Abstract
Optical nonlinearities, such as thermooptic mechanisms and freecarrier dispersion, are often considered unwelcome effects in siliconbased resonators and, more specifically, optomechanical cavities, since they affect, for instance, the relative detuning between an optical resonance and the excitation laser. Here, we exploit these nonlinearities and their intercoupling with the mechanical degrees of freedom of a silicon optomechanical nanobeam to unveil a rich set of fundamentally different complex dynamics. By smoothly changing the parameters of the excitation laser we demonstrate accurate control to activate two and fourdimensional limit cycles, a perioddoubling route and a sixdimensional chaos. In addition, by scanning the laser parameters in opposite senses we demonstrate bistability and hysteresis between two and fourdimensional limit cycles, between different coherent mechanical states and between fourdimensional limit cycles and chaos. Our findings open new routes towards exploiting siliconbased optomechanical photonic crystals as a versatile building block to be used in neurocomputational networks and for chaosbased applications.
Introduction
The longterm solutions of any one or twodimensional nonlinear system restrict to nothing more complicated than a stable/unstable fixed point or a limit cycle, their possible bifurcation types being already sufficiently well studied^{1}. If more than two dimensions are at play, trajectories may become much more complex, eventually displaying aperiodicity and extreme sensibility to initial conditions, that is, exhibiting chaos. The Lorenz equations are the paradigmatic example of how complex and rich the solution of a relatively simple deterministic set of differential equations can be^{2,3}, where only one of the three is nonlinear. Despite being five decades old, wide ranges of the parameters governing the equations are still unexplored.
Classical nonlinear dynamics have been studied quite extensively in different optomechanical (OM) architectures, where the classical version of quantum Langevin equations describes the evolution of the optical cavity field and the mechanical oscillator^{4}. A relatively standard solution of that nonlinear system is a state of wide mechanical coherent oscillations^{5}. Interesting features and tools can appear in that regime, for example, the existence of several stable mechanicalamplitude solutions for a fixed set of external parameters, which was studied theoretically in ref. 6 and demonstrated experimentally in refs 7, 8. Moreover, as predicted in ref. 9 for sufficiently high laser power, the system can eventually enter into a chaotic regime, which may be exploited for chaosbased secure data communication or sensing, among other applications. In this regard, integrated OM systems may present advantages with respect to coupled lasers^{10} or hybrid optoelectric oscillators^{11} in terms of the ease of integration, scalability and engineering of the nonlinearities^{12}. Despite their potential, only the works of Carmon et al.^{13,14} and Monifi et al.^{15} in dielectric microtoroids and by Wu et al.^{16} and our group^{17} in silicon (Si)based integrated devices have addressed chaos in integrated OM architectures.
We report here on the nonlinear dynamics of an OM cavity system, namely a Sibased onedimensional photonic crystal, mainly affected by optical nonlinearities of diverse origin: thermooptic effects, freecarrier dispersion and OM coupling^{18}. These are intercoupled through the number of intracavity photons (n), which affects and is affected by the previous mechanisms. As a consequence, new dynamics in an OM cavity are disclosed by exploring a wide area of the parameter space. The system displays a heterogeneous variety of stable dynamical solutions that, in some specific cases, coexist and give rise to bistability and hysteresis.
Results
General properties of the sample
The device investigated here is a onedimensional OM photonic crystal fabricated using standard Si nanofabrication processes (see Methods) in a silicononinsulator (SOI) wafer. As seen from the top view in Fig. 1a, the crystal lattice constant is quadratically reduced towards the centre of the beam, hereby defining highQ optical cavity modes.
To model accurately the fabricated OM system and account for the differences with respect to the nominal geometry, the inplane geometry is imported from the scanning electron microscopy (SEM) micrographs into the finite element method (FEM) solver (see Methods and Supplementary Fig. 4), the thickness being that of the top Si layer of the SOI wafer. This procedure ensured a good agreement between the measured optical modes and those extracted from simulations (Fig. 1b).
In the following, we investigate the third optical mode (Fig. 1c) of the OM photonic crystal. It displays an asymmetric field distribution with respect to the xz plane, giving rise to surprisingly high values of the singleparticle OM coupling rate (g_{o,OM}, dots of Fig. 1d) for inplane (xy plane) flexural modes (red dots of Fig. 1d). In Fig. 1d, we have indicated with an arrow the one having three antinodes along the xdirection (Ω_{m}=54 MHz), since it will be shown it is at the heart of the complex dynamics discussed in the next sections. The radiofrequency (RF) signal associated with optical transduction of thermally driven mechanical eigenmodes is also reported in Fig. 1d. The peaks positions show a good agreement with the modes predicted to display high g_{o,OM} values.
An evident discrepancy between the FEM simulations of g_{o,OM} and the actual transduced RF signal concerns outofplane flexural modes (blue dots of Fig. 1d), which should not present significant g_{o,OM} values due to symmetry considerations (see Methods and Supplementary Fig. 6). However, geometric inhomogeneities along the z axis may break the symmetry along that direction, unbalancing the contributions from the top and bottom Si–air interfaces. As it will be discussed later, among the outofplane flexural modes, the fundamental one (=5 MHz, first peak in the RF spectrum) plays a key role in the chaotic dynamics of the system.
