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Recently, the first microfluidic optomechanical device was demonstrated, capable of operating with non-solid states of matter (viscous fluids, bioanalytes). These devices exhibit optomechanical oscillation in both the 10–20 MHz and 10–12 GHz regimes, driven by radiation pressure (RP) and stimulated Brillouin scattering (SBS) respectively. In this work, we experimentally investigate aerostatic tuning of these hollow-shell oscillators, enabled by geometry, stress, and temperature effects. We also demonstrate for the first time the simultaneous actuation of RP-induced breathing mechanical modes and SBS-induced whispering gallery acoustic modes, through a single pump laser. Our result is a step towards completely self-referenced optomechanical sensor technologies.
© 2014 Optical Society of America
1. Introduction
Optomechanical microresonators enable strong-coupling between their photon modes and phonon modes through photothermal effects [1–3], radiation pressure force [4–10], optical gradient force [11–14], and electrostriction mechanisms [15–21]. This capability has been harnessed for many fundamental experiments including optomechanical cooling [5–7,19], induced transparency [22, 23], and dark modes [24]. Efforts have also been made towards sensing applications such as accelerometers [25, 26], mass sensors [27, 28], and force sensors [29, 30]. Recently, the first microfluidic optomechanical devices capable of operating with liquids and targeted towards bio-applications [18, 31] were demonstrated, enabling optomechanics with non-solid states of matter. Indeed, while individual optomechanical devices can operate at multiple oscillation frequencies [18, 21], these frequencies are discrete and the spectral gaps cannot easily be filled. Achieving complete spectral coverage with a single device can provide frequency-on-demand capability that is extremely important for frequency-hopping and cognitive oscillator applications in communication systems. Enhanced tuning also helps align optical and mechanical modes to ‘phase match’ fundamental light-matter interaction processes that employ optomechanics [19, 22–24].
Till date, continuous tuning of the discrete optomechanical oscillation frequencies is primarily achieved through direct temperature control [32] or by managing the dropped power [33,34]. However, tuning by means of temperature is often impractical since low phonon occupation and specific laser power coupling are often desirable, especially for single-phonon experiments in the quantum regime. For this reason aerostatic tuning, i.e. tuning through air pressure, is an alternative that can provide improved tuning capability with greater spectral coverage. Aerostatic tuning has been successfully employed with hollow microbubble resonators [35,36], optofluidic ring resonators (OFRR) [37], and ring resonators on flexible membranes [38]. Another deformation-based strategy involves the use of stretchable polymer microspheres and achieves THz range tuning [39]. However, these extremely-tunable devices have not yet been employed for optomechanics, and the tuning of mechanical modes in conjunction with tuning of optical modes has not been studied previously. We have developed an OFRR-derived opto-mechanofluidic resonator (OMFR) platform [21, 31] that employs optical radiation pressure (RP) and stimulated Brillouin scattering (SBS) to actuate mechanical vibrations spanning the frequency ranges of 2 MHz – 1 GHz and 10 GHz – 12 GHz. In this study, we experimentally investigate the aerostatic tuning of these OMFR devices and we show the simultaneous actuation and tuning of multiple modes and mechanisms of optomechanical oscillation.
2. Setup and working principle
Our experimental devices (Fig. 1(a)) are fabricated from fused-silica glass capillary preforms that are drawn under heating with infra-red laser light from two CO2 lasers [37,40]. The diameter of the hollow micro-device can be varied as a function of position by modulating the heating laser power, with the widest region forming a bulb about 135 μm in diameter for example (Fig. 1(a)), where acoustic and optical modes are simultaneously confined. The resonator wall thickness is about 12 μm (Fig. 1(a)) in this example. The wall can be made thinner as needed by using a thin-walled preform, HF etching, and glass-blowing [41]. One end of the device is sealed using optical glue while the other end is an open port for pressure control (Fig. 1(b)). Pressure inside the capillary is controlled by means of a syringe pump capable of micro-liter control while the ambient pressure remains constant at 1 atm (101 kPa). Continuous-wave light from a 200 kHz linewidth C-band external cavity diode laser (Newport Corporation TLB-6728) tunable between 1520 – 1570 nm (30 GHz fine-tuning range) is coupled into the ”bottle” [42] optical whispering-gallery modes (WGMs) of the device by means of a tapered optical fiber (Fig. 1(b)) [43]. The taper is < 1 μm in diameter at the narrowest point and is single-mode for the wavelength range used. An oscilloscope is used to monitor the forward transmission in the tapered fiber and to identify where the optical resonances occur. RP oscillations are observed on nearly all of the high-Q optical modes. As described below, SBS oscillations require a more stringent phase matching condition. Optical resonances that support SBS phase matching are found by scanning the laser across the optical resonances and measuring resultant oscillations on the photodetector. Input laser power in the range of 10 – 20 mW is used. Frequency-locking between the laser and the optomechanical device is automatically achieved on the red-detuned side of the optical resonance through the well-known thermal self-stability mechanism [44]. Self-stability is achieved here because of the ultra-high-Q of the optical resonances. In contrast to most optomechanics experiments, the coupling fiber is placed in contact with the resonator for improved resilience to ambient vibration and geometry change. Indeed, the taper does add some damping to the mechanical modes but this effect is very small since the mechanical Q-factors remain high. We observe the excitation of breathing mechanical oscillations in the 10 – 20 MHz span driven by radiation pressure (Fig. 1(c)), and also whispering-gallery acoustic mode (WGAM) oscillations driven by stimulated Brillouin scattering [15, 16, 18, 19, 21, 34, 45] (Fig. 1(d)). Multimode operation through simultaneous oscillations of RP and SBS modes is possible, as we show later. Both modes of operations are tunable by controlling the aerostatic pressure in the device.
