Comprehensive numerical investigation of intrinsic mechanical modes of one-dimensional optical waveguides

Abstract

Typical suspended one-dimensional waveguides, tapered optical fibers (TOFs), combine sub-wavelength optical confinement with mechanical flexibility, enabling applications in optomechanics, quantum optics, atomic physics and sensing. However, the intrinsic mechanical modes (IMMs) of TOF, including flexural, longitudinal, and torsional modes, remain underexplored. Through analytical formulations and simulations, we comprehensively and explicitly explore the classification of various mechanical modes of a TOF. The characteristics of various TOF IMMs are investigated in detail, including frequencies, displacements, and vibration patterns. Novel degenerate IMMs are confirmed, and these modes result from dynamic coupling between the modes in the tapered and waist regions. It is also revealed that the IMMs are strongly geometry-dependent. The frequencies scale with the length and diameter of the waist, whereas linear tapers exhibit stronger energy localization than exponential tapers. The elaborate design and engineering of TOFs allow for precise mechanical resonance tuning and seamless integration with nanophotonic systems, positioning them as a versatile platform for quantum precision metrology and advanced optomechanical technologies.

Graphical abstract

Keywords

one-dimensional optical waveguides / non-degenerate intrinsic mechanical modes / mode coupling

1 Introduction

Micro/nano-mechanical resonators have emerged as indispensable tools in scientific research and technological innovation [1–3]. Owing to their ultralow masses and high-quality factors, these resonators offer exceptional sensitivity for detecting a wide range of physical quantities, including force, mass, pressure and acceleration [4]. Their excellent integrability and scalability facilitate the fabrication of large-scale arrays and their incorporation into complex system architectures [5]. Consequently, micro/nano-mechanical resonators are widely utilized across diverse fields, including sensing technologies [6], optical systems [7], and quantum technologies [8].

Micro/nano-photonic one-dimensional waveguides, which possess outstanding optical properties, have attracted growing interest over the past two decades [9–13]. As a practical experimental platform, optical waveguides have been widely applied in areas such as optomechanics, waveguide quantum electrodynamics (QED), and quantum information processing [14–16]. The strong evanescent field near the waveguide surface allows effective coupling with nearby emitters, enabling light-matter interactions in systems involving atomic and solid-state emitters [17–19]. Additionally, the integration of optical waveguides with various microstructures is often used to modulate and control such interactions [20–23]. Micro/nano-photonic one-dimensional waveguides, with nanoscale mechanical properties, have the potential to develop hybrid systems that combine optics and mechanics, thereby broadening their range of applications. In the field of cavity optomechanics and information processing, mechanical modes can serve as carriers for information encoding when coupled with optical modes. This has the potential to enhance the functional complexity, precision and information-carrying capacity of the system [24, 25]. However, mechanical modes pose challenges for practical applications [26]. For instance, in systems that involve emitters or micro/nano-structures, the interaction between optical modes and mechanical vibrations can lead to instability in light-matter coupling [27, 28], thereby limiting the performance of these systems. Furthermore, in quantum systems based on micro/nano-photonic waveguides, improvements in the signal-to-noise ratio and the inherent security advantages of quantum mechanics offer additional opportunities for application [29, 30].

As a typical one-dimensional waveguide, TOF with a high aspect ratio, as the simplest suspended waveguide, has important applications ranging from optomechanical and optical sensing to quantum optics and atomic mechanics [31, 32]. The TOF structure can support IMMs, including flexural, longitudinal and torsional modes [30]. However, degenerate IMMs, which are inherent to TOFs despite their prevalence, remain underexplored both theoretically and experimentally. These modes, characterized by overlapping eigenfrequencies and spatially intertwined vibration patterns, can critically influence optomechanical coupling efficiency, energy dissipation pathways and quantum coherence lifetimes in nanophotonic systems. Notably, mechanical oscillations in the TOFs induce significant heating in adjacent atomic arrays, drastically reducing the trapping lifetime of atoms [33]. TOF torsional modes have been explored experimentally using direct electrical excitation, mechanical excitation, and optical modulation [28, 34, 35]. Flexural modes of TOFs are systematically studied and measured using the near-field scattering of the TOF evanescent field by a microfiber tip [36]. Nevertheless, all types of TOF IMM have not been understood systematically yet; thus, it is essential to conduct numerical simulations of the TOF IMMs.