Bistability and hysteresis
It is well known that large energies stored in optical resonators may give rise to strong nonlinear effects. The two most intense sources of optical nonlinearities in Sibased optical resonators are: thermooptic effects, which redshift the resonance position (λ_{r}) from its unperturbed position (λ_{o}) proportionally to the average cavity temperature increase (ΔT); and freecarrier dispersion, which induces a blueshift proportional to the free carrier population (N). The dynamics of ΔT and N can be described by a system of rate equations (see Methods) that are coupled through the number of intracavity photons:
where n=n_{o}=2P_{l}κ_{e}λ_{o}/κ^{2}hc in perfect resonance, that is, λ_{l}=λ_{r}. P_{l} and λ_{l} are the laser power and wavelength, respectively; κ_{e} and κ are the extrinsic and overall optical damping rates, respectively, the latter determining the cavity resonant linewidth (Δλ_{o}=κ/2π). The only two parameters that have been modified experimentally within this work are P_{l} and λ_{l} with a resolution below 1 μW and 1 pm, respectively. It is also worth noting the possibility of tailoring, in a much less accurate way, other parameters entering the equations, such as κ_{e}, κ and the heat dissipation rate, by adjusting the fibre/sample relative alignment. In most of the situations, the longterm dynamic solution of the {ΔT, N} system is a stable fixed point. However, for specific combinations of P_{l} and λ_{l}, the fixed point undergoes a supercritical Hopf bifurcation, transmuting into an unstable fixed point surrounded by a stable limit cycle (see Methods). The longterm solution of the {ΔT, N} system is then that called from now on selfpulsing (SP)^{19}, where the cavity resonance oscillates periodically around the laser at a frequency denoted by ν_{SP}. The total time required to complete the SP limit cycle is thus 1/ν_{SP}, although it is not necessarily drawn at a constant pace. When the SP limit cycle is active, light within the cavity is modulated in a strongly anharmonic way, creating an optical frequency comb with multiple peaks spectrally located at integers of ν_{SP}, the intensity of which decreases with frequency. An important consequence is that the intracavity radiation pressure force (F_{o}), which is proportional to n, is also modulated in an equivalent fashion. In a previous work^{18}, we demonstrated that if the Mth harmonic of F_{o} has a significant overlap with the linewidth of a mechanical mode with sufficient g_{o,OM}, the latter enters a highamplitude, coherent regime, which fulfils the requirements to be identified as phonon lasing^{4,20}.
The transition from pure SP dynamics to the coupled SP/phonon lasing counterpart is effectively that from a twodimensional to a fourdimensional coupled nonlinear system. Indeed, the mechanical oscillator is not just being unidirectionally driven, since the OM coupling provides an effective back coupling that forces ν_{SP} to be frequencylocked to ν_{SP}=Ω_{m}/M over a wide spectral range of the laser source (see Methods). Lower M values result in stronger F_{o} and higher coupling strengths between the SP and the mechanical coherent oscillator. Once the system is found in a fourdimensional limit cycle it becomes very robust, a characteristic that can be quantified to some extent from the spectral width of the frequencylocked plateau and the low phase noise of the RF signal, which displays a main RF peak linewidth of a few kHz. These features are a direct indication of ranges of laser parameters where two stable states of the system coexist, namely the twodimensional SP and the fourdimensional SP/phonon lasing. The existence of bistability enables the possibility of frequency jumps and hysteresis as the laser parameters are scanned. To check for these features we have studied the full RF spectra as a function of λ_{l} at P_{l}=2 mW. The experimental data were taken by sweeping λ_{l} from bottom to top (Fig. 2a) and vice versa (Fig. 2b). As expected, λ_{l}frequency jumps and λ_{l}hysteretic regions appear where the system undergoes state transitions. By singling out the position of the first RF harmonic (Fig. 2c), the λ_{l}bistable regions are clearly exposed, which is also observed in the numerical simulations (see Methods).
Most of the λ_{l}bistable features described above are reproduced well by numerically solving the fourdimensional coupled nonlinear system {ΔT,N,u,du/dt} (see Methods), where u and du/dt are the generalized coordinate for the displacement of the mechanical mode and its derivative, respectively. An abrupt transition between two available states occurs if the frequency difference between them (Δν) reaches a specific value Δν<Δν_{up} (Δν>Δν_{down}) for an up (down) transition, where Δν_{up}<Δν_{down}. To unveil a hysteresis cycle, Δν must fulfil the conditions for performing both an up and a down transition along the explored parameter path. In general, in our system we observe that Δν_{up} and Δν_{down} increase (that is, the frequency jumps are larger) with P_{l} and for lower M values, in response to an increasing of the coupling strength between the {ΔT,N} system and the mechanical oscillator.
Regions displaying λ_{l}bistability unfold experimentally and in the numerical simulations at different spectral windows. It is worth noting that it is not possible to simultaneously operate within two of these windows because they are associated with the same photonic resonance. However, it is viable to probe photonic resonances alternative to the one being pumped, taking care not to induce further optical nonlinearities. In that case, the dynamics of the probe transmitted signals would be coherent with that of the pump, since each one just follows the common dynamics of the effective refractive indices^{15}.
