1. Introduction

In the past decade, cavity optomechanics is an emerging research field exploring the interaction between light and mechanical vibration in a variety of macro-, micro-, and nano-optical systems [1, 2]. Useful optomechanical effects have been realized, ranging from amplifying and cooling of phonon modes [3–6], optomechanically induced transparency [7, 8], and wavelength conversion [9–11] to all-optical non-reciprocal devices [12, 13]. In practical experiments, the scalability of the cavity optomechanical systems is limited, due to the frequency mismatching of the mechanical or optical modes in the individual optomechanical system. Therefore, tuning the optical and mechanical modes of the fabricated microresonators is crucial for the potential applications of the mechanical oscillators.

Several methods to tune the optical frequencies of microresonators have been proposed and demonstrated experimentally [14–21], including aerostatic pressure, strain, magnetic and temperature methods. The tunability of the high-Q optical modes can be used for several potential applications [22–25]. For other applications, such as optomechanical frequency conversion [11] and long-distance optomechanical synchronization [26], the tunability of optomechanical modes are necessary. However, the tuning of the mechanical modes is rarely reported [27, 28]. The aerostatic pressure tuning of the mechanical modes would introduce additional air damping to damage the mechanical Q, and temperature tuning of the mechanical modes would also destruct the mechanical Q. Therefore, these two tuning methods are appropriate for optical modes and not suitable for the mechanical modes especially in the cryogenic environment. In addition, the quality (Q) factors of mechanical modes is also limited by the material absorption, structure corresponding to the material damping, and clamping damping, respectively [29, 30]. The efficient tuning of the mechanical modes is a very challenge task in a high-quality microresonator.

In this paper, we demonstrate the tuning of both optical and mechanical modes in hollow bottle-like microresonators (BLMRs), which have been proposed as highly sensitive sensors due to the evanescent fields distributed inner and outer of the thin shell [14, 15, 31–33]. We have fabricated the BLMRs with high optical (1.55 × 10⁸) and mechanical (2.5 × 10³) quality factors. By mechanically stretching the BLMRs, we demonstrate the frequency tuning of the optical modes over a free spectral range (917 GHz) and the mechanical modes of 1 MHz. During the tuning process, the Q factors of optical and mechanical modes were maintained. These properties are promising for potential applications of such optomechanical oscillators, such as tunable optical frequency conversion [9–11], tunable non-reciprocal devices [12, 13], and long-distance optomechanical synchronization [26].

2. Experimental setup

To fabricate the BLMRs, the fused-silica glass capillary with an outer diameter of 140 μm and an inner diameter of 100 μm was first pulled thinner using a hydrogen flame. Then, two counter-propagating CO₂ beams were focused on the capillary, where the heated region becomes thinner. For a bottle-like resonator, two parts along the capillary at a distance of 300 μm were heated by the appropriate laser power. The device diameter was controlled by the drawing length during the first step. This method was similar with the fabrication reported in [31]. And multiple such resonators can be built along a single capillary. After these processes, the microresonator was mounted on the piezoelectric transducer (PZT) stage for the mechanical stretch, as shown in the Fig. 1(a). Figure 1(b) illustrates the photograph of a typical BLMR, with a diameter of 76 μm and a wall thickness of 18.7 μm.

 

Fig. 1 (a) Schematic of the strain tuning optomechanical modes experiment setup. FPC: fiber polarization controller. PD: photo detector. ESA: electronic spectrum analyzer. DSO: digital oscilloscope. PZT: the piezoelectric transducer. (b) A photograph of the BMLR sample with a diameter of 76 μm. (c) The typical calculated distribution of the optical modes at the cross-section of the BLMR. (d) The typical transmission spectral of the WGM with a linewidth of about 1.6 MHz. Solid red line represents the theoretical calculation with Q₀ = 1.55 × 10⁸ and Q_ex = 20 × 10⁸.

