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Resonant fiber optic gyroscope using a reciprocal modulation and double demodulation technique

Lu Liu, Shuang Liu, Junyi Hu, Huilian Ma, and Zhonghe Jin

Open Access Open Access

Abstract

A resonant fiber optic gyroscope (RFOG) using a reciprocal modulation and double demodulation technique based on a single laser source is proposed and demonstrated. The effect of the residual amplitude modulation of the phase modulator is well suppressed thanks to the reciprocal modulation and demodulation. On this basis, the backscattering noise is also eliminated by the double demodulation process. The long-term bias stability of the RFOG is successfully improved to 0.2°/h for a test time of 45 hours.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Fiber-optic gyroscopes (FOGs) based on the Sagnac effect are important inertial rotation-rate sensors. The interferometric fiber optic gyroscope (IFOG) and the resonant fiber optic gyroscope (RFOG) are two main types of FOGs [1,2]. The Sagnac effect indicates that an optical path difference occurs between the clockwise (CW) and counterclockwise (CCW) propagating beams in a closed fiber coil or an optical fiber ring resonator (FRR) of a rotating gyroscope, which is directly proportional to the rotation rate [3]. This optical path difference is translated into a resonance frequency difference of the FRR in the RFOG. Unlike the IFOG which improves the resolution by resorting to increasing the length of the fiber coil, the RFOG enhances the rotation-induced Sagnac effect via a high-finesse FRR and the improvement directly depends on the finesse [4]. Therefore, an RFOG is able to achieve the same level of measurement accuracy as an IFOG with a much shorter optical path and has broad application values, especially in the field where requires small, light and robust gyroscopes.

After more than 40 years of development, the operating principles and error mechanisms of the RFOG are basically clear [5–7]. Similar as the IFOG, various sources of noise that degrade the RFOG performance mainly include Rayleigh backscattering, polarization fluctuation, nonlinear optical Kerr effect, and laser frequency noise [8–16]. However, the RFOG still faces many difficulties and challenges in reducing these noise factors due to the use of a narrow linewidth light source and a multi-turn cavity. When we take a measure to suppress one kind of noise, another kind of noise is always introduced. For example, a carrier suppressed separate phase modulation technique can eliminate the backscattering induced noise [8], however, the accompanying residual amplitude modulation (RAM) of phase modulators (PMs) yields bias errors at the gyro output [9–11].

It was appreciated early, by Sanders et al. that the backscattering is a dominant error source in resonant optic gyroscopes [5]. This noise source includes the backscattering intensity term and the coherent term which is the interference between the signal and the scattered wave [17]. It has been verified that the carrier suppressed separate phase modulation technique can simultaneously counter the backscattering intensity and coherent errors. However, it could not solve the accompanying RAM problem. The RAM refers to the phenomenon that the intensity of output light from a PM varies with the modulation signal at the same frequency. The intensity fluctuation will be demodulated together with the rotation-rate signal and thus yields bias errors at the gyro output [18,19]. Descampeaux et al. proposed a method to reduce the influence of RAM using a feedback loop [20]. A reciprocal modulation-demodulation technique is proposed and demonstrated to suppress both the RAM and coherent backscattering effects [9]. Besides, the effect of laser frequency noise can also be eliminated, resulting in an angular random walk (ARW) improvement for a short-time test. However, the effect of backscattering intensity still exists because of a common modulation applied on the two opposite beams, which finally limits the long-term bias stability of the RFOG. A double modulation-demodulation technique was firstly proposed by Strandjord et al., which simultaneously suppressed the backscattering noise and the influence of RAM [10]. Three lasers are employed and two additional optical phase lock loops (OPLLs) are indispensable in this three-laser detection configuration, which significantly increases the complexity of the system. At present, it still remains to be investigated how to effectively suppress both the backscattering noise and the RAM of PMs to improve the long-term stability.

In this paper, a novel method based on the reciprocal modulation and double demodulation technique using a single laser is proposed and demonstrated. First, the RAM problem is solved by applying a common primary phase modulation on the two opposite beams. Two additional non-demodulated carrier suppressed phase modulations are further added to eliminate the backscattering coherent term. A second high-frequency phase modulation is used to separate the CW and CCW beams on spectrum to suppress the error due to backscattering intensity. Double demodulating finally outputs the same demodulation signal as the traditional single modulation-demodulation process, with full suppression of the backscattering noise and the RAM problem. The influence of modulation parameters and imperfections of the second modulation signals on gyro performance are further analyzed. The optimal modulation parameters are then obtained. Measurements on the effect of the second harmonic distortion agree well with the theoretical predictions. In a long-term static test of 45 hours at room temperature, the gyro bias stability is shown to be better than 0.2°/h according to the Allan deviation analysis.

