1. INTRODUCTION

The fiber-optic gyroscope (FOG) utilizes the Sagnac effect between the two counterpropagating light beams for rotation measurement. As the core device in inertial navigation systems, FOG is indispensably adopted under many extreme environments like aerospace, drones, and wellbore surveys [1,2]. Those applications have a strong demand for the reliability and miniaturization of the gyroscope. The interferometric FOG (IFOG) is the first generation and the only mature type of FOG, in which the Sagnac effect behaves as the phase shift between the two counterpropagating light beams, and the phase shift is detected by the interference of a low-coherence light. The IFOG has a very simple structure known as a “minimal scheme” [3,4], which brings superiority in cost, system simplicity, robustness, and portability over the ring laser gyroscope [5,6] and the advantage of much higher resolution over the microelectromechanical system gyroscope [7,8].

In spite of their great success, high-resolution IFOGs usually employ kilometer-level fiber coil to enlarge the phase shift caused by rotation. The long fiber coil, however, intensifies the thermal ununiformity and leads to the Shupe effect, which would degrade the resolution [9]. By redirecting the light beam into the fiber coil, re-entrant FOG has been proposed as a size-reduced IFOG configuration to relieve the Shupe effect [10,11]. However, the re-entrant FOG is not a very effective option because its actual performance is limited by the optical power loss and the complicated demodulation method. The miniaturization of IFOG remains very challenging.

As the second generation of FOG, the resonant FOG (RFOG) is another potential solution to miniaturization, which employs a high-finesse recirculating fiber ring resonator (FRR) to enhance the Sagnac effect [12,13]. It has the same theoretical resolution as IFOG but with much shorter fiber length, and the nonreciprocity noises such as the Shupe effect can be effectively suppressed with a shorter fiber [2,14] in practice. More attractively, the technology for RFOG is also the most feasible solution to the on-chip gyroscope, in which an optical waveguide resonator replaces the FRR [15–18].

Active research has been dedicated to RFOG [19,20] over the past four decades, but progress has gone very slowly, and the best-reported RFOG has not reached the navigation grade yet [21,22]. Unlike the situation in IFOG, the Sagnac effect in RFOG behaves as the resonant frequency shift of FRR, so, up to date, all the RFOGs use narrow-linewidth lasers to detect the resonant frequency deviation, which introduces serious parasitic noises including backscattering, backreflection, polarization cross talk, nonlinear Kerr effect, and laser frequency noise [13]. In addition, at least two independent optical frequency locking loops are required to lock the laser to FRR resonances in both clockwise (CW) and counterclockwise (CCW) directions [23]. The extra elements for frequency locking further deteriorate the parasitic noises. In general, the current RFOG technologies lose the simplicity of IFOG, and the complicated system setup results in extra noises and fragility.

Here we have developed a new RFOG readout technique, in which the multibeam interferometry of the low-coherence white-light source is proposed to detect the resonant frequency deviation caused by Sagnac effect. It inherits the simplicity of IFOG with the same “minimal scheme,” and the two main problems of traditional RFOGs are avoided. A navigation-grade bias instability (BI) of 0.009°/h and an angle random walk (ARW) of 0.0093°/$\sqrt{\mathrm{h}}$ are obtained with a 100-m fiber length in the demonstrational experiment, which is the first report, we believe, of the RFOG.

2. GYRO PRINCIPLE

A. White-Light Multibeam Interferometry

The setup of the RFOG is shown in Fig. 1. According to multibeam interference, the transmission light electric field ($E$) of the add-drop FRR has the expression [24]

 

Fig. 1. RFOG setup. ASE, amplified spontaneous emission; PD, photodetector; FPGA, field programmable gate array; MIOC, multifunction integrated-optics chip; FRR, fiber ring resonator; CW, clockwise; CCW, counterclockwise; AD, analog-to-digital; DA, digital-to-analog. Different from traditional IFOGs based on the minimal scheme, the long fiber coil is replaced with a high-finesse FRR. A photograph of the RFOG setup is also provided in Fig. 8 in Appendix A.

