Dispersion-induced dynamics of coupled modes in a semiconductor laser with saturable absorption
DDispersion-induced dynamics of coupled modes in a semiconductor laser withsaturable absorption
Finbarr O’Callaghan, Simon Osborne, and Stephen O’Brien
Tyndall National Institute, Lee Maltings, University College Cork, Cork, Ireland
We present an experimental and theoretical study of modal nonlinear dynamics in a specially de-signed dual-mode semiconductor Fabry-Perot laser with a saturable absorber. At zero bias applied tothe absorber section, we have found that with increasing device current, single mode self-pulsationsevolve into a complex dynamical state where the total intensity experiences regular bursts of pul-sations on a constant background. Spectrally resolved measurements reveal that in this state theindividual modes of the device can follow highly symmetric but oppositely directed spiralling orbits.Using a generalization of the rate equation description of a semiconductor laser with saturable ab-sorption to the multimode case, we show that these orbits appear as a consequence of the interplaybetween the material dispersion in the gain and absorber sections of the laser. Our results provideinsights into the factors that determine the stability of multimode states in these systems, and theycan inform the development of semiconductor mode-locked lasers with tailored spectra.
PACS numbers: 42.55 Px, 42.65.Sf
I. INTRODUCTION
Semiconductor lasers with a saturable absorber cangenerate short and high-power optical pulses by themechanisms of self-pulsation and mode-locking [1–4].These modes of operation are typically associated withdifferent timescales determined by the relaxation oscil-lation frequency [GHz] and the round trip time in thecavity [10-100s GHz]. As self-pulsations (SPs) are oftena significant source of instability in mode-locked lasers, athorough understanding of mechanisms leading to theirappearance is desirable [5, 6].The model of a laser with a saturable absorber (LSAmodel) considers the dynamics of the total field on timescales long compared to the round trip time in the cav-ity. While it cannot therefore describe phenomena suchas mode-locking, this model has provided valuable in-sights into the origins of SPs and the factors that leadto the appearance of bistability in devices with saturableabsorbers [7–10].Quantitative dynamical models of semiconductorlasers with saturable absorbers include travelling-wavemethods that consider the spatio-temporal dynamics ofthe slowly-varying electric fields [11, 12]. A lumped el-ement time domain model has also been developed thateliminates the spatial dependence in favour of a delay-differential equation for the field variable [13]. Thesemodels are efficient tools for understanding the physi-cal origins of complex spectral and pulse shaping mech-anisms in mode-locked semiconductor lasers [14–16].For certain applications of these devices however, wemay be interested in quantities such as the frequencyand phase-noise properties of the individual locked modesrather than the pulse train generated by the device. Ex-amples include stable terahertz frequency generation andtailored comb-line emission demonstrated recently by ourgroup [17–19]. These Fabry-Perot (FP) lasers included aspectral filter to limit the number of active modes, and here it may be appropriate to formulate the problem ofdescribing the dynamics in the frequency domain. Insuch a model, each longitudinal mode of the cavity is con-sidered as an independent dynamical variable [20]. Theround-trip time in the cavity then determines the modespacing, and by including phase-sensitive modal interac-tions in the model, one can describe mode-locked statesas mutually injection locked steady-states of the system[21].In this paper we consider a device that supports twolongitudinal modes with a large frequency spacing. Inthis case, a frequency domain description based on anextension of the LSA model represents a natural startingpoint. A transition to mode-locking is not possible inthis device. Instead, we have found familiar single-modeSP dynamics, but also interesting examples of coupleddual-mode dynamics. Here we describe a transition toa multimode state where the total intensity experiencesbursts of fast pulsations. We show that in this state theindividual modes follow oppositely directed spiralling or-bits that are related to the underlying SP dynamics ofthe system. Our modeling approach is valid for smallvalues of the gain and loss per cavity round trip. Be-cause these approximations are not expected to hold ina semiconductor laser, our results are necessarily qual-itative. However, we highlight an interesting exampleof a multimode instability that can arise in two-sectiondevices with large dispersion, and our results can guidethe future development of optimized mode-locked deviceswith tailored spectra.This paper is organised as follows. In section II we in-troduce our device and experimental setup. We presentoptical and mode resolved power spectra as well as a se-ries of characteristic intensity time traces illustrating aprogression to a region of complex dual-mode dynamics.In section III we describe our proposed model equations,which are a multimode extension of the LSA model thataccounts for dispersion of gain and saturable absorptionwith wavelength in the system. In section IV we un- a r X i v : . [ phy s i c s . op ti c s ] M a y w a v e l e ng t h [ n m ] fr e qu e n c y [ GH z ]
30 40 50 60 70 80 90 device current [mA] ν ν (a)(b)(c)(c) FIG. 1: (a) Optical spectrum of the dual-mode two-sectiondevice as the device current is varied. The bias applied to theabsorber section is 0 V. (b),(c) Corresponding power spectraof the long ( ν ) and short ( ν ) wavelength primary modes. cover the bifurcation structure of the system leading tothe measured results. We conclude by discussing the im-plications of our results for future work. II. EXPERIMENT