2. Principle and analysis

2.1 Basic principle of the reciprocal modulation and double demodulation RFOG

The RFOG configuration of the reciprocal modulation and demodulation technique is shown in Fig. 1. Light from a narrow-linewidth laser diode (LD) is firstly phase modulated by a primary phase modulator PM0 at frequency fP. The primary modulation is used for resonance-center detection and the influence of RAM can be eliminated since a common modulation is applied with a same phase modulator for the input light in CW and CCW directions. Thus, the RAM-induced error is reciprocal for the two beams [9]. The reciprocal error will be automatically canceled by the laser frequency servo loop and has no effect on the rotation measurement. The output light from PM0 is then divided into two beams by a Y-branch PM. Four modulation signals are separately applied to the four modulation electrodes so that each beam is further phase-modulated twice. The modulation signals at fN1 and fN2 are used to provide carrier suppression to reduce the backscattering coherent errors [8], which will not be demodulated. The modulation signals at fS1 and fS2 are the second high-frequency phase modulation signals, of which the frequencies should be close to (n + 1/2) × FSR, where FSR is the free spectral range of the resonator and n is an integer. The second modulation serves to reconstitute the resonance curve of the FRR at the first demodulation output. fS1 and fS2 are slightly different to avoid sidebands of CW and CCW lights overlap, thus the influence of backscattering intensity can be eliminated. The two beams are further coupled to the FRR in CW and CCW directions and then detected by two photodetectors (PDs).

 figure: Fig. 1.

Fig. 1. Scheme diagram of the reciprocal modulation and double demodulation RFOG.

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The output signals of PDs are double demodulated by lock-in amplifiers (LIAs), each of which consists of a multiplier and a low-pass filter (LPF). The second modulation signals in CW and CCW directions are respectively demodulated for the first time in LIA3 and LIA1 at twice the modulation frequency, which is different from the traditional single modulation-demodulation scheme [21]. Since the intensity fluctuation caused by the RAM of the phase modulator has the same frequency as the modulation frequency, the influence can be removed by using a twice demodulation frequency. Moreover, the influence of backscattering intensity can also be addressed because of different demodulation frequencies applied to CW and CCW beams. The primary modulation is then demodulated at fP for both directions in LIA2 and LIA4. The second demodulation output of LIA2 is scaled to a rotation rate, and the output of LIA4 is used as an error signal to the laser servo loop. The central frequency of the laser is tuned by a proportional-integration (PI) controller, locking to the resonance frequency of the FRR in CW direction. In conclusion, the backscattering noise and the RAM of PMs can be fully suppressed using the reciprocal modulation and double demodulation scheme.

2.2 General expression of the demodulated signals

In order to understand the detailed operation of the reciprocal modulation and double demodulation scheme, the following analysis is devoted to calculating the outputs of the first and second demodulators, with simulation results presented.

Take the CCW light as an example, the output light electric field of the LD is expressed as

$$E(t) = {E_0}{e^{j2\pi {f_0}t}}\textrm{,}$$
where E0 is the light wave amplitude and f0 is the laser central frequency. Since the phase modulation at fN1 is not demodulated, its influence is negligible in the subsequent signal processing. The light field after the second phase modulation is
$$E(t) = {E_0}{e^{j(2\pi {f_0}t + {M_\textrm{P}}\sin 2\pi {f_\textrm{P}}t + {M_{\textrm{S1}}}\sin 2\pi {f_{\textrm{S1}}}t)}},$$
where MP and fP are the modulation index and frequency of the primary modulation, MS1 and fS1 are the modulation index and frequency of the second modulation. To simplify the expression, the insertion loss of PM0 and the Y-branch PM has been ignored. Using the Bessel function expansion, Eq. (2) becomes
$$E(t) = {E_0}\sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {{J_\mathrm{\alpha }}({M_\textrm{P}})} } {J_\mathrm{\beta }}({M_{\textrm{S1}}}){e^{j2\pi ({f_0} + \alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}})t}},$$
where Jα(Mp) refers to the first kind of Bessel function with the αth order, α is an integer representing the αth order of modulation sidebands, and Jβ(MS1) is similar.

