamic model of simultaneous passive coherent beam combining and passive mode locking for coupled fiber lasers. The presence of a saturable absorber in the composite cavity results in the generation of packets of mode locked pulse trains. Within each packet the repetition rate of the pulses is determined by the length difference between the fibers.

©2012 Optical Society of America

## 1. Introduction

Passive coherent beam combining is under intensive investigation as a means of scaling up the output power of fiber lasers [1–6]. One common approach is to use 50:50 couplers to combine fiber amplifiers pair-wise in a tree structure to form an interferometric composite cavity. The coherently phased output is the result of the system selecting the modes that minimize the overall loss in the composite cavity. To date, most of the beam combining experiments based on this approach have used cw lasers and there is very little data on their dynamic behavior under Q-switching [7] or mode locking [8] conditions. Since mode locking enables the creation of high-peak power short pulses from a single fiber laser [9], the possibility of further power scaling through passive combining of several mode locked lasers is worthy of investigation.

In this paper we present a detailed dynamical model of passive coherent beam combining in the presence of a saturable absorber. Our results show, in agreement with experiment [8], that the presence of a saturable absorber leads to the generation of packets of mode locked pulses from the coherently-combined lasers. Within each packet the periodicity of the pulse train can be controlled by varying the length difference between the fibers. Repetition rates of hundreds of gigahertz are readily obtained for short enough length differences. The combining efficiency is high for two lasers but does drop as the number of lasers is increased, a phenomenon also seen in cw beam-combining [1, 6]. Our work represents, to the best of our knowledge, the first modeling study of simultaneous beam combining and mode locking of lasers [10].

## 2. Simulation model and single fiber mode locking

The model employed here is an extension of our previous dynamic model [11, 12] to include the effects of a saturable absorber. In the geometry of Fig. 1 , the field in each fiber is described by the amplified Nonlinear Schrödinger equation:

*j*denotes each fiber. The saturated gain can be represented bywhere ${g}_{0}$ is the gain coefficient, ${P}_{sat}$ the saturation power, and $T=2n\overline{L}/c$ the round trip time over the mean fiber length $\overline{L}$. Here the E-field is already normalized so that ${\left|{E}_{j}\right|}^{2}$represents the power. Note that it is not necessary to include a rate equation for the gain dynamics because phase locking occurs on a time scale much shorter than the gain evolution [12]. A fast saturable absorber is located in the output arm of the combined lasers and is modeled as:where ${P}_{SA}$ is the absorption saturation power and ${\alpha}_{0}$ is the non-saturated loss. This model is appropriate for a semiconductor saturable absorber mirror (SESAM) with a response time of a few picoseconds. All the active fibers we used are doped with erbium as the gain medium.

Table 1 lists the parameter values used in our simulations which are based on the split-step Fourier method for solving the NLSE. The fields evolve from an initial noise distribution and we examine the output after about 2000 round trips in fiber cavities with lengths of about 8 meters. Each round trip contains 24 discrete spatial steps.

To test the basic mode locking model we first simulated a single amplifying fiber with a saturable absorber. The resulting output was a train of 8-ps pulses spaced by the cavity roundtrip time and with an average power of 26.7 mW. The spectrum is approximately Gaussian with a bandwidth of 60 GHz as determined by the loss dispersion.

## 3. Two-channel beam combining and mode locking

To study simultaneous beam combining and mode locking we follow the standard approach that uses discrete 50:50 directional couplers to create an interferometric system of coupled amplifier pairs in a composite cavity. In Fig. 1 the angle-cleaved port is a source of loss (no feedback) while the saturable absorbing (SA) mirror in Port 2 provides the mode locking mechanism and serves as the output coupler. Here the fiber amplifiers are of different lengths, arbitrarily chosen as 8.053 m and 8.071 m. The nonlinear coefficient is set to zero initially in order to focus on the essential mechanisms responsible for coherent combining and mode locking.

