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We experimentally investigate an intermittent route to chaos in a photonic integrated circuit consisting of a semiconductor laser with time-delayed optical feedback from a short external cavity. The transition from a period-doubling dynamics to a fully-developed chaos reveals a stage intermittently exhibiting these two dynamics. We unveil the bifurcation mechanism underlying this route to chaos by using the Lang-Kobayashi model and demonstrate that the process is based on a phenomenon of attractor expansion initiated by a particular distribution of the local Lyapunov exponents. We emphasize on the crucial importance of the distribution of the steady-state solutions introduced by the time-delayed feedback on the existence of this intermittent dynamics.
© 2016 Optical Society of America
1. Introduction
The development of photonic integrated circuits (PICs) has constituted a major technology advance in recent years allowing for simplifying the design and improving the reliability of optical devices [1,2]. Recently, monolithic PICs yielding optical chaos have been designed for applications in telecommunications [3,4], fast random bit generation [5–9] and all-optical self-pulsation generation [10–12]. Their characteristic compactness and high phase stability make them particularly adapted to accurate experimental analysis of fundamental physical phenomena. A major consequence of the combination of compactness and phase stability found in PICs is the possibility to carry out reliable experimental investigations of laser dynamics with optical feedback. When considering lasers with feedback, a crucial point is the time-scale contrast between the feedback time delay τext (round-trip time of the light in an external cavity) and the period of the laser relaxation oscillation τr. Long (τext > τr) and short (τext < τr) cavity regimes have been identified for which the dynamics yielded by the laser significantly differ [13, 14]. By contrast to the existence of plentiful works reporting dynamics in the long cavity regime [15–17], the studies in the short cavity regime have been until recently mostly limited to theoretical considerations due to practical implementation difficulties in free-space optics [18, 19]. The practical advantages of PICs open the way to in-depth investigations of the dynamical diversity observed in semiconductor lasers in the short cavity regime and in the transition between long and short cavity regimes [20].
Crisis-induced intermittency [21] is a dynamical phenomenon seen throughout various domains of physics [22–26] and is characterized by the sudden appearance of irregular bursts from a regular evolution when the value of a control parameter is varied. The system is then governed by an alternation of two dynamics, corresponding to intermittent evolutions on two distant sub-attractors (attractor expansion) or two coexisting attractors (attractor merging) [21]. As the value of the control parameter is varied, the balance between the two characteristic dynamics, namely laminar regions and bursts, evolves until one of them ends up by prevailing. Despite abundant interdisciplinary reports of crisis-induced intermittency, experimental studies in semiconductor laser with optical feedback are still lacking. Reorganizations of regular output signals into a fast and a slow dynamics have been observed in lasers with optical feedback as off the 1980s [27–29] and were initially termed intermittency. However, they are nowadays known as low-frequency fluctuations [30,31] or regular pulse packages [13,14] and the mechanisms at their origin are in consequence totally different from the intermittent dynamics presented here.
In this article, we present a systematic study of crisis-induced intermittency occurring in the route to chaos observed in a PIC consisting of a semiconductor laser with optical feedback. The time delay of the feedback has been chosen in order to provide suitable conditions for intermittent dynamics. We demonstrate that, under these conditions of feedback delay, the route to chaos systematically unfolds through intermittency. It starts with an initially regular period-doubling dynamics gradually interspersed by bursts of chaotic fluctuations as the feedback strength is increased. The dynamical scenario is qualitatively reproduced with the Lang-Kobayashi model and theoretical insight into the mechanism underlying the phenomenon is provided by the numerical analysis of the local Lyapunov exponents. In particular, we point out how the intermittent dynamics is originated through a phenomenon of attractor expansion induced by an increase in the values of the feedback strength.
2. Experimental set-up
Our experimental system is composed of a distributed feedback semiconductor laser bounded by a photodiode on one side and by an external cavity on the other side, as presented in Fig. 1 [6, 8]. The external cavity has a length of 3.3 mm and includes two semiconductor optical amplifiers (SOA1 and SOA2) and a passive waveguide ended by a reflector on which a high-reflectivity coating (higher than 99.9%) is applied. The output signal is provided by the photodiode, therefore optical data are not directly accessible in this chip. The laser and the SOAs are made of InGaAs/InGaAsP layers grown on InP substrate. The optical gains of the SOAs have been studied in a similar PIC [7]. We cannot measure these gains in our system due to the absence of optical output, yet we speculate that their values are of the same order as those in Ref. [7].
