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Abstract
High-performance demodulation of Sagnac effect is of great importance for rotation rate measurement in inertial navigation system. In this paper, we propose a flexible measurement of rotation rate based on a phase-controlled microwave photonic filter (MPF), which incorporates an orthogonal double-sideband (ODSB) modulator, a Sagnac loop, a linearly chirped fiber Bragg grating (LCFBG), a polarizer, and a photodetector. The ODSB modulator is used to generate optical carrier (OC) and first-order sidebands with mutually orthogonal polarizations. For the MPF, its central frequency can be tuned through changing the phase difference between the OC and first-order sidebands thanks to the dispersion of the LCFBG. Therefore, if the OC and first-order sidebands are separated by a polarization beam splitter and then travel along the Sagnac loop in opposite directions, the rotation-induced phase difference between them will lead to a shift on the frequency response of the MPF. Thus, two ways can be adopted to detect the rotation rate of the Sagnac loop for different applications: monitoring the frequency response shift of the MPF and measuring the power variation at a certain frequency. Besides, the measurement sensitivity can be easily adjusted to satisfy specific requirements by tuning a polarization controller or choosing a different operating frequency. An experiment is performed to validate the proposed scheme. The results show that the maximum frequency shift of the MPF can reach 1.7 GHz at a rotation rate of 1 rad/s, and a scale factor of 0.016 mW/(rad/s) is obtained at 4 GHz.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The demonstration of rotation rate detection with light by Sagnac in 1913 has paved the way for fiber-optic gyroscope [1,2], which has a wide range of applications due to its simple structure, high resolution, quick response, and strong immunity to vibrations [3]. On the basis that two light waves acquire a phase difference when propagating in opposite directions along a rotating optical loop, known as Saganc effect, the rotation rate can be obtained by demodulating the Sagnac phase difference precisely. Given the current maturity of electronic technology, it is essential to find ways to map the optical phase difference into the electrical intensity, phase, or frequency, so that the signal processing can be completed in the electrical domain.
Based on a Sagnac interferometer and a broadband light source, the rotation-induced phase difference can be demodulated through the interference intensity output from a photodetector (PD) [4–6]. For this interferometric-intensity-based method, the structure is simple and compact. However, since the output intensity exhibits a sinusoidal dependence on the rotation rate, which leads to a nonlinearity and limited dynamic range, a closed-loop structure with feedback control mechanism was usually adopted for biasing the system to make it operate in a linear point with maximum sensitivity [7–9]. In this way, the information of the rotation rate can be derived from the feedback control signal. Nevertheless, the instability of the control signal would deteriorate the measurement accuracy and scale factor [10,11].
Alternatively, the Sagnac phase can be converted to the phase shift of an electrical signal through a heterodyne detection [12], a pseudo-heterodyne detection [13,14], or a triangular phase modulation [15]. A significant advantage for mapping the rotation rate into the phase shift of the electrical signal is that it offers a good linearity and stability of the scale factor in a wide dynamic range. Besides, it enables the system to be free from baseband noises as well as source intensity fluctuations compared with the interferometric-intensity-based method. However, such attractive characters have not been fully demonstrated yet because of the introduced optical nonreciprocity into the Sagnac loop, which will cause a degradation of measurement performance consequently.
By using a high-Q optical resonator, the rotation-induced phase difference can be transformed to electrical frequency shift with a high sensitivity [16] due to the fact that the lightwaves can circulate many turns in the resonator. Different from the interferometric-intensity-based and phase-based measurements mentioned above, a highly coherent light source is required to realize the high-Q resonator. As a result, the noise resulting from the Rayleigh backscattering shows up and cannot be ignored. It will destroy the measurement linearity and cause a lock-in phenomenon [17]. Other special schemes have to be taken in order to reduce the backscattering-induced noise, which makes the signal processing complicated [18–20].