The transfer function of the FRR is given by [22]

$$T({f_\textrm{c}}) = h({f_\textrm{c}}){e^{\varphi ({f_\textrm{c}})}},$$
where h(fc) and φ(fc) are the amplitude and phase-delay transfer functions of the FRR when the incident light frequency is fc. For simplicity, when fc = f0 + αfP + βfS1, the amplitude and phase-delay transfer function are then given by
$${h_{\mathrm{\alpha ,\beta }}} = h({f_0} + \alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}}),$$
and
$${\varphi _{\mathrm{\alpha ,\beta }}} = \varphi ({f_0} + \alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}}).$$
After multi-turn transmission in the FRR and p ${h_{\mathrm{\alpha ,\beta }}} = h({f_0} + \alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}}),$ hotoelectric conversion of PD1, the output voltage can be written as
$$\begin{array}{l} {V_{\textrm{PD1}}}(t) = \sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\alpha ^{\prime}} ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta ^{\prime}} ={-} \infty }^{ + \infty } {c{\varepsilon _0}{K_{\textrm{PD1}}}{E_0}^2{J_\mathrm{\alpha }}({M_\textrm{P}}){J_\mathrm{\beta }}({M_{\textrm{S1}}})} } } } {J_{\mathrm{\alpha ^{\prime}}}}({M_\textrm{P}}){J_{\mathrm{\beta ^{\prime}}}}({M_{\textrm{S1}}}) \cdot \\ {h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha ^{\prime},\beta ^{\prime}}}}\cos [{2\pi (\alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}} - \alpha^{\prime}{f_\textrm{P}} - \beta^{\prime}{f_{\textrm{S1}}})t + {\varphi_{\mathrm{\alpha ,\beta }}} - {\varphi_{\mathrm{\alpha^{\prime},\beta^{\prime}}}}} ], \end{array}$$
where KPD1 is the photoelectric conversion coefficient of PD1 with a unit in V/W, c is the light velocity in vacuum and ε0 is the electric permittivity of the vacuum, α′ and β′ represent the other orders of modulation sidebands different from α and β.

The resonator output from PD1 is first demodulated by LIA1 at twice the second modulation frequency, or at 2fS1. Thus a signal at frequency 2fS1 is used as the reference signal:

$${V_{\textrm{S1}}}(t) = \cos [{2\pi (2{f_{\textrm{S1}}})t\textrm{ + }{\varphi_{\textrm{S1}}}} ],$$
where φS1 is the phase component of the reference signal. For simplicity, a unit amplitude is assumed. Then the synchronous demodulated signal is
$$\begin{array}{l} {V_{\textrm{demo\_S1}}}(t) = \frac{1}{2}\sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\alpha ^{\prime}} ={-} \infty }^{ + \infty } {{J_\mathrm{\alpha }}({M_\textrm{P}}){J_\mathrm{\beta }}({M_{\textrm{S1}}}){J_{\mathrm{\alpha ^{\prime}}}}({M_\textrm{P}}) \cdot } } } \\ \{{{J_{\mathrm{\beta } + 2}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha^{\prime},\beta +\ 2}}}\cos [{2\pi (\alpha {f_\textrm{P}} - \alpha^{\prime}{f_\textrm{P}})t + {\varphi_{\mathrm{\alpha ,\beta }}} - {\varphi_{\mathrm{\alpha^{\prime},\beta +\ 2}}} + {\varphi_{\textrm{S1}}}} ]} + \\ {{J_{\mathrm{\beta -\ }2}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha^{\prime},\beta -\ 2}}}\cos [{2\pi (\alpha {f_\textrm{P}} - \alpha^{\prime}{f_\textrm{P}})t + {\varphi_{\mathrm{\alpha ,\beta }}} - {\varphi_{\mathrm{\alpha^{\prime},\beta -\ 2}}} - {\varphi_{\textrm{S1}}}} ]} \}. \end{array}$$
As mentioned before, the function of the second modulation is to reconstitute the resonance curve of the FRR. Thus, when there is no primary modulation, Eq. (9) should be able to accurately reflect the spectral characteristic of the resonator output. To meet this requirement, the second modulation frequency should be set exactly at (n + 1/2) × FSR, where n is an integer. Figure 2 shows the simulation results of the normalized demodulation output of the first demodulator and the normal resonance curve. In the simulation, we consider an FRR of which parameters are the same as the experiment. The total fiber length of the cavity is 120 m and the finesse is 23.8. The simulation results show both the demodulation output and the resonance curve have a same spectral full width at half maximum (FWHM), which means a same finesse. In addition to resonance peaks, resonance valleys also appear at the demodulation output, and both of them can be used as the reference frequencies for laser frequency locking.

 figure: Fig. 2.

Fig. 2. Simulation results of the resonance curve and the output of the first demodulator. (a) Resonance curve. (b) Output of the first demodulator.

