Phase locking of coupled lasers with many longitudinal modes
Moti Fridman, Micha Nixon, Eitan Ronen, Asher A. Friesem, Nir Davidson
PPhase locking of coupled lasers with many longitudinal modes
Moti Fridman, Micha Nixon, Eitan Ronen, Asher A. Friesem and Nir Davidson
Dept. of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
Compiled November 7, 2018Detailed experimental and theoretical investigations on two coupled fiber lasers, each with many longi-tudinal modes, reveal that the behavior of the longitudinal modes depends on both the coupling strength aswell as the detuning between them. For low to moderate coupling strength only longitudinal modes which arecommon for both lasers phase-lock while those that are not common gradually disappear. For larger couplingstrengths, the longitudinal modes that are not common reappear and phase-lock. When the coupling strengthapproaches unity the coupled lasers behave as a single long cavity with correspondingly denser longitudinalmodes. Finally, we show that the gradual increase in phase-locking as a function of the coupling strengthresults from competition between phase-locked and non phase-locked longitudinal modes. c (cid:13)
Phase locking of two coupled lasers operating withonly one longitudinal mode was investigated over theyears [1–4]. It was shown theoretically and experimen-tally that a simple relation exist between the couplingstrength that is needed for phase locking and the fre-quency detuning between the lasers [4–6]. While a sharptransition from no phase locking to full phase lockingwhen the coupling strength exceeds a critical value is pre-dicted, the experimental results revealed a gradual tran-sition, which could be explained by introducing noise toeach laser [4,6]. For lasers with many longitudinal modesit was shown that for strong coupling strength only com-mon longitudinal modes survive, leading to full phaselocking [7–10]. Yet, the detailed behavior of phase lock-ing and the spectrum of longitudinal modes as a functionof the coupling strength between coupled lasers were sofar not reported.Here we present our investigations and results ontwo coupled fiber lasers, each operating with up to20,000 longitudinal modes. Specifically, we show how thephase locking between the two lasers and their longitu-dinal mode spectrum vary as a function of the couplingstrength which is continually and accurately controlledwith polarization elements. We find a gradual increase inthe number of longitudinal modes which are phase lockedas the coupling strength increases, leading to a gradualtransition from no phase locking to full phase lockingwithout the need to introduce noise. We support the ex-perimental results with calculations in which a modifiedeffective reflectivity model is exploited.The experimental configuration for determining thephase locking and the spectrum of longitudinal modesfor two coupled fiber lasers as a function of the couplingstrength between them is presented in Fig. 1. Each fiberlaser was comprised of a polarization maintaining Yt-terbium doped fiber, where one end was attached to ahigh reflection fiber Bragg grating (FBG), with a centralwavelength of 1064 nm and a bandwidth of about 1 nm ,that served as a back reflector mirror, the other end at-tached to a collimating graded index (GRIN) lens withanti-reflection coating to suppress any reflections back into the fiber cores, and an output coupler (OC) withreflectivity of 20% common to both lasers. The laserswere pumped with 915 nm diode lasers from the back endthrough the FBG. The two fiber lasers were forced to op-erate in orthogonal polarizations by using a calcite beam-displacer in front of a common output coupler and thecoupling strength κ between the lasers was controlled byan intra-cavity quarter wave plate (QWP). κ = sin (2 θ ),with θ the orientation of the QWP with respect to thecalcite main axes. The optical length of the cavity ofone fiber laser was 10 m while the optical length of theother was 11 . m , so each fiber has ∼ ,
