High-precision angular rate detection based on an optomechanical micro hemispherical shell resonator gyroscope

Wenyi Huang; Senyu Zhang; Jamal N. A. Hassan; Xing Yan; Dingwei Chen; Guangjun Wen; Kai Chen; Guangwei Deng; Guangwei Deng; Yongjun Huang

doi:10.1364/OE.482859

1. Introduction

As a device to measure the angular rate of moving objects, gyroscope has been widely used in various areas, such as navigation, inertial guidance, vibration detection, attitude stability, and consumer electronics [1]. Many kinds of gyroscopes have been proposed and developed in the past years, and the high-precision solid-state gyroscopes can be categorized into three main categories: hemispherical resonator gyroscope (HRG) [2], laser gyroscope [3], and fiber optical gyroscope [4]. After years of researches and developments, HRG worked by the using of Coriolis force has reached the inertial grade measurement precision [5,6]. Motivated by high performance large-scale HRG, micro hemispherical shell resonator gyroscope (MHSRG) have also been developed rapidly [7–13], due to the huge requirements for the chip-scale high-precision micro gyroscope. Specifically, the resonator structure design [7,8] and processing technology [9] have been studied in the beginning period for MHSRG and the quality factor of the uncoated resonator has reached 4.45 million [10]. Besides, progress has been made in the research of structural driving and detecting electrode technology of MHSRG [11]. However, there are certain errors in the processes of nano-fabrication, which will lead to frequency split, mode deflection and uneven damping of resonator, and correspondingly the error suppression techniques are used to improve gyroscope performance [12,13]. However, the measurement stability and accuracy are reduced because that the traditional MHSRG based on capacitance detection is affected by electromagnetic interference (EMI) and mechanically thermal noise. Moreover, due to the surface metallization of MHSRG resonator, the nano-fabrication barrier is significantly increased and the asymmetry of damping loss in the structure is enhanced.

Recently, based on the development of optical nano-fabrication process, there have been obvious advances in the field of micro-opto-electro-mechanical systems (MOEMS) gyroscopes [14–16]. Specifically, there is significant progress in the field of cavity optomechanics which gradually becomes a new solution for the photonic-integrated sensors [17,18]. Their sensing precision exceeds currently conventional limits and approaches the quantum noise limit. These optomechanical systems, which depend on the coupling between light and mechanical motion, mainly include Fabry-Perot cavity [19–21], ring and spherical cavities that support whispering gallery modes (WGMs) [16,22], and photonic crystal based nanocrystalline cavity [23]. Through optimization of structure and measuring device, the sensitivities have been reached on the order of 10⁻¹⁸ m/Hz^1/2 [24], allowing cooling [25], force sensing [26], gravity measurement [27], and inertial sensing [28]. Due to such high displacement sensitivities, optomechanical system can be used for MOEMS inertial sensors such as gyroscopes and accelerometers. To date, there has been research on analyzing feasibility and pointing out the great potential of optomechanical gyroscopes [20,21,29]; however, there is no real optomechanical gyroscope configurations reported so far.

The purpose of the paper is to propose a real and novel structure with practical values for the MHSGR using Coriolis force transduction and using the cavity optomechanical technology to drive the MHSRG and detect the angular rate. This study demonstrates a new driving and detection mechanism, which is different from the previous conventional gyroscopes, especially the MHSRG. Specifically, the resonator is no longer driven by capacitance, but driven by the opto-mechanical interaction in the optical cavity with high quality factor. The detection part is also different from the traditional capacitance detection, that is, the resonant displacement is measured by exploiting dispersive shift and dissipative losses [22] of an optical WGM resonator to waveguides. In this paper, the operating principle and overall structure of the whole new system are analyzed and designed, respectively. Especially, the mechanical harmonic oscillator structure and the optical WGM mode are optimized by using numerical Finite Element Analysis approaches, respectively. The theoretical analysis and numerical simulation results shown in this paper indicate that the optomechanical MHSRG has huge potential applications for high-precision chip-scale micro-gyroscope.

2. Operating principle

2.1 Optomechanical MHSRG device architecture design

The proposed optomechanical MHSRG is composed of a suspended micro hemispherical shell resonator which also is an optical WGM cavity, the curved optical coupling waveguides, and the underlying substrate (seen in Fig. 1(a)); and the main measurement principle is shown in Fig. 1(b). Firstly, for the MHSRG configuration, the micro hemispherical shell resonator is anchored on the quartz substrate by means of a support rod. And the resonator as a kind of WGM spherical shell microcavity has extremely high optical quality factor, small mode volume, strong optical and mechanical interaction [18]. Figure 1(c) shows the mechanical mode and optical mode of the resonator by FEM modal analysis, respectively. Specifically, the driving and sensing mechanical modes are spatially 45° apart from each other, in which the selected two mechanical wineglass modes of MHSRG are shown on the right side of Fig. 1(c). Those two mechanical modes are the typical modes of HRG.

Fig. 1. (a) (b) 3D model and operating principle of the optomechanical MSRG, (c) FEM modelled optical WGM mode and mechanical Wineglass deformation modes of the MSRG.

