Light polarization is an inherent property of the coherent laser output that finds applications, for example, in vision, imaging, spectroscopy, cosmology, and communications. We report here on light polarization dynamics that repeatedly switches between a stationary state of polarization and an irregularly pulsating polarization. The reported dynamics is found to result from the onset of chimeras. Chimeras in nonlinear science refer to the counterintuitive coexistence of coherent and incoherent dynamics in an initially homogeneous network of coupled nonlinear oscillators. The existence of chimera states has been evidenced only recently in carefully designed experiments using either mechanical, optomechanical, electrical, or optical oscillators. Interestingly, the chimeras reported here originate from the inherent coherent properties of a commercial laser diode. The spatial and temporal properties of the chimeras found in light polarization are controlled by the laser diode and feedback parameters, leading, e.g., to multistability between chimeras with multiple heads and to turbulent chimeras.

The spontaneous occurrence of a homogeneous behavior in an heterogeneous network of coupled systems remains a fascinating phenomenon. The most striking example is the synchronous collective dynamics within large populations of interacting oscillators,1 including, e.g., the synchronous flash of fireflies2 and the adjustment of rhythms of coupled pulsating lasers.3 Contrariwise, Kuramoto and Battogtokh proved in 20024 that a homogeneous system can exhibit heterogeneous behavior, i.e., a group of identical oscillators that are nonlocally coupled may experience the coexistence of coherent (synchronized) and incoherent (unsynchronized) dynamics. This new type of collective dynamics was named after the legendary Greek creature: chimera. An in-depth analysis of the underlying mechanisms leading to the emergence of chimera states might have noticeable consequences on how to study very high-dimensional network systems such as the brain5 and its asymmetric function6 that is thought to be related to some brain disorders.7 For example, partial seizure arises from the emergence of local synchronization in a group of neurons.8 

The first experimental observations of chimera states have been conducted only very recently using various types of oscillators: chemical,9,10 mechanical,11 optomechanical,12 electrical,13 or optical.14,15 In the experiment using optical devices, an external nonlinear coupling was introduced. In Ref. 14, chimeras states are observed between coupled virtual oscillators but whose dynamics result from a nonlinear coupling between the laser wavelength and the bias electrical current (i.e., an optoelectronic coupling). In Ref. 15, chimera states result from the phase locking-unlocking between multiple longitudinal modes in a laser diode coupled to a nonlinear saturable absorber. A natural question is whether optical chimeras may be observed from the inherent coherent properties of a commercial laser diode.

In this manuscript, we demonstrate that optical chimeras may be naturally observed and controlled in the light polarization dynamics. Although most of the commercially available laser diodes have a well-defined and stable light polarization direction, re-injecting light with orthogonal polarization forces periodic switching to the normally depressed orthogonal polarization direction through a coherent coupling mechanism.16 We demonstrate that increasing the feedback strength brings the laser diode into a dynamics that combines the normally coherent quiescent polarization dynamics with an incoherent pulsating polarization dynamics. A space-time analogy17 is then applied to map the temporal coherent-incoherent dynamics into spatial regions of coherent and incoherent dynamics among coupled virtual oscillators [see Fig. 1(a)]. As in Refs. 13 and 14, the equivalent nonlocal coupling at the origin of the chimera solution is not the delayed feedback itself but the impulse response of the laser occurring at much shorter time scales, thus coupling neighboring virtual oscillating nodes. This convenient representation allows the visualization of the mechanism leading to the chimera state emergence and of their growth in space and time. The resulting chimera state in light polarization dynamics shows multistability leading to multiheaded chimeras. Addition of a second weak feedback controls the number of chimera-heads. Interestingly, light polarization dynamics show the full family of chimera known from the literature, ranging from stationary chimeras to turbulent chimeras.18,19

FIG. 1.

(a) Space-time analogy method: a temporal data trace is split periodically in slices of length Σ. The data values are color-coded and vertically stacked to form a 2D spatio-temporal representation. (b) Setup of a laser diode subjected to a coherent polarization rotated optical feedback. The polarization state is schematically represented along the light path. Col., collimator lens; BS, beam splitter; FR, Faraday rotator; P, polarizer; Att., attenuator; M1 and M2, mirrors; Iso., isolator; PD, photodiode.