000 longitudi-nal modes within the FBG bandwidth. The combinedoutput power was detected by fast photo detector whichwas connected to a RF spectrum analyzer, to measurethe beating frequencies and determine the longitudinalmode spectrum at the output [5]. We also measured thephase locking between the two fiber lasers by detectingthe interference of small part of the light from each laserwith a CCD camera, and determining the fringe visibil-ity [6]. The longitudinal mode spectrum was measuredat first when θ = 0 ( κ = 0) and then sequentially re-peated such measurement, each after rotating the QWPby 1 ◦ until we reached 45 ◦ ( κ = 1). Pump Calcite crystalFiber lasers QWP OCCCD
Experimental setup
DetectorRF spectrum analyzer
HWPFBG
Fig. 1. Experimental configuration for investigating thephase locking and the spectrum of longitudinal modesof two coupled fiber lasers as a function of the couplingstrength. FBG - fiber Bragg grating. HWP - half waveplate. QWP - quarter wave plate. OC - output coupler.We developed a model for calculating the distribu-tion of longitudinal modes and phase locking for the two1 a r X i v : . [ phy s i c s . op ti c s ] D ec oupled lasers. For each laser, the effective reflectivity[11, 12] of its own reflection and the light coupled into itfrom the other laser was calculated self consistently. Thelongitudinal mode spectrum was then derived from thetotal effective reflectivity of the two lasers. The effectivereflectivity resulting from the coupling to the other laserfor each laser can be shown to be, R eff , = (cid:18) − r (1 − √ κ ) − r κe ıl , k − r (1 − √ κ ) e ıl , k (cid:19) − , (1)where k denotes the propagation vector of the light, κ the coupling strength between the two lasers, l , thelength of each laser and r the reflectivity of the outputcoupler. To account for gain competition between thelongitudinal modes we used r = 0 .
55 as a fitting param-eter, rather than our experimental value of r = 0 . R effj e ıkl j + (cid:16) R effj e ıkl j (cid:17) + . . . = (cid:16) − R effj e ıkl j (cid:17) − , (2)where j = 1 ,
2. Finally, the output laser field of the twocoupled lasers R out , namely the amplitude of the lon-gitudinal modes, is obtained as a sum of the two selfconsistent fields, as R out = (cid:16) − R eff e ıkl (cid:17) − + (cid:16) − R eff e ıkl (cid:17) − . (3)Figure 2 shows the experimental and calculated longi-tudinal mode spectrum as a function of coupling strength κ . Figure 2(a) shows the experimental results of thelongitudinal mode spectrum as a function of couplingstrength over 200 M Hz range, and Fig. 2(b) the corre-sponding calculated results. Note that within our 1 nm laser bandwidths, the 200 M Hz range would be repeatedmany times. The experimental and calculated results arealso shown in greater detail for four specific couplingstrengths ( κ = 0 , . , . , and
1) in Figs. 2(c)-(f), re-spectively. Without coupling (i.e. κ = 0) two indepen-dent sets of frequency combs exist simultaneously, onecorresponds to the 10 m long fiber laser (15 M Hz sep-aration between adjacent longitudinal mode) while theother corresponds to the 11 . m long fiber laser (13 M Hz separation), as also seen in Fig. 2(c). Each 7th longitu-dinal mode of the 10 m long laser is very close to the8th mode of the other, so they are essentially commonlongitudinal modes. When κ is increased from 0 to 0.3the longitudinal modes that are not common graduallydisappear according to their detuning while transferringtheir energy to the remaining ones via the homogenousbroadening of the gain. The longitudinal modes withthe larger detuning disappear first while the ones withsmaller detuning disappear for larger values of κ and onlythe common longitudinal mode remains, as also seen inFig. 2(d), indicating that at this coupling strength thereis full phase locking. As the coupling strength increasesabove 0 . κ . Finally, when κ approaches unity, whereby all the light from one laser istransferred to the other, new longitudinal modes appearin between adjacent longitudinal modes,as also seen inFig. 2(f), corresponding to a single combined laser cavitywhose length is the sum of the two lasers. A m p lit ud e [ db ] Frequency [MHz](c)
Frequency [MHz](d) A m p lit ud e [ db ] Frequency [MHz](e) A m p lit ud e [ db ] Frequency [MHz](f) A m p lit ud e [ db ] Frequency [MHz] (b) C oup li ng s t r e ng t h Frequency [MHz] (a) C oup li ng s t r e ng t h (c)(d)(f)(e) Fig. 2. Experimental and calculated distributions of lon-gitudinal modes for two coupled lasers as a function ofthe coupling strength κ . (a) Experimental results; (b)calculated results; (c) κ = 0; (d) κ = 0 .