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The experimental setup is illustrated in Fig. 1(a). To characterize the optical and mechanical modes in BLMRs, a tunable laser in 1550 nm band was used in the experiments. The laser was efficiently coupled into the BLMR through a tapered fiber near-field coupler. A fiber polarization controller (FPC) was used to control the polarization of input laser. The output light signal from the BLMR was detected by a 125 MHz low noise photo-detector (PD). The electronic spectrum analyzer (ESA) was used to study the mechanical modes in the microresonators, as the transmitted optical signal of the BLMR was modulated by the mechanical vibrations due to the optomechanical interaction [3,4]. A voltage was applied to a piezoelectric transducer (PZT) stage to stretch the BLMRs to tune the optical and mechanical modes in the BLMRs. The whole system was placed in a cleaning chamber to reduce the interference of the contaminants and the perturbation of the air.

3. Strain tuning optical modes

First of all, we studied the optical modes in the BLMRs. The Q factor of optical modes could be observed as high as 1.55 × 10⁸ in these BLMRs. A typical transmission spectrum of the WG mode was shown in Fig. 1(d), in which the ringing phenomenon due to sweeping laser frequency was observed [34, 35]. In addition, a series of optical modes were obtained when the wavelength was scanned from 1550 nm to 1570 nm, as shown in Fig. 2. The measured free spectral range (FSR) describing the distance between two adjacent angular modes in the same mode family is about 7.3 nm, which is greater than the calculated value based on the optical path length of the equator. This is because the tapered fiber coupling position has slightly deviated from the equator of the resonator during the experiment.

 

Fig. 2 Transmission spectra of the optical modes of the BLMR sample B for different voltage of the PZT. The transmission dips correspond to the WGM resonances. The measured free spectral range (FSR) describing the distance between two adjacent angular modes is about 7.3 nm.

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As we increased the PZT position with the step mode, the microresonator was stretched in the z-axis direction (Fig. 1(a)). Due to the hollow microbottle structure, the diameter of the microresonator can be easily changed. As a result of the geometry boundary condition variation and the refractive index change of the BLMRs determined by the photo-elastic coefficient, the resonance wavelength was tuned by the PZT. As shown in Fig. 2, the PZT position tuned from 0 to 144 μm, the wavelength of optical modes is tuned over one FSR. Therefore, the “fully tunable” regime [17] was achieved in our experiment. The resonance frequency shifting due to the change in the size and the refractive index of the microresonator can be approximated as [36]

We measured the WGMs wavelength drift with the position of the PZT, and got the tuning effect of three samples with different diameters. Here, the wavelength of the optical modes was recorded by a wavemeter. The extracted frequency shifts of optical modes against the PZT position in different BLMR samples are plotted in Fig. 3(a). The slopes of the BLMRs with diameters of 82 μm, 76 μm and 69 μm are 3.67 GHz, 5.99 GHz and 8.60 GHz per micrometer, respectively. To study the fine tuning of the BLMR, we measured the WGM shift of another sample with changing the PZT voltage with the fine mode, as shown in Fig. 3(b). For the limited resolution 5 nm of the PZT, the fine tuning resolution were around 9.2 MHz with respect to the increasing and decreasing of the PZT voltage. It was at the expense of hysteresis due to the nonlinearity of the piezo stacks providing the strain at the fine mode and the fluctuation of the laser and surrounding environment.

 

Fig. 3 (a) The WGM shift of BLMR with respect to the increasing the PZT position. The diameter of the resonators are 82μm, 76μm and 69μm with a thickness of 19.1μm, 18.7μm and 14.0μm, respectively. (b) The fine tuning of the WGM in a microresonator with diameter of 73μm with respect to the increasing and decreasing of the voltage of the PZT.

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We can calculate the force sensitivity as reported in [17]

4. Strain tuning optomechanical modes

Similar to the silica microspheres and microtoroids, there are high-Q mechanical modes in the BLMR [30]. Due to the effect of the radiation pressure of the optical field in the resonator, which induces dispersive optomechanical interaction, the mechanical mode can be parametrically excited by the laser and the mechanical vibrations modulate the optical resonance frequency. Through the tapered-fiber excitation, a CW input power far below the threshold of parametric instability was used. Figure 4 shows a typical RF power spectrum of the transmitted optical signal of sample B, which is taken by an electronic spectrum analyzer (ESA). There are several peaks in the spectrum, which indicates mechanical vibrations in the BLMR. The mechanical modes corresponding to the peaks in the experimental spectrum is identified by the finite-element method (COMSOL Multiphysics v4.2), as shown in the insets of Fig. 4.

