Synchronization of Micromechanical Oscillators Using Light Mian Zhang∗ ,1 Gustavo Wiederhecker∗ ,1, 2 Sasikanth Manipatruni,1 Arthur Barnard,3 Paul McEuen,3, 4 and Michal Lipson∗∗1, 4 1 2
School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA. CePOF, Instituto de F´ısica, Universidade Estadual de Campinas, 13083-970, Campinas, SP, Brazil. 3 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA. 4 Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA. (Dated: December 16, 2011) ∗
arXiv:1112.3636v1 [physics.optics] 15 Dec 2011
∗∗
These authors contributed equally to this work. To whom correspondence should be addressed; E-mail:
[email protected]
Synchronization, the emergence of spontaneous order in coupled systems, is of fundamental importance in both physical and biological systems. We demonstrate the synchronization of two dissimilar silicon nitride micromechanical oscillators, that are spaced apart by a few hundred nanometers and are coupled through optical radiation field. The tunability of the optical coupling between the oscillators enables one to externally control the dynamics and switch between coupled and individual oscillation states. These results pave a path towards reconfigurable massive synchronized oscillator networks.
Synchronization processes are part of our daily experiences as they occur widely in nature, for example in fireflies colonies [1], pacemaker cells in the heart [2], nervous systems [3] and circadian cycles [4]. Synchronization is also of great technological interest since it provides the basis for timing [5], navigation [6], signal processing [7], microwave communication [8], and could enable novel computing [9, 10] and memory concepts [11, 12]. At the micro and nanoscale, synchronization mechanisms have the potential to be integrated with current nanofabrication capabilities and to enable scaling up to network sizes. The ability to control and manipulate such networks would enable to put in practice nonlinear dynamic theories that explain the behaviour of synchronized networks [13–15]. Recent work on coupled spin torque [16, 17] and nanoscale electromechanical oscillators (NEMS) [18, 19] exhibit synchronized oscillation states. However, the major challenges with synchronized oscillators on the nanoscale are neighbourhood restriction and non-configurable coupling which limit the control, physical size and possible topologies of complex networks [15, 20, 21]. Here, we demonstrate the synchronization of two dissimilar silicon nitride (Si3 N4 ) optomechanical oscillators coupled only through optical radiation field as opposed to the traditional coupling through physical contact. The tunability of the optical coupling between the oscillators enables one to externally control the dynamics and switch between coupled and individual oscillation states. These results pave a path towards realizing massive synchronized oscillator networks. In optomechanical oscillators (OMOs) the mechanical oscillations can be controlled by light. These structures support tightly confined optical modes as well as long-living (high quality factor) mechanical vibrational modes [22, 23]. When an optical resonance is excited, both light and sound become localized within the cavity’s small volume, which leads to a strong coupling between the optical and mechanical fields. The optical modes are affected by the geometric distortions induced by the mechanical modes and the mechanical modes
are affected by forces on the structure exerted by the optical modes. This mutual feedback between light and sound, proposed first in the context of gravitational wave detectors [24], has been explored to amplify or cool down mechanical modes of mesoscopic structures [25]. Recent work shows that when such feedback is negative, cooling of the mechanical mode to its quantum mechanical ground-state can be achieved [26– 29]. When operating the optomechanical cavity in the positive feedback regime, the optical mode provides gain to the mechanical mode. Above a certain threshold power, when the gain is high enough to overcome the intrinsic mechanical damping, the mechanical mode starts a regenerative oscillation and behave as a free-running oscillator. The transmitted laser signal becomes deeply modulated at the mechanical frequency of the oscillator [22, 30, 31]. We demonstrate here that when two such OMOs are fabricated with slightly different dimensions (i.e. slightly different mechanical frequencies) and are coupled through the optical field, the two OMOs can achieve synchronization. By switching off and on of the optical coupling between two OMOs we demonstrate that both the individual free-running and synchronized oscillation dynamics can be achieved. Each individual OMO, shown in figure 1a,b consists of two suspended vertically stacked Si3 N4 disks with high optomechanical coupling [23, 32–34]. They are fabricated using standard e-beam lithography followed by dry and wet etching steps [see supplementary information (SI)]. The two disks are 40 µm in diameter and 210 nm in thickness while the air gap between them is 190 nm wide. Such a small gap and the relative low refractive index of Si3 N4 (n ≈ 2.0) induce a strong optical coupling between the top and bottom disks. Since the optical modes of the stacked disks depend strongly on their separation, any mechanical mode that modulates the vertical gap induces a corresponding modulation in the optical resonant frequency; a measure for the efficiency of this process is the optomechanical coupling, defined as gom = ∂ω/∂x where ω is the optical frequency and x is the mechanical dis-
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FIG. 1: Design of the optically coupled optoemchanical oscillators (OMOs). (a) Schematic of the device illustrating the mechanical mode profile and the optical whispering gallery mode. (b) Scanning electron micrograph (SEM) image of the OMOs with chrome heating pads for optical tuning by top illumination. (c,d) The symmetric (S) and anti-symmetric (AS) coupled optical supermodes. The deformation illustrates the mechanical mode that is excited by the optical field. (e) The dynamics of the coupled OMOs can be approximated by a lumped model for two optically coupled damped-driven nonlinear harmonic oscillators.
