A two color Fabry-Perot laser subjected to optical injection in both modes is examined experimentally and theoretically. The theoretical analysis predicts a bistability between locked states due to a swallow-tail bifurcation, which is unique to the dual injection system. This bistability is confirmed experimentally and used as the basis for an all optical memory element with switching times below 500 ps. (C) 2011 American Institute of Physics. (doi: 10.1063/1.3605584

Amann, Andreas

Heinricht, Patrycja

O'Brien, Stephen

Osborne, Simon

Wetzel, Benjamin

Applied Physics Letters

Patrycja Heinricht

Benjamin Wetzel

Stephen O’Brien

Andreas Amann

Simon Osborne

Crossref

Bistability in an injection locked two color laser with dual injection

O\u27Brien, Stephen

Irish Universities

Cork Open Research Archive

Title Bistability in an injection locked two color laser with dual injectionAuthor(s) Heinricht, Patrycja; Wetzel, Benjamin; O'Brien, Stephen; Amann,Andreas; Osborne, SimonPublication date 2011Original citation Heinricht, P., Wetzel, B., O’Brien, S., Amann, A. and Osborne, S.(2011) 'Bistability in an injection locked two color laser with dualinjection', Applied Physics Letters, 99(1), pp. 011104. doi:10.1063/1.3605584Type of publication Article (peer-reviewed)Link to publisher'sversionhttp://aip.scitation.org/doi/abs/10.1063/1.3605584http://dx.doi.org/10.1063/1.3605584Access to the full text of the published version may require asubscription.Rights © 2011 American Institute of Physics.This article may bedownloaded for personal use only. Any other use requires priorpermission of the author and AIP Publishing. The following articleappeared in Heinricht, P., Wetzel, B., O’Brien, S., Amann, A. andOsborne, S. (2011) 'Bistability in an injection locked two color laserwith dual injection', Applied Physics Letters, 99(1), pp. 011104 andmay be found at http://aip.scitation.org/doi/abs/10.1063/1.3605584Item downloadedfromhttp://hdl.handle.net/10468/4317Downloaded on 2018-08-23T18:47:24ZBistability in an injection locked two color laser with dual injectionPatrycja Heinricht, Benjamin Wetzel, Stephen O’Brien, Andreas Amann, and Simon Osborne,Citation: Appl. Phys. Lett. 99, 011104 (2011); doi: 10.1063/1.3605584View online: http://dx.doi.org/10.1063/1.3605584View Table of Contents: http://aip.scitation.org/toc/apl/99/1Published by the American Institute of PhysicsBistability in an injection locked two color laser with dual injectionPatrycja Heinricht,1,2 Benjamin Wetzel,1,3 Stephen O’Brien,1 Andreas Amann,1,4and Simon Osborne1,a)1Tyndall National Institute, University College Cork, Lee Maltings, Cork, Ireland2Department of Physics, University College Cork, Ireland3Institut FEMTO-ST, UMR 6174 CNRS Universite´ de Franche-Comte´, 25030 Besancon, France4School of Mathematical Sciences, University College Cork, Ireland(Received 17 May 2011; accepted 9 June 2011; published online 5 July 2011)A two color Fabry-Pe´rot laser subjected to optical injection in both modes is examinedexperimentally and theoretically. The theoretical analysis predicts a bistability between lockedstates due to a swallow-tail bifurcation, which is unique to the dual injection system. Thisbistability is confirmed experimentally and used as the basis for an all optical memory elementwith switching times below 500 ps.VC 2011 American Institute of Physics. [doi:10.1063/1.3605584]Optical injection of laser diodes gives rise to a range ofnonlinear phenomena including chaos and multistability andhas been studied extensively for the single mode laser.1Recently, dual-mode lasers, a family that includes verticalcavity surface emitting lasers (VCSELs), ring lasers, pho-tonic crystal lasers, and two color Fabry-Pe´rot (FP) lasers,subjected to a single optical injection have been studied dueto potential applications in all-optical signal processing.2–6An additional injected mode can also add functionality tomultimode FP lasers.7–9 However, the understanding of thesedual injected FP lasers is complicated by the effects of theuninjected modes.In this letter, we examine both theoretically and experi-mentally a two color FP laser subject to dual optical injec-tion. Our model