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Then the output of the first demodulator should be demodulated for the second time which is completed by LIA2 at fp, given by

$$\begin{array}{l} {V_{\textrm{demo\_P}}}(t) = \frac{1}{4}\sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {{J_\mathrm{\alpha }}({M_\textrm{P}}){J_\mathrm{\beta }}({M_{\textrm{S1}}}) \cdot } } \\ \{{{J_{\mathrm{\alpha +\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2}}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha +\ 1,\beta +\ 2}}}\cos ({\varphi_{\mathrm{\alpha ,\beta }}} - {\varphi_{\mathrm{\alpha +\ 1,\beta +\ 2}}} + {\varphi_\textrm{P}} + {\varphi_{\textrm{S1}}})} + \\ {J_{\mathrm{\alpha +\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta -\ 2}}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha +\ 1,\beta -\ 2}}}\cos ({\varphi _{\mathrm{\alpha ,\beta }}} - {\varphi _{\mathrm{\alpha +\ 1,\beta -\ 2}}} + {\varphi _\textrm{P}} - {\varphi _{\textrm{S1}}})\textrm{ + }\\ {J_{\mathrm{\alpha -\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2}}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha -\ 1,\beta +\ 2}}}\cos ({\varphi _{\mathrm{\alpha ,\beta }}} - {\varphi _{\mathrm{\alpha -\ 1,\beta +\ 2}}} - {\varphi _\textrm{P}} + {\varphi _{\textrm{S1}}})\textrm{ + }\\ {{J_{\mathrm{\alpha -\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta -\ 2}}}({M_{\textrm{S1}}}){h_{\mathrm{\alpha ,\beta }}}{h_{\mathrm{\alpha -\ 1,\beta -\ 2}}}\cos ({\varphi_{\mathrm{\alpha ,\beta }}} - {\varphi_{\mathrm{\alpha -\ 1,\beta -\ 2}}} - {\varphi_\textrm{P}} - {\varphi_{\textrm{S1}}})} \}, \end{array}$$
where φP is the phase component of the reference signal in the second demodulation. Figure 3 shows the simulation results of the second demodulation output. The modulation parameters used in the simulation have been optimized to maximize the demodulation slope which are fP = 19 kHz, MP = 1.62, fS1= FSR/2, and MS1 = 1.527. For comparison, the demodulation output of the traditional single demodulation scheme is also shown in Fig. 3. It can be seen that the two curves completely overlap each other near the resonance point where f0 is a multiple of FSR, which indicates that the second demodulation output of the double demodulating system has the same slope rate as the traditional single demodulating scheme. Thus, the double demodulating scheme has no deterioration of the RFOG rotation-rate sensitivity. An additional slope can be found where f0 is near (1/2 + n) × FSR and n is an integer, which is caused by the resonance valley in Fig. 2(b). The zero point of which can serve as another lock-in frequency for the laser with no interference to the signal processing at the adjacent points.

 figure: Fig. 3.

Fig. 3. The calculated output of the second demodulator in the double demodulating scheme and the general demodulation output in the single demodulating method.

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2.3 Optimization of key modulation parameters

In the double demodulating scheme, the modulation parameters need to be carefully optimized to improve the rotation-rate sensitivity. Take the CCW beam as an example, as previously indicated that the modulation frequency of the second modulation should be set to a multiple and a half of the FSR. The other key parameters mainly include the primary modulation frequency fP, the primary modulation index MP, and the second modulation index MS1. First, we will investigate the effect of MS1 on the first and the second demodulation output. The phase component of each demodulation signal has been optimized to maximize the amplitude of the first demodulation curve, or the slope at the resonance point of the second demodulation curve.

The first demodulation output can be used as an equivalent resonance curve of the FRR, however, it is able to reconstitute the resonance curve without distortion only when MS1 is optimized. Figure 4(a) shows the simulation result of the normalized output of the first demodulator for several values of MS1. It can be seen that the equivalent resonance curve changes significantly with the modulation index. Figure 4(b) further investigates the relationship between the demodulation slope at the resonance point of the second demodulator and MS1. Five different groups of Mp and fp are calculated. The optimal values of MS1 are the same in all cases. Thus, it is verified that the primary modulation parameters have no effect on the optimal value of MS1. Therefore, the modulation index MS1 should be set to 1.527 as much as possible regardless of other modulation parameters.

 figure: Fig. 4.

Fig. 4. The calculated output of the first and second demodulation. (a) The normalized equivalent resonance curve for several valves of MS1. (b) Relationship between the normalized second demodulation slope rate and MS1.

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The second demodulation output is used to detect the resonance frequency shift due to rotation. To maximize the rotation-rate sensitivity, we calculated the normalized slope rate of the second demodulator at different values of fP and MP, shown in Fig. 5. MS1 is 1.527 and fS1 = FSR/2. It can be seen that when fP is 19 kHz and MP is 1.62, the slope rate reaches the maximum. When MP is larger than 1, the best value almost keeps the same, corresponding to different values of fP.

 figure: Fig. 5.

