1. Introduction

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 [2] enables mechanical amplification/damping via wavelength tuning across the cavity mode [7].

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) [20]. 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.

 

Fig. 1 Integrated waveguide-DBR microcavity opto-mechanical system. (a) Schematic showing optical waveguide, Fabry-Perot cavity consisting of one fixed and a second movable distributed Bragg reflector (DBR), a suspended micro-mechanical bridge resonator with spring constant K_Spring, air gaps surrounding the movable DBR, and direction of optical forces (F-pt: photothermal pressure, F-rp: radiation pressure); the in-plane vibration of the movable DBR mirror modulates the Fabry-Perot cavity transmittance (out-of-plane beam motion can generally not be read out). (b) Fabricated silicon-on-insulator coupled opto-mechanical cavity with waveguide Fabry-Perot (red) and silicon microbridge mechanical resonator (green).

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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 L_C = 3 μm. The DBR mirrors consist of three silicon slabs of length L = 25 μm, height H = 4.8 μm, and width d_Si≈5λ/4n_Si. The silicon slabs are separated by air gaps of width d_air ≈λ/4. We modeled our microcavity using a one-dimensional transfer matrix [21]. 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 (P_opt) 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, β_i, 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 (P_DBR-N) to that in the cavity (P_Cav) 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 “P_Cav” 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 [7].

 

Fig. 2 Transfer matrix calculation of cavity optical field. (a) Cross-section of DBR mirrors and optical cavity showing calculated electric field squared profile on-resonance. (b) Calculated power flow, where P_opt. is the incident power on the input DBR, P_cav. is the cavity power, and P_DBR. is the power in the Nth- silicon slab of the DBR. The β_N-factors indicate the power in each DBR region relative to P_cav. The absorbed power in each silicon region N is found from the difference between the power flow into region N and the flow out of that region, summed for both forward and backward power flows.

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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:

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. P_DBR-2 = β₂P_Cav) 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:

2.3 Optical spring

An optical force changes a resonator’s spring constant by K_opt. = ∂F_opt./∂z. The net optical force is F_opt = F₀χ_enhL(ϕ), where F₀ is the force without a cavity, ${\chi }_{enh}=f\sqrt{T_{cav}}/\pi$is the cavity optical power enhancement on resonance, f is the finesse, T_Cav. is the net power transmitted through the cavity on-resonance (and √T_Cav 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. 3 Optical manipulation of a micro-mechanical spring. (a) Finite-element method simulation of radiation and photothermal pressure and resulting beam bending; the simulated images clearly show the opposing beam bending. (b) Calculated optical force modification of effective micro-mechanical spring constant (normalized by K_eff) as a function of laser detuning for P_opt = 370 μW. Radiation pressure and photothermal forces have opposite effect on K_opt. The solid lines in (b) are for the Fabry-Perot cavity only; the dashed lines include the effect of spurious cavities formed between the high reflectivity DBR mirrors and the cleaved waveguide facets at the sample input/output (see Fig. 4b inset).

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The resonant frequency (ν) and damping (Γ) in the presence of an optical force are [22]:

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 [20]. 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 (Q_M) 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 [23] using existing packaging technology.

 

Fig. 4 (a) Experimental setup. (b) Measured optical spectrum; inset: detail of resonance near λ₀ = 1593.75 nm and Lorentzian fit; the high frequency oscillations (0.2-0.3 nm spacing) are due to reflections from the cleaved end-facets of the input/output waveguides. (c) Measured mechanical resonance for the fundamental in-plane resonance mode (M = 0) for red and blue detuned laser (Δλ =+0.38 nm [λ = 1594.13 nm] and Δλ=−0.34 nm [1593.41 nm], respectively, both at P_opt = 590 μW with effective Q_M ranging from 4.3×10³ to >1×10⁵; insets: detail of mechanical resonance for red (ν=101,139 Hz) and blue detuning (ν = 101,176 Hz) with Lorentzian fit. The displacement amplitude is Δz_RMS≈4.0 nm in (c) (red curve, Δλ=+0.38 nm).

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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.

 

Fig. 5 Optical tuning of resonant frequency and damping. (a) Measured resonant frequency shift (ν-ν₀) vs. detuning and theoretical shift. (b) Measured mechanical resonance width (Γ) vs. cavity detuning and theoretical linewidth. All measurements were performed at P_opt = 370 μW. Frequency tuning is dominated by radiation pressure, while the linewidth Γ is determined entirely by photothermal forces. The vertical lines at Δλ = −0.34 nm and + 0.38 nm (Δλ = −0.33 nm and + 0.34 nm) indicate the detuning for power dependent measurements (theory) in Fig. 6a, 6b, and 6c. The model includes the effect of the sample input/output waveguide facets. The scattering in Fig. 5b for Δλ<0 results from the reduced oscillation amplitude due to opto-mechanical damping for blue-detuned wavelengths. The damped oscillations result in a lower signal-to-noise ratio and noisier Γ data for Δλ<0.