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Secondly, the curved optical waveguide over the substrate can be used for a well evanescent coupling of light into the resonator shown in the top panel of Fig. 1(a). So the established typical optomechanical system is used to drive and sense the aforementioned two mechanical modes (seen in Fig. 1(b)), because huge interaction force can be generated, and the dispersion and dissipation are affected by the coupling of distance between the resonator and waveguide [18]. Based on the above operating mechanism, to theoretically get maximum sensing sensitivity, four waveguides located at 0°, 90°, 180°, 270° are used to drive the mode deformation of first wineglass mode (primary motion mode). And other four waveguides at 45°, 135°, 225°, 315° are used to sense the mode deformation of the second wineglass mode (secondary motion mode).

Thirdly, in force-rebalance mode of HRG, when the shell resonator of MHSRG processes relative to the substrate due to the applied external angular rate, the optomechanical system applies the feedback force in real time, to overcome the Coriolis force and keep consistent with the drive mode of wineglass at all times. Therefore, in the end, the applied angular rate is calculated according to the displacement and feedback force. The completed operating system can be referred to Fig. 2.

Fig. 2. The completed operating system for the proposed optomechanical MHSRG.

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For the above designed optomechanical MHSRG, the biggest realization challenge is the co-fabrications of the MHSRG and the optical coupling waveguide, due to the different nano-fabrication processes and requirements. For this issue, we refer to the well-developed nano-fabrication process based on the pop-up rings method [30] and also consider the curved optical coupling waveguide. Specifically, the pop-up ring mask is formed on a silicon substrate by etching a ∼300 nm thick silicon nitride using low pressure chemical vapor deposition (LPCVD). Next, a hemisphere mold is formed by hydrofluoric, nitric, and acetic (HNA) etching of silicon using the pop-up ring mask. The HNA concentration is optimized to obtain the smoothest surface. And the pop-up rings allow excellent control of lateral and vertical etch rates to process highly symmetric hemispherical mold. Then a 1-µm thick SiO₂ structural layer is thermally grown and then patterned with reactive ion etching (RIE). And a 200-nm thick Si layer obtained by plasma-enhanced chemical vapor deposition (PECVD) is etched with electron-beam lithography (EBL) and RIE to process the coupling waveguide. After protecting the waveguide, the MHSRG is released by XeF2. The anchor of the MHSRG is controlled by a timed-release process.

2.2 Optomechanical driving based on evanescent coupling of WGM

The proposed MHSRG needs firstly to be driven by laser power based on optomechanical coupling, which is realized by coupling the light into and out of the WGMs through the optical waveguide [31], shown in the top panel of Fig. 1(a). Based on the basic coupling configuration, the transmission is correlated with the optical power coupled into the WGMs, which is governed by the extrinsic coupling κ_e, the intrinsic coupling κ_i, and the scattering component κ_s [32]. Here κ_e and κ_s are related to the coupling distance between the micro hemispherical shell resonator and waveguide. Specifically, κ_e defines light transfer from waveguide to WGM, κ_s is the light scattered at the coupling region, which would be zero in an ideal optical coupling scenario. Moreover, light will be also scattered from the WGM into the surrounding environment at a rate of κ_i, which is limited by the material and surface fabrication quality of the resonator.

Based on the classic coupled-mode theoretical framework [33], the normalized intracavity electromagnetic (EM) field, a, is expressed by:

(1)$$\frac{{da}}{{dt}} ={-} i\Delta a - \frac{\kappa }{2}a + \sqrt {{\kappa _e}} {a_{in}}$$

where Δ = ω_l – ω₀ defines the detuning of the laser frequency ω_l from the cavity resonance frequency ω₀, and a_in is the input laser amplitude from the waveguide. Then we can have κ = κ_e +κ_i +κ_s and, therefore, the optical quality factor is defined by Q_opt = ω₀/κ. Since the lifetime of photons is much shorter than the mechanical oscillation period in this paper, the steady-state condition is satisfied such that da/dt = 0. Therefore, the intracavity field can be expressed as:

(2)$$a = {{\sqrt {{\kappa _e}} {a_{in}}} / {(i\Delta a + \frac{\kappa }{2}a)}}$$

Moreover, the output field of the waveguide satisfies the input-output relationship [34],

(3)$${a_{out}} ={-} {a_{in}} + \sqrt {{\kappa _e}} a$$

Therefore, the normalized transmission through the waveguide is:

(4)$$T\textrm{ = }\; {|{{{{a_{out}}} / {{a_{in}}}}} |^2} = {\left|{1 - \frac{{{\kappa_e}}}{{{{{\kappa_i}} / 2} + {{{\kappa_s}} / 2} + {{{\kappa_e}} / 2} + i\Delta }}} \right|^2}$$

It should be noticed that κ_e and κ_s vary with coupling distance d₀ (from resonator to waveguide) due to the exponential decay of evanescent field [32]. Hence, the transmission T is highly depended on d₀. For example, when detunning Δ = 0, the on-resonance transmission T₀ changes with d₀ as shown in Fig. 3(a). If κ_s = 0, the critical coupling position (κ_e = κ_i) will be appeared and as a result T₀ will be zero, whereas, if κ_s = κ_e, there is not critical coupling position any more. Moreover, due to the change of distance d₀ between resonator and waveguide, the transmission T of WGM can be also varied with three different features as shown in Fig. 3(b), namely a dispersive and two dissipative modulations, which can be represented by three optomechanical coupling rates, g_om, γ_om, γ_s. In such figure, |g_om| represents the dispersion effect, and | γ_om| and | γ_s| represent the dissipative characteristics of the system [17,22,32].

