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Abstract
We describe curved-mirror Fabry-Perot cavities with embedded silicon nitride membranes, fabricated using a monolithic surface-micromachining process. The presence of the suspended membranes was confirmed by confocal microscopy, and their properties were verified through optical studies and thermomechanical calibration of mechanical/vibrational noise spectra measured at room temperature and atmospheric pressure. The cavities exhibit reflectance-limited finesse (F ∼ 103) and wavelength-scale mode volumes (VM ∼ 10·λ3). The short cavity length (L ∼ 2·λ) results in large optomechanical coupling, which is desirable for numerous applications in sensing and quantum information.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction and background
Cavity optomechanical systems [1] combine a mechanical resonator with an optical cavity, and are at the forefront of recent efforts to observe and exploit quantum behaviors of mesoscopic or macroscopic objects [2]. One motivating factor is the unique capability of mechanical oscillators to mediate the exchange of coherent quantum states amongst diverse physical entities [1,3]. For many proposed schemes, it is necessary that the mechanical element is first cooled to its ground state of vibrational motion – i.e. that nearly all of its thermal vibrational energy is removed, so that quantum behavior dominates [4]. Near-ground-state cooling has in fact already been achieved [5,6] in cavity optomechanics, by combining cryogenic pre-cooling with radiation-pressure-based ‘back-action’ cooling. However, there is great interest in reducing the cost and complexity of such systems, for example by removing the need for cryogenics. Operation at room temperature, as well as increased integration of optical, mechanical, and electrical elements, will enhance the scope for practical applications.
‘Membrane-in-the-middle’ (MIM) systems [7,8] have emerged as powerful enablers, in large part because they allow for the optical and mechanical resonators to be independently optimized. The MIM concept has spawned a vast and evolving research field, notably driven by their demonstrated potential for quantum-enabled sensing [9,10] and quantum transduction [3,11]. Nearly all reported MIM systems use silicon nitride (SiN) membranes, owing to their low optical and mechanical losses, with mechanical Q-factors > 106 routinely achieved for uniform ‘drumhead’ membranes [7]. Moreover, stress-engineering of tether-attached membranes (‘trampolines’) has enabled Q-factors as high as ∼108 at room temperature [9,12], and patterning of membranes with phononic crystals has enabled Q-factors > 108 at room temperature [13] and > 109 at T = 10 K [14]. With these advances, SiN-based MIM systems are a leading candidate to observe and exploit quantum phenomena at ambient temperatures [8,9,12,13].
All MIM cavity optomechanical systems to date have employed hybrid assembly, with a separately fabricated membrane chip inserted into the space between two mirrors. The individual mirrors are also fabricated separately, each on a substrate or fiber end facet. This approach offers considerable flexibility for cavity tuning and alignment, but requires a relatively complex and costly positioning and control setup, not easily scaled to large numbers of cavities. Moreover, it places practical limits on the minimum achievable cavity size (i.e., optical mode waist and mode volume). For example, a fiber-based MIM cavity ∼ 80 μm in length was reported [15], and hybrid integration of the membrane onto one of the cavity mirrors [16] might conceivably enable shorter cavities. However, to our knowledge the smallest analogous system reported to date employed a nanowire as the mechanical element [17]. Thus, MIM systems, in spite of their myriad attractive properties, to date lack the potential for extreme scaling (i.e., to large numbers) and size reduction offered by on-chip micro-toroid [2] and photonic crystal [6] cavity optomechanical devices.
Here, we describe MIM cavities fabricated using a surface-micromachining approach, which produces patterned SiN membranes suspended within curved-mirror, on-chip Fabry-Perot cavities. Theoretically predicted cavity properties were confirmed by optical measurements, and a thermomechanical calibration procedure was used to verify large optomechanical coupling between the vibrating membrane and the cavity modes. Considering feasible improvements in cavity finesse and membrane properties, we assess the potential for these integrated MIM devices to address important problems in quantum information and sensing. We also discuss challenges associated with miniaturization.
