Cavity opto-mechanics exploits optical forces acting on mechanical structures. Many opto-mechanics demonstrations either require extensive alignment of optical components for probing and measurement, which limits the number of opto-mechanical devices on-chip; or the approaches limit the ability to control the opto-mechanical parameters independently. In this work, we propose an opto-mechanical architecture incorporating a waveguide-DBR microcavity coupled to an in-plane micro-bridge resonator, enabling large-scale integration on-chip with the ability to individually tune the optical and mechanical designs. We experimentally characterize our device and demonstrate mechanical resonance damping and amplification, including the onset of coherent oscillations. The resulting collapse of the resonance linewidth implies a strong increase in effective mechanical quality-factor, which is of interest for high-resolution sensing.
©2011 Optical Society of America
It is well known that light exerts a force [1–3] on macro-scale objects. The emergence of micro-electro-mechanical systems (MEMS) makes such forces appreciable when considering the small mass and large compliance achievable in micro-scale devices [4–6]. The light intensity and resulting force is enhanced by optical cavities, e.g. linear Fabry-Perot [7–9] or circular micro-ring/disk/toroids [10–14]. In addition to amplifying the optical force, a cavity results in coupling between the optical and mechanical systems. The resulting feedback  enables mechanical amplification/damping via wavelength tuning across the cavity mode .
Following initial demonstrations of opto-mechanical interaction in micro-scale resonators [7,10], much work has focused on increasingly sophisticated experiments for achieving quantum mechanical ground state cooling [8,9,14,15]. While impressive results were achieved, the experimental systems required alignment of external optics for opto-mechanical excitation/damping and resonator motion readout [7–15]. The external optics generally limits the opto-mechanical system to a single device. Recently, integrated waveguide [4–6] cavity opto-mechanical devices were reported, in which the cavity itself forms a mechanical resonator [16–18]. This on-chip waveguide approach [4–6,16,19] paves the way for dense integration of many optical components on a single-chip.
In order to take advantage of the dense integration enabled by the monolithic integration of optical waveguides with opto-mechanical structures, we propose and demonstrate a new versatile architecture for amplification and damping in a silicon opto-mechanical integrated circuit. Our approach features the self-aligned fabrication of a bus waveguide coupled to an opto-mechanical resonator. Specifically, a silicon microbridge is intersected by a waveguide cavity (Fig. 1 ). The suspended microbridge forms a mechanical resonator. We create a Fabry-Perot cavity using silicon/air distributed Bragg reflectors (DBR’s) . One DBR is fixed; the second is etched into the microbridge center. Any in-plane bridge motion displaces the movable DBR and modulates the cavity transmittance enabling sensitive motion readout.
Our architecture has several advantages. The cavity is co-fabricated with the mechanical resonator, avoiding the need for extensive optics alignment [7–10,14,15]. Furthermore, the waveguide-DBR cavity enables the integration of many opto-mechanical components on a common platform: a single fiber coupled to the chip enables the optical interconnection of many opto-mechanical devices via on-chip waveguides without the need for precise alignment to each component. Finally, the opto-mechanical designs are decoupled enabling independent control of optical properties (resonance wavelengths, optical Q-factor, finesse, photon lifetime) and mechanical properties (resonance frequency, mechanical Q-factor, thermal time constant). Having a large design freedom is essential to exploit cavity opto-mechanics for practical applications. For example, we design the microbridge with a high compliance to enable coherent mechanical oscillations with a large increase in effective mechanical Q-factor at low optical powers.
2. Device model
2.1 Optical model
Our optical microcavity consists of two DBR mirrors separated by a waveguide of length LC = 3 μm. The DBR mirrors consist of three silicon slabs of length L = 25 μm, height H = 4.8 μm, and width dSi≈5λ/4nSi. The silicon slabs are separated by air gaps of width dair ≈λ/4. We modeled our microcavity using a one-dimensional transfer matrix . This model was based on experimentally measured or well-known material parameters (Appendix Table A1) with the exception of: (i) the exact thickness of the silicon slab in the DBR; (ii) the exact thickness of the air gap in the DBR; (iii) the diffraction losses within the air gaps of the DBR; and (iv) the material absorption loss of our silicon device layer. The two thicknesses were constrained such that their sum was equal to the nominal DBR period (944 nm), leaving only three free parameters in our transfer matrix model. These three free parameters were then chosen so that the output finesse and cavity transmission of the model matched our measured values (Appendix Table A2). We added input and output waveguides on both sides of the DBR cavity to reproduce the measured etaloning from facet reflections.
