The Sagnac effect demonstrates that the rotation of a ring interferometer induces a phase shift between counterpropagating beams [1]. Based on this effect, various optical gyroscopes have been developed, including the ring laser gyroscope (RLG) [2], interferometric fiber-optic gyroscope (IFOG) [3], resonant fiber-optic gyroscope (RFOG) [4], and Brillouin fiber-optic gyroscope (BFOG) [5]. It is noteworthy that both RLGs and IFOGs are well-established, commercially available gyroscopes that are renowned for their ultra-high precision in detecting angular rotation rates [6,7]. However, their considerable sizes and costs present a significant challenge in miniaturized applications. In response, research has been conducted into integrating sensitive structures within photonic chips to reduce their form factors. However, this approach has resulted in a reduction in sensitivity [8,9]. This is mainly due to the fact that the magnitude of the Sagnac effect is proportional to the size of the sensitive coil, especially for IFOGs in which the beam is transmitted only one turn within the sensitive ring, and the reduction in size brings about a proportional decrease in the detection sensitivity. In order to solve the problem of the theoretical sensitivity decrease caused by the size reduction, some researchers have explored quantum-photonics-based approaches [10,11], including multi-mode entanglement squeezing-enhanced FOGs and quantum photonics gyroscopes utilizing nonlinear multi-resonant interactions, both of which theoretically show potential for enhancing gyroscope detection sensitivity. In addition, rapid advances in microcavity fabrication techniques have enabled the development of integrated optical gyroscopes with high-quality (Q) microcavities, where multi-turn optical transmission within the cavity compensates for reduced sensitivity. These innovations have been experimentally validated, yielding promising results (for more details, see Appendix C) [12–14].

In all high-Q-microcavity-based schemes, highly coherent light sources with extremely narrow linewidths are used to improve the detection sensitivity, which results in the presence of various parasitic effects related to the coherence of the light sources, such as backscattering noise, polarization noise, and Kerr effect, which undoubtedly deteriorate the long-term stability of the system. What is worse, high-Q microcavities significantly enhance detection sensitivity; they also exacerbate the noise and nonlinearity within the system [12,13,15], which not only further deteriorates the long-term stability of the system, but also complicates the detection process. Furthermore, the multi-modal nature of these high-Q microcavities necessitates the integration of a Pound-Drever-Hall (PDH) locking loop to stabilize the laser frequency, which is essential for maintaining precision in angular rotation rate measurements [16]. However, the occurrence of intra-cavity mode drift and interactions may reduce the effectiveness of the locking loop, thereby leading to a degradation of the gyroscope’s performance over time. Up to now, the best bias instability for the integrated optical gyroscope based on a mm scale microcavity is only 3 deg/h [13].

The broadband-source-driven resonant optical gyroscope (ROG) scheme has been proposed in recent years as a good solution to the problems caused by the coherence of the light source [17,18]. This is because the use of a broadband light source reduces all parasitic effects associated with source coherence directly from the source. At the same time, since this scheme is based on the filtering characteristics of the resonator to achieve the detection of the angular rotation rate, it eliminates the need for an additional frequency locking loop to maintain resonance. Previous studies have shown that this detection scheme can achieve navigation-grade accuracy on a 100-m-based resonator [17,19,20]. However, the major drawback of this scheme is the low energy utilization, which is mainly due to the non-resonant light coupling out of the resonator from the two vacant ports of the resonator [19]. It has been shown that to achieve the same theoretical sensitivity, an ROG based on a broadband source requires two orders of magnitude larger input optical power than an ROG based on a narrowband source. Furthermore, as the Q value increases, optical power loss also escalates for a given resonator length.

In order to solve these problems, this paper proposes for the first time a broadband-source-driven chip-scale integrated optical gyroscope scheme based on a multi-mode co-detection technique, where all excited eigenmodes within the microcavity are utilized for rotation rate detection. This novel approach departs from the traditional reliance on single-mode transmission for angular velocity detection [12,13,21], significantly enhancing the long-term stability of the gyroscope system while also addressing the issue of low power utilization associated with broadband light sources. In this paper, we first establish a detailed theoretical and equivalent model for the multi-mode co-detection technique, thoroughly investigating its underlying mechanism. We then fabricate a four-port high-Q whispering-gallery-mode (WGM) microcavity with a 9.2 mm diameter (D), capable of supporting multiple transmission modes, and calculate the equivalent modes for all modes within the cavity. Finally, we conduct gyroscope tests using this high-Q microcavity as the core sensing element. The experiments demonstrate that the chip-scale gyroscope exhibits excellent stability under both room and variable temperature conditions, with open-loop dynamics reaching up to tens of thousands of degrees per second.

