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Abstract
In this paper, we report on the realization of optomechanics and its broad tuning in liquid crystal-filled micro-bubble resonators (LC-MBR). Acoustic anisotropy of LC has enriched optomechanical resonant modes in such whispering-gallery modes (WGM) cavities. Meanwhile anisotropic dependence of sound speed of LC on temperature strongly enhances the tuning ability of the resonances. By applying magnetic field to control re-orientation of LC in MBR, optomechanical resonant frequency changes significantly after magnetic field goes beyond Fréedericksz transition threshold. By designing such an optomachanical system, optomechanical oscillator with broad tuning range can be realized.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, optomechanics of whispering-gallery modes (WGM) cavities has attracted wide attention because of its strong coupling effects between photon modes and phonon modes [1], and such unique features in micro-resonators have been used to realize cavity-assisted laser-cooling [2], optomechanical induced transparency [3] and sensing applications such as accelerometers [4], mass sensors [5] and force sensors [6]. Opto-mechano-fluidic resonators (OMFR), including hollow capillary and liquid core, have been chosen to minimally reduce acoustical loss of fluid, and proved to be an ideal platform to realize fluid-shell’s hybrid mechanical vibration [7]. Usually, the input fluid is isotropic, so the sound speed of the acoustic wave propagation has no difference at all directions. OMFR is widely used as viscometers [8], thermo-optomechanical sensing [9], and 2-D optical/optomechanical microfluidic sensing to recognize mixed fluids with the same optical refractive index [10].
Anisotropic material, like calcium fluoride crystalline, has been proved to be an ideal medium to enable high mechanical quality factors greater than 105 [11]. Three independent elastic constants of calcium fluoride mean that by choosing different crystalline orientations, the vibration modes and their mechanical eigenfrequencies differ with each other. Liquid Crystal (LC) also shows outstanding anisotropy in acoustic and viscoelastic area [12]. Sound speed and attenuation [13,14] in LC are highly dependent on wave propagation’s directions.
Actually, LC in WGM cavities [15,16] has shown great potentials because its optical, dielectric and magnetic properties are anisotropic. In polymer matrix, nematic LC droplet will spontaneously align as a spherical WGM cavity. Then under external electric field, LC droplet will re-orient, which leads to an obvious change of refractive index, and this will consequently shift optical resonant frequency [17]. Starting from this milestone, temperature [18] and magnetic [19] tunings of optical WGM modes have been reported, while cholesteric liquid crystal has been used in such droplets as Bragg cavities to realize omnidirectional laser [20].
Tuning of mechanical resonant frequency is also challenging in optomechanics, and several methods have already been proposed. Aerostatically tunable optomechanical oscillators have been driven by both radiation pressure (RP) as well as stimulated Brillouin scattering (SBS) [21].Its simultaneous optomechanical tuning in MHz and GHz can realize self-referenced pressure sensing in harsh conditions. The strain tuning by stretching the micro-resonator can control both optical and optomechanical mode frequencies, and the geometric, photoelastic and stiffening effects have been analyzed to explain such tuning process [22]. For both of the methods above, the tuning ability of hollow MBR is largely limited by fused-silica’s weak sensitivity to external stimulus [23].
In this paper, we combined anisotropic nematic LC with hollow-core WGM resonators. LC-filled MBR with high optical (106) and mechanical (102) quality factors is obtained. Anisotropic optomechanical modes are observed and identified by temperature dependence of resonant frequencies. Moreover, the tuning range of resonant frequency is 7.5% of total, which is better than the strain (2.9%) and air pressure (5.5%) methods before.
2. Experimental setup
Hollow-core MBRs are fabricated from silica capillary by fuse-and-blow technique. Thick silica wall bubbles are prepared so that optical fields are mostly in the shell. One end of the capillary is sealed and the other end is connected to a needle cylinder to choose appropriate air pressure in the capillary-tube system. After heating the capillary by the fusion splicer, a MBR is formed. Here we use a MBR which has a size of 136 μm and thickness of 10 μm.