placement amplitude. In our device this is calculated to be gom /2π = 49 GHz/nm (see SI). The mechanical mode that couples most strongly to the optical field is also illustrated by the deformation in figure 1a which has a natural frequency of Ωm /2π ≈ 50.5 MHz. In contrast to most micro-scale synchronization demonstrations, the two OMOs are not physically connected, instead they are separated by a distance of dg = (400 ± 20) nm. Due to the evanescent optical coupling between the left (L) and the right (R) OMOs, optical supermodes are formed with a spatial
profile that spans both the L and R cavities (see fig. 1c,d). The resulting optical resonance splits into a symmetric, lower frequency mode (fig. 1c) and an anti-symmetric higher frequency mode (fig. 1d). Light coupled to any of these supermodes interacts with both the L and the R cavities and therefore we can choose to pump light only into one of the cavities (through a taper fibre); we choose the R OMO, as shown in the schematic of figure 2a. As light interacts with the R OMO above its oscillation threshold, it excites the OMO to a regenerative oscillation mode which in turn induces modulation of the light at its mechanical frequency ΩR . When light tunnels through the gap to interact with the L OMO, it forces the left cavity with resonant frequency ΩL to entrain into the right cavity frequency ΩR ; the reciprocal occurs when light interacts back with the R OMO. This mutual injection of the modulated light from the two OMOs form the basis for entrainment and the onset of synchronized oscillation [35, 36] (see SI for details). We control the degree of optical coupling between the L and R OMOs by thermo-optic tuning of the optical resonant frequencies. Note that the maximum optical coupling between the two OMOs is also limited by the gap size between them (dg = (400 ± 20) nm). This limitation however can be overcome by connecting the cavities with optical waveguides [37]. In our device, as illustrated in the schematic of figure 2a, the tuning of optical frequencies is accomplished using an out-ofplane laser beam with wavelength 1550 nm, that can be focused on either OMO. In the center of both OMOs we deposit a 200 nm layer of chrome in order to increase the heating efficiency of the laser (see fig. 2a,b); in our experiment we chose to tune the L OMO which is not directly coupled to the tapered fibre. As heat is dissipated in the chrome pads, the cavity temperature increases which in turn red shifts the optical resonance of the cavity through the thermo-optic effect [38]. A signature that the optical frequency of both OMOs is matched is given by the almost symmetric resonance dips observed in the optical transmission spectrum (fig. 2b and 2d). The degree of optical coupling between the oscillators can be controlled by the heating laser power (fig. 2c). We are able to both either maximize the coupling or completely decouple the two OMOs (see SI). Individual characteristics of the two OMOs are measured separately by switching off the coupling and exciting the optical modes of each cavity with a continuous wave (CW) laser coupled to a tapered optical fibre. As the laser frequency is swept (from a higher to a lower frequency) across the optical resonance of the OMOs, the radio-frequency (RF) spectrum of the transmitted laser signal is detected by the photodiode (PD) and recorded using a RF spectrum analyser (RSA). The results revealing the single-cavity optomechanical dynamics are shown in figure 3a,b. Due to the optical spring effect [33, 39], the natural mechanical frequencies (fL , fR ) = (ΩL , ΩR )/2π = (50.348, 50.237) MHz increase whereas the intrinsic mechanical damping is reduced. The intrinsic mechanical quality factors of the two OMOs are (QmL , QmR ) = (2.3 ± 0.2, 3.4 ± 0.3) × 103 . Above a specific laser-cavity de-
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FIG. 2: Controlling the two OMO system. (a) Schematic of the experimental setup. The pump and probe light are launched together into the cavities and are detected separately by photodiodes (PD). An erbium doped fibre amplifier (EDFA) is used to amplify the transmitted signal to increase the signal strength. (b) Transmission spectrum of the coupled cavities. The green and orange coloured optical resonances correspond to the pump and probe resonances respectively. NT: normalized transmission. (c) Anti-crossing of the optical mode as the relative temperature of the L OMO (TL ) and the R OMO (TR ) is changed through varying the tuning laser power. The tuning laser is focused on to the two OMOs respectively to obtain the negative and positive relative temperatures. (d) Transmission spectrum of the maximally coupled state indicated by the white horizontal line in (c).
tuning, indicated by the horizontal white dashed lines on figure 3a,c the mechanical losses are completely suppressed by the optomechanical gain and the OMO enters a regenerative freerunning oscillation mode, characterized by sudden linewidth narrowing and amplitude growth. This behavior have been theoretically shown to be a Hopf bifurcation of the optomechanical oscillator [20, 21, 36]. The lower amplitude peak before the bifurcation is due to amplified thermal Brownian motion of the cavity mechanical mode. It is clear that each cavity has only one mechanical mode in the frequency range of interest. Due to the slight difference in geometry, these frequencies differ by ∆f = fL − fR = (70.0 ± 0.5) kHz. We show the onset of synchronization by sweeping the pump laser across the optical resonance, analogously to the single-cavity measurements. This is performed at various power levels, starting from slightly above the estimated oscillation threshold power of the L and R OMOs, Pth−(L,R) ≈ (640, 880) nW, up to several times the threshold power (see SI). Using the heating laser, we also tune the optical coupling to its maximum value, indicated by the dashed-white line in figure 2c. At a relative low input power, Pin = (1.8±0.2) µW, the mechanical peaks at fR and fL are simultaneously observed on the RF spectrum shown in figure 3c, below the dashed-white line. When the laser frequency is closer to the optical resonant frequency, a Hopf bifurcation occurs and both OMOs start to oscillate. Since cavity R has a higher oscillation threshold, due to its lower mechanical quality factor, it requires more optical power and therefore oscillates closer to
the optical resonance. At a higher input optical power level, Pin = (11 ± 1) µW, shown in figure 3d, the first Hopf bifurcation takes place at ∆ωL /2π ≈ −0.10 GHz, and similarly to the case shown in figure 3c, the L OMO oscillates first. However, as the laser frequency further moves into the optical resonance, the two OMOs start oscillating in unison at an intermediate frequency of fS = ΩS /2π = 50.37 MHz, which is a clear sign of synchronization. At even higher optical input power, Pin = (14 ± 1) µW, the OMOs do not oscillate individually and instead go directly into synchronized oscillation after the white-dashed line in figure 3e. To confirm that the OMOs are synchronized we performed numerical simulations for each of the power level we tested based on the experimentally measured mechanical, optical frequencies and quality factors. We calculated the effective motional mass mef f and the optomechanical coupling gom from finite element simulations (FEM). The dynamics of the coupled OMOs is described by a lumped model for two dampeddriven nonlinear harmonic oscillators that are optically coupled and experience thermal noise (see SI). The simulated spectra in figure 3f,g,h exhibit all the essential dynamics and show good agreement with the corresponding measured spectra. We experimentally verify that both structures are oscillating at the synchronized frequency by probing the mechanical oscillation of each cavity individually. This verifies that the amplitude death of one of the oscillators does not occur, a known phenomenon in coupled nonlinear oscillators [40, 41].