The device we consider is a multi-quantum well IndiumPhosphide based ridge-waveguide Fabry-Perot laser withone high-reflection (HR) coated mirror. The total devicelength is 545 µ m with a saturable absorber section oflength 30 µ m adjacent to the HR mirror. The device hasa peak gain near 1550 nm, and slotted regions etchedin the ridge define a spectral filter, which is designed toselect two primary modes with a spacing of 480 GHz.Further details on the design of similar devices and theiroperating characteristics can be found in [17].Fig. 1 (a) shows the optical spectrum of the laser as thedrive current in the gain section of the device is varied.These spectra were obtained keeping a constant bias onthe short contact of 0 V and varying the pump current inthe gain section from below lasing threshold at 30 mA toa value of 90 mA. All measurements of this device werecarried out at a temperature of 16 . ° C.We label the short and long wavelength primary modesof the device as ν and ν respectively. From Fig. 1(a) we see that the long wavelength mode of the devicereaches threshold first at a drive current of 30 mA. Thetwo primary lasing modes are located near 1550 nm and1544 nm and they have a spacing of six fundamental cav-ity modes. These primary modes dominate the spectrumthroughout the parameter region of interest. The cor-responding power spectral densities for each of the pri-mary modes are shown in Fig. 1 (b) and (c). Structure appears in the power spectral density of ν at approxi-mately 35 mA, indicating the onset of dynamical modu-lation with a frequency of c. 1 . ν at threshold.The self-pulsations are initially sinusoidal and their fre-quency increases gradually with device current until afurther transition at c. 45 mA. Near this value of the de-vice current a discontinuity appears in the frequency ofthe intensity modulation, which subsequently increasesagain until a device current of 70 mA, where a dramaticswitch to a region of dual-mode dynamics is observed.In the region of dual-mode dynamics the power spec-tra become symmetric with a large range of frequenciespresent, including a signature of low-frequency modula-tion in the 100 MHz range. This region extends overa current range of approximately 10 mA, with the dy-namics switching abruptly to the short wavelength mode ν near 80 mA. Following the dual-mode region we ob-serve a single peak in the intensity power spectrum thatgradually diminishes in strength. This indicates that, forthe largest values of the device current shown, we havereached a state of constant output on the mode at shortwavelength.Representative time traces for the intensity of the longwavelength mode in the first and second regions of dy-namics are shown in Fig. 2 (a) and (b). The device time [ns] i n t e n s it y [ a r b . un it s ] time [ns](a) (b) ν ν i n t e n s it y [ a r b . un it s ] time [ns] time [ns](c)(e)(g) (d)(f)(h)T ν ν T ν ν FIG. 2: (a),(b) Measured time traces for the long wavelengthmode ν for a device current in the gain section of 44 and 59mA respectively. (c),(d) Time traces of the total intensity ata current of 75 mA in the gain section. (e)-(h) Mode resolvedtime traces at a current of 75 mA in the gain section. Thebias applied to the absorber section is 0 V in all cases. currents are 44 and 59 mA respectively. Fig. 2 (a) showscharacteristic self-pulsation dynamics, where the inten-sity reaches small values between pulses and the pulseduration is significantly less than the interval betweenpulses. The dynamics in the second region as shown inFig. 2 (b) are also strongly modulated but they are muchcloser to sinusoidal than in the region of self-pulsations.Time traces of the total intensity and of the individ-ual modes taken from the region of dual-mode dynamicsare shown in Fig. 2. (c-h). The device current in thelong contact for these measurements was 75 mA. One cansee that the total intensity experiences regular bursts offast pulsations that are modulated by a much lower fre-quency envelope. The individual modes in this dynamicalstate display a distinctive symmetric saw-tooth structure,where each mode closely follows a time-reversed trajec-tory of the other. One can see that there is a significantantiphase component to these dynamics, as the inten-sity of the individual modes reaches values close to zeroover a considerable interval, whereas the total intensityis modulated around a finite background level. III. MODELLING OF THE DEVICE RESPONSE
Modelling the dynamics of our experimental systemwhile treating each mode individually is complicated bythe relatively large number of independent parameters[12, 14]. However, we have successfully modelled the dy-namics of dual-mode devices with optical injection andfeedback in past work [22, 23], and the LSA model pro-vides a guide for extending these models to the case ofa two-section laser. This model treats the absorber sec-tion as an unpumped region with an unsaturated absorp-tion and carrier recovery time that depend on the appliedvoltage.We do not believe that undertaking a complete theoret-ical study of the bifurcation structure of the dual-modeLSA model along the lines of [8] would be practical here.Instead, our goal is to understand the physical roles of thevarious parameters of the system, and to obtain numeri-cal estimates for these parameters based on a comparisonof simulation results with the results of our experiment.In physical units the multimode extension of the LSAmodel reads˙ S m = [(1 − ρ ) ˜ G m ( N g ) + ρ ˜ A m ( N q ) − γ m ] S m ˙ N g = j − N g τ s − (cid:88) m ˜ G m ( N g ) S m ˙ N q = − N q τ q − (cid:88) m ˜ A m ( N q ) S m (1)Here S m is the photon density, and N g and N q are thecarrier densities in the gain and absorber sections respec-tively. The ratio of the absorber section length to thetotal device length is given by ρ . The total field lossesof each mode are γ m = α mir + α int + α m f , where α mir arethe mirror losses and α int are the internal losses assumed .