Recently, the significant development of microwave photonic technique has driven its application in areas of optical sensing demodulation [21,22]. In our previous work, we have successfully demodulated the rotation rate with an optoelectronic oscillator (OEO) [23,24], which is a hybrid resonator that consists of an optical path and an electrical path. The main feature of the OEO-based method is that a heterodyne Sagnac loop, which means that the two counter-propagating optical signals in the Sagnac loop are with different operating frequencies, is embedded in the OEO resonator, and thus the rotation-induced phase difference leads to an oscillating frequency shift of the OEO with a large scale factor. However, the frequency instability [23] or the free spectrum range (FSR) [24] of the oscillator has a great impact on the measurement precision. For example, the minimum rotation rate that can be detected was restricted to be 0.32 rad/s when the FSR of OEO is 5.22 MHz [24].
In order to overcome these limitations, we propose a scheme of Sagnac phase detection based on a phase-controlled microwave photonic filter (MPF) in this paper. The MPF takes advantage of the dispersion characteristics in optical fiber transmission, and its central frequency is a function of the phase difference between optical carrier (OC) and first-order sidebands. Then, by separating the OC and first-order sidebands based on the polarization property of optical signals and making them travel in opposite directions along the Sagnac loop, the rotation-induced phase difference between the OC and first-order sidebands can be converted to the frequency response shift of the MPF. Thus, by monitoring the frequency response shift of the MPF, the Sagnac phase can be detected. Furthermore, an intensity-based demodulation of the Sagnac phase can also be realized by applying a single-tone microwave signal at a proper frequency. For both demodulation methods, the measurement sensitivity can be easily adjusted through tuning the polarization controller (PC) or choosing a different operating frequency of the system to satisfy specific requirements.
In our experiment, two demodulation methods mentioned above are performed to validate the scheme. First, the frequency response of the MPF is monitored for rotation rate measurement, and the corresponding frequency shifts at the first, second, and third transmission notch are 1.7 GHz, 1.1 GHz, and 0.7 GHz, respectively when the Sagnac loop rotates at a rate of 1 rad/s. Then the intensity-based demodulation scheme is implemented. The results show that the peak power variation as a function of rotation rate is linear at 4 GHz and 8 GHz, and the scale factors are 0.016 mW/(rad/s) and 0.004 mW/(rad/s), respectively. The proposed detection scheme using an MPF shows flexibility and high sensitivity, and provides an effective way for rotation rate measurement.
2. Principle
2.1 Operation principle of the MPF
The setup of the proposed demodulation method based on MPF is shown in Fig. 1(a). A linearly polarized OC emitted from a laser diode (LD) is sent to a LiNbO3 Mach-Zehnder modulator (MZM) via PC1, which is used to align the polarization direction of OC at an angle of $\alpha$ with the principal axis of the MZM. Thanks to the polarization-dependent property of LiNbO3 crystal, the output signal of the MZM can be written as follows when the MZM operates at the minimum transmission point (MITP) by adjusting the direct current bias.
Fig. 1. (a) Setup of the demodulation scheme by monitoring the frequency response of the MPF. (b) Setup of the demodulation scheme by measuring the microwave signal power at a certain frequency. (c) Schematic spectrum evolution at different locations.
Then the ODSB modulation signal is sent to a Sagnac loop via an optical circulator (OCir1). By carefully tuning PC2, the OC and first-order sidebands can be separated by a polarization beam splitter (PBS) and travel in opposite directions along the Sagnac loop, which is formed by a length of polarization maintaining fiber (PMF). In this way, the rotation of the Sagnac loop will induce a phase difference between the OC and first-order sidebands, which can be represented as [1,2]
where R and ${L_S}$ are the radius and fiber length of the Sagnac loop respectively, c is the light velocity in vacuum, ${\lambda _c} = 2\pi c/{\omega _c}$ is the wavelength of OC, and $\Omega $ is the rotation rate of the Sagnac loop. One thing that should be noted is that both the OC and first-order sidebands travel along the slow axis of the PMF in the Sagnac loop. After reflected by a linearly chirped fiber Bragg grating (LCFBG) via OCir2, the signal is given byAfter traveling through PC3, ${E_x}$ and ${E_y}$ interfere at a polarizer (Pol.), and the generated signal can be written as
2.2 Demodulation scheme by monitoring the frequency response of the MPF
We can see from Eq. (7) that the frequency response of the MPF is a function of $\varDelta \varphi$ and ${\varphi _0}$. Thus, when PC3 is fixed, the phase difference induced by the Sagnac effect can be demodulated by monitoring the frequency response of the MPF. Obviously, the frequency at the transmission notch is
Fig. 2. (a) Frequency responses of the MPF at different Sagnac phases. (b) Frequencies at different transmission notches as a function of the rotation-induced phase shift variation.
