1. Introduction

There is ongoing interest in the coherent combining of lasers, combining multiple optical fields with a fixed phase relationship, to achieve high brightness sources. Recently, it has been shown that fiber lasers can be easily combined using intracavity 2×2 fiber couplers developed for the telecommunications industry [1, 2]. Up to ten fiber lasers have been combined using multiple 2×2 couplers with high coherence [3, 4]. Scaling to large arrays using this technique is expected to be limited by the unavoidable length mismatch between the different fiber lasers that make up the intracavity-coupled array. Linear analyses [5, 6] and a Rigrod-type analysis [7] of the intracavity coupling predict that the high coherence output is limited to the array modes that are simultaneously approximately near-resonant with the modes of the individual lasers that make up the array. Due to random length mismatches, it becomes increasingly difficult to meet this condition as the number of array elements increases [6], a factor that is complicated by unavoidable environmental fluctuations.

Recent experiments have been interpreted as showing that the linear analysis underestimates the number of lasers that can be coherently combined, but that cross mode coupling and gain saturation due to nonlinear optical processes in the optical fiber limit the efficiency of intracavity coherent coupling [3]. It has been speculated that nonlinear dynamics, the deterministic temporal fluctuations generated by the nonlinear laser coupling equations (even if no intrinsic material optical nonlinearity is included), plays a role in the coherent combining process, but this has not been confirmed [8]. Here, we present the results of investigations of the effects of the onset of intrinsic fiber optical nonlinearities on the output of intracavity coupled coherent fiber arrays. To do this, we use very long fiber laser cavities to enhance the nonlinear interaction length. We observe effects due to cross mode coupling (XMC) of the fiber array modes and to stimulated Brillouin scattering (SBS). For small arrays of up to four lasers, the onset of the nonlinearities did not significantly degrade the coherence of the laser output.

2. Experimental results and discussion

Our experiments were performed with a flexible, reconfigurable fiber array built entirely from commercially available components. We used a ring cavity configuration that allowed us to control the counterpropagating beam power through the use of unidirectional optical components. One- and four-amplifier configurations are shown in Fig. 1. Er-doped fiber amplifiers (IPG Photonics) with > 25 dB small signal single pass gain and 100 mW or 500-mW saturation power provided the gain. Frequency selective feedback was supplied by a fiber Bragg grating (FBG), with a high-reflectance bandwidth of approximately 60 GHz centered near 1557.2 nm. An intracavity polarizer and polarization rotators (Newport Corp.) were used to enforce a common linear polarization. Trees of 2×2 or 1×4 fiber couplers (Newport Corp.) were used for intracavity coupling. A fiber optic circulator (Optics For Research) was used to enforce unidirectional operation and the coherent output was coupled from the cavity with a 90/10, 90% circulating and 10% output, fiber coupler. The part of the cavity that is common to all round trip paths was varied by adding long fiber patch chords, 30–400 m in extra length, while the path length of the separate amplifier arms was varied by shorter length changes, ~0.01–30 m, using patchcords and the different lengths of the fiber pigtails of the different components. Here, we will concentrate on data from long cavities with approximately 400 m of fiber placed in the reflected arm of the circulator, between the circulator and the FBG. This created a total cavity length of approximately 820 m. As indictated in Fig. 1, the different amplifier arms are length mismatched using a 5-m patchcord in one, a 1-m patchcord in the second, and maintaining a mismatch of approximately 10 cm or less between the last two arms in the data presented here. All fiber is SMF-28 compatible single mode fiber.

 

Fig. 1. The left figure shows the single ring laser. Wavelength selectivity is provided by the FBG that follows 400 m of fiber in the arm of the optical circulator. DFB#1is for injection of a narrow line source, DFB#2 is a local oscillator for the optical spectra. The right figure shows the reconfiguration of the amplifier segment for four amplifiers. The paths differ in the amplifier arms by approximately 5 m, 1 m, and <0.1 m relative to the short arm.