Fig. 3. (a) The on-resonance transmission, T₀, as a function of coupling distance d₀due to the different κ_s and κ_e which will be varied exponentially with the increasing of d₀. The inset figure is the conventional transmission curves for the WGM cavity under three coupling conditions. (b) The original WGM mode (black) changing features governed by dispersive (red) and dissipative (green and blue) modulations. Red arrow and green/blue arrows show the resonant frequency shift and linewidth broadening, respectively. (c) The nonlinear dynamic mappings for the displacement x to an intracavity photon number | a |² based on Eqs. (1) and (6). Intersection of curves represents stable solution (full circles) and unstable (empty circles) solutions. (d) The numerical calculated output voltage of the PD to DC rotation rate Ω at different coupling distance d₀ under P_in = 3 mW.

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Then, due to the dropped-in laser energy in the WGM cavity, there is radiation pressure, F_rp, from the light around optical resonance mode onto the mechanical mode, resulting in back-action at the optical cavity boundary. The intracavity energy inside the WGM is governed by the resonance frequency, E = ħω₀| a |², so the interaction Hamiltonian depends on this frequency shift, H_int = –ħg_omx| a |². Therefore, F_rp can be expressed by [35]:

(5)$${F_{rp}} ={-} \frac{\partial }{{\partial x}}{H_{{\mathop{\rm int}} }} = \hbar {g_{om}}{|a |^2} = \hbar {g_{om}}{n_c}$$

where ħ is the Planck constant, x defines the drive displacement, and n_c is the number of photons in the cavity obtained by taking the norm-square. Hence, the mechanical equation of motion can be shown as

(6)$$m\ddot{x} + m\gamma \dot{x} + m{\omega _m}^2x = {F_{rp}}$$

where dot notation signifies a derivative with respect to time, and the resonance frequency ω_m, viscous damping rate γ, effective mass m are the basic parameters for the mechanical resonator. Figure 3(c) shows the graphical representation of solutions for the coupled optical Eq. (1) which varies with the cavity-laser detuning Δ and mechanical Eq. (6) for the intracavity energy | a |² and displacement x induced by radiation-pressure. In this figure, the arrows indicate the possible displacement and intracavity optical energy change process due to the different cavity-laser detunings. Therefore, the energy in the cavity can be strengthened by reducing the Δ and/or increasing input laser power to increase n_c, resulting in a larger drive displacement for the proposed optomechanical MHSRG.

2.3 Optomechanical sensing of displacement & angular rate

Based on the above established optomechanical coupling system and the obtained driving displacement, now we can describe the optomechanical sensing operation principle for the optomechanical MHSRG. When the mechanical wineglass mode is driven by the optical radiation pressure due the high-efficient opto-mechanical coupling, based on the classic Coriolis effect, a DC rotation rate Ω applied to the MHSGR will cause the resonator to generate small vibration in detection mode (secondary motion mode shown in Fig. 1(a)) by dy. Specifically, in the rate-sensing mode, the mechanical equation of the MHSRG is [36]:

(7)$$\left[ {\begin{array}{@{}cc@{}} 1&0\\ 0&1 \end{array}} \right]\left[ {\begin{array}{@{}cc@{}} {\ddot{x}}\\ {\dot{y}} \end{array}} \right] + \left[ {\begin{array}{@{}cc@{}} {{c_{11}}}&{{c_{12}}}\\ {{c_{21}}}&{{c_{22}}} \end{array}} \right]\left[ {\begin{array}{@{}c@{}} {\dot{x}}\\ {\dot{y}} \end{array}} \right] + \left[ {\begin{array}{@{}cc@{}} {{k_{11}}}&{{k_{12}}}\\ {{k_{21}}}&{{k_{22}}} \end{array}} \right]\left[ {\begin{array}{@{}c@{}} x\\ y \end{array}} \right] + \left[ {\begin{array}{@{}cc@{}} 0&{ - 4{A_g}\mathrm{\Omega }}\\ {4{A_g}\mathrm{\Omega }}&0 \end{array}} \right]\left[ {\begin{array}{@{}cc@{}} {\dot{x}}\\ {\dot{y}} \end{array}} \right] = \left[ {\begin{array}{@{}cc@{}} {F_1/m}\\ {F_2/m} \end{array}} \right]$$

where x, y are the generalized displacements of driving mode and sensing mode, A_g is the angular gain, F_i (i = 1,2) are the radiation pressure of each mode, c_i,j (i,j = 1,2) are damping coefficients and k_i,j (i,j = 1,2) are spring coefficients. The mechanical motion of MHSRG is affected by the above parameters and those parameters have the following relationship [36]:

(8)$$\left\{ \begin{array}{l} {k_{11}} = \frac{{\omega_1^2 + \omega_2^2}}{2} + \frac{{\omega_1^2 - \omega_2^2}}{2}\cos 2{\theta_\omega }\\ {k_{12}} = {k_{21}} = \frac{{\omega_1^2 - \omega_2^2}}{2}\sin 2{\theta_\omega }\\ {k_{22}} = \frac{{\omega_1^2 + \omega_2^2}}{2} - \frac{{\omega_1^2 - \omega_2^2}}{2}\cos 2{\theta_\omega } \end{array} \right., $$