2.3 Demodulation method by monitoring the output signal intensity from PD
Alternatively, the rotation rate can be demodulated through monitoring the output signal intensity from PD at a certain modulation frequency, as shown in Fig. 1(b). For this case, the signal intensity as a function of the rotation-induced phase difference can be written as
As can be seen from Eq. (10) that ${\omega _{ml}}$ is related to ${\varphi _0}$, that is, the linear operation point of the system can also be adjusted by tuning PC3 at a certain modulation frequency.
3. Results and discussions
An experiment based on the setup shown in Fig. 1 is performed to validate the proposed scheme. The wavelength of OC is set to be 1549.3 nm so that it can be reflected by the LCFBG, whose central wavelength and dispersion are 1549.3 nm and 2650 ps/nm, respectively. The Sagnac loop is formed by a PMF coil with a length of 560 m and a radius of 0.085 m. Both the bandwidths of the MZM and PD used in the experiment are 10 GHz.
3.1 Frequency response of the MPF versus rotation rate
According to the analyses mentioned in Section 2.2, the rotation-induced phase difference between the OC and first-order sidebands would cause a shift on the frequency response of the MPF. Therefore, by monitoring the frequency response of the MPF, the rotation rate can be demodulated. A vector network analyzer (VNA, KEYSIGHT, E5063A), which has a bandwidth of 100 kHz - 18 GHz and an output power of -15 dBm, is used in the experiment to measure the frequency response of the MPF under different rotation rates. For this instance, the output of the VNA is connected with the EA while the converted electrical signal from PD is sent to the input port of the VNA, as shown in Fig. 1(a).
Before the system starts to work, the separation of the OC and the first-order sidebands should be guaranteed at the Sagnac loop. Figure 3 shows the optical spectra at the outputs of the PBS measured by an optical spectrum analyzer (OSA, YOKOGAWA, AQ6370D), where the red dash line is the OC at one output of the PBS while the blue solid line refers to the first-order sidebands at the other output of the PBS. It is apparent from Fig. 3 that the OC and first-order sidebands are separated properly by tuning PC2. Then the Sagnac loop is closed and the frequency response of the system is obtained via the VNA, as presented in Fig. 4(a). The red dashed line in Fig. 4(a) is the simulated frequency response of the MPF with parameters of $D = 2650$ ps/nm, $\lambda_c = 1549.3$ nm, and ${\varphi _0} ={-} 0.27\pi$, while the blue solid line represents the measured frequency response by the VNA. We can see from Fig. 4(a) that the experimental result agrees well with the simulation result. The little notches as circled with black solid lines are associated with the uneven frequency responses induced by the devices used in our experiment, such as the MZM, the coaxial cables. The positions of these little notches are fixed, and thus they will have little influence on the measurement result. According to Eq. (7), the ${\varphi _0}$ caused by PC3 has the same influence on the frequency response of the MPF as the rotation-induced phase difference $\varDelta \varphi$, thus the tuning of PC3 would inevitably lead to a shift on the frequency response of the MPF, as shown in Fig. 4(b). This is crucial for the intensity-modulated measurement later, since a linear point can be achieved by tuning PC3.
Fig. 4. (a) Measured frequency response of the MPF and the simulation result. (b) The frequency responses of the MPF when tuning PC3.
