Optomechanical cavities have proven to be an exceptional tool to explore fundamental and applied aspects of the interaction between mechanical and optical waves. Here we demonstrate a novel optomechanical cavity based on a disk with a radial mechanical bandgap. This design confines light and mechanical waves through distinct physical mechanisms which allows for independent control of the mechanical and optical properties. Simulations foresee an optomechanical coupling rate g0 reaching 2π × 100 kHz for mechanical frequencies around 5 GHz as well as anchor loss suppression of 60 dB. Our device design is not limited by unique material properties and could be easily adapted to allow for large optomechanical coupling and high mechanical quality factors with other promising materials. Finally, our devices were fabricated in a commercial silicon photonics facility, demonstrating g0/2π = 23 kHz for mechanical modes with frequencies around 2 GHz and mechanical Q-factors as high as 2300 at room temperature, also showing that our approach can be easily scalable and useful as a new platform for multimode optomechanics.
© 2017 Optical Society of America
Optomechanical microcavities simultaneously confine optical and mechanical modes. The interaction between these confined modes has proven to be a very rich field of study for basic science, such as macroscopic quantum phenomena [1,2], quantum simulation of condensed matter [3,4], topological phase pattern-formation of coupled oscillators  and non-classical states of light [6,7]. Applications of optomechanical devices have been addressed as well, including weak-force sensing [8–10], ultra-sensitive accelerometers , radio-frequency sources , multi-mode  and synchronous  oscillators, reconfigurable optical filters  and hybrid systems .
Regardless of the context, the optomechanical interaction often benefits from a strong optomechanical coupling, low optical and mechanical dissipation rates (high-Qs) and the so-called resolved sideband limit, which occurs when the mechanical mode frequency is larger than the optical mode linewidth. We choose to denote the optomechanical coupling by the vacuum optomechanical coupling rate, g0, which measures the optical cavity frequency shift induced by a mechanical mode with an amplitude equivalent to the quantum harmonic oscillator at the ground-state.
The challenge in simultaneously achieving an optimum combination of these properties is that the optical and mechanical optimization are often constrained to each other [17–19]. One promising platform is based on the optomechanical crystal cavity, where both waves are confined by bandgaps in two-dimensional periodic structures . These devices were fabricated using the traditional approach of a direct write electron beam lithography. However, for massive fundamental studies and applications it is desirable to migrate the fabrication process of these devices to commercial CMOS-compatible facilities, where the photo-lithography process considerably reduces time and cost when compared to the traditional approach.
Here we propose a new optomechanical device based on a silicon microdisk with a circular mechanical grating fabricated in a CMOS-compatible foundry (Fig. 1(a)). The optical waves are confined at the disk edge in whispering gallery-like modes due to total internal reflection, whereas the mechanical modes are confined as an edge state lying within a phononic bandgap of the circular grating. We show that these two unrelated confining mechanisms relax the need for simultaneous optical and mechanical bandgaps, while keeping both optical and mechanical losses to a minimal. These relaxed requirements along with recent progress in low-loss CMOS-compatible photonic devices [21, 22] are encouraging for addressing scalable applications of optomechanical cavities. Moreover, our whispering-gallery based design is a natural candidate for addressing recent theoretical proposals concerning many optically-coupled optomechanical cavities [3–5].
Using both numerical simulations and measurements we demonstrate a tailorable mechanical bandgap up to several GHz within an experimentally accessible range of geometrical parameters. The fabricated design presented an optomechanical coupling rate of g0/2π ≈ 23 kHz for a mechanical mode in the GHz range, while simulations indicated that an order of magnitude larger optomechanical coupling rate could be achieved for an even high mechanical mode frequency. Mechanical radiation losses are also inhibited by the mechanical bandgap, which allows for a high mechanical Q-factor, essentially limited by material losses .
In order to gain some insight on our final device design, we start by analyzing a floating silicon ring resonator (Fig. 1(b)). This idealized optomechanical structure benefits from the highly co-localized optical and mechanical modes defined by its geometrical constraint, in a similar way to the optical and mechanical modes responsible for forward Brillouin scattering in silicon nanowires . However, in the floating ring structure, mechanical losses are reduced to material losses while optical losses are due both to material and to radiation (bending) loss.