allows us to identify a bistable regionbetween two locked states, which is born from a swallow-tailbifurcation. We then demonstrate that this bistability can beused as an all-optical memory element.We consider a two color FP laser of length 545 lm withan InAlGaAs/InP active region. Mode selection is achievedby perturbing the FP mode spectrum using slotted regionsetched into the ridge waveguide. The exact positions of theseslots are determined by relating the refractive index profilein real space to the threshold gain in wavenumber space andthen solving the corresponding inverse problem.10The optical spectrum of the free running two-color laseris shown in Fig. 1, where the spacing between the shortwavelength mode m2 and long wavelength mode m1 is 480GHz corresponding to a frequency separation between sixfundamental FP modes. The large frequency spacing resultsin weak coupling between the modes and therefore stablesimultaneous lasing of both modes is observed, which is incontrast to both the VCSEL and ring laser. In the following,we choose the pump current such that both lasing modeshave equal optical intensity in the absence of opticalinjection.Using two independent tunable master lasers, we nowstudy the response of the two-mode laser to simultaneous op-tical injection into both modes. We have previously shownthat a four dimensional rate equation model accuratelydescribes the dynamics of a two color laser subject to opticalinjection into one of the two modes.6 In the dual injectioncase, this model generalizes to a five-dimensional system forthe normalized complex fields, E1/2, and the normalizedexcess carrier density, n, whose dynamic evolution is givenby_E1 ¼ 12ð1 þ iaÞðg1ð2nþ 1Þ  1Þ  iDm1 E1 þ K1;_E2 ¼ 12ð1 þ iaÞðg2ð2nþ 1Þ  1Þ  iDm2 E2 þ K2;T _n ¼ P n ð2nþ 1ÞXmgmjEmj2; (1)where the nonlinear modal gain isgm ¼ g0m1 þ eXnbmnjEnj21: (2)In this system, the bifurcation parameters are the normalisedinjection strengths, K1 and K2, and the normalised angularfrequency detunings, Dm1 and Dm2. Other values used arechosen to be in agreement with experimental data withFIG. 1. (Color online) Left: the optical spectrum of the two color laser.Right: locking and unlocking boundaries in K1-K2 plane withDm1¼Dm2¼14 GHz. The points show the experimentally determinedlocking () and unlocking () boundaries, and the lines show the numeri-cally calculated saddle-node (SN) and subcritical torus (TR) bifurcations.a)Electronic mail: simon.osborne@tyndall.ie.0003-6951/2011/99(1)/011104/3/$30.00 VC 2011 American Institute of Physics99, 011104-1APPLIED PHYSICS LETTERS 99, 011104 (2011)the phase-amplitude coupling, a¼ 2.6, and the normalisedpump current, P¼ 0.4. The parameter T¼ ss/sph¼ 800 is theratio between carrier lifetime, ss, and the photon lifetimesph¼ 1.02 ps. The modal gain includes the linear modal gain,g0m ¼ 1, and the nonlinear gain taking into account the crossand self saturation, which are determined by ebnm. We choosee ¼ 0:01 and b12¼ b21¼ 2/3, and b11¼ b22¼ 1 to be consist-ent with the stability of the free running two-color laser.6We first consider the case of equal negative detuningfrequencies Dm1¼Dm2< 0. Using the numerical continuationsoftware AUTO (Ref. 11), we find that the dynamical system(1) has a stable fixed point for large injection strengths K1and K2. This region is bounded from below by a saddle-node(SN) bifurcation as shown in the right hand panel of Fig. 1.The SN line is almost circular in the K1-K2 plane and mani-fests itself experimentally as an unlocking transition from atwo-mode locked state with decreasing injection strengths ofthe master lasers. The system (1) also possesses a stable limitcycle solution for small injection strengths which isdestroyed via a subcritical torus bifurcation with increasinginjection strengths (TR line in Fig. 1). Similar to the SN line,the TR line is again almost circular in the K1-K2 plane, andboth lines are concentric around the origin. For sufficientlynegative detuning Dm1¼Dm2<12 GHz, the radius of thetorus line is larger than the radius of the SN line, whichopens up a region of bistability between the two bifurcationlines, where a stable limit cycle and a stable fixed point solu-tion coexist. Experimentally, the torus bifurcation is identi-fied by a transition from an unlocked to a locked state withincreasing injection strength. The experimental data in theright hand panel of Fig. 1 accurately confirm the predictedshape of the bistability region.In the limit of K1 ! 