Fig. 5. Normalized demodulation slope at different values of MP and fP.

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2.4 Influence of the second harmonic distortion

In the reciprocal modulation and double demodulation RFOG shown in Fig. 1, five different modulation signals are used. Generally, any imperfections in the modulation signals can produce rotation-rate errors. Among them, imperfections in PM0 can be canceled thanks to the reciprocal modulation-demodulation process, and the effect of the two modulation signals at fN1 and fN2 without involving demodulation can also be addressed by choosing appropriate modulation frequencies and bandwidth of LPFs. Therefore, we only need to make effort to deal with the imperfections in the second modulation signals at fS1 and fS2. In order to suppress the effect of backscattering intensity, the second phase modulation on CW and CCW directions must be set at slightly different frequencies, which means that at least one modulation frequency could not be set accurately at (n + 1/2) × FSR, where n is an integer. In addition, an actual single-frequency sinusoidal wave has higher harmonic distortion which also yields bias errors. Here we mainly consider the effect of the second harmonic distortion. Then the light after the second phase modulation can be expressed as

$$E(t) = {E_0}{e^{j[{2\pi {f_0}t + {M_\textrm{P}}\sin 2\pi {f_\textrm{P}}t + {M_{\textrm{S1}}}\sin 2\pi {f_{\textrm{S1}}}t + {M_\textrm{h}}\sin 2\pi (2{f_{\textrm{S1}}})t} ]}},$$
where Mh is the equivalent modulation index of the second harmonic component. Using the Bessel function expansion, Eq. (11) becomes
$$E(t) = {E_0}\sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\gamma } ={-} \infty }^{ + \infty } {{J_\mathrm{\alpha }}({M_\textrm{P}}){J_\mathrm{\beta }}({M_{\textrm{S1}}}){J_\mathrm{\gamma }}({M_\textrm{h}}){e^{j2\pi ({f_0} + \alpha {f_\textrm{P}} + \beta {f_{\textrm{S1}}} + 2\gamma {f_{\textrm{S1}}})t}}} } } ,$$
where Jγ(Mh) is the first kind of Bessel function with the γth order, γ is an integer representing the γth order of second harmonic sideband. After the same process explained in Eqs. (4) –(10), the signal after double demodulation is given by
$$\begin{array}{l} {V_{\textrm{demo\_h}}}(t) = \frac{1}{4}\sum\nolimits_{\mathrm{\alpha } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\beta } ={-} \infty }^{ + \infty } {\sum\nolimits_{\mathrm{\gamma } ={-} \infty }^{ + \infty } {\sum\nolimits_{\textrm{q} ={-} \infty }^{ + \infty } {{J_\mathrm{\alpha }}({M_\textrm{P}}){J_\mathrm{\beta }}({M_{\textrm{S1}}}){J_\mathrm{\gamma }}({M_\textrm{h}})} } \cdot } } \\ \{{{J_{\mathrm{\alpha +\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2q}}}({M_{\textrm{S1}}}){J_{\mathrm{\gamma -\ q\ +\ 1}}}({M_\textrm{h}}){h_{\mathrm{\alpha ,\beta ,\gamma }}}{h_{\mathrm{\alpha +\ 1,\beta +\ 2q,\gamma -\ q\ +\ 1}}}\cos ({\varphi_{\mathrm{\alpha ,\beta ,\gamma }}} - {\varphi_{\mathrm{\alpha +\ 1,\beta +\ 2q,\gamma -\ q\ +\ 1}}} + {\varphi_\textrm{P}} + {\varphi_{\textrm{S1}}}) + } \\ {J_{\mathrm{\alpha -\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2q}}}({M_{\textrm{S1}}}){J_{\mathrm{\gamma -\ q\ +\ 1}}}({M_\textrm{h}}){h_{\mathrm{\alpha ,\beta ,\gamma }}}{h_{\mathrm{\alpha -\ 1,\beta +\ 2q,\gamma -\ q\ +\ 1}}}\cos ({\varphi _{\mathrm{\alpha ,\beta ,\gamma }}} - {\varphi _{\mathrm{\alpha -\ 1,\beta +\ 2q,\gamma -\ q\ +\ 1}}} + {\varphi _\textrm{P}} - {\varphi _{\textrm{S1}}}) + \\ {J_{\mathrm{\alpha +\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2q}}}({M_{\textrm{S1}}}){J_{\gamma \textrm{ - q - 1}}}({M_\textrm{h}}){h_{\mathrm{\alpha ,\beta ,\gamma }}}{h_{\mathrm{\alpha +\ 1,\beta +\ 2q,\gamma -\ q\ -\ 1}}}\cos ({\varphi _{\mathrm{\alpha ,\beta ,\gamma }}} - {\varphi _{\mathrm{\alpha +\ 1,\beta +\ 2x,\gamma -\ q\ -\ 1}}} - {\varphi _\textrm{P}} + {\varphi _{\textrm{S1}}}) + \\ {{J_{\mathrm{\alpha -\ 1}}}({M_\textrm{P}}){J_{\mathrm{\beta +\ 2q}}}({M_{\textrm{S1}}}){J_{\mathrm{\gamma -\ q\ -\ 1}}}({M_\textrm{h}}){h_{\mathrm{\alpha ,\beta ,\gamma }}}{h_{\mathrm{\alpha -\ 1,\beta +\ 2q,\gamma -\ q\ -\ 1}}}\cos ({\varphi_{\mathrm{\alpha ,\beta ,\gamma }}} - {\varphi_{\mathrm{\alpha -\ 1,\beta +\ 2q,\gamma -\ q\ -\ 1}}} - {\varphi_\textrm{P}} - {\varphi_{\textrm{S1}}})} \}, \end{array}$$
where q is an integer used to fully represent all the frequency components that can be demodulated. Using Eq. (13), we can calculate the bias error due to the modulation frequency deviation from the optimal value, or (n + 1/2) × FSR.