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FIG. 3: RF spectra of the OMOs and synchronization (a, b) RF power spectra of cavity L (a) and R (b) as a function of laser frequency when the coupling is turned off. The horizontal white lines indicate the onset of self-sustaining oscillation. PSD: power spectral density. (c) When the coupling is turned on, at an input power Pin = (1.8 ± 0.2) µW cavities L and R do not synchronize and oscillate close to their natural frequencies (see SI). (d) At Pin = (11 ± 1) µW synchronization occurs after the horizontal solid white line. The synchronized frequency appears between the two cavities natural frequencies but only appear after a region of unsynchronized oscillation (between the dashed and solid white lines). (e) The system oscillate directly in a synchronized state at input optical power Pin = (14 ± 1) µW. (f,g,h) Corresponding numerical simulations for the OMO system based on the lumped harmonic oscillator model illustrated in fig. 1d. NPSD: normalized power spectral density.
Note that based solely on the transmitted pump signal, which provides global information of the coupled OMO system, one cannot clearly distinguish the individual contribution from each OMO to the synchronized signal. Using a weak probe laser, as shown in the setup in figure 2a, we excite optical resonances that are not strongly coupled between the two OMOs and therefore selectively probe the oscillation of cavity L or R. Due to their low optical quality factor (Qopt ≈ 4 × 104 ), probing these resonances also minimize perturbations to the mechanical oscillations. figure 4e shows the transmission of the probe mode where an asymmetric splitting is evident. This asymmetry is due to the difference between the optical resonant frequencies of the L and R OMO at the probe wavelength. The uneven light intensity distribution in the two OMOs due to this mode asymmetry can be directly observed by capturing the scattered light with an infrared camera (fig. 4c,d). We probe the OMOs in two pump conditions: between the 1st and 2nd Hopf bifurcation, and after the 2nd Hopf bifurcation for Pin = (11 ± 1) µW , indicated by the white lines in figure 3d. When the pump laser is in the range −0.13 < ∆ωL /2π < −0.10 GHz (between the dashed and solid line in fig. 3d), we expect that only the L OMO to oscillate. Indeed, the probe laser RF spectrum shows a strong peak at fL (red curve in fig. 4f). This same peak does show
in the spectrum when probing the R OMO (blue curve in fig. 4f) but it is 13 dB weaker. These results confirm that in this range, the oscillation state is indeed asynchronous and the L OMO oscillates with much larger amplitude. When the pump laser is in the range ∆ωL /2π < −0.13 GHz we expect from figure 3d the cavities to be synchronized and the optical probe signal to have an RF tone of similar amplitude at the synchronized frequency fS when probing individually the L and R OMOs. Indeed when the two OMOs are individually probed (blue and red curves in fig. 4g), the RF peaks at the synchronized frequency fS differs in amplitude by less than 0.5 dB. This shows that both cavities are indeed oscillating at the synchronized frequency. We have demonstrated the onset of synchronization of two optomechanical oscillators coupled through the optical radiation field. Monolithic integration and the ability to control the coupling strength are promising for realizing large oscillator networks in which the oscillators can be addressed individually. Furthermore, established and future micro-photonics techniques such as electro-optic and thermo-optic techniques can now be extended to switch, filter and phase shift the coupling of these oscillators. Here we demonstrated coupling the near field between oscillators which can be switched on and off by thermo-optical means. In order to achieve
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FIG. 4: Pump-probe measurement of the individual OMOs oscillation when coupled. The input pump power is Pin = (11 ± 1) µW as in fig. 3d. (a,b) Schematic of the pump-probe measurement principle. While the pump laser (green) is symmetrically shared between the two OMOs, the probe laser (blue for probing R and red for L) can measure each cavity selectively. (c,d) The uneven probe intensity distribution of the cavities, observed by an infrared CCD camera when the pump laser is off. (e) Normalized transmission (NT) spectrum for the probe resonances, which correspond to the orange resonances shown in fig. 2b. The red (blue) dashed line corresponds to the probe wavelength region for probing the L (R) OMO, as illustrated in (a,b). (f) The red (blue) curve is the L (R) cavity probe transmission RF spectrum, when the pump is in the asynchronous region −0.13 < ∆ωL /2π < −0.10 GHz shown in fig. 3d; a strong peak at fR is observed but with very different amplitude for two probing conditions. The right inset figures show the same curves in linear scale, emphasising the large difference between the blue and red curves. (g) Same curves shown in (f) but with the pump laser in the synchronous region ∆ωL /2π < −0.13 GHz of fig. 3d. Here both cavities have similar amplitude at fS , which can be clearly noticed in the linear scale inset.
long range coupling of mechanical oscillators, optical waveguides and optical fibres could be used enabling oscillator networks spread over large areas only limited by optical waveguide/fibre losses. Optically mediated mechanical coupling will also remove the restrictions of neighbourhood while creating 1D/2D/3D mechanical oscillator arrays [42]. Using long range, directional and controllable mechanical coupling, synchronized optomechanical systems may enable a new class of devices in sensing, signal processing and on-chip non-linear dynamical systems [15].