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59 1 . wavelength [ µ m] − . . . . g a i n [ un it s o f γ m ] ∆ A m ∆ G thrm ν ν α int − − j s ] i n s t e n s it y [ a r b . un it s ] j TC s FIG. 3: Schematic diagram of the material gain and absorp-tion in a typical semiconductor laser. The locations of thetwo primary modes of the device ν and ν are indicated. Themodel parameters that describe the dispersion of the modalgain and absorption are also highlighted. Inset: Branchesof equilibrium solutions of the single mode LSA model fortypical parameters considered here. The vertical line is thelasing threshold, j TC s , where the zero field solution becomesunstable. constant for all modes. Additional losses, α m f , due tothe action of the spectral filter are also included. Thecurrent density in the gain section is j , while the carrierlifetimes in the gain and absorber sections are τ s and τ q respectively.Typical profiles of the gain and absorption spectra ina semiconductor laser of the kind we consider are shownschematically in Fig. 3. The negative offset of the gainfunction at long wavelength gives an estimate of the back-ground losses, α int . Here we have indicated the locationsof the two primary modes of the laser. To define the dis-persion of gain and absorption, we first fix a reference car-rier density value, N thr g , in order to define the dispersionof the gain profile for our model. We take this referencevalue to be the threshold carrier density for the deviceassuming a transparent absorber section, and define themode with the largest material gain at this carrier densityas our reference mode, m . The threshold carrier densitydefines a reference value for the modal gain: ˜ G m ( N thr g ) =(1 − ρ ) − γ m , and a set of modal differential gain values,˜ g gm . The gain function for each mode can then be lin-earized around the reference value of the carrier densityso that ˜ G m ( N g ) = ˜ G m ( N thr g ) + ˜ g gm ( N g − N thr g ). Dis-persion in the modal absorption is included by defining˜ A m ( N q ) = ˜ g qm N q − A m , where the differential absorp-tion is ˜ g qm , and unsaturated losses for each mode, A m ,will be determined by the applied voltage.To derive normalized equations, we rescale time inunits of the photon decay rate of a plain FP cavity with-out spectral filtering: γ = α mir + α int . We define the nor-malized pump current, p = ( j − j thr ) /j thr ≡ j s −
1, where j thr = N thr g /τ s , and we define the normalized carrier den-sities in each section of the device: n g = ( N g − N thr g ) /N thr g and n q = ( N − N q ) /N thr g , where N = A m / ˜ g qm . Innormalized units the equations then read˙ I m = [(1 − ρ ) G m ( n g ) + ρA m ( n q ) − γ (cid:48) m ] I m T ˙ n g = p − n g − (cid:88) m G m ( n g ) I m T ˙ n q = ∆( q − n q ) + (cid:88) m A m ( n q ) I m (2)where q = N N thr g , T = γτ s , and ∆ = τ s /τ q . In theseequations the normalized gain functions are G m ( n g ) = G m ( n thr g ) + g gm N thr g n g where G m ( n thr g ) = γ − ˜ G thr m + ∆ G thr m , γ (cid:48) m = γ m /γ , and g gm = ˜ g gm /γ . Here ∆ G thr m ≡ γ − ( ˜ G thr m − ˜ G thr m ) describesthe dispersion of the reference linear gain profile. Thenormalized modal absorption functions are A ( n q ) = − g qm N thr g n q + ( g qm − g qm ) N − γ − ∆ A m where g qm = ˜ g qm /γ , and ∆ A m = A m − A m . Here thenormalized carrier density in the absorber section is de-fined so that the modal absorption is linearized aroundthe saturated value for the reference mode. Note thatthe phase space of system (2) contains two invariantthree dimensional sub-manifolds, defined by I m = 0, for m = { , } . The dynamics on each of these sub-manifoldsreduces to the single-mode LSA system, and for this rea-son we will refer to these sub-manifolds as the single-mode manifolds of the system.Based on previous estimates obtained for similar de-vices [22], we take the carrier lifetime in the gain section, τ s = 1 ns. The mirror losses of the device are calcu-lated to be α m = 13 . − , and we estimate the in-ternal losses to be α int = 9 . − . These losses de-termine the cavity decay rate for the plain FP laser tobe γ = 2 × s − , and T (cid:39) N thr g , we compared our results with threshold data froma plain two-section FP laser. With a uniform currentdensity over the full device length, the threshold currentof the FP laser was 13.5 mA. With a bias of 0 V ap-plied to the absorber section, the threshold increased to17 mA, with the peak emission at 1557.5 nm. The scaleof N g is defined so that the differential gain at thresh-old for the single section FP is equal to 1 in normalisedunits. We therefore set N thr g equal to 2.2 based on esti-mates of the increase in the carrier density necessary toreach the defined threshold level at the wavelength of thereference mode. This estimate was made by taking thelosses