One can also easily evaluate the vacuum optomechanical coupling rate, g0, following a perturbation theory approach for such structure:25] and another coming from strain-induced modifications of the refractive index (photoelastic effect) . The phase-matching condition arising from Eq. (1) to describe the frequency-shift is automatically satisfied in our case, as we focus on mechanical modes with zero azimuthal order. In the above equation ∂ω/∂x is the commonly known frequency pull parameter gOM which depends on how the physical deformation is parameterized (or equivalently, on how the mode profile is normalized). Such dependence is eliminated for g0 = gOMxzpf, where is the zero point fluctuation, ℏ is the reduced Planck’s constant, Ωm the mechanical mode’s frequency and meff the effective motional mass for the chosen parameterization  — indeed, experiments to probe the optomechanical coupling turn out to be sensitive to g0 instead gOM .
The main feature captured by this simple model is the ability to tailor the mechanical frequency from a couple to tens of GHz, while raising the optomechanical coupling rate from tens to hundreds of kHz. This is achieved by adjusting only wring while keeping R fixed. In Fig. 1(c) we also highlight the lowest order dilatational mode (red dots); larger markers account for larger g0. Such high g0 is dominated mainly by the photoelastic contribution (see Appendix F). As an example, a device with a ring radius of R = 8 µm and ring width of wring = 1 µm would have a net optomechanical coupling rate of g0/2π = 88 kHz at a mechanical frequency of 4.5 GHz (throughout the text, the silicon-on-insulator device layer was kept fixed and equal to 220 nm). Both values are one order of magnitude larger than the radial-dilatational modes in a Si disk with the same radius — a back-of-the-envelope calculation gives frequency , where vL = 8433 m/s is the longidutinal acoustic velocity in silicon, and vacuum coupling rate , dominated by the moving boundary contribution .
On the other hand, the optical mode is nearly unaffected by changes in the ring width. The ring optical mode is similar to the mode of a disk (not shown) with same radius resulting in an essentially constant optical resonance frequency for wring > 1 µm as shown in Fig. 1(d). Indeed, the simulation shows that the fundamental TE mode changes by only a few percent for 0.5 µm < wring < 1.0 µm, mainly because the mode profile begins to slightly spread out of the ring region. The highly different azimuthal wavevectors for the optical and mechanical modes explains this opposite behavior – while a typical whispering gallery optical mode has azimuthal wavevectors equal to m/R, where m is on the order of 83 for modes at the 1550 nm wavelength, the mechanical mode has zero azimuthal wavevector.
The frequency pull parameter gOM weakly depends on the cavity radius (R) since the optical and mechanical mode profiles do not significantly alter for radii much larger than the wavelengths. Because the mechanical mode has no longitudinal dependence, its frequency should also be the same. Therefore, g0 changes only through ; the smaller R is made, the larger g0 gets. For the real bullseye cavity, however, the smallest R size is also bounded to the size of the desired circular grating. Despite idealized, this model gives an upper boundary for the vacuum optomechanical coupling rate g0 as well as an insight for the edge state mechanical mode on our real structure.
In order to mimic the optomechanical properties of the floating ring in an anchored device, we propose the bullseye optomechanical resonator. Our design consists of a silicon disk supported by a central silica pedestal and possessing a phononic shield in the form of a bullseye radial Bragg grating (see Fig. 1(a)). Circular gratings have been successfully used previously to confine photonic [29–31] and plasmonic [32–34] modes. Therefore, extending this circular grating confinement mechanism to acoustic waves should enable ring-like modes in the edge of the bullseye disk.
Similarly to the idealized ring resonator, the rather wide wring in our bullseye structure (≈ 1 µm) and high optical azimuthal numbers ensure that the optical modes are whispering-gallery like. Although higher radial order optical modes penetrate deeper inwards, for wring < 1 µm, the relevant lowest order radial mode are weakly influenced by the radial grating structure. This behavior is confirmed by the optical frequency curves shown in Fig. 2(a) and allows us to optimize the mechanical grating design without disturbing the optical properties.
The mechanical waves reflected by this circular grating (Fig. 3(a)) can be understood using the linear crystal analogue shown in Fig. 3(b). This approximation is quite precise since the interfaces interacting with a cylindrical wave traveling through the circular grating are similar to those interacting with a plane wave in a linear crystal (see Appendix D). The linear crystal approximation therefore allows for a simpler design of the mechanical bullseye Bragg grating.
We are interested in creating a phononic bandgap that confines the radial dilatational mechanical modes at the disk edge. These modes arise from longitudinal waves (L) propagating along the radial direction, which are mixed with shear-vertical (SV) waves due to reflection at the free-boundaries . Therefore, a phononic shield with a partial bandgap for L and SV waves is enough to confine the ring-like dilatational mode. Figure 3(c) shows a typical band structure for the equivalent linear crystal; blue lines represent x-polarized (L) or z-polarized (SV) modes, whereas the red lines represent y-polarized (Shear-Horizontal, SH) modes. A partial bandgap (gray shades) is then formed between 3.2 GHz and 5 GHz.