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Acknowledgements
This work was supported in part by the National Science Foundation under grant 0928552. The authors gratefully acknowledge the partial support from Cornell Center for Nanoscale Systems which is funded by the National Science Foundation and funding from IGERT: A Graduate Traineeship in Nanoscale Control of Surfaces and Interfaces (DEG0654193). This work was performed in part at the Cornell Nano-Scale Science & Technology Facility (a member of the National Nanofabrication Users Network) which is supported by National Science Foundation, its users, Cornell University and Industrial users. We acknowledge Richard Rand and Steven Strogatz for fruitful discussion about our results.
Methods
Device Fabrication The two 210 nm thick stoichiometric Si3 N4 films are deposited using low-pressure chemical vapour deposition (LPCVD). The 190 nm SiO2 layer is deposited by plasma-enhanced chemical vapour deposition (PECVD). The underlying substrate is a 4 µm SiO2 formed by thermal oxidation of a silicon wafer. The OMOs are defined by electron beam lithography which is then patterned by reactive ion etching. The heater pads are subsequently defined by photolithography lift-off process. After defining the circular pads with lift-off resist, 200 nm of chrome is deposited on the device using electron beam evaporation and the residual chrome is liftoff afterwards. In order to release the structure, the device is immersed in buffered hydrofluoric acid (6 : 1) for an isotropic etch of the SiO2 in between the disks and the substrate layer. The device is then dried with a critical point dryer to avoid
7 stiction between the two Si3 N4 disks. Experimental setup The schematic for testing the OMO system is illustrated in figure 2a. Two tunable external cavity diode lasers are combined using a 3dB directional coupler to an optical fiber that is fed into a vacuum probe station. Inside the vacuum chamber, the tapered fiber is positioned close to the OMO of interest to allow evanecent coupling using a micropositioning system. The output light is then splitted by a WDM splitter to a New Focus 1811 (125 MHz bandwidth) photodetector. Since the power level we use to test for our device is low, an erbium doped fibre preamplifer is used to
amplify the output signal and improve signal-to-noise ratio in the detector. The electronic signal from the detector is split and fed to an oscilloscope to observe the time waveform and to RSA for the frequency spectrum. To obtain the RF map, the laser is configured to sweep from the blue side of the resonance to the red side in a stepwise fashion by applying an external voltage to the laser cavity piezo-transducer. At each frequency step, a snap-shot of the RF spectrum is recorded with 1 kHz resolution bandwidth and 100 Hz video bandwidth.
Supplementary Information Detailed experimental setup
We measure the optomechanical transduction of the coupled OMOs using the setup shown in figure S1. The green (red) line indicates the pump (probe) laser path. The probe is only used when taking the pump-probe measurements. The radio frequency (RF) spectral maps shown in the main text and in figure S1 are obtained with the probe laser off. Both the pump and the probe laser are fibre-coupled, tunable, near-infra (IR) lasers (Tunics Reference and Ando AQ4321D). Their optical power is controlled using independent variable optical attenuators. The pump and probe light are individually sent to a polarization controller and combined with a 50 : 50 directional fiber coupler. A fraction of the power is monitored by a power meter which indicates the equivalent input optical power to the system. To prevent the back scattered light from entering the laser, an optical isolator is used before feeding the laser into a vacuum probe station (Lakeshore TTPX) operating at a pressure of 10−5 mT. The light is evanescent coupled to the OMOs through a tapered optical fibre waveguide by using a micro positioning system. A small portion of the transmitted light (10%) is also monitored by a power meter. The remaining transmitted light is split with a wavelength division multiplexing coupler to separate the pump and the probe laser. Since the pump power used is low, especially for sub-threshold measurements, the pump light is optionally amplified with a low noise erbium pre-amplifier (EDFA, Amonics AEDFA-PL-30) before coupling to a 125 MHz bandwidth photodiode (New Focus 1181). An additional detector (Thorlabs PDB150C-AC) can be switched on when the probe measurement is necessary. Half of the detected signal is sent to an oscilloscope and the remaining is coupled to a radio-frequency spectrum analyser (RSA, Agilent E4407B). The heating light source is provided by another near-IR laser (JDS SWS16101), operating at 1550 nm, and amplified by a high power EDFA (Keopsys KPS-CUS-BT-C-35-PB-111-FA-FA) that can provide a maximum power of 2 W. The light is sent to the microscope optics which focus the light on to the device. Typically, 50 mW of laser power is needed to achieve the desired tuning range, details of tuning aspect can be found in section .
Measurements
The RF spectral maps are obtained by detuning the laser from blue to red into the optical resonance in a stepwise fashion, as controlled by a voltage applied to laser’s external cavity piezo; the laser used has a tuning coefficient of 1.1 GHz/volt. For each voltage step, the RF spectrum is recorded. Therefore, the step size determines the vertical resolution of the RF spectra map (see Fig.3 main text) whereas the resolution bandwidth of the RSA determines the horizontal resolution. Here we used a detuning step size of 3 MHz and a resolution bandwidth of 1 kHz (100 Hz video bandwidth). This allows us to obtain a high resolution map while keeping the data collection time reasonable (≈ 20 minutes).
