Mechanical pressure exerted by circulating radiation on a mirror of a Fabry-Perot resonator or on the wall of a whispering gallery resonator couples optical fields to mechanical vibrations. Radiation pressure induced opto-mechanical coupling can lead to a regenerative oscillation of the mechanical modes. This parametric instability, which was first proposed and analyzed by Braginsky [1], has recently been demonstrated experimentally in silica microresonators, including silica toroids and silica microspheres [2–4]. Radiation pressure induced coupling has also been used to cool mechanical vibrations of optical resonators [5–8], including silica toroids. Whispering gallery modes (WGMs) with ultra high optical finesse make silica microresonators a unique system for the exploration and application of radiation pressure induced opto-mechanical effects.

WGMs formed through total internal reflection can be excited through evanescent waves. Techniques for evanescent excitation include frustrated total internal reflection in a high index optical prism and the use of evanescent waves from a tapered optical fiber. Studies of optomechanical effects in silica microresonators have thus far used a fiber taper for the excitation of WGMs. The fiber taper approach [9] is highly efficient and near unity coupling efficiency has been achieved. There are, however, also some limitations. The gap between the fiber taper and the resonator, which can be difficult to control in certain environments, affects the quality factor as well as the coupling efficiency.

WGMs can also be excited via direct free-space optical coupling. The free-space approach takes advantage of enhanced evanescent escape in local regions of WGMs in slightly deformed and non-axi-symmetric silica microspheres [10–12]. In this paper, we report experimental studies of opto-mechanical effects in these deformed silica microspheres, in which WGMs are excited via direct free-space evanescent coupling with a coupling efficiency as large as 45 %. The efficient coupling has enabled the realization of radiation-pressure induced mechanical oscillation.

 

Fig. 1. SEM images of a deformed microsphere viewed along the direction of short axis (a), long axis (b), and fiber stem (c). (d) Schematic for free-space evanescent excitation of WGMs near the equator. (e) Transmission spectrum obtained with free-space evanescent excitation. The deformed microsphere used has a diameter of 36 6 µm and a deformation near 2 %.

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Non-axi-symmetric silica microspheres used in our studies were fabricated by fusing together two microspheres with similar sizes with a focused CO₂ laser beam. To retain high optical finesse, the degree of deformation, defined as ε=r_a/r_b-1, where r_a and r_b are the radius for long and short axis, respectively, was gradually reduced through repeated heating to approximately 1~2 %. We monitor the degree of deformation by measuring the quality factor of the WGMs and also by measuring r_a and r_b directly. Figures 1(a)–(c) show scanning electron micrograph (SEM) of a deformed microsphere viewed along three different directions. The deformation can also be induced in a single microsphere with a pulsed CO₂ laser beam [13].

In a deformed optical resonator, the angle of incidence is no longer conserved. The evanescent decay length as well as the evanescent tunneling rate increases exponentially when the angle of incidence approaches the critical angle of incidence. As shown in earlier studies, for a nearly spherical but non-axisymmetric silica microsphere, the angle of incidence becomes closest to the critical angle in regions 45° away from either the long or short axis [10, 11]. As a result, WGMs in these resonators can feature strong and directional evanescent escape from these local regions.

For the excitation of WGMs in a deformed silica microsphere, a laser beam was focused to a region on the equator 45° away from either the long or short axis with the focal point just outside the sphere surface, as shown schematically in Fig. 1(d) [12]. A frequency-stabilized tunable Ti:Sapphire laser with λ~800 nm was used. The transmission of the laser beam was monitored, while the laser beam orientation and the laser focal spot size were adjusted in order to optimize the coupling efficiency. Figure 1(e) shows a transmission spectrum, in which we have subtracted from the input laser power the wavelength-independent scattering background of the laser beam by the microsphere (about 15% of the incident laser power). The absorption dip in the transmission spectrum indicates a maximum coupling efficiency as large as 45 %. The asymmetry in the transmission lineshape arises from interference between the transmitted laser beam and the emission from the excited WGM. Note that the transmission exceeds unity at frequency slightly above the WGM resonance since the wavelength-independent scattering background has been subtracted. The coupling efficiency is limited by the mode matching between the excitation laser beam and the directional mode pattern of the WGM. The directional far-field emission pattern from the WGM shows complex amplitude and phase variations. Further improvement in the coupling efficiency is possible by tailoring the amplitude and phase variations of the excitation laser beam.

