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Directmodulationofsemiconductorringlasers:Numericalandasymptoticanalysis

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DOI:10.1364/JOSAB.29.001983

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Direct modulation of semiconductor ring lasers:numerical and asymptotic analysis

Sifeu Takougang Kingni,1,2,* Guy Van der Sande,1 Lendert Gelens,1 Thomas Erneux,3 and Jan Danckaert1

1Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium2Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Department of Physics,

Faculty of Science, University of Yaoundé I, Po. Box 812, Yaoundé, Cameroon3Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine, Code Postal 231, 1050 Bruxelles, Belgium

*Corresponding author: Sifeu.Takougang.Kingni@vub.ac.be

Received March 14, 2012; revised May 25, 2012; accepted May 29, 2012;posted June 1, 2012 (Doc. ID 162968); published July 17, 2012

We investigate numerically the dynamical behavior in a semiconductor ring laser (SRL) subject to a periodic mod-ulation of the injection current. By varying the amplitude and frequency of the modulation at a fixed bias current,different dynamical states including periodic, quasi-periodic, and chaotic states are found. At frequencies com-parable to the relaxation oscillation frequency, the intensities of the counterpropagating modes of the SRLs mayexhibit in-phase chaotic motion similar to single mode semiconductor lasers. However, antiphase chaotic oscilla-tions in the modal intensities are observed for modulation frequencies significantly lower than the relaxation os-cillations frequency. We show that this antiphase chaotic regime does not involve carrier dynamics and is a resultof the underlying symmetry of the SRL. We derive a two-dimensional asymptotic model valid on time-scales long-er than the relaxation oscillations, which reproduces the observed dynamical behavior. In a further simplification,we can link this reduced set of equations to Duffing-type oscillators. © 2012 Optical Society of America

OCIS codes: 140.1540, 140.5960.

1. INTRODUCTIONThe interaction between counterpropagating modes in ringlasers has attracted a lot of attention since the first conceptionof a ring laser in 1965 by Aronowitz [1]. In the 70s, 80s, and90s, ring lasers were studied in the context of He-Ne ringlasers [2], CO2 lasers [3,4], and solid-state lasers [5–7].Matsumoto and Kumabe reported the first semiconductor ringlaser (SRL) in 1977 [8]. An important limitation of their designwas that light could not be coupled out of the ring cavity, suchthat lasing action had to be observed from reflections of thedevice sidewalls. In 1980 a SRL including an output waveguidewas designed and fabricated [9]. In recent years, SRLs haveagain become the topic of intense study due to their uniquefeature of directional bistability, which opens up the pos-sibility of using them in systems for all-optical switching, gat-ing, wavelength-conversion functions, and optical memories[10–15]. Also, the bidirectionality of the SRL has been widelyexploited for rotation sensing applications [16,17]. SRLs donot require cleaved facets or gratings for optical feedbackand are thus particularly suited for monolithic integration[18]. Moreover, SRLs have been recognized to be ideal opticalprototypes of nonlinear Z2-symmetric systems [19], appearingin many fields of physics. SRLs are interesting because of theinteraction between the two counterpropagating fields sup-ported by the ring. Both modes are coupled nonlinearlythrough gain saturation effects and linearly due to localizedand/or distributed backscattering. This interaction has beenstudied in many papers theoretically [17,20–25] and experi-mentally [18,22,26–30].

In this work, we will focus on the nonlinear dynamicalbehavior of externally driven SRLs. This topic is of great in-terest both for fundamental and applied sciences [24,25]. In

the literature, the nonlinear dynamics of externally drivensemiconductor lasers have been widely investigated becauseof the important roles semiconductor lasers play in conven-tional and chaotic optical communications. In particular,the study of the dynamical properties of semiconductor laserssubject to current modulation has received a lot of attention.Depending on the modulation parameters and the internal la-ser parameters, the laser may exhibit ultrafast sharp pulses inthe emitted power, but also complex time-dependent dy-namics such as period doubling and chaotic pulsating behav-ior have been observed [31–43]. The majority of works on thedirectly modulated semiconductor lasers have been per-formed on conventional edge-emitting lasers [31–38] and ver-tical cavity surface-emitting lasers [39–43]. However, to thebest of our knowledge, the nonlinear dynamical behavior ofa current-modulated SRL remains unaddressed. Recently,Cai et al. [44] reported preliminary results on the experimentalinvestigation of direct modulation of 30 μm radius SRL witherror-free operation up to 1.5 Gb ∕ s. They found that the biascurrent is a key parameter with large impact on the dynamicproperties of SRLs when operating in a regime with multiplelongitudinal modes, leading them to numerically analyze mod-els for multimode operation of SRLs wavelengths [45]. Here,we shall focus purely on the single-longitudinal and single-transverse mode regime of the SRL. We will analyze thedynamics of a current-modulated SRL both at modulation fre-quencies close to the relaxation oscillation frequency and atmodulation frequencies that are smaller.

The paper is organized as follows. The system under studyis presented in Section 2. The two following sections are de-voted to the dynamical behavior of a current-modulated SRLin two different frequency ranges. Section 5 is devoted to an

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asymptotic interpretation of the observed behavior. Finally,the conclusion is given in Section 6.

2. MODEL AND OPERATING REGIMES OF ASEMICONDUCTOR RING LASERThe dynamical behavior of a SRL operating in a single-longitudinal, single-transverse mode can be adequatelymodelled by a two-mode model. Such a model has been suc-cessfully applied to describe SRLs as gyroscopes [46] andexplained the regime of alternate oscillations, stochasticmode-hopping between multiple stable attractors and excit-ability as experimentally observed in SRLs [28–30,47]. Therate equations for the slowly varying complex amplitudesof the electric fields of the counterpropagating modes Ecw

and Eccw and the carrier number N read [47,48]:

dEcw;ccw

dt0� κ�1� iα��N�1 − sjEcw;ccwj2 − cjEccw;cwj2� − 1�

× Ecw;ccw − keiϕkEccw;cw (1)

dNdt0

� γ�μ − N − N�1 − sjEccwj2 − cjEcwj2�jEccwj2

− N�1 − sjEcwj2 − cjEccwj2�jEcwj2�; (2)

where t0 is the time. k �����������������k2d � k2c

qis the amplitude and ϕk �

arctan �kc ∕ kd� the phase of the backscattering coefficient,which is an explicit linear coupling between the two counter-propagating fields typical for semiconductor ring and disk la-sers. kd � 0.0327 ns−1 represents the dissipative coupling andkc � 0.44 ns−1 the conservative coupling. The linewidth en-hancement factor α � 3.5 represents the amplitude-phasecoupling. The nonlinear gain saturation due to, for example,spatial-hole burning or carrier heating effects, is modelled byself- and cross-saturation coefficients s � 0.005 and c � 0.01,respectively. κ � 100 ns−1is the field decay rate, γ � 0.2 ns−1

is the decay rate of population inversion, and μ is the normal-ized injection current, modulated according to

μ � μdc � μm sin�2πf mt0�; (3)

where μdc is the dc bias injection current, μm is the modulationamplitude, and f m is the modulation frequency.