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For two counterpropagating single-frequency laser beams ($E_{\mathrm{CW}}$ and $E_{\mathrm{CCW}}$), their interferometric signal at the photodetector (PD) is ${|E_{\mathrm{CW}}+E_{\mathrm{CCW}}|}^2/2$. Considering the low-coherence white-light source as the combination of numerous lasers operating at different frequences, its multibeam interferometric signal ($P_{\mathrm{PD}}$) is the integration of all the frequency components and is expressed as [26]

B. Simulation and Experiment Demonstration

In the simulation, the spectral profile $S(\upsilon )$ is assumed to be rectangular for simplicity. The spectral bandwidth is set to 4 THz, centering at 193 THz, much larger than the $\mathrm{\Delta }{\upsilon }_{\mathrm{FSR}}$ of the FRR. Under several values of finesse ($\mathcal{F}$), the relationship curves between the normalized $P_{\mathrm{PD}}$ and the $f_{\mathrm{sag}}$ are computed, as plotted in Fig. 2(a). The simulation curve appears as a Lorentzian resonance, and it is the result of the proposed white-light multibeam interferometry. It has to be pointed out that it is not the actual resonance of FRR. Despite the similar appearance, its finesse is found to be a half of the actual FRR finesse ($\mathcal{F}$). Because of only focusing on a small range of $f_{\mathrm{sag}}$ [$|f_{\mathrm{sag}}|<\mathrm{\Delta }{\upsilon }_{\mathrm{FSR}}\ll$ bandwidth of $S(\upsilon )$], it is found that the simulated curves are not affected by the spectral profile and bandwidth of the light source.

 

Fig. 2. (a) Simulation analysis and (b) experimental demonstration of Eq. (4). They are the outputs of the proposed white-light multibeam interferometry and are defined as the response curves of the proposed RFOG.

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Afterwards, an experiment is carried out to demonstrate the proposed white-light multibeam interferometry. In the experiment, the FRR in Fig. 1 is composed of two 98:2 couplers and a fiber coil. The fiber length and diameter of the coil are $\sim 100\mathrm{m}$ and 140 mm, respectively. The round-trip loss of the cavity is measured to be 0.268 dB. The RFOG is based on panda-type polarization-maintaining (PM) fiber to minimize polarization-related noises. By using an acousto-optic frequency shifter and a narrow-linewidth laser to scan the FRR spectrum, the full width at half-maximum (FWHM) and the $\mathrm{\Delta }{\upsilon }_{\mathrm{FSR}}$ of the FRR resonance are measured to be 30.5 kHz and 1.918 MHz, respectively, corresponding to a finesse of 63.

A multifunction integrated-optics chip (MIOC) is adopted here for light splitting and combining, polarization filtering, and phase modulation. The MIOC is based on ${\mathrm{LiNbO}}_3$, and its half-wave voltage ($V_{\pi }$) is 3.99 V. The light source of the system is an amplified spontaneous emission (ASE) source based on erbium-doped fiber. Its bandwidth is $\sim 35\mathrm{nm}$, centering at 1550 nm. Its power is set to 100 mW in the following experiments. In navigation-grade IFOGs, it is a most commonly used source because of its high stability and power. The PD (Thorlabs, PDA10CS-EC) is based on InGaAs and is working at the C band. Its bandwidth ranges from 0 Hz to 1 MHz, and it has adjustable gain.

By driving the MIOC with opposite sawtooth waveforms at its two arms, an equivalent $f_{\mathrm{sag}}$ can be introduced between the CW and CCW directions (more details on the $f_{\mathrm{sag}}$ introducing scheme can be found in Appendix A). Therefore, an experimental curve of Eq. (4) can be obtained by setting the $f_{\mathrm{sag}}$ at a series of discrete values and acquiring the corresponding $P_{\mathrm{PD}}$, as shown by the red dots in Fig. 2(b). The blue curve connects the adjacent dots and outlines the curve profile. Its FWHM is 60.9 kHz, corresponding to a halved finesse of 31.5 versus the real FRR finesse. The experiment and the simulation results are in good accordance.