(9)$$\left\{ \begin{array}{l} {c_{11}} = \left( {\frac{{{\omega_1}}}{{2{Q_1}}} + \frac{{{\omega_2}}}{{2{Q_2}}}} \right) + \left( {\frac{{{\omega_1}}}{{2{Q_1}}} - \frac{{{\omega_2}}}{{2{Q_2}}}} \right)\cos 2{\theta_\tau }\\ {c_{12}} = {c_{21}} = \left( {\frac{{{\omega_1}}}{{2{Q_1}}} - \frac{{{\omega_2}}}{{2{Q_2}}}} \right)\sin 2{\theta_\tau }\\ {c_{22}} = \left( {\frac{{{\omega_1}}}{{2{Q_1}}} + \frac{{{\omega_2}}}{{2{Q_2}}}} \right) - \left( {\frac{{{\omega_1}}}{{2{Q_1}}} - \frac{{{\omega_2}}}{{2{Q_2}}}} \right)\cos 2{\theta_\tau } \end{array} \right., $$

(10)$$m = \int\limits_V {\rho ({\phi _{x1}}^2 + {\phi _{y1}}^2 + {\phi _{z1}}^2)} dV = \int\limits_V {\rho ({\phi _{x2}}^2 + {\phi _{y2}}^2 + {\phi _{z2}}^2)} dV, $$

(11)$${A_g} = {{\int\limits_V {\rho |{{\phi_{x1}}{\phi_{y2}}} |} dV} / m}, $$

where θ_ω is the angular difference between spring coordinate system and main coordinate system, θ_τ is the angular difference between damping coordinate system and main coordinate system, ω₁ is the frequency of driving mode, ω₂ is the frequency of sensing mode, Q₁ and Q₂ are the quality factor of driving and sensing mode, respectively; and ϕ_xi, ϕ_yi and ϕ_zi (i = 1, 2) are the shape function of wineglass modes, ρ is the density [37]. Usually, Q₁ = Q₂ = Q_m, θ_ω = θ_τ = 0.

As discussed before, in the driving mode, changing the coupling gap d₀ will cause dispersive and/or dissipative modulations, which affect the intracavity power and output transmission T. This feature is also satisfied for the sensing mode. Therefore, by setting one cavity-laser detuning Δ, any motion changing dy generated in sensing mode will cause a changing in transmission T considering dispersive and/or dissipative transduction, which can be easily detected by a photoelectric detector (PD). Here the output optical power P_m modulated by the mechanical motion can be written by [22,38]

(12)$${P_m} = {P_{in}}\frac{{dT}}{{dy}}\eta y(t) = {P_{in}}\eta \left|{{g_{om}}\frac{{\partial T}}{{\partial \Delta }} + {\gamma_{om}}\frac{{\partial T}}{{\partial {\kappa_e}}} + {\gamma_s}\frac{{\partial T}}{{\partial {\kappa_s}}}} \right|y(t)$$

where η represent some realistic losses from the cavity to the detector and P_in is the laser input power, and |a_in|² = P_in/ ħω_l. For the WGM we know that [22]:

(13)$$\frac{{\partial T}}{{\partial {\kappa _\textrm{e}}}} ={-} \frac{{4({{\kappa_\textrm{i}} + {\kappa_\textrm{s}}} )({4{\mathrm{\Delta }^2} - \kappa_\textrm{e}^2 + {{({{\kappa_\textrm{i}} + {\kappa_\textrm{s}}} )}^2}} )}}{{{{({4{\mathrm{\Delta }^2} + {\kappa^2}} )}^2}}}$$

(14)$$\frac{{\partial T}}{{\partial \mathrm{\Delta }}} = \frac{{32\mathrm{\Delta }{\kappa _\textrm{e}}({{\kappa_\textrm{i}} + {\kappa_\textrm{s}}} )}}{{{{({4{\mathrm{\Delta }^2} + {\kappa^2}} )}^2}}}\,$$

(15)$$\frac{{\partial T}}{{\partial {\kappa _\textrm{s}}}} ={-} \frac{{4{\kappa _\textrm{e}}({4{\mathrm{\Delta }^2} + \kappa_\textrm{e}^2 - {{({{\kappa_\textrm{i}} + {\kappa_\textrm{s}}} )}^2}} )}}{{{{({4{\mathrm{\Delta }^2} + {\kappa^2}} )}^2}}}$$

The interaction between those parameters governs the scale-factor at each d₀. Figure 3(d) illustrates the output voltage linearly varies by DC rotation rate under different scale factors, which will be discussed in detail in Sec. 3.2.

It should be noticed that for the proposed MHSRG, the effects of Sagnac effect on radiation pressure F_rp should be considered somehow. Theoretically, due to the Sagnac effect, the frequency of the optical mode experiences a Sagnac shift in the WGM cavity, i.e.,

(16)$$\Delta = \Delta + \Delta {\omega _s}$$

here $\Delta {\omega _s} = 4\pi R\Omega /{n_g}\lambda $, and R is the cavity radius, n_g is the group index, and λ is the operating wavelength. However, the Sagnac shift Δω_s are approximately 142 Hz and 1183 Hz in the WGM cavity when angular rate Ω are 12 °/ s and 100 °/ s, respectively. They are much smaller than cavity-laser detuning Δ which is about 70 MHz, so the Sagnac effect in the MHSRG can be ignored.