Fig. 1 Experimental overview (a) Colorized scanning electron micrograph of a hollow-core fused-silica optomechanical resonator with radius modulated by design. Resonator wall thickness can be varied as needed. (b) 1.5 μm light is coupled to the ultra-high-Q optical modes by means of a tapered fiber placed in contact to minimize vibrational issues. The optical pump and the scattered light are made to interfere on high speed photodetectors in both forward and backward directions, thus generating beat notes at the mechanical vibration frequency. (c) Radiation pressure (RP) drives “breathing mode” optomechanical oscillators (OMOs) that generate both upper and lower sidebands of the input optical signal. (d) Stimulated Brillouin scattering (SBS) excites traveling whispering gallery acoustic modes (WGAMs) that generate only a single Stokes shifted sideband in the backscattering direction.
Centrifugal radiation pressure induced optomechanical oscillations (OMOs) (Fig. 1(c)), described previously in [4, 46, 47], result in the actuation of ‘breathing’ mechanical eigenmodes ΩRP of the resonator and the generation of modulation sidebands to the pump. We measure these sidebands when they interfere with the pump on a photodetector in the forward propagation direction on the tapered fiber.
Stimulated Brillouin scattering is an acousto-optical nonlinearity [48–51] that has been recently used for generating high frequency acoustic waves in various resonant and nonresonant systems [15–21, 34, 40, 45, 52] (Fig. 1(d)). SBS is caused by photoelastic scattering of input (pump) light from acoustic perturbations in the material, and coherently amplifies these acoustic perturbations through positive feedback from electrostrictive pressure generated by the scattered light and the pump light [50]. This process is ‘phase matched’ when the energy and momentum of the two photon modes (pump mode and scattered ‘Stokes’ mode) are separated precisely by the energy and momentum of the phonon mode (acoustic mode) under consideration. This phase matching occurs near 11 GHz for a 1550 nm pump laser in silica material. However, we note that the optical mode separation in a resonator strongly affects the precise frequency obtained [16]. By monitoring back-scattered light through a circulator, we can measure the temporal interference of the pump laser and the scattered Stokes optical signal which occurs at the acoustic frequency ΩSBS (Fig. 1(d)).
Multiple phenomena can contribute to the pressure response of the RP- and SBS-driven OMOs in a hollow optomechanical system, namely (1) geometric effect, (2) stress effect, and (3) temperature effect, as shown in Fig. 2. The deformation of the resonator geometry by the application of differential pressure ΔP = Pint − Pamb, will affect the mechanical frequency ΩM directly. This can be modeled via the resonant frequency equation for the breathing mechanical modes of an annular resonator [53] of radius R.
where, E is the modulus of elasticity and ρ is the mass density. The stiffness of the resonator decreases as the radius of the annulus increases. For a radius R of 67.5 μm and a wall thickness of 12 μm (Fig. 1(a)), the change of radius of the resonator with respect to ΔP can be calculated through finite element analysis in COMSOL Multiphysics [54] to provide dR/dΔP = 3.82 × 10−12 m/kPa. Similarly, the mechanical frequency due to geometry perturbation can also be calculated to get dΩM/dR = −0.19 × 1012 Hz/m. Here, we define a new quantity, the pressure coefficient of frequency, PCf = dΩM/dΔP = dΩM/dR × dR/dΔP = −0.736 Hz/kPa caused by geometry change alone.Fig. 2 Aerostatic tuning mechanisms: Increasing the internal aerostatic pressure (Pint) causes geometry change (radius, dR) and increases the stress (S) in the resonator shell, both of which cause the mechanical frequency to shift. Geometry and stress effects also modify the optical modes, which changes the laser power coupling and thus energy dissipation in the resonator, modifying the device temperature. Since the mechanical modulii of a material are temperature-dependent, this also induces a change in the mechanical frequency of the OMO.