According to the results in Figs. 2(a) and 2(b), the PD receives maximal power $P_{\mathrm{PD}}$ if no rotation is applied to the FRR, and it declines sharply if the FRR rotates. The curve in Fig. 2(b) is defined as the response curve of the RFOG. Different from traditional RFOGs, where CW and CCW directions need independent and complicated interrogation systems, the proposed RFOG has a much simpler working scheme. Also unlike the cosine response curve of IFOG [2], the response curve of the RFOG appears as a sharp peak due to multibeam interference, and thus it has improved sensitivity to rotation.

C. RFOG Modulation and Demodulation

The RFOG readout is based on utilizing the property of the response curve in Fig. 2(b). Because of the symmetry of the response curve, modulation techniques are necessary to discriminate between the positive/negative $f_{\mathrm{sag}}$, so as to distinguish the rotation direction. The details of the modulation process are shown in Fig. 3. If there is no modulation and rotation, $P_{\mathrm{PD}}(f_{\mathrm{sag}})=P_{\mathrm{PD}}(0)\approx 20\mathrm{\mu W}$, corresponding to the peak value of the response curve in Fig. 2(b). Because the frequency is the derivative of the phase with time, if the MIOC is driven with upward sawtooth wave (frequency, 21 kHz; peak amplitude, $V_{\pi }$) at the CW arm and the opposite at the CCW arm, a positive frequency bias ($f_{\mathrm{mod}}=21\mathrm{kHz}$) is introduced to the $f_{\mathrm{sag}}$, and the output of the PD decreases to $P_{\mathrm{PD}}(f_{\mathrm{mod}})$. Afterwards, opposite sawtooth waves are applied to introduce a negative frequency bias $-f_{\mathrm{mod}}$ and the output of the PD becomes $P_{\mathrm{PD}}(-f_{\mathrm{mod}})$.

 

Fig. 3. Modulation signals and the corresponding $P_{\mathrm{PD}}$ in different gyro states. $V_{\mathrm{MIOC}}$, voltage applied on the MIOC; $V_{\pi }$, half-wave voltage; $f_{\mathrm{mod}}$, modulated frequency bias.

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In the modulation, the $f_{\mathrm{mod}}$ alternately changes between $\pm 21\mathrm{kHz}$ with a rate of 500 Hz. The variation of $P_{\mathrm{PD}}$ during the $f_{\mathrm{mod}}$ switching is defined as the error signal $e=P_{\mathrm{PD}}(f_{\mathrm{mod}})-P_{\mathrm{PD}}(-f_{\mathrm{mod}})$. If there is no rotation, considering the axial symmetry of $P_{\mathrm{PD}}$, $e$ is equal to 0. If a rotation rate is applied and an $f_{\mathrm{sag}}$ is introduced, the error signal becomes

In the gyro uses, rotation-induced $f_{\mathrm{sag}}$ is usually very small ($f_{\mathrm{sag}}\ll \mathrm{FWHM}$). Therefore, Eq. (5) can be simplified as

A field programmable gate array (FPGA, NI 7855R) module with integrated analog-to-digital (AD) and digital-to-analog (DA) converters is adopted for $P_{\mathrm{PD}}$ acquisition, $f_{\mathrm{sag}}$ modulation, and $e$ calculation, as shown in Figs. 1 and 4(a). The FPGA multiplies the $P_{\mathrm{PD}}$ and the modulation square ($\pm f_{\mathrm{mod}}$) with a mixer. The mixer output passes through a low-pass filter (LPF, 20 Hz), and the error signal $e$ is generated referenced to the average input power $P_{\mathrm{in}}$. After these algorithm operations, $e$ is irrelevant to the power level, and it has no unit. Similar to Fig. 2(b), at some discrete values of $f_{\mathrm{sag}}$, $e$ is recorded, as shown by the red dots in Fig. 4(b). This curve is antisymmetric and it crosses zero if no rotation exists, so the rotation direction can be distinguished. At the central region, $e$ is proportional to $f_{\mathrm{sag}}$, with an $e$-to-$f_{\mathrm{sag}}$ coefficient of 1/(7600 Hz). By using a high-precision rotating platform, the $\mathrm{\Omega }$-to-$f_{\mathrm{sag}}$ coefficient is tested to be 3.4°/(h · Hz). After the calibration of these scaling factors, the rotation signal can be successfully recovered from $P_{\mathrm{PD}}$, and the RFOG gets ready for measurement.