Single and coupled cavities measurement
The single cavity data are obtained by coupling the tapered fiber either to the L or R OMO. When one OMO is tested, the remaining one is heated by the heating laser with high power (∼ 50 mW) to ensure that they are completely decoupled. The coupled cavity data are obtained by coupling to the R OMO with the tapered fibre. In this case we use the external heating laser to fine tune the coupling so that their split spectrum is symmetric.
8
Heating laser Tunable laser (pump)
Switch
PD
Spectrum Analyzer Tunable laser (probe)
Power meter
PD
Vacuum chamber
Oscilloscope
Power meter
Variable optical attenuator
Optical isolator
Fiber polarization controller
90:10 beam splitter
50:50 beam splitter
Wavelength division multiplexing splitter
PD
Photodiode Switch
FIG. S1: Detailed experimental setup. See SI text for more details.
Pump probe measurement
The pump probe measurements provide direct evidence for the synchronization of the two OMOs. The individual probe of each cavity, as shown in Fig. 4 main text, relies on the asymmetric coupling of one the higher order optical supermodes. This asymmetry arises due to their different optical resonant frequency (See section ) which stems from the slight difference in the geometry of the two OMOs. This leads to a different mode splitting for the higher and lower order optical modes. In the devices we have tested, the majority of them show similar non-identical mode splitting. Due to its lowers optical quality factor (Q) and reduced optomechanical coupling gom , the threshold power for regenerative oscillations [22, 43] of the probe resonance is Pthprobe ≈ 20 mW, which is roughly 20, 000 times larger than the pump resonance threshold optical power Pthpump ≈ 1 µW. We used a probe power of Pp = (20 ± 2) µW, ensuring a low-noise detected probe signal without affecting the cavity oscillation dynamics.
Optical and Mechanical modes
To obtain the optical and mechanical modes of the optomechanical disk cavity we rely on finite element simulations using COMSOL®. From these numerical simulations we derive parameters for the lumped model that describes the optomechanical dynamics, such as the effective motional mass mef f , and the optomechanical coupling rate gom . The optical modes are sought by solving the Helmholtz vector wave equation with an ansatz E(r, z, φ) = E(r, z) exp(imφ). In the table S1 we show the mode radial electric field profile for the lowest order optical transverse-electric (T E) modes. The mechanical displacement field is sought by enforcing complete cylindrical symmetry, u(r, φ, z) = u(r, z), the mode profiles are also shown on table S1. From the sought eigenmodes, the optomechanical coupling coefficients for the supported optical modes are calculated using boundary perturbation theory [44, 45],
gom ≡
∂ω ω0 = ∂x 2
R
2 2 (U · n ˆ ) ∆12 E · tˆ + ∆−1 ˆ | dA 12 |D · n , R 2 |E| dV
(S1)
where the dimensionless displacement field is defined as U ≡ u/ max |u|, the relative permitivity differences are given by ˆ ∆12 = 1 − 2 and ∆−1 ˆ indicate the tangential and normal components of the vectors. 12 = 1/1 − 1/2 , the unit vectors t and n
9 The effetive motional mass is calculated as, Z mef f =
a
2
ρ |U | dV.
(S2)
20 µm 230 nm 190 nm 2.37 µm 230 nm 5 µm
Mechanical mode
Profile (|E · rˆ|)
Ωm 2π
(MHz) mef f (pg)
50.5
110
28.7
194
n Mode T Em λ0 (nm) gom /2π(GHz/nm)
1 T E115
1582.28
49.4
2 T E110
1584.87
11.3
3 T E106
1582.31
17.9
4 T E101
1591.01
10.6
TABLE S1: Optical and mechanical modes parameters. (a) Geometry of the optomechanical cavity used to calculate the modes and parameters shown in the tables. For the optical modes profiles, it is shown the modulus of the radial electric field |E · rˆ|; gom is calculated using Eq. (S1). whereas for the mechanical modes it is shown the displacement amplitude |u| as colors and the deformation represents the normalized displacement.
Top illumination thermal tuning
The coupling between the cavities is controlled by changing their resonant frequencies through the thermo-optic effect. We choose to use 200 nm thick chrome pads as the heating element since they absorb 25% of 1550 nm light at normal incidence, taking into account its reflectivity. Chrome is also resistant to buffered oxide etch which follows in the fabrication steps. The 1550 nm laser is amplified with an EDFA, coupled to the imaging microscope and focused on the chrome pads. The heat absorbed by the chrome pads induces a temperature change ∆T = Rth Pabs , where Rth = ∂∆T /∂Pabs ≈ 5.2 × 103 K/W is the simulated effective thermal resistance of our device. Due to thermo-optic effect, the temperature frequency shift rate is given by the perturbation expression, R ∂ωT ω0 α(r, z)Trel (r, z)|E|2 dV R gth = =− (S3) ∂∆T 2ng |E|2 dV
10 where 0 < Trel (r, z) < 1 is the dimensionless relative temperature distribution of the device, α isRthe material-dependent thermoR optic coefficient, and ng is the optical mode group index. If we define the overlap integral Γ = SiN |E|2 / all E|2 , Eq. (S3) is (SiN)
approximately given by gth ≈ −(Trel )ω0 αSiN Γ/(2ng ). In Fig. S2 we show the simulated relative temperature field Trel (r, z), (SiN) at the edge of the disk Trel = Trel ≈ 0.83. From these results we can estimate the top illumination laser power needed to tune the cavity’s optical frequency by ∆ωT , ∆ωT 2ng ∆ωT Pabs = ≈ (S4) (SiN) gth Rth ω0 Rth (Trel )αSiN Γ For our device, tuning of δλ ≈ 0.2 nm is sufficient to completely decouple the two cavity modes. Using ng ≈ 1.8, αSiN = 3 × 10−5 K−1 , and Γ ≈ 0.59, Eq. (S4) gives a tuning efficiency gth /2π ≈ −256 MHz/K, therefore a laser power of P = Pabs /25% ≈ 24 mW is needed to control the optical coupling between the cavities (see section ).This value is in reasonable agreement with the experimental power range.