Circumferential stress (S) is also developed on the resonator when pressure is applied. Stress-induced stiffening effects are well-understood, and have been previously studied in the case of beams as described by equation (5.144) of [53] and in the case of curved beams and annular resonators in [55]. In general, higher tensile stress (such as that caused due to pressure exerting an outward force on a hollow tube) causes mechanical resonance frequencies to increase [55].
Finally, we note that the optical modes also shift in response to the applied pressure, by the optomechanical coupling coefficient dω/dR. For a fundamental optical WGM at free-space wavelength λ0 = 1550 nm on our R = 67.5 μm device, the azimuthal wavenumber of the modes is M = 2πRn/λ0 = 396. Therefore, we can analytically calculate the couping coefficient dω/dR = −cM/(2πR2n) for this simplified situation, which is estimated at −2.85 GHz/nm for the considered device. Employing the above analysis for mechanical modes, the optical mode tuning to applied pressure is estimated as dω/dΔP = dω/dR × dR/dΔP = −0.011 GHz/kPa. In our ultra-high-Q resonator (Qopt ≈ 108) where the optical losses are extremely low and the mode linewidth is narrow, these shifts in the optical mode can significantly affect power coupling into the device, and thus affect the optical power being dissipated in the silica. The relationship between the net heat dissipated to the cavity with the optical resonance shift and temperature change has been revealed in [44,56], which also examined the microcavity dynamical thermal behavior in detail. The temperature tuning of optical modes in silica resonators is known to be 6 ppm/K [44]. Since silica has a large temperature coefficient of Young’s modulus of 183 ± 29 ppm/K [57] while the thermal expansion coefficient of silica is about 2.6 ppm/K [57], the resulting temperature coefficient of mechanical frequency is aaproxiamately 90 ppm/K and is dominated by the Young’s modulus change. Thus a minute shift of the optical mode can significantly temperature-tune the oscillation frequency, even when the optical mode is over-coupled due to taper-device contact. This is challenging to theoretically model as the thermal pathways in the system are complex. However, we can control the power coupling into the resonator to be constant by tracking the optical resonance when it shifts under applied pressure. Since the temperature of the resonator is related to the power dissipated in the resonator, feedback control of the coupled power maintains a constant temperature in the device and eliminates any additional temperature-related stress or radius effects.Such feedback control will successfully arrest the unpredictability of the RP-driven OMO. However as we explain later, the SBS-driven OMO cannot be controlled in this way as it depends on two optical modes that typically have slightly different temperature coefficients.
3. Experimental results
Our experiment is capable of testing over a wide differential pressure range (ΔP = −90 kPa to 1000 kPa). However, in order to avoid optical and mechanical mode hopping, we experimentally measure the behavior of the RP- and SBS-driven OMOs over a smaller pressure range. As described above, the PCf is the fractional shift of oscillation frequency for a change in ΔP. As shown in Fig. 3, the mechanical vibration frequency is tuned through internally applied pressure. The sensitivity is measured to be PCf = −3.1 Hz/kPa without any control applied to the coupled optical power. As we described above, when pressure is increased inside the resonator, the geometric deformation and the temperature effect both act to decrease the mechanical frequency. However, the stress effect simultaneously acts to increase the mechanical frequency. A study of these three effects separately is necessary to verify our hypothesis. However, decoupling the geometric effect and stress effect is not feasible. Nevertheless, one can apply feedback control on the coupled optical power in order to at least eliminate the temperature effect. With such feedback applied (Fig. 3), the PCf without the temperature effect is measured to be +4.4 Hz/kPa. Since the geometrical effect is small as calculated (PCf = −0.736 Hz/kPa), this positive PCf verifies the strong influence of the additional stress-tuning effect. The reversed sign of PCf indicates that the temperature effect is a major influence in determining sensitivity of this high-Q OMO.
Fig. 3 Aerostatic tuning of a 13.07 MHz RP-driven OMO. We characterize aerostatic tuning of the OMO using a fixed-frequency pump laser, observing a negative pressure coefficient of frequency (PCf) caused by lowering of the temperature of the device due to the optical mode shift. When feedback control is applied on the laser (to track the shifting optical mode) the amount of power coupling into the device does not change. Thus the temperature effect is eliminated and the net positive PCf indicates that the OMO frequency is dominated by the increasing stress in the resonator shell.