The MBR is injected with DMOAP (N,N-dimethyl-n-octadecyl-3-aminopropyl-trimethoxysilyl chloride, wt. 1% in water) aqueous solution for 30 min to generate homeotropic alignment layer. After washed with deionized water to remove redundant surfactant, we inject 4-Cyano-4'-pentylbiphenyl (5CB) nematic LC into MBR at 313 K, and then let the cell cool down. Formation of homeotropic-aligned LC is confirmed by observing bright-field and crossed-polarized micrographs in Figs. 2(a) and 2(b) [24].
A continuous-tunable laser (Anritsu Tunics Plus CL) at around 1550 nm is used as the pump source. We optically pump the MBR via a tapered fiber. The transmitted light is sent to photodiode and signals are collected by oscilloscope (Tektronix TDS3012) and electronic spectrum analyzer (ESA, Tektronix RSA306B) separately. The optomechanical mode information can be obtained by analyzing the power spectra of the transmitted light. When light is coupled to a specific optical whispering gallery mode, vibration of the resonator leads to a periodic transmitted light power fluctuation that can be detected by ESA.
The fiber-coupled MBR is put on a temperature-controlled heater. We fine-adjust the position of temperature sensor near MBR to calibrate the temperature accuracy by monitoring nematic-isotropic (N-I) phase transition of LC at 307.8 K. Before measurement, we package the MBR-taper coupling system in Fig. 1(a) to reduce environmental disturbance to the optical modes and mechanical vibration modes [25]. A glass slide that covers upon four glass spacers is not shown in Fig. 1(a). A coating layer of aluminum foil is set around the system to reduce heat exchange with surroundings [26].
Fig. 1 (a) Schematic representation of LC-MBR with fiber taper coupling and packaging. (b) Equatorial cross-section of LC-MBR (The blue lines present the direction of LC orientation).
3. Results and discussions
We use a polarized optical microscope to check the LC alignment in MBR. Because the radial nematic LC droplet situation has been studied thoroughly [27], we can take nematic LC droplet as a reference. In nematic droplet with homeotropic anchoring, LC molecules align along the radial direction and the anisotropic droplet forms a hedgehog defect at the center of the droplet, which is visible as a dark spot [28]. Figure 2(a) shows the bright-field micrograph of LC-filled MBR under non-polarized light at 298 K. We can also find a convergent hedgehog defect at the center of the bubble. When LC-filled MBR is observed under crossed polarizers, a typical cross can be seen in Fig. 2(b) like the case in nematic droplet.
Fig. 2 Light field and polarized micrographs of LC-MBR. (a) bright-field, (b) crossed polarized, (c)-(f) bright field micrographs with phase transition, (g)-(j) polarized micrographs with phase transition.
Figures 2(c)–2(f) are the bright-field micrographs while Figs. 2(g)–2(j) are the crossed polarized micrographs when we first heat the bubble over the clearing point of 5CB (307.8 K), then let LC cool down and return to nematic phase again. The series of diagrams show the alignment of LC is repeatable when temperature goes up and down.
Large amounts of high-Q (≥107) optical modes can be observed in hollow-core MBR when we fine-sweep the laser frequency from 1550 nm to 1560 nm. The typical optical mode, which can induce shell-fluid hybrid optomechanical modes of LC-MBR, has the optical Q factor of 5.5 × 106. Figures 3(a), 3(b), 3(c) respectively show optomechanical spectra of the hollow-core, isotropic methylbenzene (MB)-core and anisotropic LC-core MBR with similar geometry at 20 mW laser-pump. At the same pump condition, hollow MBR only has one optomechanical peak, MB-MBR emerges one main peak and a low power peak, while LC-MBR comes out four different peaks. We numbered them sequentially for convenience. Note that the filling of LC or fluid will increase the effective mass of MBR, this will lead to a reduction of mechanical resonant frequencies. Higher harmonic vibrations, which are caused by the nonlinear transfer characteristic of the taper fiber-resonator coupling system, can also been recognized in these spectra [1].
Fig. 3 Optomechanical spectra of MBR when core is filled with (a) air, (b) methylbenzene (MB), (c) LC.