In this work, we systematically simulated and analyzed the degenerate IMMs of a TOF with an ideal geometric structure. Based on analytical mathematical formulations, we verified several novel IMMs that had not been previously identified. We analyze the relationship between degenerate IMMs and mode orders, revealing that the mechanical behavior of TOFs is strongly governed by their geometric structure. Using finite element method (FEM) simulations, we further characterize the dependence of displacement and eigenfrequencies on mode order, demonstrating excellent agreement between numerical results and theoretical predictions. By tuning the TOF geometry, specifically by adjusting the waist length and diameter, numerical simulations show that the frequencies of the IMMs can be precisely tailored. Additionally, the shape of the tapered region significantly influences the spatial distribution of vibrational displacements, with exponential tapers exhibiting broader energy localization compared to linear profiles. The simplicity of the TOF structure facilitates its straightforward fabrication and seamless integration with other micro/nanophotonic components. These simulation results can contribute to the mechanical design of TOFs and other waveguide-based systems, including integrated photonic and MEMS devices. This work is both feasible and practical for engineering one-dimensional waveguides that can effectively control the vibration frequencies and amplitudes. Having a comprehensive understanding of one-dimensional waveguide IMMs holds significant potential for applications in atomic physics, quantum precision measurement and optomechanics [27, 36–39].

2 Structural model and typical vibration modes

Figure 1 schematically illustrates the structural diagram of a typical TOF, featuring a cylindrical cross-section. The TOF consists of three main regions: standard fiber, tapering, and waist regions. Experimentally, the TOF profile can be precisely controlled by adjusting the fabrication parameters. Considering both the adiabatic conditions required during fabrication and practical application constraints, two typical taper profile structures are designed with linearly or exponentially varying radii. The TOF waist has a uniform radius that is smaller than the wavelength used in experiments.

FEM is employed to numerically simulate and extract the IMMs of the TOF. The typical IMMs of TOF include intrinsic flexural mechanical modes (IFMMs), intrinsic longitudinal mechanical modes (ILMMs), and intrinsic torsional mechanical modes (ITMMs). The eigenfrequencies of the three types of IMMs can be determined. In the experiment, the suspended TOF are fixed at two standard fiber regions. The displacement of the IMMs is determined by the TOF radius. By using the equipartition theorem, ⟨\delta z_i^2⟩=k_{B}T/k_i, the amplitude of the TOF can be obtained, where k_B is the Boltzmann constant, T is the temperature, and k_i is the stiffness of the TOF. The radius of the TOF waist is three orders of magnitude smaller than that of standard fibers. Thus, the IMMs are concentrated in the waist and tapered regions. Due to the above reason, a simplified model is adopted for simulations. The TOF model comprise only the waist and tapered regions. This model can further reduce the simulation time. The TOF waist is a uniform cylinder with a region of length L and radius r, where L\gg r.

The IFMMs primarily originate from the nonuniform distribution of transverse shear forces along the longitudinal direction of the TOF. Using the Euler−Bernoulli beam theory, The n order IFMM eigenfrequencies of TOF with leghth L and uniform radius r are given by

(1)f_F^n=\sqrt{{\displaystyle \frac{E}{\rho }}}{\displaystyle \frac{{(2n+1)}^2\pi r}{16L^2}},n=1,2,3\cdots .

where E is the Young’s modulus and \rho is the mass density of the TOF.

The ILMMs originate from the longitudinal strain along the length of the TOF. By applying classical dynamical equations, the vibration behavior of these longitudinal modes can be approximated using a one-dimensional model. The vibration frequency of the n order ILMM can be expressed as

(2)f_L^n=\sqrt{{\displaystyle \frac{E}{\rho }}}{\displaystyle \frac{n}{2L}},n=1,2,3\cdots .

For the ITMMs, considering the tangential force distribution in the TOF, the eigenfrequency of the n order ITMM can be described as follows:

(3)f_T^n=\sqrt{{\displaystyle \frac{E}{\rho }}}{\displaystyle \frac{n}{2L}},n=1,2,3\cdots .

The waist radius r is set to around 250 nm and the waist length is set to L=4\mathrm{m}\mathrm{m}. The radius of the tapered region varies linearly along the TOF axis, and the tapered angle is 1.97 mrad. The material of the TOF is silica with a refractive index of 1.45. The Young’s modulus is E=73\mathrm{G}\mathrm{P}\mathrm{a} and the mass density is \rho =2320\mathrm{k}\mathrm{g}/{\mathrm{m}}^3.