Fig. 1 Photonic integrated circuit. PD: photodiode, DFB laser: distributed feedback laser, SOA: semiconductor optical amplifier.
The external cavity frequency of 11.8 GHz (τext = 85 ps) is higher than the relaxation oscillation frequency (3 ~ 7 GHz, corresponding to 467> τr >143 ps), indicating that the laser operates in the short cavity regime. The laser threshold current Jth is equal to 12.0 mA. The feedback strength is tuned by changing the injection current in SOA1 (JSOA1) while the current in SOA2 is kept constant (JSOA2 =5.0 mA).
3. The experimental intermittency route
Figure 2 presents the cascade of dynamics observed in the PIC as the feedback strength JSOA1 is increased. The laser is biased with an injection current equal to 3.0 Jth. The dynamical scenario begins with an initial stationary state. As JSOA1 is increased, bifurcations are undergone revealing a first periodic dynamics pulsing at 9.4 GHz, followed by a period doubling dynamics [Figs. 2(a.1)–2(b.1)]. In the routes to chaos observed in common laser systems, a succession of period doubling bifurcations giving way to chaotic fluctuations is expected from this stage [32]. By contrast, in the present case the transition from period doubling to chaos yields temporal waveforms exhibiting a particular double dynamics organized in laminar regions and bursts, as illustrated in Figs. 2(a.2)–2(a.3). The laminar regions of low amplitude correspond to the regular period-doubling dynamics while the bursts of high amplitude are made of complex chaotic fluctuations [Figs. 2(c)–2(d)]. This alternation of laminar regions and bursts is characteristic of intermittency.
Fig. 2 Experimental observation of intermittency as the feedback strength JSOA1 increases for a value of the injection current J equal to 3.0Jth. (a) Temporal waveforms and (b) RF spectra. (a-b.1) JSOA1=10.8 mA, (a-b.2) JSOA1=12.93 mA, (a-b.3) JSOA1=13.12 mA, (a-b.4) JSOA1=13.23 mA. (c) and (d) are zooms on the dashed boxes in traces (a.2) and (c), respectively.
As the feedback strength increases, the balance between the laminar regions and the bursts is altered. The bursts are first few and the waveform is mostly made of regular fluctuations [Fig. 2(a.2)]. The corresponding radio-frequency (RF) spectrum [Fig. 2(b.2)] suggests a dynamics highly governed by the period doubling dynamics, as sharp frequency peaks dominate the spectrum while chaos content is low. Yet, as JSOA1 is increased, the bursts spread and yield a richer chaos content. The general level in the spectrum increases and the amplitude contrast with the peaks corresponding to the frequencies of the period doubling lowers [Fig. 2(b.3)]. As JSOA1 further increases, the laminar regions shrink until the whole temporal waveform is changed into chaos [Fig. 2(a.4)]. In the corresponding spectrum, the initial periodic frequency signature has disappeared to the benefit of a broad frequency content [Fig. 2(b.4)].
Figure 3(a) presents an experimental bifurcation diagram corresponding to the scenario in Fig. 2. The intermittency region (labeled I) appears as a transition between the fully regular period doubling state (region P2) and the fully chaotic state (region C), exhibiting both dynamics (P2+C). Figure 3(b) shows a two-dimensional bifurcation diagram obtained by varying JSOA1 from 0 to 15.0 mA for values of the normalized laser injection current J/Jth between 1.0 and 5.0. The phenomenon of intermittency is observed whenever a route to chaos is undergone and is not restricted to particular values of injection currents or feedback strengths. It is therefore a systematic mechanism, inherent to the process of generation of a chaos attractor from an initial limit cycle. It is worth noting that the intermittency regions indicated in Fig. 3(b) include a variety of different dynamics, where the laminar parts can be stationary, periodic or quasi-periodic solutions of low amplitude, while the burst parts can show periodic, quasi-periodic or chaotic oscillations of high amplitude. Although the bifurcation scenario and the succession of dynamics are robust and repeatable, we experimentally noticed that the precise values of JSOA1 for which they occur are likely to show small drifts when operating under different initial conditions. In consequence, differences of the order of a few mA can be noticed when comparing Figs. 3(a) and 3(b).
Fig. 3 (a) Experimental bifurcation diagram of the scenario in Fig. 2 (J/Jth=3.0). (b) Experimental two-dimensional bifurcation diagram showing evidence of intermittent dynamics in a wide range of parameters.