0, the limit cycle solution corre-sponds to a two-mode equilibrium solution of the four-dimen-sional single mode injection model. In this case, thebistability in Fig. 1 reduces to the previously reported bistabil-ity between single mode locked and two mode equilibriumstate.6 However, this bistability involves partly unlockedstates and the switching times are limited to values > 2 ns.In order to obtain faster switching, it is therefore desira-ble to use a bistability between states in which both modesare locked. This can be achieved by using unequal detunings,Dm1=Dm2. Figure 2 shows the calculated SN bifurcation inthe K1-K2 plane for various Dm1 and fixed Dm2. The SN lineis deformed from its circular form with equal detuning, andat a swallow-tail bifurcation, a pair of cusps are born aroundDm1¼ 12 GHz. The cusps, which indicate the presence ofhysteresis,12 form two corners of a triangular shaped regionwith two stable equilibria which we label by F21 and Fs2. Wenote that this bistable region is a unique feature of opticalinjection in both modes. In contrast to the bistability inFig. 1, it does not extend to the single mode injection limitand is a consequence of the SN bifurcation only, withoutinvolving the torus bifurcation.Let us now fix K1 at a value which intersects the bistableregion. Increasing K2 from zero, the laser is initially lockedto the Fs1 state and remains in this state until the SN curve isreached. At this point, the stable Fs1 disappears and the laseris locked to the Fs2 state. Likewise, now decreasing K2, thelaser remains locked in the Fs2 state until the lower part ofthe triangular region is reached and Fs2 collides with anunstable fixed point. Then, the system returns to the Fs1 state.This predicted hysteresis behavior is confirmed experimen-tally in the inset of Fig. 3, where a typical hysteresis loop forthe intensity of mode 1 is shown for varying K2.FIG. 2. (Color online) The SN bifurcation for Dm2¼18 GHz and variousvalues of Dm1 in the K1-K2 plane. For Dm1>12 GHz, a bistable regionappears as shown by the shaded region with two cusp bifurcations in the cor-ners. The inset shows the bifurcation diagram for Dm2¼18 GHz andDm1¼4 GHz, including two Bogdanov-Takens points (BT) and the associ-ated Hopf (HB) and homoclinic (P) bifurcation lines. The black shadedregion, which is bounded by the Hopf bifurcation line and the horizontalsaddle node bifurcation line, indicates the presence of a stable limit cycle. Inthe black and gray striped region within the triangle bounded by saddle nodebifurcation lines, a stable limit cycle and a stable fixed point coexist.FIG. 3. (Color online) Partition of the K1-K2 plane for Dm1¼8 GHz andDm2¼18 GHz into three regions: I contains no locked states; II comprisesof one locked state and one unlocked state; and region III consists of onlylocked states. The inset shows hysteresis of the linear power in mode 1 atK1¼ 0.01 for up (P:) and down (P;) sweep in K2 as indicated by the arrowsin the main panel. The color of region III represents the contrast ratio P: toP; (in dB) between these up and down sweeps, where the dark shading indi-cates a large contrast of optical power in mode one.011104-2 Heinricht et al. Appl. Phys. Lett. 99, 011104 (2011)From Fig. 2, the size of the bistable region can beincreased by simply increasing the ratio Dm2:Dm1. However,as the detuning of m1 becomes comparable to the relaxationoscillation ( 5 GHz), Bogdanov-Takens bifurcations appearalong the SN bifurcation line, which result in the appearanceof a Hopf bifurcation (HB) and a homoclinic bifurcation (P)as shown in the inset of Fig. 2. Between the HB and P bifur-cation lines, we observe stable limit cycle dynamics. Whilethese bifurcations are interesting from a dynamical point ofview, they limit the region of bistability between lockedstates and do not contribute to improve switching. For thecurrent letter, we