The measured transmission spectra are matched to the exact level of transmission at resonance in the model by adding input and output sample coupling factors. These factors (≈0.32) account for both coupling losses and Fresnel reflections from the vacuum cell windows. They are then used to convert the incident laser power on the vacuum cell to the forward optical power in the input waveguide (Popt) that is incident on the DBR cavity.
The model finds an exact solution for the optical fields at various wavelengths and positions in the cavity. It also enables us to find the various cavity penetration factors, βi, which describe the decrease in field intensity with optical penetration into the DBR mirror. Figure 2a shows the calculated square of the electric field with the wavelength set to the cavity resonance. In order to obtain the cavity penetration factors, we calculate the forward and backward optical power flow in the structure (Fig. 2b). The ratio of the power flowing in a particular DBR layer N (PDBR-N) to that in the cavity (PCav) gives the penetration factors: β1 ≈ β2 = 0.28, β3 ≈ β4 = 0.08, β5 ≈ β6 = 0.02, and β7 = 0.005. (We note that a similar definition also holds for a single DBR mirror, where “PCav” is replaced by the incident power hitting the DBR). The penetration factors are critical for proper scaling of the optical force – the DBR mirror makes our structure and its analysis fundamentally different from previous Fabry-Perot opto-mechanical structures .
2.2 Optical forces
The opto-mechanical forces arise from radiation- and photothermal pressure. In the absence of a cavity, the radiation pressure on the DBR microbridge is:Fig. 2). We note that in this equation β1>>β7 and the forward and backward power flows at the first DBR slab (PDBR-1) are almost identical indicating that the mirror reflectivity is R≈100%. A full treatment, however, considers all terms (ΣPDBR-N), which is what we have done in our analysis.
The photothermal pressure results from nonuniform optical absorption and heating in the DBR due to the limited penetration of the field into the mirror (see Appendix Fig. A2). From Fig. 2b it is clear that most of the optical power is absorbed in the cavity and in the first silicon slab (i.e. PDBR-2 = β2PCav) and that the other silicon slabs (β4 and β6) contribute only minimally. In the absence of a cavity, absorption-induced heating, thermal expansion and beam bending produce an equivalent force:Eq. (2) emphasizes that the optical power drops significantly in the first DBR silicon slab compared to the power in the main part of the cavity (Fig. 2 and Appendix A2). Any heating in the outermost DBR silicon slab (β6) produces an opposite force and its contribution must be subtracted from the net photothermal force. Furthermore, the contribution of the first DBR silicon slab dominates, since β2>>β6 (i.e. the heating of the inner DBR slab is much greater than the heating of the outer DBR slab). The contribution of the middle slab (β4) can be neglected since any thermal expansion will not result in beam bending due to symmetry. For the optical powers considered here any absorption and heating in the cavity is minimal and does not lead to significant thermo-optic tuning of the optical resonances. We note that absorption, however, may limit the ultimate cavity finesse in any photothermal opto-mechanical device.
2.3 Optical spring
An optical force changes a resonator’s spring constant by Kopt. = ∂Fopt./∂z. The net optical force is Fopt = F0χenhL(ϕ), where F0 is the force without a cavity, is the cavity optical power enhancement on resonance, f is the finesse, TCav. is the net power transmitted through the cavity on-resonance (and √TCav relates the power coupled into the cavity to the incident power), L(ϕ) is the normalized cavity lineshape, and ϕ is the phase of the cavity optical field. The optically induced change in spring constant for a Lorentzian lineshape is:Fig. 3a ) resulting in opposing Kopt (Fig. 3b).