The measurement of the ration rate based on the frequency shift of the MPF is then validated by mounting the 560-m Sagnac loop on a rotation table. Figure 5(a) provides the recorded frequency responses of the MPF by the VNA when the table rotates at different rates from 0 to 1.276 rad/s in a clockwise direction. We can see from Fig. 5(a) that the frequency response of the MPF drifts to a larger frequency with the increase of the rotation rate. It should be noted that the shift direction of the frequency response of the MPF depends on the rotation direction of the Sagnac loop, as well as the transmission direction of the OC or the first-order sidebands along the Sagnac loop. Besides, it can be seen from Fig. 5(a) that the measurement sensitivities to rotation rates at different transmission notches are not exactly the same. For example, when the Sagnac loop rotates at a certain rate, the frequency shift at the first transmission notch is larger, which is consistent with the theory. Therefore, a low-bandwidth system is enough for a highly-sensitivity measurement in real situations. Figures 5(b), (c), and (d) illustrate the measurement results (black dots) on the frequency shift at the first, second, and third transmission notch, respectively. As can be seen from the results that the frequency shifts at 1 rad/s for three cases are 1.7 GHz, 1.1 GHz, 0.7 GHz, respectively. The minimum rotation rates that can be measured for these cases are 6.2${\times} $10−6 rad/s (1.3 °/h), 9.8${\times} $10−6 rad/s (2.0 °/h), and 1.4${\times} $10−5 rad/s (3.4 °/h), respectively, when the resolution of VNA is set to be 10 kHz. A simulation result (red dots) is also given based on the parameters mentioned above. Both the measurement data and the simulation data are firstly fitted with a linear function, and then with a nonlinear function of $y = A\left( {\sqrt {x + B} - \sqrt B } \right)$ according to the analyses mentioned in Section 2.2, which are presented in the insets. Compared with these two fitting results, we can find that the measured notch frequency shift has a near-linear dependency on the rotation rate within the measurement range, and the slope decreases with the transmission notch number. It is worthy to note that the measurement sensitivity at each transmission notch can be adjusted by tuning PC3 for a change of ${\varphi _0}$. Therefore, a measurement with suitable sensitivity and dynamic range can be designed for the requirements in real situations through tuning PC3 or choosing a proper transmission notch. The results shown in Figs. 5(b), (c), and (d) also reveal that the measured notch frequencies have a similar trend of variation with the simulation data. The difference between them may result from the inaccuracy of the parameters in simulation, the imperfect separation of the OC and first-order sidebands during measurement, and so on.
Fig. 5. (a) Measured frequency responses of the MPF at different rotation rates. (b) The relationship between the frequency shift at the first transmission notch and rotation rate. (c) The relationship between the frequency shift at the second transmission notch and rotation rate. (d) The relationship between the frequency shift at the third transmission notch and rotation rate.
3.2 Intensity-based detection of rotation rate at a certain modulation frequency
If a single-frequency microwave signal is applied on the MZM, then the shift on the frequency response of the MPF will finally lead to a signal power variation at the output of PD, as depicted by Eq. (9). Thus, the rotation rate can be easily demodulated by a power meter, which provides a fast response since frequency scanning is avoided in this way. This intensity-based method may also be preferable for measurement scenarios concerning about the simplicity and cost. An experiment is performed to validate this. Instead of using a VNA, a tunable microwave signal generator (MSG) is adopted to generate the modulation signal applied on the MZM, and the output signal from PD is monitored by an electrical spectrum analyzer (ESA, Agilent N9010A), as shown in Fig. 1(b). The rotation rate of the Sagnac loop can be obtained through monitoring the peak power variation of the electrical spectrum detected by the ESA.