2. Fabrication and morphological characterization
We previously reported a ‘buckling self-assembly’ process for fabricating arrays of high-finesse, curved-mirror microcavities [18,19]. Here, we adapted these processes to embed free-standing silicon nitride trampoline resonators inside the buckled cavities as depicted in Fig. 1(a). First, a 3-period a-Si/SiO2-based Bragg mirror, centered at 1550 nm wavelength, was deposited by magnetron sputtering onto a double-side-polished silicon wafer. This mirror was followed by a nominally 30 nm thick SiO2 layer, which functions as an etch blocking layer, and a nominally 200 nm thick a-Si layer, which functions as the sacrificial etch layer. On top of the sacrificial layer, a nominally 150 nm thick, stoichiometric Si3N4 layer was deposited in a PECVD chamber with the substrate temperature fixed at 325 °C. The as-deposited Si3N4 layer has compressive stress of ∼ 200 MPa, but attains a tensile stress of ∼ 770 MPa after a 120 minute anneal at 600 °C in a nitrogen atmosphere.
Fig. 1. (a) Schematic cross-sectional view (not to scale) of a buckled dome microcavity with an embedded free-standing Si3N4 membrane. For the devices discussed here, the sacrificial layer is ∼ 200 nm thick, and sets the spacing between the membrane and the bottom mirror. (b) Microscope image of a completed membrane-in-cavity device. The green ‘flower’ shape is the suspended membrane, and the circular interference fringes arise from the buckled profile of the upper mirror. As is evident, many of these first-generation devices suffered from imperfect alignment between the etch hole pattern (thus the suspended membrane) and the buckled dome microcavity. The small blue dots inside the membrane area were visible on approximately half of the fabricated devices, and are believed to be pinholes in the thin etch blocking layer, which allow the XeF2 to locally attack the underlying mirror. (c) Microscope image of a suspended SiN membrane fabricated independently of the optical cavity process. (d) Microscope image of a purposely detached membrane fabricated independently of the optical cavity process, showing that the XeF2-based sacrificial etching of a-Si leaves the etch-stop layer surface (SiO2, green in the image) clean and smooth. Note, however, the ‘scalloping’ faintly visible on the central region of the membranes in parts c. and d. (see the main text).
On top of the nitride layer, a low adhesion ‘teflon-like’ layer was deposited and patterned by lift-off, followed by sputtering deposition of a 4.5-period Ta2O5/SiO2 Bragg mirror with ∼300 MPa compressive stress. As described elsewhere [19], a subsequent heating step was used to promote loss of adhesion over the ‘teflon’ features, resulting in buckling delamination of the compressively stressed upper mirror, and creating curved-mirror microcavities that support high-quality Laguerre-Gaussian modes. An ICPRIE process was then used to create vertical holes through the top sputtered layers and the membrane below [20]. These etched ‘access holes’ expose the sacrificial Si layer located below the silicon nitride membrane. A gas-phase XeF2 etch [21,22] was then used to selectively remove Si and release the silicon nitride membrane. Finally, 3 additional periods of Ta2O5/SiO2 were sputtered in order to increase the finesse and optical symmetry of the cavities. A microscope image of a typical completed device is shown in Fig. 1(b).
The XeF2 membrane release etch was chosen due to its high selectivity for etching a-Si compared to the top mirror (Ta2O5, SiO2) and membrane (Si3N4) materials. Furthermore, the dry (i.e., gas phase) nature of the XeF2 process avoids complications associated with stiction. This etching was performed in a custom-built system, using a pulsed two-chamber design similar to that described elsewhere [21]. The two chambers (an ‘etch’ chamber and an ‘expansion’ chamber) are first pumped down to a base pressure of 150 mTorr. The expansion chamber is then filled with gaseous XeF2, created through the sublimation of commercially sourced XeF2 crystals, for 1 minute. The gas from the filled expansion chamber is subsequently moved into the etch chamber containing the sample, where (for another 1 minute) it etches any exposed silicon surfaces within the wafer. The etch chamber is finally pumped down and the process is repeated. 10 cycles were performed for the devices presented here, with the required number varying depending on the access hole configuration, and desired membrane size.
The capability for the XeF2 etch to produce intact membranes was first characterized on samples containing only a 3-layer structure, as follows. First, a-Si ‘sacrificial’ and SiN ‘membrane’ layers were deposited as described above, on top of a thermally oxidized Si wafer. Next, ring-shaped arrays of holes were patterned and transferred into the SiN layer using ICPRIE, and the wafer was then exposed to 10 cycles of XeF2 to locally remove a-Si. A typical ‘drumhead’ resonator [23,24] produced by this process is shown in Fig. 1(c). For further insight, we used a profilometer tip to remove the suspended membrane from some devices, as shown for example in Fig. 1(d). It was observed that the XeF2 slightly etched the silicon nitride layer, as evidenced by the ‘scalloping’ pattern on the membrane visible in Figs. 1(c) and 1(d). The number of ‘steps’ in these patterns matches the number of cycles of XeF2 to which the device was subjected, consistent with a gradual migration of the etch front with each cycle, from the access holes outward. Such undesired etching might cause optical scattering loss and will likely need to be addressed in future work, perhaps through further tuning of the properties of the PECVD film [21]. In any case, membrane removal [Fig. 1(d)] revealed a smooth and uniform oxide surface, indicating that the process was capable of simultaneously producing a completely removed sacrificial layer and properly released membrane. The optical results discussed below further support this conclusion.