It has been known that continuous single-mode propagation is not a prerequisite for optical gyroscopes. A short segment of single-mode optical fiber or waveguide at the common input and output ports is sufficient to maintain the reciprocal operation of the gyroscope. This property is exploited in the design of a chip-scale resonant integrated optical gyroscope based on a multi-mode co-detection technique. This design measures the angular rotation rate by exploiting the multi-modal capabilities of the microcavity and maintaining system reciprocity through the application of ingenious system structure design. This innovative approach not only enhances the detection sensitivity but also significantly improves the system’s overall stability. Figure 1(a) depicts the schematic of the monolithically integrated resonant optical gyroscope. A multi-mode microcavity is employed as the rotation-rate sensing element, which is interrogated with a broadband super-luminescent diode (SLD). Light from the SLD is directed into the microcavity via a 2×2 coupler and a push-pull Y-branch phase modulator, which circulates in both clockwise (CW) and counterclockwise (CCW) directions. This multi-turn circulation enables the generation of complex interference patterns at the input port of the Y-branch phase modulator. The light resulting from the interference is then transformed into an electrical signal by a photodetector (PD). Subsequent synchronous demodulation and filtering provide an accurate angular rotation rate. The straightforward architecture and efficient signal processing of this system significantly enhance its practical applicability.

The use of a broadband source in the resonant optical gyroscope offers a number of significant advantages [17–20], although it does suffer from a primary limitation in the form of power utilization. The implementation of a multi-mode co-detection technique represents an effective means of overcoming this limitation. When a broadband source is employed, the high-Q microcavity acts as an effective spectral filter. Consequently, the spectral line shape of the light emitted from the transmission port is significantly altered, as depicted in Fig. 1(b). The original broad spectrum of the light is reshaped into a distinct frequency comb, featuring sharp peaks corresponding to the microcavity’s resonance frequencies. As the cavity contains numerous non-degenerate modes, the final spectrum of light emitted from the transmission port is the sum of the response spectra for all modes within the cavity, as depicted in Fig. 1(c). For clarity, different colors are used to distinguish the various non-degenerate modes. It is evident that the response spectra exhibit considerable variation among different modes. This is mainly due to the fact that each mode has its own distinct coupling coefficients and transmission losses. Furthermore, the presence of mode dispersion leads to small differences in the equivalent transmission paths between modes, which gives rise to disparate free spectral ranges (FSRs) among the modes. The optical field response function for each mode within the microcavity can be expressed as HE,i(f)=ki(1−αi)14ejπffFSRi(1−ki)(1−αi)12ej2πffFSRi,(1)with the FSR for the ith mode being fFSRi=cniLi,(2)where i represents the number of different modes within the microcavity, ki is the coupling efficiency of mode i,αi is the loss coefficient of mode i for circulating one turn within the microcavity, c is the velocity of light in vacuum, Li is the length of the path that mode i transmits for one turn within the cavity, and ni is the effective refractive index of the mode i within the cavity.

Figure 1(d) demonstrates the fundamental detection mechanism employed by the multi-mode co-detection technique. In a stationary state, the spectra of beams traversing the microcavity from the CW and CCW directions coincide exactly, resulting in a peak in light intensity due to the phenomenon of perfect interference. Upon rotation, the Sagnac effect alters the resonance frequencies of the beams transmitting in opposite directions, causing their spectra to separate. This separation results in a reduction of the optical power of the interfering light. The frequency shift induced by the rotation is then quantified by monitoring the variations in the optical power of the interfering light, thereby providing precise angular rotation rate measurements. It is important to note that the frequency difference induced by rotation differs among the light modes, which can be expressed as Δfi=Dini·λi·Ω,(3)where Di is the effective diameter of the microcavity for mode i,Ω denotes the angular rotation rate, and λi is the resonance wavelength associated with mode i. Consequently, the expression for the relationship between the detected optical power and the resonance frequency difference induced by the rotation can be expressed as P(Δfi)=∫f1f2P0(f)∑i=0Nmod|[HE,i(f+Δfi)+HE,i(f−Δfi)]|2df,(4)where P0(f) is the input power at frequency f,Δfi represents the shift in the resonance frequency for mode i induced by rotation, Nmod is the total number of modes within the microcavity, and f1 and f2 are the start and end frequencies of the broadband light source, respectively.