FIG. 1.

(a) Space-time analogy method: a temporal data trace is split periodically in slices of length Σ. The data values are color-coded and vertically stacked to form a 2D spatio-temporal representation. (b) Setup of a laser diode subjected to a coherent polarization rotated optical feedback. The polarization state is schematically represented along the light path. Col., collimator lens; BS, beam splitter; FR, Faraday rotator; P, polarizer; Att., attenuator; M1 and M2, mirrors; Iso., isolator; PD, photodiode.

Close modal

The experiment consists of a laser diode subjected to a polarization-rotated optical feedback (PROF). The experimental setup, sketched in Fig. 1(b), comprises a commercial edge-emitting laser (EEL) JDS SDL-5400 emitting at 852 nm. We operate at J = 85 mA (six times the threshold current) and at 25 °C. The free-running laser emits in its transverse electric (TE) mode with a suppression ratio of 44 dB of the orthogonal transverse magnetic (TM) mode. The beam is collimated and sent into three arms with two 30:70 beam splitters. In the polarization rotated arm, a Faraday rotator is inserted in the external cavity and turns both TE and TM polarizations by 45°. The TM mode is then filtered with a polarizer, while the mirror reflects the TE mode to the rotator, which turns again the polarization by 45°. As a result, the TE mode is injected in the laser with the same direction as the TM mode with a delay set to τ1 = 8 ns and a feedback ratio set to 28.5%. The feedback ratio is deduced from the calibration of the attenuation wheel and from the knowledge of the transmittance of the beam splitter. In the detection arm, we measure the dynamics of the TM mode with a 12 GHz Newfocus 1554-B photodiode and a 36 GHz LeCroy oscilloscope. The delayed feedback is used here to bifurcate from a stationary polarization state to a pulsating polarization dynamics.20,21 As will be detailed and evidenced in the following, the polarization rotation in the feedback loop allows for observing periodically repeating sequences of either stationary polarization states (coherent state) or chaotically pulsating polarization states (incoherent state), which we later relate to chimera states. The time-period of such alternating polarization states is fixed by the delay in the feedback loop. Furthermore, as we demonstrate later, a second isotropic feedback with delay τ2 can be used to control the periodicity of the resulting polarization dynamics, hence controlling the chimera states.

Under the influence of the PROF and for that level of feedback strength, the lasers show periodical polarization switchings between both the transverse electric (TE) and transverse magnetic (TM) polarizations but with chaotic oscillations on the upper state of the plateaus, as depicted in Fig. 2(a.1). The periodicity of the switchings is of Σ = 2τ1 + ε = 16.3 ns with ε = o(τ1) = 0.3 ns, a small time-lag induced by the internal time scales of the EEL. The switching is therefore controlled by the time delay in the feedback loop. In Fig. 2(a.2), we use a space-time representation for the evolution of the TM field intensity ITM. This analogy between time-delay systems and spatio-temporal systems was first introduced in Ref. 17. The time evolution of ITM is divided in slices of duration Σ. Each slice stands for a virtual space Sv composed of an infinite number of virtual oscillators positioned in between [0, Σ]. The slices are incrementally stacked vertically and represent the time evolutions of the virtual oscillators. This representation shows a chimeralike time evolution of a spatially extended systems with two distinct regions: a homogeneous region where all virtual oscillators behave coherently and a heterogeneous region where the oscillators behave incoherently from each other. This chimerical behavior has been found to be stationary; i.e., it lasts experimentally over at least tens of minutes with the same spatial partition between both regions. The stationary chimera state observed experimentally can be considered as “virtual” since it is through the space-time analogy that virtual oscillators are positioned in our single laser system and are nonlocally coupled. The spatial extension of the “virtual oscillators” in the space-time analogy is determined by the time-delay value τ1 of the polarization-rotating optical feedback, and the nonlinear coupling between the “virtual” oscillators originates from the incoherent pulsating dynamics in the plateaus and its corresponding decorrelation time. As will be shown in the following, the resulting chimera shows however all the features of chimeras observed in truly spatially extended networks of oscillators including merging, death, multistability among higher order chimera states, and the route to turbulent chimeras. Moreover, the analogy we draw between our time-delayed laser diode and a set of spatially coupled virtual oscillators has been tested recently in the context of recurrent neural network also called reservoir computing, in which was demonstrated the training of the same polarization rotating laser system to complex data classification and signal restoring.22 

FIG. 2.