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Fig. 6 Laser power dependent tuning of micro-mechanical resonators (M = 0). (a) Measured and theoretical resonance shift vs. P_opt for red- (Δλ=+0.38 nm) and blue- (Δλ = −0.34 nm) detuned laser. (b) Measured and theoretical mechanical resonance linewidth (Γ) for red- and blue-detuning; the vertical lines in (a,b) correspond to the power used in the detuning measurements in Fig. 5a and 5b. The shaded area indicates the measurement 1 Hz resolution limit of the electronic spectrum analyzer. (c) Extracted vibration power for red-detuning showing clear onset of coherent oscillations and sharp increase in amplitude as Γ goes to zero; inset: resonance spectra showing linewidth narrowing for increasing laser power. (d) Vibration power for device 2 with a similar mechanical design, but a higher finesse resulting in lower P_th; inset: resonance spectra at powers below and above threshold. For (c,d) the lines are linear fits to the pre-threshold and post-threshold data, respectively. The resonance frequency for device 1 in (c) is ν₀≈101,150 Hz; for device 2 in (d) it is ν₀≈101,560 Hz.

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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 Q_M. 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 τ_pt-fast>>τ_rp.

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 (τ_pt.slow/fast>>τ_rp) 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 [9], 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 P_opt 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 P_opt 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 P_opt 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 (P_th≈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 [24].

Our modeling suggests a 1/Q_M dependence for P_th, similar to previous work [11]. We further expect a P_th~1/f² dependence [24]. To test our model, we measured a second device with similar mechanical parameters (Q_M=2.5×10⁴) but with a slightly modified DBR and a large increase in finesse (f = 380, Appendix Fig. A4). We observe P_th=32 μW (Fig. 6d), a reduction by 10 compared to the previous device (Fig. 6c, where Q_M=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.

4. Discussion

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 Q_opt (>10⁸) microtoroid resulted in self-oscillation for 20 μW power [11]. Using m_eff=3.3×10⁻⁸ kg and ν=4.4 MHz [11], we extract a spring constant K_{eff, microtoroid} = 2.5×10⁷ N/m. Our cavity is modest in comparison (Q_opt=5×10³). Nonetheless, we can observe coherent mechanical oscillations with P_th=32 μW (Fig. 6d), which is among the lowest threshold powers experimentally observed. Therefore, while our Q_opt is much smaller than previous work [11], self-oscillation is possible at low optical power since our mechanical structure is significantly more compliant (K_eff=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~w_bridge/L_bridge² and K_eff~t_bridge(w_bridge³/L_bridge³) [25]. By scaling t_bridge, w_bridge and L_bridge, we can further reduce K_eff and P_th without affecting ν₀ or changing the cavity design. Conversely, ν₀ can be tailored while leaving K_eff 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 P_th (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 [28]. The device required electrostatic actuation and exhibited a 4.6 picogram mass-loading resolution, limited by the frequency noise (Allan variance) [26,27] σ_A~1/(ΔzQ_M^1/2). Our coherent mechanical oscillator exhibits a sharp increase in Δz and Q_M 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 Δz_RMS≈0.059 nm at P_opt = 74 μW and increases sharply beyond threshold with Δz_RMS≈4.0 nm at P_opt=450 μW implying a 68-fold reduction in the Allan variance based on Δz alone. The calculated frequency noise at P_opt=450 μW is σ_Aω≈2π×(3.9×10⁻⁴ Hz) which will enable sub-attogram mass-loading resolution with only slight changes in the device design (Appendix A5).

5. Conclusion

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 [28]. Besides mass-loading [29], our device may be used for high resolution inertial sensing, where the acceleration of a proof-mass attached to the microbridge is measured via $\overrightarrow{a}={\omega }_0{}^2\overrightarrow{z}$[30]. Magnetic field sensors can be realized by coating the microbridge with a ferromagnetic film [31] 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

Table A1. Transfer matrix input parameters.

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Table A2. Transfer matrix output parameters.

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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 K_eff (the mechanical response to a steady-state distributed force), K_p (the mechanical response to a localized force at the DBR), and dz/dP_abs (the steady-state displacement of the DBR microbridge in response to a heat load at the waveguide). The parameter dz/dP_abs 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, τ_pt-fast and τ_pt-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, P_cav, 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.

Table A3. Input material parameters for Comsol [35]

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Table A4. Output values from Comsol thermal-mechanical model.

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Fig. A2 (a) Illustration of localized DBR heating due to absorption. The asymmetric heat load implies a thermal expansion such that the beam bends in a direction that shortens the cavity length. (b) Calculated temperature change at the heated DBR vs. time following a 1 μW heat impulse at t = 0. The DBR moves towards the optical cavity in response to this heat load.