4. Numerical analysis
To give insight into the origin of intermittency, we carry out a numerical study of the system using the Lang-Kobayashi equations [15, 33, 34]. This numerical study aims at understanding the evolution of the system in the phase space and thus to bring an interpretation to the temporal evolution based on attractor considerations. The purpose is to clarify the birth of the jumps from an initial regular evolution to the complexity of the intermittent dynamics and to identify the orbits relative to the laminar regions and the bursts. The choice of the Lang-Kobayashi model is motivated by its simplicity and universality. It may yet be far from reproducing the experimental complexity, to which models taking into account the nonlinear dynamics of the SOAs and particularities related to short cavity configurations would be more appropriate. Our objective is to find a set of parameters compatible with the experimental ones for which the Lang-Kobayashi model qualitatively reproduces the observed intermittent dynamics. The equations write as follows:
In these equations N, E and Φ are respectively the carrier density, the electric field amplitude and the electric field phase. τext and κ represent the feedback delay and strength. α is the linewidth enhancement factor, J is the laser injection current (Jth is the threshold current), GN is the gain coefficient, N0 is the carrier density at transparency, τp and τs are the photon and carrier lifetimes. ϵ is the gain saturation coefficient. We use the parameter R to measure the feedback strength. R can be seen as a reflectivity coefficient for the electric field amplitude and is defined by . In this formula, τin and R2 are respectively the photon round-trip time in the laser internal cavity and the amplitude reflectivity coefficient of the laser output facet. The values of the parameters are given in Table 1.
Table 1. Parameter values in simulations
The succession of dynamics leading to chaos through intermittency is presented in Fig. 4. For increasing values of the feedback strength R, the evolution of the laser intensity and the optical frequency shift versus time are given. In parallel, phase trajectories show the corresponding attractors developed around the external cavity modes and anti-modes in the phase space defined by the carrier density and the optical frequency shift, representing the shift from the optical frequency of the solitary laser. Modes and anti-modes correspond to originally stable and unstable steady-state solutions born from saddle-node bifurcations in the Lang-Kobayashi equations [30], respectively. The bifurcation cascade begins with the destabilization of an initial limit cycle corresponding to a period doubling dynamics and located on the maximum gain mode (lowest value of carrier density). As R increases, this limit cycle is changed into a small chaos attractor [Fig. 4(b.1)]. When a critical value is reached (R = 0.0964), intermittency occurs and the temporal waveform [Fig. 4(a.2)] displays a double dynamics made of bursts and laminar regions. The corresponding phase trajectory reveals that the initial attractor has expanded so that the global attractor is now organized in two sub-attractors [Fig. 4(b.2)]. The temporal evolution of the optical frequency shift shows that bursts and laminar regions are each related to one of the sub-attractors, which can then be identified to a laminar sub-attractor and a burst sub-attractor. The jumps between them cause the alternation of bursts and laminar regions seen in the temporal waveform [Fig. 4(a.2)]. As both sub-attractors grow on, they eventually end up by merging into one final attractor [Fig. 4(b.3))] At that point, there are no more laminar regions or bursts and the dynamics is fully chaotic [Fig. 4(a.3)].
Fig. 4 Numerical scenario showing intermittency as the feedback strength R increases. (a) Temporal waveforms, and (b) attractors. (a.1–b.1) R = 0.0963, (a.2-b.2) R = 0.0964, (a.3–b.3) R = 0.120. The temporal waveforms present the evolution of the laser intensity (black traces at the top) and the optical frequency shift (red traces at the bottom). In trace (a.2), the evolution of the optical frequency shift reveals that the succession of laminar regions (L) and bursts (B) in the time trace corresponds exactly to jumps between the two sub-attractors.