 

Fig. 4 Displacement power spectrum of the mechanical modes obtained from sample B. The insets show the numerically simulated mechanical modes of the BLMR with corresponding frequencies. These fundamental mechanical modes can be excited in the microbottle resonator used in our experiments.

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Since the mechanical resonance frequency depends on the geometric effect and the stress induced stiffening effects [27, 42], we can also expect the effective tuning of the mechanical modes. We choose the highest Q mechanical mode for mechanical modes tuning experiments, with the frequency ω_m/2π = 34.1 MHz and the linewidth γ_m/2π = 15 kHz. The spectrum of this mechanical mode under different PZT displacement are shown in Fig. 5. The tuning range of 0.98 MHz was obtained for sample B with the PZT position tuned from 0 to 156 μm. The tuning range was almost 3 orders of magnitude greater than the results shown by the aerostatic pressure induced optomechanical modes shift in [27]. Compared with the tuning of optical modes that the frequency tuning range is about 0.46% of the mode frequency, the relative tuning range of mechanical mode is about 2.9% and is about 6 times higher. Since the stiffening effects of the BLMRs would change the mechanical frequency as mentioned in [27], our experiment results indicate that the influence of stiffening effects in the resonator shell for the mechanical frequency shift is greater than the geometric effect, as discussed below.

 

Fig. 5 The mechanical power spectrum of the optomechanical mode versus different voltage of the PZT with a interval of 20μm.

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More results of mechanical frequency tuning in different samples are shown in Fig. 6. The frequency shift and linewidth of mechanical modes were recorded every 4 μm of the PZT movement until the microresonator was broken. As shown in Fig. 6(a), the slopes of the mechanical frequency versus the position of PZT are 7.75, 6.4, 9.15 kHz/μm, respectively. By the numerical simulation of the frequency shift against the radius, we found that d(ω_m/2π)/da = −0.94, −1.11, −1.36 MHz/μm for sample A, B and C, respectively. Since we can not directly measure the cavity radius for different PZT position, we approximately estimate the radius change as da = −adv/v and deduce the geometric deformation induced mechanical frequency shift against the PZT position as $\frac{d({\omega }_m/2\pi )}{dz}=-\frac{d({\omega }_m/2\pi )}{da}\times \frac{a{k}_o}{\nu }$, where k_o = dν/dz is the slopes in Fig. 3. By this method, we can estimate the pure geometry effect induced d(ω_m/2π)/dz as 0.74, 1.31, 2.10 kHz/μm, respectively. Comparing to the experimental results in Fig. 6(a), the estimated geometry effect is about one order of magnitude smaller. Therefore, we can conclude that the stiffening effect is dominant over the geometric deformation when tuning the mechanical mode frequency. In addition, the linewidth is not affected obviously during the tuning process, as depicted in Fig. 6(b). In other words, the optomechanical quality factors remain high which is one of the great strengths for our tuning method.

 

Fig. 6 (a) The frequency tuning of the mechanical modes in the BLMR sample A, B, C. (b) The linewidth of the mechanical modes during the tuning process.

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5. Conclusions

We have experimentally demonstrated and characterized the high quality factor optical modes and mechanical modes in the silica hollow bottle-like microresonators, which are tuned by the strain over 917 GHz(7.3 nm) and 1 MHz, respectively. We found that the geometric, photo-elastic and stiffening effects play different roles in the controlling of optical and mechanical modes frequency. The strain tuning method can provide frequency-on-demand, which is critical in the light-mater interaction processes with optomechanical oscillators. This method can find applications in optomechanical frequency conversion [11] and long-distance optomechanical synchronization [26].