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With the development of new applications using semiconductor ring lasers (SRLs) subject to optical feedback, the stability properties of their outputs becomes a crucial issue. We propose a systematic bifurcation analysis in order to properly identify the best parameter ranges for either steady or self-pulsating periodic regimes. Unlike conventional semiconductor lasers, we show that SRLs exhibit both types of outputs for large and well defined ranges of the feedback strength. We determine the stability domains in terms of the pump parameter and the feedback phase. We find that the feedback phase is a key parameter to achieve a stable steady output. We demonstrate that the self-pulsating regime results from a particular Hopf bifurcation mechanism referred to as bifurcation bridges. These bridges connect two distinct external cavity modes and are fully stable, a scenario that was not possible for diode lasers under the same conditions.
© 2017 Optical Society of America
1. Introduction
Semiconductor Ring Lasers (SRLs) have attracted recent attention for their possible applications in photonic integrated circuits because their cavities do not require cleaved facets or gratings [1]. The circular geometry of the active cavity allow SRLs to operate in two possible directions, namely, the clockwise (CW) and the counterclockwise (CCW) modes [2,3]. Many theoretical and experimental investigations concentrate on the dynamics and interplay of these two modes [4–7]. Recently, SRLs subject to optical feedback have been studied for their nonlinear response [8–11] or as potential candidates for new applications [12,13]. Optical feedback allows the control of multi-wavelength emissions of the laser [14,15] for which a stable output is required. Furthermore, semiconductor lasers are capable of generating tunable self-pulsations (SPs) with frequencies above 20 GHz which are desired for a number of signal-processing applications. For conventional single mode semiconductor lasers, a short external cavity and a high feedback rate are necessary for obtaining stable SPs [16]. As we shall demonstrate in this paper, SPs are possible for semiconductor ring lasers subject to optical feedback under much smoother conditions.
The onset of high frequency intensity oscillations in semiconductor lasers results from a particular Hopf bifurcation mechanism involving the beating between two external cavity modes (ECMs). As the feedback rate is progressively increased from zero, ECMs appear in pairs through saddle-node bifurcations. One branch is always unstable and is called “antimode” while the other branch is stable and is called “mode” [17]. The latter may however change stability through a Hopf bifurcation and lead to a branch of solutions that terminates at another Hopf bifurcation located on a nearby ECM. Those branches of pulsating oscillations are called “bifurcation bridges” exhibiting frequencies proportional to the inverse of the round-trip time [17–19]. These bridges have motivated a series of experimental and theoretical studies and have raised the question of their stability [20,21]. For single mode semiconductor lasers, higher order instabilities appearing as the feedback rate increases seriously limit the observation of either a stable ECM or stable time-periodic intensities. In this case, bridges mostly connect a stable mode and an unstable antimode. Consequently, the stability of the fast oscillations is limited to the vicinity of the first Hopf bifurcation. Only for low values of the linewidth enhancement factor α < 1, we may expect fully stable bridges between stable modes [22]. We will show that fully stable bridges are however possible for SRLs for arbitrary values of α. This is because of backscattering that favors bidirectional regimes for moderate nonlinear gain saturation allowing CW and CCW modes to oscillate in antiphase. Our main objective is to find numerically and analytically the best conditions for the observation of either stable ECM or stable antiphase oscillations. In case of no feedback, Sorel et al investigated in [4] the onset of CW and CCW oscillations which were called alternate oscillations. We show that they are naturally part of the first bifurcation bridge as we increase the feedback rate. By following the Hopf bifurcation points in parameter space, we will highlight the preponderant role of the feedback phase on the bifurcation bridges as well as the effect of backscattering on the oscillatory frequencies.
Our theoretical study is motivated by recent experiments performed with a SRL with on-chip filtered optical feedback [23]. Here, we show how stable bifurcation bridges can be anticipated from these experimental results.
The plan of the paper is as follows. We start, in Sec. 2, by describing the experimental results motivating our analysis. In Sec. 3, we introduce the dimensionless rate equations describing a SRL subject to feedback and we compute the ECMs. Section 4 is devoted to the bifurcation bridge mechanism and the characterization of the oscillatory solutions. In Sec. 5, we analyze the effect of the feedback phase on the bridges and we confirm our analytical predictions with numerical continuation techniques. Section 6 makes the link between our results and the well-known alternate oscillations by analyzing the effect of the pump. In Sec. 7, we discuss our main results.