Due to the exploratory nature of the process, the mask set was designed to produce domes of varying base diameter (50-100 μm) and with etch-hole patterns of varying ‘ring’ diameters and numbers of holes. As shown in Fig. 1(b), imperfect alignment between the dome pattern mask and the etch-hole pattern mask resulted in the majority of devices showing some misalignment between the optical cavity and the embedded membrane. This was attributable to practical issues associated with the mask aligner system employed, and should be easily addressed in future work. A large number of well-aligned devices were nevertheless identified and characterized, and we focus most of the remaining discussion on the representative ‘case-study’ device shown in Fig. 2(a). This device combines a dome of base diameter ∼ 74 μm and peak (buckled mirror) height ∼3.2 μm with an etch array pattern comprising 5 equally spaced holes (each 4 μm in diameter on the mask) placed on a ‘ring’ of diameter ∼ 30 μm. Due to the isotropic nature of the XeF2 etch, this hole pattern resulted in a ‘flower-shaped’ suspended membrane with 5-fold symmetry. A confocal microscope was used to characterize many devices on the wafer, and this study revealed consistently intact membranes at the expected depth [see for example Figs. 2(b) and 2(c)]. Note that the membranes lie ∼ 3.3 μm below the planar surface (outside the buckled regions) of the wafer, corresponding to the thickness of the 7.5-period Ta2O5/SiO2 mirror [see Fig. 1(a)]. The confocal microscope was also used to measure the profile of the upper (buckled) mirror for many domes, and these profiles were confirmed to be consistent with elastic buckling theory. Specifically, for a circular delamination buckle: Δ(r) = δ·[0.2871 + 0.7129·J0(μ·r)] [25], where Δ is the vertical displacement, r is the radial coordinate normalized to the base radius (a) of the dome, δ is the peak height, J0 is the Bessel function of the first kind and order zero, and μ = 3.8317. As an example, Fig. 2(d) shows the measured profile for the ‘case study’ dome compared to the predicted profile with δ = 3.2 μm and a = 37 μm. The excellent agreement indicates that neither the membrane nor the etch holes had a significant impact on the mirror shape, consistent with our previous results [20].
Fig. 2. Images of the ‘case-study’ MIM cavity described in the main text. (a) Standard microscope image; the concentric interference fringes arise from the profile of the buckled upper mirror and the ‘flower-shaped’ region in the center is the sacrificial etched cavity and suspended membrane. (b) Confocal microscope image with the focus set at the top surface of the upper mirror (outside the buckled regions). (c) Confocal image with the focus set ∼ 3.3 μm below the top surface, where the membrane layer is located. (d) Profile of the buckled dome (upper mirror of the cavity) as determined using the confocal microscope (blue symbols), and fit to the profile predicted for a circular delamination buckle (red solid line).
3. Optical properties
3.1 Predictions
As we have shown previously [18,19], the small height-to-width aspect ratio of our buckled dome cavities implies that their basic optical properties can be well-approximated using planar models and (initially) neglecting transverse confinement effects. Figure 3 shows transfer-matrix predictions for a planar-equivalent model of the ‘case-study’ dome from Fig. 2. Here, a 150 nm thick membrane is assumed to be separated from the bottom and top mirrors by 200 nm and 3.2 μm thick air gaps, respectively. The SiN membrane was assigned a real refractive index of 2, and the other materials were modeled using dispersion relations described previously [19]. As shown in Fig. 3(a), (a) single resonant mode with FWHM linewidth ∼0.29 nm (Q ∼ 5000) and centered at ∼1550 nm is predicted within the stop-bands of the mirrors.