The laser parameter space has been further explored by monitoring the system response to changes on P_{l} for fixed values of λ_{l}. The analysed domain lies in the vicinity of the widest bistable region of Fig. 2c (the precise values of λ_{l} under analysis are highlighted with dashed lines and called from now on I, II, III and IV, respectively). The results are reported in Fig. 3 by plotting the spectral position of the first RF harmonic peak.
Regions displaying P_{l}bistability and P_{l}hysteresis appear only if λ_{l} is above the λ_{l}frequency jump of the forward λ_{l} scan. If λ_{l} lies well within the λ_{l}bistable region, such as in I, the system can start from either of the two stable states shown in Fig. 3 (red and black curve). However, within the studied P_{l} range along I, Δν only satisfies the requirements for performing a downtransition (Δν>Δν_{down}), being unable to accomplish an uptransition to a M=1 state. To drive the system to a M=1 state, λ_{l} has to be increased and, subsequently, decreased.
A P_{l}bistable region unwraps for λ_{l} values in the close vicinity of the λ_{l}frequency jump, the width of which decreases with λ_{l} (see green and blues curves at II and III). It fully closes at IV where the system stays robustly in the M=1 plateau. Interestingly, P_{l}bistability involves two different fourdimensional SP/phonon lasing states, namely the pervasive Ω_{m}=54 MHz mode with M=1, and a 198 MHz mode (associated with an inplane flexural mode having seven antinodes along the xdirection, see Fig. 1d and Methods) with M=4, that is, the first RF peak appears at 49.5 MHz.
State switching within the P_{l}bistable region would be enabled straightforwardly by applying power pulses/dark pulses with pulse heights dependent on the specific λ_{l} in a range from a few μW to 100 s of μW. The mechanical lifetime and frequency of the modes involved would currently limit switching speeds to the megahertz range, but this could be further improved by pushing ν_{SP} to the gigahertz range with a smart tailoring of the heat dissipation and freecarrier recombination rates.
Regions of the laser parameters space supporting more than two stable states could also be present. Indeed, it is plausible that the P_{l}bistable region reported in Fig. 3 could support three stable states, namely two different fourdimensional SP/phonon lasing states and twodimensional SP. Unfortunately, we have been unable to reach all of them by just scanning P_{l}.
Period doubling bifurcation of cycles and onset of chaos
The dynamical solutions of the system become increasingly complex for specific ranges of P_{l} and λ_{l}, giving rise to perioddoubling cycles and the onset of chaos. For continuous dynamical systems the Poincaré–Bendixson theorem states that period doubling bifurcation of cycles or bifurcations towards strange attractors can only arise in three or more dimensions^{1}. Therefore, this kind of bifurcations can occur, in principle, if the OM device is found in a fourdimensional SP/phonon lasing state and P_{l} and λ_{l} are varied from there. In fact, for high enough P_{l} values, new complex dynamics appear at the sides of the plateaus. Figure 4 illustrates the most striking case, covering both sides of the M=1 plateau while pumping at P_{l}=4 mW, that is, an intracavity stored optical energy (calculated at perfect laser/cavity resonance) of 50 fJ, which is slightly lower than the energy required to disclose a chaotic regime in ref. 16 (60 fJ). In our case, the transition between the M=2 and M=1 plateaus does not involve pure SP states, contrary to the observation at P_{l}=2 mW (see Fig. 2). Along the transition, the dimension of the system is, at least, four, following a route along which the attractor undergoes subsequent perioddoubling bifurcations (see lower part of Fig. 4a–c). A much broader linewidth of the Ω_{m}/2 peak of the M=1 first period doubling enables its discrimination from the M=2 coherent state (red and black curves of the inset to Fig. 4c). Subsequent period doubling bifurcations appear before entering into the M=1 coherent regime, with its peak intensity decreasing as the period doubling level increases. On the other side of the M=1 plateau the system suffers another bifurcation in which it abruptly enters into a chaotic regime within a strange attractor volume in the phase space. The RF spectra below and above that bifurcation are shown in Fig. 4d (black and red curves, respectively). A chaotic behaviour is also foreseen by our numerical simulations for high enough λ_{l} values at high P_{l} (see Methods and Supplementary Figs 9–11). The simplest version of our model, which considers a single mechanical mode at Ω_{m}, predicts: the onset of chaos in the ΔT and N space by a smooth perioddoubling route and a coherent highamplitude oscillation of the mechanical mode. This would lead to a flat broadband RF spectrum with sharp peaks only at the frequencies of the Ω_{m} harmonics. In the experiment, the onset of chaos is abrupt and leads to a broadband RF spectrum as well, but with a much richer structure of sharp peaks (red curve of Fig. 4d). The most intense ones are associated to the Ω_{m} mode (main peak plus harmonics) lasing in the plateau, similarly to the numerical simulation (see Methods and Supplementary Fig. 10). The remaining peaks are ascribed to the activation of the fundamental outofplane flexural mode described before (=5 MHz), providing peaks at , at its harmonics and at sidebands of the Ω_{m}associated peaks. On the basis of these experimental results, and as predicted, we conclude that a chaotic regime is established by the SP/single mechanical mode lasing state, thus creating a broad band F_{o}. However, this is not the stable solution of the system since, in response to F_{o}, other mechanical modes displaying large enough g_{o,OM} can be activated. The nonlinear system thus increases its dimension by a number equal to twice the number of mechanical modes involved. We have confirmed this hypothesis by including a second harmonic oscillator at in our model (see Methods and Supplementary Fig. 11) driven by F_{o}, predicting that the chaotic dynamics are only present in ΔT and N, while the two mechanical modes oscillate coherently at incommensurate frequencies. It is worth noting that the only additional mode to be amplified is an outofplane flexural one having a large g_{o,OM} value, while the other inplane ones, although having large g_{o,OM} values as well, are probably damped because of mechanical mode competition^{21}.
