## Abstract

We have studied the effect of geometry deformation on the mechanical frequencies and quality factors for different modes in the Whispering Gallery Mode (WGM) microresonators, that is unavoidable in the practical fabrication. The subsidence of the sphere and a more general condition with fewer symmetries and complex deformation of eccentricity, subsidence, and offset are first modeled in this paper, which could tune the mechanical frequency in a much wider spectral range than the pillar-diameter-induced perturbation. we also show that the mechanical quality factors for the non-whispering-gallery mechanical mode could be increased in the order of 4 magnitudes at a specific subsidence, and form a mechanical bound state in the continuum (BIC) which is induced by the symmetry breaking and reveals new mechanisms to confine radiation. A much broader BIC window width with higher mechanical quality factor could be achieved, which is of great importance in both fundamental research and scientific applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The optomechanical interactions in whispering gallery mode (WGM) microcavities such as microtoroids [1–3], microspheres [4–7], and microdisks [8–11] have been studied in the past two decades. Benefitting from the ultrahigh optical quality factor and small mode volume of the WGM microcavities, the optomechanical interactions between co-located optical and mechanical modes in the same resonator are strongly enhanced. Mechanical resonators have been widely utilized in optical information processing, such as optomechanical light storage [12,13], coherent optical wavelength conversion [14,15], and optomechanically induced non-reciprocity [16,17]. Moreover, mechanical resonators can serve as a universal transducer to mediate various quantum systems in distinct different spectral range with tunable coupling in quantum information processing [18–20]. For identical and scalable cavity optomechanics systems, frequency synchronization [21,22] and ultrahigh intrinsic mechanical quality factor are highly desirable, which requires isotropic homogenous symmetry of the resonator. However, the practical fabrication process always brings in geometry deformation of the mechanical resonator, which breaks the isotropic homogeneity and perturbs the mechanical vibrations. So it is vital to characterize the geometry-deformation induced perturbation for both fundamental research and scientific applications. In this paper, we study the geometry-induced perturbation on the mechanical frequencies and intrinsic quality factors dominated by the clamping loss for different spheroidal modes in the WGM microresonators. Finite-element method (FEM) is employed to simulate the mechanical motion in suitable boundary conditions. The pillar geometry such as pillar height and diameter, as well as the sphere deformation including eccentricity, subsidence and offset, which are defined as the relative deviation from the sphere, the distance change of the sphere center to the top of the pillar, and the offset of the sphere center to the pillar axis respectively, are well considered. Especially, more sophisticated geometry but general condition with complex deformation and few symmetry is first modeled in this paper. We found that the symmetry breaking generated by deformation in the system induces the formation of mechanical BICs in the resonator. Benefitting from its wider tuning range in mechanical eigenfrequency and much broader BIC window width with higher mechanical quality factor, the off-centered subsided microspheroid shows importance and potential in both fundamental research and scientific applications.

The mechanical vibration modes in an isotropic homogeneous elastic sphere with free boundary conditions was first formulated by H. Lamb [23] and later described with equations [24]

where**is the displacement vector,**

*u**ρ*is the mass density,

*λ*=

*σE*/[(1 +

*σ*)(1-2

*σ*)] and

*μ*=

*E*/[2(1 +

*σ*)] are Lamb constants with

*E*denoting the Young’s modulus and

*σ*denoting the Poisson ratio of the material. The general solution of Eq. (1) is written as

*j*is the spherical Bessel function,

_{l}*Y*is the spherical harmonic function,

_{l}^{m}*V*is the longitudinal sound velocity for

_{k}*k*= 1 and the transverse sound velocities for

*k*= 2, 3.

*ν*is the vibrational frequency characterized by the mode numbers (

_{n,l,m}*n*,

*l*,

*m*) where

*n*(

*n*= 1, 2, …) is the radial mode number,

*l*(

*l*= 0, 1, 2, …) is the angular momentum mode number and

*m*(

*-l ≤ m ≤ l*) is the azimuthal mode number. It is noted that the vibrational modes (

*n*,

*l*,

*m*) are (2

*l*+ 1)-fold degenerate for each

*l*. Two classes of modes could be inferred from Eq. (2). For optomechanical application, only one class of modes with optical path length change can be excited by radiation pressure, which are recognized as spheroidal modes. The other class of modes are those that only involve shear stress other than radial displacement or volume change, which are recognized as torsional modes and could not be excited by radiation pressure. It has been proved that the eigenvalues of spheroidal modes (

*n*,

*l*) are inversely proportional to the diameters (

*D*) of the elastic spheres by numerical simulation [4,25]. For more sophisticated geometries such as a deformed silica sphere supported by a silicon pillar, namely the deformed chip-based silica microsphere, it is difficult to obtain analytical solutions for the eigenvalues. In this paper, Finite-element method (FEM) is employed to simulate the mechanical motion in suitable boundary conditions.