This phenomenon of attractor expansion inducing long stretches of time on distant sub-attractors shows features similar to crisis-induced intermittency, reported in various dynamical models [21–26]. We managed to find a set of parameters for which intermittent dynamics can be numerically reproduced by increasing the feedback strength. The simulated results presented in this paper correspond to a feedback delay time of 140 ps, which is equivalent to an external cavity length of 5.3 mm (τext = 2nLcav/c, where Lcav is the external cavity length and n=3.96 is its refractive index). However, in our PIC, the external cavity length is 3.3 mm and the resulting feedback delay time is 85 ps. This deliberate discrepancy between the numerical and experimental time delays is motivated by a concern of making the bifurcation scenario and the temporal waveforms obtained in each case easy to understand with clear illustrations. We noticed that in the 85 ps case the jumps were fewer, causing the number of laminar regions and bursts to be too few to illustrate properly the experimental observations of Fig. 2. Therefore we present here numerical evolutions obtained with a longer delay time (140 ps), for which the temporal attractor characteristics are closer to the experimental ones. We carried out further experimental studies on a 5.3-mm external cavity PIC, with a feedback delay time equal to 140 ps, which revealed intermittency with dynamical features similar to the ones presented in Figs. 2 and 3 but with fewer manifestations of routes to chaos and in consequence fewer examples of intermittent dynamics. We explain this tendency by the fact that a longer feedback delay time induces a reduction in the number of bifurcations to chaos. Similarly, we also carried out numerical studies by setting the feedback delay time τext to 85 ps to match the experiment. In this configuration, intermittent route to chaos is visible as in the case of τext = 140 ps presented in Fig. 4. Yet we choose to keep τext at this value of 140 ps since the evolutions of the time traces and, in particular, the sub-attractors are better suited as general illustrations to the phenomenon of intermittency we describe in this paper. It is worth mentioning that the properties of intermittent dynamics depend on the value of the delay time. We will dedicate deeper attention to this fact in a future study, yet we can observe that there are optimal conditions on τext for which intermittent dynamics exist.
5. Statistical characteristics
5.1. Characterization with local Lyapunov exponents
The local stability of the attractor can be analyzed with the maximum local Lyapunov exponents (LLE) of the system [35]. The study of the local Lyapunov exponents is motivated by identifying a criterion based on the local stability of the attractors that would explain the process leading to the phenomenon of attractor expansion presented in the previous section. Figure 5 presents the distribution of the LLE in the attractor at the point where intermittency is about to start (R = 0.0963) and at its first occurrence (R = 0.0964). The attractors are constituted of multitudes of dots, each dot being a LLE calculated with an averaging time of 50 ps. Figure 5(a) reveals an inhomogeneous distribution of the LLE with a high concentration of large values at the vicinity of the closest anti-mode. This concentration suggests that the system is getting prepared to generate a burst from this region of the attractor, where the values of the LLE are the highest. For a slightly higher value of R, the initial attractor expands inducing a more complex distribution of the LLE, as illustrated in Fig. 5(b).
Fig. 5 Distribution of the local Lyapunov exponents (LLE) in the phase space (a) without and (b) with intermittency (see Visualization 1 for a detailed evolution when the feedback strength increases). (c) Distribution of the LLE in the temporal evolution of the optical frequency shift.
Figure 5(c) presents the evolution of the optical frequency shift versus time in the colored scale of the LLE values. An evolution on the laminar (burst) sub-attractor corresponds to values fluctuating around -8 GHz (between -4 and 8 GHz). The dots shaping the connection from the laminar sub-attractor to the burst sub-attractor (jump-up) correspond to high values of LLE, suggesting that a jump from the laminar state to a burst is associated with high instability. By contrast, a jump back from the burst sub-attractor to the laminar one (jump-down) only involves low values of LLE, suggesting that the recovery to the solution closer to the maximum gain mode is easier.
We understand from Fig. 5 that intermittency is a matter of interaction between two counteracting phenomena. The first is the natural expansion of the initial attractor to neighboring modes, originally stable solutions of saddle-node bifurcations (nodes) undergone by the increase of the feedback strength. The second is the repelling effect of the anti-mode, the unstable counterpart (saddle), that inhibits to some extent the attractor expansion. The coexistence of these two opposed effects yields a distribution revealing a high density of large values of LLE in the portion of the attractor close to the anti-mode, illustrating high instability [Fig. 5(a)]. It is worth noting that local Lyapunov exponents are not invariant against coordinate transformation and that discussing distances between modes and orbits of high-dimension attractors represented in a two-dimensional phase space may not be relevant. Nonetheless, we propose the interpretation that the distance between modes and anti-modes is a crucial factor in the balance between these two behaviors. If they are too close to each other, like in the long cavity regime, the initial attractor can easily reach neighboring modes and give way to chaotic dynamics including low-frequency fluctuations [30,31]. By contrast, a too large mode spacing makes the expansion of an initial attractor to neighboring modes difficult, such as in the case of very short cavity regimes where chaos is scarce and the dynamics are mostly periodic [10, 12]. In configurations suitable for intermittency, the destabilizing action of the anti-mode starts by pushing back the expanding orbit towards the maximum gain mode. As the feedback strength increases, the attractor gradually resists this repelling effect and its orbit expands in the phase space. At a given point, the anti-mode has the possibility to propel the orbit in the direction of a neighboring mode, triggering thus the intermittent dynamics.