To confine the first order mechanical dilatational mode at any desired frequency (chosen via wring), we further investigate the dependence of the partial bandgaps on the grating parameters, a, w and t (Fig. 3(d)). We also restrict the design parameters within the range accessible through Si-photonics foundries. The left graph in Fig. 3(d) shows the main bandgaps (colored regions) as a function of w/a for different values of a keeping t = 70 nm, while the right graph shows the bandgaps dependence on t/a while maintaining w/a = 0.25. As expected, raising a both shrinks the total bandgap and moves it towards lower frequencies. It is also worth noting that the bandgaps around the region of interest widens as t/a is decreased, a consequence of the larger acoustic velocity contrast between the thicker and thinner regions.
One can easily find a suitable design of a bullseye disk with larger optomechanical coupling for a given mechanical frequency using the bandgap maps in Fig. 3(d), and the relation between the mechanical frequency and the ring width from Fig. 1(c). For example, choosing a 4.0 GHz mechanical mode frequency, we determine from Fig. 1(c) that the ring width should be around wring = 1.1 µm and from the bandgap maps that a = 1 µm with w/a = 0.2 and t/a = 0.07.
In order to further evaluate this optimized design, we use (axisymmetric) Finite Element Method (FEM) simulations for the optical and mechanical modes of the whole bullseye structure to calculate the optomechanical coupling rate, g0, and the radiation limited mechanical quality factor (Qm). The results are shown in Fig. 4(a) superimposed on a mechanical density of states map of the corresponding linear crystal; higher density of states (DOS) are represented by darker regions. We choose R = 8 µm for the simulations as it is the smallest radius size that can support gratings with 5 periods confining modes down to 1 GHz (corresponding to a ≈ 1.5 µm.
Two mechanical mode types give notoriously high optomechanical coupling: for wring = 1.1 (2.0) µm, which tunes the mechanical frequency to approximately 4.0 GHz (2.0 GHz), the coupling rates are (33 kHz) and (14 kHz). Similarly to the floating ring, these couplings are dominated by the photoelastic effect due to the large strain overlap to the localized fundamental transverse electric (TE) optical mode within the outermost ring.
The type-I mode shown in Fig. 4(b) is related to the first order dilatational mode of the ring structure when its frequency is within the bandgap of the analogue linear crystal. On the other hand, the type-II mode is related to a high order flapping mode. In the floating ring structure such mode has nearly zero optomechanical coupling due to the opposite parity between the optical and mechanical modes. However, in the bullseye disk this parity is broken due to the position of the slab connecting each grating ring, resulting in a surprisingly large g0 for these flapping modes.
Since radial waves within the bandgap cannot propagate towards the pedestal, anchor losses are drastically reduced. This anchor loss suppression mechanism is manifested in the FEM simulation (using perfectly matched layers described in Appendix H) results shown in Fig. 4(c), where the mechanical quality factor grows exponentially with the number of periods in the grating. The simulation is done by following the type-I mode (for wring = 1 µm) as the number of periods is reduced from 6 to 0.
In the few GHz frequency range, the ring region strongly hybridizes to an almost continuum of modes of the inner disk limited by the groove, these hybrid modes all radiate through the pedestal. The bullseye grating indeed prevents the coupling of the outer ring mode to the inner disk continuum of modes and ensures high mechanical Q modes.
3. Experimental demonstration
We experimentally demonstrate the bullseye design in devices fabricated by a commercial foundry (Fig. 5(a)). Using a CMOS commercial foundry has several advantages towards coupled optomechanical cavity arrays as well as implementing high volume and low cost on-chip circuitry and sensing. The fabrication is based on a deep-UV optical lithography (see Appendix A) with nominal resolution of 130 nm.
In order to evaluate the effects of the bullseye grating structure we fabricated devices with wring = 1.5 µm and 2.0 µm, corresponding to first order mechanical dilatational modes around 3.0 GHz (see Appendix E, Fig. 12) and 2.0 GHz respectively, while varying the bullseye pitch from a = 650 nm to 1450 nm. We have not fabricated devices with higher frequencies by this point to avoid being too close to the Foundry’s nominal resolution.
The devices are probed by a tunable laser that is evanescently coupled to the cavities through a tapered optical fiber (Fig. 5(c)) with the fiber polarization matching the bullseye’s TE mode. A lorentzian fit to the DC optical transmission signal yields intrinsic optical quality factors of 4 × 104 (κi/2π ≈ 5 GHz and free spectral range FSR ≈ 10 nm), as shown in Fig. 5(d). Although very low-loss straight silicon waveguides  and high-Q photonic crystal cavities  were demonstrated with similar fabrication processes, our optical quality factors are similar to those previously reported on other circular optical cavities, e.g. ring resonators [36, 37]. Moreover, disks without grooves with radius between 5 and 20 µm were patterned in the same chip and their optical Q-factors never exceeded 5 × 104.