In contrast to the RP-driven OMO, the SBS-driven 11 GHz OMO requires two optical modes and one acoustic mode that are mutually phase matched (this can also be understood as a momentum and energy conservation requirement) [21,31]. At such high frequencies, the WGAMs are part of a continuum, and thus do not restrict a specific operational frequency. As a result, pressure tuning of SBS-driven OMOs is highly sensitive to the separation between the two optical modes involved, which are known to exhibit a wide variety of slopes in their dispersion behavior [58, 59]. Our experimental data provides evidence for this argument by demonstrating both positive (Fig. 4(a)) and negative (Fig. 4(b)) PCf for different 11 GHz OMOs on the same resonator. We find that for any specific optomechanical coupling mode the SBS-based sensor demonstrates very large absolute pressure sensitivity (up to |PCf| = 55 kHz/kPa) and is repeatable.
Fig. 4 Aerostatic tuning of 11.2 GHz SBS-driven OMOs: Both (a) negative and (b) positive PCf are observed in multiple trials, exhibiting large absolute pressure sensitivity. The red line is a linear fit to the data.
Finally, we note that on a low stiffness shell-type resonator that we explore here, it is easy to actuate SBS-driven oscillation simultaneously with RP-driven oscillation. A representative experimental spectrogram demonstrating such dual-mode operation is shown in Fig. 5. The SBS oscillation (ΩSBS ≈ 11.2 GHz) is modulated by the RP oscillation (ΩRP ≈ 16 MHz) resulting in sidebands that are separated by ΩRP. This modulation occurs because the resonator geometry is being modified by the radiation pressure oscillation, resulting in coupling between the two oscillation modes. As the internal pressure is changed from 1 atm (∼101 kPa) to 5 atm (∼500 kPa), both ΩRP and ΩSBS are tuned (Fig. 5). The tuning of the two modes can be quantified accurately by interferometric measurement of the forward and backward scattered optical signals as explained previously. A representative result is shown in Fig. 6. The frequency shift data are presented in parts-per-million (ppm) since the absolute sensitivity of the SBS oscillator is four orders-of-magnitude greater than the RP oscillator. Such multimode operation enables the future possibility of self-referencing the pressure sensor without need for additional amplitude or wavelength references.
Fig. 5 RP and SBS oscillations can be simultaneously actuated on a single device as evidenced in this spectrogram. Experimentally, the ΩRP = 11 GHz oscillation shows multiple sidebands separated by ΩRP = 16 MHz. As internal pressure is changed from 1 atm (∼101 kPa) to 5 atm (∼500 kPa), both the 11 GHz OMO and the 16 MHz OMO are tuned. CF, center frequency.
Fig. 6 Aerostatic tuning experiment with simultaneous RP (15.2 MHz) and SBS oscillation (11 GHz) on a device. Fractional mechanical frequency shift is recorded in parts-per-million (ppm) for convenient comparison since the SBS OMO absolute sensitivity is four order-of-magnitude higher.
4. Conclusions
We have experimentally demonstrated and characterized the first aerostatically-tunable optomechanical oscillators driven by both radiation pressure as well as stimulated Brillouin scattering. Potential applications of this system include pressure sensing in extremely harsh conditions, particularly in high temperature environments. Understanding the pressure-sensitivity of these deformable devices is also important for future optomechanics experiments with liquid-phase media such as viscous liquids, bioanalytes [21, 31], and superfluids [60]. We note that multimode oscillators with different coefficients-of-frequency have been employed widely in quartz [61] and MEMS [62] resonator technology to provide reference-free operation for timing reference and sensor applications. Finally, we note that this coupled system involves a MHz regime mechanical oscillator, a GHz regime mechanical oscillator, and a 200 THz regime optical oscillator. This presents a unique and exciting opportunity to explore coupled oscillator dynamics over extremely broad timescales, and to study the coupling of fields and signals spanning radio-frequency, microwave, and optical regimes.
Acknowledgments
Funding for this research was provided through a University of Illinois Startup Grant. We would like to acknowledge stimulating discussions and guidance from Prof. Tal Carmon, Prof. Xudong Fan, Prof. William P. King, Prof. Randy Ewoldt, Prof. Taher Saif, Prof. Rashid Bashir, Prof. Kimani Toussaint, Prof. Lynford Goddard, and Sandeep Anand.
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