28; (e) κ = 0 . κ = 1. Solid (blue) curves denote experimental resultsand dotted (red) curves denote calculated results.Figure 2 reveals a good quantitative agreement be-tween the experimental and calculated results. In partic-ular, the observed gradual disappearance of non-commonlongitudinal modes as the coupling is increased, theirgradual reappearance when the coupling is further in-creased and finally the doubling of the frequency combat near unity coupling strength are all accurately recon-structed by our model.The results of Fig. 2 can be exploited to produce thefull phase diagram of the longitudinal modes behavior,as shown in Fig. 3. Here the behavior is presented as afunction of the coupling strength between the lasers andthe detuning between adjacent longitudinal modes. Thephase diagram includes four regions. In the first regionof weak coupling the longitudinal modes are not phaselocked for any finite detuning between them. In the sec-ond region of moderate to strong coupling and small de-tuning, the longitudinal modes are phase locked, indi-cating that the coupling is strong enough to overcomethe detuning. In the third region where the couplingstrengths is not sufficient to overcome the larger detun-ing between adjacent longitudinal modes, these modes2annot lase. Finally, in the fourth region, where the cou-pling strength approaches unity, the two coupled lasersbehave as a single and longer laser with denser longitu-dinal modes. As seen, for detuning smaller than about1 M Hz there is a direct transition from no phase lock-ing to phase locking as the coupling strength is increased.For larger detunings the transition from no phase lockingto phase locking is interrupted by the no lasing region. C oup li ng s t r e ng t h Phase locking No lasingNo phase lockingSingle laser
Fig. 3. Experimental and calculated phase diagram ofthe longitudinal modes behavior as a function of the cou-pling strength between two coupled lasers and detuningbetween adjacent longitudinal modes. Solid curves de-note experimental results. Dotted curves denote calcu-lated results.We also measured directly the phase locking (i.e. fringevisibility) between the two coupled lasers as a functionof the coupling strength. The results, presented in Fig. 4,reveal a gradual increase in phase locking as the couplingstrength become stronger. Fig. 4 also shows the ratio ofthe power in the common longitudinal mode over that ofall longitudinal modes for each coupling strength, meas-ured from the results of Fig. 2(a). The good agreementbetween this ratio and the direct measure of phase lock-ing verifies that the gradual increase in phase lockingcorresponds to the gradual disappearance of the non-common longitudinal modes which are not phase-locked.Finally, Fig. 4 also shows the ratio of the power in thecommon longitudinal mode over that of all longitudinalmodes for each coupling strength, calculated from the re-sults of Fig. 2(b) which are also in good agreement withboth measurements.To conclude, we presented how phase locking betweentwo coupled lasers which operate with many longitudinalmodes depends on the coupling strength and the detun-ing between the modes. We found that there is a grad-ual transition from no phase locking to full phase lock-ing with increasing coupling strength due to the gradualdisappearance of longitudinal modes that are not phaselocked. The experimental results were confirmed withcalculations using a modified effective reflectivity modelfor coupled lasers that we developed. Our results providea fairly complete picture for elucidating the behavior of
Coupling strength P h a s e l o c k i ng Fig. 4. Experimental and calculated phase locking as a function of the coupling strengthbetween two coupled fiber lasers. Dots denote the directly measured phase locking; starsdenote the phase locking calculated from experimental results in Fig 2(a); solid curve denotesthe phase locking calculated from the results in Fig. 2(b).11
Fig. 4. Experimental and calculated phase locking as afunction of the coupling strength between two coupledfiber lasers. Dots denote the directly measured phaselocking; stars denote the phase locking calculated fromexperimental results in Fig 2(a); solid curve denotes thephase locking calculated from the results in Fig. 2(b).the spectrum of the longitudinal modes in coupled lasersand its effect on phase locking.This research was supported in part by the USA-IsraelBinational Science Foundation.
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