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Resonant fiber optic gyroscope with three-frequency differential detection by sideband locking

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Abstract

A new scheme of three-frequency differential detection with a sideband locking technique is firstly proposed to suppress backscattering noise for improving the accuracy of resonator fiber optic gyroscope (RFOG). In the system we proposed, one light path is divided into three paths and sinusoidal wave modulations of different frequencies are respectively applied to generate the sideband. The first-order sidebands of the three channels of light in the cavity are locked to the adjacent three resonance peaks by sideband locking technique. The carrier and the remaining sidebands of the three channels of light are moved to a position away from the resonance peak, thereby achieving the purpose of being suppressed by the cavity itself. As a result, the frequency difference between the CW light and the other two CCW lights reaches one free spectral range (FSR), eliminating the expected backscattering noise. The experimental results demonstrate that the RFOG has a bias stability 0.9°/h based on the Allan deviation, and the corresponding angular random walk (ARW) 0.016°/√h, which validate that our scheme can effectively suppress backscattering noise to promote performance of RFOG in practical applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

RFOG is a new type of optical inertial sensor based on the resonance frequency difference generated by Sagnac effect to test the angular velocity of rotation [13]. Compared with ring laser gyro (RLG) and interferometric fiber optic gyroscope (IFOG), the advantage of RFOG is the recirculation of light around the fiber optic path. So RFOG can achieve the same signal-to-noise ratio performance [4,5] and the high-precision inertial angular rate measurement with less fiber, which reduces the cost of gyroscopes. As a result, RFOG has unique advantages of high precision and miniaturization. Therefore it is one of the promising development tendencies of optical gyros [6], which has attracted the attention of many scholars in the field of inertial navigation [710].

The coupling between the backscattering light and the main light in the cavity of the RFOG causes backscattering noise, which is an important factor to limit the accuracy of the gyro [11]. The most common mitigation method has been the employment of a carrier-suppressed phase or frequency modulation on the resonator input light to reduce the interference between the primary signal light and backscattering light from the light propagating in the opposite direction [12,13]. The backscattering noise can be effectively suppressed by phase difference traversal (PDT) technology [14,15] and hybrid phase-modulation technology (HPMT) [16], and is theoretically and experimentally analyzed [17] for the first time. The carrier-suppression phase modulation (CSPM) technology can reduce the backscattering noise and ultra-high carrier suppression is crucial [18,19]. It can be seen from [1219] that there are optimum modulation indexes for PDT, HPMT and CSPM technologies to realize optimum suppression. However, for most gyro systems, many modulation schemes such as sinusoidal wave modulation [20], triangular wave modulation [21], serrodyne wave modulation [22], trapezoidal wave modulation [23] and their combinations [24,25] have been proposed. Although the carrier suppression method can reduce the backscattering noise, phase modulator is obviously sensitive to temperature, which requires high accuracy of control voltage. In practical applications, none of all the schemes given above can acquire the modulation index effectively, and it is difficult to totally suppress the magnitude of carrier. In view of the problem above, the backscattering light will still interfere with the main light propagating in the cavity of the RFOG. Thus the backscattering noise is introduced, which is a key problem urgently to be solved.

An attractive countermeasure for addressing the rotational sensing error from interference of backscattering light is to use the three-frequency differential detection based on sideband locking technique to suppress the backscattering noise. In this method, one light path is divided into three paths, and sinusoidal wave modulations of different frequencies are respectively applied to generate the sideband. The first-order sidebands of the three channels of light in the cavity are locked to the adjacent three resonance peaks by sideband locking technique. The carrier and the remaining sidebands of the three channels of light are moved to a position away from the resonance peak, thereby being suppressed by the cavity itself. At the same time, the frequency difference between the CW light and the other two CCW lights is equal to one free spectral range (FSR). In this way, if the either light is backscattered into the other direction, the erroneous signals caused by backscatter interference will occur at a frequency difference of clockwise (CW) and counterclockwise (CCW). The frequency difference is many orders of magnitude (e.g., 10 MHz for a resonator of 20.5 m length) above the frequency band of the rotation measurement (typically, DC to ≈100 Hz). Therefore, it is easy to filter out errors caused by backscatter interference from the gyro signal, which can eliminate the expected backscattering noise and furtherly improve the performance of RFOG in practical applications.

2. Problem description

Backscattering noise is one of the main noises of a RFOG [2628]. In the fiber cavity, if we set the light field of the CW and CCW main lights as ECW and ECCW, the frequency of the CW and CCW lights as ωCW and ωCCW, the transfer function of the resonant cavities from the CW and CCW lights as FCWCW) and FCCWCCW), and the backscattering coefficient as Rb, then the field of the CW total light will be a superposition of the main light and backscattering light:

$${E_{\textrm{CWT}}}{\kern 1pt} \textrm{ = }{E_{\textrm{CW}}}{F_{\textrm{CW}}}\textrm{(}{\omega _{CW}}\textrm{) + }\sqrt {{R_b}} {E_{\textrm{CCW}}}{F_{\textrm{CCW}}}\textrm{(}{\omega _{CCW}}\textrm{)}$$

This is a simplified model. The light of high order backscattering from CW also exists. Considering the small backscattering coefficient, the high order term is ignored. From the Eq. (1), the light intensity ICW in the CW direction of the cavity can be obtained:

$$\begin{aligned} {I_{\textrm{CW}}}{\kern 1pt} &= {E_{\textrm{CWT}}}{E^ \ast }_{\textrm{CWT}}\textrm{ = }{|{{E_{\textrm{CW}}}} |^2}{H_{sg}}\textrm{ + }{R_b}{|{{E_{\textrm{CCW}}}} |^2}{H_{bc}}\textrm{ + }\\ & \textrm{ }\sqrt {{R_b}} |{{E_{\textrm{CW}}}{E_{\textrm{CCW}}}} |{H_{ch}} = {I_{\textrm{CW0}}}\textrm{ + }{N_1}\textrm{ + }{N_2} \end{aligned}$$
where Hsg, Hbc, and Hch are the conjugate products of transfer function. ICW0 in the Eq. (2) indicates that the gyro effective signal is used for angular velocity detection, and the rest shows the backscattering noise which should be eliminated. The backscattering noise is mainly divided into two parts. N1 represents the influence of the backscattering light itself on the gyro signal. It can be suppressed by applying different modulation frequencies to the CW and CCW lights, and the corresponding demodulation and filtering; N2 means the interference of backscattering light and the main propagating light. Currently carrier suppression is often applied to the optical path to reduce the effects of noise [19,23,29].