due to spectral filtering α m f = 10 cm − , and usingan approximate model for the semiconductor susceptibil-ity [24]. This model allowed us to account for the largeblue-shift of the gain peak from its position in the FPlaser at threshold. We note that the measured increasein threshold of the dual-mode device, the placement ofthe spectral filter and the large separation between the selected modes are all factors that suggest large unsatu-rated absorption and enhanced dispersion of the modelparameters.The carrier lifetime in the absorber section can bemuch shorter than in the gain section, with a strong de-pendence on the applied bias [25]. We fix the carrier life-time in the absorber section to be 50 ps, so that ∆ = 20.However, provided ∆ is not close to one, we have foundour results are not dependent on the precise value of thisquantity. In order to complete the model we must spec-ify the values of the linear gain, unsaturated absorption,and the differential gain and absorption for each primarymode of the device. Because of the large size of thisparameter space, we begin by considering the dynamicsof two coupled modes with similar parameters. Guidedby the known dispersive properties of the semiconductorsusceptibility, and by the observed behavior of the device,we then make a series of further adjustments to these pa-rameters until we have obtained satisfactory agreementwith measured data. IV. BIFURCATIONS OF A DUAL-MODESEMICONDUCTOR LASER WITH ASATURABLE ABSORBER
For our numerical simulations, the parameters describ-ing the gain function at the position of the referencemode, ν , are fixed. To begin we assume a flat gain andabsorption curve and we examine the effects of dispersionin the differential gain and absorption on the dynamicsof the coupled system. From Fig. 1 we see that in ourexperiment the device begins to lase with constant inten-sity output on the long wavelength mode, before enteringa region of SP. At the largest values of the pump currentthe intensity switches to short wavelength. With equallinear gain and unsaturated absorption, the mode withthe largest differential gain will reach threshold first. Onthe other hand, a larger differential absorption will meanthat a mode will saturate its losses more quickly abovethreshold and thereby dominate at larger pump values.To reproduce this behaviour, we set the differential gainand absorption of mode ν in normalised units to be 1 . . ρ − respectively. The ratio of these quantities formode ν is then s ≡ g q /g g = 22. We set the differentialgain and absorption of ν to be 1 . . ρ − respec-tively so that s = 7.8. The remaining parameters valueswe choose to begin are A [1,2] = 0 . γ and ∆ G thr m [1,2] = 0.Note that the differential gain in normalised units is likelyto be less than unity given the higher current density atthreshold in the dual-mode device. We have decided notto make this correction in order to make the comparisonof the various model parameters more transparent. Wehave confirmed that our results are largely independentof the precise values chosen for g g , provided the otherdifferential quantities are scaled accordingly.While the LSA model can predict the appearance ofself-pulsations immediately at threshold [7, 8], our chosen i n t e n s it y [ a r b . un it s ] fr e qu e n c y [ GH z ] scaled current density [ j s ] scaled current density [ j s ] H H λ ⊥ λ ⊥ (b)(d)(f)(a)(c)(e) FIG. 4: Left panels: Simulated bifurcation diagrams. Rightpanels: Power spectral densities. (a),(b): Single-mode dy-namics. (c),(d): Long wavelength mode ν . (e),(f): Shortwavelength mode ν . parameter values are consistent with the observation ofa narrow region of constant output at threshold in ourexperiment. If we consider a single mode system, thezero field equilibrium solution of these equations is stableuntil a transcritical bifurcation at a threshold value of thepump p TC = ρA m (1 − ρ ) g g γN thr g . The inset of Fig. 3 shows the branches of single modeequilibrium solutions of Eqn (2) taking the model pa-rameters for mode ν . Here ∆ exceeds a minimum valuegiven by ∆ = s q (cid:48) q (cid:48) . where q (cid:48) o = γ − ρA m . This condition leads to constantoutput at threshold, as the upper branch of equilibriumsolutions takes physical values after exchanging stabilitywith the zero field solution. A further bifurcation to SPswill then occur provided there is sufficient saturable ab-sorption in the system. We will find that the stabilityof the single-mode equilibria plays a fundamental role inorganising the dynamics in our device, and we will there-fore present numerical bifurcation diagrams for each ofthe single-mode solutions of our model as we vary themodel parameters for each mode.Numerical bifurcation diagrams and intensity powerspectra obtained with our first set of parameters areshown in Fig. 4. Figs 4 (a) and (b) describe the