3. Sensing performance analysis for the optomechanical MHSRG

3.1 Optical mode and mechanical mode demonstrations

In order to understand these interaction terms in Eqs. (1) and (6), it is necessary to analyze and simulate the optical and mechanical modes of the micro shell resonator by FEM modal analysis. Firstly, since the optical mode of the traditional spherical WGM cavity is mainly concentrated in the outer part of the sphere, the micro hemispherical shell resonator can also form the optical microcavity with high optical quality factor. Based on the parameters shown in Table 1, the two-dimensional symmetry method is used to effectively simulate the micro shell resonator in COMSOL. When the excited WGM modes are of the first order (the field distribution is shown on the left side of Fig. 1(c)), the optical resonant frequency is 288.22 THz, and the optical quality factor can reach 5.4 × 10⁷. Because the shape and size of resonator will affect the optical properties of the cavity [39], the optical resonance frequency and quality factors of WGM are then studied by parameter scanning according to the thickness of micro hemispherical shell resonator. The result shown in Fig. 4(a) finds that when the thickness of the micro hemispherical shell resonator is above 3 µm, the resonance frequency and quality factor are almost not changed too much. Whereas, when the micro hemispherical shell resonator is too thin, the optical resonance frequency will increase and the optical quality factor will decrease.

Fig. 4. Features of optical and mechanical mode by FEM modal analysis respectively. (a) The optical resonance frequency (blue) and optical quality factors (red) varied by the resonator thickness Th. (b) The mechanical frequency governed by the resonator radius R (blue) and thickness Th (red). (c) the optical resonance frequency (blue) and quality factor (red) varied by the resonator radius R.

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Table 1. Parameter of MSRG used for FEM numerical calculation

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The size of the micro hemispherical shell resonator not only affects the optical properties of the cavity, but also affects the mechanical resonant frequency. Regularly, the inertial sensors need lower mechanical frequency to get much precision sensing performance, but the external environment vibration loses are hard to reduce. Therefore, the specific mechanical resonance frequency should be considered carefully. In this paper, in order to study the feasibility of the designed system and the influence of different structure sizes on the mechanical resonant frequency, we use the solid mechanics module in COMSOL to simulate the mechanical wineglass modes of the micro hemispherical shell resonator. In simulations, all the parameters shown in Table 1 are fixed, except the resonator sizes including the thickness and radius. As shown in Fig. 4(b), the smaller radius of the hemispherical shell will lead to the rise of the mechanical resonant frequency, especially when it is less than 30 µm, the resonant frequency rises rapidly. While the thickness affects the mechanical resonant frequency linearly. Finally, the mechanical resonance frequencies for the driving mode and sensing mode are optimized to 333.71 kHz and 333.72 kHz, respectively, based on the FEM simulation. Moreover, we also numerically investigated the effects of the resonator radius on the optical resonance frequency and quality factor. As can be seen in Fig. 4(c), when the radius is increased from 20 µm to 100 µm, the optical resonance frequency and quality are, respectively, down shifted and increased near-linearly, in an acceptable range. Specifically, the quality factor will be increased ∼3 × 10⁷ to ∼3.4 × 10⁸.

3.2 Performance characterization of optomechanical MHSRG

As mentioned in Sec. 2, due to the decay rate of the evanescent field is highly related to the coupling distance because of the dispersive and dissipative modulations (seen in Fig. 2(b)), here we further show that all the parameters κ_e, κ_s, and Δ are exponentially varied with the coupling distance d₀. In the paper, the optical decay rates are chosen as κ_e = 1.013 × 10⁹exp(-8.8 × 10⁻⁶d₀) Hz, κ_s = 0.117 × 10⁹ exp(-13.5 × 10⁻⁶d₀) Hz, κ_i = 6 × 10⁶Hz, and the red-side detuning is Δ = 0.649 × 10⁹ exp(-13.5 × 10⁻⁶d₀) Hz [22]. Based on the above relationship, the results are shown in Fig. 5(a). Then, the dissipative losses will broaden the WGM linewidth which given by γ_om and γ_s, where | γ_om| = dκ_e/dd₀ and | γ_s| = dκ_s/dd₀. Besides, the dispersive opto-mechanical coupling rate given by g_om = dΔ/dd₀, describe the optical resonance shift of WGM due to variation of local effective refractive index [40]. Considering the small displacement of MSRG around a fixed d₀, the parameters g_om, γ_om, and γ_s can be linearized, where g_om = dΔ(d₀)/dd₀, γ_om = dκ_e(d₀)/dd₀ and γ_s = dκ_s(d₀)/dd₀ [17,32,41]. As a result, the magnitudes of those parameters are shown in Fig. 5(b). It can be known that, with the changing of the distance d₀, all the coupling rates are reduced quickly in the small range, for example, d₀< 0.3 µm, which means in such distance range we can get large sensitivity.

Fig. 5. Theoretically calculated optical frequency shift, decay rates, and optomechanical rates in WGM. (a) The frequency shift Δ, extrinsic rate κ_e, and scatter rate κ_s as the functions of d₀. Intrinsic coupling κ_i is limited by material of the resonator. (b) The magnitude of the optomechanical rates varied exponentially with coupling distance d₀.