There are other two remarkable features reported in Fig. 4a,c: the absence of hysteresis in the transition between M=2 and M=1 and the hysteresis cycle involving chaos and the SP/single mode lasing limit cycle. The former is in contrast with what is reported in Fig. 2, leading us to conclude that the hysteresis cycles reported above only appear when the transition is through twodimensional SP, that is, at low P_{l}. The second feature is an indication of two types of coexisting attractors: fixed points and strange attractors. It also means that a large enough instantaneous perturbation can knock the coherent oscillation into chaos or vice versa. The complete insight of the latter bistability exceeds the scope of this work, but we suggest it is a direct consequence of the coupling of the two mechanical modes through n.
The time evolution of the transmitted signal below and above the abrupt bifurcation delimiting the onset of chaos is reported in Fig. 5. As expected, above the bifurcation, the signal is aperiodic while below the bifurcation the signal is coherent. Aiming to reconstruct the embedding state in a projection on a threedimensional space, in Fig. 5b,d we plot the transmitted signal against itself delayed 10 ns and itself delayed 20 ns. A limit cycle is reconstructed for the coherent case, while the described trajectory in the chaotic case tends to fill a restricted volume. It is worth noting that, by weakly probing another photonic resonance in addition to the one being pumped, the chaotic dynamics of the pump signal will be transferred to the probe, since each one just follows the common dynamics of the effective refractive indices^{15}.
Finally, we applied the Rosenstein algorithm^{22} to 10μslong transmission registers taken with a temporal resolution of 10 ps (see Methods and Supplementary Fig. 12). The output is the temporal evolution of the average divergence of initially close statespace trajectories (Fig. 5e). The applied algorithm starts converging for an embedding dimension m≥6. The latter must be at least equal to the dimension of the system^{22}, which agrees with the dimension of our model considering two mechanical modes. Largest Lyapunov exponents (LLE) quantify the exponential behaviour of the divergence and estimate the amount of chaos in the system. The sensitive dependence on initial conditions, quantified by LLE, preclude longterm predictions, but it promises improved shortterm predictions^{23}. From the exponential fit of the blue curve of Fig. 5e we extract an LLE value of 1.3 × 10^{5} s^{−1}. This value is about one order of magnitude lower than and two orders of magnitude lower than Ω_{m}, which are the typical frequencies of the system. Obviously, the divergence growing trend saturates at some point, since trajectories cannot go any further than the scale of the attractor volume, which is of the order of the amplitude of the transmission oscillations. To check for the validity of the used parameters, we applied the same algorithm to the coherent signal (red curve in Fig. 5e), the result being that the trajectories stay parallel at a divergence value that is almost an order of magnitude smaller than in the chaotic case.
An estimation of the LLE in OM systems has been realized in refs 15, 16. In the former reference, its value is provided without units, preventing any comparison with our work. In the case of ref. 16, the LLE exceeds by three orders of magnitude the one reported here and is much larger than the typical frequencies of the system. In that work, the LLE is calculated using the Grassberger–Procaccia method proposed in ref. 24, which, as stated in a later ref. 22, is sensitive to variations of its input parameters (number of data points, embedding dimension, reconstruction delay) and it is usually unreliable except for long, noisefree time series. In our case, we apply the Rosenstein algorithm, which is robust where Grassberger–Procaccia is not and is widely acknowledge to be accurate because it takes advantage of all the available data and works well with small data sets.
Discussion
We have reported a rich set of complex dynamical solutions, including chaos, experimentally observed in a single and compact Sibased OM nanobeam, optically pumped with a continuouswave, lowpower laser source. The reported features are very robust and stable for hours, in spite of being achieved at atmospheric conditions of temperature and pressure.
We have provided a glimpse of the dynamical map of the system by exploring the laser parameters (power and wavelength) in different senses. In particular, at low laser powers, we have demonstrated hysteresis and bistability of two kinds involving: a twodimensional SP and a fourdimensional SP/phonon lasing state and two fourdimensional SP/phonon lasing states involving two mechanical modes. The potential application, for instance for OM memories, of the reported P_{l}bistability is probably higher than the λ_{l}bistability, since state switching would be achieved by applying power pulses.
At high laser powers, we have observed that the transition between two consecutive fourdimensional SP/phonon lasing states is through a perioddoubling route. In addition, we have demonstrated the onset of a sixdimensional chaos displaying bistability and hysteresis with a fourdimensional SP/phonon lasing state. We envision the use of many of those chaotic optical sources on an SOI chip in a multichannel approach. Each device will eventually show chaotic dynamics that are dependent on the specific geometry of the OM crystal and of the parameters of the laser used to pump the optical resonance, effectively behaving as a multichannel chaotic source that could be multiplexed into a single optical fibre.
