The high quality factors and small mode volumes inherent to whispering gallery mode (WGM) microresonators^[1] confer an excellent advantage in the fabrication of micro-sensing devices across various categories^[2–5]. Compared to the traditional diabolic points (DPs) encountered in Hermitian systems, the splitting caused by perturbation in the vicinity of exceptional points (EPs) of non-Hermitian systems can be significantly enhanced because of the typical eigenvector collapse, a square root relationship around the second-order EP, for example^[6]. This hypothesis has been substantiated in various detection schemes, including but not limited to nanoparticle detection^[7], temperature sensing^[8], mass detection^[9], and biosensing^[10].

Among numerous detection schemes, the optical microresonator gyroscopes for angular velocity detection based on the Sagnac effect have garnered increasing attention because of their significant applications in aerospace, military, and industrial domains^[11–14]. Many methods have been explored to improve gyroscope performance, like using bound states in the continuum for further loss reduction in integrated platforms^[15]. The remarkable sensitivity enhancement of EPs has also been demonstrated through deliberately tailoring the gain–loss equilibrium on both theoretical analysis and experimental validation^[13,16–20]. However, accurate manipulation of multiple parameters is necessary to attain the desired EP conditions. In addition, the isolation of the EP causes the detection system to be highly susceptible to the influence of external irrelevant noise and, thus, deviate from the EP state. To overcome the aforementioned challenges, a proposed solution has been put forward, which extends the EPs constrained by coupled parameters to exceptional surfaces (ESs) that satisfy all coupled parameters^[21,22]. In comparison to EPs, the ES system displays enhanced robustness against specific disturbances, such as fabrication discrepancies and environmental interference. In the past few years, ES-based microresonator sensors have demonstrated remarkable potential across numerous applications^[23–27].

The optomechanical interaction mediated through radiation pressure in WGM microresonators has been an important platform for rich physical research and complicated applications^[28–32]. Among these, optomechanically induced transparency (OMIT)^[33] is shown as one sharp transparent window upon the wide absorption dip^[34]. By adjusting multidimensional parameters, OMIT can generate multiple available line shapes, which is highly beneficial for various applications including light delay and storage^[35], slow light realization^[36], and single-photon routers^[37]. Importantly, the controllable and sharp transmission characteristics of OMIT make it promising for precision sensing.

In this work, we theoretically propose a novel optomechanical approach based on the ES for angular velocity sensing, which simultaneously achieves high sensitivity and stability. Compared to the traditional gyroscope based on the Sagnac effect, this gyroscope has achieved an improvement of more than 3 orders of magnitude within a wide range of angular velocities up to hundreds of radians per second. The narrow linewidth of the mechanical modes is expected to enhance the resolvability of frequency detuning within the desired ultra-low rotational velocity range. Furthermore, the coupling parameters have been optimized for better detection. This model may pave a new path for the design and optimization of highly sensitive micro-optical gyroscopes.

The gyroscope we propose is based on a ring microresonator with an embedded S-shaped waveguide, as depicted in Fig. 1(a). This S-shaped waveguide selectively couples the counter-propagating modes, which satisfies the unidirectional coupling condition from clockwise (CW) to counterclockwise (CCW) modes required for ES construction^[38,39].

Based on the coupled-mode equations, the Hamiltonian can be expressed in the basis space of CW and CCW modes, $$\frac{d}{dt}\left[\begin{array}{c}{\tilde{a}}_{\mathrm{cw}}\\ {\tilde{a}}_{\mathrm{ccw}}\end{array}\right]=H_0\left[\begin{array}{c}{\tilde{a}}_{\mathrm{cw}}\\ {\tilde{a}}_{\mathrm{ccw}}\end{array}\right],H_0=\left(\begin{array}{cc}{\omega }_0-i\gamma & 0\\ {\kappa }_1& {\omega }_0-i\gamma \end{array}\right).$$(1)

Herein, ${\tilde{a}}_{\mathrm{cw},\mathrm{ccw}}$ are the field amplitudes of the CW and CCW modes, ${\omega }_0$ represents the resonance frequency, and the total loss $\gamma =({\gamma }_0+{\gamma }_1+{\gamma }_2+{\gamma }_{RS}+{\gamma }_{SR})/2$, where ${\gamma }_0$ indicates the intrinsic loss rate, and ${\gamma }_1$ and ${\gamma }_2$ describe the loss rate at the coupling points of the ring waveguide with the upper and lower straight waveguides, respectively. ${\gamma }_{RS}$ and ${\gamma }_{SR}$ refer to the coupling loss between the annular waveguide and the S-shaped waveguide, where we assume ${\gamma }_{RS}={\gamma }_{SR}$. The unidirectional coupling term ${\kappa }_1=-i·2\sqrt{{\gamma }_{RS}{\gamma }_{SR}}{e}^{i\varphi }$ arises from the S-bend waveguide structure, where the CW mode of the main ring resonator comes across the S-shaped waveguide coupling area twice and finally transforms as the CCW mode according to the coupled-mode theory. Here, $\varphi$ represents the propagating phase shift along the S-bend, and the existence of 2 results from these two structural paths (through the right contact spot first and then the left contact spot, and vice versa).