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Then, we discuss the sensing sensitivity (or scale factor) and the noise limit (or angular random walk (ARW)) of the proposed MHSRG. When the gyroscope works in rate detection mode, the continuous external driving force is applied to keep the gyroscope vibrating in the driving direction and the displacement in the sensing direction is maintained to be zero. In general, large scale factor helps to improve signal to noise ratio (SNR) and bias stability. Besides, the ARW performance of the gyroscope studied in this paper is affected by the mechanical thermal noise caused by the Brownian motion of the resonator (ARW_mech) [42], the detection noise of PD (ARW_NEP), shot noise (ARW_SN) and back-action noise (ARW_ba) from the laser [38].

The scale factor is the linear correlation coefficient between the output voltage V_out and the input angular rate Ω of the MHSRG. The total scale factor, SF, is mainly composed of three parts: the mechanical scale factor of the resonant structure, the optomechanical scale factor in the WGM system and the photoelectric conversion scale factor of the PD.

(17)$$SF = \frac{{{V_{out}}}}{\Omega } = {S_{mech}} \cdot {S_P} \cdot {S_V}$$

Firstly, the mechanical scale factor, S_mech, is defined as the ratio of the amplitude of displacement C_y in the sensing mode to the external input rotation rate Ω. Secondly, the optomechanical scale factor, S_p, represents the ratio between the output power P_m and C_y. Thirdly, the photoelectric conversion scale factor, S_V, is determined by the PD, and S_V = 2800 V/W for U2T BPRV2125A can be chosen in this paper for the theoretical calculation. When the MHSRG is perfectly mode-matched, based on Eq. (7), S_mech is expressed by:

(18)$${S_m}_{ech} = \frac{{{C_y}}}{{{\Omega _z}}} = \frac{{4{A_g}{q_{drive}}}}{{{c_{22}}}}$$

where q_drive is the driving amplitude by optical radiation pressure. The relationship between output power and displacement is detailed in 2.3, so the S_p can be obtained based on Eq. (12):

(19)$${S_p} = \frac{{{P_m}}}{y} = {P_{in}}\frac{{dT}}{{dy}}\eta = {P_{in}}\eta \cdot \left|{{g_{om}}\frac{{\partial T}}{{\partial \Delta }} + {\gamma_{om}}\frac{{\partial T}}{{\partial {\kappa_e}}} + {\gamma_s}\frac{{\partial T}}{{\partial {\kappa_s}}}} \right|$$

Based on the above equations, we can easily get the final output voltage V_out with the input rotation rate Ω and obtain the SF, as shown in Fig. 3(d) in Sec. 2.

The ARW is mainly used to characterize the short-term accuracy of the MHSRG. The ARW_mech of the MHSRG worked in force-rebalance mode and perfect mode-matched can be written by [36]

(20)$$AR{W_{mech}} = \frac{1}{{2{A_g}{q_{drive}}}}\sqrt {\frac{{{k_B}{T_e}}}{{m{\omega _m}{Q_m}}}} \cdot \frac{{180}}{\pi } \cdot 60\left( {{^\circ{/} {\sqrt h }}} \right)$$

where Q_m is the mechanical quality factor, k_B is the Boltzmann constant, and T_e is the temperature. The detection noise of PD is electronical noise which is quantified by noise-equivalent-optical-power (NEP), and NEP = 2.8 pW/Hz^1/2 for the classical Newport 2117 detector can be used. Therefore, the equivalent rotation rate measurement noise can be obtained by converting the equivalent noise of the detector output noise through the scale factor:

(21)$$AR{W_{NEP}} = \frac{{NEP}}{{({P_{in}}\eta \left|{{g_{om}}\frac{{\partial T}}{{\partial \Delta }} + {\gamma_{om}}\frac{{\partial T}}{{\partial {\kappa_e}}} + {\gamma_s}\frac{{\partial T}}{{\partial {\kappa_s}}}} \right|) \cdot {S_{mech}}}} \cdot \frac{{180}}{\pi } \cdot 60\left( {{^\circ{/} {\sqrt h }}} \right)$$

Meanwhile, the light field has quantum shot noise, which obeys Poisson statistics:

(22)$$AR{W_{SN}} = \frac{{\sqrt {2\hbar {\omega _l}{P_{\det }}{\eta _{qe}}} }}{{{P_{in}}\eta \left|{{g_{om}}\frac{{\partial T}}{{\partial \Delta }} + {\gamma_{om}}\frac{{\partial T}}{{\partial {\kappa_e}}} + {\gamma_s}\frac{{\partial T}}{{\partial {\kappa_s}}}} \right|\cdot {S_{mech}}}} \cdot \frac{{180}}{\pi } \cdot 60\left( {{^\circ{/} {\sqrt h }}} \right)$$

where P_det and η_qe are optical power and quantum efficiency of detector, respectively. So the shot noise power $\sqrt {2\hbar {\omega _l}{P_{\det }}{\eta _{qe}}} $= 6 pW/ Hz^1/2. Moreover, the shot noise in WGM will exert a radiation pressure causing the resonator to move and generate back-action noise. And the ARW_ba is obtained also by converting the back-action noise through S_mech:

(23)$$AR{W_{ba}} = \frac{{\hbar {g_{om}}}}{{2m{\omega _m}{A_g}{q_{drive}}}}\sqrt {\frac{{2{n_c}}}{\kappa }} \cdot \frac{{180}}{\pi } \cdot 60\left( {{^\circ{/} {\sqrt h }}} \right)$$