Besides the specific applications that some of the previous features may find, namely for memories, switches, sensing or secure data transmissions, the straightforward access to a large part of the parameter space makes natural the exploitation of this device as a test bed for investigating uncommon complex nonlinear dynamics such as static and dynamic bifurcations, different routes to chaos and so on. The nonlinear system can easily increase its complexity, for instance, by pumping several optical modes simultaneously or by weakly coupling several selfsustained OM devices, from pairs to networks. Thus, we propose our device as a versatile building block to develop mechanical neurocomputational architectures^{25}.
It is worth noting that a recent ref. 16 reports on a twodimensional integrated geometry showing similarities to our system in terms of some of the observed dynamical solutions. In addition to the claim of chaotic dynamics, which has been discussed in comparison with our findings in the previous section, ref. 16 reports pure SP and several phonon lasing regimes, each one activated using different harmonics of the optical force. Those latter features are in agreement with what was reported by us previously in ref. 18. A comprehensive comparison of the works of both groups, including an analysis of the input parameters at which those regimes are triggered, is included in the Supplementary Discussion.
Methods
OM crystal geometry
The investigated device is an OM photonic crystal, the unit cell of which contains a hole in the middle and two symmetric stubs on the sides (see Supplementary Fig. 1). The peculiarity of this geometry is a full phononic bandgap ∼4 GHz (ref. 26). The investigation of highfrequency mechanical modes is reported elsewhere^{27}. We describe here the geometrical parameters of our device. In a defect region consisting of 12 central cells the pitch (a), the radius of the hole (r) and the stubs length (d) are decreased in a quadratic way towards the centre. The maximum reduction of the parameters is denoted by Γ. A 10 period mirror is included at both sides of the defect region. The nominal geometrical values of the cells of the mirror are a=500 nm, r=150 nm and d=250 nm. The total number of cells is 32 and the whole device length is ∼15 μm.We have fabricated a set of devices in which Γ has been varied from Γ=64% to Γ=83% of the original values. All the results presented in this work correspond to the structure with Γ=83%, but the same effects have been observed for other values of Γ.
OM crystal fabrication
The devices were fabricated in standard SOI Soitec wafers with silicon layer thickness of 220 nm (resistivity ρ∼1–10 Ω cm^{−1}, pdoping of ∼10^{15} cm^{−3}) and a buried oxide layer thickness of 2 μm. The pattern was written by electron beam lithography in a 100nmthick polymethylmethacrylate resist film and transferred into silicon by reactive ion etching. Application of buffered hydrofluoric acid removed the buried oxide layer and released the beam structures.
Measurements
The experiments are performed in a standard setup for characterizing optical and mechanical properties of OM devices. A tunable infrared laser covering the spectral range between 1,460–1,580 nm is connected to a tapered fibre. The polarization state of the light entering the tapered region is set with a polarization controller. The thinnest part of the tapered fibre is placed parallel to the OM photonic crystal, in contact with an edge of the etched frame (top right photo of Supplementary Fig. 2a,b). The gap between the fibre and the structure is about 200 nm. A polarization analyser is placed after the tapered fibre region. The long tail of the evanescent field and the relatively high spatial resolution (∼5 μm^{2}) of the tapered fibre locally excited the resonant optical modes of the OM photonic crystal. Once in resonance, the mechanical motion activated by the thermal Langevin force causes the transmitted intensity to be modulated around the static value (Supplementary Fig. 2c). To check for the presence of a radiofrequency modulation of the transmitted signal an InGaAs fast photoreceiver with a bandwidth of 12 GHz was used. The radiofrequency voltage is connected to the 50 Ω input impedance of a signal analyser with a bandwidth of 13.5 GHz. All the measurements were performed in an antivibration cage at atmospheric conditions of air pressure and temperature.
Tapered fibre characteristics and fabrication
The experiments are carried out with tapered optical fibres having diameters in the smallest section of ∼1.8 μm (Supplementary Fig. 3a), which is commensurate with the wavelength of interest (∼1.5 μm) and ensures an evanescent field tail of several hundreds of nanometres. For the fibre fabrication, we used a homemade setup in which an SMF28 optical fibre is stretched in a controlled way using two motorized stages. The central part of the fibre is placed in a microheater where the temperature is about 1,180 °C (ref. 27). The fibre transmission at a wavelength of 1.5 μm is monitored during the pulling procedure (Supplementary Fig. 3b). The signal is subjected to a short time fast Fourier transform algorithm, so that the frequency components associated with inference between different supported modes are measured (Supplementary Fig. 3c). The singlemode configuration is achieved when all those frequency components disappear (Supplementary Fig. 3d). Using two rotating fibre clamps, the tapered fibre is twisted twice around itself. The two fibre ends are gently brought closer over several hundreds of micrometres so that a looped structure forms in the tapered region. The two fibre ends are afterwards pulled apart to reduce the loop size down to a few tens of microns. In that process, the two parts of the fibre at the loop closing point (upper part of the loop on the top left photo Supplementary Fig. 3) slide smoothly in opposite senses. The microlooped shape provides functionalities similar to those of dimpled fibres. For the fibre loop photonic structure, the dispersion relation is linear and the group refractive index (n_{g}) is equivalent to the effective refractive index (n_{eff}). We have calculated n_{eff} using the beam propagation method taking the material refractive index of the cladding of the initial fibre to be n=1.468 at λ_{l}=1,515 nm, resulting in a value of n_{eff}=1.373.