dynamicsof both of the primary modes restricted to their respec-tive single mode manifolds, obtained by setting the inten-sity of the inactive mode to zero for the time evolution of Eqn (2). In Fig. 4 (a), as expected, the single mode dy-namics of both modes exhibit threshold behaviour similarto the observed behaviour of the long wavelength mode inour experiment, with a region of constant output foundafter the zero field solution becomes unstable. Followingthis region, they enter a region of SPs at the location ofthe first Hopf bifurcation, and the SP region is boundedin each case by a second Hopf bifurcation at larger pumpcurrent. In Fig. 4, dashed and dotted lines labelled H and H indicate the second Hopf bifurcation points thatbound the SP region at larger pump values for each mode.Mode resolved numerical bifurcation diagrams andpower spectra for the two modes in the full coupled modesystem are shown in Fig. 4 (c)-(f). Chosen parametersensure that the long wavelength mode reaches thresh-old first, and because the second mode is initially sup-pressed, mode ν reproduces the dynamics found in thesingle mode system. However, before the region of singlemode SP ends at the second Hopf bifurcation shown inFig. 4 (a), the dynamics become dual-mode, with the SPintensity gradually switching across to the shorter wave-length mode ν as the pump is increased further. Thisdual-mode region comes to an end shortly after a pumpvalue of j s = 3 where the system enters a region of singlemode SP on ν . The region of SP dynamics finally endsat the subcritical Hopf bifurcation of ν , and the dynam-ics switch to constant output in mode ν for large valuesof the pump current.In the power spectrum of Fig. 4 (d) we see that at theonset of SPs, they occur with a frequency of c. 500 MHz,with a linear increase in each interval of single or dual-mode dynamics thereafter, and reaching a value of c. 5GHz at j s = 4 .
25. We can compare this evolution withthe dependence of the relaxation oscillation frequency,which, neglecting the effects of saturable absorption, isgiven by ω RO = (cid:114) (1 − ρ ) g g N thr g pT . (3)At j s = 4 .
25, the result is approximately 6 GHz, which isa reasonable estimate of the SP frequency in this model.Note however that the close to linear increase of the nu-merical SP frequency contrasts with the square-root de-pendence of the above expression.If we compare the numerical variation of the SP fre-quency with our experiment, we see that the measuredSP frequency appears with a large value of c. 1.5 GHz,and that it then remains relatively constant. Our simula-tions therefore underestimate the SP frequency at onset,and overestimate its rate of increase. One can also seethat the extent of the SP region that we find numeri-cally is much larger than the measured value. While themeasured variation of the SP frequency may be partlydue to an uncharacteristic behavior of our device, itshould be noted that we cannot expect to obtain quanti-tative agreement with experiment using the LSA model.We have confirmed this by comparing the results of nu-merical simulations made using the LSA model and thedelay-differential model of [13]. For example, using ex-perimentally calibrated parameters appropriate to a self-pulsating FP laser, we have found that the LSA modelwill in general predict a far larger SP region, with a largerSP frequency than the delay-differential model. In ad-dition, while we can adjust unknown parameters suchas the absorber recovery time to match numerical andexperimental results in the case of the delay differentialmodel, this is in general not possible when using the LSAmodel. This comparison emphasizes the added impor-tance of accounting for the large changes in gain and lossthat can occur in two-section semiconductor lasers, whereSP dynamics involve strong saturation of the absorptionfor typical parameters.Time traces taken from the center of the dual-moderegion with j s = 3 are shown in the left panel of Fig. 5,while the right hand panel of Fig. 5 shows a phase spacerepresentation of the dynamics for three pump values,taken before, during and after the transition from ν to ν . We see that the intensity shifts continuously from onemode to the other through a region of in-phase SP. Asexpected, the gradual transition of the dynamics from ν to ν that we observe in these simulations leads toagreement with the experimental measurements of Fig.1 near threshold and at large values of the pump.Valuable insight into what factors can lead to betteragreement with experiment can be gained from a closeexamination of the time traces presented in Fig. 2 (c-h).From these figures, we see that for the majority of theorbit duration the intensity is close to the single modemanifold of one mode or the other. In Fig. 2 (f) wesee the large intensity oscillations of ν decaying towarda state with almost