By changing the distance from the fiber taper to the optical cavity we could optimize the optical coupling and achieve almost critical coupling (not shown, κe ≈ κi). We also used a similar technique to ensure the excitation of edge-localized optical modes as described in the Appendix B.
Room temperature thermal fluctuations excite the mechanical modes that modulate the phase of the intra-cavity optical field. The cavity resonance dispersion converts the phase-modulation into an amplitude modulation of the transmitted light, which can be measured using a radio-frequency (RF) spectrum analyzer. Despite the high mechanical frequencies of the fabricated device, the rather low optical quality factor (apparently limited by the foundry fabrication of circular elements) place the optomechanical cavity barely in the resolved sideband limit (κ/2 ≈ ωm). Therefore, the laser is tuned to the maximum slope of the optical mode transmission to ensure the most efficient conversion from phase to amplitude modulation.
In order to calibrate the optomechanical coupling rate, g0, the input laser beam is modulated at a frequency close to the mechanical resonance with a calibrated phase modulator . Figure 5(e) shows a typical RF spectrum around the mechanical modes of a device with wring = 2.0 µm and a = 1400 nm. The modes around 2.15 GHz (red) are identified as the type-I mechanical modes whereas those around 2.00 GHz (blue) are type-II modes. The RF-calibration tone is also shown near 2.0 GHz (purple line). The flat noise-floor is dominated by intrinsic laser noise; no excess laser phase noise is observed in this frequency span.
In the RF spectrum, four or more peaks are observed for each mode type, instead of a single peak expected from axisymmetric isotropic silicon simulations — any other mechanical mode other than the type-I and type-II would be very lossy and/or have very low g0 in this isotropic approximation. We further investigate this behavior using three-dimensional numerical simulations that take into account both the in-plane anisotropy of silicon’s  and the fluctuations in wring throughout the cavity’s perimeter. We found that the material anisotropy together with a tiny variation of 10 nm (0.5%) in wring is enough to account for the typical 10 MHz splitting between peaks within a given mode family (see Appendix F).
The mechanical mode fitting and calibration give typical mechanical quality factors of Qm = 2300 at room temperature which are also confirmed by pump-probe experiments (see Appendix E), and optomechanical coupling rates as high as g0/2π = 23 kHz for the type-I mode shown in Fig. 5(e). The g0 values are in good agreement with the simulated values when silicon’s anisotropic elasticity is taken into account (see Appendix F, Fig. 9(b)). Figure 5(f) shows the corresponding spectral density of displacement fluctuations (Sxx) for the type-I mode with largest transduction, revealing a displacement sensitivity, given by the background noise, on the order of for a typical input optical power of Pin = 250 µW.
In order to understand how the bullseye grating structure is preventing mechanical loss, we compare the measured mechanical spectra — obtained from a series of devices with varying values of a — with the mechanical modes obtained from the axisymmetric numerical model of the bullseye cavity. Figure 6(a) shows the normalized (see Appendix C) measurement results for both type-I and type-II modes as well as the calculated band-edge frequency related to the third and fourth bands on Fig. 3(d) (dashed lines). The corresponding linear crystal’s DOS is superimposed to the calculated mechanical modes on Fig. 6(b). Neither the axisymmetric nor the linear model capture the in-plane anisotropy of silicon or any wring azimuthal fluctuations, nonetheless those simulations reproduce most of the features observed in the measured spectra; inside the bandgap the calculated optomechanical coupling is as high as 30 kHz and the mechanical mode is mostly confined within the wring (Fig. 6(c)), resulting in high mechanical quality factors. Outside the bandgap, the once confined mechanical mode can couple to grating band modes and leak energy through the pedestal. This effect is clearly seen for a between 1.2 and 1.3 µm in Fig. 6(a); despite the large calculated optomechanical coupling of 27 kHz, the mechanical mode spreads inside the circular grating resulting in a larger mechanical leakage and a small signal to noise ratio transduction signal. On the other hand, even inside the high DOS region (dark shaded regions of Fig. 6(b)), the mechanical modes might not couple to the grating bands (due to polarization mismatch) and result both in high optomechanical coupling and large mechanical quality factors; the a = 1.0 µm spectra is one such example. Deeper into the grating bands, however, any mechanical polarization travels through the grating and the mechanical Q-factor drops drastically down to a few hundreds as observed for a = 0.85 µm (Qm ≈ 400) and a = 0.8 µm (Qm ≈ 100). These results show a path to design a device with high mechanical quality factor and optomechanical coupling by adopting a simple one-dimensional model for the band structure and floating rings to infer the mechanical modes frequencies.