Figure 6 shows the simulation result of the second demodulator output when the laser frequency is linearly swept. Mh is set to 0.01527 corresponding to a second harmonic distortion of -40 dB, MP and MS1 are 1.62 and 1.527, respectively. In the simulation, five different values of fS1 are calculated. The indicated laser frequency where the demodulation output is zero increases linearly while the modulation frequency deviation increases. As Fig. 6(b) indicates that the frequency-offset is about 21 Hz when fS1 is deviated by 600 Hz, and the offset is proportional to the frequency deviation of fS1 from the optimal value of 0.5 × FSR, which finally results in a bias error at the gyro output. The measured gyro output at different values of fS1 is shown in Fig. 7. As can be seen that the bias error increases from 20.73°/h to 177.52°/h as the modulation frequency deviation increases from 600 Hz to 2.4 kHz, and is changed by about 50°/h when fS1 is increased by 600 Hz. In practice, since both Mh and fS1 are quite stable during RFOG operation, the bias error is also stable which can be compensated based on the theoretical analysis above.

 figure: Fig. 6.

Fig. 6. Demodulation curves of the second demodulator under different fS1. (a) The demodulation curves. (b) Amplification near zero point.

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 figure: Fig. 7.

Fig. 7. Measured gyro outputs at different fS1.

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3. Experiment and discussion

3.1 Errors due to the intensity of the backscattered light

Figure 8 shows the schematic diagram of the RFOG using the reciprocal modulation-demodulation technique [9]. Single demodulating the transmission output of the FRR after a low-pass filter is used as the feedback loop error signal, or the gyro output signal. fP is the frequency of common modulation, the reciprocal configuration can eliminate the influence of the RAM. fN1 and fN2 are the frequencies of two non-demodulated signals, and the non-demodulated modulation is used for suppressing the interference term of backscattering noise. However, the effect of backscattering intensity can still affect the bias stability of the gyro in this scheme, especially in an environment with unstable temperatures. By disconnecting the input light in CCW direction at point A in Fig. 8, we can observe the effect of backscattering intensity from the CW direction at the gyro output. Figure 9 shows the tested gyro output in the single demodulation system using an FRR with diameter, length and finesse of 16.7 cm, 120 m and 23.8. The peak-to-peak fluctuation is measured to be approximately 10 °/h. For comparison, the output measured from the double demodulation system based on the same FRR is also shown. There is no obvious drift observed. It is indicated that the influence of the backscattering intensity has been well suppressed by using the double demodulation technique.

 figure: Fig. 8.

Fig. 8. Schematic diagram of the reciprocal modulation-demodulation RFOG.

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 figure: Fig. 9.

Fig. 9. Test of the intensity term of the backscattering induced rotation-rate error.

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3.2 Improvement of long-term bias stability

The RFOG using the reciprocal modulation and double demodulation technique depicted in Fig. 1 is setup and tested. The semiconductor laser works at 1550 nm with a linewidth of 3 kHz. The sensing element is an FRR with twin 90° polarization-axis rotated splices. The length, diameter and finesse of the resonator are 120 m, 16.7 cm and 23.8, respectively. The primary modulation frequency fP is 19 kHz, the second modulation frequencies fS1 and fS2 are 860 kHz and 860.15 kHz, respectively. The non-demodulated modulation frequencies fN1 and fN2 are 1.5 kHz and 940 Hz, respectively. Figure 10(a) shows the gyro output for a test time of 45 hours with an integration time of 10 s and the corresponding Allan deviation analysis is shown in Fig. 10(b). The bias stability of the RFOG is better than 0.2 °/h at an integration time of about 10 hours.

 figure: Fig. 10.