In the broadband-source-driven ROG scheme, a transmission-type resonator is required to improve the signal-to-noise ratio of the gyroscope output. Therefore, we fabricated a dual-side-coupled four-port WGM microcavity. It is important to note that dual-side coupling inherently results in higher coupling losses compared to single-side coupling. Therefore, to maintain a high Q value for the WGM microcavity, we fabricate the microcavity by manual polishing [22–24]. The detailed manufacturing process is as follows.

First, a centimeter-sized MgF2 wafer purchased from a commercial crystal company is affixed to a stainless-steel rod using UV-curing glue. Then, the rod is precisely mounted onto a ball-bearing lathe and rotated by a motor with adjustable speed. Aluminum oxide sandpapers with varying grits of 1200, 1500, 2000, and 5000 are successively employed to shape the resonator until the desired size is achieved without noticeable scratches on the sample surface. Afterwards, the rim surface of the sample is manually polished using polishing cloths assisted by diamond suspensions with different particle sizes. In our fabrication procedure, 3 μm, 1 μm, 0.25 μm, 0.1 μm, and 0.05 μm suspensions are used subsequently to achieve an extremely smooth rim. Each step lasts for about 30 min. After each step, the sample should be carefully cleaned with anhydrous ethanol and deionized water. Finally, a thorough cleaning procedure with a 3 min water flow and acetone soaking is applied before packaging. Figure 2 shows physical photographs of the high-Q microcavity.

The resonance curves of the microcavity are measured using a swept narrow linewidth laser with a center wavelength of 1550 nm, and the practical test results are shown in Fig. 3(a). The resonance curve exhibits a clear periodic structure, with each period containing several distinct resonance peaks, each corresponding to a mode with a unique Q value. It is estimated that there are approximately 67 non-degenerate modes with transmittances above 0.1 within the microcavity.

The high-resolution scans of three of the modes within the microcavity, labelled A, B, and C, are presented in Fig. 3(b) with quality factors of 7.33×108, 4.69×108, and 2.42×108, respectively. The final output of the gyroscope is the superposition of the responses of all modes. In accordance with the principle of power conservation, the equivalent spectrum after multi-mode superposition is also shown in Fig. 3(b), where its equivalent Q (Qe) and the equivalent maximum transmittance (ρe) are noted as 2.08×108 and 0.62, respectively. According to Eq. (4) and Eq. (A6) (see Appendix A) and the coupling and loss parameters for each mode obtained from Fig. 3(a), the final response and demodulation curves at the maximum slope points for the multi-mode co-detection scheme are calculated and presented in Figs. 3(c) and 3(d). For comparison, the corresponding simulation curves for single-mode detection are also shown, corresponding to a Q of 7.33×108. This comparison demonstrates a 211-fold enhancement in power utilization when employing multiple modes within the cavity to sense the angular rotation rate. Moreover, the utilization of the multi-mode co-detection technique results in a 60-fold enhancement of the slope of the demodulation curve. This substantial enhancement indicates that, under the same input power conditions, the theoretical sensitivity of the multi-mode co-detection scheme is 60 times greater than that of the single-mode detection scheme. It can be observed that the improvement in demodulation slope is slightly less pronounced than the increase in power utilization. This discrepancy can be attributed to the fact that the effective Q of all modes combined is reduced in comparison to the maximum Q. The results show the profound advantages of multi-mode co-detection in enhancing both the efficiency and sensitivity of optical gyroscopes that utilize high-Q microcavities.

To fully assess the performance of the broadband-source-driven integrated optical gyroscope based on the multi-mode co-detection technique, the gyro system is placed on a single-axis rotating platform, and is tested under stationary and rotation situations. The basic schematic diagram of this integrated optical gyroscope is shown in Fig. 4. After passing through a circulator (CIR), light from an SLD source at 1550 nm is divided into two equivalent beams by a pull-push Y-branch phase modulator (PM), and then enters the microcavity from CW and CCW directions. The two opposite waves from the microcavity interfere with each other at the input port of the Y-branch PM. A PD converts the optical power returning from CIR into an electrical signal, which is then collected to the field-programmable gate array (FPGA) for digital signal processing via an analog-to-digital (A/D) converter. The angular rotation rate signal can be acquired after synchronous demodulation and low-pass filtering processes within the FPGA.