2. Experimental motivations
We use a semiconductor ring laser (SRL) with on-chip filtered optical feedback (FOF). The fabrication of the device was done in the framework of JePPIX [24] on an InP wafer on which both active and passive components are integrated. In the device, which is schematically depicted in Fig. 1, a directional coupler is used to couple the SRL output to the feedback section which consists of two arrayed waveguide gratings (AWGs) to split/recombine light into four different wavelength channels. The AWG channel spacing is 1.336 nm and the free spectral range of the AWG is 5.65 nm. Four semiconductor optical amplifiers (SOAs) are located between the two AWGs in the feedback section to control the feedback. Each SOA gate can be independently pumped electrically. Changing the currents injected in the SOA gates modifies the strength (and phase) of the feedback. By properly adjusting the feedback strength in each wavelength channel, we have demonstrated controlled wavelength tuning [15] and fast wavelength switching [12].
Fig. 1 Schematic representation of the device. The abreviations used are AWG: Arrayed Waveguide Grating, LF: Lensed Fiber, Det: fast detector, OSA: Optical Spectrum Analyzer, Iring: bias current of the SRL, and Igate: pump current to one of the SOA gate.
Without feedback, the SRL is multimode. However, when only one SOA gate is pumped in such a way that the laser becomes single mode, the setup is equivalent to a single mode SRL subject to a conventional feedback [23]. This is because feedback only acts on the lasing wavelength. In this case, experimental results indicate that for some ranges of the feedback strength and phase values, the SRL output exhibits antiphase oscillations as shown in Fig. 2(a). Moreover, when the feedback strength is progressively increased, these oscillations first grow in amplitude, then decrease and finally disappear. This is illustrated by the electrical spectra of Figs. 2(b)–2(d) recorded from the laser output for three increasing feedback strengths (i.e. three increasing pumping current of one SOA gate). These spectra have a strong peak at the frequency of the antiphase oscillations and the height of the peak measures their amplitude. From Fig. 2(b) to Fig. 2(c), the oscillations amplitude increases and then decreases from Fig. 2(c) to Fig. 2(d). These preliminary experimental investigations suggest the possibility of observing a stable bifurcation bridge. In the next section, we formulate the dimensionless rate equations appropriate for a single mode SRL subject to feedback. As mentioned before, pumping one single SOA gate allows the laser to operate in single mode conditions.
Fig. 2 (a) Time trace of the SRL output subject to optical feedback. The CW and CCW modes are in mauve and green respectively. They oscillate in anti-phase. The CCW mode is offseted in the vertical direction for clarity. (b), (c), and (d) Electrical spectra of the laser output with respect to the experimental setup’s noise background for three increasing feedback strengths.
3. Formulation
We consider a SRL operating in a single transverse and longitudinal mode where the two directional modes (CW and CCW) are subject to a delayed feedback. The response of the SRL is analyzed using the set of rate equations formulated in [2] supplemented by terms that take into account the feedback [25]. It consists of two complex equations for the slowly varying electric field Ecw,ccw of the two counter-propagating modes and one real equation for the carrier density N. In dimensionless form, they are given by
In Eqs. (1)–(3), time is measured in units of the photon lifetime (τphoton = 5ps). α is the linewidth enhancement factor, γ is the carrier decay rate and μ is the renormalized pump parameter. The feedback parameters are its strength η, its delay τ, and its phase θ. kd and kc are the real and imaginary parts of the backscattering coefficient. Backscattering of the counterpropagating modes has been the subject of several investigations [2,26,27]. In the common phenomenological description used here, kd is called the dissipative backscattering and kc is called the conservative backscattering. Conservative backscattering originates from localized steps in the refractive index of the passive cavity while dissipative backscattering originates from localized steps in the absorption coefficient (losses) of the passive cavity. In SRLs, steps in the refractive index can be due to the waveguide sidewall roughness, the presence of output couplers, and reflections from the facets of the output waveguides. Backscattering is device dependent but, for a given device, kd and kc remain essentially constant. Nevertheless, small variations could be due, for example, to Mie scattering from a dust particle in the ring cavity. Experimental measurements have shown that backscattering in SRLs is mainly of conservative nature [2], i.e. kd/kc << 1. Along with the gain saturation terms Gcw and Gccw, backscattering is necessary to describe mode competition in SRLs [4,5]. Gain saturation is phenomenologically described by where s and c are small real parameters describing the self- and cross-saturation between the directional modes, respectively. Table 1 shows the values of the dimensionless parameters used in the simulations (and their values in physical units).