Before studying the effect of current modulation, weshortly revisit the device operation without current modula-tion in Fig. 1 (μm � 0.0), which depicts the orbit diagramsof Eqs. (1) and (2), taking the extremes of the modal intensi-ties ICCW � jECCWj2 and ICW � jECWj2 as a function of the dcbias injection current μdc for two exemplary values of thebackscattering phase ϕk.

At the threshold current, μdc ≈ 1.0, laser action starts. Asillustrated by the vertical dashed lines of Fig. 1, five distinctoperating regimes of SRL can be identified for increasing thevalues of the dc injection current μdc: (I) bidirectional contin-uous wave operation (bi-CW), (II) tristable continuous waveoperation (tri-CW), (III) bistable unidirectional operation (bis-UNI), (IV) bidirectional with alternate oscillations (bi-AO),and (V) tristable operation with alternate oscillations (tri-AO). Full details concerning the description of this successionof different dynamical regimes are given in [24,29,47,48].

It is important to highlight the characteristic time scales ofthe solitary SRL. The principal time scale in any semicon-ductor laser is the relaxation oscillations (RO) timescale.For Eqs. (1) and (2), the relaxation oscillation frequencycan be approximated by f ROjμdc ≈

��������������������������2�μdc − 1�γκ

p∕ 2π. In the

case of SRLs, there is also a second important time scale:the alternate oscillations (AO) timescale. It characterizesthe bi-AO regime of the solitary SRL in which two couplingmechanisms, the cross-gain saturation and the backscattering,compete with each other, inducing intensity oscillationsat a frequency f AOjϕk≈1.5 ≈ k

���������������������������������������������������− cos�2ϕk� − α sin�2ϕk�

p∕ π ≈

96.45 MHz [23,47,48].The following sections deal with the theoretical investiga-

tion of the current-modulated SRL in different operatingregimes. In this study, we shall vary the modulation param-eters μm and f m. Since the solitary SRL has two characteristictimescales, the RO frequency and the AO frequency, it is in-teresting to study the dynamical behavior of directly modu-lated SRL at frequencies comparable to the RO frequency(high frequency) and frequencies comparable to the AOfrequency (low frequency).

3. HIGH FREQUENCY OF MODULATIONIn this section, we investigate the dynamics of directly modu-lated SRL in the bis-UNI regime (corresponding to μdc � 1.704and ϕk ≈ 1.5), with the modulation frequency of its injectioncurrent comparable to f ROjμdc�1.704 ≈ 845 MHz. Figure 2 showsorbit diagrams depicting the local extrema of the counterpro-pagating mode intensities ICCW ∕CW as a function of modulationamplitude μm for two frequencies of modulation, smaller(a) and larger (b) than f ROjμdc�1.704.

Considering f m ∕ f ROjμdc�1.704 ≈ 0.866 [see Figs. 2(a1)and2(a2)], one can see that for low values of μm the output powersstart to oscillate around the former steady states (in this casethe SRL was initialized with the highest power in the clock-wise mode). When increasing μm, the amplitudes of oscillationgrow and eventually the SRL switches mode at μm � 0.87 to a

Fig. 1. The extremes of the modal intensity versus the dc bias injec-tion current μdc (with μm � 0.0 and f m � 0.00 MHz) for specificvalues of ϕk: (a) ϕk � 0.2 and (b) ϕk ≈ 1.5. The maxima (minima) ofICCW � jECCWj2 are denoted by open black squares (circles). The max-ima (minima) of ICW � jECWj2 are denoted by gray crosses (dots). Bi-furcation diagrams are obtained by scanning upwards and downwardsin μdc. The roman numbers indicate different operating regimes.

1984 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Takougang Kingni et al.

predominantly oscillating counterclockwise emission. Notethat this is still a bistable situation where oscillations can alsooccur predominantly in the clockwise mode. With a furtherincrease of μm a period-doubling route to chaos takesplace. For a modulation frequency above f ROjμdc�1.704, i.e.,f m ∕ f ROjμdc�1.704 ≈ 1.326, the results from Figs. 2(b1) and2(b2) can be characterized as follows: a period 1 oscillationexists when μm ≻ 0.0 until μm � 1.04 where a bifurcationoccurs for higher μm; the diagrams indicate again a period-doubling route to chaos. The chaotic behavior is followedby a multiperiodic dynamics.

It is clear in the bifurcation diagrams of Fig. 2 that chaosoccurs when the current modulated SRL is strongly excited,i.e., for high μm. The chaotic behavior shown in Fig. 2 isfurther detailed in Fig. 3, which shows the time traces of

ICCW ∕CW and the corresponding phase portrait for specificvalues of μm and f m. As can be seen in Figs. 3(a1) and 3(b1),the chaotic oscillations of the counterpropagating mode inten-sities are in-phase and constitute large spikes with randomlydistributed amplitudes, alternatively followed by irregularburst of smaller amplitude. This two-frequency structure isalso foreshadowed in the phase portraits of Figs. 3(a2) and3(b2), where one can notice that carrier dynamics is heavilyinvolved in the observed behavior. Also, the chaotic attractorsintrinsically possess the characteristic notch of double-periodic oscillations. The observed dynamics in correspon-dence with the shape of the chaotic attractors and the specificroutes to chaos allows us to conclude that the chaotic regimesshown in Fig. 2 have the same origin than the one observedin directly modulated single-mode semiconductor lasers[31–33,37,39]. Specifically, it has been shown that single-modesemiconductor lasers exhibit period-doubling routes to chaosas the modulation current is increased only with modulationfrequency of the same order or higher than the relaxationfrequency [31–33,37,39].

4. LOW FREQUENCY OF MODULATIONAt frequencies of modulation significantly lower than therelaxation oscillations frequency, single-mode semicon-ductor lasers are known to exhibit only periodic behavior[31–33,37,39]. Nevertheless, we will show here that the SRLcan indeed exhibit periodic behavior and complex dynamicswhen the modulation frequency is in the range of the SRL’ssecond timescale: the AO timescale. In this section, we willexamine the effect of low frequency current modulation onthe different dynamical regimes established in Fig. 1 (bi-CW,tri-CW, bi-AO, and bis-Uni).