Fig. 10. Test result of the RFOG using the reciprocal modulation and double demodulation technique. (a) Gyro output for a test time of 45 hours. (b) Allan deviation analysis of the gyro data collected in 45 hours.

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The experiment result shows the RFOG based on the reciprocal modulation and double demodulation technique has a good long-term bias stability thanks to the well suppressed backscattering noise. It is very difficult for the reciprocal modulation-demodulation system [9] due to the unsuppressed intensity term of the backscattering as shown by the red curve in Fig. 9. The measured bias stability of 0.06°/h for half an hour shows the reciprocal modulation-demodulation scheme has lower random noise because of reciprocity [9], however, the reciprocity will be destroyed for a long-term test. The long-term bias stability of the RFOG is determined by both random noise and long-term variations, therefore, the single demodulation technique is not suitable for practical RFOG applications.

Unfortunately, the measured ARW of the RFOG with the proposed novel technique is worse than the theoretical photon shot noise. To suppress the backscattering induced error, the second modulation frequencies applied on the CW and CCW waves are slightly different. Thus, the laser frequency noise is a nonreciprocal effect which significantly deteriorates the ARW of the RFOG [16], resulting in the measured bias stability being no better than 0.02°/h demonstrated by the three lasers-driven RFOG [10]. However, the use of multi-frequency light source inevitably leads to an increase in system complexity and cost. Especially, for narrow-linewidth lasers working at about 1550 nm, the central wavelength difference between each laser should be less than about 0.9 pm in order to make sure that the frequencies of the beat signals generated by the master laser and each of the slave lasers are within the bandwidth of OPLLs. The simplicity of the optical system in this paper is beneficial for the integration and miniaturization of the RFOG. To further improve the gyro performance, we can reduce the effect of laser frequency noise by using a laser source with lower white noise, or by using the optical filtering technique [23].

4. Conclusions

In this paper, a reciprocal modulation and double demodulation technique has been proposed for an RFOG based on a single laser source. The mechanism of system operation and noise reduction are presented. Both the theoretical and experimental results show that the backscattering noise and the RAM of phase modulators are all well suppressed. The influence of modulation parameters on gyro performance is further investigated and optimized. Experimentally, a long-term bias stability of 0.2°/h is successfully obtained over a 45-hour gyro performance test. Only one single laser is used in this configuration, which ensures the simplicity of the system and has great value in the integration and miniaturization of the RFOG.

Funding

National Natural Science Foundation of China (62175218, 61675181); State Key Laboratory of Advanced Optical Communication Systems and Networks (2022GZKF008).

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.

References

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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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Figures (10)
Fig. 1.
Fig. 1. Scheme diagram of the reciprocal modulation and double demodulation RFOG.

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Fig. 2.
Fig. 2. Simulation results of the resonance curve and the output of the first demodulator. (a) Resonance curve. (b) Output of the first demodulator.

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Fig. 3.
Fig. 3. The calculated output of the second demodulator in the double demodulating scheme and the general demodulation output in the single demodulating method.

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Fig. 4.
Fig. 4. The calculated output of the first and second demodulation. (a) The normalized equivalent resonance curve for several valves of MS1. (b) Relationship between the normalized second demodulation slope rate and MS1.

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Fig. 5.
Fig. 5. Normalized demodulation slope at different values of MP and fP.

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Fig. 6.
Fig. 6. Demodulation curves of the second demodulator under different fS1. (a) The demodulation curves. (b) Amplification near zero point.

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Fig. 7.
Fig. 7. Measured gyro outputs at different fS1.

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Fig. 8.
Fig. 8. Schematic diagram of the reciprocal modulation-demodulation RFOG.

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Fig. 9.
Fig. 9. Test of the intensity term of the backscattering induced rotation-rate error.

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Fig. 10.
Fig. 10. Test result of the RFOG using the reciprocal modulation and double demodulation technique. (a) Gyro output for a test time of 45 hours. (b) Allan deviation analysis of the gyro data collected in 45 hours.