Table 1. Values of the model parameters used in the simulations
An ECM is a single frequency solution of Eqs. (1)–(3) of the form
where A, Icw, Iccw, ϕ and ω are real constants. Numerical simulations suggest to look for ECMs exhibiting Icw = Iccw. Substituting (6)–(8) into (1)–(3) leads to conditions for the constant coefficients. Defining I = Icw = Iccw, ϕ = ϕcw − ϕccw, and separating the real and imaginary parts, we obtain Equations (9) and (11) imply sin(ϕ) = 0 meaning that two distinct family of solutions are possible, namely ϕ = 0(mod2π) and ϕ = π(mod2π). Using Eqs. (10) and (12), we obtain equations for ω. They are given by I(ω) and A(ω) can then be obtained by solving Eq. (9) together with Eq. (13). Note that without backscattering (i.e. with kd = kc = 0), we recover the frequency of the ECMs of the single-mode diode laser subject to feedback. We refer to the solution with ϕ = 0 as In-phase ECMs and with ϕ = π as Out-phase ECMs, respectively.4. Bifurcation bridges
In order to have a first insight on the ECMs, Fig. 3 Bottom shows the intensity of the ECMs as a function of η. We use parameters values that typically allow to reproduce numerically experimental results [14], see Table 1. The In- and Out-phase ECMs are shown in green and mauve, respectively. The first In- and Out-phase branches emerging at η = 0 are modes. As the feedback strength increases other ECMs successively appear through saddle-node bifurcations (at η = 0.031 and η = 0.045 in Fig. 3). The upper parts of these branches (above the saddle-node bifurcation) are modes and may be stable. The lower parts are anti-mode and are always unstable. In order to predict a bridge between two ECMs, we look for a crossing between branches [17]. A fundamental property of the ECMs in SRLs is the possibility to have one (or more) crossings between an In-phase and an Out-phase mode as illustrated in Fig. 3. This strongly suggests the presence of stable bridges of periodic solutions. This prediction is confirmed by a bifurcation diagram obtained by numerical integration of Eqs. (1)–(3), see Fig. 3 Top. For very small values of the feedback, the Out-phase mode is stable but quickly destabilizes through a Hopf bifurcation as the feedback strength is increased. The emerging oscillations grow in amplitude then decrease and finally disappear at a Hopf bifurcation now located on the In-phase mode. This bifurcation bridge is located around the first crossing of modes. Along this bridge, the two modes oscillate in anti-phase, see Fig. 4. A second bifurcation bridge emerges around the second crossing (close to η = 0.01). The two modes also oscillate in anti-phase. As the feedback is further increased, a new Hopf bifurcation appears at η ≃ 0.018 (see Fig. 3 Top) leading to in-phase oscillations at the relaxation oscillations frequency. This bifurcation is equivalent to the first Hopf bifurcation observed for conventional semiconductor lasers. A cascade of secondary bifurcations follows before the In-phase branch becomes stable again. Finally, a third bifurcation bridge associated with a new crossing appears near η = 0.05.
Fig. 3 Top: bifurcation diagram of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)–(3). Parameters values are μ = 1.2, and θ = 3.712. Bottom: Power of the CW mode as a function of η for the In-phase (mauve) and Out-phase (green) ECMs given by Eqs (14) and (15), respectively. The arrows indicate the crossings between the modes, corresponding to stable bifurcation bridges in the laser output. Along these bridges the CW and CCW modes oscillate in antiphase.
Fig. 4 Power of the CW mode (full line) and the CCW mode (dashed line) along the first bridge of Fig. 3, for η = 0.002. The two directional mode oscillate in anti-phase with a period T = 140.