In order to suppress the backscattering noise, the phase modulator is generally spliced at the input end of the CW and CCW optical path, and the CW and CCW lights is modulated by applying different frequency by the phase modulator. At the gyro output end, the corresponding frequency demodulation is used for backscattering noise separation. As is shown in Fig. 1, we set the applied phase modulation frequencies are ΩCW and ΩCCW. Respectively, the phase modulation coefficients are β and β´, then the CW optical field and backscattered optical field generated by CCW optical path in the fiber cavity can be expanded by the Bessel function to

$${E_{\textrm{CW}\_ \textrm{M}}} = {\kern 1pt} {E_0}{e^{i({\omega _{CW}}t + \beta \sin \,\,{\Omega _{CW}}t)}} = {E_0}\sum\limits_{n ={-} \infty }^{ + \infty } {{J_n}(\beta )} {e^{i({\omega _{CW}}t + n\,{\Omega _{CW}}t)}}$$
$$\sqrt {{R_b}} {E_{\textrm{CCW}\_ \textrm{M}}} = \sqrt {{R_b}} {E_0}\sum\limits_{n^{\prime} ={-} \infty }^{ + \infty } {{J_{n^{\prime}}}({\beta^{\prime}} )} {e^{i({\omega _{CCW}}t + n^{\prime}{\Omega _{CCW}}t)}}$$
where Jn is the n-order Bessel coefficient. Substituting Eqs. (3) and (4) into Eq. (2) yields CW direction light intensity ICW_M with sinusoidal modulation:
$$I_{\textrm{CW}\_ \textrm{M}} = I_{\textrm{CW0}\_ \textrm{M}} + N_{1\_ \textrm{M}} + {N_{2\_ \textrm{M}}}$$

 figure: Fig. 1.

Fig. 1. Illustration of RFOG system and the spectra of CW and CCW.

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The expression of each component in Eq. (5) is:

$$I_{\textrm{CW0}\_ \textrm{M}} = {E_0}^2\sum\limits_{n ={-} \infty }^{ + \infty } {{H_{sg\_0}}{{|{{J_n}} |}^2}} \textrm{ + 2}{E_0}^2\sum\limits_{k = 1}^{ + \infty } {\sum\limits_{n ={-} \infty }^{ + \infty } {{H_{sg\_k}}|{{J_{n - k}}{J_n}} |\bullet \cos ({k{\Omega_{CW}}t} )} }$$
$${N_{1\_M}} = {R_b}{E_0}^2\sum\limits_{n = - \infty }^{ + \infty } {{H_{bc\_0}}{{\left| {{J_n}} \right|}^2}} \textrm{ + 2}{R_b}{E_0}^2\sum\limits_{\textrm{k} = 1}^{ + \infty } {\sum\limits_{n = - \infty }^{ + \infty } {{H_{bc\_k}}\left| {{J_{n - k}}{J_n}} \right| \bullet \cos \left( {k{{\Omega }_{CCW}}t} \right)} }$$
$$\begin{array}{l} {N_{2\_M}} = 2\sqrt {{R_b}} {E_0}^2\sum\limits_{n = - \infty }^{ + \infty } {\sum\limits_{n' = - \infty }^{ + \infty } {{H_{ch\_nn'}}{J_n}\left( \beta \right){J_{n'}}\left( {\beta '} \right)} } \bullet \\ \cos \left[ {\left( {n{{\Omega }_{CW}} - n'{{\Omega }_{CCW}}} \right)t + \left( {{\omega _{CW}} - {\omega _{CCW}}} \right)t} \right] \end{array}$$

The modulated CW optical signal ICW_M is demodulated by the frequency of ΩCW and then passed through low-pass filtering, so Eq. (5) leaves only one DC quantity term. Since the Sagnac frequency difference is usually a low-frequency signal and is usually less than the filtering bandwidth, the gyro effective signal in Eq. (6) and backscattering noise in Eqs. (7) and (8) can be corrected to

$${I_{\textrm{CW0}\_ \textrm{M}}} = {E_0}^2\sum\limits_{n = - \infty }^{ + \infty } {{H_{sg\_1}}\left| {{J_{n - 1}}{J_n}} \right|}$$
$${N_{1\_{\textrm{M}}}} = 0$$
$${N_{2\_{\textrm{M}}}} = \sqrt {{R_b}} {E_0}^2{H_{ch\_10}}{J_1}\left( \beta \right){J_0}\left( {\beta '} \right)\cos \left[ {{{\Omega }_{CW}}t + \left( {{\omega _{CW}} - {\omega _{CCW}}} \right)t} \right]\cos {{\Omega }_{CW}}t$$

It can be seen from the low-pass filtered gyro signal in Eq. (9) and the backscattering noise in Eqs. (10) and (11) that the backscattering light represented by N1_M itself affects the gyro signal. As is shown in Fig. 1, by applying different modulation of the frequency ΩCW and ΩCCW to the CW and CCW lights, the frequency difference is ΔΩ=|ΩCW-ΩCCW|. The backscattering noise can be suppressed when the frequency difference ΔΩ is larger than the filtering bandwidth. However, the interference noise N2_M of the backscattering light and the signal light cannot be completely eliminated. The magnitude of the interference noise N2_M in Eq. (11) is related to the product of the Bessel function. So, to bring the Bessel function to zero, the carrier suppression method suppresses N2_M by selecting the appropriate β. Although the carrier suppression method can reduce the backscattering noise, it is obviously affected by temperature and requires high control voltage accuracy. In practical applications, it is difficult to totally suppress the magnitude of carrier.

3. Principle of the scheme

The Eq. (11) of the backscattering noise N2_M indicates the magnitude and frequency characteristics of the noise. Not only N2_M is related to the phase modulation coefficient β, but also its frequency is equal to the frequency difference of the CW and CCW lights. If the frequency spacing between the CW and CCW lights can be increased under the premise of obtaining the Sagnac frequency difference, the backscattering error can be eliminated by low-pass filtering. It provides a new idea on backscattering noise suppression. This paper proposed a scheme of three-frequency differential detection based on sideband locking in order to suppress backscattering noise of gyro.