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Equations (13)

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E ( t ) = E 0 e j 2 π f 0 t ,
E ( t ) = E 0 e j ( 2 π f 0 t + M P sin ⁡ 2 π f P t + M S1 sin ⁡ 2 π f S1 t ) ,
E ( t ) = E 0 ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ J α ( M P ) J β ( M S1 ) e j 2 π ( f 0 + α f P + β f S1 ) t ,
T ( f c ) = h ( f c ) e φ ( f c ) ,
h α , β = h ( f 0 + α f P + β f S1 ) ,
φ α , β = φ ( f 0 + α f P + β f S1 ) .
V PD1 ( t ) = ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ ∑ α ′ = − ∞ + ∞ ∑ β ′ = − ∞ + ∞ c ε 0 K PD1 E 0 2 J α ( M P ) J β ( M S1 ) J α ′ ( M P ) J β ′ ( M S1 ) ⋅ h α , β h α ′ , β ′ cos ⁡ [ 2 π ( α f P + β f S1 − α ′ f P − β ′ f S1 ) t + φ α , β − φ α ′ , β ′ ] ,
V S1 ( t ) = cos ⁡ [ 2 π ( 2 f S1 ) t  +  φ S1 ] ,
V demo\_S1 ( t ) = 1 2 ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ ∑ α ′ = − ∞ + ∞ J α ( M P ) J β ( M S1 ) J α ′ ( M P ) ⋅ { J β + 2 ( M S1 ) h α , β h α ′ , β +   2 cos ⁡ [ 2 π ( α f P − α ′ f P ) t + φ α , β − φ α ′ , β +   2 + φ S1 ] + J β −   2 ( M S1 ) h α , β h α ′ , β −   2 cos ⁡ [ 2 π ( α f P − α ′ f P ) t + φ α , β − φ α ′ , β −   2 − φ S1 ] } .
V demo\_P ( t ) = 1 4 ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ J α ( M P ) J β ( M S1 ) ⋅ { J α +   1 ( M P ) J β +   2 ( M S1 ) h α , β h α +   1 , β +   2 cos ⁡ ( φ α , β − φ α +   1 , β +   2 + φ P + φ S1 ) + J α +   1 ( M P ) J β −   2 ( M S1 ) h α , β h α +   1 , β −   2 cos ⁡ ( φ α , β − φ α +   1 , β −   2 + φ P − φ S1 )  +  J α −   1 ( M P ) J β +   2 ( M S1 ) h α , β h α −   1 , β +   2 cos ⁡ ( φ α , β − φ α −   1 , β +   2 − φ P + φ S1 )  +  J α −   1 ( M P ) J β −   2 ( M S1 ) h α , β h α −   1 , β −   2 cos ⁡ ( φ α , β − φ α −   1 , β −   2 − φ P − φ S1 ) } ,
E ( t ) = E 0 e j [ 2 π f 0 t + M P sin ⁡ 2 π f P t + M S1 sin ⁡ 2 π f S1 t + M h sin ⁡ 2 π ( 2 f S1 ) t ] ,
E ( t ) = E 0 ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ ∑ γ = − ∞ + ∞ J α ( M P ) J β ( M S1 ) J γ ( M h ) e j 2 π ( f 0 + α f P + β f S1 + 2 γ f S1 ) t ,
V demo\_h ( t ) = 1 4 ∑ α = − ∞ + ∞ ∑ β = − ∞ + ∞ ∑ γ = − ∞ + ∞ ∑ q = − ∞ + ∞ J α ( M P ) J β ( M S1 ) J γ ( M h ) ⋅ { J α +   1 ( M P ) J β +   2 q ( M S1 ) J γ −   q   +   1 ( M h ) h α , β , γ h α +   1 , β +   2 q , γ −   q   +   1 cos ⁡ ( φ α , β , γ − φ α +   1 , β +   2 q , γ −   q   +   1 + φ P + φ S1 ) + J α −   1 ( M P ) J β +   2 q ( M S1 ) J γ −   q   +   1 ( M h ) h α , β , γ h α −   1 , β +   2 q , γ −   q   +   1 cos ⁡ ( φ α , β , γ − φ α −   1 , β +   2 q , γ −   q   +   1 + φ P − φ S1 ) + J α +   1 ( M P ) J β +   2 q ( M S1 ) J γ  - q - 1 ( M h ) h α , β , γ h α +   1 , β +   2 q , γ −   q   −   1 cos ⁡ ( φ α , β , γ − φ α +   1 , β +   2 x , γ −   q   −   1 − φ P + φ S1 ) + J α −   1 ( M P ) J β +   2 q ( M S1 ) J γ −   q   −   1 ( M h ) h α , β , γ h α −   1 , β +   2 q , γ −   q   −   1 cos ⁡ ( φ α , β , γ − φ α −   1 , β +   2 q , γ −   q   −   1 − φ P − φ S1 ) } ,
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