Figure 2(a) shows the axial distribution of vibrational displacements on the TOF surface for the 1^{\mathrm{s}\mathrm{t}}- to 5^{\mathrm{t}\mathrm{h}}-order IFMMs. The vibration displacement of the TOF peaks in the waist region due to the abrupt reduction in radius. Owing to the strong radius dependence of flexural rigidity, IFMMs are predominantly localized in the ultra-thin waist region, where reduced stiffness enables large transverse displacements. As the TOF transitions into the tapered regions with gradually increasing radius, the displacement rapidly decays to negligible levels. Furthermore, for the n^{\mathrm{t}\mathrm{h}} order IFMM, the waist exhibits n antinodes for the displacement peaks. Figure 2(b) shows the 3D schematic diagrams of second-orders degenerate IFMMs of TOF profiles which vibrate along two orthogonal directions [40, 41]. The TOF supports two degenerate IFMMs arising from orthogonal vibrational polarizations along the principal axes of its cross-section.

Figure 3(a) illustrates the variation in frequencies as a function of the mode order for the IFMMs [40]. Ideally, the two degenerate modes exhibit identical eigenfrequencies and displacement. However, the modes emerge minor splitting due to geometric asymmetry or fabrication-induced anisotropy in experiments [40]. The frequencies exhibit a quadratic increase with respect to the mode order. Summarized from the data in Fig. 2(a), the maximum displacement as a function of the mode order is plotted in Fig. 3(b), which shows an exponential decay.

Figure 4 shows the distribution of the maximum displacement on the TOF surface for the 1^{\mathrm{s}\mathrm{t}}−5^{\mathrm{t}\mathrm{h}}-order ILMMs along the TOF axis and the corresponding 3D schematic diagrams illustrating the mechanical mode profiles, including the displacement directions (red arrows) and nodal structures in the TOF. The first type of mode is mainly localized in the waist region of the TOF, called localized ILMMs. The localized ILMMs are represented by light-colored solid curves in Fig. 4(a), and the corresponding 3D schematic diagrams illustrating the 2^nd-order modes are shown in Fig. 4(b). In the waist region, the n^{\mathrm{t}\mathrm{h}} order localized ILMMs exhibit exactly n antinodes with high displacements. In the tapered region, the displacements of the antinodes are much smaller than those in the waist region, which can be negligible. The second kind of mode is distributed across both the tapered and waist regions, called non-localized ILMMs. Furthermore, the non-localized ILMMs can be further categorized into two distinct mode groups based on the different vibration patterns, as shown in Fig. 4(a).

Localized and non-localized ILMMs exhibit different characteristics in the tapered region. In contrast, non-localized ILMMs exhibit a spatially extended energy distribution across the entire TOF structure. The two non-localized ILMMs are represented by dark-colored solid curves (ILMM 1) and dotted curves (ILMM 2) in Fig. 4(a), whereas the corresponding 3D schematic diagrams illustrating the 2^{\mathrm{n}\mathrm{d}}-order modes are shown in Figs. 4(c) and (d). Similar to localized ILMMs, in the waist region, the non-localized ILMMs exhibit distinct antinodes with high displacements. However, in the tapered region, non-localized ILMMs displayed significant displacements, which can be attributed to the gradual radius transition in the waist-taper interface. This distinction highlights the interplay between the geometric confinement in the waist and the gradual radius modulation in the tapered regions, which collectively shape the mode-specific displacement distribution. For the two non-localized ILMMs, two distinct displacement patterns in the waist region emerge based on the symmetry of the tapered region motion: i) Co-moving tapers (same vibration direction at both tapers): maximum displacement occurs at the waist center owing to constructive wave interference, concentrating energy in the waist center regions. ii) Counter-moving tapers (opposite directions at both tapers): The minimum displacement at the waist center results from destructive wave interference, redistributing energy to the two end regions of the waist. Notably, localized modes have greater displacement than non-localized modes. Moreover, the phase inversion between adjacent nodes reflects the formation of a standing wave. These modes share identical eigenfrequencies but exhibit spatially distinct vibrational patterns, reflecting the interplay between the geometric symmetry and mechanical boundary conditions.