The resonant frequency (ν) and damping (Γ) in the presence of an optical force are :Fig. A2 and Tables A3 and A4). The fast time constant describes the DBR slab heating (τpt-fast = 3 μs) while the slow time constant corresponds to the heat flow in the silicon microbridge, which has a larger thermal mass and responds slower (τpt-slow = 162 μs). The large discrepancy in time constants means that optically-induced changes to ν are dominated by radiation pressure, while changes to Γ are dominated by the photothermal pressure, predominantly the fast component (τpt-fast).
2.4 Device fabrication
The devices are fabricated from silicon-on-insulator wafers with 4.8 μm silicon device layer and 1 μm SiO2 buried oxide, similar to Ref . We pattern the DBR mirrors and silicon microbridge resonator using a single electron-beam lithography exposure and subsequent development followed by cryogenic etching (SF6/O2) through the silicon device layer. The rib waveguides are patterned with contact lithography and a shallow cryogenic etch to a depth of ≈0.5 μm. We then thin the samples and cleave the waveguide facets for optical coupling. The silicon microbridge is released by etching the SiO2 layer in BHF followed by CO2 critical point drying to prevent stiction.
3. Experimental characterization
3.1 Experimental setup
Our experimental setup in Fig. 4a uses a custom vacuum cell with sapphire windows and a rough pump to obtain a base pressure of 20 mTorr (measured with a thermocouple gauge). All measurements are performed at room temperature. The device is characterized in vacuum to minimize damping – the mechanical Q-factor (QM) is approximately 10 at one atmosphere and is >1×104 at 21 mTorr (Appendix Fig. A3). Light from a tunable laser is coupled to our sample using a NIR 100x long working distance objective. After passing through the device and vacuum cell windows, the light is collected with a 50x objective and measured with a custom InGaAs photodetector (2 MHz bandwidth) with an integrated transimpedance amplifier (gain=5×105 V/A) coupled to a digital multimeter. Although we use high-NA objectives to couple light to our device, future devices can be fiber-coupled with low loss  using existing packaging technology.
First, the Fabry-Perot cavity modes are mapped by sweeping the laser wavelength and measuring the transmittance (Fig. 4b). Next, the mechanical resonances are measured by fixing the laser wavelength near the λ0=1593.75 nm cavity mode and detecting the modulated signal as the DBR vibrates using an electronic spectrum analyzer with 1 Hz resolution (16×averaging to eliminate noise) as the microbridge oscillates. The mechanical resonance spectra show a pronounced wavelength-dependence indicating a strong opto-mechanical interaction (Fig. 4c). Although we can detect several mechanical resonance modes (Appendix Fig. A3), we perform more detailed measurements on the λ0=101 kHz fundamental in-plane (M=0) mechanical resonance (Fig. 5 and Fig. 6 ). For both wavelength detuning and power dependence measurements we perform a Lorentzian curve fit to obtain ν and Γ.
3.2 Wavelength detuning measurements: fundamental in-plane mode (M=0)
We performed experiments on the fundamental in-plane mode (M=0) in vacuum (P~20 mTorr) since this maximizes QM. To characterize our device we fixed the laser power and stepped λ across the Fabry-Perot mode. The measurements indicate a resonant frequency increase for blue detuning (Δλ<0) and a decrease for red detuning (Δλ>0), in agreement with a radiation pressure based effect (Fig. 5a). While the photothermal pressure decreases ν slightly for blue detuning, this contribution is minimal since τpt-fast>>τrp.
We observe an increase in Γ for blue detuning and a decrease for red detuning (Fig. 5b). From Eq. (5) and the slow photothermal response (τpt.slow/fast>>τrp) we expect that Γ is dominated by photothermal pressure. Indeed, a radiation pressure interaction suggests an increase in Γ for red- and a decrease for blue wavelength detuning , in contrast to our measurements. There is some oscillation in Fig. 5a and 5b (0.2-0.3 nm wavelength spacing), which is due to etaloning from the DBR mirrors and the uncoated waveguide facets (see the Fabry-Perot optical spectrum in Fig. 4b inset).