FEM simulations
COMSOL FEM simulations of a complete structure are used to determine the fundamental cavity mode frequencies, effective masses of the mechanical eigenmodes and singleparticle OM coupling rates (g_{o,OM}). To model the OM crystal system, the contour of the of the asfabricated structure SEM image was extracted graphically and then imported in the FEM solver (Supplementary Fig. 4). Among the many optical and mechanical modes supported by the OM crystal, there are specifically one optical and three mechanical that are discussed in the main text. Those are the third optical mode appearing at 197 THz (Supplementary Fig. 5a) and the mechanical modes appearing at =5 MHz, Ω_{m}=54 and 198 MHz (Supplementary Fig. 5b–d, respectively), whose effective masses are 12 × 10^{−12}, 13 × 10^{−12} and 11 × 10^{−12} g, respectively. Singleparticle OM coupling rates between optical and mechanical modes are estimated by taking into account both the photoelastic (PE) and the moving interface (MI) effects^{28,29,30}. The PE effect is a result of the acoustic strain within bulk silicon, while the MI mechanism comes from the dielectric permittivity variation at the boundaries associated with the deformation. The calculation of the MI coupling coefficient g_{MI} is performed using the integral given by Johnson et al.^{28}:
where Q is the normalized displacement (maxQ=1), n is the normal at the boundary (pointing outward), E is the electric field and D the electric displacement field. The dielectric permittivity is denoted by ɛ, Δɛ=ɛ_{silicon}−ɛ_{air}, Δɛ^{−1}=−. The speed of light in vacuum is denoted by c, is the reduced Planck constant, m_{eff} is the effective mass of the mechanical mode and Ω_{m} is the mechanical mode eigenfrequency, so that is the zeropoint motion of the resonator. A similar result can be derived for the PE contribution^{29,30}:
where δ_{ij}=ɛ_{air}n^{4}p_{ijkl} S_{kl}, being p_{ijkl} the PE tensor components, n the refractive index of silicon and S_{kl} the strain tensor components. The addition of both contributions results in the overall singleparticle OM coupling rate:
It is worth noting that the stringlike modes studied in this work have very low PE contribution in comparison with the MI counterpart. From the analysis of the MI surface density (the integrand of equation (2)) of the =5 MHz mode (Supplementary Fig. 6) it is possible to conclude that the contribution provided by the top and bottom Si–air interfaces are both quite large, very similar in absolute value (about g_{MI}/2π=1 MHz) but with opposite signs. As a consequence of the latter, a rather low value of g_{MI}, namely g_{MI}/2π=7 kHz, is calculated. However, geometric inhomogeneities in the real structure along the z axis may break the symmetry along that axis, unbalancing the contributions from the top and bottom Si–air interfaces. On the contrary, the MI surface density to the Ω_{m}=54 MHz mode (see Supplementary Fig. 7) allows concluding that the main contribution comes from the lateral Si–air interface. The outer stub surface and the inner hole surface contribute with different signs, but the overall value is rather large, namely g_{MI}/2π=650 kHz.
Selfpulsing and phonon lasing
This section addresses the general mechanism of SP limit cycles in optical resonators and the specific case of phonon lasing driven by SP occurring in OM cavities. An extended discussion of these phenomena can be found in ref. 18.
We consider the two main sources of optical nonlinearities in siliconbased cavities, which are the thermooptic effect and freecarrier dispersion^{19}. In the former effect, there is an associated redshift of the optical resonance as a function of the effective temperature of the cavity (ΔT), while in the latter effect the spectral shift is towards the blue as a function of the density of free carriers (N). Since ΔT is directly related with nonradiative recombinations of free carriers, the differential equations governing both optical nonlinear effects are linked, and can be expressed in the following terms:
where V_{o} is the optical mode volume and n_{r} is the refractive index. Twophoton absorption is considered as the main freecarrier generation mechanism in the freecarrier density differential equation (equation (5)), where β is the tabulated twophoton absorption coefficient. Equation (5) also includes a surface recombination term governed by a characteristic lifetime τ_{FC}. We consider freecarrier absorption as the main heating mechanism in equation (6), where α_{FC} is defined as the rate of temperature increase per photon and unit freecarrier density. The heat dissipated to the surroundings of the cavity volume is quantified by the characteristic lifetime τ_{T}. Importantly, is the cavity resonant wavelength including firstorder nonlinear effects. Since κ^{−1} is much smaller than τ_{T} and τ_{FC}, it is possible to consider that the response of the optical cavity to freecarrier dispersion and thermoopticinduced refractive index changes is adiabatic. Experimentally we have access to both laser parameters (P_{l} and λ_{l}), which are included in the dynamic equations implicitly in n, thus impacting the generation rates. Figure 6 illustrates the four different solutions that equations (5) and (6) may show depending on λ_{l} (please note that we have kept P_{l} fixed at 2 mW). For λ_{l} values close to λ_{o} (Fig. 6a), the system displays two fixed points, one stable (solid circle) that attracts neighbouring trajectories and one unstable (empty circle) that repels trajectories within its vicinity. In the experiment, for λ_{l} values close to λ_{o}, the system is found at the stable fixed point. Thermal or mechanical disturbances are confined in the attractor phase surface associated with the fixed point so that {N,ΔT} is found in its close vicinity. If driven away from that point, the system settles down to equilibrium through exponentially damped oscillations. By gradually increasing λ_{l} the system eventually experiences a supercritical Hopf bifurcation (Fig. 6b). Just at the bifurcation the fixed point is still stable, although it is a weak one, and the system decays to it algebraically if found in its vicinity. Slightly above the bifurcation, the stable fixed point transforms into an unstable fixed point surrounded by a stable limit cycle (Fig. 6c). If the system is driven away from the limit cycle, it will rapidly converge to it again. This limit cycle is defined as the SP limit cycle and is the stable dynamical solution of equations (5) and (6) for a wide range of laser parameters. It was first demonstrated experimentally by Johnson et al.^{19} in a silicon microdisk. In the SP limitcycle regime, the pair of values {N,ΔT} oscillate periodically with a frequency denoted by ɛ_{SP}, although the closed trajectory is not described at a constant pace. In the experiment, the SP limit cycle induces an anharmonic periodic modulation of the transmitted optical signal and, hence, of n. By further increasing λ_{l} the SP increases ɛ_{SP}, which is a consequence of the higher average temperature of the system. Another important effect of increasing λ_{l} is that the second unstable fixed point gets closer to the limitcycle trajectory. The SP limit cycle is wiped out when that unstable fixed point intercepts the limitcycle trajectory (Fig. 6d), deflecting out its path and driving the system down to a situation in which ΔT=0, where the cavity resonance is far away from λ_{l}.