Fig. 3. Transfer-matrix predictions for the planar-equivalent model of the cavity shown in Fig. 2. (a) Transmittance spectrum showing a resonant mode at ∼ 1550 nm with linewidth ∼0.29 nm (Q ∼ 5000). (b) Field intensity (E·E*) profile for the resonant mode from part a. The SiN membrane essentially acts as an additional layer in the bottom mirror, and is roughly centered on a field anti-node. (c) Variation in resonant wavelength with change in membrane position. (d) Change in resonant frequency versus membrane displacement. The predicted linear optomechanical coupling strength is G/(2π) ∼ 25 GHz/nm.
Note that in the ‘MIM’ devices described here, the membrane is actually not near the middle of the cavity. As shown in Fig. 3(b), the resonant mode can be viewed as having longitudinal order of 4 or 5, depending on whether the SiN membrane (and underlying etched air gap) is viewed as part of the cavity or as part of the bottom mirror. For example, if one views it as part of the cavity, then the effective cavity length is L ∼ 5·λ ∼ 3.9 μm, which is the sum of the air gaps, the optical thickness of the SiN membrane (∼0.3 μm), and the penetration depths into the upper and lower Bragg mirrors.
A fundamental figure of merit for cavity optomechanical systems is the ‘frequency pull parameter’, or optical resonant frequency shift per mechanical displacement, G = dω/dz. In the MIM cavity, the mirrors are generally assumed to be rigidly fixed, and the frequency shift is related to displacements (i.e. along the cavity axis) of the membrane. For conventional MIM systems, with a membrane assumed to be much thinner than the overall cavity length and positioned near the middle of the cavity, analytical expressions have been derived and predict a sinusoidal variation of the cavity resonance frequency with membrane position [7]. From these expressions, a maximum frequency pull parameter (for optimal membrane placement mid-way between field nodal and anti-nodal positions) can be predicted as Gmax = 2·|rm|·ω/L. Here, rm is the field amplitude reflection coefficient of the suspended membrane, for which analytical expressions are also available [26]. For example, for our case-study device above, |rm| ∼ 0.57 and Gmax/2π ∼ 55 GHz/nm.
Our structure deviates significantly from these assumptions, in that the membrane is not centrally placed and its thickness is not vanishingly small compared to the cavity length. In fact, it more closely resembles the “membrane-at-the-end” (MATE) system [16], for which Gmax can actually exceed the previous estimate. In any case, a more exact treatment is afforded by direct transfer-matrix solutions obtained for varying membrane position. As shown in Fig. 3(b), the ‘resting’ position of the membrane is near a field anti-nodal position. Thus, we can expect the pull parameter to be lower than the optimal value above, as verified by the plots shown in Figs. 3(c) and 3(d), which predict G/(2π) ∼ 25 GHz/nm or Gλ = dλ/dz ∼ 0.185 for the case-study device.
A more fundamental figure of merit is the single-photon optomechanical coupling strength [1], g0 = G·xZPF, where xZPF = (ħ/2·meff·Ωm)1/2 is the zero-point-fluctuation amplitude of the mechanical oscillator, related to its effective mass meff and mechanical resonance frequency Ωm. Using the values meff ∼ 0.1 ng and Ωm/2π ∼ 10 MHz discussed below (i.e. for the fundamental vibrational mode) gives xZPF ∼ 2.5 fm and g0/2π ∼ 0.1 MHz for our case study device. This g0 value is already significantly higher than typically reported for MIM systems [1,7,14], and could likely be increased in future process iterations as discussed further below.
3.2 Measurements
Optical measurements were obtained using a custom microscope setup. Light from a 1550 nm-range tunable laser (Santec TSL-710) was coupled into the bottom of the samples (i.e. through the silicon substrate) with the use of a parabolic mirror collimator and a standard microscope objective to focus light onto individual dome cavities. The device substrate was placed on a microscope stage, and transmitted light was collected from the top with a 50x infinity-corrected objective lens. Output light was imaged using an infrared camera (Raptor).