In summary, we present a new design for an optomechanical bullseye cavity with independent confinement approaches for optical and mechanical modes. Such strategy allows for tailorable mechanical modes up to 8 GHz and with g0/2π as large as 200 kHz. We experimentally demonstrate the bullseye optomechanical cavity using a standard optical lithography fabrication technique in a commercial CMOS foundry, resulting in modes between 2 GHz and 3 GHz with mechanical quality factors as high as 2300 at room temperature. The whispering gallery optical modes presented quality factors as high as 4 × 104 and optomechanical coupling rate on the order of g0/2π = 23 kHz, close to what is expected at these frequencies. Coupling between a single mechanical mode and multiple optical modes is also shown (Appendix E) and could be applied to efficient wavelength conversion . Such applications could achieve commercial feasibility by substituting the taper by integrated waveguides with grating couplers [40, 41] along with hermetic packaging.
Our results in the bullseye disk can be compared to forward Brillouin scattering in silicon nanowires [24, 42, 43], since the optical and mechanical modes involved are quite similar — indeed, the silicon nanowire can be seen as a bullseye cavity with infinite radius. Our results show that the bullseye circular grating presents a clear improvement of an order of magnitude in the mechanical Q-factor when compared to the silicon nanowire for the same mechanical mode profile.
On the other hand, the optomechanical interaction demonstrated for wring = 0.45 µm nanowires (equivalent to g0/2π ≈ 200 kHz according to recently proposed conversion formulas  if the nanowire is bended into a 12 µm radius ring) is an order of magnitude stronger than in our fabricated bullseye cavities with wring = 2.0 µm. Interestingly, this equivalent nanowire g0 matches our bullseye simulations for wring = 0.45 µm, strongly indicating that further experiments regarding the bullseye design shall achieve an order of magnitude enhancement in the optomechanical coupling rate by shrinking the ring width.
As compared to other high mechanical frequency silicon disks proposals  whose high mechanical Q-factor relies on the smallness of the pedestal, our design has the advantage of mechanical frequency tailorability without perturbing the optical frequency. Comparing to optomechanical crystals such as the nanobeam  and the snowflake  designs, the bullseye disk leads to similar mechanical frequencies and should achieve a g0 value comparable to the snowflake’s, although still an order of magnitude lower than the nanobeam. However, our design still has an advantage in tailorability, for it uncouples the optical and mechanical designs, without increasing mechanical losses.
Finally, the CMOS flexibility allied with the ability to independently tailor both optical and mechanical modes, along with improvements in fabrication to enhance the optical Q-factor, could then be explored in the design of large arrays of coupled oscillators; whispering-gallery optical modes turn out to be a more natural approach for optically-coupled cavities . This would allow for coupling between many non-colocalized optical modes enabling a novel route towards the investigation of multimode optomechanical phenomena, in a approach essentially different from, yet very competitive to, those based on optomechanical crystals.
A. Sample fabrication
The devices were fabricated through the EpiXfab initiative at IMEC on a silicon-on-insulator wafer (top silicon layer of 220 nm over 2 µm of buried silicon oxide). Simple disks were first patterned on the top silicon layer through deep UV lithography and plasma etching. A similar (aligned) patterning cycle was then performed to make the circular grooves, although now the plasma would only etch 150 nm of silicon instead of the whole 220 nm layer. Finally, an in-house post-process step was performed to selectively and isotropically remove the buried oxide using diluted hydrofluoric acid in order to mechanically release the bullseye cavities.
It should be noticed that alignment errors affect performance only by producing an effective ring eccentricity, resulting in the same effects described in Appendix F. However the foundry’s overlay accuracy is better than 30 nm, which is unlikely to couple the clockwise and counterclockwise optical modes.
B. Experimental protocol
In order to couple light to the optical modes confined at the bullseye’s outer ring, the taper-cavity distance was slowly decreased through a sub-micron positioning system while the optical transmission was simultaneously monitored in a broad wavelength range (≈ 100 nm). The reduced overlap between the taper’s guided mode and inner bullseye modes ensures that we are not accessing inner modes when the taper is far from the cavity, since all optical modes have alike optical quality factors. After such mode identification, the taper is positioned closer to the cavity to increase the optical coupling and hence the phase/frequency to amplitude transduction provided by the cavity.