The off-diagonal terms in the matrix expression of the Hamiltonian represent the energy conversions between two states. In Eq. (1), the upper-right off-diagonal element of the Hamiltonian constructed by this model is 0, while the lower-left off-diagonal element is a non-zero term, which corresponds to the Hamiltonian characteristic of a second-order EP. In this case, the eigenvalues of the system remain unchanged and are equal to ${\omega }_0-i\gamma$, with only one linearly independent eigenvector ${\mathrm{\Phi }}_{\mathrm{EP}}={(0,1)}^T$. Regardless of how the coupling parameters change, the Hamiltonian matrix always remains in Jordan normal form, implying the construction of the ES in the parameter space. When detecting in such an ES system, deviation in coupling parameters caused by environmental noise can only transit the system from one EP state to another EP state, thus ensuring the robustness of sensitivity enhancement.

This model introduces a microcavity coupled with dual waveguides (add–drop filter structure) as the auxiliary cavity optomechanical system. After setting the control laser at the red sideband, the optomechanical interaction causes the weak probe light to be blocked by this auxiliary cavity when no rotation exists, while if the system rotates at a rate of ${\mathrm{\Omega }}_s$, these two counterpropagating modes experience opposite Sagnac frequency shifts $\mathrm{\Delta }{\omega }_s$, leading to an additional coupling from the CCW to CW mode of the main microring through the auxiliary cavity optomechanics. This ensures the square root dependence of frequency splitting on the rotational velocity, as shown in Fig. 1(b).

In the auxiliary cavity system, the OMIT phenomenon is achieved under the coupling with a mechanical radial breathing mode. Here the drop end of the add–drop filter structure is introduced as the detection output to meet the construction condition of ES, that is, there is no feedback light through this auxiliary system without rotation. The drop efficiency $D$ is defined as the ratio of the detected power at the drop port to the input detection power, $$D={|d_P|}^2={\left|\sqrt{{\kappa }_{\text{ex}1}{\kappa }_{\text{ex}2}}\frac{1+if(\mathrm{\Omega })}{\kappa /2-i(\mathrm{\Omega }+\overline{\mathrm{\Delta }})+2\overline{\mathrm{\Delta }}f(\mathrm{\Omega })}\right|}^2,$$(2)with $$f(\mathrm{\Omega })=2m_{\mathrm{eff}}{\mathrm{\Omega }}_m{g}^2\frac{\chi (\mathrm{\Omega })}{[i(\overline{\mathrm{\Delta }}-\mathrm{\Omega })+\kappa /2]},$$(3)$$\chi (\mathrm{\Omega })=\frac{1}{m_{\mathrm{eff}}({\mathrm{\Omega }}_m^2-{\mathrm{\Omega }}^2-i{\mathrm{\Gamma }}_m\mathrm{\Omega })}.$$(4)

Herein, ${\kappa }_{\text{ex}1}$ and ${\kappa }_{\text{ex}2}$ represent the loss rates at the coupling points of the upper and lower straight waveguides, respectively. $\kappa$ indicates the total loss of the auxiliary cavity. $g$ is the optomechanical coupling ratio. $\overline{\mathrm{\Delta }}$ represents the detuning between the control light and the resonance frequency of the auxiliary cavity, which equals the probe light without rotation. Here the control laser is fixed as $\overline{\mathrm{\Delta }}=-{\mathrm{\Omega }}_m$. ${\mathrm{\Omega }}_m$ indicates the resonance frequency of the mechanical oscillator, which characterizes the mechanical modes along with the damping rate ${\mathrm{\Gamma }}_m$, effective mass $m_{\mathrm{eff}}$, and magnetic susceptibility $\chi (\mathrm{\Omega })$. Here we set the parameter as $g=2\times {10}^6$, ${\mathrm{\Omega }}_m/\pi =120\mathrm{MHz}$, ${\mathrm{\Gamma }}_m=1.3\mathrm{kHz}$, $m_{\mathrm{eff}}=20\mathrm{ng}$, ${\kappa }_{\text{ex}1}/\pi ={\kappa }_{\text{ex}2}/\pi =25\mathrm{MHz}$, and $\kappa /\pi =55\mathrm{MHz}$.