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Various high-speed detectors were used to monitor the output from the output coupler, the various output ports of the intracavity 2×2 couplers, and the leakage through the FBG. Light from the cavity output coupler, and/or the loss ports, was passed through optical isolators to various detectors for monitoring either by a digital voltmeter for average output power, an oscilloscope for temporal characteristics with a bandwidth up to 1 GHz, or a microwave spectrum analyzer (MSA). The output could be combined with the output from a single mode distributed feedback laser diode (DFB#2 in Fig. 1) so that the heterodyne measurement yielded the optical spectrum. We generated optical spectra by setting the frequency of the MSA to some nominal value, typically in the range of 10–100 MHz, and sweeping the temperature of the DFB laser diode so that its output scanned the range of frequencies reflected by the FBG. This yielded a relatively coarse optical spectrum that could be augmented by comparison with the direct detection power spectrum. Because of the scanning nature of the spectral measurements, power spectra take fractions of a second and the optical scans take approximately 20 seconds, temporal fluctuations in the output yielded random fluctuations in the amplitude of the individual features of the spectra. A second DFB laser diode (DFB#1) could be used to inject narrow-band light into the cavity to enforce single-mode or near single-mode output. As discussed below, this was done to enhance SBS to demonstrate that the spectral broadening of the free-running system suppressed the SBS.

All connections between fiber pigtails were made with FC/APC fiber connectors to allow easy reconfiguration. Backscatter is negligible from these connectors when they are clean. However, the connectors easily became fouled/damaged even at the relatively modest (<1 W average power) intracavity powers that were generated. The etalons formed by the imperfect connectors proved to be a major limiting factor in the apparatus. Our experience indicates that all intracavity connections should be replaced by fusion splices in future studies.

The unidirectional ring cavity configuration of Fig. 1 is useful because there is one segment, between the optical circulator and the FBG, where there are counterpropagating laser beams in the cavity. Because SBS is a backscatter beam, it will be preferentially suppressed except in this long arm by the optical isolator and circulator elements. We observed that evidence for SBS was limited to a very narrow range of operating conditions, as we discuss more fully below.

Starting first with a single amplifier, we varied the circulating power in the long cavity by changing the power of the pump laser diodes. Figure 2 shows the changes in the optical spectra of a single amplifier configuration, as in Fig. 1(a), as a function of the circulating power. The key spectral change is a broadening of the spectrum from a small number of modes to a dense set of cavity modes. Significant broadening is observed at circulating power levels on the order of 10 mW. Using a canonical value of 1/(W·km) for the fiber nonlinear coupling coefficient [9], one observes the broadening when the cross-mode coupling is on the order of 1%. There is some broadening at lower power levels but it tends to be highly structured and temporally varying. For the most part, it is associated with the cavity imperfections due to parasitic intracavity reflections. Also, over a relatively narrow range of power levels below the onset of significant broadening, we observe a peak offset by the 10.8 GHz Brillouin shift [10]. An example is the 5 mW optical power spectrum of Fig. 2. At these pump levels, the SBS feature could be enhanced and additional SBS peaks induced by using the DFB laser diode (DFB#1 in Fig. 1) to inject narrow-band light into the cavity and injection lock the laser. We do not show those results here but the spectrum under injection locking with the DFB laser consisted of a peak at the injection frequency and peaks offset by the SBS shift and its harmonics. However, without external input, at higher power levels there was no obvious SBS feature and the power in individual modes was considerably lower. The broadening at higher powers is consistent with cross-phase coupling between the cavity modes induced by the intrinsic χ ³ fiber nonlinearity [9].

 

Fig. 2. Optical spectra (a) and power spectra (b) of the output from the single amplifier ring configuration of Fig. 1(a) for different circulating power levels. Shown with the optical spectra is the low transmission/high reflection band of the FBG.

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When additional lasers were added to form a coherent array, the spectrally broadened output gained additional structure that corresponded to the various path-length mismatches. Fig. 3 shows two different optical spectra corresponding to the four amplifier configuration of Fig. 1. There is now a periodic sequence of spectral features that run across the high-reflection bandwidth of the FBG with a spacing determined by the smallest path length difference among the four amplifiers. There are two different optical spectra in Fig. 3 and power spectra in Figs. 4(a)–4(b). The arrays differed in these spectra by a changed path length difference for the two amplifier paths with the smallest length mismatch, and this change caused the change in the frequency spacing of the spectral features.

 

Fig. 3. Optical spectra corresponding to the four-amplifier ring configuration of Fig. 1. The optical spectra are superposed on the transmission spectrum of the FBG that defines the bandwidth of the ring cavity.