 

Fig. 4. (a) Demodulation process in the FPGA; (b) measured error signal versus $f_{\mathrm{sag}}$. LPF, low-pass filter.

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3. GYRO TEST

The proposed RFOG is first adopted for the rotation test with the high-resolution rotating platform (TD-320), whose nominal rotation resolution and range are 0.0001°/s and >500°/s, respectively. A sinusoidal rotation rate (10°/h, 0.01 Hz) is applied to the RFOG. At the same time, the FPGA records the error signal and recovers the rotation. The rotation data are downsampled to 1 Sa/s via averaging and are plotted in Fig. 5(a), where the sine rotation rate is well recovered. Then the rotation rate is decreased to 1°/h, and Fig. 5(b) is obtained. The recovered sinusoidal curves prove that the resolution of the proposed RFOG is better than 1°/h in the bandwidth of 1 Hz.

 

Fig. 5. Test results of the RFOG. Gyro readout under sinusoidal rotation of (a) 10°/h, (b) 1°/h, and (c) 0°/h; (d) moving average of the static test data in (c) with a time window of 1000 s; (e) spectral power density of the results in (a)–(c); (f) Allan deviation of the static test data in (c).

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The RFOG is then horizontally placed in a sealed box for a static test, where the temperature fluctuation is about 0.4 K/h. In this case, the RFOG is only subjected to the Earth’s rotation, which is 7.8°/h at the testing site (Shanghai, China). The gyro data recorded in 35,000 s are shown in Fig. 5(c). The output has a bias around −4.5°/h, which may be attributed to the nonreciprocity of couplers [27]. When the RFOG is turned over to sense the opposite of the Earth’s rotation, the bias changes from −4.5°/h to −20.1°/h. The bias difference is twice the Earth’s rotation rate. As shown in Fig. 5(c), the dominating gyro noise has the white-noise characteristics. After averaging with a moving window of 1000 s, the long-term bias drift emerges, as shown in Fig. 5(d). Figure 5(e) plots the power spectral density of the gyro data in Figs. 5(a)–5(c). The peaks correspond to the applied rotation signals, and the noise floor is generally around 0.69°/(h · Hz^0.5).

The overlapping Allan deviation of the 35,000-s gyro data is calculated and is shown in Fig. 5(f) [28]. An ARW of $0.0093°/\sqrt{\mathrm{h}}$ and a BI of 0.009°/h can be read out from it, which demonstrates that the resolution of the proposed RFOG has reached the navigation grade. In this demo experiment, the RFOG works in an open-loop configuration, so the linear region of the error signal curve in Fig. 4(b) restrains the measurement range of $f_{\mathrm{sag}}$ to be about $\pm 15\mathrm{kHz}$, corresponding to a rotation rate of ±14°/s. Table 1 lists all the key parameters of the RFOG during the test. This RFOG is also compatible with a closed-loop operation. By adding a feedback control loop to adjust the frequency shift of the MIOC, the rotation-induced $f_{\mathrm{sag}}$ can be well compensated so as to null the error signal. The closed-loop configuration can further extend the measurement range and stabilize the gyro bias [29].

Table 1. Parameters of the RFOG

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4. NOISE ANALYSIS

A. Shot-Noise Limited Resolution

The resolution of RFOG is ultimately limited by the shot noise at the detector, which is given by [16,30]

B. RIN-Induced ARW Noise

The noises of gyros are typically classified into two types: short-term ARW noise and long-term nonreciprocity-induced bias drift. In Fig. 5(f), the turning point of Allan deviation appears at an averaging time of $\sim 3000\mathrm{s}$, proving that short-term ARW noise is dominating in this RFOG. Therefore, noise analysis on the ARW is necessary to explore the full potential of the proposed RFOG. Since the rotation signal is read from light intensity at the PD, the relative intensity noise (RIN) of $P_{\mathrm{PD}}$ is the main contributor to ARW noise and is investigated. Usually, relative to the fundamental shot noise, the RIN induced by the ASE source (know/n as excess RIN) is much higher [31].