We first consider modulation around an injection currentμdc � 1.201 and work at a backscattering phase ϕk ≈ 1.5.For these parameters, the solitary SRL operates in the bi-CW regime. When a sinusoidal time modulation is added tothe steady injection current, the system behaves like a periodi-cally nonlinear oscillator. So the system can be expected toexhibit periodic behavior, quasi-periodic behavior and chaosdepending to the values of the modulation parameters. In or-der to identify the dynamical regimes in modulation param-eter space, we have constructed two parameters (μm, f m)bifurcation diagrams by examining the Lyapunov exponentsand time traces for each cell. Such a diagram is useful whenthe system is examined experimentally, as it can serve as aguide for appropriate operating parameters to target a parti-cular dynamical regime. We have studied the two parameters(μm, f m) bifurcation diagrams for μdc � 1.201 (correspondingto f ROjμdc�1.201 ≈ 450 MHz) and ϕk ≈ 1.5 by varying μm from 0 to0.2 and f m from 0 MHz to 400 MHz (not shown). We have ob-served that in this bi-CW regime the directly modulated SRLonly exhibits periodic behavior. Similarly, in the tri-CW regime(μdc � 1.87, ϕk � 0.2 and corresponding to f ROjμdc�1.87 ≈

939 MHz), we found by studying the two parameters (μm,f m) bifurcation diagrams with μm ∈ �0; 0.87� and f m ∈

�0 MHz; 400 MHz� (not shown), that the current-modulatedSRL displays only periodic behavior.

In Fig. 4, we present a two parameters bifurcation diagramillustrating the dependence of the behavior of current-modulated SRL on the parameters μm and f m for ϕk ≈ 1.5and specific values of μdc corresponding to the bi-AO regime

Fig. 2. Bifurcation diagrams of the modal intensities ICCW, ICWversus μm for specific values of f m: (a) f m � 731.40 MHz(f m ∕ f ROjμdc�1.704 ≈ 0.866) and (b) f m � 1.12 GHz (f m ∕ f ROjμdc�1.704 ≈

1.326) with μdc � 1.704 (corresponding to f ROjμdc�1.704 ≈ 845 MHz).Black (gray) shaded areas indicate local maxima (minima).

Fig. 3. In panel (1) we plot time traces of modal intensities ICCW(black), ICW (gray) while in panel (2), we depict the correspondingphase portraits modal intensity vs. carrier density for specific val-ues of μm and f m in Fig. 2 with μdc � 1.704 (corresponding tof ROjμdc�1.704 ≈ 845 MHz): (a) μm � 1.50 and f m � 731.40 MHz and(b) μm � 1.58 and f m � 1.12 GHz.

Takougang Kingni et al. Vol. 29, No. 8 / August 2012 / J. Opt. Soc. Am. B 1985

(μdc � 1.4 and corresponding to f ROjμdc�1.4 ≈ 637 MHz) andthe bis-Uni regime (μdc � 1.704 and corresponding tof ROjμdc�1.704 ≈ 845 MHz). From Fig. 4, we see that in theseregimes the system can show periodic, quasi-periodic andchaotic behavior. In the bi-AO regime as shown in Fig. 4(a),chaotic regions are observed for frequencies around 20 MHzand 100 MHz, while, in the bis-Uni regime, chaos is observedfor frequencies around 100 MHz, 200 MHz, and 350 MHz[see Fig. 4(b)]. In order to know the route to chaotic behaviorexhibited by the system, we have plotted in Fig. 5 the bifurca-tion diagrams depicting the local extrema of the counter-propagating modes intensities as a function of μm forspecific values of μdc and f m (indicated by the arrowson Fig. 4).

In Figs. 5(a1) and 5(a2), the following transitions are ob-served in the bi-AO regime for a modulation frequency of110 MHz. As μm increases from zero, the dynamics of theSRL moves from a larger quasi-periodic domain to a period3 oscillation, which exists until μm � 0.136 where the bifurca-tion diagrams present a small quasi-periodic window. Thiswindow leads to a period doubling bifurcation to the onsetof chaos at μm � 0.33. The chaotic sea breaks down at μm �0.37 and multiperiodic dynamics is born. For f m � 240 MHzand μdc � 1.704 (the bis-Uni regime), the bifurcation diagramsobtained in Figs. 5(b1) and 5(b2) exhibit a period-doublingtransition to chaos and show that the system is chaotic whenμm varies between 0.156 and 0.246. Overall, the SRL exhibitscomplex dynamics, even at modulation frequencies lowerthan the RO frequency, frequencies at which a directly modu-lated single-mode semiconductor lasers would merely showperiodic oscillations. Similar behavior as shown in Fig. 5can be found at modulation frequencies around 20 MHz forμdc � 1.4 and around 100 MHz and 350 MHz for μdc � 1.704.

5. ASYMPTOTIC INTERPRETATIONIn order to gain more insight in the dynamical behavior ofcurrent-modulated SRL at low modulation frequencies, wepropose an asymptotic simplification of Eqs. (1) and (2) byadopting the same methods used to derive a two-dimensionalphase space model for a solitary SRL in [23]. Such a two-dimensional phase space model has allowed for understand-ing different dynamical regimes observed in the SRL such

Fig. 4. (Color online) Regions of dynamical behavior in the parameter space spanned by modulation frequency f m and amplitude μm for ϕk ≈ 1.5and specific values of μdc: (a) μdc � 1.4 (in bi-A0 regime and corresponding to f ROjμdc�1.4 ≈ 637 MHz) and (b) μdc � 1.704 (in bis-Uni regime andcorresponding to f ROjμdc�1.704 ≈ 845 MHz). Periodic oscillations are in black, quasi-periodic oscillations are in yellow and chaos is in red.

Fig. 5. Bifurcation diagrams of modal intensities ICCW, ICW versus μmfor specific values of μdc and f m: (a) f m � 110 MHz and μdc � 1.4 (cor-responding to f ROjμdc�1.4 ≈ 637 MHz) and (b) f m � 240 MHz and μdc �1.704 (corresponding to f ROjμdc�1.704 ≈ 845 MHz). Black (gray) indi-cates local maxima (minima).