Spectral scans were obtained by focusing the optical mode onto the camera, and summing the intensity of all pixels within the region of the modes while taking care to avoid saturation. Comparative measurements were made between the ‘case study’ device from Fig. 2 and an un-etched ‘control’ dome of similar size (peak height ∼ 3.0 μm), with results shown in Fig. 4. Note that the low-amplitude, periodic ripple in both scans is due to interference effects between the surfaces of the silicon substrate, and could be eliminated in future work by the addition of an anti-reflection coating on the bottom surface. Aside from this, the scans are consistent with theoretical predictions for a half-symmetric Fabry-Perot cavity [18,19]. For example, the upper mirror has effective radius of curvature RC0 ∼ 100 μm here [see Fig. 2(d)], suggesting a fundamental mode spot size radius w0 ∼ (λ/π)1/2·(L·RC0)1/4 ∼ 3.1 μm and transverse mode spacing ΔλT ∼ λ3/(2·π2·w02) ∼ 20 nm, both of which are in good agreement with the measured results. Higher-order transverse modes exhibit non-degeneracy (multiple sub-peaks), especially for the etched dome, which can be attributed to slight deviations from spherical symmetry for the buckled mirror. Consistent with this, while the control dome exhibited clear Laguerre-Gaussian (LG) modes [see Fig. 4(a) insets], the ‘case study’ dome tended to exhibit Hermite-Gaussian (HG) modes.
Fig. 4. (a) Spectral transmission scan for a dome without etch holes, thus with no released membrane. The insets show mode-field intensity profiles imaged at the resonant wavelengths indicated. (b) Spectral transmission scan for the case study dome from Fig. 2. The insets show selected mode-field intensity profiles, as in part a.
For the case study dome, the measured linewidth of the fundamental TE00 resonance is ∼ 0.3 nm (Q ∼ 5000), in excellent agreement with the transfer-matrix predictions above. Similar results were obtained on a large number of other devices. Note that this corresponds to a finesse F ∼ 103, comparable to values we have reported in earlier work [19]. Since we neglected membrane-induced scattering/absorption losses in the transfer matrix treatment, we can conclude that this finesse is limited mainly by the low period count of the mirrors (i.e. reflectance-limited finesse). This lends further evidence to the good optical quality of the surfaces left behind by the XeF2 sacrificial etch, consistent with the microscope images in Section 2. We anticipate that significantly increased finesse should be possible with improved Bragg mirrors.
4. Thermomechanical calibration
As discussed above, confocal microscope studies were used to identify a large number of cavities with intact, free-standing membranes. Devices with buckle height suitable to exhibit optical resonances in the near-1550 nm wavelength region were then selected for further study. They were optically coupled using the setup described in Section 3, but with a portion of the transmitted light tapped off into a fiber-coupled, high-speed photodetector (Resolved Instruments DPD80). The position of the fundamental LG00 resonance was first identified by scanning the laser wavelength, and then the laser wavelength was fixed at a wavelength slightly red-detuned from this resonance. This so-called ‘tuned-to-slope’ technique [27] imprints the thermal Brownian motion of vibrating cavity elements onto a time-varying optical power signal received by the photodetector [2,7]. Signals were captured and digitally processed (FFT) to obtain a power spectral density (PSD) plot (W2/Hz) in the 0 to 40 MHz range. Due to the relatively low cavity Q, it was not necessary to lock the laser wavelength to the cavity resonance here. Rather, the laser wavelength was manually adjusted to yield maximum-amplitude features in the FFT spectra. The extracted data was found to be stable and repeatable, with multiple measurements performed over the course of many days producing similar results.
Figure 5 shows representative spectra obtained for both the reference and case study domes discussed in Section 3. As detailed elsewhere [18], buckled cavities without a suspended membrane inside already exhibit thermal vibrational features due to the movement of the top mirror. The fundamental resonance at ∼ 7 MHz in Fig. 5(a) is in very good agreement with theoretical predictions for a dome of this size [18]. A weaker peak associated with a higher-order mirror resonance is also visible, at ∼ 21 MHz. Very similar spectra were observed for a large number of membrane-free domes. As shown in Fig. 5(b), dramatically different vibrational spectra were observed for cavities containing suspended SiN membranes. For the case study dome, and many other similar devices, the vibrational features associated with the buckled mirror (e.g., at ∼ 7 MHz) are still present as expected, but are superimposed with several other resonant lines that we attribute to the thermal vibrational motion of the SiN membrane. For example, if the flower-shaped membrane is treated very approximately as a circular membrane of diameter ∼ 40 μm (and the etch holes are ignored), well-known analytical expressions [27] predict a fundamental vibrational frequency ∼ 10 MHz given the tensile stress (∼770 MPa) mentioned above. Thus, we postulated that the large feature at ∼10.5 MHz in Fig. 6(b) was attributable to a fundamental vibrational resonance (analogous to the (0,1) mode of a circular membrane [27]), and that the higher-frequency resonances could similarly be attributed to higher-order vibrational modes (analogous to (m,n) modes of a circular membrane). Moreover, we noted that the stronger resonance peaks at ∼22 and ∼32 MHz lie at approximately the expected frequencies for the (0,2) and (0,3) modes of a circular membrane having the aforementioned properties. Since only the (0,n) vibrational modes lack a nodal point at the center of the membrane [27], the higher amplitudes of these peaks can be explained by the increased overlap between these mechanical modes and the fundamental optical mode field [8], which is nearly centered on the membrane for this cavity. Finally, it is also worth noting that the quality factor (Qm ∼ 50) of these membrane-attributed vibrational modes is consistent with values reported for similar SiN membrane resonators at atmospheric pressure [23], with Qm limited mainly by acoustic radiation into the surrounding air medium.