constant output, and the intensitythen quickly switching to a similar state in ν from which i n t e n s it y [ a r b . un it s ] time [ns] T ν ν (a)(b)(c) FIG. 5: Left: Simulated time traces of the total intensity(upper panel), and of the individual modes (center and lowerpanels). The pump current value is j s = 3. Right: Phasespace diagrams for three values of the pump current as shown. -0.010.00.01 λ ⊥ scaled current density [ j s ] -0.010.00.01 λ ⊥ A (a)(b) g q A H H λ ⊥ λ ⊥ (i) (ii)(iii)(iv)(i)(iii)(ii) FIG. 6: Effect of including dispersion in model parame-ters on the transverse Lyapunov exponent and the locationof the second Hopf bifurcation of each single-mode equilib-rium of the model. Dashed and dotted lines indicate Hopfbifurcations of ν and ν respectively. Solid lines are thetransverse Lyapunov exponents of each mode as indicated.(a) (i) Parameters as in Fig. 4, (ii)-(iv) A = 0 . γ and∆ G thr m = [(ii)0 , (iii) − . , (iv) − . γ − . (b) (i) Parameters asin (iv) of (a), (ii), (iii) g q = 1 .
5, and A = (ii)0 . γ, (iii)0 . γ . the oscillations grow again. The clear observation of thisnear single mode state with constant output at the begin-ning and end of these bursts of pulsations suggests thatthe single mode equilibrium states of (2) may be playingan important role in organising the observed dynamics.In particular, given the presence of symmetric invariantsub-manifolds in the phase-space, we should examine thedependence of the transverse stability of the single modeequilibrium solutions on the model parameters.The transverse stability of the single-mode equilibriumsolution for mode ν i , I i , is determined by the sign of itstransverse Lyapunov exponent λ ⊥ i = (1 − ρ ) G j ( I i ) + ρA j ( I i ) − γ j (4)where { i, j } = { , } and mode ν i is transversally stablefor negative values of λ ⊥ i . For illustration purposes, wehave plotted and labelled solid lines in Fig. 4 that indi-cate where the changes in transverse stability occur forthe chosen parameter values. Note that in this case ν be-comes transversally unstable while ν becomes transver-sally stable with increasing j s . The impact of dispersionof linear gain and unsaturated absorption on the dynam-ics of our model can be illustrated by a plot of the loca-tions of the Hopf bifurcations and changes in transversestability of the single mode equilibrium solutions as therelevant parameters are varied. In Fig. 3, the physi-cal dispersion of the unsaturated absorption suggests alarger value of A for ν , and stability changes for bothmodes obtained with A increased to a value of 0 . γ ,and ∆ G thr m ranging from 0 to − . γ − are shown in Fig.6 (a). In these figures vertical dashed and dotted linesindicate the locations of the second Hopf bifurcation for ν and ν respectively. Curved solid lines plot the valueof the transverse Lyapunov exponent for each mode as in-dicated, with sign changes of these quantities indicatingchanges in transverse stability. One can see the impactthat a change of only 0 . − to the linear gain profilehas on the transverse stability properties of these modes.The cumulative net effect of these changes is that thereis now a much larger separation between the second Hopfbifurcation points of both modes. In addition, the signchanges of the transverse Lyapunov exponents for eachmode now occur between the pair of Hopf bifurcations.Numerical bifurcation diagrams obtained for parame-ter set (iv) of Fig. 6 (a) are plotted in in the left handpanels of Fig. 7. Here, A = 0 . γ , ∆ G thr m = − . γ − ,and all other parameters are unchanged from Fig. 4.The increased unsaturated losses mean that mode ν is suppressed for longer and, in contrast to the resultsof Fig. 4, mode ν now completes a region of single-mode SP bounded by two single mode Hopf bifurcations.Following the SP region, the system enters a region ofsingle-mode constant output. However, as we increasethe pump current still further we find the dynamics aredramatically “blown-out” from the single mode manifoldand we enter a region of coupled dynamics in both modes.The location of the blow-out is determined by the loss oftransverse stability of ν , and this point is located shortlyafter ν has become transversally stable. The region ofdual-mode dynamics is bounded at larger values of thepump current by a subcritical Hopf bifurcation of mode ν .In order to compare the measurements of Fig. 2 inthe dual-mode region with the simulation results of Fig.7, we have plotted mode-resolved and total intensitytime traces with j s = 5.1 in the right-hand panels ofFig. 