 

Fig. 2. (a) Power spectrum of directional optical emission from a WGM measures the frequency and intrinsic linewidth of a mechanical resonance in a deformed silica microsphere. The excitation power used is far below the threshold pumping power for parametric instability. (b) Dots and squares are the experimentally obtained mechanical resonance frequencies in deformed microspheres of varying sizes. Solid, dotted and dashed lines are the calculated resonance frequency as a function of sphere size for mechanical modes with (n,l)=(1,0), (1,1), and (1, 2), respectively.

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To characterize mechanical modes that couple to WGMs in a deformed silica microsphere, we have measured the noise or power spectrum of the optical transmission at an excitation power far below the threshold pumping power, th P, for parametric instability. Thermal vibrations of the mechanical modes lead to corresponding changes in the optical round-trip path length. The effects of the thermal vibration or Brownian motion are thus imprinted on the power spectrum of the optical transmission of a WGM. Figure 2(a) shows as an example a power spectrum featuring a mechanical resonance frequency of 87.17 MHz and a linewidth of 5.6 kHz, which corresponds to a mechanical quality factor, Q_m=15,600. For this measurement, the frequency of the excitation laser beam was fixed and was tuned to slightly above the WGM resonance and the excitation power used was ~1% of the threshold pump power of parametric instability where the narrowing of mechanical linewidth due to radiation pressure is negligible [14]. Figure 2(b) plots mechanical resonance frequencies obtained for microspheres with diameters ranging from 25~40 µm. The mechanical quality factors for these mechanical modes range from 3,000 to greater than 10,000.

The fundamental mechanical modes in a homogenous and isotropic sphere with free-surface boundary conditions can be calculated with the well-established theory developed by Lamb [15]. Figure 2(b) plots the theoretical result of the lowest three fundamental modes with (n,l)=(1,0), (1,1), and (1, 2), respectively, where n and l are the radial and angular mode numbers, respectively. The material parameters used for the calculation are a mass density of 2.2 g/cm ³, a Young’s module of 73.1 GPa, and a Poisson ratio of 0.17 for silica. Overall, the mechanical frequencies are inversely proportional to the diameter of the sphere and are in good agreement with the experimentally obtained mechanical resonance frequencies. This, along with the observation of high mechanical quality factor, indicates that the fusion of two spheres and the small deformation do not adversely affect the properties of the mechanical modes. The small deformation, however, leads to a mechanical mode splitting in the range of 0.2~0.4 MHz for the l=2 mode [3,4]. Below threshold, the two frequencies are simultaneously observed. However, with increasing pump intensity, only one mode, which has a higher mechanical quality factor, goes in the regime of parametric instability. Note that the (1,1) mode is not optically active because there is essentially no optical path length change for this mode [4].

Under a continuous wave optical excitation, radiation pressure exerted by the circulating optical power in a WGM couples to relevant mechanical modes of the resonator, leading to the generation of Stokes and anti-Stokes optical sidebands [2] with frequencies given by ωS=ω-ω_m and ω_AS=ω+ω_m, where ω and ω_m are pump frequency and mechanical vibration frequency, respectively. The generation of the Stokes sideband corresponds to the excitation or amplification of the mechanical mode by the circulating optical beam, while the generation of the anti-Stokes sideband can lead to radiation-pressure-induced damping or cooling of the mechanical mode. The amplification or the damping is also accompanied by a frequency shift in the mechanical resonance. This opto-mechanical coupling can be theoretically described by the coupled equations of the mechanical displacement and the intra-cavity optical field. Using the rotating wave and the slowly varying envelope approximations for the field amplitude, the optomechanical damping rate, γ_optomech, and the shift of the mechanical frequency, Δω_m, induced by the opto-mechanical coupling are given, in stead state, by [2, 7],

where Δω=ω-ω ₀ with ω ₀ being the cavity resonance frequency, τ is the cavity decay time, n is the refractive index of the silica microsphere, κ is the external coupling efficiency, P is the input optical power, m is the effective mass of the mechanical oscillator, and R is the radius of the microsphere. κ is unity in the case of perfect mode matching [16].