The frequency as a function of the mode order for the ILMMs is shown in Fig. 5(a). The red open squares correspond to localized ILMMs, whereas the blue solid squares and yellow circles represent the two non-localized ILMMs. Localized and non-localized ILMMs exhibit two distinct frequency-scaling behaviors: i) A linear increase with mode order for all ILMM types, consistent with the theoretical prediction for high-aspect-ratio waveguides. ii) Localized ILMMs consistently displaying higher frequencies than non-localized ILMMs. iii) The frequencies of the two distinct nonlocalized ILMMs are degenerate. The displacement of the localized ILMM decreases as the increasing mode order increases, as indicated by the red open squares in Fig. 5(b). However, the displacement exhibited a fluctuating trend with increasing mode order, as indicated by the yellow and blue solid squares in Fig. 5(b). This trend results in the interference between the two degenerate non-localized ILMMs, which is more pronounced for TOFs with shorter waists. This will be presented in the next section. This hierarchical classification reveals how asymmetric tapering introduces additional degrees of freedom for tailoring mechanical mode dynamics, with implications for optomechanical energy transfer and resonant control in nanophotonic systems. This stratification is hypothesized to arise from mechanical mode coupling between the waist-taper region, where the interplay of geometric constraints and radius-dependent stiffness modulates the energy distribution throughout the structure.

Figure 6 shows the distribution of the maximum displacement for the 1^{\mathrm{s}\mathrm{t}}−5^{\mathrm{t}\mathrm{h}}-order ITMMs along the TOF axis and the corresponding 3D schematic diagrams illustrating the mechanical mode profiles, including the rotating directions (red arrows) and nodal structures in the TOF. Similar to ILMMs results from the axial motion of the TOF, that torsional motion along the azimuthal direction generates ITMMs. These ITMMs also exhibit a localized ITMM and two degenerate nonlocalized ITMMs, analogous to their longitudinal counterparts. The localized mode and the two non-localized modes are represented by the light-colored solid curves, dark-colored solid curves (ITMM 1) and dotted curves (ITMM 2) in Fig. 6(a), while the corresponding 3D schematic diagrams illustrating the 2^{\mathrm{n}\mathrm{d}}-order modes are shown in Figs. 6(b)−(d). As the mode order increases, the vibrational displacements of the non-localized ITMMs gradually migrate toward the waist region, eventually overlapping spatially with the waist-region modes. The rotating direction in the tapered regions on both tapers of the TOF gives rise to two degenerate, non-localized ITMMs. Depending on whether the tapers rotate in the same or opposite directions, the vibrational behavior changes significantly: co-rotating tapers lead to maximum displacement at the waist center owing to constructive wave interference, whereas counter-rotating tapers result in minimal displacement at the waist center as destructive wave interference redistributes the energy toward the tapered regions. The difference in behavior arises from the interaction between rotational symmetry and geometry: co-rotating tapers strengthen mode localization, whereas counter-rotating tapers cause energy to spread out. Such tunable torsional dynamics are critical for optomechanical torque transduction and spatially selective phonon routing.

Figure 7(a) shows the frequency dependence of the mode order for the ITMMs. The red open squares correspond to localized ITMMs, whereas the blue solid squares and yellow circles represent the two degenerate non-localized ITMMs. The ITMMs exhibit frequency-scaling behaviors similar to those of the ILMMs, with the displacement showing a fluctuating trend as the mode order increases. This fluctuation is clearly observed in Fig. 7(b), where the displacement of the localized ITMM (red open squares) decreases with increasing mode orders. This hierarchical classification reveals that non-localized ITMMs exhibit mode hybridization, where tapering induces coupling between co- and counter-rotating torsional waves. The interplay between rotational symmetry and geometric phase matching leads to the splitting of degenerate frequencies and redistribution of vibrational energy.

3 The relationship between IMMs and TOF’s geometrical parameters

TOF profile sensitively determines the characteristics of the IMMs. It is necessary to investigate how the geometrical parameters of the TOF influence the IMMs. The interplay between the reduced radius of the tapered region and the geometric confinement of the waist region results in richer vibration mode profiles than those of uniform fibers. This structural heterogeneity gives rise to mode-localization effects, frequency splitting, and spatially modulated displacement patterns that are highly sensitive to dimensional variations. In applications such as optical or mechanical sensing, these IMMs can be used to amplify sensitivity. These nuanced vibrational characteristics generate multidimensional datasets, encompassing frequency shifts and amplitude gradients, which enable the detection of sub-nanometer-scale perturbations. Importantly, this deeper understanding of waist-dominated mode behavior provides a pathway for optimizing sensor architectures through tailored mechanical impedance matching and mode-selective transduction. By correlating the geometrical parameters with the mechanical response, our findings offer a framework for enhancing both the precision and reliability of optomechanical sensing.

