Two-mode injection locking in vertical-cavitysurface-emitting lasers

M. SciamannaSupélec and Laboratoire Matériaux Optiques, Photonique et Systèmes, Centre National de la Recherche Scientifique,

Unité Mixté de Recherche 7132, Unité de Recherche Commune Supélec et Université de Metz,2 Rue Edouard Belin, F-57070 Metz, France

K. PanajotovDepartment of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium

Received June 9, 2005; accepted July 15, 2005

We report on the dynamics that accompany polarization switching induced by orthogonal optical injection ina vertical-cavity surface-emitting laser (VCSEL). As the injection strength increases, the VCSEL bifurcatesto injection-locked steady states and time-periodic and possibly chaotic dynamics. Of particular interest is atwo-mode injection-locking solution, i.e., locking of the two VCSEL polarization modes. A detailed stabilityanalysis unveils new bistability mechanisms in optical injection problems. © 2005 Optical Society of America

OCIS codes: 250.7260, 190.3100, 260.5430.

The vertical-cavity surface-emitting laser (VCSEL)emits linearly polarized (LP) light but may switch be-tween two orthogonal LP modes as we vary the tem-perature or the injection current.1 Injection of LPlight orthogonal to that of the free-running VCSELmay control the slave VCSEL and induce locking tothe master laser frequency and polarization,2,3 henceleading to applications in reconfigurable opticalinterconnects,4 all-optical flip-flop operation,5 and op-tical memory.6 However, to the best of our knowledgethe polarization dynamics that accompanies this op-tical injection-induced polarization switching (PS) inVCSELs has never been studied. We report here anin-depth stability analysis of the laser system andbring new light to this optical injection problem.

Our model extends the model of Martin-Regaladoet al.1 to y LP optical injection and is written as

Fx = ��1 + i���DFx + idFy − Fx� − i��p + ���Fx − �aFx,

�1�

Fy = ��1 + i���DFy − idFx − Fy� + i��p − ���Fy + �aFy

+ �injEinj0 , �2�

D = − �eD − �eD�Fx�2 − �eD�Fy�2 + �e�

− i�ed�FyFx* − FxFy

*�, �3�

d = − �sd − �ed�Fx�2 − �ed�Fy�2 − i�eD�FyFx* − FxFy

*�.

�4�

Fx,y are the slowly varying x and y LP componentsof the electric field. D and d are two carrier inversionnumbers.1 Parameters are chosen such that the free-running VCSEL exhibits a stationary x LP state; i.e.,the injection is LP and orthogonal to the free-runningVCSEL polarization: �p=30 rad/ns, �a=0.5 ns−1, �s=50 ns−1, �=3, �=300 ns−1, �e=1 ns−1, and �=1.5.�inj is the injection rate, Einj

0 is the injected-field am-

plitude, and �� is the detuning: ��=�inj−�th, where�th= ��x+�y� /2. �x,y are the frequencies of the twoVCSEL LP modes: �x,y= ��p±��a. �inj is fixed to�inj=�, which corresponds to the optimal case of amode-matched injected input beam.1

Equations (1)–(4) successfully reproduce the PSwith bistability that has been reportedexperimentally.2,3 We focus here on the laser dynam-ics and its bifurcations. Figure 1 shows a bifurcationdiagram of Ix,y��Fx,y�2 as a function of the normalizedinjected power Pinj /I0 and for ��=−�p. Pinj�Einj

0 2

and I0 is the total intensity without injection. As weincrease the injection strength, the intensity of thenormally depressed y LP mode increases, and theVCSEL successively bifurcates to qualitatively differ-ent dynamics. Figure 2 shows time traces of intensityof the x LP mode ��a1�– �a2�� and in the y LP mode��a1�– �h2�� corresponding to the qualitatively differ-ent dynamics indicated by (a)–(h) in Fig. 1. For verysmall injected power, the two LP mode intensities ex-hibit anticorrelated time-periodic dynamics (a1) and(a2) at �0.48 GHz, which is much smaller than therelaxation oscillation (RO) frequency �fRO��2��e��−1��1/2 / �2��2.77 GHz or the beat frequency be-tween the two LP modes �2�p / �2��9.5 GHz�. An op-tical spectrum analysis (not shown) unveils a wavemixing mechanism between the master and the slavefields. The frequency offset between the two fields is

Fig. 1. (Color online) Bifurcation diagram of Ix andIyversus Pinj /I0.