3.3 Power dependence measurements: fundamental in-plane mode (M=0)
Next, we set the laser to a red- and blue-detuned wavelength and varied the optical power. The resonant frequency shows a linear dependence on Popt with blue detuning (Δλ=−0.34 nm) resulting in an increase in ν and red detuning (Δλ=+0.38 nm) showing a decrease (Fig. 6a). Our model is in good agreement with experimental results.
The linewidth change in Eq. (5) predicts a linear Popt dependence for Γ and shows good agreement with the experimental results for blue detuning (Fig. 6b). In contrast, the red detuning measurements show a strong nonlinearity. The behavior can be explained by considering the oscillation amplitude (the area of the Lorentzian fit normalized by the laser power, Fig. 6c). The data show a clear threshold beyond which the amplitude increases rapidly. Equation (5) is linearly decreasing with Popt and is only physically meaningful for Γ>0. The power at which the linewidth collapses (Γ→0) is the threshold at which coherent mechanical oscillations are observed (Pth≈300 μW in Fig. 6c). Our instrument resolution is 1 Hz, so Γ<1 Hz is interpreted as exceeding threshold (Fig. 6b shaded area). The data provide clear evidence of coherent mechanical oscillations with (i) pre-threshold behavior (Γ decreases); (ii) the threshold condition (Γ→0); and (iii) linear gain (Γ≈0 with a rapid increase in amplitude) similar to a “mechanical” laser .
Our modeling suggests a 1/QM dependence for Pth, similar to previous work . We further expect a Pth~1/f2 dependence . To test our model, we measured a second device with similar mechanical parameters (QM=2.5×104) but with a slightly modified DBR and a large increase in finesse (f = 380, Appendix Fig. A4). We observe Pth=32 μW (Fig. 6d), a reduction by 10 compared to the previous device (Fig. 6c, where QM=1.7×104 and f=140), in general agreement with our model. We also note that there is essentially no change in ν0 with change in the cavity design (ν0≈101,150 Hz for device 1 in Fig. 6c; ν0≈101,560 Hz for device 2 in Fig. 6d); this emphasizes our architecture’s large design flexibility in tailoring the optical parameters independently from the microbridge.
In our device, the mechanical parameters are governed by the microbridge geometry, while the cavity properties are determined by the DBR’s, which occupy a very small fraction of the overall microbridge volume. The advantage of decoupling the opto-mechanical designs becomes apparent by considering previous opto-mechanical microcavities [11–13]. Radiation pressure in an ultra-high Qopt (>108) microtoroid resulted in self-oscillation for 20 μW power . Using meff=3.3×10−8 kg and ν=4.4 MHz , we extract a spring constant Keff, microtoroid = 2.5×107 N/m. Our cavity is modest in comparison (Qopt=5×103). Nonetheless, we can observe coherent mechanical oscillations with Pth=32 μW (Fig. 6d), which is among the lowest threshold powers experimentally observed. Therefore, while our Qopt is much smaller than previous work , self-oscillation is possible at low optical power since our mechanical structure is significantly more compliant (Keff=2.75 N/m).
We demonstrated the ability to modify the optical design with little change in mechanical parameters (Fig. 6c and 6d). A guide for additional mechanical design changes are the relations ν0~wbridge/Lbridge2 and Keff~tbridge(wbridge3/Lbridge3) . By scaling tbridge, wbridge and Lbridge, we can further reduce Keff and Pth without affecting ν0 or changing the cavity design. Conversely, ν0 can be tailored while leaving Keff unchanged. This shows the flexibility of our opto-mechanical architecture for practical applications in which accurate control of ν0 (large ν0 reduces frequency noise [26,27]) or Pth (to reduce power requirements) is desired.