The structure principle is shown in Fig. 2. A tunable narrow-linewidth fiber laser is used as an optical carrier signal. When the three frequency differential gyro is working, the laser output light is divided into three lights. The first light is a CW light, and two fixed modulation frequencies (Ω1H, Ω1L) are applied by the phase modulator PM1. Their relationship is: Ω1H>>Ω1L. Among them, the high modulation frequency Ω1H is used to generate the sideband; the low modulation frequency Ω1L is used to generate the Pound-Drever-Hall (PDH) error signals at Ω1L. After the first light dropped out from the cavity, the optical signals were converted into an analog signals by the PD1, and then the analog signals were converted into a digital signals by the ADC, and then the digital signals were demodulated to be PDH error signals at Ω1L by the demodulation module. The sideband generated by Ω1H is locked to the resonance peak by adjusting the laser PDH RTL. The second light is a CCW light, and two modulation frequencies (Ω2H2D, Ω2L) are applied by the phase modulator PM2. Their relationship is: Ω2H2D >>Ω2L. Among them, the high modulation frequency Ω2H2D is an adjustable frequency (Ω2H is a fixed bias frequency and Ω2D is an adjustable frequency) for generating sidebands; the low modulation frequency Ω2L is a fixed frequency for generating PDH error signals at Ω2L. When the DDS is adjusted according to the SBL PDH RTL, it produces an adjustable frequency of Ω2D, which in turn locks the sidebands produced by the adjustable frequency Ω2H2D to adjacent resonant peaks. The third light is a CCW light, and two modulation frequencies (Ω3H3D, Ω3L) are applied by the phase modulator PM3. Their relationship is: Ω3H3D >>Ω3L. Among them, the high modulation frequency Ω3H3D is an adjustable frequency (Ω3H is a fixed bias frequency and Ω3D is an adjustable frequency) for generating sidebands; the low modulation frequency Ω3L is a fixed frequency for generating PDH error signals at Ω3L. When the DDS is adjusted according to the SBL PDH RTL, it produces an adjustable frequency of Ω3D, which in turn locks the sidebands produced by the adjustable frequency Ω3H3D to adjacent resonant peaks. The second light and the third light are simultaneously injected into and dropped out from the cavity. After the optical signals were converted into an analog signals by the PD2, the analog signals were converted into a digital signals by the ADC, and then the digital signals were demodulated to be PDH error signals at Ω2L and Ω3L by the demodulation module.

 figure: Fig. 2.

Fig. 2. Schematic diagram of three-frequency differential gyro. (laser, tunable narrow-linewidth fiber laser; PD, photodetector; PM, phase modulator; C1, C2, couplers; Demod, demodulation module; SBL, sideband locking; RTL, resonance-tracking loop; DDS, direct digital synthesis; err. @ Ω1L, Ω2L, Ω3L, PDH error signals at Ω1L, Ω2L, Ω3L; ADC, analog-to-digital converter; DAC, digital-to-analog converter; FPGA, field-programmable gate array.)

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3.1 Scheme of sideband locking

The scheme of sideband locking is same as standard PDH locking except that the error signal used for locking is the sideband’s error signal rather than the carrier's error signal. This method requires modification of the standard PDH technique to achieve adjustable carrier frequency. In this technique, two modulation frequencies are applied to the phase modulator. Just like the standard PDH locking, one of the modulation frequency is a fixed frequency which is just like the PDH frequency locking, and the other one is an adjustable frequency to achieve a tunable sideband produced by the phase modulator. This tunable sideband is just like a carrier. As is shown in Fig. 3(b), this research used CW light to study. Compared with Fig. 3(a), two modulation frequencies are applied to the CW light by the phase modulator PM1. Their relationship is: Ω1H>>Ω1L. Among them, the high modulation frequency Ω1H is used to generate the sideband; the low modulation frequency Ω1L is used to generate the PDH error signals, and the sideband ω + Ω1H generated by Ω1H is locked to the resonance peak by adjusting the laser. The resonant cavity can be taken as a band-pass filter. The carrier frequency component is moved to the stop-band away from the resonance peak, thereby achieving the suppression of the carrier by the resonance peak.

 figure: Fig. 3.

Fig. 3. Modulation structures for traditional and tunable modulation/demodulation locking. (a) Pound Drever Hall (PDH); (b) Sideband locking (SBL). (For SBL only the upper half of the modulation structure is shown. The solid curve represents |F(ω)|2 and the dashed curve represents ∠F(ω), where F(ω) is the amplitude transmission coefficient of the cavity. For the frequency-tunable cases, the arrow labeled tune indicates the frequency spacing that is adjusted to tune the carrier, denoted by a thick line.)

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In the three-frequency differential detection scheme, two sinusoidal modulation frequencies are applied to each light. The research use CW light to study. The incident light E0 is modulated by the phase modulator PM1 to obtain an incident electric field:

$${E_{\textrm{in}}} = {E_0}{e^{i(\omega t + {\beta _1}\sin \,\,{\Omega _{1H}}t\textrm{ + }{\beta _1}^{\prime} \sin \,\,{\Omega _{1L}}t)}}$$

Equation (12) is expanded with the Bessel function as:

$$\begin{aligned} {E_{\textrm{in}}}{\kern 1pt} \approx &{E_0} \times \{ {J_0}({\beta _1})[{{J_0}({\beta_1}^{\prime} ){e^{i\omega t}} + {J_1}({\beta_1}^{\prime} ){e^{i(\omega + {\Omega _{1L}})t}} - {J_1}({\beta_1}^{\prime} ){e^{i(\omega - {\Omega _{1L}})t}}} ]\\ &\textrm{ + }{J_1}({\beta _1})[{{J_0}({\beta_1}^{\prime} ){e^{i(\omega + {\Omega _{1H}})t}} + {J_1}({\beta_1}^{\prime} ){e^{i(\omega + {\Omega _{1H}}\textrm{ + }{\Omega _{1L}})t}} - {J_1}({\beta_1}^{\prime} ){e^{i(\omega + {\Omega _{1H}} - {\Omega _{1L}})t}}} ]\\ & - {J_1}({\beta _1})[{{J_0}({\beta_1}^{\prime} ){e^{i(\omega - {\Omega _{1H}})t}} + {J_1}({\beta_1}^{\prime} ){e^{i(\omega - {\Omega _{1H}}\textrm{ + }{\Omega _{1L}})t}} - {J_1}({\beta_1}^{\prime} ){e^{i(\omega - {\Omega _{1H}} - {\Omega _{1L}})t}}} ]\} \end{aligned}$$

The modulation process is demonstrated in Eq. (13) above. As is shown in Fig. 3, the incident light is modulated by Ω1H to generate upper sideband ω + Ω1H and lower sideband ω - Ω1H, and then the carrier and the generated sideband are modulated by Ω1L to generate sub-bands ω ± Ω1L, ω + Ω1H ± Ω1L and ω - Ω1H ± Ω1L. The amplitude of the carrier is determined by J01) J01´). Therefore, after using dual frequency modulation, the carrier can be furtherly suppressed by selecting the appropriate modulation depths β1 and β1´.