Stationary chimera in space-time representation. Time evolution of the field intensity ITM in the time interval [0, Σ] obtained (a.1) experimentally and (b.1) numerically. (a.2) and (b.2) are the corresponding space-time representation over 50 periods. ITM intensity level is color-coded. Numerical simulation are initialized with 2θ1-periodic sin function.

FIG. 2.

Stationary chimera in space-time representation. Time evolution of the field intensity ITM in the time interval [0, Σ] obtained (a.1) experimentally and (b.1) numerically. (a.2) and (b.2) are the corresponding space-time representation over 50 periods. ITM intensity level is color-coded. Numerical simulation are initialized with 2θ1-periodic sin function.

Close modal

The experimental observations are qualitatively well reproduced considering a modified rate-equation model.16,23 The model writes

Ė1=(1+iα)NE1+η2E1(sθ2)exp(iϕ)+n1,
(1)
Ė2=(1+iα)k(Nβ)E2+η1kE1(sθ1)
+η2E2(sθ2)exp(iϕ)+n2,
(2)
TṄ=PN(1+2N)[|E1|2+|E2|2],
(3)

where E1,2 are the TE and TM fields, respectively. s is the time normalized by the photon lifetime τp. For simplicity, we consider τp = 1 ps. N is the carrier density. The model simulates the time s dependency of E1 (TE) and E2 (TM) when accounting for the PROF delay θ1 = τ1/τp and the second isotropic feedback delay θ2 = τ2/τp and their respective feedback strength η1 and η2. α is the linewidth enhancement factor, T is the ratio of carrier to cavity lifetime, k is the gain coefficient ratio between the TM and TE modes, β is the TM mode additional losses, and P is the pump parameter above threshold. In the following, the simulated field intensity ITM = |E2|2 is filtered with a 12 GHz low-pass filter to match the experimental bandwidth. For more realistic simulations, spontaneous noise emission is also considered through the addition of njRξj (j = 1, 2), which are two white Gaussian noise sources with variance R and zero mean.

Simulations have been performed using a 4 steps Runge-Kutta algorithm with a time step of 1 ps. The value θ1 = 8000 is chosen to match the experimental time-delay value (8 ns). In agreement with the experiment, we consider a pumping current value well above threshold P = 1. We refer to our previous work in Ref. 24 for a detailed theoretical analysis of that rate equation model and the corresponding bifurcations. The bifurcation scenario observed when increasing η1 remains qualitatively the same for a large range of values of α and T. We therefore consider realistic values for the linewidth enhancement factor α = 5 and the carrier to photon lifetime T = 150. The chimera states as discussed here are observed for a moderately large value of feedback rate η1. As is well known in the delayed feedback experiment using lasers, the feedback rate is difficult to estimate from the experiment since we measure only the feedback ratio and have no indication of the coupling efficiency of the feedback beam into the laser cavity. In the following, we consider η1 = 0.12. Although the noise is not required to observe the chimera states theoretically, we consider a weak noise level with R = 10−12 to keep the simulations as close as possible to realistic conditions of the experiment and to have a clearer insight into the stability of the chimera states in the presence of noise. The remaining parameters regarding TE and TM modes are taken as k = 0.95 and β = (1 − k)/(2k) to account for a laser diode lasing mostly in the TE mode in the absence of isotropic feedback.

Figure 2(b.1) shows the numerical time evolution of ITM without isotropic feedback and for the parameters listed above. The model is initialized with a 2θ1-periodic sin function. As in the experiment, we observe a quiescent state on the lowest plateau, whereas the upper one has a chaotic dynamic. The space-time representation is also very similar to the one obtained experimentally with a stationary chimera state divided into a coherent region and an incoherent region.