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0146-9592/05/212903-3/$15.00 © 2005 Optical Society of America

given by foff��a / �2��=0.24 GHz. The wave mixingis linked to a pulsation of the population inversion,which in turn yields an intensity modulation at �2 foff. For a slightly larger injection amplitude the LPmode intensities still exhibit a slow time-periodic dy-namics, but now they relax with faster, in-phase os-cillations at RO frequencies (b1) and (b2). For a stilllarger amount of optical injection, the two LP modesexhibit a stationary behavior [(c1) and (c2)]. Both LPmodes are locked in frequency to the master laser;i.e., this steady state corresponds to two-mode injec-tion locking, which we call also elliptically polarizedinjection locking (EPIL). The EPIL steady state un-dergoes a Hopf bifurcation to time-periodic, in-phasedynamics at the RO frequency in the two LP modes(d1) and (d2). As we increase the injection strengthfurther, the time-periodic dynamics bifurcates to pe-riod two [(e1) and (e2)], period four [(f1) and (f2)], etc.until the chaotic regime [(g1) and (g2)] with irregularbursts of pulses that extend over a large range of in-tensity values is reached. Finally, the bifurcation cas-cade leads to PS induced by optical injection with aswitch-off of the x LP mode and injection locking ofthe y LP modes [(h1) and (h2)]. As shown in Fig. 1,the VCSEL exhibits bistability between the y LPinjection-locked steady-state and the time-periodic,period-doubled, or even chaotic regimes that emergesfrom the EPIL steady state. This bistability is foundfor injection-strength values between that which cor-responds to the saddle-node bifurcation of the y LPinjection-locked solution (see the bifurcation analysisof Fig. 3) and that which corresponds to the PS point.It is worth mentioning that elliptically polarizedsteady states are also found in the PS of a free-running VCSEL with increasing current and exhibitbistability with the x LP state.7 However, their bifur-cation is very different from that of the EPIL solu-tions.

To gain insight into the bifurcations in the EPILsteady state, we have used the packageDDE-BIFTOOL,8 which allows one to continuebranches of steady states and time-periodic solutions

irrespective of their stability. In Fig. 3 we plot indarker (lighter) points and curves the values of Ix�Iy�that correspond to the injection-locked solutions as afunction of Pinj /I0 and to different values of ��.Stable (unstable) parts of the branches are indicatedby solid (dashed) curves. Saddle-node, transcritical,and Hopf bifurcations are labeled by squares, aster-isks, and diamonds, respectively. Filled diamondsmark the supercritical Hopf bifurcations, i.e., thosethat modify the stability of the steady states.

In Fig. 3(d), ��=�x=−�p+��a. As we increase theinjected power from zero, a stable EPIL steady stateappears with IxIy. For larger Pinj /I0, it undergoes asupercritical Hopf bifurcation. For a still larger valueof Pinj /I0, the only stable steady state is the y LPinjection-locked solution that appears from a saddle-node bifurcation. The node corresponds to the branchwith the largest Iy�0 and Ix=0. The branches ofEPIL and y LP injection-locked solutions cross at atranscritical bifurcation. For larger negative detun-ings [Figs. 3(e) and 3(f)], the EPIL solution suddenlyunfolds. A saddle-node bifurcation leads to twobranches of solutions with different Ix,y. The branchwith the largest (smallest) Ix �Iy� values is a stable so-lution (node). The second branch corresponds to asaddle-type solution. The node undergoes a super-critical Hopf bifurcation. Interestingly, the saddle-node bifurcation on the EPIL steady state might belocated at a larger injection strength than the saddlenode on the y LP injection-locked solution [Fig. 3(f)],hence leading to bistability between the twoinjection-locked solutions. Figure 3(f) thereforecomplements the bistability shown in Fig. 1 [orequivalently in Fig. 3(e)]. For smaller negative detun-ings [Figs. 3(a)–3(c)], a similar unfolding mechanismoccurs on the EPIL solution. In Fig. 3(c) a stablebranch of EPIL is still possible, but, as we decrease��, the stability of the EPIL solution is lost as soon

Fig. 3. Steady states and their stability for �� �rad/ns�=(a) −16, (b) −20, (c) −28, (d) −28.5, (e) −30, (f) −34. Ix �Iy� isplotted in darker (lighter) points and curves as a functionof Pinj /I0. Symbols are defined in the text. Inserts, enlarge-ments around branches of EPIL solutions.

Fig. 2. Time traces of Ix,y for Pinj /I0= (a1), (a2) 0.03%; (b1),(b2) 0.16%; (c1), (c2) 0.18%; (d1), (d2) 0.35%; (e1), (e2)0.47%; (f1), (f2) 0.51%; (g1), (g2) 0.54%; and (h1), (h2)0.55%.