1986 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Takougang Kingni et al.

as alternate oscillations, stochastic mode-hopping betweenmultiple stable attractors and excitability. All these regimeshave also been experimentally observed in SRLs [28–30,47].The numerical investigations of Eqs. (1) and (2) in the para-meter region of Figs. 4 and 5 show that the quantity (N − 1)remains small on dynamical time scales longer than the re-laxation oscillations due to the large ratio of carrier (1 ∕ γ)to photon (1 ∕ κ) lifetimes. The simulations further show thatthis is the case as long as the saturation coefficients s and c areof the same order or smaller than this ratio, while the back-scattering parameters kd and kc are smaller than the photondecay rate κ. It has been demonstrated that with such param-eter values the numerical models (1)–(2) predict the dynami-cal behavior experimentally observed in SRLs [47,48]. Theseparameter values suggest to investigate the limit ρ � γ ∕ κ → 0assuming N − 1, s and c ≪ 1 and kd and kc≪ κ. In order todetermine the leading approximation of Eqs. (1) and (2),we use the above guidelines to define new variables andparameters as

ρ � γ ∕ κ; N − 1 � ρn; s � ρS; c � ρC;

kκ� ρK; t � γKt0; ω � 2π

f mγK

;(4)

where n is assumed to be O�1� and ρ � γ ∕ κ is a small param-eter. The parameters S, C, and K are of O�1�. After substitut-ing Eq. (4) into Eqs. (1) and (2) and taking the limit ρ → 0, weobtain a constraint on the total intensity

jEccwj2 � jEcwj2 � �μdc − 1� � μm sin�2πf mt ∕ γK� > 0 (5)

and the dynamics of current-modulated SRL can be describedby the following reduced set of equations:

dθdt

� −2 sin ϕk sin ψ � 2 cos ϕk cos ψ sin θ

� J�t� cos θ sin θ (6)

cos θdψdt

� αJ�t� sin θ cos θ� 2 cos ϕk sin ψ

� 2 sin ϕk cos ψ sin θ; (7)

where θ � 2 arctan�jEcwj ∕ jEccwj� − π ∕ 2 ∈ �−π ∕ 2; π ∕ 2� repre-sents the relative modal intensity, ψ ∈ �0; 2π� is the phasedifference between the counter-propagating modes, andJ�t� � �C − S���μdc − 1� � μm sin�2πf mt ∕ γK�� ∕K is the nor-malized injection current. We refer to Appendix A for the de-rivation of the reduced model (6) and (7) from the full set ofrate equations (1)–(2) by using asymptotic methods.

To check the validity of the asymptotic reduction and rela-tion (5) in particular, we present in Fig. 6 time traces of modalintensities and total intensity inside the cavity of the SRL[obtained by numerically solving Eqs. (1) and (2)] togetherwith the temporal evolution of the term [μ�t� − 1] for two ex-emplary values of modulation parameters. While very differ-ent dynamics are observed in the dynamics of the modalintensities, in-phase period 1 behavior in Fig. 6(a2), and chaosin Fig. 6(b2), the respective total intensity [depicted inFigs. 6(a1) and 6(b1)] oscillates steadily at the modulationfrequency in both cases. Figures 6(a1) and 6(a2) show thatrelation (5) holds by comparing total intensity and the term�μ�t� − 1�, which both clearly oscillate together at the same fre-quency. Yet, a small difference between the total intensity andthe term �μ�t� − 1� exists, which can be explained by the factthat relation (5) is obtained after taking the limit ρ → 0. In theasymptotic theory, higher order corrections can be derived toaccount for these small differences. In Fig. 6(a2), the twocounterpropagating modes intensities display in-phase period1 oscillation at the modulation frequency, while in Fig. 6(b2),the two counterpropagating mode intensities depict antiphasechaotic behavior, while we will study in more detail later.

Similar as in the previous section, we will investigate thedynamics in the current-modulated SRL around ϕk ≈ 1.5,and compare the results from Eqs. (1) and (2) to the onesof the asymptotic reduction of Eqs. (6) and (7). In Fig. 7,we present time traces of the modal intensities obtained bynumerical integration of Eqs. (1) and (2) (left panels) and timetraces of the relative modal intensity θ ∕ π obtained by numer-ical integration of the asymptotic reduction of Eqs. (6) and (7)(center panels). We make the comparison (right panels) byprojection of the numerical results from Eqs. (1) and (2) onthe corresponding two-dimensional �θ;ψ� phase portraits ofthe asymptotic results from Eqs. (6) and (7). In Figs. 7(a)–7(c),time traces and phase portraits are plotted for a period 1oscillation, a period 3 oscillation, and quasi-periodic behavior,which take place at (μdc, μm, f m): (1.704,0.1,240 MHz),(1.4,0.25,110 MHz), and (1.4,0.05,110 MHz), respectively.The two counterpropagating modes intensities can displayin-phase period 1 oscillation at the modulation frequency[see Fig. 7(a1)]. The corresponding time trace of the relativeintensity θ ∕ π also presents a period 1 oscillation at the mod-ulation frequency [see Fig. 7(a2)]. Comparing the phase por-trait of both, it is clear that the two limit cycles match verywell [see Fig. 7(a3)]. An in-phase period 3 oscillation is de-picted in Fig. 7(b). Although both time traces of the modalintensity in Fig. 7(b1) and of the relative modal intensity inFig. 7(b2) do not show the period 3 oscillation very clearly,the period 3 oscillation is more apparent in the dynamicsof the phase difference ψ of the counterpropagating modes

Fig. 6. (Color online) Temporal evolution of the term �μ�t� − 1.0�(red)and time traces of modal intensities ICCW (black), ICW (gray) and thetotal intensity Icw � Iccw (green) inside the cavity of the SRL whenoperating in the periodic regime (a) μdc � 1.704, μm � 0.05, f m �240 MHz and in the chaotic regime; (b) μdc � 1.704, μm � 0.30,f m � 270 MHz.

Takougang Kingni et al. Vol. 29, No. 8 / August 2012 / J. Opt. Soc. Am. B 1987

(not shown). Nevertheless the period-3 oscillation is evi-denced in the phase portraits of Fig. 7(b3). Notice that theself-intersection of the trajectory is an artifact of the projec-tion onto the �θ;ψ� plane. The intensity of the two counterpro-pagating modes exhibit anti-phase quasi-periodic dynamics inFig. 7(c1). The time trace of the relative intensity θ ∕ π presentsa quasi-periodic dynamics [see Fig. 7(c2)]. Both time tracesshow two frequencies, which are not proportional. The phaseportrait of both [see Fig. 7(c3)] corresponds to tori, whichneatly overlap. We can conclude from the comparisons madein all the right panels of Fig. 7 that results obtained fromEqs. (1) and (2) and those obtained with the asymptotic theoryof Eqs. (6) and (7), agree very well.

