Superharmonic resonances in a two-dimensional non-linear photonic-crystal nano-electro-mechanical oscillator

The phenomenon of superharmonic resonances of an oscillator in response to a periodic forcing function occurs in most non-linear oscillators.¹ For driving frequencies Ω_d in the vicinity of a rational fraction of the oscillator natural frequency $Ω1st$ (⁠$Ωd≃Ω1st×1/n$⁠), the non-linearity adjusts the frequency of the free-oscillation to exactly n times the frequency of the forcing, i.e., $nΩd$⁠. Each resonance occurs within a tongue, axes of which are the driving frequency and driving amplitude. Within this tongue, amplitude and phase, respectively, feature a resonance and a shift. Such resonances have already been implemented in NEMS, by use of harmonic driving^2–4 as well as, more scarcely, subharmonic driving⁵ without observation of the phase evolution across the tongue in the subharmonic excitation regime. At the same time, significant advances have been achieved in Nano-Electro-Mechanical Systems (NEMS), technology by hybridizing optoelectronic/photonic devices in NEMS architectures to form hybrid nano-opto-electromechanical systems (NOEMS).^6–10 In this article, resonances under subharmonic driving are observed in an individual NEMS, formed by a suspended photonic crystal mirror. The phase evolution is, in particular, investigated: the power-dependence as well as the excitation frequency-dependence of the phase trajectories for several non-linear orders up to n = 8 are measured; these phase portraits reveal the linear evolution of the phase shift between the oscillator and the initial drive while “crossing” the resonance regions at constant force amplitude.

The NEMS device under study is a deformable mirror, formed by a two-dimensional photonic-crystal suspended InP membrane,^11,12 integrating interdigitated electrodes underneath (see Fig. 1(a)) to avoid mechanical quality factor degradation.¹³ The external driving scheme relies on dielectric transduction.^4,6,14 The suspended membrane experiences a force resulting from an electric field gradient created by biasing the periodic set of electrodes, allowing for an actuation of its motion in the MHz range. InP suspended membrane is actuated by applying an AC voltage V(t) = V_AC cos(2πΩ_dt) with a frequency Ω_d (cf. Fig. 1(a)). This modulated voltage induces a force that drives the out-of-plane oscillation of the dielectric resonator around its equilibrium position. This oscillation translates into a modulation of the mutual capacitance C(x) between the IDE's array and the hovering membrane; x denotes here the out-of-plane displacement of the membrane from its equilibrium position. The resulting position-dependent capacitive force corresponds to the first derivative of the charging energy $U=C(x)×V2/2$ for a given bias voltage V: $F=−∂U/∂x=−G×V×C(x)/2$ with $G=C′(x)×V/C(x)$⁠.⁶ This static force, which is proportional to the field gradient along the x direction, leads to an energy exchange between electronic and mechanical subsystems at a rate G. For the quantitative characterization of our electrostatic actuator, finite element modeling (FEM) is carried out yielding $G−1$ of 4 nm/V for a membrane-electrode distance of 400 nm.

The displacement of the forced NEMS resonator is probed using a standard optical interferometry setup: The photonic crystal membrane is placed in a vacuum chamber pumped down to $10−4$ mbar and acts as the end mirror in one of the arms of the interferometer as depicted in Fig. 1(b). The light from a He-Ne laser of wavelength 632.8 nm and of power 1.5 mW is sent into the interferometer and focused on the membrane with a 0.4 NA microscope objective, down to a beam waist of about 0.6 μm. The phase shift of the laser light reflected by the membrane surface (with a reflectivity R of about 50%) is measured at the interferometer output, by recording its oscillatory components with a photodetector and a lock-in amplifier. All measurements are done at room temperature. The measured mechanical spectrum is shown in Fig. 1(c) for the actuated membrane at a 5 V AC drive, highlighting the excitation of several mechanical modes.

In a first series of experiments, we perform forward and downward frequency sweeps around the fundamental frequency $Ω1st$ (i.e., harmonic excitation) at different AC drive amplitudes. Amplitude and phase (denoted $φ$⁠) evolutions are recorded simultaneously in an homodyne detection configuration for upward (red, right triangle) and downward (blue, left triangle) frequency sweeps at 1 V and 10 V AC drive voltage (see Fig. 2). In the following, $φ1st$ will denote the phase at the mechanical resonance frequency $Ω1st$⁠.

For a low actuation amplitude (V_AC = 1 V, see Figs. 2(a) and 2(c)), the amplitude and phase spectra are identical for bi-directional sweeps. In this regime, the eigen-frequency of the fundamental drum mode $Ω1st$ is found at 2.753 MHz and $φ1st=0°$⁠; its full width of $Γ1st=0.9$ kHz is obtained by measuring the ring-down decay rate. This measurement is done at low V_AC to avoid extra non-linear damping.¹⁴ We calibrate a displacement of 2.08 nm at the beam's center using the method described in Ref. 15, in fair agreement with a FEM computation yielding 4 nm.