It has been demonstrated mathematically [17] that these bifurcation bridges admit mixed ECM solutions of the form
with I1 = 0 (I2 = 0) at the Hopf bifurcation point of mode 2 (mode 1). The amplitudes I1 and I2 are function of η. The intensities |Ecw|2 and |Eccw|2 are now pulsating with a frequency given by |ω1 − ω2|, meaning a beating between the In- and Out-phase ECMs, and with a phase given by ϕcw1 − ϕcw2, and ϕccw1 − ϕccw2, respectively. Moreover, the two modes oscillate in anti-phase because |(ϕcw1 − ϕcw2) − (ϕccw1 − ϕccw2)| = |(ϕcw1 − ϕccw1) − (ϕcw2 − ϕccw2)| = |0 − π| = π. We can estimate |ω1 − ω2| from Eqs. (14) and (15) at the crossing points. For the first bridge, the beating frequency is in first approximation. This contrasts with the conventional single mode semiconductor laser where the beating frequency is proportional to τ−1. Here the delay has only little effect on the oscillations frequency. For the parameters values of Fig. 3, Eq. (18) gives |ω1 − ω2| = 0.042, i.e. a period T = 149 close to T = 140 obtained numerically. Equation (18) highlights the dominant role of the backscattering parameters kc and kd.Because bifurcation bridges appear close to the crossings of ECM branches, we next examine how they depend on parameters. Two parameters that can be varied in experiments, namely the feedback phase θ and the pump P play an important role as discussed in the next sections.
5. Effect of the feedback phase
The feedback phase θ is a key parameter because it dramatically changes the relative positions of the ECMs. The crossings between the ECMs in function of the phase can be obtained with Eqs (9)–(13) and are shown by the red lines in Fig. 5. In order to verify and illustrate the bridge mechanism we have determined numerically the Hopf bifurcation points from the rate equations (1)–(3). Specifically, we use a path continuation technique that allows us to detect and follow the Hopf bifurcation points in parameter space. These Hopf bifurcations are supercritical and are connected by branches of periodic solutions confirming the bridge mechanism. The Hopf bifurcation points are shown by the black lines in Fig. 5. The ECMs solutions are unstable in the shaded region (bifurcation bridges region). By increasing θ, in the range of small to moderate feedback strengths, we successively find 0, 1, 3 or 2 feedback strength values at which the In-phase and Out-phase modes intersect (full red lines in Fig. 5). We have verified that a stable bifurcation bridge appears around each of these crossings, as illustrated by Fig. 3 Top. The dashed red line indicates crossings between a mode and an anti-mode (one In-phase and the other one Out-phase). The corresponding Hopf bifurcations are unstable. The arrow indicates the point where the Hopf bifurcation lines and the red crossing line simultaneously intersect. At this codimension-two double Hopf bifurcation point the crossing between ECMs coincides with the limit point of the Out-phase branch. As this limit point is associated with a change of stability of the Out-phase ECM, this consequently explains the change of stability of the Hopf bifurcations. The bifurcation diagram of Fig. 6(a) illustrates the dynamics for a feedback phase θ = 4.3, i.e. slightly higher than the double Hopf point. The second bridge (around η = 0.025) is located on an unstable ECM branch and is therefore not seen in the bifurcation diagram. It is worth noting that the Hopf bifurcation at the relaxation oscillations frequency no longer appears at this phase. The output intensity is then either constant or periodic.
Fig. 5 Hopf bifurcations lines (black) and crossings between In-phase and Out-phase ECMs (red) in the feedback strength versus feedback phase diagram. Full red lines indicate mode-mode crossings while the dashed red line indicates crossings between a mode and an anti-mode. Full and dashed black lines indicate stable and unstable bifurcations, respectively. Anti-phase oscillations appear in the grey region (bifurcation bridge region). The arrow indicates the point where the Hopf lines cross each other and the red line. Fixed parameters are the same as in Fig. 2.
Fig. 6 Bifurcation diagrams of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)–(3) for different values of the feedback phase: (a) θ = 4.3, (b) θ = 3.6, and (c) θ = 3.5. Fixed parameters are the same as in Fig. 3.
When the phase is decreased from θ = 3.712 (Fig. 3), the first two bridges get closer and then merge as illustrated in Fig. 6(b). If the phase is further decreased, the bridge becomes smaller [Fig. 6(c)] and finally disappears.
It is clear that for a given set of laser parameters, the critical feedback strength where the oscillatory instabilities appear strongly depends on the feedback phase.