With the help of numerical finite-element simulation in fluid-shell MBR [31], we try to match the simulated eigen frequency of specific mode order with the experimental results, so as to identify the measured optomechanical modes. By precisely adjusting the geometry of MBR structure and boundary conditions, we find that the peak in hollow MBR corresponds to (1,1,0) breathing mode, while MB Peak I and II corresponds to (1,1,0) and (1,2,0) breathing mode respectively. This consequence is consistent with the previous fluid-shell simulations [7,22]. On the other hand, there is a less than 10% simulation deviation if we suppose that Nematic Peak #2 belong to the (1,2,0) breathing mode in LC-MBR. Other peaks in Fig. 3(c) should be related to the complex wineglass modes with non-zero azimuthal order. We would show solid proof that these modes are coming from the acoustic anisotropy of LC in the temperature tuning section afterwards.
3.1 Temperate tuning
The LC-MBR is heated within 298.0~306.0 K and 311.0~319.0 K with 2.0 K interval. After the temperature is stable for 10 min, optomechanical spectrum is recorded. Figure 4 shows the optomechanical spectra at different temperature when MBR is pumped at 20 mW. The left column is mechanical spectra of nematic LC-MBR with four main vibration modes, while the right column is the mechanical spectra of isotropic LC-MBR with only one vibration mode. Around the clearing point of LC, it is hard to excite optomechanical modes at the pump power of 20 mW.
Fig. 4 Optomechanical spectra of a LC-MBR when temperature changes (left side-nematic, right side-isotropic).
The dependence of the mechanical resonances on temperature is shown in Fig. 5(a). When temperature rises from 298.0 K to 306.0 K, the four resonances in nematic LC-MBR have different change slope (shift rate is from 4.7%-7.5%). The widest tuning range is 0.54 MHz and the smallest tuning range is 0.29 MHz. All data are summarized in Table 1. Figure 5(b) presents mechanical resonances of peak 2 at high spectral resolution. The quality factor of the mechanical resonance is 9.4 × 102, 8.1 × 102, 8.3 × 102, 8.9 × 102, and 7.8 × 102 respectively. Also shown in Fig. 5(a) is the dependence of mechanical resonance on temperature when LC is in isotropic phase. Mechanical frequency change is 0.30 MHz, and its shift rate is 4.3%.
Fig. 5 (a) Optomechanical mode’s peak shifts with rising temperature. (b). Frequency shift of nematic Peak 2 with fine resolution.
Table 1. Comparison for tuning ability of LC or methylbenzene-filled MBR (Temperature range: 298~306 K)
Based on the fundamental Lamb’s theory for spherical droplet [29], we can have the mechanical vibration frequency = [30], in which is density, R is resonator radius, and bulk modulus of medium B is proportional to the square of sound speed c2. For more complicated hybrid fluid-shell cavity, is still proportional to sound speed c. On the other hand, the relation of on fluid density is not that straightforward, it highly depends on the shape of the cavity [31].
For sound speed-temperature relationship (c-T) of 5CB LC, sound speed c has an acceleration reduction process before the clearing point, and reaches to a minimum value before undergoing the phase change (details in Appendix. A. Fig. 9) [32]. Density of LC decreases linearly with temperature (-T) [33], but its decreasing rate (≤1%) is quite smaller than sound speed’s decreasing rate (≈5%) within 298.0~306.0 K. So the line type of -T is dominant by sound speed-temperature relationship (c-T). The similarity between Fig. 5(a) and c-T relationship supports our opinion. The tuning ratio (7.5%) is greater than aerostatic pressure (2.9%) or stretching (5.5%)-induced vibration modes shift [21,22].
The reason that mechanical resonance around 307.8 K can hardly been observed mainly comes from the fact that sound attenuation increases steeply near clearing point [32] (details in Appendix. B Fig. 10). LC director will intensely fluctuate and the ordered alignment will rapidly vanish at this point. The huge dissipation of vibration energy leads to a significant rise on driving power to observe mechanical resonance in LC-MBR. Moreover, sound speed in LC [34,35] is highly anisotropic. Thus in Fig. 5(a), LC-induced axial anisotropic acoustic wave propagation in LC-MBR contributes to the multiple peaks. Based on Table 1, we know that frequency shifts Δ for these peaks under the same temperature interval of 298.0~306.0 K are quite different. This implies that these vibration modes experience different sound speeds [36].