A coherent opto-mechanical oscillator has many practical applications. We previously demonstrated a chemical sensor, using a similar structure, but without opto-mechanical effects. The sensor was based on analyte sorption, mass loading and mechanical resonance shift . The device required electrostatic actuation and exhibited a 4.6 picogram mass-loading resolution, limited by the frequency noise (Allan variance) [26,27] σA~1/(ΔzQM1/2). Our coherent mechanical oscillator exhibits a sharp increase in Δz and QM above threshold, which leads to an enhanced sensing resolution due to the increased signal-to-noise ratio. Indeed, the vibration amplitude in Fig. 6c is ΔzRMS≈0.059 nm at Popt = 74 μW and increases sharply beyond threshold with ΔzRMS≈4.0 nm at Popt=450 μW implying a 68-fold reduction in the Allan variance based on Δz alone. The calculated frequency noise at Popt=450 μW is σAω≈2π×(3.9×10−4 Hz) which will enable sub-attogram mass-loading resolution with only slight changes in the device design (Appendix A5).
We proposed and experimentally demonstrated a versatile cavity opto-mechanics architecture. Our approach has the benefits of an on-chip coupling waveguide, which enables future large-scale integration of many opto-mechanical devices on a single-chip (e.g. sensor arrays). Furthermore, both optical and mechanical designs are decoupled enabling precise tailoring of device properties for practical applications. We demonstrated photothermal damping and amplification of the mechanical resonances, including a threshold condition beyond which the mechanical resonance linewidth collapses and the oscillation amplitude and effective mechanical Q-factor increase sharply.
The opto-mechanical self-oscillation demonstrated here eliminates the need for on-chip electrical power required in our previous chemical sensors . Besides mass-loading , our device may be used for high resolution inertial sensing, where the acceleration of a proof-mass attached to the microbridge is measured via . Magnetic field sensors can be realized by coating the microbridge with a ferromagnetic film  or by measuring the Lorentz force [32,33] for current flowing through the microbridge. The ability to individually engineer the opto-mechanical properties in a chip-scale integrated waveguide opto-mechanical system will enable a wide variety of sensors and sensor arrays with unprecedented resolution.
Appendix A1. Transfer matrix parameters and opto-mechanical coupling
Appendix A2. Photothermal pressure: DBR microbridge thermal-mechanical modeling
We modeled the local heating in our microstructure (Fig. A2a) using Comsol Multiphysics. The overall length, width (in-plane) and thickness (out-of-plane) of the DBR-microbridge are 400 μm, 1.8 to 2.12 μm (different sections of the beam have slightly different widths), and 4.2 μm, respectively. The DBR occupies the central portion of the microbridge, and is comprised of three silicon slabs separated by two air gaps. The DBR section is 26 μm long. Each silicon slab is nominally 694 nm wide separated by air gaps that are nominally 250 nm wide. The material properties are listed in Table A3.
The thermal-mechanical response of the DBR-microbridge is modeled by mechanically clamping its end faces (corresponding to attachment to the silicon substrate), and setting those faces in equilibrium with a 295 K thermal bath. A heat load is applied to the innermost silicon DBR slab (β2). The heat load is (arbitrarily) set to be 1 μW, so the results are normalized by this value. The heat load is applied instantaneously at t=0 to enable computation of the thermal response times. The load is applied in only the central 4 μm of this DBR slab, to approximate the width of the optical mode.
The finite-element solver computes the temperature, thermal strain, and displacement induced in the structure as a function of time. The primary steady-state outputs are Keff (the mechanical response to a steady-state distributed force), Kp (the mechanical response to a localized force at the DBR), and dz/dPabs (the steady-state displacement of the DBR microbridge in response to a heat load at the waveguide). The parameter dz/dPabs is essentially an optical absorption-displacement gain coefficient that is found from simulation assuming an asymmetric heat distribution across the DBR silicon slabs. By examining the temporal dependence of the temperature at the DBR after an impulsive heat load is applied at the DBR at t=0, the thermal response time of the DBR microbridge is found.
Our model gives a bi-exponential temporal response of the temperature, which we denote with two time constants, τpt-fast and τpt-slow, characterized by respective weights W(fast) and W(slow) (Table A4). This bi-exponential response is due to the fast heat flow out of the short DBR slab section of the microbeam, followed by a slower heat flow out of the microbridge into the supports. The temperature change vs. time at the loaded DBR is shown in Fig. A2b.