Hereafter, we consider the case of an OM cavity with mechanical degrees of freedom that can be driven by optical forces. When the SP limit cycle solution is active the radiation pressure force (F_{o}) is modulated in the same way as n, since they are related in a linear way, that is, . The differential equation governing the generalized coordinate for the displacement of a mechanical mode (u) is that of a damped linear harmonic oscillator driven by F_{o}:
where m_{eff}, k_{eff} and Ω_{m} are its effective mass, spring constant and eigenfrequency, respectively. The nonlinear resonance position includes now the effect of the mechanical motion when evaluating n, that is, . Importantly, the response of n to u is also adiabatic since κ is a few orders of magnitude smaller than Ω_{m}. Equations (5)–(7), , describe the dynamics of a fourdimensional {N,ΔT,u, u′} nonlinear system that is coupled through n. One of the consequences of the coupling of the two systems is that selfsustained mechanical motion is achieved if one of the low harmonics of the SP main peak at ν_{SP} is resonant with the mechanical oscillations (Mν_{SP}=Ω_{m}, where M∈)(Supplementary Fig. 8). The experimental features of this coupled system can be well reproduced with the couple set of equations (5)–(7), , , the only free parameters being τ_{FC} and τ_{T} (Supplementary Fig. 8b). Those are extracted by fitting the temporal trace of the transmitted signal in an unlocked configuration using equations (5) and 6)^{18}.
It is worth noting that the phonon laser limit cycle is finally wiped out in a similar fashion as in the pure SP case, that is, the second unstable fixed point intercepts the limitcycle trajectory and drives the cavity out of resonance.
Modelling of hysteresis and chaos with one mechanical mode
We have numerically solved the system of four nonlinear equations composed by equations (5)–(7), , . The simulations are done sequentially for increasing (decreasing) values of λ_{l} using initial conditions provided by one point {N,ΔT,u,u′} of the stable state calculated for the preceding λ_{l} value. The obtained results are summarized in Fig. 7, where we plot the fast Fourier transform of the transmitted signal. The model predicts, on the one hand, bistability and hysteresis between twodimensional SP and fourdimensional SP/phonon lasing, thus displaying some of the features reported in the main text. The model also accounts for the onsetting of chaos on (N,ΔT), while the mechanical mode oscillates coherently (sharp peaks appear at Ω_{m}=54 MHz and its harmonics). However, the predicted route is much smoother than the experimental and through subsequent period doubling bifurcations instead of through an abrupt transition. It is worth noting that, again here, the period doubling dynamics is only present in (N,ΔT), while the mechanical mode oscillates in a coherent fashion. Simulated trajectories associated to a fourdimensional SP/phonon lasing and to chaos are shown in Supplementary Fig. 9. The chaotic dynamics are robust along a wide spectral range, and are only wiped out when the second unstable fixed point shown in Fig. 6 gets close enough to the strange attractor volume, effectively deflecting down the trajectory of the system (like in Fig. 6d). The cavity then relaxes down to the initial cold situation at ΔT=0, where the cavity resonance is far away from the laser line. In the measurements shown in Fig. 4 we have not reached that condition, since by doing so we would have lost the possibility of performing the backscan measurement of Fig. 4b. The model described by equations (5)–(7), , captures most of the features described along the main text. However, it does not incorporate three main experimental observations, namely: the period doubling bifurcation route between M=2 and M=1, the presence of bistability and hysteresis between fourdimensional SP/phonon lasing and chaos and the specific peak structure of the chaotic RF signal. The first observation remains a subject for further studies, since we predict much smaller highorder plateaus that what measured experimentally, which eventually prevents reaching a similar situation to that reported in Fig. 4c. In the following section we address the former observations.