Figure 8 illustrates the dependence of the IMM frequencies on the waist length of the TOF. Figure 8(a) shows a schematic of the TOF profile for the seven waist lengths: 1, 3, 5, 7, 9, 12, and 15 mm, with a fixed waist radius of 250 nm. The TOF features an adiabatic taper with an angle of 1.97 mrad, ensuring smooth radius reduction and minimizing modal coupling losses while preserving the phase-matching conditions across the transition region. Figure 8(b) shows the vibration frequencies and displacement as functions of waist length for the {50}^{\mathrm{t}\mathrm{h}}-order IFMMs. This indicates that the TOF with a longer waist supports a greater number of IFMMs within the same frequency range, which is a direct consequence of the expanded mechanical degrees of freedom. Figures 8(c) and (d) show the vibration frequencies as functions of waist length for the {20}^{\mathrm{t}\mathrm{h}} ILMM and the {10}^{\mathrm{t}\mathrm{h}} ITMM, respectively. Figures 8(e) and (f) further illustrate the displacement trends as functions of the waist length of the {20}^{\mathrm{t}\mathrm{h}} ILMM and the {10}^{\mathrm{t}\mathrm{h}} ITMM, respectively. For ILMMs and ITMMs, longer waist lengths enable larger effective wavelengths while supporting lower mode orders within the same physical structure. The results show a negligible change in displacement with increasing waist lengths, which is attributed to the reduced structural stiffness and elongated vibration wavelengths. However, for longer waist lengths, the periodicity of the displacement modulation decreases with the mode order, indicating stronger wave interference effects in these structures. This trend underscores the enhanced spatial complexity and mechanical adaptability of the vibrational modes in extended TOFs, which arise from the distributed strain energy and boundary condition effects.

The interplay of geometric scaling reveals that longer TOFs inherently enable the broader tunability of mechanical resonance characteristics. The ability to modulate the IMMs frequencies through the waist length coupled with the emergence of densely packed high-order resonances offers a pathway to optimize optomechanical systems for applications such as ultrasensitive displacement sensing, resonant filtering, and broadband energy harvesting. These findings highlight the critical role of structural parameters in the dynamics of ITMMs for advanced photonic-mechanical hybrid technologies.

Figure 9 presents the frequency dependence of the IMMs on the waist radius for the TOFs. Figure 9(a) schematically illustrates the waist radius investigated: 50, 150, 250, 350, 450, 1000, and 2000 nm, with a fixed waist length of 4 mm. The TOF features an adiabatic taper with an angle of 0.87 mrad, ensuring a gradual radius transition. Figures 9(b)−(d) present the vibration frequencies (red open circle) and displacement (purple solid circle) as functions of waist radius for the {50}^{\mathrm{t}\mathrm{h}}-orders IFMMs, {10}^{\mathrm{t}\mathrm{h}}-order ILMMs, {10}^{\mathrm{t}\mathrm{h}}-order ITMMs. Notably, the IFMMs exhibit a monotonic frequency decrease with increasing waist radius across all mode orders. In contrast, the ITMMs and ILMMs show a consistent upward trend in frequency under identical geometric scaling. Thus, TOFs with smaller waist radii support a greater number of IFMMs but fewer ILMMs and ITMMs within the same frequency range.