Fig. 5. FFT spectra extracted from ‘tuned-to-slope’ measurements for (a) a ‘regular’ cavity with no etch holes, and thus no released membrane, and (b) the case-study cavity shown in Fig. 2. Both were captured with time-averaged power at the detector P0 ∼ 23 μW. The resonances in part a. are attributable to vibrational modes of the buckled upper mirror of the cavity, and are also present in part b. The additional resonances in part b. are attributed to the vibrational modes of the released SiN membrane inside the cavity.
Fig. 6. (a) Comparison between the experimental vibrational spectra (with P0 ∼ 8 μW, see main text) and predicted (0,1), (0,2), and (0,3) resonance frequencies obtained using a finite-element numerical solver (COMSOL). The red dashed lines indicate frequencies predicted using an areal spring model for trapped gas beneath the membrane, and the black dotted lines indicate frequencies predicted using the COMSOL fluid physics modules for this trapped gas. The insets show the predicted vibrational mode shapes. (b) The curves show the results of a thermomechanical fitting procedure (see main text), performed on the experimental vibrational resonance at ∼ 10.5 MHz (i.e., the (0,1) mode). The displacement spectral density extracted from the experimental data (blue solid line) is compared to that predicted for a damped harmonic oscillator subject to Brownian motion (red dashed line).
To gain further insight, mechanical properties were modeled using a finite element analysis software package (COMSOL). The suspended SiN feature was modeled as a membrane, with an assumed shape (including holes) based on the confocal microscope image shown in Fig. 1(c). Clamped boundary conditions were applied to membrane edges, while the hole peripheries were treated as free boundaries. For the PECVD SiN, we assumed a density ρ = 3100 kg/m3, Young’s modulus E = 270 GPa, Poissons Ratio ν = 0.27, and in-plane tensile stress of T = 770 MPa (as measured during fabrication).
Aside from its elastic properties, the vibrational motion of the membrane is expected to be affected by squeeze film effects [23] in the ∼ 200 nm thick air layer below. The high resonant frequencies of our devices places them in the so-called ‘elastic damping’ regime [23], wherein the gas is unable to flow far within the period of oscillation but rather is compressed between the membrane and ‘substrate’ (i.e., the lower mirror in our case), thus imparting an additional spring constant to the motion. We incorporated this effect in two different ways. First, we employed the analytical approximation [23] kg = Pa A/d, where Pa is the ambient pressure, A is the area of the resonator, and d is the distance from the membrane to the SiO2 etch stop layer below (∼ 200 nm). Since this spring is caused by local gas compression, it is highly dependent on mode shape. We therefore introduced it into the COMSOL model as an areal spring, kA = kg/A ∼ 5 × 1011 N/m3, distributed uniformly over the SiN membrane. This allows for modes with shapes of varying displacement profiles to be proportionally impacted. Second, for corroboration and comparison, we removed this areal spring and instead introduced an air layer at atmospheric pressure into the COMSOL model, which allows direct numerical analysis of the physics of the fluid layer below the membrane.
As shown in Fig. 6(a), both of these approaches provided very good fits between the predicted (0,n) modal frequencies and the three strongest vibrational resonances in the experimental scan, consistent with the discussion above. In this figure, simulated resonance frequencies for the areal spring model (red dashed line) and the coupled membrane-fluid simulation (black dotted line) nearly overlap. The lower-amplitude resonances in the experimental scan can be associated with (m,n) vibrational modes (see supplemental information) where m ≠ 0, all of which have a nodal point at the membrane center and thus exhibit lower coupling to the fundamental optical mode. The COMSOL model reproduced the number of observed modes [for example, four additional modes between (0,1) and (0,2)] but there was less agreement between simulated and observed frequencies in these cases. We believe this is due to significant overlap between the etch holes and the vibrational profiles for these modes (i.e. these modes ‘live’ closer to the etch holes). The simple COMSOL model employed does not capture possible local variations in the membrane stress near the etch holes [12], nor does it capture likely reductions in the local ‘gas spring’ effects near the etch holes due to a well-known gas escape mechanism [28]. Nevertheless, the excellent high-level agreement between the models and the experimental data suggests that the most critical details are understood.