Modelling of chaos with two mechanical modes
As explained briefly in the main text, most of the sharp peaks that are present in the chaotic RF spectrum reported in Fig. 4 are associated to the Ω_{m} mode (main peak+harmonics) that was lasing in the plateau, similarly to what is reported in the simulation of the previous section. The remaining peaks are associated with the activation of the fundamental outofplane flexural mode described before (=5 MHz), providing peaks at , at its harmonics and at sidebands of the Ω_{m}associated peaks. To account for that mode in the model, we have included a second harmonic oscillator in addition to equations (5)–(7), , . The force driving the second oscillator is very likely radiation pressure , because that particular mode displays the stronger transduced signal below the threshold for SP, where thermal Langevin forces are dominant. We rule out pure mechanical nonlinear interactions because, if they were to be accounted for, other mechanical modes should also be active when the system is in a plateau, which is not the case. The additional equation thus reads:
The g_{o,OM} value for this specific mode has been taken by scaling the g_{o,OM} value corresponding to the Ω_{m}=54 MHz mode by the RF signal ratio between the two modes. The obtained results are very similar to those of the previous section. Within the M=1 plateau, the second harmonic oscillator is off (see Supplementary Fig. 10b). However, when the system enters into the chaotic regime, it gets activated as a consequence of the chaotic dynamics of . Through OM coupling its oscillations modulate λ_{r}. As a result of the latter, the fast Fourier transform of the transmitted signal displays sharp peaks at Ω_{m} and (see Supplementary Fig. 10a). It is also possible to distinguish sidebands at Ω_{m}±. The first harmonic oscillator oscillates coherently also in the chaotic regime, which again is only present in (N, ΔT). In Supplementary Fig. 11, we report temporal series spanning 1 μs of the different magnitudes described by the nonlinear system of equations (5)–(7), , and the resulting transmitted signal.
Application of the Rosenstein algorithm
We have used the Rosenstein method for calculating the LLE from our experimental series^{22}. The output is the temporal evolution of the average divergence of initially close statespace trajectories. The method follows directly from the definition of LLE and is accurate because it uses the whole available set of data. Although it is suitable for relatively small data sets, the temporal series has to be well resolved and span over a temporal range that should be several times greater than the predictability horizon of the system. Otherwise, the results will not be trustworthy and the extracted LLE values will be orders of magnitude greater than the real ones, that is, the trajectories coming from ascloseaspossible initial states will almost immediately diverge up to the scale of the attractor volume. Indeed, this was our case when applying the Rosenstein algorithm to short chaotic time series (few hundreds of nanoseconds), since, given a reference state on the time series, it was not possible to find a good ascloseaspossible state in the remaining set of data. Another critical input value is the embedding dimension of the system (m). It is imperative to evaluate the algorithm using different values for m. In Supplementary Fig. 12 it is apparent that satisfactory results are obtained only when m is, at least, six, which is indeed the dimension of the system. This is due to the fact that chaotic systems are effectively stochastic when embedded in a phase space that is too small to accommodate the true dynamics. Finally, it is worth noting that the superposition of two sinusoids (for instance, two harmonic oscillators) with incommensurate frequencies is a nonchaotic system, but is quasiperiodic and deterministic. Although the time series would look chaotic, applying the Rosenstein algorithm to such a signal would lead to a flat divergence curve^{22}, leading to a null LLE. In our case the divergence is clearly growing with time, which is a clear signature of chaos.
Data availability
All data used in this study are available from the corresponding author on request.
Additional information
How to cite this article: NavarroUrrios, D. et al. Nonlinear dynamics and chaos in an optomechanical beam. Nat. Commun. 8, 14965 doi: 10.1038/ncomms14965 (2017).
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Acknowledgements
This work was supported by the European Comission project PHENOMEN (H2020EU713450), the Spanish Severo Ochoa Excellence program and the MINECO project PHENTOM (FIS201570862P). DNU, PDG and MFC gratefully acknowledge the support of a Ramón y Cajal postdoctoral fellowship (RYC201415392), a Beatriu de Pinos postdoctoral fellowship (BPDGR 2015 (B) and a Severo Ochoa studentship, respectively. We would like to acknowledge Jose C. Sabina de Lis, J.M. Plata Suárez, A. Trifonova and C. Masoller for fruitful discussions.
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D.N.U. performed the experiments. D.N.U. and N.E.C. developed the model and analysed the data. M.F.C., P.D.G., M. S., F.A. and C.M.S.T. participated in the experiments at different stages. A.G. and A.M. fabricated the samples. All authors contributed to the interpretation of the results and writing of the manuscript.
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Supplementary information
Supplementary Information
Supplementary Figures, Supplementary Discussion and Supplementary References (PDF 2839 kb)
Supplementary Movie 1
Transition from M=2 to M=1. The movie shows the transition between M=2 to M=1 plateaus through subsequent period doubling bifurcations. (MOV 13291 kb)
Supplementary Movie 2
Transition to chaos. The movie shows the transition from a coherent signal to a chaotic signal by just changing the laser wavelength. We sweep that magnitude back and forth a couple of times to illustrate the robustness of the system. (MOV 3248 kb)
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NavarroUrrios, D., Capuj, N., Colombano, M. et al. Nonlinear dynamics and chaos in an optomechanical beam. Nat Commun 8, 14965 (2017). https://doi.org/10.1038/ncomms14965
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