#### 3.1 CW beam combining

When the SA mirror is replaced by a normal partially reflecting mirror, the output is quasi-cw with a spectrum that shows a beating between the individual frequency combs of the individual lasers. As seen in Fig. 2 , the spectrum consists of array modes spaced approximately by the beat frequency $\Delta f=c/(2n\Delta l)=5.56$GHz, where $n$ is the index and $\Delta l$ is the length difference between the two channels. Each of the frequency “spikes” is actually an envelope of about 100-MHz width (FWHM) that contains a few single-cavity longitudinal modes. The phase difference between two adjacent longitudinal modes is randomly distributed from 0 to 2π, indicating that there is no phase locking among the lasing modes. In the time domain we see the high frequency oscillation representing the complex beating between these modes with random phases. The combining efficiency is very high, with 99.97% of the power emerging from the output Port 2 and only 0.03% from the loss Port 1. The beam combining process selects the modes with minimum loss.

#### 3.2 Beam combining and mode-locking with saturable absorber

In the presence of a saturable absorber, the absorption in the system is low at high power and high if the power is below the saturation power. The system will thus discriminate against the low power cw state with random phases in favor of the solution in which the phases of adjacent modes are locked in a fixed relationship to generate high peak power pulses. Figure 3 shows the spectrum of the output of Port 2 in the presence of the saturable absorber. It can be seen that the spectrum in this case is much wider, with a bandwidth (FWHM) of about 100 GHz compared to the 20 GHz bandwidth under cw operation. The spectrum contains a large number of “lines”, each of which, upon magnification reveals a width of about 0.6 GHz. These lines are spaced by the cw beat frequency $\Delta f=c/(2n\Delta l)$, which in this example is 5.56 GHz as determined by the length difference.

For mode locking, the adjacent array modes need to be locked with a fixed phase relationship in order to generate a mode-locked pulse instead of cw output. Figure 3 also shows the phase difference between two adjacent array modes. For those frequencies around the array modes where there is a spike, the phase difference is fixed. Hence those phase-locked modes will constructively interfere with each other, leading to the mode-locked pulse output.

In the time domain (Fig. 4 ), the output consists of packets of pulses at the round trip period ${\rm T}=2n\overline{L}/c,$, with each packet containing a train of mode locked pulses spaced by $\delta {\rm T}=1/\Delta f=2n\Delta l|c$. For our chosen parameters each packet has a temporal width of 1 ns, which corresponds to the inverse of the 0.6 GHz spectral with of each spike in Fig. 3. Each of the mode locked pulses in a packet has a width of 5 ps determined by the inverse of the 100 GHz bandwidth. Clearly the separation of pulses in each packet can be tuned by changing the fiber length difference $\Delta l$.

The average power out of Port 2 is 49.70 mW, representing 98.02% of the total power. Thus the combining efficiency remains high under mode locking conditions.

#### 3.3 Comparison with experimental results

To compare with experiment, we apply our model to the work of Lhermite et. al. [8], in which packets of gigahertz pulse trains were generated through the coherent combining of mode locked fiber lasers. Their experiment employed the Mach-Zehnder ring cavity configuration shown in Fig. 5 .

The interferometer modulates the individual laser spectrum with a beat frequency $\Delta f=c/\Delta L$ thus creating pulse trains with interval $\delta T=\Delta L/c$, where $\Delta L$is the roundtrip optical path length difference. Each of the active fibers is 10m long and a delay line is inserted in one arm to permit a variable optical path-length difference $\Delta L$which we set at 5.1 mm, 3.5 mm, and 2.1 mm in accordance with the experimental values. (Note: According to the authors the smallest $\Delta L$used in the experiment was 2.1 mm, not 2.3 mm.) These lengthdifferences predict pulse repetition periods of 17 ps, 11.67 ps, and 7 ps, respectively. Figure 6 shows the simulated pulse packets. There is one of these packets within a round trip time of 120 ns. It is seen that the period of the pulse train within the packet decreases from 17 ps to 7 ps as the length difference is decreased, in agreement with the theoretical expression $\delta T=\Delta L/c$. For these parameters the pulse heights within the train are not uniform.