First, the modulation frequency is set to be 4 GHz, which locates on the falling edge of the first passband of the MPF. Figure 6(a) presents the electrical spectra under different rotation rates, and the inset is a zoom-in view of the peak powers. The relationship between the peak power and rotation rate is given in Fig. 6(b). As observed, the signal power (with a unit of dBm) tends to be constant when the rotation rate increases to a certain value, owing to the fact that the frequency of 4 GHz comes to the top of the first passband of the MPF. By transforming the unit of peak power from dBm to mW, which is shown with red dot in Fig. 6(b), we can see that the peak power changes almost linearly with the rotation rate and the slope is 0.016 mW/(rad/s). From this result we can calculate the minimum rotation rate that can be demodulated is about 6.3${\times} $10−8 rad/s (0.013 °/h) when the system noise floor is -90 dBm. Then we tune the modulation frequency to 8 GHz, which locates on the falling edge of the second passband of the MPF. The measurement results are shown in Figs. 6(c) and 6(d). It can be seen that the sensitivity becomes lower at the modulation frequency of 8 GHz as the scale factor becomes 0.004 mW/(rad/s), which refers to the minimum rotation rate that can be measured is about 2.6${\times} $10−7 rad/s (0.054 °/h). Note that the signal power as a function of rotation rate depends on several factors: (1) the modulation frequency used for demodulation and the ${\varphi _0}$ caused by PC3; (2) the bandwidth of the passband of the MPF; (3) the measurement range for rotation rate. The first two factors also contribute to the measurement sensitivity.
Fig. 6. (a) Measured electrical spectra for the 4-GHz modulation signal at different rotation rates. (b) The relationship between the peak power at 4 GHz and rotation rate. (c) Measured electrical spectra for the 8-GHz modulation signal at different rotation rates. (d) The relationship between the peak power at 8 GHz and rotation rate.
Note that the maximum rotation-induced phase difference that can be demodulated is limited to $\pi $, which is determined by the period of the frequency response of MPF. According to Eq. (2), we can derive that the maximum measurable ration rate is ${\Omega _{\max }} = \pi {\lambda _c}c/({\textrm{4}\pi R{L_S}} )$ and it is calculated to be 2.44 rad/s for our experiment setup. Table 1 shows the performance comparison between the proposed method and other demodulation techniques. As can be seen from Table 1, the proposed scheme has a comparable performance with other demodulation techniques, but it becomes simpler and more flexible.
Table 1. Performance comparison between the proposed scheme and other demodulation techniques
3.3 Discussions
Reciprocity is a key feature for rotation measurement based on Sagnac effect because it greatly reduces the influence of some deleterious non-reciprocal effects [28,29]. For our proposed system, although the two lights travelling along the Sagnac loop in opposite directions have different wavelengths, the fiber length of the Sagnac loop could be designed as short since the measurement scale factor is large (∼GHz/(rad/s)) so that the influence of the wavelength-induced nonreciprocity could be reduced. Besides, the Sagnac loop can be placed before MZM, as shown in Fig. 7. In this way, the two counter-propagating light waves have the same wavelength without deteriorating the system performance [23].
Fig. 7. (a) Setup of the improved scheme to eliminate the wavelength-induced nonreciprocity. (b) Schematic spectrum evolution at different locations.