PΩ1L is the final intensity expression corresponding to Ω1L, which is obtained by the conjugate multiplication of the incident electric field in Eq. (13). It represents the light intensity of the modulated incident light at various frequency components, and the final intensity expression for Ω1L is described as:

$$\begin{aligned} {P_{\Omega 1L}} = & 2{J_0}({\beta _1}^{\prime} ){J_1}({\beta _1}^{\prime} ){P_0} \times \{ {J_0}{({\beta _1})^2}\{ {\mathop{\rm Re}\nolimits} [T(\omega )]\cos {\Omega _{1L}}t\textrm{ + }{\mathop{\rm Im}\nolimits} [T(\omega )]\sin {\Omega _{1L}}t{\} }\\ &\textrm{ + }{J_1}{({\beta _1})^2}\{ {\mathop{\rm Re}\nolimits} [T(\omega \textrm{ + }{\varOmega _{1H}})]\cos {\Omega _{1L}}t\textrm{ + }{\mathop{\rm Im}\nolimits} [T(\omega \textrm{ + }{\varOmega _{1H}})]\sin {\Omega _{1L}}t{\} }\\ & \textrm{ + }{J_1}{({\beta _1})^2}\{ {\mathop{\rm Re}\nolimits} [T(\omega - {\varOmega _{1H}})]\cos {\Omega _{1L}}t\textrm{ + }{\mathop{\rm Im}\nolimits} [T(\omega - {\varOmega _{1H}})]\sin {\Omega _{1L}}t{\} \} } \end{aligned}$$
where T(ω)= F(ω) F*(ω+Ω1L)- F*(ω) F(ω-Ω1L), F(ω) is the transfer functions of the resonant cavity, and P0 is the incident optical power. By demodulating the cos(Ω1L) term, as is shown in Fig. 4, we can see a linear region near each frequency component and the carrier is effectively suppressed. In this study, only the information of the first-order sidebands is needed, and the remaining sidebands will not be described here.

3.2 Three-frequency differential principle based on sideband locking

As is shown in Fig. 5, when the gyro is stationary, the first-order sidebands of the three lights are respectively locked to the adjacent three resonance peaks, and the frequency difference between the CW light and the other two CCW lights is one FSR. The sidebands ω±Ω1H are generated by applying a high modulation frequency Ω1H to the phase modulator PM1, and the lower sideband ω1H is locked to the resonance by adjusting the laser. Taking advantage of the characteristics of the phase modulator’s continuously adjustable frequency, the frequency of Ω1H is changed to move the carrier to a position away from the resonance peak, thereby achieving the suppression of the carrier and upper sideband ω1H by the resonance peak. The sidebands ω±2H2D) are generated by applying a high modulation frequency Ω2H2D to the phase modulator PM2, and the frequency of Ω2D is generated by adjusting the DDS. And then the upper sideband ω+Ω2H2D is locked to the adjacent right resonance peak. The lower sideband ω-Ω2H2D is moved to the position away from the resonance peak, thereby achieving the suppression of the lower sideband ω-Ω2H2D by the resonance peak. The sidebands ω±3H3D) are generated by applying a high modulation frequency Ω3H3D to the phase modulator PM3, and the frequency of Ω3D is generated by adjusting the DDS. And then the lower sideband ω3H3D is locked to the adjacent left resonance peak. The upper sideband ω3H3D is moved to the position away from the resonance peak, thereby achieving the suppression of the upper sideband ω3H3D by the resonance peak.

 figure: Fig. 4.

Fig. 4. Demodulation of cos(Ω1L) term when β1=2.405, β1´=2.405, Ω1H = 1.3 MHz, Ω1L=100 KHz.

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 figure: Fig. 5.

Fig. 5. Illustration of carrier suppression by locking the sideband to the cavity resonance. (The sidebands for CW (blue) locking and CCW (red) locking. Illustrated are three resonances separated by one FSR of 10 MHz.)

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Figure 6 is the resonance curve of the fiber ring resonator when the gyro is rotated. When the gyro is stationary, the Eq. (15) is true.

$${f_{PM2}} + {f_{PM3}} = 2FSR$$

 figure: Fig. 6.

Fig. 6. Spectrum diagram of three-frequency lights under CW rotation. (Gyro resonance modes for CW (blue) and CCW (red) lights. Illustrated are three resonances separated by one FSR of 10 MHz, and a rotation induced frequency shift of the counter-propagating modes determines.)

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Since the Sagnac effect causes the frequency of the CW and CCW light to shift, once the gyro rotates CW, the relationship between the frequency intervals of the three lights in the fiber cavity is obtained from Fig. 6. As is shown in Eqs. (16) and (17):

$${f_{PM2}} + {f_{PM1}} = FSR\textrm{ + }\Delta f$$
$${f_{PM3}} - {f_{PM1}} = FSR - \Delta f$$
fPM1 is the frequency corresponding to the angular frequency Ω1H of the phase modulator PM1, fPM2 is the frequency corresponding to the angular frequency Ω2H2D of the phase modulator PM2, fPM3 is the frequency corresponding to the angular frequency Ω3H3D of the phase modulator PM3, and Δf is the Sagnac frequency difference generated by rotation. The two equations are subtracted.
$${f_{PM2}} + 2{f_{PM1}} - {f_{PM3}} = 2\Delta f$$

When environmental factors such as temperature are changed, the frequency of the laser will produce drift, and also fPM1, fPM2 and fPM3 will change to the same amplitude and direction. The Sagnac frequency difference is constant, so Δf will not be affected by the laser frequency drift. When the temperature is changed, the FSR of the resonant cavity will change. As can be seen from the Eq. (18), Δf is not affected by the change of FSR. The angular velocity expression can be obtained by combining the formula [6] of the resonant fiber optic gyro Sagnac frequency difference Δf:

$$\varOmega \textrm{ = }\frac{{({{f_{PM2}} + 2{f_{PM1}} - {f_{PM3}}} )n\lambda }}{{2D}}$$

It can be seen from the equation above that the gyro angular velocity is proportional to the Sagnac frequency difference. Carrier suppression is also applied to the three-frequency differential gyro. Even if the temperature or control voltage fluctuates, the backscattering noise increases. Since the three modulation frequencies are different and the N2_M frequency is approximately equal to 10 MHz, which is much higher than the modulation frequency, the backscattering noises N1_M and N2_M can be suppressed by corresponding demodulation and filtering.

4. Experiments and results

4.1 SPR of cavity based on sideband locking

The fiber laser used in the experiment has a wavelength of 1550 nm and a line width of 100 Hz. The research reduces polarization-fluctuation induced drift in RFOG by using single-polarization fiber. The single-polarization fiber cavity has a length of 20.5 m. The diameter of the fiber cavity is 0.1 m. The refractive index of the core is 1.45. The full width at half maximum (FWHM) is 378 KHz. The measured finesse is about 26.7. The LiNbO3 PM was used in the experiment, and its insertion loss was 2.6 dB. The half-wave voltage of PM1 is Vπ=3.36 V. The half-wave voltage of PM2 is Vπ=3.46 V. The PM3 half-wave voltage is Vπ=3.65 V. Figure 7 shows the tested cavity suppression ratio (SPR).

 figure: Fig. 7.

Fig. 7. Measurement of the cavity SPR.