This good agreement motivates us to use the phase-space ofthe asymptotic model Eqs. (6) and (7) without modulation tointerpret the chaos found in the SRL at low frequency of mod-ulation. In Fig. 8, we study chaotic dynamics obtained by si-mulating Eqs. (1) and (2) at (μdc, μm, f m): (1.4,0.35,110 MHz)and (1.704,0.233,240 MHz), respectively. We also comparestable and unstable manifolds of the saddle point solution ob-tained from Eqs. (6) and (7) (without current modulation)with Poincaré sections of the chaotic dynamics. In Figs. 8(a1)and 8(b1), we show two situations of antiphase chaotic inten-sity oscillations. The first antiphase state has mode intensityvariations that mainly retain the characteristics of the single-mode chaos. The antiphase relationship between the modes issuch that individual chaotic mode intensity pulsations alter-nate [see Fig. 8(a1)]. This type of antiphase dynamics has beenobserved in optically pumped NH3 bidirectional ring lasers aswell [49]. However, in Fig. 8(b1), the antiphase relationshipcorresponds to intensity variations of both counterpropagat-ing modes that are in the opposite sense, which has also beenobserved in [49] and has been found theoretically in opticallyinjected SRL [25]. A more rigorous confirmation of the anti-phase dynamics can be given by the time trace of the total

power inside the SRL, which remains approximately constantthroughout the simulation and oscillates at the modulation fre-quency f m (not shown). The strange attractor of these chaoticstates can be seen in the Poincaré section obtained by collect-ing a point from the numerical integration of Eqs. (1) and (2)at each period of the modulation 1 ∕ f m and at a specific valueof the modulation current μ�t� [μdc for Figs. 8(a2)–8(b2) andμdc � μm for Figs. 8(a3)–8(b3), corresponding to a phase ofmodulation of 0 and π ∕ 2, respectively] and projecting thatpoint on the 2D phase space of the asymptotic model Eqs. (6)and (7). In Figs. 8(a2)–8(b2) and 8(a3)–8(b3), we show thesePoincaré sections (black dots) together with the invariantmanifolds of the saddle point solution as predicted by Eqs. (6)and (7) without current modulation at the specific value of thecurrent for which the Poincaré section was constructed. Thestable (unstable) manifold is indicated with light (dark) graydots. The relevance of these invariant manifolds is motivatedby the fact that at the moment we plot each point of the Poin-caré section, the system approximately finds itself in thephase-space of Eqs. (6) and (7) corresponding to the instan-taneous current value at that time. We indeed observe that thestrange attractor of the chaotic state tends to broaden andtwist as we construct its Poincaré section corresponding tohigher values of the current. This is to be expected as the re-duced phase-space �θ;ϕ� looks qualitatively different at thoseinstants: The two unidirectional solutions lie further apart andthe unstable manifold of the saddle (connecting the saddleand those unidirectional solutions) twists. This antiphasechaos can be also found at modulation frequencies around20 MHz for μdc � 1.4 and around 100 MHz and 270 MHz forμdc � 1.704 for specific values of μm.

One can also notice the topological resemblance ofFigs. 8(a2)–8(b2) and 8(a3)–8(b3) to the Poincaré section

Fig. 7. Time traces of modal intensities ICCW (black), ICW (gray) cal-culated using Eqs. (1) and (2), the relative modal intensity θ ∕ π calcu-lated using the reducedmodel and the corresponding phase portrait ofdifferent pulsing states obtained by projection of the numerical resultsfrom Eqs. (1)–(2) on the corresponding two-dimensional �θ;ψ� phaseportraits of the asymptotic results from Eqs. (6)–(7) and the oneobtained from the numerical integration of Eqs. (6)–(7) (gray):(a) period-1-oscillation; (b) period-3-oscillation and (c) quasi-periodicity states, where (μdc, μm, f m): (1.704,0.1,240 MHz),(1.4,0.25,110 MHz) and (1.4,0.05,110 MHz), respectively.

Fig. 8. Time traces of modal intensities ICCW (black), ICW (gray)calculated using Eqs. (1) and (2) and the corresponding Poincarésections (black dots) [obtained by projection of the numerical resultsfrom Eqs. (1)–(2) on the corresponding two-dimensional �θ;ψ�phaseportraits of the asymptotic results from Eqs. (6)–(7)] togetherwith the stable and instable manifolds of the saddle point solutionof the reduced system Eqs. (6)–(7) (without current modulationat the specific value of the current) with (μdc, μm, f m):(a) (1.4,0.35,110 MHz) and (b) (1.704,0.233,240 MHz), respec-tively. The stable (unstable) manifold is indicated with light (dark)gray dots.

1988 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Takougang Kingni et al.

of the chaotic behavior of a periodically forced Duffingoscillator:

x� δ _x − βx� x3 � f 0 sin�2πf mt�; (8)

where β>0 and f 0 > 0. In the Duffing oscillator there also ex-ist two symmetric stable states separated by a saddle point,corresponding to the central maximum in the double wellpotential V�x�:

V�x� � −βx2

2� x4

4: (9)

In the periodically forced Duffing oscillator, it is wellknown that an increase of the forcing term can lead to chaoswhen the stable and unstable manifold of the saddle becometangent and subsequently intersect transversely. It was alsoshown that the closed attracting sets for the Duffing equationequal the closure of the unstable manifold [50]. Such align-ment of the attracting set with the unstable manifold can alsobe observed in Figs. 8(a2)–8(b2) and 8(a3)–8(b3) in the caseof the SRL. We do remark, that the invariant manifolds plot-ted in Figs. 8(a2)–8(b2) and 8(a3)–8(b3) are for the Eqs. (6)and (7) without modulation. In view of the resemblance in ob-served dynamics in the SRL and the Duffing oscillator, weconjecture that the way the directly modulated SRL evolvesto the antiphase chaos is topologically similar to the waythe periodically forced Duffing oscillator evolves to chaos.Therefore, the chaos found in Fig. 5 and analyzed in Fig. 8is purely due to the bistable character of the SRL andthe typical semiconductor lasers timescales as relaxationoscillations are not involved in the onset of chaos, in con-trast to other current modulated semiconductors lasers[31,32–33,37–43].