We test both the long-term and short-term performance of the gyroscope. Figure 5(a) illustrates the gyroscope’s response to Earth’s rotation. The orange curve represents the gyroscope’s central axis, which is oriented vertically upward from the horizontal plane, while the green curve depicts it as oriented vertically downward. The results closely align with the anticipated rotational rate of the Earth, which is 7.5 deg/h at a latitude of 30°N in Hangzhou, China. Subsequently, the gyroscope was subjected to further testing in a laboratory setting. Figure 5(b) presents the gyroscope’s output over a 36 h period without additional temperature control conditions, while Fig. 5(c) shows the corresponding Allan deviation analysis of 6000 s gyro output data. The angle random walk (ARW) and bias instability can be calculated to be 0.8deg/h1/2 and 1 deg/h, respectively. These values indicate that this gyroscope has achieved tactical-grade performance.

A noteworthy advantage of this detection scheme is its exceptional thermal stability. Figure 5(d) illustrates the output of the gyroscope in the presence of rapid temperature changes. Over a 22 min period, the temperature of the microcavity increased from 31.2°C to 45.8°C, with the fastest rate of temperature change exceeding 10°C/min. Throughout this period, the gyroscope exhibited a stable output, with occasional spikes potentially resulting from external electrical perturbations. This stability is primarily attributed to the fact that the scheme relies on the aggregate behavior of all peaks and modes within the spectral range of the broadband source rather than on specific resonance peaks or modes. Consequently, this approach minimizes the impact of intra-cavity mode interactions resulting from temperature variations. Furthermore, the system’s reciprocal design ensures that any changes in the FSR due to temperature are symmetrically reflected in the light transmitting in opposite directions, thereby preserving the operational stability of the gyroscope’s operating point.

Figure 6(a) depicts the gyroscope’s response to sinusoidal oscillations of the platform. Three sinusoidal amplitudes were employed, namely, 15 deg/h, 0.5 deg/s, and 5 deg/s. Figure 6(b) shows the output of the gyroscope in response to the rotating platform at speeds ranging from ±1deg/s to ±1000deg/s. The test results are compared with the simulation results in Fig. 6(c) (simulation details are included in Appendix A), and it can be seen that they are in good agreement, which confirms the excellent linearity of the gyroscope response and its high dynamic range, which can be up to tens of thousands of degrees per second, under open-loop conditions.

The above test results demonstrate that this novel gyroscope scheme exhibits notable advantages in terms of long-term stability and operating dynamic range, which are highly beneficial for the practical application of gyroscopes. To further clarify the limiting factors of the detection sensitivity of the current system and to identify the future direction of improvement, the noise composition of the current system is discussed and analyzed. First, based on the detection power of 8 μW at PD, the shot-noise-limit sensitivity of the current system can be calculated to be 0.177deg/h1/2 [25], which is much lower than the practical detection sensitivity illustrated in Fig. 5(c). It can therefore be concluded that other noise sources limit the detection sensitivity of the current system. Previous studies have demonstrated that relative intensity noise (RIN) is typically the primary noise source that limits the short-term detection sensitivity of resonant optical gyroscopes when using broadband light sources [17,26]. Moreover, the magnitude of RIN itself varies at different modulation frequencies. Figure 7(a) illustrates the proportions of the various noise sources present in the current system at different modulation frequencies (for more details, see Appendices A and B). The green squares represent the measured ARW induced by shot noise, the yellow triangles depict the ARW induced by electrical noise measurements, and the red circles show the test results of ARW induced by RIN. The black dashed lines correspond to the simulation results of the RIN, where both the electrical noise and RIN are inserted into the error bars. It can be seen that using a higher modulation frequency within the current system can reduce the error induced by the RIN. However, the higher modulation frequency also necessitates more rigorous circuit performance. An alternative approach is to expand the spectral width of the broadband source, which can directly reduce the RIN-induced error. Furthermore, a wider spectral width ensures coverage over a greater number of resonance peaks, thereby enhancing the overall stability of the system.

In this gyroscope, which employs a multi-mode co-detection scheme, the theoretical sensitivity is determined by the product of the equivalent quality factor, Qe, the maximum equivalent transmittance, ρe, and the number of non-degenerate modes within the microcavity. For any mode within the microcavity, its optical power transfer function at the transmission port can be expressed as [25,27] HP,i(f)=(Ai2Bi)2(1−Bi2Bi)2+[sin(πnifLic)]2,(5)with Ai and Bi as substitute variables $$\begin{gathered} Ai=ki(1−αi)14, \\ Bi=(1−ki)(1−αi)12. \end{gathered}$$

Given that the line shape of Eq. (5) agrees with the Lorentzian line shape, the specific values of Ai and Bi can be obtained by scanning the resonance curve of the microcavity.