Interestingly, although we observe regular switchings between a quiescent state and a chaotic state in the time evolution of the TM-mode, as shown in Fig. 3, the TE-mode only exhibits switchings between two quiescent states: when the TM-mode is off, the TE-mode has a nonzero stationary emission and when the TM-mode shows chaotic oscillations, the TE-mode is off. Thus, this dynamic cannot be considered as a chaotic switching between two different polarization states. As a result, the chimera state could also be observed from the total output power of the laser Itot = ITE + ITM as switchings between quiescent and chaotic states are observed, as shown in Fig. 3(b).

FIG. 3.

Time evolution of both polarization modes and the total intensity. (a) Time evolutions of the field intensities ITE (orange) and ITM (blue) obtained numerically for identical parameters as in Fig. 2(b.1). (b) Time evolution of the total intensity Itot = ITE + ITM.

FIG. 3.

Time evolution of both polarization modes and the total intensity. (a) Time evolutions of the field intensities ITE (orange) and ITM (blue) obtained numerically for identical parameters as in Fig. 2(b.1). (b) Time evolution of the total intensity Itot = ITE + ITM.

Close modal

Analytically, it is possible to demonstrate that a simple PROF system accepts polarization switching solutions at a periodicity of (2θ1)/(2n + 1) with n as a positive integer that arises from a cascade of Hopf bifurcations.25 However, these harmonic solutions appear to be unstable in numerical simulations and have been stabilized recently in the experiment by adding a second weak isotropic optical feedback.25 As demonstrated in Refs. 26 and 27, the addition of a time delay has a substantial impact on chimera states by creating and maintaining them. Phenomenologically, the addition of a second delay plays here two main roles: it is used (i) as a perturbation to pull the laser out from the strong attraction of the 2θ1-periodicity solution and (ii) as a buffer that memorizes the current solution and as a force that keeps the system on it. In the following, we have fixed the isotropic feedback ratio to only 0.9% which is 31 times weaker than the PROF feedback. The experimental coupling of the feedback light into the laser cavity being unknown, we chose numerically to set even weaker isotropic feedback strength η2 = 0.002 which is 60 times weaker than η1.

In Fig. 4(a.1), when setting the isotropic delay τ2 to 9.4 ns, the TM intensity ITM dynamics shows consecutive switchings in the time interval Σ = 2τ1 + ε = 16.5 ns. Unlocking the third harmonic of the fundamental 2τ1-periodicity leads to a change in the spatio-temporal pattern in Fig. 4(a.2). We observe experimentally a stationary chimera state split into three regions of incoherence. We later refer to this type of dynamics as a multiheaded chimera state, as in Ref. 14. Here, we have a 3-headed chimera state. Numerical simulation reproduces the experimental findings when setting the isotropic delay θ2 in between [10 000, 11 800] (i.e., [10, 11.8] ns considering τp = 1 ps). We show an example with θ2 = 10 000 in Fig. 4(b.1). Here, the simulation is initialized with two antiphased 2θ1/3-periodic sin functions for the variable E1 and E2. The amplitude of the sin-function is set to the square root of the injection current as it is the value taken by E1 for a free-running laser.

FIG. 4.

Experimental and numerical multiheaded chimera. 3-headed chimera: (a.1) temporal and (a.2) space-time representation of the experimental data and (b.1) Temporal and (a.2) space-time representation of the numerical data. 7-headed chimera: (c.1) temporal and (c.2) space-time representation of the experimental data and (d.1) Temporal and (d.2) space-time representation of the numerical data.

FIG. 4.

Experimental and numerical multiheaded chimera. 3-headed chimera: (a.1) temporal and (a.2) space-time representation of the experimental data and (b.1) Temporal and (a.2) space-time representation of the numerical data. 7-headed chimera: (c.1) temporal and (c.2) space-time representation of the experimental data and (d.1) Temporal and (d.2) space-time representation of the numerical data.