The measured spectra start to exhibit an asymmetry around 2 V. For amplitudes larger than this critical drive voltage, amplitude (Fig. 2(b)) and phase (Fig. 2(d)) response curves become bistable between upward and downward sweeps. The observed hysteresis reveals spring hardening effect (see Fig. 2(b)), which is ascribed to mechanical nonlinearities predominantly due to the tensions in the tethers induced by their transverse displacements.¹⁶ This non-linear response can be understood by use of the non-linear differential equation¹⁷ 

where ω₀ (⁠$=2πΩ1st$⁠) is the angular frequency of the membrane. h and Ω_d, respectively, denote the amplitude and frequency of the external drive and β_i the reduced ith-order nonlinearity coefficients. To account for our NEMS device nonlinearity, the measured hysteresis is fitted to the steady-state solution of Eq. (1), as plotted by the green line in Fig. 2(b). From the fit retaining solely the cubic-order non-linearity, we obtain a value of $β3=1.4×1018 m−2$⁠.

Forcing the fundamental mechanical mode at frequencies $Ωd≃Ω1st/n$ either leads to quasi-periodic oscillations, or, when the forcing amplitude is sufficiently high, to periodic oscillations at a frequency $n×Ωd$⁠. In the latter case, the resonant driving of the oscillator by the periodic force occurs within regions with a tongue shape, the so-called resonance tongues. Such regions are shown in Fig. 3, plotting the evolution of the noise spectrum of the fundamental mechanical mode $Ω1st$ while modulating the electric load in proximity of $Ω1st/n$ at various AC drive amplitudes. All 2D surface plots, up to the 8th-order of subharmonics, exhibit a characteristic “tongue” shape with a widened linewidth $ΔΩres$ for increased driving amplitude. The threshold bias V_th above which forced periodic oscillations occur (see Fig. 3(b)) increases for increasing values of n, due to lower non-linear mechanical coefficients. The width of the tongues $ΔΩres$ for a constant AC drive amplitude decreases with n (see Fig. 3(b) for $VAC=10 V$ corresponding to a bias in the saturation regime for every investigated n values). This reduction of the tongue widths may arise from the emergence of higher orders of perturbation,¹⁸ dissipation due to temperature rise^19,20 or weaker non-linear coefficients. These two former physical phenomena also explain the observed saturation of the tongue width at high voltage (see inset Fig. 3(b)).

The phase dynamics of the membrane can be described as slowly varying quadratures x(t) = Xcos(2πΩ_1stt) + Ysin(2πΩ_1stt). The solution is then pictured as a trajectory in the X-Y phase plane. The corresponding phase and quadrature components of position can be experimentally measured during the ring-down time, as shown in Fig. 4(b). The external drive of the NEMS resonator is switched off at frequencies $Ω1st/n$⁠. In each ring-down time, the in-phase and quadrature components of the position are recorded, following a single trajectory from the initial stationary state to the origin with 300 phase points.

For harmonic and subharmonic driving, the phase difference between the oscillator and the driving force is bounded between $±π/2$ (see Figs. 2(c) and 2(d)) and depends on the excitation frequency detuning. If the resonance tongue is “crossed” along a line of constant AC drive, i.e., if the amplitude of the force is kept constant whereas its frequency is varied, the amplitude response goes across a maximum while a phase shift is expected to vary by π. To investigate this phase shift across the tongue, ring-down measurements were performed at different frequency detuning (see inset Fig. 3(b)) for different subharmonic excitation orders. Figure 4(a) plots the measured phase difference for the 2nd- and the 6th-order subharmonic excitations, respectively, at 2 V and 10 V AC drive and at various detuning around the corresponding superharmonic resonances for the same initial drive phase. Some phase portraits for n = 2 and n = 6 are shown in Figure 4(b) at some specific detuning. As expected, the phase shift is found to remain bounded within the width the resonance tongue between $±π/2$⁠. It exhibits a linear behavior, varying by π while “crossing” the tongue. Near the border of the tongue, the phase-position is almost merged to the origin in the phase space diagram, yielding an uncertainty contribution in the derived phase. In the middle of the resonant tongue (⁠$ΔΩd=0$⁠), zero degrees of phase shift are measured within 4% error.

In this article, the non-linear dynamics of a fully integrated NEMS under subharmonic excitation is experimentally investigated. Wedge-like resonance domains are observed in the phase plane spanned by the forcing frequency and amplitude; the linear phase shift between the oscillator and the force across the resonance region is measured. Such resonances, in a regime of self-sustained oscillation, could be used for implementing frequency synchonization and chaotic motion in a NOEMS device. Combined to the scalability of the platform and the possible integration of other on-chip functionalities,²¹ they may open the way to nano-opto-electromechanical oscillator networks, enabling the investigation of many-body dynamics in clustering systems^22,23 or the implementation of neurocomputing^2,5 and high precision sensing.^24–26 

We thank S. Barbay as well as K. Makles, T. Briant, S. Delglise, P.-F. Cohadon, and A. Heidmann for fruitful discussions and G. Hwang for his help with the critical point drying step. This work is supported by the “Agence Nationale de la Recherche” programme MiNOToRe, the French RENATECH network, and the Marie Curie Innovative Training Networks (ITN) cQOM.