The results demonstrate a clear increase in displacement with longer waist lengths. For IFMMs, a smaller waist radius enhances the bending rigidity, directly elevating the resonant frequencies. Conversely, the ILMMs and ITMMs are governed by axial and torsional stiffness, which increase more slowly than the inertial mass, resulting in lower frequencies for larger radii. Furthermore, a reduced waist radius induces mechanical fragility. Sub-micron waist radius compromises structural robustness, increasing susceptibility to buckling or fracture under external stress. This geometric constraint modifies the vibrational characteristics by localizing the strain energy near the waist-taper interface. To systematically investigate how tapered profile geometries influence the mechanical mode characteristics of TOFs, we compared two types of TOF with linear and exponential tapered regions. Both linear and exponentially tapered fibers are frequently processed in experiments [42–44]. As illustrated in Fig. 10(a), both types share identical global parameters: a waist redius of 250 nm, waist length of 4 mm, and total taper length of 20 mm. The distinction lies solely in the functional form of the radius transition, that is, linear and exponential taper, enabling a direct comparison of the geometric effects on the vibration dynamics. Figures 10(b)−(d) show the displacement distributions along the TOF axis for the {25}^{\mathrm{t}\mathrm{h}} order IFMM, {50}^{\mathrm{t}\mathrm{h}} order ITMM, and {35}^{\mathrm{t}\mathrm{h}} order ILMM. For IFMM and ITMM, the linear-tapered TOF consistently localizes more vibrational energy in the waist region than the exponential taper. For the ILMM, while the peak displacements are comparable, the displacement of the linear taper exhibits a sharper decay along the TOF axis in the tapered region, indicating a stronger mode confinement. This comparative study demonstrates that linear-tapered profiles inherently suppress vibration displacements through enhanced mechanical rigidity and a spatially constrained energy distribution. Such behavior is advantageous for applications requiring robustness against environmental perturbations, such as ultra-stable optical trapping platforms and vibration-isolated photonic sensors. Conversely, exponential tapers offer superior displacement flexibility, making them preferable for resonant energy harvesting and high-sensitivity transduction systems. These findings provide a design framework for tailoring the TOF mechanical responses through geometric engineering.

4 Conclusions

In summary, we present a systematic investigation of IMMs, including IFMMs, ILMMs, and ITMMs, and further elucidate their dependence on key geometric parameters of TOFs: waist length, radius and taper profile. By analyzing the interplay between the structural features and mode characteristics in TOFs, we present strategies for achieving precise control over IFMMs, ILMMs and ITMMs, enabling tailored resonance properties for optomechanical applications. FEM simulations reveal that the dynamic coupling between the tapered regions and the central waist structure gives rise to degenerate IMMs, where distinct physical mechanisms govern each IMMs. For instance, IFMM degeneracy arises from two orthogonal transverse vibrations perpendicular to the fiber axis, whereas ILMMs exhibit degenerate pairs owing to co-directional and counter-directional axial motion along the fiber. Similarly, ITMMs originate from co-rotating and counter-rotating torsional waves encircling the fiber axis. The dependence of the vibrational displacement on the mode order reveals a periodic oscillation between the degenerate modes, directly confirming the mode coupling. Notably, TOFs with co-directional taper motion or rotation exhibit maximum displacement at the waist center, whereas counter-directional configurations produce minimal displacement there, which is consistent with constructive and destructive wave interference, respectively. It can be precisely controlled by varying the waist length, waist radius, and taper profile of the TOFs. Increasing the waist length enhances the IFMM displacement owing to the distributed strain energy but reduces the ILMM and ITMM displacements, whereas all IMMs exhibit lower eigenfrequencies as the structural stiffness decreases with extended vibration wavelengths. Conversely, larger waist radii elevate the IFMM frequencies by increasing the bending rigidity but suppress the ILMM and ITMM frequencies owing to the dominant inertial mass effects, alongside reduced displacement amplitudes across all modes from diminished mechanical compliance. Additionally, linear tapers yield higher IFMM frequencies and smaller displacements compared to exponential profiles, where broader energy localization mitigates stiffness-induced suppression.

This structure can be fabricated using simple micro/nanofabrication techniques and equipment. Advancements in micro/nano fabrication technology have enabled the high precision manufacturing of the TOF shape using flame brushing and electric heating methods [45, 46]. Various TOF structures have been proposed and widely applied in experimental settings by precisely controlling the heating length, stretching length, and stretching speed [42–44]. For instance, this TOF structure can be designed to overlap with atomic clouds, facilitating the preparation of atomic arrays [47]. Single atoms, atomic arrays and other micro/nano devices can be effectively coupled with micro/nano optical waveguides, facilitating information coupling between optical frequencies and microwaves [48, 49], and hold significant potential for applications in quantum information processing and precision quantum measurements [50]. However, the lifetimes of these atomic arrays are fundamentally constrained by residual mechanical vibrations in the system, resulting in lifetimes that are orders of magnitude shorter than those achieved in free-space setups. These vibrations originate from the mechanical vibration-induced heating of atomic ensembles, and the impact of different mechanical modes on this heating were analyzed [33]. Controlling the shape of the TOF provides effective modulation of its IMMs, and the use of a piezo allows for fine-tuning of the produced TOF length, thereby achieving regulation of the mechanical modes. This proposal is feasible and holds significant importance for further research on micro/nano-structures based on TOFs.

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