Close modal

It is possible to control the number of incoherent cores in multiheaded chimera by carefully adjusting the delay of the isotropic feedback. We show in Figs. 4(c.1) and 4(c.2) the experimental temporal evolution of ITM and its spatio-temporal representation when increasing τ2 up to 9.8 ns. In Fig. 4(c.1), the system shows a polarization switching periodicity of 2.34 ns ≈ 2τ1/7 that, therefore, gives rise to a 7-headed chimera in Fig. 4(c.2). In order to obtain the same result numerically, the simulation has been initialized with a 2θ1/7-periodic sin function. We show in Figs. 4(d.1) and 4(d.2) the numerical temporal and spatio-temporal representation of ITM showing a 7-headed-chimera for θ2 = 11 500. Without this specific initialization, the laser polarization dynamics ends into a 1-head chimera only, hence suggesting multistability between chimera states with multiheads.

To unveil this multistability property of the system, the number of chimera-heads is color-mapped in Figs. 5(a) and 5(b) while scanning the delay parameters θ1 and θ2 and keeping other parameters fixed. Constant initialization leads to only one-head chimera solution (not shown here). By contrast, in Fig. 5(a), the simulations are initialized with a 2θ1/3-periodic sin function. Although the one-head chimera is the predominant solution, regions of 3 and 5-headed chimera can be achieved. Even more complicated chimera states come out when initializing with higher order sin function. In Fig. 5(b), the simulations are initialized with a 2θ1/5-periodic sin function. New regions of chimera states appear over those already unveiled in Fig. 5(a), hence demonstrating the multistability property of the multiple-head chimera states. Multiheaded chimeras are also obtained when the system is initialized with random values, hence demonstrating that both the noise and time-delay play a role in selecting the multiheaded chimera solution.

FIG. 5.

Mapping of the number of chimera-heads in (θ1, θ2) space for (a) 2θ1/3-periodic sin function initialization and (b) 2θ1/7-periodic sin function initialization. (c) Dynamic in spatio-temporal representation when crossing the region of the 5-headed chimera (9000, 11 200) to the 3-headed chimera (9000, 11 300) indicated by an arrow in (a).

FIG. 5.

Mapping of the number of chimera-heads in (θ1, θ2) space for (a) 2θ1/3-periodic sin function initialization and (b) 2θ1/7-periodic sin function initialization. (c) Dynamic in spatio-temporal representation when crossing the region of the 5-headed chimera (9000, 11 200) to the 3-headed chimera (9000, 11 300) indicated by an arrow in (a).

Close modal

The maps of Fig. 5 tells us that chimera states may bifurcate between solutions with different complexity (i.e., different numbers of heads). In Fig. 5(c), we show the spatio-temporal representation of ITM when transitioning the system from a 5-headed chimera region to a 3-headed chimera region (by changing θ2 from 11 200 to 11 300). The onset of the transition is marked by a white dashed line. While the system settles in a 5-headed chimera state, pulling it to a 3-head chimera dynamics leads to a slow spatial drift of the chimera heads followed by a merging of the (1.5) and (2.5) heads, the death of the (4.5) head, and the survival of the (3.5) and (5.5) heads.

In the vast majority of experimental studies, a careful seeding is mandatory to observe the chimera states9–12 or to unlock high-order multiheaded chimera.13,14 In our experiment, although the first order chimera state (1-head chimera) is inherent to the PROF setup and therefore does not need any initialization, we have demonstrated a simple method to control the order of the multiheaded chimera states without the use of seeding: varying the delay of the coherent isotropic feedback, we are able to drive the system into different chimera states showing different numbers of heads.