In the excess RIN analysis, modulation on the MIOC is turned off, and $P_{\mathrm{PD}}$ is adjusted to 16 μW. The RIN of $P_{\mathrm{PD}}$ is recorded by the FPGA and is plotted as the red curve in Fig. 6. The excess RIN mostly originates from the random beat of different frequencies of the broadband light [32,33]. Because of the spectral shaping effect of the FRR, the RIN of $P_{\mathrm{PD}}$ rises around the frequency of multiple free spectral range (such as 0 Hz and 1.918 MHz). For comparison, the blue curve depicts the RIN of input ASE light ($P_{\mathrm{in}}$), which has a flat noise floor in the frequency domain. Due to the spectral shaping of the FRR, the red curve has higher noise at low frequency and lower noise at high frequency. At the RFOG working frequency of 21 kHz, the rising RIN floor of $P_{\mathrm{PD}}$ is responsible for the ARW noise in Fig. 5(f) ($0.0093°/\sqrt{\mathrm{h}}$). Since the RIN floor of $P_{\mathrm{in}}$ (blue curve) is about 9 dB lower, a lower ARW of $0.0033°/\sqrt{\mathrm{h}}$ can be obtained, if without the unwanted spectral shaping effect.

 

Fig. 6. RIN of $P_{\mathrm{PD}}$ and $P_{\mathrm{in}}$. The working frequency of the RFOG is 21 kHz.

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On the other hand, the spectral shaping effect also has positive utilization to suppress the ARW noise [33,34]. By inserting another fiber resonator after the ASE source to reshape the spectra of $P_{\mathrm{in}}$ and $P_{\mathrm{PD}}$, a series of RIN notches can be introduced to reduce the RIN floor at the point of working frequency [33]. Except for the spectral shaping method, many other RIN suppression techniques have been developed for IFOG to suppress the ARW noise, such as optical or electrical subtraction [32,35], quadrature demodulation [36], and light depolarization before photodetection [37]. Most of them are compatible with this novel RFOG. So, better ARW performance can be expected to approach the shot-noise limited resolution.

C. Nonreciprocity Noises

Other than ARW noise, nonreciprocity-induced long-term bias drift is another important factor of BI. These nonreciprocities originate from unwanted backreflection, backscattering, polarization cross talk, the Kerr effect, the Shupe effect, etc. [13]. These nonreciprocity noises significantly degrade the performance of traditional RFOG systems because of the employment of high-coherence lasers. However, they are well mitigated in this paper by the use of low-coherence light source and the minimal scheme.

The polarization cross talk in the proposed RFOG is suppressed in the following actions: first, the adoption of PM fiber minimizes the polarization fluctuations while light circulates inside the FRR. In addition, the MIOC has integrated polarizers to ensure the polarization consistency between the input and output light. The high polarization extinction ratio of the PM fiber and the MIOC makes the whole system operate stably in single polarization. Further, thanks to the low-coherence nature of the broadband light, the residual interference noise between two orthogonal polarizations is well mitigated [38]. The efficiency of these polarization-suppression actions (polarizer, PM fiber, and low-coherence source) has been widely demonstrated in numerous navigation-grade IFOGs.

The nonlinear Kerr effect is proportional to the intensity imbalance between the CW and CCW light waves. According to the analysis in Ref. [9], it can also be substantially weakened through the utilization of a low-coherence source.

The Shupe effect originates from thermal ununiformity along the long fiber coil. In most IFOG systems, the fiber coil is commonly fabricated with elaborate winding and packaging techniques to mitigate the thermal ununiformity [39,40]. The residual Shupe effect, however, limits the further resolution improvement of IFOG. Here, the equivalent length of the RFOG is quite long ($\sim 6300\mathrm{m}$) considering its finesse of 63. However, the RFOG “packages” the 6300-m equivalent length into an actual size of only $\sim 100\mathrm{m}$ in the way of optical recirculation and resonance. Compared with the elaborate winding of IFOG coil, it is obviously a more effective approach. Therefore, the RFOG has inherent insensitivity to the thermal-related Shupe effect due to its compact actual size.