In addition, $\mathrm{\Omega }={\omega }_p-{\omega }_c$ represents the frequency detuning between the probe laser and the control laser, and the sharp transmission window of the transmission spectrum of the probe field in OMIT appears near $\mathrm{\Omega }={\mathrm{\Omega }}_m$. The drop efficiency exhibits a drastic change versus detuning as a typical OMIT effect [Fig. 2(a)]. A narrow valley emerges in the drop spectrum near $\mathrm{\Omega }={\mathrm{\Omega }}_m$, reaching its minimum value of zero. As there is no light from the auxiliary cavity when ${\mathrm{\Omega }}_s=0$, the ES state described by Eq. (1) is maintained. The inset further depicts the relationship between drop efficiency D and frequency detuning $\mathrm{\Delta }$ when the probing laser frequency is changed, where $\mathrm{\Delta }={\omega }_p-{\omega }_0$. When the main cavity gyroscope rotates at an angular velocity ${\mathrm{\Omega }}_s$, there is a frequency detuning of $\mathrm{\Delta }=\pm \mathrm{\Delta }{\omega }_s$ (depending on the rotation direction). From this, we can obtain the plot of drop efficiency as a function of ${\mathrm{\Omega }}_s$ in Fig. 2(b). In the case of ${\mathrm{\Omega }}_s\ne 0$, the drop efficiency deviates from zero due to the frequency detuning caused by rotation, resulting in the emergence of additional coupling ${\kappa }_2({\mathrm{\Omega }}_s)=-i\sqrt{{\gamma }_1{\gamma }_2}{t}_0{e}^{i\phi }·d_P$, where $t_0$ and $\phi$ are the attenuation and phase accumulation terms from the feedback waveguide and optical isolator. As a consequence, the ES state of the microcavity gyroscope breaks with a square root response to rotational velocity.

According to the above principles, ideally, the disturbance Hamiltonian brought by the rotation should be expressed as $$H_I=\left(\begin{array}{cc}\mathrm{\Delta }{\omega }_s& -i\sqrt{{\gamma }_1{\gamma }_2}{t}_0{e}^{i\phi }·d_P\\ 0& -\mathrm{\Delta }{\omega }_s\end{array}\right).$$(5)

The Sagnac frequency shift $\mathrm{\Delta }{\omega }_s=4\pi R{\mathrm{\Omega }}_s/n_g\lambda$ is proportional to the rotational velocity. $R$ and $\lambda$ represent the resonant cavity radius and operating wavelength, respectively. At this time, the total Hamiltonian is jointly contributed by the stationary Hamiltonian and the disturbance Hamiltonian, $$H=H_0+H_I=\left(\begin{array}{cc}{\omega }_0+\mathrm{\Delta }{\omega }_s-i\gamma & -i\sqrt{{\gamma }_1{\gamma }_2}{t}_0{e}^{i\phi }·d_P\\ -i·2\sqrt{{\gamma }_{RS}{\gamma }_{SR}}{e}^{i\varphi }& {\omega }_0-\mathrm{\Delta }{\omega }_s-i\gamma \end{array}\right).$$(6)

The corresponding eigenfrequencies are calculated as $${\omega }_{\pm }={\omega }_0-i\gamma \pm \sqrt{2i^2\sqrt{{\gamma }_{RS}{\gamma }_{SR}}\sqrt{{\gamma }_1{\gamma }_2}{t}_0{e}^{i(\varphi +\phi )}·d_P+\mathrm{\Delta }{\omega }_s^2}.$$(7)

For a small rotational velocity range, the second-order term under the square root can be ignored. In particular, in the ideal case of $t_0=1$, by adjusting the phase accumulation term $\phi$ in the feedback waveguide to satisfy the condition of $\varphi +\phi =(2n+1)\pi$, where $n$ is an integer, the imaginary component of the complex eigenfrequency splitting can be eliminated. As a result, the microcavity gyroscope system exhibits genuine frequency splitting. At this point, $$\mathrm{\Delta }{\omega }_{ES}\approx 2\sqrt{2\sqrt{{\gamma }_{RS}{\gamma }_{SR}}·\sqrt{{\gamma }_1{\gamma }_2}·d_P}.$$(8)

The square root response results in the enhancement of detection sensitivity in ES-based gyroscopes compared to traditional gyroscopes operating in the DP state (where the complex eigenfrequency splitting is directly proportional to the rotational velocity, i.e., $\mathrm{\Delta }{\omega }_{DP}=2\mathrm{\Delta }{\omega }_S$). In addition, the aforementioned frequency splitting expression reduces the requirement for cavity dimensions in the gyroscope, from centimeter scale to micrometer scale specifically, which promotes the development of integrated micro-optical gyroscopes.