As a check to our use of the penetration factors to model the heating distribution, we have also performed simulations in which the heat load is distributed asymmetrically among all DBR silicon slabs (not just at the innermost DBR silicon slab), consistent with our optical cavity modeling in Fig. 2 that shows that 28% of the total cavity optical power, Pcav, flows in the slab closest to the cavity (β2), 8% is in the center silicon slab (β4), and 2% is in the silicon slab furthest from the cavity (β6). The results showed that the contribution of the center DBR slab can be ignored due to symmetry, while heating and thermal expansion in the outer most slab (β6) produces a force that counter acts the bending produced by the inner most slab (β2). We account for the opposing forces by including β2 and β6 in Eq. (2) for the photothermal force.
Appendix A3. Mechanical resonance modes and pressure dependence
We used Comsol Multiphysics simulations to determine the mechanical resonance modes of our device. The total beam length is L=400 μm (incl. DBR mirror section), height H=4.8 μm, and width W=1.8-2.12 μm. The first four in-plane mechanical resonances (M=0, 2, 3, 5) are shown in Fig. A3a along with their corresponding resonance frequencies. Out-of-plane modes (M=1, 4) also exist, but these cannot be measured using our in-plane Fabry-Perot cavity. Of the in-plane modes, only M=0, 3 result in a DBR displacement that tunes the Fabry-Perot cavity. Modes M=2, 5 rotate the DBR mirror and are difficult to measure.
We initially performed experiments at atmospheric pressure and room temperature (Fig. A3b). The results show two mechanical resonances corresponding to the M=0, 3 modes – these modes give a linear displacement of the DBR mirror. We also performed the same measurement at <50 mTorr pressure. As expected, the resonances narrow with a strong increase in amplitude due to decreased air damping. We performed more detailed pressure measurements on the M=0 resonance (f0=101 kHz) to ascertain the damping mechanism. The results indicate an inverse-linear relationship between pressure and mechanical Q-factor (QM) over the range 20 mTorr – 2000 mTorr (Fig. A3b, inset). This suggests that we are operating in a damping regime dominated by viscous flow  and/or squeeze-film damping [37,38], where QM~gap/pressure. Both viscous and squeeze-film damping can be minimized by increasing the air gap. Our devices are made of single-crystal-silicon, which is an excellent mechanical material with low intrinsic material loss. For this reason, we have not measured a QM saturation at low pressure (20 mTorr, Fig. A3b inset). This is important for sensing applications, where a large QM is critical for maximizing sensing limits of detection.
Appendix A4. Optical and mechanical properties of device 2 (Fig. 6d)
We measured the threshold power for onset of coherent oscillations in a second device (Fig. 6d) with similar microbridge parameters as the first device (Fig. 5ab and Fig. 6abc). The optical design, however, is modified slightly to give a significantly larger finesse with slightly lower cavity transmission. We keep the cavity length at LC=3 μm, but modify the DBR so that the designed (nominal) silicon slab width is dSi=0.644 μm and dAir=0.300 μm. The previous device (Fig. 5ab and 6abc) had designed silicon slab width dSi=0.694 μm and dAir=0.250 μm (actual dimensions obtained via transfer matrix calculations were dSi=0.651 μm and dAir=0.293 μm as in Table A1). The measured Fabry-Perot spectrum in Fig. A4 shows a resonance peak at λ0=1600.78 nm with finesse f=380, a significant increase compared to the first device. The fundamental mechanical resonance (M=0 mode) is centered at ν0≈101,560 Hz (Fig. 6d), which compares well with the resonance of the first device, where ν0≈101,150 Hz (Fig. 6c).
In this second device we demonstrated the ability to change our optical cavity without affecting the mechanical parameters substantially. We have also shown via simple design equations how we can modify the microbridge design without affecting the cavity. The other parameters that can be modified independently are the force time constants, τopt(N). The radiation pressure time constant, τrp, can be modified via the optical cavity photon lifetime. The photothermal time constant, in particular τpt-fast, can be tailored by simply varying the length of the DBR silicon slab. Currently, the length is set (arbitrarily) to 26 μm. This length can be reduced substantially to decrease the photothermal time constant. Indeed, we have previously shown that the thermal time constant has a length dependence τ~length2 . This enables us to tailor the photothermal time constant to achieve maximum opto-mechanical interaction and thereby develop efficient sensors.