5. CONCLUSION

In summary, we present the first navigation-grade RFOG with the simplest readout scheme to date. As shown in Table 2, compared with traditional IFOGs and RFOGs, the proposed RFOG utilizes a low-coherence white-light source, high-finesse FRR, and the minimal scheme. It combines the key advantages of conventional IFOGs and RFOGs such as ultralow complexity and compact size. In the experiment, an ARW of 0.0093°$/\sqrt{\mathrm{h}}$ and a BI of 0.009°$/\mathrm{h}$ have been achieved with a 100-m fiber coil. It shows great potential in both high resolution and on-chip integration.

Table 2. Comparison between the Proposed RFOG and Traditional FOGs

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APPENDIX A

The MIOC integrates two phase modulators at the CW and CCW arms, respectively, and they are driven with opposite voltage to work in the push–pull mode. By applying sawtooth waveforms to the MIOC, the MIOC works as two frequency shifters, since frequency is the derivative of phase. Figure 7(a) shows the applied sawtooth modulation waveforms. The sawtooth slope is $V_{\pi }{f}_{\mathrm{mod}}$ at the CW arm and $-V_{\pi }{f}_{\mathrm{mod}}$ at the CCW arm, which would introduce $\pm \frac{f_{\mathrm{mod}}}{2}$ frequency shifts at the CW and CCW arms, respectively.

 

Fig. 7. Scheme of introducing equivalent Sagnac frequency via sawtooth modulation. (a) Modulation waveforms at two arms of the MIOC; (b) modulation process and scheme.

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Fig. 8. Photograph of the RFOG system. PC, personal computer.

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The modulation process and scheme of the MIOC are shown in Fig. 7(b). The input light electric field is $E_0(\upsilon )$. After the light splitting and the CW frequency shifter ($+\frac{f_{\mathrm{mod}}}{2}$), $E_A$ (the light electric field at point A) has the expression $E_A=\frac{\sqrt{2}}{2}{E}_0(\upsilon -\frac{f_{\mathrm{mod}}}{2})$. Then, after passing through the FRR, $E_B$ (the light electric field at point B) has the expression $E_B=F_{\mathrm{CW}}{E}_A=\frac{\sqrt{2}}{2}{F}_{\mathrm{CW}}(\upsilon )E_0(\upsilon -\frac{f_{\mathrm{mod}}}{2})$. Further, $E_B$ passes through the CCW frequency shifter ($-\frac{f_{\mathrm{mod}}}{2}$) and the light combiner, and the CW light field ($E_{\mathrm{CW}}$) has the expression

(A1)$$E_{\mathrm{CW}}=\frac{1}{2}{F}_{\mathrm{CW}}(\upsilon +\frac{f_{\mathrm{mod}}}{2})E_0(\upsilon ).$$

For the same reason, the CCW light field ($E_{\mathrm{CCW}}$) has the expression

(A2)$$E_{\mathrm{CCW}}=\frac{1}{2}{F}_{\mathrm{CCW}}(\upsilon -\frac{f_{\mathrm{mod}}}{2})E_0(\upsilon ).$$

Equations (A1) and (A2) indicate that an equivalent Sagnac frequency $f_{\mathrm{mod}}$ can be introduced between the CW and CCW directions by driving the MIOC with the sawtooth waveform plotted in Fig. 7(a). It is the theoretical basis of the RFOG modulation in this paper.

Funding

National Key Research and Development Program of China (2018YFC1503703); National Natural Science Foundation of China (61620106015, 61875121, 61975116).

Acknowledgment

S. X. Zhao proposed the method, performed the experiment, and wrote the draft of the paper; Q. W. Liu, H. L. Ma, and Z. Y. He edited the paper and assisted in improving the sensor performance; S. X. Zhao and Y. Y. Liu analyzed the data; Z. Y. He supervised the project.

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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