6. Effect of the pump
The pump current is known to change the dynamical regime of the solitary ring laser [2]. When increasing the pump from threshold, the laser output first exhibits a stable intensity steady-state which then destabilizes at a Hopf bifurcation and leads to alternate oscillations [2,4] (i.e. anti-phase oscillations in the two propagating directions). The Hopf frequency can be determined analytically [5]
These oscillations are very similar to the oscillations along the bifurcation bridges. The ratio kd/kc is typically small (kd/kc = 0.014) and the kd/kc << 1 limit of the Hopf frequency (19) gives the approximate frequency along the first bridge (18). Further analysis of Eqs (9)–(13) reveals that the pump current has little effects on the bifurcation bridges except at small feedback rates. When the pump current is increased, the Hopf bifurcation branch close to η = 0 extends and suddenly expands to the whole phase range, slightly before μ = 1.3. Consequently the bridges region surrounds η = 0, see Fig. 7. Anti-phase oscillations are now present for any value of θ and η close or equal to 0. Bifurcation diagrams of Fig. 8 illustrate the emergence of a bridge around η = 0 when μ is increased from 1.2 [Fig. 8(a)] to 1.3 [Fig. 8(b)] while the phase is kept constant at θ = 0. When η = 0 (no feedback) the system still exhibits antiphase oscillations in [Fig. 8(b)].Fig. 7 Hopf bifurcations lines (black) in the feedback strength versus feedback phase diagram. Full and dashed lines indicate stable and unstable bifurcations, respectively. The pump parameter value has been increased to μ = 1.3. Anti-phase oscillations appear in the grey region (bifurcation bridge region) which now surrounds η = 0.
Fig. 8 Bifurcation diagrams of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)–(3) for different values of the pump: (a) μ = 1.2, and (b) μ = 1.3. The phase is θ = 0.
Mathematically, this bifurcation bridge is delimited by another Hopf bifurcation line in the negative feedback strength range. This line can be obtained by applying to the stability diagram the transformation η → −η, θ → θ − π which let Eqs. (1)–(3) invariant. The alternate oscillations without feedback can thus be seen as part of the oscillatory solution branch constituting a bifurcation bridge starting at a Hopf point at a negative feedback strength and ending at another Hopf point at a positive feedback strength.
7. Discussion
In this paper, we have studied the onset of stable bifurcation bridges between ECMs in SRLs subject to a delayed feedback. The periodic oscillations along these bridges result from a beating between two adjacent ECMs and their frequency mainly depends on the backscattering between the two directional modes. By contrast to conventional diode lasers [22], SRLs exhibit fully stable bifurcation bridges for any realistic values of the linewidth enhancement factor. In the range α = 2 − 6, the location in parameter space of these stable bridges smoothly depends on α. Along with the first experimental insights discussed in Sec. 2, this motivates new experiments on the bridges dynamical behavior using a ring laser setup.
In our experiments or in on-chip applications, the delay τ of the feedback is constant. Still, it is interesting to discuss its effect as its constant value depends on the device design and, if necessary, a longer delay can be obtained by increasing the length of the feedback loop. From Eqs. (14) and (15), we note that if ωτ = O(1), as it is the case for our bifurcation bridges, the effects of the feedback strength η are enhanced when τ increases. Moreover, the number of ECMs is known to increase with τ [28]. Consequently, for the same range of η, there will be more crossings between ECMs implying more bifurcation bridges. But the dynamics induced by the bifurcation bridges remains qualitatively the same. We have verified that our main conclusions hold for delay values ranging from τ = 5 to τ = 100. In particular, changing the feedback phase strongly modifies the crossing points between ECMs allowing to control the critical feedback strength where anti-phase oscillations appear. Our analysis concentrates on steady and periodic states but, of course, the delay plays an important role in the complex dynamics (quasi-periodicity, chaotic behaviors,...) found in other ranges of parameter values.
For some values of the solitary laser parameters, it has been found that the output exhibits antiphase oscillations without feedback [2, 4]. These oscillations are similar, including in frequency, with the oscillations along the bifurcation bridges. We have shown that these alternate oscillations actually belong to the first bifurcation bridge.
Our results highlight the role of the feedback phase on the apparition of the oscillatory instabilities. It is the key parameter for the control of the oscillatory instabilities. For applications requiring a stable output, phase control sections will be needed in the feedback loops of new integrated SRLs.
Funding
Belgian FRIA; FWO; Hercules Foundation; Research Council of the VUB; Damascus University; Belgian FNRS; Belgian Science Policy Office (IAP-7/35 “photonics@be”).
Acknowledgments
The authors thank X.J.M. Leijtens and J. Bolk from Eindhoven University of Technology for device fabrication in the framework of JePPIX.
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