We understand therefore that a necessary condition to the existence of intermittency is an intermediate mode spacing, short enough to yield the creation of a new sub-attractor but large enough to allow the orbit to spend long stretches of time on each sub-attractor. Since the mode spacing is determined by the feedback delay time τext [30], its value is a key parameter in the mechanism. By changing the value of τext in our numerical analysis, we find that intermittency exists for delay times equivalent to external cavity lengths ranging few millimeters. This order of values corresponds to the short cavity regime and such systems can be designed with PICs.
Figure 6(a) displays the histogram of the LLE values. We notice that the distribution gets qualitatively re-shaped when the feedback strength increases as it starts from a symmetrical curve, followed by a gradual emergence of a long-tailed broader distribution developed for large LLE values. This has been explained for general intermittent dynamics as the consequence of the orbit interpolating the two sub-attractors [36].
Fig. 6 (a) Distribution of the local Lyapunov exponents and (b) evolution of the values of the average and the standard deviation with the feedback strength R.
We observe in Fig. 6(b) that the values of both the average and the standard deviation of the LLE gradually increase as the systems goes through intermittency until stabilizing to a maximum value. This smooth evolution illustrates the fact that the final large attractor needs time to be built, and that intermittency is a gradual process of construction of a final chaos attractor starting from the expansion of an initial small attractor.
5.2. Evolution of the laminar times
As visible in Fig. 2(a), the balance between laminar regions and bursts and their duration depend on the feedback strength. We analyze the evolution of the average duration of the laminar regions in the intermittency transition. It has been reported in systems exhibiting intermittency that the evolution of the average laminar times T versus the control parameter P follows the law: T ∝ |P − Pc|−γ [21,37]. In this formula, Pc is the critical value of the control parameter P for which the phenomenon of intermittency starts, γ is termed critical exponent and represents the scaling factor obtained when plotting T and |P – Pc| in a logarithmic scale. In our case, the experimental control parameter P is the feedback strength, represented by JSOA1 in the experiment and by R in the simulations. Evaluating γ in systems showing crisis-induced intermittency is a means to compare their temporal properties and gives insight into attractor characteristics. In other words, γ illustrates how the system behaves and recovers from intermittent transitions.
Figure 7 presents the evolution of the average laminar times with the feedback strength obtained in the experimental scenario of Fig. 2 and in the simulated scenario of Fig. 4. The graphs depict in a Log-Log representation the distribution of the average laminar times (squares) aligned on the straight line corresponding to their fitting curve T ∝ |JSOA1(R) − JSOA1c (Rc)|−γ. The important point is that the distribution of the average laminar times follows the power law distribution. This property seems to be acknowledged as a characteristic of crisis-induced intermittency throughout many domains of physics [21-26]. We insist on the fact that the evolution of the average laminar times follows the power law evolution, which is often observed in systems showing intermittency.
Fig. 7 (a) Experimental and (b) numerical evolutions of the average laminar times in the intermittency transitions of Figs. 2 and 4. The critical values above which intermittency begins are JSOA1c = 10.9 mA and Rc = 0.0963. The squares are measured data, the straight lines are curve fittings.
The values of γ of 1.80 for the experimental case and 0.50 for the simulated case are estimated from Fig. 7. In the various regions in which intermittency is experimentally seen in the route to chaos in Fig. 3(b), we find that the distribution always yields a scaling law as in Fig. 7 with values of γ varying between 0.52 and 4.60. We understand from the interval of values of γ that all intermittent transitions are not the same and that the parameters (injection current, feedback strength) have an impact on the distribution of the laminar times.
6. Conclusion
We reported experimental and numerical studies of a systematic intermittent route to chaos in a laser with optical feedback in a photonic integrated circuit. We highlighted the fact that the scenario unveils a state in which a double dynamics is seen in the transition from period doubling oscillations to chaos. The characteristic intermittent dynamics organized in laminar regions and bursts is explained by a succession of jumps between two sub-attractors. Under appropriate conditions of the feedback delay time, intermittency occurs when an anti-mode repels the orbit of the attractor towards different neighboring modes. This phenomenon is characterized by a particular distribution of the local Lyapunov exponents.
The technology of photonic integrated circuits allows for exploring laser systems in the millimeter scale and therefore to unveil new dynamical phenomena. The route to chaos through intermittency observed in the present PIC is an example of dynamics revealed by an appropriate value of the feedback time delay.
Funding
Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP24686010) and Management Expenses Grants from the Ministry of Education, Culture, Sports, Science and Technology in Japan.
Acknowledgments
The authors thank Ingo Fischer for fruitful discussions and suggestions.
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