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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 (Q_M) over the range 20 mTorr – 2000 mTorr (Fig. A3b, inset). This suggests that we are operating in a damping regime dominated by viscous flow [36] and/or squeeze-film damping [37,38], where Q_M~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 Q_M saturation at low pressure (20 mTorr, Fig. A3b inset). This is important for sensing applications, where a large Q_M is critical for maximizing sensing limits of detection.

 

Fig. A3 Simulation and measurement of mechanical resonance modes. (a) Finite-element-method simulation of first four in-plane mechanical resonances; modes M = 1 (254.0 kHz) and M = 4 (718.0 kHz) are out-of-plane resonances (not shown). (b) Measured mechanical resonance modes M = 0 (101 kHz) and M = 3 (605 kHz) under vacuum (P<50 mTorr) and ambient conditions (atmosphere); inset: measured pressure dependent mechanical Q-factor for the M = 0 mode showing an inverse pressure relationship (Q_Mechanical = 1/pressure^0.96) over the range 20 mTorr – 2 Torr. The wavelength was λ = 1593.350 nm in (b) and λ = 1593.845 nm (inset).

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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 L_C=3 μm, but modify the DBR so that the designed (nominal) silicon slab width is d_Si=0.644 μm and d_Air=0.300 μm. The previous device (Fig. 5ab and 6abc) had designed silicon slab width d_Si=0.694 μm and d_Air=0.250 μm (actual dimensions obtained via transfer matrix calculations were d_Si=0.651 μm and d_Air=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, τ_opt(N). The radiation pressure time constant, τ_rp, can be modified via the optical cavity photon lifetime. The photothermal time constant, in particular τ_pt-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 [39]. This enables us to tailor the photothermal time constant to achieve maximum opto-mechanical interaction and thereby develop efficient sensors.

 

Fig. A4 (a) Measured Fabry-Perot optical spectrum for device 2 in Fig. 6d. The inset shows a detail of the resonance peak at λ₀ = 1600.78 nm with Lorentzian fit to obtain δλ = 0.32 nm, FSR = 120 nm, Q_opt = 5,000 and finesse f = 380. (b) Measured linewidth narrowing and collapse for device 2. The shaded area indicates the 1 Hz measurement resolution of our electronic spectrum analyzer, implying that measurements below 1 Hz can be interpreted as Γ → 0.

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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 ${\sigma }_A=3.8\sqrt{k_{B}T/\left(m_{eff}{\omega }^3\Delta z^2{Q}_{M}N/BW\right)}$ [40], where k_B is Boltzman’s constant, T is the temperature, m_eff≈6.8×10⁻¹² kg is the effective mass, ω≈2π×101,100 rad/s is the mechanical resonant frequency, Δz is the displacement, Q_M≈20,000 (for P_opt=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 Q_M.

To obtain the frequency noise, we first find the displacement Δz at P_opt=74 μW (Fig. 6a, 6b, and 6c). Using the approach in [28], 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 L_C=3 μm, the mode number is j=12. The measured signal amplitude is 3×10⁻⁹ V²/Hz (P_opt=74 μW in Fig. 6c). We thus obtain a displacement Δz_RMS≈0.059 nm. Since Δz scales with our measured signal, we can obtain the displacement at P_opt=450 μW (above threshold) to be Δz_RMS≈3.8 nm (Fig. 6c) and at P_opt=590 μW to be Δz_RMS≈4.0 nm (Fig. 4c). The frequency noise is then found as σ_Aω= 2π×(2.5×10⁻²) Hz (below threshold, P_opt=74 μW) and σ_Aω=2π×(3.9×10⁻⁴) Hz (above threshold, P_opt=450 μW). These calculations assume Q_M≈2×10⁴; however, beyond threshold Γ →0 and Q_M→∞ 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 Q_M≈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 [28]) by setting Δω =ωσ_A and using the relation $\Delta \omega /\omega =\Delta m/2m_{eff}$ [26] to obtain Δm=2.1×10⁻¹⁷ kg for the pre-threshold case (P_opt=74 μW), assuming Q_M≈2×10⁴. Beyond threshold (P_opt=450 μW), we obtain Δm=3.3×10⁻¹⁹ kg, again assuming Q_M≈2×10⁴. However, we emphasize that beyond threshold Q_M>>2×10⁴, since we are instrument limited in our measurement of the mechanical resonance linewidth Γ; using Q_M~1×10⁶ we obtain σ_Aω=2π×(5.5×10⁻⁵) Hz. By decreasing our sensor’s mass by a factor of ten (m_eff) 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 m_eff 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 Q_M. In practice, however, chemical sensors are operated at ambient pressures, where Q_M is much smaller. Even so, by tailoring the microbridge resonator to be very compliant (for example, by lengthening it) and hence decreasing K_eff, 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.

Acknowledgments

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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