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The transmissive single-polarization fiber ring resonator can be regarded as a band-pass filter. By changing the frequency of Ω1H to move the carrier away from the resonance peak, the intensity of the carrier is suppressed by the cavity itself, thereby achieving the suppression of the carrier by the resonance peak. When the modulation frequency Ω1H is 1.3 MHz, the SPR will reach 41 dB. The SPR increases with the improvement of the cavity’s finesse. According to the Bessel function, β1=2.405, J01) =0. In the actual experiment, the control precision of Vpp will cause 1% noise, and finally we will get the carrier with the intensity suppression ratio SPR = 20·lg {J0 (0)/ J01 (1 ± 1%)]} ≈38 dB. When the high modulation frequency Ω1H and the low modulation frequency Ω1L are simultaneously applied to the phase modulator PM1 when β11´=2.405, according to the formula (13), the SPR will reach 76 dB. With the suppression of the cavity when Ω1H>2π·3.44·FWHM, the total suppression ratio SPR > 76 + 41 = 117 dB.

4.2 Experiments of three-frequency differential detection based on sideband locking

A three-frequency differential RFOG is constructed according to the scheme shown in Fig. 2. In the experiment, the incident optical power of the fiber laser was set to 45 mW. The splitting ratio of the coupler used is 40:30:30.” When the gyro is working, the frequency difference between the CW light and the other two CCW lights in the cavity is about 1 FSR.

As is shown in Fig. 8, the three light paths are modulated. The first light is a CW light, which is applied with two sinusoidal modulation frequencies of Ω1H=2π·1.3 MHz and Ω1L=2π·100KHz by the phase modulator PM1. The J-11) sideband generated by Ω1H is locked to the resonance peak by adjusting the laser. The second light is a CCW light, and two sinusoidal modulation frequencies of Ω2H=2π·8.7 MHz and Ω2L=2π·100KHz are applied thereto by the phase modulator PM2. The frequency of Ω2D is generated by adjusting the DDS, thereby locking the J12) sideband generated by Ω2H2D to the adjacent resonance peak. The third light is a CCW light, and two sinusoidal modulation frequencies of Ω3H=2π·11.3 MHz and Ω3L=2π·170KHz are applied thereto by the phase modulator PM3. The frequency of Ω3D is generated by adjusting the DDS, thereby locking the J-13) sideband generated by Ω3H3D to the adjacent resonance peak. Carrier suppression is applied to the three-frequency differential detection. The modulation frequencies of three ways are different and the N2_M frequency is approximately equal to 10 MHz, which is much higher than the modulation frequency. Therefore, even if the temperature or control voltage fluctuations causes the increases of backscattering noise, the three-frequency differential RFOG still can suppress backscattering noises N1_M and N2_M by demodulation and low-pass filtering.

 figure: Fig. 8.

Fig. 8. PD signals of CW and CCW and their demodulations by low frequency. (the intensity output detected by PD1 for CW (green) light; the intensity output detected by PD2 for CCW (orange) light; the demodulation (blue) of the cosine term of Ω1L; the demodulation (red) of the cosine term of Ω2L; the demodulation (red) of the cosine term of Ω3L.)

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The error curve demodulated by the low modulation frequency has many linear regions whose directions are the same. So in order to judge the specified first-order sideband instead of the remaining sidebands locked on the resonance peak, necessary auxiliary lock flags should be adopted [30]. As is shown in Fig. 9, we take the rising part of the swept signal for driving laser piezoelectric ceramic (PZT) as an example. The CW light is modulated by a high modulation frequency Ω1H=2π·1.3 MHz and a low modulation frequency Ω1L=2π·100KHz. In the experiment, the Ω1H term and the 1L term are simultaneously demodulated as auxiliary lock information, and a specific demodulation curve can be obtained by adjusting the respective demodulation phases. The L1 line indicates that the first-order sideband above the critical value is at resonance, and the demodulation error curve is in the linear region. We will be able to exclude other sidebands from locking to the resonance peak. With the L1 auxiliary lock flag, both first-order sidebands may be locked. The L2 line provides additional locking information to ensure that the first-order lower sideband is locked to the resonant peak. The falling part of the swept signal for driving laser PZT works the same as the rising part and will not be described here.

 figure: Fig. 9.

Fig. 9. Scheme of locking the first-order sideband to cavity resonance. (a) the swept signal for driving laser PZT; (b) the intensity output detected by PD; (c) the demodulation of the cosine term of 2Ω1L;(d) the demodulation of the cosine term of Ω1H; (e) the demodulation of the cosine term of Ω1L.

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Furtherly, we conduct the measuring accuracy experiment of RFOG to verify the effectiveness of our three-frequency differential scheme based on sideband locking technology. As is shown in Fig. 10(a), a bias stability of 0.9°/h is obtained with an integration time of 10s over 100s test. Moreover, the ARW is 0.016°/√h. By contrast, we perform the Allan deviation analysis of double closed-loop control scheme. As is shown in Fig. 10(b), the bias stability is about 5.5°/h, and the ARW is 0.243°/√h. Therefore, compared with traditional double closed-loop control scheme, our three-frequency differential scheme based on the sideband locking technology obtains better performance in bias stability and ARW.

 figure: Fig. 10.

Fig. 10. Allan deviation of gyroscope output. (a) Allan deviation of three-frequency differential RFOG based on the sideband locking output; (b) Allan deviation of double closed-loop RFOG output.

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Finally, the experiment of measuring power spectral density (PSD) in RFOG with three-frequency differential scheme based on sideband locking technology (TFDSBL) and traditional double closed-loop control scheme (DCL) are conducted, which validates the effectiveness of our three-frequency differential scheme based on sideband locking technology.

As is shown in Fig. 11, in the lower frequency range of the PSDs, the PSDs of the two cases are in the same order. Since the two schemes adopt the same single-polarization fiber ring resonator, the long-term drift introduced by the polarization error is effectively eliminated, and the long-term stabilities of the gyro corresponding to the two schemes are in the same order. However, in the higher frequency range of the PSDs, the PSD of the TFDSBL is lower than the DCL by one to two orders, which means that the backscattering noise is effectively suppressed, and the ARW is also increased by one order.

 figure: Fig. 11.

Fig. 11. PSDs of the gyro outputs in case of the DCL and case of the TFDSBL.

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5. Conclusion

Backscattering noise is an important factor to restrict the accuracy of RFOG. In order to improve the accuracy of RFOG, this paper proposes a new scheme of three-frequency differential detection with sideband locking technique to suppress backscattering noise. Firstly, we establish a theoretical model of the backscattering light in the cavity, and derive the formation mechanism of backscattering noise. Secondly, in view of the frequency characteristics of backscattering noise, we establish the system of three-frequency differential detection based on sideband locking technique to suppress backscattering noise. Finally, the experimental results show that, the Allan deviation is about 0.9°/h, and the ARW is 0.016°/√h. The proposed scheme can effectively suppress the backscattering noise and promote the performance of RFOG in practical applications compared to the traditional double closed-loop control scheme. The result of our research is of great significance in improving the wide application of RFOG in inertial navigation.

Funding

National Natural Science Foundation of China (61973019).