7. The similar nature of the two dynamical statesis clear, with the numerical results reproducing the ob-served bursts of fast pulsations in the total intensity andalso the switching sequence with the intensity rising onthe short wavelength mode before switching and fallingon the long wavelength mode. However, the frequency ofthe bursts that we find numerically is much too low ataround 10 MHz. In addition, the numerical bifurcationsequence in not in full agreement with our measurements.We find a large region of constant output between the sec-ond Hopf bifurcation of ν and the onset of coupled modedynamics and we also do not see any evidence of a dis-continuity in the frequency of the intensity modulationat intermediate values of the pump.By further adjustment of parameters we can shift thepoint where the transverse stability of the equilibriumstate of ν changes at smaller values of the pump. Thiswill result in a narrowing of the region of constant out-put in agreement with experiment. Fig. 6 (b) illustratesthe effect of an increase in g q combined with a further i n t e n s it y [ a r b . un it s ] scaled current density [ j s ] time [ns] (b)(d)(f)(a)(c)(e)H H λ ⊥ λ ⊥ FIG. 7: Left panels: Simulated bifurcation diagrams. Rightpanels: Intensity time traces. (a) Single-mode dynamics. (b)Total intensity. (c),(d) Long wavelength mode ν . (e),(f)Short wavelength mode ν . increase of A . The net effect of these adjustments isthat the positions of the Hopf bifurcations remain largelythe same, but the changes of transverse stability happenmuch closer in pump current to the second Hopf bifurca-tion of ν . Numerical bifurcation diagrams and intensitypower spectra with g q = 1 . A = 0 . ν is the same as in Fig. 7. However, instead ofa dramatic transition to a region of complex coupled dy-namics, in this case we find a very narrow region where adual-mode equilibrium state of the system is stable. Thisstate appears here for the first time in our simulations,and it appears because the order of the changes in trans-verse stability of the single-mode equilbiria has been re-versed compared to the previous example. We find thatthe dual-mode equilibrium state quickly evolves into adual-mode limit cycle at a Hopf bifurcation point. Thisdual-mode limit cycle is unusual in that the amplitude of ν over the cycle is very weak to begin. With a furtherincrease in the pump current, the dual mode limit cycleloses stability, and we observe a dramatic transition to aregion of complex coupled dynamics in both modes. Asin the previous example, the region of complex dynamicsis bounded at a large pump current by a subcritical Hopfbifurcation in ν .A plot of the mode-resolved and total intensity timetraces taken from the region of complex coupled dynam-ics in Fig. 8 with j s = 5.2 is shown in the left-hand panelsof Fig. 9. When compared to our experimental results,there is a greater degree of asymmetry between the dy-namics of the two modes in this example. Unlike theprevious example however, the frequency of the burstsof fast pulsations of the total intensity is now accuratelymatched to our experimental results. We note also thatthe observed bifurcation sequence provides an explana-tion for the dynamics we found at intermediate values ofthe pump current in our experiment. We can now seethat the discontinuity in the frequency of the intensitymodulation of ν was due to the momentary appearanceof a stable dual-mode steady state in the system, leadingto the absence of any structure in the intensity powerspectrum over this interval. In addition, we must rein-terpret the region after the discontinuity as a dual-modestate with strong intensity modulation, but where theamplitude of the component in ν is very weak.The time-traces of Fig. 9 are depicted in a phase-spacerepresentation on the right of Fig. 9. If we consider tra-jectories close to the single-mode equilibrium state of ν ,these trajectories are attracted towards the single modemanifold as the single-mode equilibrium state has a nega-tive transverse Lyapunov exponent. Because this state isunstable in the single-mode manifold, as the trajectoryapproaches the manifold it is repelled into a spirallingorbit towards the SP limit cycle, which is stable withinthe ν manifold. This is the origin of the fast pulsa-tions that grow from the quasi-single mode steady stateof ν . Once the trajectory approaches the SP limit cy-cle, it feels the transverse instability of this limit cycleand is ultimately ejected from the region near the single-mode manifold, undergoing a large amplitude excursionwhere both fields have large intensity. This large excur-sion leads the trajectory to enter the slow region nearzero intensity, where it is drawn towards the single modemanifold of ν , where the single mode equilibrium stateis stable within the manifold. This leads to the cycle offast pulsations that decay towards the single mode equi-librium state of ν . Finally, as the trajectory approaches i n t e n s it y [ a r b . un it s ] fr e qu e n c y [ GH z ] scaled current density [ j s ] scaled current density [ j s ] (b)(d)(f)(a)(c)(e)H H λ ⊥ λ ⊥ FIG. 8: Left panels: simulated bifurcation