As shown in Eq. (1), the Stokes or the anti-Stokes process can be resonantly enhanced when the relevant sideband is resonant with the cavity resonance. For blue detuning (Δω>0), the Stokes process overwhelms the anti-Stokes process and the optomechanical damping rate γ_optomech is negative. In this case, radiation pressure leads to an overall amplification of the mechanical oscillation. Above the threshold, i.e., when |γ_optomech|>γ, where γ is the intrinsic mechanical damping rate, the mechanical vibration enters the regime of parametric instability. Conversely, for red detuning (Δω<0), the optomechanical damping rate γ_optomech is positive and radiation pressure leads to an overall cooling of the mechanical oscillation. These processes were first analyzed by Braginsky in the adiabatic limit τω_m≪1 and were also discussed in the context of dynamic back actions [1, 6].

 

Fig. 3. The squares are the experimentally obtained linewidth (a) and the frequency (b) of a mechanical mode as a function of the detuning between the pump laser (with a power 0.75 P_th) and the cavity resonance. The mechanical mode features an intrinsic frequency of 86.8 MHz and intrinsic linewidth of 21 kHz. The deformed microsphere used has a diameter of 36 µm and a cavity linewidth is 56 MHz. The solid lines are result of a theoretical calculation discussed in the text.

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Figure 3(a) and (b) show the effective linewidth and frequency of a (1, 2) mechanical mode obtained as a function of Δω with an optical pumping power of 0.75 P_th. For the microsphere used in this measurement, the intrinsic mechanical frequency and linewidth are 86.8 MHz and 21.0 kHz, respectively and the optical cavity linewidth is 56 MHz. The experimental results are in general agreement with the theoretical calculation based on Eqs. (1) and (2), where we have used n=1.45, κ=0.1, m=5.4×10^-11 kg, and R=8 µm. Note that, for Δω<0, radiation pressure induced cooling has also been observed. The cooling process is complicated by optical bistability and will be discussed elsewhere.

Above the threshold pumping power for the parametric instability, the regenerative oscillation of the mechanical mode results in strong periodic oscillations in the optical transmission from the WGM, as shown in Fig. 4. For Figs. 4(a) and (b), the oscillation in the optical emission occurs at 118 MHz, corresponding to the regenerative oscillation of the (1, 0) mechanical mode. The pumping power used is 2.0 P_th with 15 P_th=mW. For Fig. 4(c) and (d), the oscillation in the optical transmission occurs at 98 MHz, corresponding to the regenerative oscillation of the (1, 2) mechanical mode. The pumping power used is 1.3 P_th with P_th=7.2 mW. The intrinsic mechanical quality factor for the (1, 0) and (1, 2) modes obtained at a pumping power far below P_th are 9800 and 9100, respectively. In these measurements, the pump laser is blue-detuned from the WGM resonance such that ω_S=ω-ω_m is nearly resonant with the WGM. Note that by setting the appropriate amount of pump detuning, regenerative oscillations of either the (1, 0) or (1, 2) mode can be realized. The results shown in Fig. 4 demonstrate that radiation pressured induced mechanical oscillations can be realized with free-space evanescent excitation.

 

Fig. 4. The temporal dependence and power spectrum of optical transmission from a WGM with the excitation laser power above P_th. For (a) and (b), the deformed microsphere has a diameter of 39 µm and the cavity linewidth is 120 MHz. For (c) and (d), the deformed microsphere has a diameter of 31 µm and the cavity linewidth is 98 MHz.

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In summary, we show that opto-mechanical effects including radiation pressured induced mechanical oscillations can be efficiently realized with free-space evanescent excitation of WGMs in a slightly deformed silica microsphere. The simplicity, the robustness, and the relatively high coupling efficiency of the free-space evanescent coupling approach should enable explorations and applications of radiation pressure induced opto-mechanical effects under a variety of environments, for example, for carrying out opto-mechanical studies in a cryogenic environment for the cooling of a macroscopic mechanical oscillator toward its quantum ground state [17–19].