Acknowledgments

The authors would like thank our colleagues for supporting our experiments, as well as the reviewers from OE for their thoughtful comments.

Disclosures

The authors declare no conflicts of interest.

References

1. F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014). [CrossRef]  

2. H. Jiao, L. Feng, N. Liu, and Z. Yang, “Improvement of long-term stability of hollow-core photonic-crystal fiber optic gyro based on single-polarization resonator,” Opt. Express 26(7), 8645–8655 (2018). [CrossRef]  

3. H. Ma, J. Zhang, L. Wang, and Z. Jin, “Development and Evaluation of Optical Passive Resonant Gyroscopes,” J. Lightwave Technol. 35(16), 3546–3554 (2017). [CrossRef]  

4. L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012). [CrossRef]  

5. J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015). [CrossRef]  

6. H. J. Arditty and H. C. Lefevre, “sagnac effect in fiber gyroscopes,” Opt. Lett. 6(8), 401–403 (1981). [CrossRef]  

7. L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015). [CrossRef]  

8. H. Ma, X. Lu, L. Yao, X. Yu, and Z. Jin, “Full investigation of the resonant frequency servo loop for resonator fiber-optic gyro,” Appl. Opt. 51(21), 5178–5185 (2012). [CrossRef]  

9. L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993). [CrossRef]  

10. X. Yu, H. Ma, and Z. Jin, “Improving thermal stability of a resonator fiber optic gyro employing a polarizing resonator,” Opt. Express 21(1), 358–369 (2013). [CrossRef]  

11. N. Nakazawa, “Rayleigh backscattering theory for single mode optical fibers,” J. Opt. Soc. Am. 73(9), 1175–1180 (1983). [CrossRef]  

12. G. A. Sanders, M. G. Prentiss, and S. Ezekiel, “Passive ring resonator method for sensitive inertial rotation measurements in geophysics and relativity,” Opt. Lett. 6(11), 569–571 (1981). [CrossRef]  

13. Y. Zhi, L. Feng, J. Wang, and Y. Tang, “Reduction of backscattering noise in a resonator integrated optic gyro by double triangular phase modulation,” Appl. Opt. 54(1), 114 (2015). [CrossRef]  

14. J. Wang, L. Feng, Q. Wang, X. Wang, and H. Jiao, “Reduction of angle random walk by in-phase triangular phase modulation technique for resonator integrated optic gyro,” Opt. Express 24(5), 5463–5468 (2016). [CrossRef]  

15. J. Wang, L. Feng, Q. Wang, H. Jiao, and X. Wang, “Suppression of backreflection error in resonator integrated optic gyro by the phase difference traversal method,” Opt. Lett. 41(7), 1586–1589 (2016). [CrossRef]  

16. L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52(8), 1668–1675 (2013). [CrossRef]  

17. M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990). [CrossRef]  

18. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000). [CrossRef]  

19. H. Ma, Z. He, and K. Hotate, “Reduction of Backscattering Induced Noise by Carrier Suppression in Waveguide-Type Optical Ring Resonator Gyro,” J. Lightwave Technol. 29(1), 85–90 (2011). [CrossRef]  

20. H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006). [CrossRef]  

21. D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008). [CrossRef]  

22. M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997). [CrossRef]  

23. J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015). [CrossRef]  

24. L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014). [CrossRef]  

25. Q. Wang, L. Feng, H. Li, X. Wang, Y. Jia, and D. Liu, “Enhanced differential detection technique for the resonator integrated optic gyro,” Opt. Lett. 43(12), 2941–2944 (2018). [CrossRef]  

26. K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990). [CrossRef]  

27. K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999). [CrossRef]  

28. K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999). [CrossRef]  

29. T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991). [CrossRef]  

30. N. Liu, Y. Niu, L. Feng, H. Jiao, and X. Wang, “Suppression of backscattering-induced noise by sideband locking based on high and low modulation frequencies in ROG,” Appl. Opt. 57(26), 7455–7461 (2018). [CrossRef]  

References

  • View by:

  1. F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
    [Crossref]
  2. H. Jiao, L. Feng, N. Liu, and Z. Yang, “Improvement of long-term stability of hollow-core photonic-crystal fiber optic gyro based on single-polarization resonator,” Opt. Express 26(7), 8645–8655 (2018).
    [Crossref]
  3. H. Ma, J. Zhang, L. Wang, and Z. Jin, “Development and Evaluation of Optical Passive Resonant Gyroscopes,” J. Lightwave Technol. 35(16), 3546–3554 (2017).
    [Crossref]
  4. L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
    [Crossref]
  5. J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
    [Crossref]
  6. H. J. Arditty and H. C. Lefevre, “sagnac effect in fiber gyroscopes,” Opt. Lett. 6(8), 401–403 (1981).
    [Crossref]
  7. L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015).
    [Crossref]
  8. H. Ma, X. Lu, L. Yao, X. Yu, and Z. Jin, “Full investigation of the resonant frequency servo loop for resonator fiber-optic gyro,” Appl. Opt. 51(21), 5178–5185 (2012).
    [Crossref]
  9. L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993).
    [Crossref]
  10. X. Yu, H. Ma, and Z. Jin, “Improving thermal stability of a resonator fiber optic gyro employing a polarizing resonator,” Opt. Express 21(1), 358–369 (2013).
    [Crossref]
  11. N. Nakazawa, “Rayleigh backscattering theory for single mode optical fibers,” J. Opt. Soc. Am. 73(9), 1175–1180 (1983).
    [Crossref]
  12. G. A. Sanders, M. G. Prentiss, and S. Ezekiel, “Passive ring resonator method for sensitive inertial rotation measurements in geophysics and relativity,” Opt. Lett. 6(11), 569–571 (1981).
    [Crossref]
  13. Y. Zhi, L. Feng, J. Wang, and Y. Tang, “Reduction of backscattering noise in a resonator integrated optic gyro by double triangular phase modulation,” Appl. Opt. 54(1), 114 (2015).
    [Crossref]
  14. J. Wang, L. Feng, Q. Wang, X. Wang, and H. Jiao, “Reduction of angle random walk by in-phase triangular phase modulation technique for resonator integrated optic gyro,” Opt. Express 24(5), 5463–5468 (2016).
    [Crossref]
  15. J. Wang, L. Feng, Q. Wang, H. Jiao, and X. Wang, “Suppression of backreflection error in resonator integrated optic gyro by the phase difference traversal method,” Opt. Lett. 41(7), 1586–1589 (2016).
    [Crossref]
  16. L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52(8), 1668–1675 (2013).
    [Crossref]
  17. M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
    [Crossref]
  18. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
    [Crossref]
  19. H. Ma, Z. He, and K. Hotate, “Reduction of Backscattering Induced Noise by Carrier Suppression in Waveguide-Type Optical Ring Resonator Gyro,” J. Lightwave Technol. 29(1), 85–90 (2011).
    [Crossref]
  20. H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
    [Crossref]
  21. D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
    [Crossref]
  22. M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
    [Crossref]
  23. J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015).
    [Crossref]
  24. L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
    [Crossref]
  25. Q. Wang, L. Feng, H. Li, X. Wang, Y. Jia, and D. Liu, “Enhanced differential detection technique for the resonator integrated optic gyro,” Opt. Lett. 43(12), 2941–2944 (2018).
    [Crossref]
  26. K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
    [Crossref]
  27. K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
    [Crossref]
  28. K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999).
    [Crossref]
  29. T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
    [Crossref]
  30. N. Liu, Y. Niu, L. Feng, H. Jiao, and X. Wang, “Suppression of backscattering-induced noise by sideband locking based on high and low modulation frequencies in ROG,” Appl. Opt. 57(26), 7455–7461 (2018).
    [Crossref]