To further verify our understanding, we used a thermomechanical calibration procedure [27] to assess the optomechanical coupling between the optical mode and individual membrane vibrational modes. The measured (one-sided) PSD of a resonator mode subject to thermal Brownian motion may be represented by [27]:
Here, Sww(f) is the measured optical power PSD (W2/Hz), SwwW is a constant offset accounting for approximately white noise in the detection equipment, α is a measurement sensitivity factor (in units of (W2/m2) in our case), and Szz(f) is the membrane’s displacement PSD (m2/Hz), which can be modeled by a Lorentzian line-shape function:For our system, α can also be predicted directly as:
It is also worth noting that we performed a similar analysis on the fundamental vibrational mode of the buckled mirror, centered at ∼ 6.7 MHz. From transfer-matrix calculations (not shown), Gλ ∼ 0.37, approximately double that for the membrane motion. Using Eq. (3) with η ∼ 1 then predicts α ∼ 1.6 × 108 W2/m2. However, the mirror vibrational mode has an effective mass > 100x higher [18] than that of the membrane vibrational mode. Thermo-mechanical fitting (not shown) yielded meff ∼ 16 ng, Q ∼ 200, and α ∼ 1.8 × 108 W2/m2, in excellent agreement with these predictions. Note that the lower amplitude of the mirror vibrational mode (in spite of its higher Gλ) is due to its higher mass and thus lower displacement amplitude. From the equipartition theorem [18,27], the mean-square displacement amplitude is < an2(t) > = kB·T/meff,n·Ωm2. For example, this predicts < a1(t) > ∼ 3.2 pm and ∼ 0.4 pm at room temperature for the fundamental vibrational modes of the membrane and mirror, respectively.
From another point of view, the analysis above is essentially equivalent to comparing the experimentally observed and theoretically predicted optomechanical coupling coefficients (G) for our system. The reasonable agreement (i.e., within a factor of 2) obtained provides confidence that the observed vibrational features can are in fact attributable to the motion of a suspended SiN membrane inside our buckled cavities, and that our models have captured the essential physical details of the system. This should in turn provide a good basis for the optimization of fabrication processes and cavity parameters in future work.
5. Discussion and conclusions
Compared to other compact MIM systems [15,16] in the literature, the monolithic system described has potential to be a more robust and stable platform. Moreover, the process can easily be scaled to produce thousands of lithographically aligned devices per square centimeter, and in the future could leverage well-established MEMS-actuation strategies for tuning of individual devices. However, there are at least two noteworthy challenges associated with our approach. First, the distance between the membrane and the bottom mirror is fixed by the thin-film processes, so that there is less flexibility (post-fabrication) for positioning the mechanical element within the optical mode field. Having said that, it is worth pointing out that we have previously demonstrated thermal tuning of cavities mediated by changes in the buckle height with temperature [18], and we expect that other MEMS-actuation mechanisms (e.g. electrostatic, piezoelectric, etc.) could also be employed. The second challenge is really more of a trade-off (discussed further below); the short length of these cavities, while resulting in large optomechanical coupling, also increases the photon loss rate κ. This in turn places extreme requirements on the quality of the cavity mirrors, for example to achieve the widely sought ‘resolved sideband’ regime, Ωm > κ, where Ωm is the mechanical resonance frequency for the vibrational mode of interest and κ = ω/Q is the photon decay rate of the optical mode.