diagrams. Rightpanels: Power spectral densities. (a),(b): Single-mode dy-namics. (c),(d): Long wavelength mode ν . (e),(f): Shortwavelength mode ν . i n t e n s it y [ a r b . un it s ] time [ns] T ν ν (a)(b)(c) FIG. 9: Left: Simulated intensity time traces. (a) Total in-tensity. (b) Long wavelength mode ν . (c) Short wavelengthmode ν . Right: Phase space diagram the equilibrium, it is repelled on account of the posi-tive transverse Lyapunov exponent of this state. Thisrepulsion drives the trajectory towards the transversallyattracting equilibrium state of ν and the cycle beginsagain. The switch in intensity from ν to ν is along acorkscrew type trajectory that is wound around the lineconnecting the equilibrium states in the two single modemanifolds. This winding may be a signature of the dual-mode limit cycle present before the region of complexdynamics.To investigate this question and to shed more light onthe bifurcation structure in this system, in Fig. 10 wehave plotted numerical continuation results obtained us-ing AUTO [26]. These results indicate that the instabilityof the dual-mode limit cycle leading to complex dynam-ics is due to a supercritical Torus bifurcation (T). We canalso see that the dual-mode limit cycle is present as anunstable object throughout the region of complex dynam-ics, and that there is evidence for further bifurcations ofinterest involving this object beyond the boundary of theregion of complex dual-mode dynamics. In particular, wecan see that the unstable dual-mode limit cycle collideswith the unstable branch of single-mode SPs of mode ν in a transcritical bifurcation (TC). This results in the un-stable branch of single-mode SPs becoming transversallystable for decreasing pump values until the subcriticalHopf bifurcation of ν is reached. This narrow region oftransverse stability for this unstable branch of SPs mayexplain the sharp feature near the relaxation oscillationfrequency found in the power spectral data of Fig. 1.The diminishing strength of this feature with increasingcurrent could be a result of the growth of the limit cycleaway from the location of the stable equilibrium.Finally, we note that although a complete discussionof the bifurcation structure that organises the dynam- . . . . . . . scaled current density [ j s ] i n t e n s it y [ a r b . un it s ] H LPTCTTME SME i n t e n s it y [ a r b . un it s ] H TCTTME SME TC H TM TME TC H TM (a) ν (b) ν FIG. 10: Bifurcation diagrams of ν (upper panel) and ν (lower panel). Single and dual-mode equilibrium states arelabeled SME j and TME respectively. Bifurcation points aresingle-mode Hopf bifurcations (H), transcritical (TC), dual-mode Hopf bifurcations (H TM ), Torus bifurcations (T) andlimit point (LP). Solid and dashed lines indicate stable andunstable objects respectively. The results of numerical simu-lations of the coupled system are included as a guide. Solidvertical lines bound the region of complex dual-mode dynam-ics. ics of Fig. 9 is beyond the scope of the current paper, we can highlight a number of interesting parallels withprevious work on optically injected dual-mode devices[22, 27]. Mathematically, this system is also four dimen-sional, but it features a single, three dimensional invari-ant manifold corresponding to the single-mode injectedsystem. Despite the lower symmetry of the dual-modeinjected system, in the example of [22], we also found asaw-tooth structure characterized by symmetric but op-positely directed trajectories of the two modes. Thesedynamics originated in a torus bifurcation of a dual-modeperiodic orbit. On the other hand, [27] considered an ex-ample of bursting dynamics from the region of the single-mode manifold. These dynamics appeared near a cusp-pitchfork bifurcation of limit cycles and a curve of globalsaddle-node heteroclinic bifurcations. We will present asimilar two-parameter bifurcation study of the currentsystem and explore these connections further in futurework. V. CONCLUSIONS
We have presented an experimental and theoreticalstudy of the dynamics of a dual-mode semiconductorlaser with a saturable absorber. The device was a spe-cially engineered Fabry-Perot laser designed to supporttwo primary modes with a large frequency spacing. Fix-ing the voltage applied to the absorber section, we per-formed a sweep in drive current in the gain section of thedevice. We found that the dynamics evolved from fami-lar self-pulsations in a single mode of the device into acomplex dynamical state of both modes. By extendingthe well-known rate equation model for the semiconduc-tor laser with a saturable absorber to the multimode case,we were able to reproduce the observed dynamics, and toshow the fundamental role played by material dispersionin both sections of the device in governing their appear-ance.
Acknowledgments.
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