2018 (3)

2017 (1)

2016 (2)

2015 (4)

Y. Zhi, L. Feng, J. Wang, and Y. Tang, “Reduction of backscattering noise in a resonator integrated optic gyro by double triangular phase modulation,” Appl. Opt. 54(1), 114 (2015).
[Crossref]

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015).
[Crossref]

J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015).
[Crossref]

2014 (2)

L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
[Crossref]

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

2013 (2)

2012 (2)

H. Ma, X. Lu, L. Yao, X. Yu, and Z. Jin, “Full investigation of the resonant frequency servo loop for resonator fiber-optic gyro,” Appl. Opt. 51(21), 5178–5185 (2012).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

2011 (1)

2008 (1)

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

2006 (1)

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

2000 (1)

1999 (2)

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[Crossref]

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999).
[Crossref]

1997 (1)

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

1993 (1)

L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993).
[Crossref]

1991 (1)

T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[Crossref]

1990 (2)

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
[Crossref]

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

1983 (1)

1981 (2)

Arditty, H. J.

Armenise, M. N.

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

Cardarelli, D.

T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[Crossref]

Ciminelli, C.

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

Dell’Olio, F.

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

Ding, C.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Ezekiel, S.

Feng, L.

H. Jiao, L. Feng, N. Liu, and Z. Yang, “Improvement of long-term stability of hollow-core photonic-crystal fiber optic gyro based on single-polarization resonator,” Opt. Express 26(7), 8645–8655 (2018).
[Crossref]

Q. Wang, L. Feng, H. Li, X. Wang, Y. Jia, and D. Liu, “Enhanced differential detection technique for the resonator integrated optic gyro,” Opt. Lett. 43(12), 2941–2944 (2018).
[Crossref]

N. Liu, Y. Niu, L. Feng, H. Jiao, and X. Wang, “Suppression of backscattering-induced noise by sideband locking based on high and low modulation frequencies in ROG,” Appl. Opt. 57(26), 7455–7461 (2018).
[Crossref]

J. Wang, L. Feng, Q. Wang, X. Wang, and H. Jiao, “Reduction of angle random walk by in-phase triangular phase modulation technique for resonator integrated optic gyro,” Opt. Express 24(5), 5463–5468 (2016).
[Crossref]

J. Wang, L. Feng, Q. Wang, H. Jiao, and X. Wang, “Suppression of backreflection error in resonator integrated optic gyro by the phase difference traversal method,” Opt. Lett. 41(7), 1586–1589 (2016).
[Crossref]

Y. Zhi, L. Feng, J. Wang, and Y. Tang, “Reduction of backscattering noise in a resonator integrated optic gyro by double triangular phase modulation,” Appl. Opt. 54(1), 114 (2015).
[Crossref]

L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015).
[Crossref]

J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015).
[Crossref]

L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
[Crossref]

L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52(8), 1668–1675 (2013).
[Crossref]

Harumoto, M.

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

Hayashi, G.

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999).
[Crossref]

He, Z.

Hirose, A.

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
[Crossref]

Ho, W.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

Hotate, K.

H. Ma, Z. He, and K. Hotate, “Reduction of Backscattering Induced Noise by Carrier Suppression in Waveguide-Type Optical Ring Resonator Gyro,” J. Lightwave Technol. 29(1), 85–90 (2011).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[Crossref]

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999).
[Crossref]

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
[Crossref]

Jia, Y.

Jiao, H.

Jin, Z.

Kaiser, T. J.

T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[Crossref]

Kyuma, K.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Lefevre, H. C.

Lei, M.

Li, H.

Liu, D.

Liu, H.

Liu, N.

Lu, X.

Ma, H.

Nakazawa, N.

Niu, Y.

Ohnstein, T.

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

Prentiss, M. G.

Qiu, T.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

Sanders, G. A.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993).
[Crossref]

G. A. Sanders, M. G. Prentiss, and S. Ezekiel, “Passive ring resonator method for sensitive inertial rotation measurements in geophysics and relativity,” Opt. Lett. 6(11), 569–571 (1981).
[Crossref]

Smiciklas, M.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

Song, W.

L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015).
[Crossref]

Strandjord, L. K.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993).
[Crossref]

Suzuki, K.

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[Crossref]

Tai, S.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Takahashi, M.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Takiguchi, K.

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[Crossref]

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
[Crossref]

Tang, Y.

Tatoli, T.

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

Walsh, J. G.

T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[Crossref]

Wang, J.

Wang, L.

Wang, Q.

Wang, W.

Wang, X.

Wu, J.

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

Yang, Z.

Yao, L.

Ying, D.

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

Yu, X.

Zhang, J.

Zhang, X.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Zhi, Y.

Appl. Opt. (4)

Electron. Lett. (1)

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[Crossref]

IEEE Photonics Technol. Lett. (1)

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 2(1), 75–77 (1990).
[Crossref]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. (1)

JEOS:RP (1)

F. Dell’Olio, T. Tatoli, C. Ciminelli, and M. N. Armenise, “Recent Advances in Miniaturized Optical Gyroscopes,” JEOS:RP 9(1), 14013 (2014).
[Crossref]

Opt. Commun. (1)

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

Opt. Eng. (1)

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

Opt. Lett. (5)

Proc. SPIE (6)

L. Feng, H. Jiao, and W. Song, “Research on polarization noise of hollow-core photonic crystal fiber resonator optic gyroscope,” Proc. SPIE 9679, 967919 (2015).
[Crossref]

L. K. Strandjord and G. A. Sanders, “Performance improvements of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 1795, 94–104 (1993).
[Crossref]

L. K. Strandjord, T. Qiu, J. Wu, T. Ohnstein, and G. A. Sanders, “Resonator fiber optic gyro progress including observation of navigation grade angle random walk,” Proc. SPIE 8421, 842109 (2012).
[Crossref]