Motivated by this topological resemblance of the SRL withthe Duffing oscillator, we try to write Eqs. (6) and (7) withμm � 0 in a similar form. The dynamical behavior of theSRL that resembles the one in the Duffing oscillator is foundclose to ϕk ≈ π ∕ 2 (ϕk ≈ 1.5 in our simulations). Previous workson SRLs have indeed demonstrated that the SRL asymptoticphase-space has a particular shape at this value of the linearcoupling phase [24,29,30]. This has been attributed to the pre-sence of a nearby Takens-Bogdanov (TB) codimension twopoint (see [24]) at

JTB2 � �4 cos ϕTB2k (10)

sin2�ϕTB2k � � 1 ∕ 2� α ∕ 2

��������������α2 � 1

p: (11)

This TB point is located close to π ∕ 2 and even tends toϕk → π ∕ 2 in the limit of large α. Hereafter, we takeϕk � π ∕ 2, in an effort to describe the dynamics close tothe TB point. Note we do not yet take the limit of α large.The Eqs. (6) and (7) reduce to the following system:

dθdt

� −2 sin ψ � J cos θ sin θ (12)

dψdt

� αJ sin θ� 2 cos ψ tan θ: (13)

As the SRL dynamics for ϕk � π ∕ 2 takes place around ψ ≈ π[see, for example, Figs. 7 and 8], we introduce ϕ � ψ − π,expanding all trigonometric functions near �θ;ϕ� � �0; 0�:

dθdt

� 2�ϕ − ϕ3 ∕ 6�…� � J�θ − 2θ3 ∕ 3�…� � H:O:T: (14)

dϕdt

� αJ�θ − θ3 ∕ 6�…� − 2�1 − ϕ2 ∕ 2�…��θ� θ3 ∕ 3�…�� H:O:T:; (15)

where H. O. T. stands for high order terms. We now considertwo different scaling of the variables and parameters to re-duce the set of Eqs. (14) and (15) to a single differentialequation:

(a)

θ � ε2y; ϕ � εx; J � εI; α � υ ∕ ε: (16)

(b)

θ � εy; ϕ � ε2x; ε2 � αJ − 2; J � ε3I;

s � εt;(17)

with ε small, x, y, I, υ are of O�1�.In the case of the first scaling (a), we substitute Eq. (16) into

Eqs. (14) and (15). We find

εdydt

� 2�x − ε2x3 ∕ 6� � ε2Iy� H:O:T: (18)

dxdt

� ε�υI − 2�y� H:O:T: (19)

Eliminating y from Eqs. (18) and Eq. (19), we get

d2xd2t

� εIdxdt

� 2�υI − 2�x −ε2�υI − 2�

3x3 � H:O:T: (20)

Equation (20) is nothing else than the expression for the Duff-ing oscillation [see Eq. (8)] [50]. In view of the resemblance ofthe reduced SRL model with that of the Duffing oscillator inthe parameter region where the TB point is located close toπ ∕ 2, we conjecture that the knowledge about the dynamics inthe Duffing oscillator can be very useful in interpreting theobserved dynamics in the SRL.

For the second scaling (b), we substitute Eq. (17) intoEqs. (14) and (15). We find

dyds

� 2x� ε2Iy� H:O:T: (21)

dxdt

� y − �ε2 � 6�y3 ∕ 6� ε2x2y� ε2�ε2 − 30�y5 ∕ 15� H:O:T:

(22)Combining Eqs. (21) and (22), we get

d2yd2s

� 2y − 2y3 � ε2�Idyds

−y3

3�

�dyds

�2 y2−y5

2

�� H:O:T:

(23)

Takougang Kingni et al. Vol. 29, No. 8 / August 2012 / J. Opt. Soc. Am. B 1989

We can recognize a weakly perturbed conservative system.From Eq. (23), one can readily write down the first integralof the leading conservation equation (ε � 0):

12

�dyds

�2− y2 � y4

2� E; (24)

where the E ≥ 0 is the constant of integration.We have thus shown that the first scaling (a) exactly gives

the Duffing equation in the leading order, while the secondscaling (b) results in a differential equation that is a weaklyperturbed conservative system. This conservative system re-presents the classical movement in a double potential well.One of the perturbation terms is the classical Duffing frictionterm, while the other ones are different higher order con-tributions to the potential well and an additional nonlinearfriction term.

6. CONCLUSIONIn this work we have theoretically studied the nonlineardynamics of SRLs under current modulation via numericalsimulations and analytical calculations. For modulation fre-quencies of the same order or higher than the relaxation os-cillation frequency, we have shown that by increasing themodulation amplitude, the counterpropagating modes of aSRL exhibit a period doubling route to in-phase chaos similarto single mode semiconductor lasers. The in-phase chaos ischaracterized by large spikes with random distributed ampli-tudes, and the intensity variations of both counterpropagatingmodes are in the same sense. Since a solitary SRL has a sec-ond timescale, namely the frequency of alternate oscillations,which is significantly lower than the relaxation oscillation fre-quency, we have constructed a detailed map of dynamicalresponse of current-modulated SRL at lower modulation fre-quency in the amplitude and frequency of modulation param-eter space. This has been done for specific values of biasinjection current by systematically characterizing the dy-namical behavior in terms of time traces and Lyapunov expo-nents. When the value of bias injection current is located inbidirectional-continuous wave or tristable—continuous waveregimes, current-modulated SRL displays only periodic behav-ior, while for the value of bias injection current located inbidirectional-alternate oscillations or bistable-unidirectionalregimes, the SRL presents periodic, quasi-periodic, and chao-tic behaviors depending on the value of amplitude and fre-quency of modulation. Bifurcation diagrams show a perioddoubling transition to antiphase chaos. Two situations of anti-phase chaotic intensity variations of the counterpropagatingmodes are found. The confirmation of the antiphase chaosis given by the total power emitted inside the SRL, which re-mains constant and oscillates at the frequency of modulationcurrent. This antiphase chaotic regime differs from chaoticregimes found in other current-modulated semiconductorslasers because it does not involve any carrier dynamics andis purely due to the bistable character of SRL. In order toget a more complete understanding of the dynamics of SRLwith low modulation frequency, we have reduced the five-dimensional full rate-equation model to a two-dimensionalmodel by using asymptotic methods. Thanks to this asympto-tic simplification, we have noted the topological resemblanceof the Poincaré section of current-modulated SRL in the

antiphase chaotic regime to the ones of chaotic behavior ofa periodically forced Duffing oscillator. We have also derivedsingle differential Duffing-type equations from the reducedSRL equations by assuming two different scalings of the vari-ables and parameters.