C. Mechanical spectra normalization
The detected current spectra is related to the mechanical noise spectra as :Figure 6(a) shows a series of mechanical spectra which were normalized by the phase modulator calibration tone (not shown) such that each curve is proportional to . After normalization each spectra is plotted in the same vertical dB scale.
D. The linear crystal approximation
In this section we further justify the linear crystal approximation. The main idea behind it is that the curvature of cylindrically symmetric grating interfaces will not affect small wavelength radially-polarized waves and will be scattered like plane waves by plane interfaces. In order to better understand such approximation, we compare both plane and circular gratings infinite along the z-direction, like shown in Figs. 7(a) and 7(b). These structures may be connected to the proposed slab-like ones by an effective medium approximation: the bulk longitudinal, vL, and shear, vS, velocities of the infinite media are rescaled to reproduce the corresponding slab velocities of the etched grating regions. Such procedure is analogue to the effective index approximation often used in optical waveguides [45, 46]. We shall focus on longitudinal waves since these are more closely approximate the bullseye’s dilatational modes.
Elastic normal modes in an isotropic solid may be described by the particle velocity field, , and its eigenvalue equation Eq. (3) is
Both and are of the form Avin+Bvout, the coefficients A and B depending on boundary conditions while vin and vout describe counter-propagating waves. Furthermore, the solutions have constant value (by construction) along interfaces matching their symmetry. Therefore, a transfer-matrix approach may be used to calculate the transmission of elastic waves through a finite number of homogeneous layers such as in Figs. 7(a) and 7(b) . Such spectrum is suitable for locating the structure’s bandgaps because waves within it will be strongly reflected even by a finite grating. Our goal is then reduced to evaluating the range of parameters for which the linear and circular gratings have similar transmission spectra.
The transfer-matrix approach relies on writing the velocity field in every medium (labeled j) as vj = Ajvin + Bjvout and matching the boundary conditions (continuity of particle velocity and normal force ) on each interface. For example, such procedure allows us to express A2 and B2 as linear combinations of A1 and B1,
Extending this argument to N media,
Assuming loss less media and denoting the incident and reflected waves by vin and vout, respectively, energy conservation further requires ; BN = 0 is expected since there can be no reflected wave traveling within the last medium. The transmission spectrum is then calculated by evaluating the ratio for every frequency of interest.
Figure 7c) shows the transmission spectra calculated by this transfer-matrix approach for the linear and circular grating. As the radius of the smallest interface, r0, decreases, it is clearly seen that the linear approximation starts to fail. On the other hand the transmission through the linear and circular gratings always disagree for low frequencies even for higher r0 values. These results summarize our initial thesis: the linear grating suitably approximates the circular one as long as the wavelength is small compared to the layers’ radius.
From Eq. (4) and (5), such conclusions can be made more quantitative. The validity of the linear approximation depends mainly on how well cylindrical waves approximate the Hankel functions that analytically solve the problem. Therefore, the linear crystal approximation is expected to work better for higher values of ωr/vL.
Calculation of transfer-matrices
Here we show the details about the calculations of the transfer-matrices for linearly and cylindrically symmetric problems. Such calculations rely on the boundary conditions of continuity of the velocity field, , and of the normal force given by , T representing the stress tensor.
The stress tensor may be expressed as a function of the·generalized Hooke’s law as T = C : S, where C is the 4-th rank stiffness tensor and S is the strain tensor, also given by the symmetric gradient of the displacement field, . For normal modes, the harmonic temporal behavior implies that .
For the planar interface described by vrect (Eq. (4)), the boundary conditions become continuity of vrect and of at a interface plane, x = w, between media 1 and 2. We therefore get the transfer-matrix
Now for the cylindrical symmetric problem, we use the continuity of vcirc from Eq. (5) and the corresponding normal force, at a cylindrical surface r = a separating media 1 and 2, to obtainEq. (7). This can be understood from the evaluation of the strain field, since the gradient of the radial displacement field also has a φ-component for these cylindrical symmetric solutions.
E. Pump-probe mechanical spectroscopy
A way to characterize an harmonic oscillator is by looking at its response due to a harmonic driving force. In optomechanical systems, this can be done to probe the mechanical modes with improved signal-to-noise ratio . To do so, we used a pump-probe scheme as illustrated in Fig. 8(a). A strong amplitude modulated pump laser drives the mechanical mode through radiation pressure. The mechanical oscillation turns into modulation of the optical resonance frequency, which can be measured in the transmission of a weak probe laser tuned to an optical mode with similar transverse profile but a few free spectral ranges away from the pumped mode; this strategy is necessary to isolate the probe signal from amplitude modulation coming directly from the pump laser.