J. Wu, M. Smiciklas, L. K. Strandjord, T. Qiu, W. Ho, and G. A. Sanders, “Resonator fiber optic gyro with high backscatter-error suppression using two independent phase-locked lasers,” Proc. SPIE 9634, 96341O (2015).
[Crossref]

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation-method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 121 (1999).
[Crossref]

T. J. Kaiser, D. Cardarelli, and J. G. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Illustration of RFOG system and the spectra of CW and CCW.
Fig. 2.
Fig. 2. Schematic diagram of three-frequency differential gyro. (laser, tunable narrow-linewidth fiber laser; PD, photodetector; PM, phase modulator; C1, C2, couplers; Demod, demodulation module; SBL, sideband locking; RTL, resonance-tracking loop; DDS, direct digital synthesis; err. @ Ω1L, Ω2L, Ω3L, PDH error signals at Ω1L, Ω2L, Ω3L; ADC, analog-to-digital converter; DAC, digital-to-analog converter; FPGA, field-programmable gate array.)
Fig. 3.
Fig. 3. Modulation structures for traditional and tunable modulation/demodulation locking. (a) Pound Drever Hall (PDH); (b) Sideband locking (SBL). (For SBL only the upper half of the modulation structure is shown. The solid curve represents |F(ω)|2 and the dashed curve represents ∠F(ω), where F(ω) is the amplitude transmission coefficient of the cavity. For the frequency-tunable cases, the arrow labeled tune indicates the frequency spacing that is adjusted to tune the carrier, denoted by a thick line.)
Fig. 4.
Fig. 4. Demodulation of cos(Ω1L) term when β1=2.405, β1´=2.405, Ω1H = 1.3 MHz, Ω1L=100 KHz.
Fig. 5.
Fig. 5. Illustration of carrier suppression by locking the sideband to the cavity resonance. (The sidebands for CW (blue) locking and CCW (red) locking. Illustrated are three resonances separated by one FSR of 10 MHz.)
Fig. 6.
Fig. 6. Spectrum diagram of three-frequency lights under CW rotation. (Gyro resonance modes for CW (blue) and CCW (red) lights. Illustrated are three resonances separated by one FSR of 10 MHz, and a rotation induced frequency shift of the counter-propagating modes determines.)
Fig. 7.
Fig. 7. Measurement of the cavity SPR.
Fig. 8.
Fig. 8. PD signals of CW and CCW and their demodulations by low frequency. (the intensity output detected by PD1 for CW (green) light; the intensity output detected by PD2 for CCW (orange) light; the demodulation (blue) of the cosine term of Ω1L; the demodulation (red) of the cosine term of Ω2L; the demodulation (red) of the cosine term of Ω3L.)
Fig. 9.
Fig. 9. Scheme of locking the first-order sideband to cavity resonance. (a) the swept signal for driving laser PZT; (b) the intensity output detected by PD; (c) the demodulation of the cosine term of 2Ω1L;(d) the demodulation of the cosine term of Ω1H; (e) the demodulation of the cosine term of Ω1L.
Fig. 10.
Fig. 10. Allan deviation of gyroscope output. (a) Allan deviation of three-frequency differential RFOG based on the sideband locking output; (b) Allan deviation of double closed-loop RFOG output.
Fig. 11.
Fig. 11. PSDs of the gyro outputs in case of the DCL and case of the TFDSBL.

Equations (19)

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E CWT  =  E CW F CW ( ω C W ) +  R b E CCW F CCW ( ω C C W )
I CW = E CWT E CWT  =  | E CW | 2 H s g  +  R b | E CCW | 2 H b c  +    R b | E CW E CCW | H c h = I CW0  +  N 1  +  N 2
E CW _ M = E 0 e i ( ω C W t + β sin Ω C W t ) = E 0 n = + J n ( β ) e i ( ω C W t + n Ω C W t )
R b E CCW _ M = R b E 0 n = + J n ( β ) e i ( ω C C W t + n Ω C C W t )
I CW _ M = I CW0 _ M + N 1 _ M + N 2 _ M
I CW0 _ M = E 0 2 n = + H s g _ 0 | J n | 2  + 2 E 0 2 k = 1 + n = + H s g _ k | J n k J n | cos ( k Ω C W t )
N 1 _ M = R b E 0 2 n = + H b c _ 0 | J n | 2  + 2 R b E 0 2 k = 1 + n = + H b c _ k | J n k J n | cos ( k Ω C C W t )
N 2 _ M = 2 R b E 0 2 n = + n = + H c h _ n n J n ( β ) J n ( β ) cos [ ( n Ω C W n Ω C C W ) t + ( ω C W ω C C W ) t ]
I CW0 _ M = E 0 2 n = + H s g _ 1 | J n 1 J n |
N 1 _ M = 0
N 2 _ M = R b E 0 2 H c h _ 10 J 1 ( β ) J 0 ( β ) cos [ Ω C W t + ( ω C W ω C C W ) t ] cos Ω C W t
E in = E 0 e i ( ω t + β 1 sin Ω 1 H t  +  β 1 sin Ω 1 L t )
E in E 0 × { J 0 ( β 1 ) [ J 0 ( β 1 ) e i ω t + J 1 ( β 1 ) e i ( ω + Ω 1 L ) t J 1 ( β 1 ) e i ( ω Ω 1 L ) t ]  +  J 1 ( β 1 ) [ J 0 ( β 1 ) e i ( ω + Ω 1 H ) t + J 1 ( β 1 ) e i ( ω + Ω 1 H  +  Ω 1 L ) t J 1 ( β 1 ) e i ( ω + Ω 1 H Ω 1 L ) t ] J 1 ( β 1 ) [ J 0 ( β 1 ) e i ( ω Ω 1 H ) t + J 1 ( β 1 ) e i ( ω Ω 1 H  +  Ω 1 L ) t J 1 ( β 1 ) e i ( ω Ω 1 H Ω 1 L ) t ] }
P Ω 1 L = 2 J 0 ( β 1 ) J 1 ( β 1 ) P 0 × { J 0 ( β 1 ) 2 { Re [ T ( ω ) ] cos Ω 1 L t  +  Im [ T ( ω ) ] sin Ω 1 L t }  +  J 1 ( β 1 ) 2 { Re [ T ( ω  +  Ω 1 H ) ] cos Ω 1 L t  +  Im [ T ( ω  +  Ω 1 H ) ] sin Ω 1 L t }  +  J 1 ( β 1 ) 2 { Re [ T ( ω Ω 1 H ) ] cos Ω 1 L t  +  Im [ T ( ω Ω 1 H ) ] sin Ω 1 L t } }
f P M 2 + f P M 3 = 2 F S R
f P M 2 + f P M 1 = F S R  +  Δ f
f P M 3 f P M 1 = F S R Δ f
f P M 2 + 2 f P M 1 f P M 3 = 2 Δ f
Ω  =  ( f P M 2 + 2 f P M 1 f P M 3 ) n λ 2 D

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