2904 OPTICS LETTERS / Vol. 30, No. 21 / November 1, 2005

as the Hopf bifurcation coincides with the saddle-node bifurcation. For detuning values beyond thiscodimension-two bifurcation, the Hopf bifurcationthat determines the stability of the EPIL steady stateis now on the saddle branch, and the EPIL solution isthen completely unstable [Figs. 3(a) and 3(b)].

Figure 4 shows the saddle-node, transcritical, andHopf bifurcations of the injection-locked solutions inthe plane �Pinj /I0 , ���. Bifurcations of the y LPinjection-locked solution are labeled Hy

1, Hy2, and Hy

3

for the Hopf bifurcations and Fy1 for the fold bifurca-

tion. Bifurcations on the EPIL solution are labeledHxy

1 and Hxy2 for the Hopf bifurcations and Fxy

1 and Fxy2

for the fold bifurcations. Supercritical (subcritical) bi-furcations are indicated by solid (dashed) curves. T isa transcritical bifurcation.

The stable y LP injection-locked solution appearsfrom saddle-node bifurcation Fy

1 and destabilizes withsupercritical Hopf bifurcation Hy

1, which then leads toa stable time-periodic dynamics with only the yLP mode lasing. A codimension-two Gavrilov–Guckenheimer bifurcation9 is located at G3, wherethe Hopf and saddle-node bifurcations merge. PointG3 is located very close to ����p−��a, i.e., for �inj��y. This detuning value is also the one for which PSrequires the minimum injected power, in agreementwith experiments.2 By contrast, the EPIL solution ex-ists only in a small range of detuning values centeredabout ��=−�p+��a (i.e., for ����x). Except for thatdetuning value, the EPIL solution appears from asaddle-node bifurcation Fxy

1 or Fxy2 . As we increase the

injected power, the EPIL solution destabilizes withHopf bifurcation Hxy

1 . The range of detunings inwhich one can find a stable EPIL solution is delim-ited by two codimension-two Gavrilov–Guckenheimer

points, G1 and G2. The saddle-node bifurcation onEPIL solution Fxy

1 and the saddle-node bifurcation ony LP mode solution Fy

1 intersect at point B. For de-tuning values between B and G2 a stable EPIL solu-tion coexists in a small range of injection strengthwith the y LP injection-locked steady state [bistabil-ity shown in Fig. 3(f)]. Figure 4 also shows how thedetuning and injection strength coordinates of theEPIL stability boundaries are modified when �p val-ues (and therefore wx and wy are changed��b1� and �b2��). This confirms that the EPIL solutionis always found for �� close to �x. Moreover, asshown by curves (c1)–(c3), the EPIL solution is ro-bust against large variations of �s: An increase of �sstill gives the possibility for a stable EPIL but in asmaller range of �� and Pinj /I0. Our conclusions aretherefore not specific to those VCSELs whose dynam-ics are strongly influenced by spin-relaxationmechanisms.1

In summary, orthogonal (y LP) optical injection inan x LP VCSEL may lead to PS with injection locking(y LP), but, interestingly, another, two-mode injectionlocking (EPIL) may appear before the PS and desta-bilize to time-periodic and possibly chaotic dynamics.A detailed stability analysis shows that (1) a stableEPIL is found for �� close to the resonance condition����x, (2) the EPIL’s stability is delimited by twoGavrilov–Guckenheimer codimension-two bifurca-tions, and (3) stable EPIL may coexist with y LP in-jection locking, a new case of bistability in optical in-jection problems.

K. Panajotov is also with the Institute of SolidState Physics, 72 Tzarigradsko Chaussee Boulevard,1784 Sofia, Bulgaria. We thank T. Erneux for manystimulating discussions. We acknowledge the supportof COST 288. M. Sciamanna’s e-mail address ismarc.sciamanna@metz.supelec.fr.

References

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Fig. 4. Top, mapping of the steady-state stability bound-aries. Bifurcations on the y LP (two-mode) injection-lockedsolution are plotted in lighter (darker) curves. Symbols andlabels are defined in the text. Bottom, comparison of the bi-furcation boundaries on EPIL for (b1) �p=−30 rad/ns and(b2) �p=−40 rad/ns or for (c1) �s=50 ns−1, (c2) �s=100 ns−1, and (c3) �s=300 ns−1, with the other parametersremaining the same. Inserts, enlargements around severalparts of the map.

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