Relative Temperature
Chrome heating pad
1.0 0.5 0.0
Heat sink
FIG. S2: Thermal tuning of optical resonances. Simulated temperature (∆T = T − T0 ) profile of the optical micro cavity. The bottom boundary act as a heat reservoir with constant temperature T0 = 300 K. In the mirroring edge, where the optical modes are localized, the temperature is T ≈ 0.83∆T
Coupled mode equations
The optical modes a1 and a2 of each optical cavity are coupled through the optical near-field. Due to scattering, there is also coupling between the clockwise (cw) and counter-clockwise (ccw) optical modes, therefore we need to consider four optical (cw,ccw) (cw,ccw) modes, a1 and a2 . The coupled equations satisfied by these modes are given by [46, 47] , γ iβ iκ a˙ cw 0 1 acw − 21 − iω1 1 1 2 2 ccw ccw √ γ1 iβ iκ − − iω 0 a 0 a˙ 1 1 2 2 2 (S5) 1cw + γ1 ηc s1 (t) cw = γ2 iβ iκ a˙ 2 0 − − iω a 0 2 2 2 2 2 iβ iκ − γ22 − iω2 a˙ ccw 0 accw 0 2 2 2 2 where ωm are optical resonance angular frequencies, γm is total damping rate, κ/2 is the inter-cavity optical coupling rate, and ηc = γe /(γi1 + γe ) is the criticality factor, where γe is the external loss rate (due to the bus waveguide) and γi is the intrinsic damping rate[48]. The system of Eqs. (S5) can be diagonalized exactly, each eigenvector is governed by an equation of the form √ κ γ1 ηc s1 (t) , for m = 1, 2, (S6) b˙ (m,±) = [−i (¯ ω + (−1)m ξ/2 ± β/2) − γ¯ /2] b(m,±) (±)m 2ξ p where ω ¯ = (ω1 + ω2 )/2, γ¯ = (γ1 + γ2 )/2 and ξ = κ 1 − (δ/κ)2 , where δ = (γ1 − γ2 )/2 + i(ω2 − ω1 ). The original fields acw,ccw can be recovered from the eigenvectors through the relation, 1,2 acw −κb(1,−) + (ξ + iδ)b(2,−) 1 ccw 1 a1 −κb(1,+) + (ξ + iδ)b(2,+) (S7) cw = a2 2ξ κb(1,−) + (ξ − iδ)b(2,−) accw κb(1,+) + (ξ − iδ)b(2,+) 2
11 cw where b(m,±) = (accw m ± am ), Eq. (S7) will be used to calculate the optical transmission function in the section below.
Steady-state transmission
To obtain the low-power steady-state optical transmission spectrum, we assume that the laser driving term in Eq. (S5) is oscillating at ω, i.e., s1 (t) = s1 eiωt . Eq. (S6) can be written in a rotating frame c(m,±) (t) = c˜(m,±) (t)eiωt . The resulting equations will be of the form, √ ˜b˙ (m,±) = i∆(m,±) − γ¯ /2 ˜b(m,±) (±)m κs1 γ1 ηc , for m = 1, 2, 2ξ
(S8)
where ∆(m,±) = ω − (¯ ω + (−1)m ξ/2 ± β/2) is the laser-cavity frequency detuning for each of the optical supermodes. The steady-state solution to (S8) is given by ˜b(m,±) = (∓)m
√ κs1 γ1 ηc , for m = 1, 2. 2ξ i∆(m,±) − γ¯ /2
(S9)
The driving laser excites directly only the mode acw 1 , therefore the steady state optical field transmitted through the bus waveguide is given by, sout 1 (ωl ) = s1 −
√
γ1 ηc acw 1
(S10)
where the optical field a1 (ωl ) is given by Eq. (S7). The normalized field transmission, t(ω) = sout 1 (ω)/s1 is given by, γ1 η c κ X ξ + iακ (−1)j κ t=1−i + 2ξ 2 j=1,2 (−1)j β + iγm + 2∆m + ξ (−1)j β + iγm + 2∆m ξ
(S11)
2
the normalized power transmission is obtained from the relation T (ω) = |t(ω)| . In Fig. S3 we show the transmission T (ω) using the best-fit parameters (ω1,2 /(2π) = 188.442 THz, γ1,2 ¯ /2π = 299 MHz, (κ, β)/2π = (1700, 298) MHz, and ηc = 0.65. The optical frequency scale is centered at ω1,2 = 188.442 THz. To obtain the thermal tuned transmission of our device, we use Eq. (S11) together with the results described in section . The resonant frequency of the cavities, when the top-illumination is on, is given by is given by ωm (T ) = ωm0 + gth ∆T , where gth /2π ≈ −256 MHz/K (see section ).
b Relative Temperature (K)
a 1.0
NT
0.8 0.6 0.4
0.2
-2
-1 0 1 Relative laser frequency (GHz)
2
10 0 -10
-4 -2 0 2 4 Relative laser frequency (GHz)
FIG. S3: Optical transmission. (a) Best-fit steady-state normalized optical transmission (red-line), calculated using equation (S11), and measured transmission spectrum (blue circles). The fit parameters are described in the text. (b) Optical transmission showing the thermal tuning of the coupled cavities, the false-color scale indicates the transmission. This map is obtained from (S11) using ω1 (T ) = ω10 + gth ∆T , in good agreement with Fig. 2 in the main text.