APPENDIX AWe start our asymptotic analysis by applying amplitude/phasedecomposition. Let

Ecw;ccw � Qcw;ccw Exp�iϕcw;ccw�; (A1)

where Qcw;ccw�t0� is the amplitude and ϕcw;ccw�t0� is the phaseof Ecw;ccw�t0�. Carrying out this transformation in Eqs. (1) to(2) gives

dQccw

dt0� κ�N�1 − sQ2

ccw − cQ2cw� − 1�Qccw − kQcw cos�ψ � ϕk�

(A2)

dQcw

dt0� κ�N�1 − sQ2

cw − cQ2ccw� − 1�Qcw − kQccw cos�ψ − ϕk�

(A3)

dψdt0

� ακN�c − s��Q2cw − Q2

ccw� � kQccw

Qcwsin�ψ − ϕk�

� kQcw

Qccwsin�ψ � ϕk� (A4)

dNdt0

� γf�μdc � μm sin�2πf mt0�� − N

− N�1 − sQ2ccw − cQ2

cw�Q2ccw

− N�1 − sQ2cw − cQ2

ccw�Q2cwg; (A5)

where ψ � ϕcw − ϕccw is the relative phase. The reduced mod-el approach is based on the large ratio of carrier (τs � 1 ∕ γ) tophoton (τp � 1 ∕ κ) lifetimes. It causes the quantity (N − 1)(i.e., the difference between the carrier density and the thresh-old value) to remain small on dynamical time scales longerthan the relaxation oscillations. To be able to determinethe leading approximation of Eqs. (A2) to (A5), we introducethe new variables and parameters defined as

ρ � γ ∕ κ; N − 1 � ρn; s � ρS; c � ρC;

k ∕ κ � ρK; t � γKt0; ω � 2πf m ∕Kγ; (A6)

where n is assumed to be O�1� and ρ � γ ∕ κ is a small param-eter. The parameters S, C and K are of O�1�. By substitutingEq. (A6) into Eqs. (A2) to (A5), we obtain a dimensionlessform of our equations with terms containing different powersof ρ. After taking the limit ρ → 0, we obtain the following lead-ing order system:

KdQccw

dt��n−SQ2

ccw −CQ2cw�Qccw −KQcw cos�ψ�ϕk� (A7)

KdQcw

dt� �n− SQ2

cw −CQ2ccw�Qcw −KQccw cos�ψ −ϕk� (A8)

1990 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Takougang Kingni et al.

Kdψdt

� α�C − S��Q2cw − Q2

ccw� � KQccw

Qcwsin�ψ − ϕk�

� KQcw

Qccwsin�ψ � ϕk� (A9)

Q2ccw � Q2

cw � �μdc � μm sin�ωt�� − 1: (A10)

Equations (A7) to (A9) are a simplified asymptotic versions ofthe original rate Eqs. (A2) to (A4) whereas Eq. (A10) is not arate equation but a constraint on the total intensity that is validon the slow time scale t and for μm≺μdc − 1 with μdc≻1. Thederivation toward the time t of the conservation relation (A10)yields after substitution of Eqs. (A7) and (A8) to

n�Qccw;ψ ; t� �1

μ�t� − 1

�ωμm2 cos�ωt� � S�Q4

ccw � Q4cw�Qccw; t��

�2CQ2ccwQ2

cw�Qccw; t� � 2KdQccwQcw�Qccw; t� cos�ψ��

(A11)

with Qcw�Qccw; t� ����������������������������������μ�t� − 1 − Q2

ccw

pand μ�t� � μdc�

μm sin�ωt�.Substituting Eqs. (A11) into Eqs. (A7) and (A9), we obtain a

closed set of two equations:

KdQccw

dt� �n�Qccw;ψ ; t� − SQ2

ccw − CQ2cw�Qccw; t��Qccw

− KQcw�Qccw; t� cos�ψ � ϕk� (A12)

Kdψdt

� α�C − S��Q2cw�Qccw; t� − Q2

ccw�

� KQccw

Qcw�Qccw; t�sin�ψ − ϕk�

� KQcw�Qccw; t�

Qccwsin�ψ � ϕk�: (A13)

Using the conservation law of Eq. (A10), Eqs. (A12) and (A13)can be rewritten in a more simple form by defining a dynami-cal variable θ as a measure for the relative modal intensity:

Qccw ������������������������������������������������μdc � μm sin�ωt�� − 1

pcos��θ� π ∕ 2� ∕ 2� (A14)

Qcw �������������������������������������������������μdc � μm sin�ωt�� − 1

psin��θ� π ∕ 2� ∕ 2� (A15)

with θ � 2 arctan�Qcw ∕Qccw� − π ∕ 2 ∈ �−π ∕ 2; π ∕ 2�.The reduced equations now read:

dθdt

� −2 sin ϕk sin ψ � 2 cos ϕk cos ψ sin θ

� J�t� cos θ sin θ (A16)

cos θdψdt

� αJ�t� sin θ cos θ� 2 cos ϕk sin ψ

� 2 sin ϕk cos ψ sin θ: (A17)

ACKNOWLEDGMENTSThis work has been partially funded by the European Unionunder project IST-2005-34743 (IOLOS) and PHOCUS

(EU FET-Open grant: 240763) and by the Research Founda-tion-Flanders (FWO). This work was supported by the BelgianScience Policy Office under grant photonics@be. S.T.K.thanks Prof. Paul Woafo (University of Yaounde I, Cameroon)for his fruitful advice. GV and LG are postdoctoral fellows ofthe FWO.

REFERENCES1. F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys.

Rev. A 139, 635–646 (1965).2. L. N. Menegozzi and W. E. Lamb, “Theory of a ring laser,” Phys.

Rev. A 8, 2103–2125 (1973).3. E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R.

Tredicce, “Spatiotemporal dynamics of lasers in the presence of animperfect O�2� symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).

4. R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce,“A mode–mode interaction for a CO2 laser with imperfect O�2�symmetry,” Phys. Rev. A 47, 500–509 (1993).

5. H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L.Lippi, and T. Mello, “Bidirectional ring laser: stability anal-ysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).

6. M. Sargent, III, “Theory of a multimode quasi-equilibrium semi-conductor laser,” Phys. Rev. A 48, 717–726 (1993).

7. C. Etrich, P. Mandel, N. B. Abraham, and H. Zeghlache,“Dynamics of a two-mode semiconductor laser,” IEEE J.Quantum Electron. 28, 811–821 (1992).

8. N. Matsumoto and K. Kumabe, “AlGaAs-GaAs semiconductorring lasers,” Jpn. J. Appl. Phys. 16, 1395–1398 (1977).

9. A. Liao and S. Wang, “Semiconductor injection lasers with a cir-cular resonator,” Appl. Phys. Lett. 36, 801–803 (1980).

10. M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H.den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, andM. K. Smit, “A fast low-power optical memory based on coupledmicro-ring lasers,” Nature 432, 206–209 (2004).