In Fig. 7 we compare our simulated intensity autocorrelation results with the experimental measurements. Initially we found a systematic discrepancy with the published pulse periods. This discrepancy was resolved with new data supplied by the authors, which replaces the published periods with the values 17.19 ps, 11.88 ps, and 6.98 ps.

From the simulations about 90% of the total power of 100 mW emerges from the output port, indicating high combining efficiency. For very small length differences the pulse interval $\Delta T$ becomes comparable to the pulse duration and the mode locking breaks down. From the spectral viewpoint, such a small $\Delta L$ leads to a $\Delta f$that is comparable to the effective bandwidth. As a result, only one array mode is left in the spectrum and thus there are not enough modes to be locked. We note that there is a relatively high background in the experimental autocorrelation compared to the background-free results of the simulation. We will consider a possible cause in the next section.

#### 3.4 The influence of the nonlinearity

In order to elucidate the basic features of mode locking in coherently combined fiber lasers we neglected the nonlinear refractive index in the above simulations. We found very good agreement with the experimental results, except that our simulated autocorrelation plots do not show the high background levels seen in the experimental results. It is well known however, that at high pump levels the nonlinear index can lead to phenomena such as multiple pulsing in mode locked fiber lasers. Here we include the Kerr nonlinearity and examine its effect on the spectral and temporal characteristics of mode locking. Figure 8
shows the time series as well as the autocorrelation traces for $\gamma $ = 0.004 m^{−1}W^{−1} and three values of length difference. The time series shows pulse splitting as well as the presence of irregular subpulses in between the main pulses. The autocorrelation trace displays a significant rise in background which can be attributed to the presence of these sub-pulses of essentially random heights. The exact nature of the pulse substructure is strongly dependent on the values chosen for nonlinearity, dispersion, and fiber length difference. The sub-pulse instability due to nonlinearity is worse for short fiber length differences.

Figure 9 shows the spectra and the spectral phase distribution for three length differences in the presence of nonlinearity. The uniform phase indicates that adjacent modes are successfully locked with a fixed phase difference even though the time series is highly irregular.

The rise in background with increasing nonlinearity is consistent with the increased self-phase modulation and frequency spread which makes it difficult for the fields in the two channels to maintain the phase relationship needed to minimize loss at the 50:50 coupler [12]. The deleterious effects of nonlinearity seen here are in contrast to the predictions by others of nonlinearity-enhanced beam combining in arrays with global coupling [13]. The contrasting roles of nonlinearity in the two different systems is an issue that still needs to be resolved.

## 4. Four-channel beam combining

To explore the possibility of mode locking larger arrays we extend our simulations to the 4-element fiber array shown in Fig. 10 . With four elements, there are 3 independent length differences and the array mode spacing is determined by the greatest common divisor (GCD) of these length differences [12]. Strictly speaking the concept of a greatest common divisor is defined only for integers. Since the length differences are real numbers, we express them as integer multiples of the resolution of the length measurement, here taken to be 1 mm. Thelength differences are said to be commensurate when they have a non-trivial GCD. When the length differences are incommensurate, the allowed modes do not have any discernible periodicity.

#### 4.1 CW operation

We first consider cw operation without saturable absorber for a 4-element array with fiber lengths randomly chosen as 8.000 m, 8.011 m, 8.024 m and 8.041 m. The length differences are *L _{AB}* = 11 mm,

*L*= 17 mm,

_{CD}*L*= 13 mm, for which there is no GCD except the trivial one, which in this case is 1 mm, the resolution of the length measurement. The implied mode periodicity is given by $\Delta f=c/(2n\Delta {L}_{\mathrm{gcd}})$ = 100 GHz. This frequency separation is comparable to the bandwidth of the two-channel fiber laser array in Section 3.2. This means that for the chosen parameters only one array mode will exist within the net gain bandwidth under cw operation. Figure 11 shows the simulation result and it is clear from the spectrum that there is only one principal array mode, the other modes being at least 10 orders of magnitude lower. In the time domain the output is close to a sinusoid with a slow modulation.