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Because of their low resolution, the optical spectra did not yield much detail. Using the power spectra of the laser output, the relatively broad features of the optical spectra can be resolved into additional structure, as shown in Fig. 4. Figure 4(a) shows the low resolution power spectra, and the other spectra in Fig. 4 show parts of those spectra with increasingly higher resolution. Note that in Fig. 4(a) there is little or no spectral power in the features at the expected SBS offset frequency of 10.8 GHz. The features are centered at periodic frequency offsets with no obvious influence from SBS. The features separated by ~4.8 GHz are due to the ~5.1 cm shortest path length mismatch of that array configuration while the ~17 GHz spacing corresponds to ~1.4 cm mismatch. The features are seen to be made up of dense substructure. In Fig. 4(b) the spacing between these components are shown to be ~200 MHz and ~40 MHz, corresponding to ~1-m and ~5-m length mismatches with the other two amplifier paths. Each of these peaks is further shown to be made up of a dense set of lines offset by the 0.24-MHz cavity mode frequency spacing, as shown in Fig. 4(c). This is clear evidence of the mode by mode coherence in the 4-amplifier coupled array. There are only individual peaks, not separate sets of peaks for each of the amplifier paths. These mode peaks also have structure, as shown in Fig. 4(d). However, this structure does match the expected frequency shifts for the path mismatches from the individual amplifier paths. Further, the structure is not evident for peaks at low harmonics of the cavity frequency. Note that the mode peak that is expanded in Fig. 4(d) is approximately 10 MHz from the center of the broad feature of Fig. 4(c). There is no obvious spectral structure to the peaks at the lower harmonics around 10 MHz. The spectral structure of individual modes generally becomes broader as the frequency becomes larger. Possibly, it is due to polarization mode dispersion.

 

Fig. 4. Power spectra of the photodetected output of the laser array in the configuration corresponding to the optical spectra in Figs. 3.(a) and 3(b) show spectra for both configurations while (c) and (d) correspond to the ~4.1 cm smallest path difference. The horizontal bar in (a) shows the frequency range of (b), and the * in (b) and (c) mark the spectral feature that is resolved in (c) and (d), respectively.

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The high coherence of the combined laser output is shown by the detailed periodicity of the spectral structure and by comparing the power directed into the 90/10 output coupler with the output from the “loss” port of the final 2×2 coupler of the tree of couplers.. Table 1 summarizes a typical result as additional amplifiers are turned on, and also compares the power with the signal leaks through the FBG. The latter is a measure of the power outside of the FBG high reflection band. All outputs are normalized to the signals with a single amplifier operating in the four-amplifier configuration where approximately 0.5 mW is coupled out of the cavity through the 10% arm of the fiber coupler. The actual power that leaked through the grating was more than two orders of magnitude less than the Output or Loss Port powers for one amplifier. With only one amplifier operating approximately equal power exits through each of the two ports of the fiber coupler. The two amplifier configuration was set up so that the beams were coupled in this 2×2 coupler so that half the output from each amplifier is lost in the two couplers directly after the amplifiers. Now, however, the useful Output dominates the Loss Port. The ~8x increase in useful Output power from one to two amplifiers indicates that the losses from the extra fiber couplers make the configuration quite close to threshold at

Table 1. Relative output power in the four-laser configuration normalized to the values with one amplifier operating. Output is the value exiting the port of the final 2×2 coupler in the tree of couplers after the amplifiers that is directed to circulate around the laser cavity, and Loss Port is the value out of the other port of the coupler. Grating is the value measured out of the FBG and is more than two orders of magnitude smaller than the other two.

View Table

this pump level when only one amplifier is turned on [7]. The coherent coupling reduces intracavity losses. Increasing from two to four amplifiers yields a 5x power increase that is closer to the 4x increase expected in the limit of pumping well above threshold. To understand this behavior, consider a laser with relatively small output coupling (10% in our case) but large circulating losses. The small output coupling is not a requirement, but it simplifies the formulas. For a single powered amplifier the round trip transmission from amplifier output to amplifier input will be less than 2^-n, where n is the number of 2×2, 50/50 couplers in the loop. In our 4-amplifier configuration n=4 and there are additional losses due to the output coupler, circulator, polarization control elements and fiber connectors. Using the standard Rigrod formalism [11] and the small output coupling to circulating losses ratio, the output power can be expressed as:

where T _circ is the circulating power transmission of the combined elements of the cavity except the 2×2 couplers, T _oc is the outcoupling transmission of the output fiber coupler, I _sat is the saturation power, and G ₀ is the amplifier gain. Coherently combining with a second identical amplifier arm eliminates the loss at one fiber coupler into (because both amplifiers use the input) and out of (because of the coherent combining) the amplifiers and doubles the power (two amplifiers instead of one) into the coupler so that:

and one calculates a ratio of 8 when G ₀=2^(1+n/2) T ^-1/2 _circ. Similarly, one can use this value for the gain to calculate the ratio between 4 and 2 lasers and calculate an expected value of 6, similar to our measured value of 5.2, with component variability explaining the discrepancy. The limiting value of 4 for the ratio is only achieved when the number of unmatched couplers in a round trip path is small enough so that the small signal gain is large compared to the roundtrip losses.