11. J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidir-ectional operation of waveguide diode ring lasers,” Appl. Phys.Lett. 70, 1192–1194 (1997).

12. Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu,“Integrated small-size semiconductor micro-ring laser withnovel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20,99–101 (2008).

13. M. Waldow, T. Plotzing, M. Gottheil, M. Först, J. Bolten, T.Wahlbrink, and H. Kurz, “25 ps all-optical switching in oxygenimplanted silicon-on-insulator microring resonator,” Opt.Express 16, 7693–7702 (2008).

14. A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari,“An inhomogeneous theoretical model for analysis of PbSequantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).

15. S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact andlow power consumption optical switch based on microrings,”IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).

16. S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-lasergyroscope without the lock-in phenomenon,” Phys. Rev. A 78,053822 (2008).

17. A. Mignot, G. Feugnet, S. Schwartz, I. Sagnes, A. Garnache, C.Fabre, and J.-P. Pocholle, “Single-frequency external-cavity semi-conductor ring-laser gyroscope,” Opt. Lett. 34, 97–99 (2009).

18. T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation ofsemiconductor ring lasers,” Electron. Lett. 26, 2095–2097(1990).

19. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed.(Springer-Verlag, 2004).

Takougang Kingni et al. Vol. 29, No. 8 / August 2012 / J. Opt. Soc. Am. B 1991

20. L. Lachinova and W. Lu, “Pattern formation and competition in anonlinear ring cavity,” J. Opt. B: Quant. Semiclass. Opt. 2, 393–398 (2000).

21. C. Born, M. Sorel, and S. Yu, “Controllable and stable mode se-lection in a semiconductor ring laser by injection locking,” IEEEJ. Quantum Electron. 41, 261–271 (2005).

22. L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J.Danckaert, “Multistable and excitable behavior in semiconduc-tor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58,197–217 (2010).

23. G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J.Danckaert, “Two-dimensional phase-space analysis and bifurca-tion study of the dynamical behavior of a semiconductor ringlaser,” J. Phys. B 41, 095402 (2008).

24. L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ringlasers,” Phys. Rev. E 79, 016213 (2009).

25. W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J.Danckaert, “Optical injection in semiconductor ring lasers,”Phys. Rev. A 81, 033802 (2010).

26. S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Mod-al structure, directional and wavelength jumps of integratedsemiconductor ring lasers: experiment and theory,” Appl. Phys.Lett. 93, 251109 (2008).

27. G. Giuliani, R. Miglierina, M. Sorel, and A. Sciré, “Linewidth,autocorrelation and cross-correlation measurement of coun-ter-propagating modes in GaAs-GaAlAs semiconductor ringlasers,” IEEE J. Sel. Top. Quantum. Electron. 11, 1187–1192(2005).

28. S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M.Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in opticalsystems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).

29. L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J.Danckaert, and G. Verschaffelt, “Exploring multistability insemiconductor ring lasers: theory and experiment,” Phys.Rev. Lett. 102, 193904 (2009).

30. S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt,A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological in-sight into the non-Arrhenius mode hopping of semiconductorring lasers,” Phys. Rev. Lett. 101, 093903 (2008).

31. C. H. Lee, T.-H. Yoon, and S.-Y. Shin, “Period doubling and chaosin a directly modulated laser diode,” Appl. Phys. Lett. 46, 95–97(1985).

32. Y. C. Chen, H. G. Winful, and J. M. Liu, “Subharmonic bifurca-tions and irregular pulsing behavior of modulated semiconduc-tor lasers,” Appl. Phys. Lett. 47, 208–210 (1985).

33. M. Tang and S. Wang, “Simulation studies of bifurcation andchaos in semiconductor lasers,” Appl. Phys. Lett. 48, 900–902(1986).

34. Y. Hori, H. Serizawa, and H. Sato, “Chaos in a directly modulatedsemiconductor laser,” J. Opt. Soc. Am. B 5, 1128–1133 (1988).

35. H.-F. Liu and W. F. Ngai, “Nonlinear dynamics of a directlymodulated 1.55 pm InGaAsP distributed feedback semiconduc-tor laser,” IEEE J. Quantum Electron. 29, 1668–1675 (1993).

36. C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera,“Theory of main resonances in directly modulated diode lasers,”IEEE J. Quantum Electron. 38, 260–269 (2002).

37. E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski,“Type-I intermittency and crisis-induced intermitency in a semi-conductor laser under injection current modulation,” NonlinearDyn. 27, 185–195 (2002).

38. Y. Chembo and P. Woafo, “Stability analysis for the synchroni-zation of semiconductor lasers with ultra-high frequency currentmodulation,” Phys. Lett. A 308, 381–390 (2003).

39. A. Valle, L. Pesquera, S. I. Turovets, and J. M. Lopez, “Nonlineardynamics of current-modulated vertical-cavity surface-emittinglasers,” Opt. Commun. 208, 173–182 (2002).

40. M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov,“Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).

41. A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamicsof the polarization of multitransverse mode vertical-cavitysurface-emitting lasers under current modulation,” Phys. Rev.E 76, 046206 (2007).

42. J. H. Talla Mbé, K. S. Takougang, and P. Woafo, “Chaos andpulse packages in current-modulated VCSELs,” Phys. Scr. 81,035002 (2010).

43. S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamicsin VCSELs driven by a sinusoidally modulated current and Röss-ler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).

44. X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon,and S. Yu, “Direct modulation of bistable semiconductor ringlasers,” in CLEO/Europe and EQEC Conference Digest (OSA,2011), paper CB2_5.

45. X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu,“Frequency-domain model of longitudinal mode interaction insemiconductor ring lasers,” IEEE J. Quantum Electron. 48,406–418 (2012).

46. T. Numai, “Analysis of signal voltage in a semiconductor ringlaser gyro,” IEEE J. Quantum Electron. 36, 1161–1167 (2000).

47. M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R.Miglierina, and S. Donati, “Alternate oscillations in semiconduc-tor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).

48. M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn,and S. Donati, “Operating regimes of GaAs-AlGaAs semiconduc-tor ring lasers: experiment and model,” IEEE J. Quantum Elec-tron. 39, 1187–1195 (2003).

49. D. Y. Tang and N. R. Heckenberg, “Anti-phase dynamics of achaotic multimode laser,” Phys. Rev. A 56, 1050–1052 (1997).

50. J. Guckenheimer and P. Holmes, “Nonlinear oscillations,dynamical systems and bifurcations of vector fields,” 3rd ed.(Springer-Verlag, 1990).

1992 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Takougang Kingni et al.