The time-dependent temperature gradient along the fiber also causes nonreciprocity of Sagnac loop since the two counterpropagating lights traverse the same region of the fiber at different times, which is known as Shupe effect [30]. The phase error caused by the thermal-induced nonreciprocity can be expressed as [30]
As the PMF has a large birefringence, the error caused by the polarization coupling between two counterpropagating lights can be ignored. However, there might be residual OC and first-order sidebands in the Sagnac loop, as shown in Fig. 8, which can result from the bias drift of MZM, the limited extinction ratio of PBS, and polarization fluctuations of the signal. To make the analysis easier to understand, the situation when there is only residual OC is considered, as shown in Fig. 8(a), which occurs when the bias of the MZM drifts from MITP state. Then there will be OCs both in clockwise (CW) and counterclockwise (CCW) along the Sagnac loop, which can be denoted by the OC-cw and OC-ccw, respectively. When the Sagnac loop rotates, there will be a phase difference $\varDelta \varphi $ between the OC-cw and OC-ccw. Then they will interfere at the polarizer, as shown in Fig. 8(b), where $\delta$ is the angle of the polarization direction of OC-cw relative to the principal axis of the polarizer. The final obtained OC is the vector sum of OC'-cw and OC'-ccw, where OC'-cw and OC'-ccw are the projections on the polarizer of OC-cw and OC-ccw, respectively. Figure 9 represents the phasor diagram of the final OC signal for different amplitudes of the OC'-cw, which depends on the operating point of the MZM when $\delta$ keeps constant. It can be seen from Fig. 9 that the phase of the OC after the polarizer is related to the amplitude of the OC'-cw. The lower the OC'-cw amplitude is, the smaller $\sigma$ is, where $\sigma$ is the phase difference between OC'-ccw and the obtained interference OC. The relationship of $\sigma$ and $\varDelta \varphi$ can be expressed as
Fig. 8. (a) Schematic spectra of the signals travelling in CW and CCW along the Sagnac loop and (b) schematic spectra of the signals after polarizer when there is residual OC. (c) Schematic spectrum of the signals travelling in CW and counterclockwise CCW along the Sagnac loop and (d) schematic spectrum of the signals after polarizer when there is residual OC and first-order sidebands.
Figure 10 shows the simulated phase error as a function of rotation-induced phase difference at different values of $\delta $. As can be seen from Fig. 10(a) that when the amplitude of OC-cw, which is the undesired part, is small compared to that of OC-ccw, the value of $\varDelta \varphi $ has little influence on the measurement phase error except for the case where the polarization direction of OC-cw is tuned to be close to the principal axis of the polarizer. However, if the amplitude of OC-cw is equal to that of OC-ccw, the measurement phase error can be greatly affected by the value of $\varDelta \varphi $, as depicted in Fig. 10(b). Under this condition, unless $\delta $ is $\pi /4$ by tuning PC3, the phase error firstly increases and then decreases along with the increase of $\varDelta \varphi $ in the 0 ∼$\textrm{}\pi $ range.
Fig. 10. Simulated phase error as a function of the rotation-induced phase difference at different values of $\delta $ when the amplitude of OC-ccw is (a) five times or (b) equal to that of OC-cw.
The analysis mentioned above is also suitable when there are both residual OC and first-order sidebands, as shown in Figs. 8(c) and (d). The phase error under this condition can be represented by
The frequency stability of the proposed MPF has a great influence on the measurement performance, such as the bias stability of the system. Besides the Shupe effect mentioned above, the factors affecting the frequency stability are mainly as follows: a) the bias drift of MZM, and b) the polarization state change of the optical signal. Therefore, adopting a temperature control scheme, replacing all the single mode fibers with PMFs, and taking a bias control board for the MZM are also helpful to improve the frequency stability of the MPF.
4. Conclusion
A phase-controlled MPF has been proposed in this paper for rotation rate measurement. By taking advantage of the polarization property of optical signal, the rotation-induced phase difference was mapped into the frequency response shift of the MPF. As a result, two demodulation schemes are available to meet the needs of different scenarios: by monitoring the frequency shift at the transmission notch of the MPF with a VNA for high-precision measurement, or by detecting the variation of output signal intensity at a certain modulation frequency for low-cost measurement. What is more, when there is a specific requirement for the measurement, a large dynamic range for example, the sensitivity can be easily adjusted for both methods through tuning PC3 or choosing a different operating frequency. Therefore, the proposed MPF is flexible for demodulation of the rotation-induced phase difference. In our experiment, these two demodulation schemes have been performed. The results show that the frequency shift at a transmission notch of about 3.5 GHz is up to 1.7 GHz at a rotation rate of 1 rad/s, and the scale factor at a modulation frequency of 4 GHz is 0.016 mW/(rad/s). Actually, the dispersion of the LCFBG used in the system can also be designed with a proper value so that the system can fit in more applications.
Funding
National Natural Science Foundation of China (61775015, U2006217); Fundamental Research Funds for the Central Universities (2021JBZ103).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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