The gain in signal relative to the thermally driven mechanical spectrum can be quantified by comparing the displacement noise spectrum in both cases. The thermal spectrum is given by the well known expression 
Now to find the displacement noise spectrum for the optically driven case, we solve the equations of motion for the pumped optical mode’s amplitude, a, and for the excited mechanical mode’s displacement, x,Eq. (9) accounts for the modulated pump laser, where |ain|2 is the incident photon flux (the input pump power reads where ωp is the pump laser frequency), Ωmod the modulation frequency, δ describes the modulation depth and κe is the extrinsic loss rate. The coupling terms describe the optical frequency shift in Eq. (9) and radiation pressure in (10). The remaining parameters concern the modes regardless of optomechanical interaction: Δ is the pump laser’s detuning to the optical resonance, κ is the total optical loss rate, Ωm is the mechanical resonance frequency, γ the mechanical loss rate, meff the mechanical mode’s effective mass and ħ is the reduced Planck constant.
By keeping the pump power low so that dynamical backaction may be ignored and by absorbing static optical resonance shifts in Δ, one gets
The displacement noise spectrum associated to this solution would show an infinitely narrow peak at Ωmod, which when integrated should lead to the driven variance
Apart from transduction constants, the actually measured peak has its width limited both by the modulator’s linewidth and by the spectrum analyzer’s Effective Noise BandWidth (ENBW). In this experiment we used an electro-optical modulator driven by a sub-Hz linewidth radio-frequency signal generator, whereas an ENBW of ≈ 100 kHz was enough to resolve the mechanical resonances. Therefore, the peak’s width was always limited by ENBW and such peak intensity as a function of Ωmod reads
It is worth noting that, although similar experimental setups are used for probing the thermal and driven noises, the driven noise acquisition requires using the spectrum analyzer actually as a demodulator tuned to the modulator’s frequency Ωmod.
It immediately follows from Eq. (17) that the gain should benefit both from a high input pump power and from a large modulation depth. It also follows that the pump-probe measurement is most effective when the pump laser is tuned to the half-linewidth of the pumped resonance (see Fig. 8(a)) and that it is limited to the cavity’s bandwidth κ.
Figure 8b) shows a plot of the gain as a function of the input power and modulation depth for cavity parameters matching our devices (see Fig. 5 and main text). The black dashed line shows the curve of unity gain, above which the pump-probe method is indeed advantageous.
We performed pump-probe measurements following the setup of Fig. 8(c). An electro-optical modulator driven by a RF-signal generator modulates the pump laser which is then guided towards the cavity. In order to avoid the thermo-optical instability, the pump laser is actively locked to the blue side of an optical resonance at half-linewidth. The counter-propagating probe laser is launched from the opposite direction and tuned to the half-linewidth of an optical mode of the same transverse family but a few free-spectral-range (FSR) apart (≈ 20 nm) — notice that, in constrast to similar pump-probe configurations for studying Brillouin scattering , there is no optomechanically backscattered light since the mechanical mode has zero azimuthal number. Optical circulators and a bandpass filter ensure that the photocurrent seen by the electrical spectrum analyzer is only due to the probe transmission. Under typical experimental conditions, pump power around 100 µW and modulation depth of 5 %, the gain would be approximately 25 dB according to Fig. 8.
The measured pump-probe spectrum is a bit more complicated than Eq. (16) due to other optical nonlinearities. Broadband nonlinear effects such as Kerr, free carrier dispersion and free carrier absorption interfere with the narrow-band optomechanical effect into a Fano lineshape (Fig. 8(d)). These extra nonlinearities may be lumped into a frequency independent complex parameter kNL, so that we may fit the Fano resonances to the expressionFig. 5 of the main text.
The main advantage of the pump-probe method in these devices is the much larger extinction near mechanical resonances compared to the thermal spectrum, as shown in Figs. 8(d) and 8(e). Such large extinction allows for easily locating the mechanical modes which can be further studied (including g0 measurements) in the direct detection setup of Fig. 5(b).
F. Effects of eccentricity and anisotropy on the bullseye’s mechanical modes
In this section we show how silicon’s elastic anisotropy and possible fluctuations in wring lead to a more thorough comprehension of the mechanical spectra shown in Fig. 5.