## 2. Mechanical motion of silica microspheres

For the ideal case, free silica microspheres without supporting structure, the FEM results agree well with the analytic calculations, that the eigenfrequencies of spheroidal modes are inversely proportional to the diameters of the elastic spheres, as *ν*_{(1,0)} = 4563 m/s/D for a radial breathing mode and *ν*_{(1,2)} = 3150 m/s/D for a quadrupole mode by FEM, with a deviation less than 0.1% compared with numerical results [26]. Results of *ν*_{(1,1)} = 3886 m/s/D, *ν*_{(1,3)} = 4647 m/s/D, *ν*_{(1,4)} = 5924 m/s/D, and *ν*_{(1,5)} = 7122 m/s/D could also be obtained by FEM. Figure 1 shows a series of eigen modes with their eigenfrequencies νn,l and mode shapes for *n* = 1, 2, *l* = 0, 1, 2, 3, 4, 5, and *m* = 0 obtained from FEM.

However, considering that the practical silica microspheres are usually fabricated by melting the tip of a single mode optical fiber with CO_{2} laser or high-voltage arc generated with a fusion splicer, as well as by melting the chip-based silica microdisks into spheres with CO_{2} laser, the microspheres are all attached to the ends of pillars. The geometry-induced perturbation should be well considered, such as the relative pillar height and diameter, the deformation of the resonator itself such as offset, eccentricity, subsidence, or more complicated deformation with fewer symmetries. Besides, the existing pillar can also transfer heat out of the sphere and reduce the laser-induced heating, which is another issue to be considered in experiments. Considering the identical vibrating mechanism for the chip-based and fiber-stem-attached microspheres, except for a subtle difference in vibrational frequencies and mechanical quality factors caused by the Poisson ratio and the Young’s modulus of different supporting pillar materials [27], but much lower threshold optical power for heating-induced optical bistability of the chip-based silica microsphere and its potential application in a cryogenic vacuum environment [7], the chip-based silica microsphere model with a silica microsphere smoothly connected to a silicon pillar on top of a perfectly matched layer is built and characterized based on finite element method, as shown in Fig. 2. This structure is parameterized by the diameter of the silica sphere, the diameter and height of the silicon pillar. According to the fabrication process, the silica volume or mass of the sphere are equivalent to the disk if ignoring the tiny silica loss during the reflowing process with CO_{2} laser. And the silicon pillar is rotationally symmetric with a height similar to the radius difference of the disk and the pillar, due to the isotropic nature of the silicon etching with XeF_{2} or HNA.

## 3. Geometry-induced perturbation

#### 3.1 Pillar-geometry-induced perturbation

In this paper, we study the geometry-induced perturbation on the mechanical frequencies and quality factors dominated by the clamping loss for different spheroidal modes in WGM resonators. First, pillar-geometry-induced perturbation are considered. Here, Δν is the eigenfrequency detuning compared with the free sphere, and *Q*_{m} is the intrinsic mechanical quality factor only dominated by the clamping loss. The relative pillar height *h* and diameter *d* are defined as the ratio of the pillar height and pillar diameter to the sphere diameter. In the silicon etching process, the pillar height changes with the mask or disk radius, while keeping the sphere diameter unchanged by varying the thickness of the silica layer. For the radial breathing mode (RBM), the radiation pressure couples strongly to the pillar in the axial direction with a periodic oscillating condition for the pillar height, which explains the oscillation of mechanical frequencies and quality factors for the fundamental RBMs (1, *l*, 0) with *l* = 0, 1, 2, 3, 4 in Figs. 3(a) and 3(b). For *l* ≥ 5, the mechanical quality factor of RBMs is smaller than 100. While for the non-RBMs (*m* ≠ 0), the vibration of the sphere could couple to the pillar in both the axial and transverse directions with *m* = 1 or decouple from the pillar with *m* ≥ 2, resulting in negligible changes in Figs. 3(c) and 3(d). The oscillating perturbation on RBMs suggests that the ratio of the pillar height to the disk radius should be adjustable, where anisotropic etching such as the KOH wet etching could be introduced [28].