With all four amplifiers operating, greater than 95% of the optical power is directed to the useful port of the 2×2 coupler for feedback and extracted power (approximately 1% is lost at each of the 2×2 couplers directly following the amplifiers). It is important to remember that this is occurring even with the high frequency fluctuations in laser output due to the simultaneous output from the many laser modes generated by the nonlinear mixing in the long cavities. If the 90/10 output coupler is replaced by a 50/50 coupler, or even more dramatically when the 90/10 coupler is configured to yield 10% cavity throughput and 90% output, coherence is maintained in the four-laser configuration, even though when only one amplifier is powered the laser exhibits amplified spontaneous emission and there is no wavelength or polarization control of the output.

The spectra shown in Figs. 3 and 4 required operation at relatively high circulating power levels and with long cavities, >50 W-m circulating power-path length product. This product is greater than the peak levels used in the experiments of Shirakawa, et al. [4,5] where the line narrowing effects of coherent combining are emphasized When the power is reduced or the cavity length is shortened, <10 W-m power-path length product, we observed that the spectra then consist of only a sparse set of features that vary from scan to scan over a period of seconds. The good coherence, however, is independent of whether there is a sparse or dense optical spectrum. It is generally maintained even though the optical spectrum of the laser is varying over time with the changes in laboratory conditions (i.e. temperature). We did not observe degraded performance that could be attributed to the nonlinear optical properties of the fiber (particularly the fiber gain medium) as in [3], but our experiments were at a lower circulating power level. Over several hours of operation, the degree of coherence slowly drops but it can be restored by polarization adjustment. When the coherence dropped and could not be restored, we would observe evidence of parasitic etaloning in the cavity indicating that one of the connectors had been damaged.

To observe four laser spectra like those in Figs. 3 and 4 we had to be careful about the length mismatches between the different paths. The ~1.4 cm path mismatch was the shortest that yielded this type of spectra, and in all cases the other mismatches had to be significantly longer, one on the order of a meter and the other several meters. If there were two pairs of paths with length mismatches on the order of ten centimeters or less, then the spectra observed were quite erratic, varying on a scan by scan basis. Over time, low resolution power spectra such would average to patterns similar to Fig. 4(a), but the fluctuations of individual modes washed out more detailed spectral features. There was still significant, though reduced, coherence to the output under these conditions. Unfortunately, examination of the details of these spectra always showed evidence of the parasitic etaloning due to damage of the fiber couplers. Therefore, we have been unable to unambiguously separate these parasitic effects from the intrinsic characteristics of the coupled array.

These spectra clearly demonstrate that highly coherent spectra are generated and that XMC is the optical nonlinearity that has a significant effect on the output spectra. The spectral broadening suppresses the SBS. This is true for the ring configuration even with the long fiber path placed between the optical circulator and the FBG so that there is significant optical power counterpropagating over long path lengths. To observe SBS it was necessary to limit the pump power to a level below the onset of significant XMC in the single amplifier case, and the SBS could be suppressed completely in the multi-amplifier case. The presence of XMC did not significantly reduce the coherence of the laser output.

In order to obtain an estimate of the distribution of longitudinal modes for the system of four coupled fiber cavities, we derive a consistency relation for a round-trip in the ring cavity. The analysis is an extension of the formalism previously used to describe the coupling of two cavities [7]. The coupled cavity is modeled as having a common length, L, with output coupler transmittance, T, and feedback, R, N separate arms of length L _i each with the same single pass small signal gain g _o l, and all beam combining/splitting is through lossless, 50/50 couplers. The total combined field at the output coupler E ₀(L) after the N amplifiers is described by [7]:

where K is the propagation constant and A(l) is the identical amplitude in each of the amplifier arms following the gain medium of length l. Here we are considering a symmetric case where 50/50 couplers require N to be a multiple of 2. Since E ₀(L) and A(l) are related and can be determined from a Rigrod analysis for the amplitude,

we obtain the consistency relation:

Therefore, from the real and imaginary parts, with l _n=L+L _n, we obtain

where the first equation determines the wavevector/frequency of the longitudinal modes, and the second gives the effective combining-induced losses for each mode. Notice that C _n scales as N for kl _n=mπl _n for all n. The equation that determines the output intensity of the coupled system for each independent longitudinal mode is given by

with $b_n={\left(\frac{T}{N}{\left(\frac{1}{2}\right)}^{\frac{N}{2}}{C}_n\right)}^2$. Additionally, in order to determine the relative intensity of the individual mode we define the following useful quantity

with b=(T/2)². In the single configuration we assume that only one of the amplifiers is active. The calculation is carried out by defining the quantity x=kL ₁, where L ₁ is the longest or shortest length of an amplifier arm, finding the zeroes of the sine sum above, and determining S for those values of the wavevector. The longitudinal modes appear with a separation of approximately 2π. The longitudinal mode may be above threshold if the losses are smaller than the gain, $b_n>e^{-2g_{0}l}$, otherwise no mode appears because lasing is not sustained.

Figure 5 shows the relative mode intensity, S, as a function of the phase shift due to a change in the optical frequency/wavenumber. The normalized mismatches of the lengths used for the calculation are approximately the same as in the experiment. The calculations are done using a cavity length of 800 m for the shortest array element, and 801 and 805.1 m for the two longest array elements. The second shortest array element is either 800.014 or 800.051 m as marked. A small signal, single pass gain of 12, and output coupler transmission, T²=R²=0.5, were used for the calculation. Fig. 5(a) approximately corresponds to the range of the optical spectrum of Fig. 3, highlighting the features determined by the smallest length mismatch, while 5(b) gives finer detail corresponding to the spectra of Fig. 4(b). There is a strong qualitative similarity between the calculated and observed spectra of the long cavity array.

For shorter cavity lengths and lower circulating power levels, the optical spectrum consists of a sparse set of modes that fluctuate with environmental conditions. There is much less relative fluctuation in the beat frequencies among the modes recorded by the photodiode power spectra when the XMC spectral broadening is established. The nonlinear coupling spreads the circulating power among many array modes, countering the winner-take-all type output of a homogeneously broadened medium, and mimicking the gain profile of the modes of the coherently coupled array.

 

Fig. 5. Calculated relative mode power, S, using a Rigrod-type cavity analysis of a four-laser array. The four array elements have lengths of 800, 805.1, 801, and either 800.045 or 800.014 m as marked. A single pass gain of 12 and output coupler transmission, T²=R²=0.5, were used for the calculation. (a) corresponds to a range similar to Fig. 3 and (b) is a detail, similar to Fig. 4(b).

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3. Conclusion

Our results emphasize a different regime than the enforced spectral narrowing observed by Shirakawa, et al., [4], and the regime of coherent coupling limited by nonlinear optical effects [3]. Here, many modes oscillate simultaneously. Due to the small number of amplifiers and long cavity length, even with the broadening, our results remain consistent with linear analysis that emphasizes the fundamental importance of simultaneous near-resonance of the individual amplifier path modes with the array modes. However, the XMC provides a mechanism to adjust the round trip phase of individual modes. While individual spectral features fluctuate, the overall spectral structure remains stable. Do to our problems with optical damage, we could not distinguish between fluctuations due to random factors, e.g. thermal and mechanical, and deterministic fluctuations due to the nonlinear interactions. Our results demonstrate that when XMC is the dominant nonlinear optical interaction, it does not destroy the coherence and the capacity for coherent output from the spectrally broadened array. We do not observe the deleterious performance attributed to optical nonlinearities in [3]. It appears that as the availability of low loss/high coherence modes decreases, the ability to achieve coherence may not decrease as rapidly as predicted by the linear analyses, but that the coherent output, and in particular its spectra, fluctuate more, unless there is always a single dominant mode present as in [4]. In this case, intensity spiking could be inducing the damage at our connectors. This would be consistent with the observed coherence in 8-amplifier laser arrays without intracavity FC/APC connectors previously reported [3]. Further work is needed to clarify this situation, the possible role of nonlinear dynamics [8], and the effects of optical nonlinearities such as SBS and gain saturation that induce spectral shifts distinct from harmonics of the array mode spacing and path length mismatch criteria.