12 Mechanical equations and optomechanical coupling
The mechanical degrees of freedom of each cavity x1 , x2 follows the usual optomechanical equations [22, 30, 49, 50], gom cw 2 2 x ¨1 = −Γ1 x˙ 1 − Ω21 x1 − (1) |a1 | + |accw + F1T (t), (S12a) 2 | mef f ω0 gom cw 2 2 |a1 | + |accw + F2T (t), (S12b) x ¨2 = −Γ2 x˙ 2 − Ω22 x2 − (2) 2 | mef f ω0 (i)
where Ωi , Γi , mef f represent the mechanical resonant frequency, dissipation rate, and effective motional mass. FT (t) is the ther
(i) mal Langevin random force with expectation value FiT = 0 and correlation function FiT (t)FiT (t + τ ) = 2kB T mef f Γi δ(τ ), where kB is the Boltzmann constant and δ(τ ) is the Dirac delta function. In contrast to the phonon-laser regime[31], we ignore terms which couples, through the mechanical displacement field, the optical modes b(±,1) with b(±,2) ; this is justified because κ ΩL,R . The full optomechanical dynamics is obtained by solving simultaneously Eqs. (S12) and (S5), such dynamics is discussed in detail in section . It is however instructive to analyze how a prescribed mechanical motion of the two mechanical oscillators is read-out through the optical modes, also how the optical force term in Eqs. (S12) couples to the two of them. Optical transduction of mechanical oscillations
To account for the mechanical effect on the optical transmission we first assume that the mechanical motion is independent of the optical fields [21], which is equivalent to ignoring the dynamical back-action. Therefore we can use Eqs. (S6) for the optical eigenvectors and simply replace the optical cavity’s resonant frequency by ωi → ωi + gom xi , where xi is the mechanical displacement amplitude for each cavity. The resonant frequency of each eigenmode b(m,±) will be given by, ω(1,±) (xi , xj ) = ω ¯ (xi , xj ) ± ξ(xi , xj )/2 ± β/2,
(S13a)
ω(2,±) (xi , xj ) = ω ¯ (xi , xj ) ± ξ(xi , xj )/2 ± β/2, (S13b) p where ω ¯ (xi , xj ) = [ωi (xi ) + ωj (xj )] /2, ξ(xi , xj ) = κ (1 − [δ(xi , xj )/κ]2 and δ(xi , xj ) = (γi − γj )/2 + [ωj (xj ) − ωi (xi )]. Due to the nonlinear ξ(xi , xj ) dependence on the mechanical displacement amplitudes x1,2 , The usual analytical approach to derive the optomechanical transduction coefficient does not apply[21]. However we can get insight into the problem if we consider the strong optical coupling limit, i.e., δ(xi , xj )/κ = gom (xi − xj )/κ 1 which means that the optical frequency splitting between the cavities is large compared to the mechanically induced frequency shift, therefore ξ(xi , xj ) ≈ κ+O(δ 2 /κ2 ). To further simplify the analysis we assume that the two cavities share identical optical optical properties, i.e., ω1 (x1 = 0) = ω2 (x2 = 0) = ω0 and γ1 = γ2 = γ0 . In this case Eq. (S13) is approximated by, ω(m,±) (x1 , x2 ) ≈ ω0(m,±) + gom (x1 + x2 )
(S14)
where ω0(m,±) = ω0 + (−1)m+1 κ/2 ± β/2. Combined with the above relations, Eq. S6 yields the following equation for the optical eigenmodes b(m,±) , √ iωt ˙b(m,±) = −iω0(m,±) − igom (x1 + x2 ) − γ¯ /2 b(m,±) (±)m γ1 ηc1 s1 e , for i=1,2. (S15) 2 The equations above (S15) can be formally integrated for a prescribed mechanical motion (xi = Ai cos(Ωi t + φi )). The homogeneous solutions (s1 = 0) decay exponentially and does not contribute after the initial transients. To find a particular solution satisfying (S15) we employ a common approach relying on the Jacobi-Anger expansion[21, 36], exp [iµ1 cos(Ω1 t + φ1 ) + iµ2 cos(Ω2 t + φ2 )] =
∞ X
im+n Jm (µ1 )Jn (µ2 )ei(mΩ1 +nΩ2 )t+i(φ1 +φ2 ) ,
(S16)
m,n=−∞
where µi = gom Ai /Ωi is the optomechanical modulation depth. Inserting Eq. (S16) in (S15) and solving the resulting equations gives, √ X im+n Jm (µ1 ) Jn (µ2 ) ei(mΩ1 +nΩ2 )t (±)m s1 γ1 ηc1 i[ωl t+P j=1,2 µj cos(Ωj t+φj )] , b(m,±) (t) = e (S17) γ0 2 2 + i ∆0(m,±) + mΩ1 + nΩ2 m,n
13 where the sum over m, n extends over [−∞, ∞], and ∆0(m,±) = ω0(m,±) − ωl . From Eq. (S17) we can clearly see the that cavity field exhibit tones at combinations of the mechanical frequencies (mΩ1 + nΩ2 ) of the two cavities. Optically mediated mechanical coupling
The optical force driving terms in Eqs. (S12) can be written in terms of the diagonal modes b(m,±) from Eq. (S17) by using Eqs. (S7). As in , for large optical coupling the terms are only resonant with the driving laser one at a time. Therefore, the driving force in each oscillator is proportional to |b(m,±) |2 , Pin γ1 ηc1 |b(m,±) |2 = 4
X im+n J (µ ) J (µ ) ei(mΩ1 +nΩ2 )t 2 m 1 n 2 , m,n γ20 + i ∆0(m,±) + mΩ1 + nΩ2
(S18)
which contains both DC terms and oscillatory terms. The terms oscillating at nΩ1 (nΩ2 ) will be responsible for entraining the mechanical mode at Ω2 (Ω1 ) and eventually synchronize the two oscillators. A detailed analysis on the synchronization of such systems have been recently discussed [36]. Synchronization Simulation Simulation approach