Meanwhile, the mechanical vibrations could also be modulated by the relative diameter of the silicon pillar *d*. The pillar-diameter-induced perturbation on the mechanical frequencies and quality factors depend on the vibration shape. As shown in Figs. 4(a) and 4(c), the mechanical frequencies increase monotonically with the increased relative pillar diameter except for RBM (1,4,0) and non-RBM (1,3,1). In the upper zoomed-in view of Fig. 4(c), the spheroidal mode (1,3,1) is perturbed by a nearby torsional mode, while the crossing trend due to their different dependence on pillar diameter is actually avoided, and exhibits a bounce in mechanical frequency and quality factor. As to the mechanical quality factor, it decreases dramatically with the increased pillar diameter for non-perturbed RBMs in Fig. 4(b), and a long-diameter-period oscillation for acoustic whispering gallery modes with *m* ≥ 2 in Fig. 4(d). It is noted that a singular peak for non-RBMs with *m* = 1 exists, and improves the mechanical quality factor in about three orders of magnitude in Fig. 4(d), indicating the formation of a mechanical bound state in the continuum (BIC) in the chip-based silica microsphere system [27,29–32]. Due to the symmetry breaking induced by the pillar, the elastic energy loss through the pillar is forbidden. BICs provides new mechanisms to confine radiation, which is of great significance for both fundamental research and applications.

#### 3.2 Sphere-deformation-induced perturbation

For the acoustic whispering gallery modes such as Brillouin modes, larger sphere diameter is usually preferred for a higher mechanical quality factor and denser optical modes for mode matching. As the sphere size increases, asymmetric geometries such as spheroid and subsided spheroid are unavoidable in fabrication, and it is vital to characterize the sphere-deformation-induced perturbation on mechanical vibrations. Here, as shown in Figs. 5(a)-5(c), deformed sphere geometries have been created in Comsol Multiphysics with the deformation of eccentricity, subsidence and offset, which are defined as the relative deviation from the sphere, the distance change of the sphere center to the top of the pillar, and the offset of the sphere center to the pillar axis, respectively.

Unlike the deformed fiber-stem microsphere by fusing two spheres together into one [33], the chip-based microsphere is naturally melted from a microdisk. Since the original radius of a microdisk is much larger than its thickness, the pre-existing spatial heterogeneity induces the transformation of an expected sphere into a spheroid with identical equatorial radius and a distinct polar radius. The relative eccentricity *ε* is defined as $\epsilon =\frac{{R}_{a}-R}{R}$, where *R* is the radius of the expected sphere and *R _{a}* is the equatorial radius of the spheroid in Fig. 5(a), thus the polar radius

*R*equals to

_{b}*R*/(1 +

*ε*)

^{2}by the mass conservation of silica. Actually, the relative eccentricity describes the extension of the expected sphere in the equatorial plane and compress in the polar direction. Simulations in Fig. 6(a) suggest that the eigenfrequencies of a free spheroid are non-degenerate while (2

*l*+ 1)-fold degenerate for a free sphere mentioned above. Besides, the inversely proportional relationship between eigenfrequency and diameter in Fig. 1(a) didn’t fit in a spheroid, where the mechanical frequency could be monotonically increasing or decreasing for different azimuthal mode number

*m*with the same angular momentum mode number

*l*in Fig. 6(c). Figure 6(d) is the sphere-eccentricity-induced perturbation on mechanical quality factor. For adjacent modes with higher

*l*such as

*l*= 4 and

*l*= 5 in Fig. 6(b), the mechanical frequency of different

*m*is intersecting, which narrows the energy gaps between adjacent modes and provides a broadband mechanical spectrum with relative high mechanical quality factor.

For the chip-based silica microspheroid, when the gravity is comparable to the surface tension, it subsides during the reflowing process and eventually turns into a subsided microspheroid as shown in Fig. 5(b). The relative subsidence *s* is defined as $s=\frac{{H}_{0}-H}{{H}_{0}}$, where *H*_{0} and *H* are the distance from the top of the pillar to the center of the spheroid before and after subsidence. Here, the conditions of *s* > 0 and *s* < 0 are visually named as “mushroom shape” and “balloon shape”, respectively. The spheroid-subsidence-induced perturbation on mechanical frequency and quality factor are shown in Figs. 7(a)-7(d). All mechanical frequencies monotonically increase with the increased subsidence in a wider range of tens MHz, such as 27 MHz per unit subsidence for mode (1,4,1), which is about 17.3% of the mode frequency, and 9 times higher than the pillar-diameter-induced perturbation. Since the relative subsidence could efficiently tune the mechanical eigenfrequency with a broadband tuning range, it could potentially be used in frequency modulation and synchronization.