_{BC}It should be noted that since the actual fiber length differences are generally not rational numbers, the GCD is only approximate and thus the array modes will only be approximate modes with some coupling loss [12]. The closer the actual difference is to a multiple of the approximate GCD, the less coupling loss there will be, and vice versa. Since the laser system will be stabilized at the state with minimal loss, an optimal approximate GCD will be reached.

#### 4.2 Mode locking in four-channel array: incommensurate length differences

In the simulation below, we use the same randomly selected fibers lengths as in the cw case: 8.000 m, 8.011 m, 8.024 m and 8.041 m. All parameters remain the same as in the simulation for Fig. 11, except that the partial mirror at Port 4 is replaced by a saturable absorber partial mirror. The nonlinearity is 0.003 m^{−1}W^{−1}. Figure 12
plots the spectral and temporal outputs. In the presence of the saturable absorber the spectrum exhibits a large number of modes as required for the formation of short pulses with the intensity needed to saturate the absorption. In the time domain, it is seen that a train of pulses with non-uniform separation is generated. The shortest separation between sub-pulses is about 10ps, corresponding to the 100 GHz array mode spacing in the cw case. The duration of each pulse is around 2.5 ps. The combining efficiency was computed to be 78%.

#### 4.3 Four-channel combining and mode locking for fibers with commensurate lengths

With randomly chosen lengths we obtained quasi-random mode locked pulse trains from the 4-element fiber laser array. We now consider an array in which the elements are chosen such that their length differences are commensurate, i.e. they have a non-trivial greatest common divisor. Also, the length differences are chosen to be in the centimeter range so that the array mode spacing of roughly $c/(2n\Delta l)$ will be in the gigahertz range. This way there will be many more array modes within the gain bandwidth which should lead to improved mode locking. For example, if we select 8.050 m, 8.260 m, 8.200 m and 8.530 m, the differences 210 mm, 60 mm, and 330 mm will have a greatest common divisor of 30 mm, which leads to an array mode separation of 3.33 GHz. As a result there will be many more array modes within the bandwidth range to be locked to produce short mode locked pulses. The simulation result for this case is plotted in Fig. 13 . It is seen that the pulses within a packet are uniformly spaced with a repetition rate of 3.3 GHz. The calculated combining efficiency of 91% is higher than in the incommensurate case.

In the commensurate 4-element case the Gaussian shape of the pulse packet in Fig. 13 is very similar to the result for 2-fiber beam combining with mode locking. There also we found a Gaussian shape and a uniform separation between adjacent frequency and temporal spikes. Two coupled fibers with a length difference of 3 cm will yield a similar spectrum as four coupled fibers with a $\Delta {L}_{gcd}$ of the same value. In practice it is not easy to precisely control the fiber lengths and thus one may expect that for arrays larger than two elements the generated pulse trains will be irregular. The combining efficiency is also reduced below that of the CW case.

## 5. Conclusions

In conclusion, we have carried out simulations of the mode locking behavior of passively combined fiber lasers in the presence of a saturable absorber. The results show, in agreement with experiment, that a pair of coherently combined fiber lasers will produce packets of mode locked pulses whose repetition rate can be tuned by varying the length difference between the fibers. For larger arrays the pulses are random unless the fiber lengths are chosen carefully such that their differences are commensurate. With careful design the mode locked array could be a useful source of tunable pulse trains for synchronization and clock applications.

## 6. Acknowledgments

The authors thank Alain Barthelemy for useful discussions and for supplying the revised data on their experiment. Partial support for this work was provided by provided by the Office of Naval Research under grant No. N00014-07-1-1155.

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