Based on Eq. (5), the maximum transmittance can be written as ρi=Ai2(1−Bi)2,(6)and the corresponding finesse (F) can be written as Fi=πBi1−Bi.(7)

According to the basic relationship between Q and F, the equation for Q can be expressed as Qi=niπLiBiλi(1−Bi),(8)where λi is the resonance wavelength corresponding to mode i.

According to Eq. (4), the equivalent mode of the microcavity is obtained by summing the power of each mode within the microcavity. As a result, the equations for ρe and Qe are in the same formats as those presented in Eq. (6) and Eq. (8), respectively.

Based on the theoretical sensitivity formula used for broadband-source-driven resonant optical gyroscopes, the theoretical sensitivity of the current gyroscope with a multi-mode co-detection scheme can be similarly expressed as δΩshot=2·cLeDe·λFeRD·PDhv·τint,(9)where Fe is the equivalent finesse of all the modes within the cavity, Le and De are the average length and diameter of all the modes within the microcavity, h is the Planck constant, v is the frequency of light, RD is the responsivity of the PD, PD is the detected optical power at the PD, and τint is the integration time.

Furthermore, when using a broadband source, the detected power at the PD can be written as [25] PD=4Pinγ0NmodA21−B2,(10)where Pin is the incident optical power at the entrance of the cavity, γ0 is the loss coefficient of the gyroscope system except for the microcavity, and A and B are the coefficients of the transfer function corresponding to the equivalent mode.

As a result, by substituting Eqs. (6), (8), and (10) into Eq. (9), the relationship between theoretical sensitivity and Qe×ρe, can be calculated as follows: lg(δΩshot)≈12lg(2nhvλc2D2γ0RDτintL)−12lg(PinNmod)−12lg(Qeρe).(11)

Assuming an entry cavity power of 1 mW and the presence of 67 non-degenerate modes within the cavity, Fig. 7(b) then illustrates the calculated theoretical sensitivity at different Qe×ρe products. The square dot indicates the theoretical sensitivity achievable with the current microcavity parameters, while the triangle represents the Qe×ρe value necessary to reach a theoretical sensitivity of 0.01deg/h1/2. It is seen that when the product Qe×ρe exceeds 3.8×1010, the gyroscope’s theoretical sensitivity will be below 0.01deg/h1/2, which represents a significant advantage among miniature gyroscopes. An increase in the entry cavity power or the number of non-degenerate modes within the cavity results in a reduction in the requirement for the product Qe×ρe.

In conclusion, a novel angular velocity detection scheme for chip-scale micro-gyroscopes has been developed, which implements a multi-mode co-detection technique in a resonant optical gyroscope for the first time. This advancement has enabled the achievement of bias stability of 1 deg/h using a high-Q microcavity with a 9.2 mm diameter, which represents a significant breakthrough in the field of gyroscopes of this scale. The proposed multi-mode detection scheme not only enhances the stability of the system by leveraging the characteristics of the broadband source but also ingeniously utilizes the multi-mode properties of the high-Q cavity to address the issue of low power utilization of the broadband source. This effectively demonstrates the significant advantages of the multi-mode detection scheme in miniature gyroscopes and breaks the tradition of optical gyroscopes that achieve angular rotation rate detection based on a specific mode within the cavity. Another advantage of this novel scheme is that its theoretical sensitivity is only limited by the shot noise and not by the quantum backaction noise induced by the nonlinear effect when using coherent light. This implies that if the Q value of the microcavity can be further increased, the scheme will achieve an exceptionally high theoretical sensitivity, as illustrated in Fig. 7(b). Previous reports have indicated that a microcavity with a Q value exceeding 1011 has been fabricated [28], which suggests that chip-scale optical gyroscopes with navigation-grade performance are likely to be available in the near future. Moreover, the entire optical system is highly simplified, and the detection circuit comprises only two well-established and stable modules: the modulation-demodulation module and the low-pass filtering module. This allows for the integration of all system components onto the chip. Based on these advancements, we believe the proposed novel detection scheme will facilitate the practical implementation of chip-scale integrated optical gyroscopes at an accelerated pace.