As for the mechanical quality factor, the subsidence slightly perturbs RBMs while greatly spoils the acoustic whispering gallery modes with *m* ≥ 2, about four orders of magnitude worse. Besides, the mechanical BICs with singular peak for *m* = 1 non-RBMs also appear. Compared with the perfect sphere, the subsidence breaks the isotropy of the resonator itself, but still maintains the rotational axial symmetry of the spheroid-pillar system, and supports the formation of the mechanical bound state in the continuum (BIC) in the chip-based subsided microspheroid system. Furthermore, the displacement of the mechanical BICs at the sphere-pillar cross-section is even symmetric in *x* and *y* direction but odd symmetric in *z*-direction as shown in Fig. 8(a), compared with a non-BIC mode such as the RBM where the displacement is odd symmetric in *x*-direction, tiny in *y*-direction and even symmetric in *z*-direction as shown in Fig. 8(b). The mismatching of the displacement-symmetry between the mechanical mode and the radiation pressure prevents the mode from coupling to the pressure wave in continuum, thus forms a mechanical bound state in the continuum with ultrahigh mechanical quality factor.

#### 3.3 Symmetry-broken-deformation for mechanical BIC

Besides the symmetric deformation analyzed above, the symmetry-broken deformation such as an off-centered pillar also occurs and perturbs the mechanical vibration significantly, especially for the mechanical BIC. In the fabrication process, the sphere could be dislocated on top of the pillar with offset due to the unbalanced control of the reflowing CO_{2} laser. The relative offset of the sphere center to the pillar axis *p* is defined as the ratio of the offset distance to the sphere radius. The off-centered sphere breaks the rotational symmetry in the whole sphere-pillar system but maintains it for the sphere resonator itself, slightly perturbs RBMs but greatly spoils the mechanical quality factors of the acoustic whispering gallery modes with *m* ≥ 2 as shown in Figs. 9(a)-9(d). It is noted that the azimuthal degeneracy has not been broken by the off-centered sphere and the mechanical BIC could not form by purely varying the relative offset.

When pillar offset is introduced in the subsided microspheroid, the rotational axial symmetry degenerates into an inversion symmetry. This complex deformation geometry is first created in this paper with parametric surface in COMSOL Multiphysics as shown in Fig. 5(c). The pillar offset together with the subsidence deformation barely affect the azimuthal degeneracy for |*m*| ≥ 2 modes, but break the azimuthal degeneracy with hundred kHz frequency difference (δν) for the originally degenerate vibrational modes (*n*, *l*, | ± 1|) in Fig. 10(a). In Fig. 10(b), multiple peaks are discovered for the mechanical quality factor when varying the subsidence value with a fixed offset, indicating multiple windows for mechanical BIC modes with ultrahigh quality factor. Here, we define the window width for mechanical BIC modes as the range of the independent variables with the mechanical quality factor *Q*_{m} higher than 10^{4}. The mechanical BICs in the centered subsided spheroid have a close window width to the off-centered one, but could reach higher maximum mechanical quality factors. Compared with the chip-based microsphere in Fig. 4(d), the subsided spheroids with and without offset both have much broader BIC window width, about 10 times broader window width than the sphere, respectively. Thus, it is much easier to support a mechanical BIC mode in the subsided spheroid, which makes it a preferred candidate for the BIC application.

## 4. Summary

In summary, we have studied the geometry-induced perturbation on the mechanical frequencies and intrinsic quality factors dominated by the clamping loss for different spheroidal modes in the WGM microresonators. The subsidence of the sphere and a more general condition with fewer symmetries and complex deformation such as an off-centered subsided microspheroid are first modeled in this paper. The symmetry breaking generated by deformation in the system induces the formation of mechanical BICs in the resonator, which reveals new mechanisms to confine radiation. Benefitting from its wider tuning range in mechanical eigenfrequency and broader BIC window width with higher mechanical quality factor, the off-centered subsided microspheroid shows importance and potential in both fundamental research and scientific applications.

## Funding

National Key Research and Development Program of China (2017YFA0303700); the National Natural Science Foundation of China (20171311628).

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