Finally, the total angular random walk ARW_total is expressed by

(24)$$AR{W_{total}} = \sqrt {AR{W_{mech}}^2 + AR{W_{NEP}}^2 + AR{W_{SN}}^2 + AR{W_{ba}}^2} $$

Based on the above theoretical calculation, now we can discuss the specific sensitivity and lose limit of the proposed MHSRG. In this paper, we lock the red-shift of Δ = –κ/2 away from the shaded area shown in inset of Fig. 6(a), where the transduction is more stable [38]. As shown in Fig. 6(a) and (b), because the optomechanical scale factor S_p varied with d₀, so the total sensitivity SF and ARW decays exponentially as d₀ increases. It is found that the scale factor and ARW are relatively excellent when d₀ < 0.4 µm. In order to get better performance, the rest of our analysis is based on d₀ = 0.2 µm. Specially, ARW_mech is affected by the coupling distance due to driving amplitude produced by optomechanical interaction. And ARW_NEP, ARW_SN, and ARW_ba are much smaller than ARW_mech and can be ignored. Therefore, the ARW_total is almost the same as ARW_mech as shown in Fig. 6(b). Moreover, the frequency split Δω, which is the difference between driving mode ω₁ and sensing mode ω₂, will lead to smaller sensing mode displacement under the same driving amplitude, so S_mech and SF are inversely proportional to the frequency split, as shown in Fig. 6(c) and (d). However, increasing the input laser power (Fig. 6(c)) or mechanical quality factor (Fig. 6(d)) can eliminate the SF performance deterioration due to the frequency split.

Fig. 6. The numerical calculations of optomechanical MHSRG performance in different structure parameter for typical parameters of a silica resonator as shown in Table 1. Scale factor and ARW vary by (a) (b) coupling distance and (c) (d) frequency split. The bottom panel of (a) is the derivative of the transmission T with respect to cavity-laser detuning Δ where dT/dΔ in the shaded area is unstable for real experiments.

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Moreover, if the device is designed and manufactured with fixed coupling distance and frequency split, we can still dynamically adjust the performance of optomechanical MHSRG by changing the input laser power and locking in different laser detuning. One example has shown in Fig. 7(a) and (c). More specially, the input laser power coupling into the cavity controls the intracavity optical mode amplitude like Eq. (1). The scale factor and driving amplitude increase with the adjustment of input light power, for instance, the driving amplitude can even reach 100 nm under ∼100 µW input laser power, as shown in Fig. 7(a). Besides, even though the ARW_total is still dominated by the thermal noise but it can obtain lower value at higher power, for example, reducing to 0.0555 °/ h^1/2 when laser input power is ∼3 mW as shown in Fig. 7(b). Aside from increasing the laser power, the changing of cavity-laser detuning Δ can also be used to control sensing performance. However, the Δ cannot be in the shaded areas which will result in small and unstable transmission T in real experiments as shown in the inset of Fig. 2(a). As shown in Fig. 7(c), the S_mech decreases as Δ increases and SF and S_p are almost at their maximum when Δ = –κ/2. Assuming Δ stay away from unstable areas, the ARW_total is small and other noise influences are much less than thermal noise as shown in Fig. 7(d).

Fig. 7. Calculated performance of optomechanical MHSRG in different input laser conditions for typical parameters of a silica resonator. (a) Scale factor, drive amplitude, and (b) ARW are governed by laser input power. (c) The normalized SF, S_M, S_P and (b) the magnitude of the ARW are changed by cavity-laser detuning Δ. The performance of shaded areas in (c) and (d) can not reach due to unstable optomechanical phenomenon shown in the bottom panel of Fig. 6(a).

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For the proposed high-precision optomechanical MHSRG in this paper, the change of temperature will lead to the frequency variation of the driving and sensing mechanical mode, resulting in frequency split. As shown in Fig. 8(a), it is found that the driving and sensing frequency decrease with the increase of temperature between -20 °C and 60 °C based on the FEM simulation, leading to performance deterioration. In particular, the difference of diving and sensing mode jumps from about 10 Hz to 70 Hz when the temperature is above 53 °C, which greatly reduces SF but increased ARW. Moreover, due to the nanofabrication error, the frequency split and uneven damping of the micro hemispherical shell resonator are inevitable. And the different temperature condition (found in Fig. 8(a)) or air pressure condition may also result in the difference of driving and sensing frequencies. Therefore, we propose an improved structure of MHSRG, which consists of 16 waveguides to suppress frequency split and maintain ultra-high accuracy, as shown in Fig. 8(b). Specifically, there are 8 waveguides at 22.5°, 67.5°, 112.5°, 157.5°, 202.5°, 247.5°, 292.5° and 337.5° used to balance the frequency of the driving and sensing mode [36] because the optomechanical interaction can change the frequency and effective damping rate of the mechanical mode base on the optical spring effects [38].

Fig. 8. (a) Thermal effect on the mechanical frequency. (b) Improved structure to balance frequency split. Calculated (c) linear measurement dynamic range, and (d) ARW of optomechanical MHSRG in different frequency split.