Appendix A5. Sensing applications
In many applications, the resolution of a sensor is limited by the frequency noise or Allan variance [26, 27] of a mechanical resonator. The Allan variance for a microbridge resonator is , where kB is Boltzman’s constant, T is the temperature, meff≈6.8×10−12 kg is the effective mass, ω≈2π×101,100 rad/s is the mechanical resonant frequency, Δz is the displacement, QM≈20,000 (for Popt=74 μW) is the mechanical quality factor, N=200 is the number of points for our measurement, and BW=1 Hz is the measurement bandwidth for our electronic spectrum analyzer. The two parameters primarily affecting the Allan variance are Δz and QM.
To obtain the frequency noise, we first find the displacement Δz at Popt=74 μW (Fig. 6a, 6b, and 6c). Using the approach in , the measured signal amplitude can be related to the slope of the Fabry-Perot resonance (slope=0.20 V/nm, where nm is the wavelength). Based on our cavity length LC=3 μm, the mode number is j=12. The measured signal amplitude is 3×10−9 V2/Hz (Popt=74 μW in Fig. 6c). We thus obtain a displacement ΔzRMS≈0.059 nm. Since Δz scales with our measured signal, we can obtain the displacement at Popt=450 μW (above threshold) to be ΔzRMS≈3.8 nm (Fig. 6c) and at Popt=590 μW to be ΔzRMS≈4.0 nm (Fig. 4c). The frequency noise is then found as σAω= 2π×(2.5×10−2) Hz (below threshold, Popt=74 μW) and σAω=2π×(3.9×10−4) Hz (above threshold, Popt=450 μW). These calculations assume QM≈2×104; however, beyond threshold Γ →0 and QM→∞ so that the frequency noise will actually be much smaller for the above threshold case. We also note that experiments using a digital sampling oscilloscope with 0.25 Hz resolution give a measured Γ~0.2 Hz (instrument limited) beyond threshold, implying a QM≈5×105. We expect the true post-threshold mechanical Q-factor to exceed 1×106.
We can find the minimum detectable mass change (e.g. due to absorption of analytes in a chemical sensor ) by setting Δω =ωσA and using the relation  to obtain Δm=2.1×10−17 kg for the pre-threshold case (Popt=74 μW), assuming QM≈2×104. Beyond threshold (Popt=450 μW), we obtain Δm=3.3×10−19 kg, again assuming QM≈2×104. However, we emphasize that beyond threshold QM>>2×104, since we are instrument limited in our measurement of the mechanical resonance linewidth Γ; using QM~1×106 we obtain σAω=2π×(5.5×10−5) Hz. By decreasing our sensor’s mass by a factor of ten (meff) and increasing the resonant frequency by a factor of ten (ω), a mass-loading resolution in the sub-attogram range (Δm<1×10−21 kg) appears possible. We note that both meff and ω can be readily changed using the design equations given in the main text without affecting the optical design. Increasing the cavity finesse will change the Fabry-Perot slope, which also enhances the mass-loading resolution.
Magnetic field and inertial sensors can be operated in vacuum with large QM. In practice, however, chemical sensors are operated at ambient pressures, where QM is much smaller. Even so, by tailoring the microbridge resonator to be very compliant (for example, by lengthening it) and hence decreasing Keff, it should be possible to reach threshold and drive the microbridge resonator into coherent mechanical oscillations at atmospheric pressure. This will enable chemical sensors with extremely high resolution.
MWP thanks the NRL Nanoscience Institute (NSI) staff for device fabrication assistance. The authors thank L.D. Epp for designing and machining various components. MWP, THS and WSR acknowledge the support of the U.S. Office of Naval Research (ONR). JBK acknowledges the support of ASEE and ONR through the Summer Faculty Research Program.
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