Figure 3(a) demonstrates the eigenfrequency splitting of the system near the ES as a function of the rotational velocity and contrasts it with the classical Sagnac frequency shift in the DP state. Here the working wavelength is set to $\lambda =1550\mathrm{nm}$, $n_g=1.44$, and $R=177.31\mathrm{\mu m}$. ${\gamma }_1$ and ${\gamma }_2$ are of the same values as ${\gamma }_{RS}$ and ${\gamma }_{SR}$, set to $1.5\times {10}^8\mathrm{Hz}$. For intuitive comparison, the frequency splitting value under the DP is enlarged by a thousand times. It is evident that the frequency shift (depicted by the blue hollow circles) under the ES follows a square root behavior in response to the rotational velocity, consistently exceeding the frequency shift at the DP (depicted by the red hollow diamonds). This indicates that within the rotational velocity range of 150 rad/s, the detection response of the ES system with mechanical mode assistance is at least 3 orders of magnitude stronger than that of traditional gyroscopes. Additionally, as the rotational angular velocity decreases, the slope of the curve under the ES condition becomes steeper, meaning that the frequency shift responds more dramatically to changes in rotational velocities. These findings demonstrate that the ES system can achieve better detection performance at lower rotational angular rates. It is noted that this enhanced resolution of frequency splitting benefits from two key issues: the square root relationship of the ES and the sharp drop of the OMIT behavior.

The superior performance of the ES system can be further characterized by the enhancement factor as $$\eta =\left|\frac{\mathrm{Re}(\mathrm{\Delta }{\omega }_{\mathrm{ES}})}{\mathrm{\Delta }{\omega }_{DP}}\right|.$$(9)

Figure 3(b) presents the enhancement factors of rotational angular velocities at various orders of magnitude. Consistent with the results shown in Fig. 3(a), an enhancement of more than 3 orders of magnitude can always be obtained. In many applications, precise measurements of small rotational velocities are required. Interestingly, when the rotation angular velocity is in a smaller order of magnitude ($<{10}^{-3}\mathrm{rad}/\mathrm{s}$), the frequency splitting achieved by our proposed gyroscope scheme is more than 6 orders of magnitude larger than that of the traditional gyroscope, as shown in the inset in Fig. 3(b). The computational results indicate that under the influence of an ultra-narrow linewidth assisted by mechanical modes, the detection performance of ES responses is better for sufficiently small rotational velocities.

To optimize the detection performance, we theoretically calculated the variation of the enhancement factor with respect to the structural coupling parameters and summarized it in a two-dimensional parameter space, as shown in Fig. 4(a). Here, the intrinsic loss is used as a benchmark to describe the values. When there is no S-shaped waveguide (${\gamma }_{RS}=0$) inside the WGM ring resonator, the system is in a DP degeneracy state, as indicated by the white dashed line in the diagram. With the existence of the S-shaped waveguide, the system moves to an EP degenerate state. All the EPs corresponding to varieties of non-zero ${\gamma }_{RS}$ form the ES, that is the sheet excluding the white dashed line in Fig. 4(a). The red dashed line takes ${\gamma }_1=10{\gamma }_0$ as an example, which clearly shows that the enhancement factor in the EP state is significantly improved compared to DP degeneracy (red solid circle). At a fixed rotational velocity, the gyroscope demonstrates superior rotational detection advantages with higher structural coupling coefficients. However, the detection of frequency splitting under small perturbations is limited by the finite linewidth of resonance. To give more insights here, the total loss dependence has been studied, as shown in Fig. 4(b). The computational results demonstrate that, in the absence of external gain, the system exhibits the most detectable frequency splitting enhancement around ${\gamma }_1={\gamma }_{RS}$. Besides, the detection effect can be further enhanced by increasing the ratio of structural coupling loss to intrinsic loss. The structural coupling loss can be adjusted by setting different coupling gap values in the experiment. With the aid of optical amplification to decrease intrinsic loss, further detection sensitivity enhancement can be envisioned.

In summary, we theoretically propose an optical microcavity gyroscope in ESs. The system exhibits a real frequency splitting enhancement of more than 3 orders of magnitude, which results from the novel system structure rather than typical device size enlargement. The building of ESs endows this model with not only splitting enhancement but also reliable robustness. Specifically, for sufficiently small perturbations, the sharp transmission characteristics generated by OMIT in the cavity-optomechanical system can elevate the sensitivity of the gyroscope to a new level. With these advantages, this approach holds promising potential for producing novel highly sensitive optical gyroscopes.