To simulate the synchronization dynamics and obtain the results shown in Fig. S4, we numerically integrate the system of equations (S5), including the displacement dependent optical resonant frequencies, i.e. ω1,2 (x) = ω1,2 + gom x1,2 , together with the two harmonic oscillator equations (S12). This is accomplished using the NDSolve function in the commercial software Mathematica®. In the absence of the random thermal noise force in Eq. (S12), it is numerically challenging to capture the dynamics before the regenerative oscillation threshold is reached, this is because the steady-state is a static one, i.e., x˙ 1,2 = 0. To overcome this issue we add a weak (low-temperature T = 1 K) noise that prevents the dynamics to reach such static equilibrium. T Since NDSolve is a deterministic solver we include the thermal drive by assigning to F1,2 (t) the outcome of a random variable with with expectation value and correlation function given by
T Fi = 0 (S19)
T (i) T Fi (t)Fi (t + τ ) = 2kB T mef f Γi δ(τ ), (S20) where kB is the Boltzmann constant. The discontinuity of this random driving term can lead to instabilities in N DSolve, to overcome this we smooth out thenoise term by interpolating the random force with a correlation time tc = (2π/Ωi )/30. Such short correlation time ensures that the noise power spectrum density (PSD) is white within the frequency range of interest.The reliability of this approach is confirmed by verifying that for weak pump powers (P Pth ), the integrated power spectrum 2 density Sxi (Ω) = |xi (Ω)| satisfy the fluctuation-dissipation theorem [51]. Z ∞
2 1 kB T x (Ω) = Sxx (Ω)dΩ = (S21) (i) 2π 0 2mef f Ω2i A complete analysis of the noise in synchronized systems is beyond the scope of this work, since an accurate numerical noise dynamics will require the simulation of the coupled non-linear stochastic dynamics of the optomechanical cavities[52, 53]. The computational complexity of such systems is also high due to the requirement for slow convergence, first order, fixed time step simulation [54–56]. Simulation results
The simulation also allows us probe not only the optical transmission PSD, but also the mechanical displacement PSD and time series of each OMO. The complete simulation results for the pump laser powers described in the main text are shown in fig. S4. The only parameter we adjusted to obtain the maps shown in figures 3 (f,g,h) in the main text and S4 was the optical pump power.
14 0 NT (dB Hz-1)
-20
-40
NDP 0 (dB Hz-1)
-60
-40
-80
-120
Displacement -0.3
20
6
4
10
-0.2
2 0
0 -2
-0.1
-10
fL
fR
-6
-20 -1.0
-0.0 -0.20
-4
-0.5
0.0
0.5
1.0
fS
30
-10
0
10
20
-0.15
-20 40
20
20
10
-0.10
(pm)
Relative laser frequency ∆ωL/2π (GHz)
0 -10
-0.05
-20
-20 -30
-0.00
-2
-0.15
-1
0
40
1
-40 -40 40
2
-0.00 50.2
-20
-40
50.4
50.5
50.2
50.3
50.4
Frequency (MHz)
50.5
50.2
50.3
50.4
50.5
-40
40
-20
50.3
20
20
0
-0.05
0
20
-0.10
-20
-20
0
20
40
-40 -40
-20
0
20
40
(pm)
FIG. S4: Numerical simulation of the coupled oscillation dynamics. From a to e: transmission RF spectra, displacement power RF spectra of the L and the R OMOs, and the displacement phase diagram of the L and the R OMOs, for input powers at (A)Pin = 4.9 W, (B)Pin = 15.8 W and (C) Pin = 17.9 W. xL (xR ): displacement of the L and R OMOs.
In figures S4A (Pin = 4.9 µW), the mechanical power spectrum of the oscillators (fig. S4A(b,c)) shows that for (−0.25 < ∆ω/2π < −0.13 GHz), only L OMO is oscillating; the R OMO is forced to oscillate at the L OMO’s frequency but have not yet reached its oscillation threshold. This is illustrated by the displacement state space figures shown in fig. S4A(e) for ∆ω/2π = −0.21 GHz (blue dashed line in fig. S4A(a)), note that |xL | is about 20 times larger than |xR |. At ∆ω/2π = −0.25 GHz, marked by the red-dashed line in fig. S4A(a), the situation changes and the R OMO oscillates with larger amplitude (|xR | ≈ 3.5|xL |) but at different frequencies; the result is a Lissajous figure that fills in the whole state space. In figures S4B (Pin = 15.8 µW), in the asynchronous region, indicated by the blue dashed line, the L OMO oscillates with an amplitude roughly 15 times of the R OMO in agreement with the measured RF spectrum and the pump probe measurement. In the unified frequency region, for both power levels Pin = 15.8 µW and Pin = 17.9 µW in fig. S4C, the phase diagram shows the two oscillators are synchronized and their amplitude differ less than 20% in agreement to the pump-probe measurements. The synchronization phase for figs. S4C(d-e) is roughly φ = 160◦ , also all the simulations for our system resulted in phase differences close to π, in agreement with the discussion in [36] that the anti-phase synchronization is a more stable state when the oscillations amplitude xL , xR are not identical.