For the first-generation prototypes described here, our main goal was to establish a workable fabrication process, and in that vein relatively modest performance was targeted for the optical and mechanical resonators. For example, the case-study cavity possesses a relatively large photon loss rate κ / 2π = f / Q ∼ 40 GHz due to the low number of periods employed in the Bragg mirrors. Also, we used a relatively thick SiN membrane (in the interest of higher yields for the experimental process), while membranes as thin as ∼ 20 nm have been reported [14]. Most notably, we have yet to obtain measurements under high vacuum conditions, so that the observed mechanical quality factor (Qm < 100) is entirely limited by viscous damping. Measurements in vacuum will be pursued in future work. Nevertheless, it is interesting to make some projections on the basis of feasible (although admittedly challenging) optical and mechanical parameters:
- i. The effective temperature needed for ground-state cooling of a mechanical vibrational mode depends on frequency, as quantified by the time-averaged thermal occupation number, NTH = (kBTeff/ħΩm), where Teff is the effective temperature of the mechanical mode of interest. In order for quantum fluctuations to dominate over thermal noise, NTH << 1 is required, and this is obviously achieved at higher Teff for higher-frequency oscillators. Lithographic alignment enables us to embed a smaller membrane into our cavities than is typically used in hybrid systems [15], such that the effective vibrational mass estimated above (∼ 90 pg) is already much lower than typically reported [7,8]. Moreover, a membrane ‘pad’ [9,14] as small as ∼ 10 μm in diameter should be sufficiently large given the small waist radius of the optical modes in our cavities. This projects to fundamental mode resonance frequencies as high as fm* = Ωm*/ 2π ∼ 50 MHz, and effective masses in the few-pg range. Another advantage of this size reduction is the relative sparsity of vibrational modes, which should make it easier to isolate interactions between light and a particular vibrational mode of interest.
- ii. Typically, ground-state cooling is only possible in the fully resolved sideband regime defined by Ωm >> κ [1]. The short length of our cavities places extreme requirements on the cavity finesse in order to achieve low κ. In the literature, F ∼ 106 has been achieved using high-period-count Ta2O5/SiO2 mirrors [29,30], and we have recently proposed [31] that hydrogenated amorphous silicon might enable similar performance with far fewer layers. Assuming F = 106 as a best-case scenario projects to Q* ∼ 5 × 106 and κ* ∼ 40 MHz for the cavity described here. Thus, Ωm* ∼ κ*/(2π) and the fully-resolved sideband regime would appear to be an unlikely goal for cavities of this size. Nevertheless, many interesting applications might still be possible in the non-resolved or partially resolved sideband regimes [7,8,15], including schemes for ground-state cooling [32] and quantum transduction [33].
- iii. Radiation-pressure back-action effects can be used to approach ground-state cooling in a room-temperature environment, provided that Qm > NTH,room = (kBTroom/ħΩm) [1,8,9]. Using fm* from above implies that this condition would be satisfied for Qm > 105. Projections about mechanical quality factor are admittedly speculative until measurements in vacuum can be completed. Nevertheless, we reiterate that similar SiN membranes have been reported to have room-temperature Qm > 106 [7,8,15,24]. Moreover, we believe that our surface-micromachining process, with judicious design of etch hole sizes and patterns, might be used to implement either trampoline-style [9,12] or phononic-crystal-isolated [14,34] central pads (see Fig. 2), and both of these strategies have reportedly enabled Qm > 107. Thus, we conservatively project that Qm* = 106 and Γm* = Ωm* / Qm* = 50 [Hz] are feasible goals.
- iv. Some of the most powerful envisioned schemes in optomechanics are predicated on nonlinear interactions at the single photon level [1], which become significant in the strong-coupling regime defined by g0 > κ, Γm. Other interesting nonlinear effects are manifested when g0 > Ωm [17]. For our cavities, assuming reductions in membrane size/mass as mentioned above, we can project that xzpf* ∼ 5 fm and g0*/(2π) ∼ 0.5 MHz are maximum feasible values. Thus, even in the most optimistic scenario, g0* < κ*, Ωm*. However, they are of similar magnitude, suggesting that the less-restrictive strong-coupling condition g = g0·ncav1/2 > κ, Γm, which can enable observation of hybrid optical-mechanical states and optomechanically induced transparency (OMIT) [1,32], should be possible at relatively low drive powers (i.e. small time-averaged number of photons circulating in the cavity mode, ncav). Finally, it is interesting to note that very high single-photon cooperativities, C0* = 4·(g0*)2/( κ*· Γm*) > 100 might be feasible for these micro-scale systems. C0 is an important measure of the strength of interaction between the light and mechanics at the single-photon level [1,9], which has recently been confirmed as a relevant figure of merit for quantum optomechanics [35].
Funding
Natural Sciences and Engineering Research Council of Canada (CREATE 495446-17, STPGP 494024).
Acknowledgments
The authors gratefully acknowledge useful discussions with C. Potts, C. Doolin, H. Ramp, and Dr. J. Davis. G. J. H. acknowledges the support of an Alberta Innovates Graduate Student Scholarship.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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