To do so, we first perform three dimensional finite element simulations of the ring structure (Fig. 9(a)) focusing on quasi-radially polarized modes. In order to clearly understand how the anisotropy affects the modes, we write silicon’s stiffness tensor:
The three-dimensional isotropic simulation (η = 0) shows 4 modes with large optomechanical coupling to the outer ring optical mode: the azimuthally symmetric dilatational mode (type A) and three extra ones that break the azimuthal symmetry, two of which (type B and B*) are degenerate and connected by π/2 rotations. As η is raised to 1, the 4 modes get more hybridized and their splitting actually becomes smaller; also, the twofold degeneracy is not broken since the anisotropic stiffness (or more generally, silicon’s cubic structure) is still symmetric under π rotations (Fig. 9(a)). On the other hand, the optomechanical coupling rate for the Type A mode gets closer to the measured value g0/2π = 23.1 ± 0.2 kHz as η approaches 1 due to the further localization of mechanical strain around the ring region, which suppresses the photoelastic effect (Fig. 9(b)). However, the B, B* and C modes still have very small g0.
There remains the problem of understanding the multiple peaks within each family. A quick estimation would show that a splitting of 10 MHz in Ωm ≈ 2 GHz could be caused by a fluctuation of 10 nm on wring = 2 µm; controlling such geometric fluctuations is far beyond the limits of deep UV lithography.
To further investigate whether such wring fluctuations are a reasonable explanation for the observed 10 MHz splitting, one has to show not only that the splitting arises from such fluctuations, but that these also cause more modes to have a high g0. Therefore, we performed another set of three-dimensional finite element simulations of ring structures, now assuming the more realistic anisotropic stiffness tensor C(η = 1) and varying the eccentricity of the devices as shown in Fig. 10 by keeping a = wring and varying b; this is the most simple way of simulating such fluctuations without recurring to more complicated stochastic methods while still getting a lot of physical insight.
Figure 10(a) shows the ring dispersion as a function of b/a. It is clearly seen that the mode hybridization caused by eccentricity leads to splittings on the order of 10 MHz. Additionally, from the highlighted mode profiles, it would be expected two modes with large g0 (those where expansions or compression are always in phase) instead of only one for the perfectly circular device.
Such conclusions are reproduced when investigating the actual bullseye disk as shown in Fig. 10(b), whose simulations were performed for half-bullseye with symmetric boundary conditions along a diameter. The Type A mode basically breaks up in two branches whose separation reaches 12 MHz as b/a approaches 1.005. Again, both branches should show similar optomechanical couplings. Finally, such considerations can be extended to explain a higher number of peaks within each family by recalling that the wring variations intrinsic to the fabrication process are of a random nature.
G. Scalable and tailorable bullseye optomechanical cavities
In order to check whether the actual dimensions match the nominal values, we fitted SEM images of the bullseye devices. The images were acquired in a NOVA 200 Nanolab Dual Beam (FIB-SEM) System and later processed to correct residual astigmatism. We rescale each SEM image to ensure that the diameter measured along the image’s x- and y-directions are equal. The validity of such scaling is confirmed by comparing the eccentricity impact on the mechanical mode splitting (Fig. 10), suggesting that diameter fluctuations should be lower than 1% of the wring width (20 nm). Also, we calibrate the absolute distance scale by assuming that each disk diameter is equal to the nominal one 24 µm.
All size measurements are within the nominal value to the fitting error — our fitting algorithm has an uncertainty of ±15 nm (±2 pixels), which is much larger than statistical errors for every fitted image. These results, shown in Fig. 11, demonstrate the scalability of bullseye optomechanical disks to a precision of 15 nm in CMOS-compatible Foundries, an important step towards commercial applications of optomechanical resonators.
Finally, we demonstrate mechanical tailorability by exploring the dependence of the mechanical frequency on wring. To do so, we characterized a set of bullseye devices with wring = 1.5 µm which are expected to have mechanical resonances related to Type I and Type II modes slightly below 3 GHz according to Fig. 1. Figure 12 shows pump-probe measurements for such set of devices over the linear crystal DOS. Like for those devices with wring = 2.0 µm, the linear crystal DOS and mechanical resonances agree (except for a small displacement in a) and again two main families of resonances are noticed, corresponding to Type I and Type II mechanical mode profiles.
H. Perfectly matched layer parameters in FEM simulations
When simulating the bullseye’s radiation loss (Fig. 4(c)), we used a semi-spherical perfectly matched layer (PML) below the device’s pedestal. Inside the PML, the spherical radial coordinate r is stretched by Δr according to:
Sao Paulo State Research Foundation (FAPESP) (2012/17610-3, 2012/17765-7, 2013/06360-9); National Counsel of Technological and Scientific Development (CNPq) (550504/2012-5); Coordination for the Improvement of Higher Education Personnel (CAPES).
The authors would like to acknowledge Paulo Dainese for fruitful discussions and CCS-UNICAMP for providing the electron microscopy infrastructure.
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