therefore focus on the analysis of the dy-namics in the region where only locked states exist.To explore this bistability experimentally, we fix Dm1,Dm2, and K2, and sweep K1 up and down measuring the opti-cal power in both modes. By monitoring the rf power spec-trum, we are able to distinguish between locked andunlocked regions. This allows us to partition the K1-K2 planeinto three regions as shown in Fig. 3. In region I no lockedstates exists, while in region II, bistability between lockedand unlocked states is observed. Here, we focus on regionIII, where only locked states exist. The predicted triangularshape of the bistability region is confirmed experimentallyby the shape of the colored region in Fig. 3. In this region,the contrast ratio of the power in mode 1 reaches values upto 8 dB.We exploit this bistability to form an all-optical memoryelement which is triggered by modulating the optical injectionin mode 2 while keeping the injection in mode 1 fixed at aconstant value K1. As shown in the upper panel of Fig. 4, themodulation consists of a 10 ns set pulse with 100 ps risetimeand a 10 ns reset pulse with 5 ns risetime. This modulation isalso indicated by the arrows in Fig. 3. In the middle and lowerpanels of Fig. 4, we observe that the laser is switched betweentwo stable locked states with different modal intensities. Thecontrast ratio in m1 is 8 dB and the fall time is < 500 ps. Thisis faster than in the single injection case,6 where the switchingspeed is limited by large transients of the order of several ns,and the emitted optical power is of the order several mWs.Switching times of < 20 ps have been reported; however, opti-cal powers are limited to 100 s of lW.13In conclusion, we have studied a two-color FP laser sub-ject to dual injection and demonstrated a bistability betweenlocked states. We have explained the bistability as originat-ing from a swallow-tail bifurcation. Although, we havefocused our analysis on a two-color FP laser, we note thatthis bistability will be present in other dual injectionsystems.This work was supported by Science Foundation Ire-land. The authors thank Eblana Photonics for fabrication oflasers.1S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, Phys. Rep.416, 1 (2005).2I. Gatare, M. Sciamanna, M. Nizette, and K. Panajotov, Phys. Rev. A 76,031803 (2007).3L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert,and G. Verschaffelt, Phys. Rev. Lett. 102, 193904 (2009).4C.-H. Chen, S. Matsuo, K. Nozaki, A. Shinya, T. Sato, Y. Kawaguchi, H.Sumikura, and M. Notomi, Opt. Express 19, 3387 (2011).5S. V. Zhukovsky and D. N. Chigrin, Opt. Lett. 34, 3310 (2009).6S. Osborne, K. Buckley, A. Amann, and S. O’Brien, Opt. Express 17,6293 (2009).7S. Yamashita and J. Suzuki, IEEE Photon. Technol. Lett. 16, 1176 (2004).8J. Ho¨rer and E. Patzak, IEEE J. Quantum Electron. 33, 596 (1997).9Y. D. Jeong, J. S. Cho, Y. H. Won, H. J. Lee, and H. Yoo, Opt. Express14, 4058 (2006).10S. O’Brien, S. Osborne, K. Buckley, R. Fehse, A. Amann, E. P. O’Reilly,L. P. Barry, P. Anandarajah, J. Patchell, and J. O’Gorman, Phys. Rev. A74, 063814 (2006).11E. J. Doedel, A. R. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Oldeman,R. Pfaffenroth, B. Sandstede, X. Wang, and C. Zhang, “AUTO-07P: Con-tinuation and bifurcation software for ordinary differential equations,”Technical Report (Concordia University Montreal, Canada, 2007).12Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed.(Springer, New York, 2004).13L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. J. Geluk,T. De Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier,Nat. Photonics 4, 182 (2010).FIG. 4. The intensity times traces of the modulated injection K2 (top), andthe modes m1 (middle) and m2 (bottom) for detunings as in Fig. 3. The modu-lation range is indicated by the arrows in Fig. 3. The inset shows the fallingedge of mode m1 switching from on to off states, where the 90:10 switchingtime is < 500 ps.011104-3 Heinricht et al. Appl. Phys. Lett. 99, 011104 (2011)

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Bistability in an injection locked two color laser with dual injection

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