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Due to the large sensing sensitivity, the linear measurement dynamic range is limited, as shown in Fig. 2(d). Also, as can be found in Fig. 6(c) and (d), the sensitivity is dropped quickly when the frequency split appears and as a result the ARW will also be deteriorated. However, this frequency split can benefit to broaden the linear measurement dynamic range. In the end of the paper, as presented in Fig. 8(c) and (d), we simply show that the changing features for the dynamic range, sensitivity and also the ARW with different frequency split.

4. Comparative study

In this section, we compare the performance of previously reported high-precision MOEMS gyroscopes and conventional MHSRG with our proposed optomechanical MHSRG, as summarized in Table 2. Specifically, there are two frame-type MOEMS gyroscopes [14,15] with electrostatic comb drive and optical output to detect sensing displacement. The performance is limited by electron noise and asymmetry between driving and sensing mode. And quality factor of mass is much less than hemispherical shell resonator, leading to the low mechanical scale factor. The performance of conventional MHSRG [8,35] is difficult to continue to breakthrough due to surface metallization and electron noise. As a new kind of MOEMS gyroscope, the proposed optomechanical MHSRG has shown theoretically excellent performances, including the largest sensitivity and lowest noise, under 3 mW input laser power. Moreover, such new MHSRG exhibits more advantages such as low cost, extremely small size and ultra-light test-mass. In addition, the traditional MHSRG uses electrode driving and capacitance detection, which requires surface metallization. The designed optomechanical gyroscope in this paper does not require surface metallization, which can reduce the fabrication complexity and improve the mechanical quality factor of the MHSRG, so as to further improve the performances. And the optical detection will not be affected by EMI and has almost no electrical noise and a high signal-to-noise ratio with small amount of quantum noise.

Table 2. Performance comparison of MOMES and MSRG (*denote the theoretical/numerical results)

View Table | View all tables in this article

5. Conclusion

An optomechanical MHSRG to detect rotation rate is proposed in this paper, which is composed of micro hemispherical shell resonator, coupling waveguide, and supporting substrate. The optomechanical system of the device can provide driving force when used as driving part, and can improve the detection sensitivity and reduce noise as the detection part. The influences of the size of the micro hemispherical shell resonator on the optical and mechanical modes are studied by means of finite element simulation. Under the appropriate coupling distance, the optomechanical MHSRG can achieve SF = 414.8 mV/(°/ s) and ARW_total = 0.0555 (°/ h^1/2) by numerical calculation in the force-rebalance mode when P_in is 3 mW and m is just 98 ng. The surface metallization of the resonator is no longer required, which can reduce the machining complexity and improve the Q_m and the use of optical detection is not affected by (EMI) with a high SNR, which shows the great potential for high performance gyroscopic applications.

Funding

National Natural Science Foundation of China (61971113, U2230206); National Key Research and Development Program of China (2018AAA0103203, 2018YFB1802102); Fundamental Enhancement Program Technology Area Fund (2021-JCJQ-JJ-0667); Joint Fund of ZF and Ministry of Education (8091B022126); Guangdong Provincial Research and Development Plan in Key Areas (2019B010141001, 2019B010142001); Sichuan Provincial Science and Technology Planning Program of China (2021YFG0013, 2021YFH0133, 2022YFG0230, 2023YFG0040); Joint Project of China Mobile Research Institute & X-NET (2022H002); Fund Project of Intelligent Terminal Key Laboratory of Sichuan Province (SCITLAB-1015).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Symbol	Parameter	Value
E	Young’s module of silica	76700 Mpa
ρ	Density of silica	2500 kg/cm³
µ	Poisson ratio	0.17
n	Refractive index, real part	1.4457
k	Refractive index, imaginary part	1 × 10⁻¹⁰
R	Radius of resonator	60 µm
Th	Thickness of resonator	1 µm
r	Radius of supporting rod	1 µm
h	Height of supporting rod	31.5 µm
m	Effective mass of resonator	98 ng
Q_m	Mechanical quality factor	5 × 10⁶
Ag	Angular gain	0.4
ω_m	Mechanical resonance frequency	2π 300 kHz
ω_l	Laser frequency	2π 390 THz
Te	Temperature	290 K

Symbol	Parameter	Value
E	Young’s module of silica	76700 Mpa
ρ	Density of silica	2500 kg/cm³
µ	Poisson ratio	0.17
n	Refractive index, real part	1.4457
k	Refractive index, imaginary part	1 × 10⁻¹⁰
R	Radius of resonator	60 µm
Th	Thickness of resonator	1 µm
r	Radius of supporting rod	1 µm
h	Height of supporting rod	31.5 µm
m	Effective mass of resonator	98 ng
Q_m	Mechanical quality factor	5 × 10⁶
Ag	Angular gain	0.4
ω_m	Mechanical resonance frequency	2π 300 kHz
ω_l	Laser frequency	2π 390 THz
Te	Temperature	290 K

High-precision angular rate detection based on an optomechanical micro hemispherical shell resonator gyroscope

Abstract

1. Introduction

2. Operating principle

2.1 Optomechanical MHSRG device architecture design

2.2 Optomechanical driving based on evanescent coupling of WGM

2.3 Optomechanical sensing of displacement & angular rate

3. Sensing performance analysis for the optomechanical MHSRG

3.1 Optical mode and mechanical mode demonstrations

3.2 Performance characterization of optomechanical MHSRG

4. Comparative study

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (24)

Optics Express