As a comparison, we choose methylbenzene, a typical isotropic liquid, to do the same thermo-optomechanic experiment. Methylbenzene-filled MBR has a diameter of 141 μm and thickness of 10 μm. Optomechanical vibration peaks shift gradually as is shown in Fig. 6. Frequency shift Δ for methylbenzene(MB) Peak 1 and 2 are 0.062 and 0.066 MHz, which are at almost the same rate as is summarized in Table 1.This agrees with the fact that temperature dependence of methylbenzene’s sound speed is isotropic [37].
3.2 Magnetic field tuning
In a planar LC cell, when the magnetic field rises to some threshold point, LC will undergo a second-order structural phase transition, known as Magnetic Fréedericksz transition (MFT). LC molecules will reorient to satisfy minimum free energy condition. MFT threshold magnetic field is Hth = (π/d) (K / χa)1/2, in which d is LC cell’s thickness, K is bend elastic constant, χa is the magnetic anisotropy of LC [14].
In our experiment, magnetic field is applied parallel to axial direction of MBR as the illustrations of Fig. 7. The intensity of magnetic field increases gradually until it reaches the MFT threshold. Figure 8(a) is the experimental results of LC-MBR optomechanical spectra at different magnetic field intensity. The transition threshold is around 840 Oe. Before the threshold, mechanical resonance does not shift along with the magnetic field (the frequency shift before threshold is ≤ 0.01 MHz). After the threshold, the resonant frequency shifts swiftly, forming a clear phase transition behavior. The observed shifts of the two peaks are 0.38 MHz and 0.31 MHz respectively.
Fig. 7 Schematic drawings of magnetic field setup and the alignment of 5CB LC molecules at different levels of the external magnetic field. (a) No magnetic field or before the MFT threshold. (b) Just beyond the MFT threshold. (c) Far above the MFT threshold.
Fig. 8 (a) Optomechanical spectra with magnetic tuning. (b) Optomechanical mode’s peak shifts with magnetic tuning.
The threshold phenomenon implies that our DMOAP-alignment layer has a strong anchoring effect on the interface between LC and fused silica, and this effect prevents the rotation of LC molecules until magnetic field is large enough to reach MFT threshold. After that point, the LC in MBR re-orient. Because long axis of LC molecule has the maximal sound speed [38], magnetic field will lead to a smaller sound speed propagation in LC-MBR, therefore optomechanical resonance shifts to lower frequency in Fig. 8(b).
If we further increase the magnetic field above 860 Oe, no mechanical resonances can be observed in the regime of 1-100 MHz at 20 mW pump power. This is because wave propagation along the long axis of LC possesses the least attenuation coefficient [13]. The complex re-orientation of LC [39] in MBR over MFT threshold will cause greater loss and dissipation. In addition, if magnetic field is applied perpendicular to the axial direction of the microcavity, no optomechanical resonances can be found when magnetic field intensity is over MFT threshold. The main reason is that the applied magnetic field would cause LC to break the uniformity along the radial direction as soon as re-orientation of LC occurs, and hence generates even larger dissipation.
4. Conclusion
In summary, we successfully realized optomechanics in anisotropic LC-filled MBR. Acoustic and viscoelastic anisotropy of LC shows obvious effects to improve tuning capability of MBR in MHz regime. The mechanical resonances are highly dependent on temperature and external magnetic field, due to the high anisotropy of sound speed, attenuation and Magnetic Fréedericksz transition of LC. Potential applications of this system include precise control of optomechanical frequency-on-demand and tunable optomechanical frequency conversion [40].
Appendix A Temperature dependence of sound speed and density of 5CB LC
Fig. 9 (a) Temperature dependence of 5CB LC long axis sound speed [32]. (b) Temperature dependence of 5CB LC density [33].
Appendix B Sound attenuation of 5CB near clearing point
Funding
National Natural Science Foundation of China (NSFC) (11874122, 11474070, 61327008, 11074051) and Specialized Research Fund for the Doctoral Program of Higher Education (20130071130004).
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