As reported in Ref. 24, an additional way to control the outcome dynamic is to vary the feedback strength of the PROF. The bifurcation diagram of ITM as a function of η1 is given in Fig. 6(a) considering the same parameters as in Fig. 2(b.1). Several regimes can be discriminated: undamped oscillations (η1 < 0.05), chaotic fluctuations (0.05 < η1 < 0.15), square-wave modulation with regular modulation on the upper state plateau (0.22 < η1 < 0.425), and pure square-wave modulation (η1 > 0.425). Among the chaotic fluctuation regime (0.05 < η1 < 0.15), regular switchings between quiescent and chaotic states are observed for 0.085 < η1 < 0.15 giving rise to the stationary chimera state reported in Fig. 2 at η1 = 0.12. However, when decreasing the feedback strength in between 0.0815 < η1 < 0.085, the stationary chimera bifurcates into an erratic chimera state as illustrated theoretically and experimentally in Figs. 6(b.1), 6(b.2), 6(c.1), and 6(c.2) where regions of coherence and incoherence evolve randomly over time between the oscillators. Using the mathematical classification g0 introduced in Ref. 28, we find that this type of chimera corresponds to a so-called turbulent chimera. Their classification technique relies on the analysis of the spatial partition dynamic g0 between the regions of coherence and incoherence. While in the stationary case, g0 remains fixed over time, it randomly fluctuates when dealing with turbulent chimeras. In Fig. 6(d), we show the time evolution of g0 corresponding to the case of a stationary chimera [Fig. 2(b.2)] and to the case of a turbulent chimera [Fig. 6(c.2)]. While the spatial partition g0 remains almost constant for the first case (i.e., confirming the stationary behavior), it evolves erratically for the second case (turbulent chimera). Turbulent chimeras have been recently predicted in semiconductor laser arrays19 and, interestingly, are found here experimentally from bifurcations arising in laser nonlinear polarization dynamics.

FIG. 6.

Bifurcation diagram and turbulent chimera states. (a) Bifurcation diagram of ITM when varying η1. The arrows indicate the corresponding figures or the corresponding dynamic. Here, the time series from which the bifurcation diagram is computed are not low-pass filtered. [(b) and (c)] Turbulent chimera: time evolution of the field intensity ITM obtained (b.1) experimentally for a PROF feedback ratio of 25.6% and (c.1) numerically for η1 = 0.082. (b.2) and (c.2) are the corresponding space-time representation. We consider here Σ equal to twice the delay. The isotropic feedback is not taken into account, and no specific initialization is applied numerically. (d) Evolution of the spatial partition g0 computed from Fig. 2(b.2) (η1 = 0.12) and (c.2) (η1 = 0.082).

FIG. 6.

Bifurcation diagram and turbulent chimera states. (a) Bifurcation diagram of ITM when varying η1. The arrows indicate the corresponding figures or the corresponding dynamic. Here, the time series from which the bifurcation diagram is computed are not low-pass filtered. [(b) and (c)] Turbulent chimera: time evolution of the field intensity ITM obtained (b.1) experimentally for a PROF feedback ratio of 25.6% and (c.1) numerically for η1 = 0.082. (b.2) and (c.2) are the corresponding space-time representation. We consider here Σ equal to twice the delay. The isotropic feedback is not taken into account, and no specific initialization is applied numerically. (d) Evolution of the spatial partition g0 computed from Fig. 2(b.2) (η1 = 0.12) and (c.2) (η1 = 0.082).

Close modal

To summarize, we have demonstrated the capability of a commercial laser diode to carry out chimera states through its polarization dynamics. Furthermore, depending on the laser and PROF feedback parameters, our system can exhibit either turbulent chimera or stationary chimera. The addition of a second feedback arm governs the obtained chimera pattern with different numbers of heads making this experiment very flexible to study the physics of chimeras, their creations, evolutions, and destructions. Finally, although not discussed here, we find that the gain ratio between the laser polarization modes is a crucial parameter to facilitate the observation of chimera states. Extended numerical simulations show that having a gain ratio close to unity, i.e., a similar gain for both polarization modes, appears to favor the onset of chimeras. Therefore, we expect that chimera states would be more easily obtained from VCSELs known for their strong polarization competition. The capability of a laser diode system to spontaneously and periodically switch between a quiescent dynamics and a chaotically pulsating dynamics—which is here referred to an optical chimera state—also shows new interest in the context of long range chaotic lidar systems that so far make use of an externally modulated laser diode with optical feedback.29 

The Chaire Photonique is a project funded by the Fondation Supélec, Région Grand-Est, Metz Métropole, Département de la Moselle, Airbus-GDI Simulation, Ministère Enseignement Supérieur et Recherche through the Préfecture de Région Grand-Est.

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